Mathematical statements are truths in a mathematical structure, a consistent structure established from a sound and reasonable set of axioms verified through the principles of logic. Although all things are abstract in the structure, the consistency of the structure and the nature of the axioms, thought to have some sense and meaning to reality somehow represents abstract formations of  reality and reason than to fictions whose characters are created from unreal imaginations.

Fictional characters of today will never be true at any time of the future, unless the character is that of a past event.

Mathematical truths are tested valid in the sciences today and those abstract theories which do not have direct applications today will have a meaningful application in the future.  The logical consistency of mathematics is the very reason that it is the language in which we study the ultimate truth - nature.

Lobachevsky: 'No part of mathematics however abstract it might be that will not be in the service of humankind" - very true statement. 

Dear Mr. Ramon Quintana,

In my opinion, we have two scientific cognitions for “apple”: one is the abstract concept which exists in our mind as the abstract picture, the abstract imagination (the abstract apples which is not eatable, not touchable,…) while another is the carrier of the abstract concept (real objective apples which are eatable, touchable,…). “The abstract concept” and “the real objective carriers of the abstract concept” can not be confused and mixed up.

In our mathematical cognizing activities, we should be aware that there are characteristic differences between “abstract concepts” (mathematical statements) and “carriers of abstract concepts”. It is true that in our science （mathematics）, how to unify “objective world” and “subjective world” has been one of the toughest works for scientists.

So, cognizing, connecting and translating “things in human science（mathematical statements）and things in natural world”, “sets in mathematics and sets in natural world”, “infinitesimals in mathematics and infinitesimals in natural world”, “abstract mathematical entities and physical realities”, … are essential and important work for scientists though very difficult.  With or without “real objective carriers” is one of the demarcations for science or non-science.

My best regards, Geng

Dear Ramon Quintana, Dear Geng Ouyang

Thank you for the interesting dialogue! I would like to add a piece of background:  in the (attached) Tanner Lecture on Human Values (Delivered at The University of Michigan April 7, 1978) Karl Popper presented a "pluralist view of the universe" which involved "3 worlds". In his own words:

"There is, first, the world that consists of physical bodies. I will call this physical

world ‘world 1’. There is, secondly, the mental or psychological world, the world of our feelings of pain and of pleasure, of our thoughts, of our decisions, of our perceptions and our observations; in otherwords, the world of mental or psychological states or processes, or of subjective experiences. I will call it ‘world 2’. [...] My main argument will be devoted to the defence of the reality of what I propose to call ‘world 3’. By world 3 I mean the world of the products of the human mind, such as languages; tales and stories and religious myths; scientific conjectures or theories, and mathematical constructions; songs and symphonies; paintings and sculptures".

I wonder if Popper´s view provides an alternative way of referring to the present problem...

Mathematics is both the medium and the message/massage.It is the medium or analytical instrument by which much of the physical/social sciences is advanced and formalized.When that formalization, which initially was epistemological in nature, reaches a certain degree of respectability, the very instrumentation of science becomes an object of study and becomes an ontology in its own right. 

Witness this in economics and biology where the prefix "mathematical" is often attached to these disciplines.And of course countless other domains which are going through a similar development trajectory.Another parallel can be drawn with computation which has triumphed in its ability to surpass the narrow bounds of its epistemological utility and fashion for itself ontological "in-roads" and footholds over much of our human existence.

http://www.amazon.com/Pablo-Triana/e/B001R1ZDLM

https://www.wolframscience.com

Mathematical statements are truths in a mathematical structure, a consistent structure established from a sound and reasonable set of axioms verified through the principles of logic. Although all things are abstract in the structure, the consistency of the structure and the nature of the axioms, thought to have some sense and meaning to reality somehow represents abstract formations of  reality and reason than to fictions whose characters are created from unreal imaginations.

Fictional characters of today will never be true at any time of the future, unless the character is that of a past event.

Mathematical truths are tested valid in the sciences today and those abstract theories which do not have direct applications today will have a meaningful application in the future.  The logical consistency of mathematics is the very reason that it is the language in which we study the ultimate truth - nature.

Lobachevsky: 'No part of mathematics however abstract it might be that will not be in the service of humankind" - very true statement. 

Ramon and Dejenie -

Shall I take it then that your view, Ramon, is something of a rejection of Pythagorean reality, whereas yours, Dejenie, is at least a partial acceptance? (Regardless, Dejenie, your argument was refreshing.) The world today seems a very physical world, as we think we understand that, but our perceptions are rather Newtonian, so though a Pythagorean reality may seem easily dismissible in today's everyday world, and the way we think, it may not be entirely without merit.

Jim

A good way to answer the question would be to look for instances where either mathematics is falsified by apparent physical reality (APR), or reversely where apparent physical reality is falsified by mathematics. If either Math or APR is consistently falsified by the other, and never the other way around, then the question answers itself.

Mathematics fully falsifies APR in Bell's inequality, and also in delayed choice experiments, and arguably in other instances as well (e.g. in the derivation of e=mc², which is purely mathematical but works in the real world.) There is no known example of a mathematical derivation falsified by APR (as would be the case, for instance, if Bell's inequalities did not work and hence no "spooky actions at a distance" existed.)

Therefore, the only possible conclusion at this time is that mathematics is ontological. Interestingly, mathematics then goes on to falsify itself in the realm of infinities (Cantor's anomaly), which can be interpreted in different ways, none of which however can belie or gainsay the primacy of math over APR.

I think this whole issue reduces to one's theory of concepts. My view is based on Aristotle and esp. Rand. Take number. To form number concepts you abstract out quantity from the particular concretes (e.g., 5 can be five of anything). So the number is real in that it is tied to reality and ultimately based on perception but the concept itself is a mental entity which is an integration of perceptual material and/or ultimately reducible to it. The danger, caused by Plato, is to view abstractions as divorced from the senses and inhabit another separate world--this always ends up in mysticism-cf the book; The Cave and the Light which explores the history of the world as a duel between Plato and Aristotle.

Abstract (away from track) mathematical entities emerge causally (relatively-observed) from, while remaining intrinsic to, natural laws/rates (or ratios -- part to whole, whole to part; theoretically, the golden ratio?) of physical 'motions' along an absolute continuum or whole (curved space; also at the self-similar golden ratio rate?).  You could say that mathematics is therefore a language; however, that could be said about any thing or object, 'language' itself comprising in binary fashion not only syntax -- procedural, structure, signs, parts of speech, etc., but necessarily foremostly meaning/semantics/symbolism/substance/awareness as well, objective or otherwise, all symbols, even actuated physical objects, e.g., a ball, etc., all nonetheless subjectively/relatively pointing back abstractly to a certain concretized or reified meaning/original spirit, intent, context ('with text'), or purpose/repurpose of an object to which only then is a posteriori language attached.  Obviously in forming a uniquely different conclusion, this stance departs 180 degrees from Badiou's default conclusion (or unproven/tentative hopefully? null hypothesis) that: the one [or whole] is not -- promptly once again, in violation of Godel's incompleteness theorem, directing Badiou down the all too now well-trodden and rutted, unfortunate circular path of ZFC set theory as fundamental.

Dr, Nev: Sorry I cannot respond--I have no idea what you are talking about.

Dr. Locke, In relation to your rejection of Plato's Idos, or separation of an object from its number or essence or object-ness or category, it is to me as Plato avoided a fallacy of composition to declare a part or object to be one and the same as its whole, which negates any need for a whole undistinguished from its parts, leaving a mere unidentifiable jumble.  For instance, your name is not Name, your name is a name but not Name, etc.  This I understand to be the distinction between an object and its antecedent Plato's Idos preserves.

Ramon, you seem to have perhaps misunderstood my post - please read it again.

1- I certainly do not say anywhere that mathematics proves the existence of infinities (in actuality, rather than in a vague conceptualization.) It does not, neither does it disprove it: infinities are non-observables. Either they exist or they don't: they cannot be generated from finite environments (because of combinatorics) nor can they ever be reduced to finiteness if they exist.

2- You seem to be somehow conflating pure and applied mathematics, which are two very different animals. The talk here is of pure mathematics

3- Hidden variables cannot exist (Gleason theorem). If however in some cases Gleason's theorem were non applicable, then it does not matter: assuming for the sake of discussion that hidden variables do exist, Renato Renner and Roger Colbeck in Zürich have convincingly demonstrated that no possible hidden variables hypothesis could possibly improve the significant outcomes produced by current theories.

Kind regards

Chris

First, I think we must refine the ontological criteria by which we define "fictional" in mathematics (or even more, debate whether ontology is the only field to search for such criteria of fictionality). If we only stay with "objective reference" as ontological criterion, as you seem to suggest, we would have entities which are more or less fictional (for instance, a natural number is less fictional than a complex number). If we accept the Fregean account of semantics of mathematics as second-order reference (mathematics entities refer to concepts about physical concepts; mathematical laws are laws of the laws of the physical world), then we have no straight answer to your question (you may say that all mathematical entities are fictional or all are non-fictional). Besides, as one answerer already put it, natural language may posit the same issue if we take reference as pointing to a concept and not to a physical entity.

All of you here seem to assume it is true that mathematics is not an empirical discipline, without presenting any arguments to support that belief. Viewed empirically, numbers are symbols used for counting objects in the world. We have all used the "llll" "/" system to count things in groups of 5 before. "5" is just an abstract symbol for those groupings, as "1" is a symbol for a group of "l". The arithmetic functions of adding, subtracting and multiplying are just formal procedures for counting and grouping things we find in the world that have spatial relations to one another, that is, things that can be counted. 

Only when we get to division do we encounter mathematical objects that are of questionable "reality". The numbers 7 and 22 can be added, subtracted and multiplied without any "ontological" problems arising. Even 7/22 is not a problem.  Division is the creation of ratios of numbers from integers with can be rational or irrational numbers. Since the quotients of these ratios, their square roots, are either terminal or repeating only for perfect squares, most real numbers are irrational, in this sense. But, even for irrational numbers, the rules of arithmetic are still consistent. So there is no more reason to doubt the reality of mathematical objects like the ratio π of a circle's circumference to its diameter, Euler's number e, the golden ratio φ, and the square root of two.

So, rather than a formal, abstract system, as it is usually interpreted today, mathematics (including set theory) is better viewed as being empirically grounded in geometry and the spatial relations things must have in order to be counted in the first place.   

In this way, I would argue that mathematical statements are ontological in the literal sense of being true about “what there is in the world”. This ontology is itself empirical, so it might be wrong. It's only true if it is the best possible explanation of what math is. 

Replies are welcome and I hope this sheds some new light on this old question.

DCD

Ramon, In reply to your last paragraph:

"If Gödel could single-handedly demolish the Frege, Hilbert, Russell program of reducing mathematics to an axiomatic system, what prevented him from establishing the truth of mathematical realism with his same method? Given his effort to prove God’s existence, he must have been motivated to prove realism. What prevented him? Did he recognize that there is semantic paradox that prevents such proofs from the outset? Is there a logical or other limit that prevents mathematics from establishing the truth of its very existence? And if so, what is the nature of that logic? I am asking now as a layperson, seeking answers from the experts in the audience."

Gödel's Incompleteness Theorem establishes the tautology of attempting to prove or establish anything on terra firma, whether mathematical realism or what have you, with that thing, similar to the basic unproductive circular tautology of defining a term using the term.  Therefore, one may deduce from this that going outside of the subject area to establish its truth is the only logical avenue to take.  Being a mathematician and logician, Gödel himself may have been limited to go outside those areas of focus, thereby preventing him from affirming mathematical realism.  In fact due to the presumed efficiency of specialization, this ability is the problem or limiting factor for most researchers, even for those in philosophy these days, the grave inability to see the forest, or mathematical realism, for the trees.  Once that is accomplished and I have a thesis out this year which undeniably, nontrivially and uniquely identifies that forest, mathematical realism among other things shall be unparadoxically firmly established.  

Ramon, Thank you for your interest, my pleasure to provide it. : )

sorry but things in the world are real-- sense perception gives is our first direct knowledge of the world- when you cross the street if you do not look a real thing will kill you--if you have no money the bank, your checks will bounce-- if you eat poison you will sicken and maybe die--if-if you area Kantian (the noumenal world is unknowable) then you are doomed from the start--there is nothing to discuss--in counting we re dealing with numbers of things, but by abstraction we can look at number divorced from things--but there is still a tie to reality--as you get to more abstract math concepts you are getting farther from direct perception (e.g., force, volume, pressure, temperature, mass) but there is always a tie to reality or the numbers would not work--try calculating what is needed to send a rocket into space without numbers that tie to reality--

Luis Vernich:

I disagree that " .. when we are doing mathematics operations we are not dealing with amounts, we are dealing with  ...  with numbers’ names and the rules we have created."

It makes much more sense to say we discover geometrical and mathematical relations in the world. We don't create them, as you assume. There would be no numbers without natural language and the tremendous evolutionary advantages our species gained by evolving the neurons needed to execute linguistic behaviors. We can speak now in abstractions only because we could first perceive and reason about objects in space, how they move and how to manipulate them to our advantage. 

You and Popper can imagine your 2 or 3 "worlds" but I'll stick with the one we all find objectively, when we just observe closely and reason carefully. Mathematical and other true statements are true only in as much as they correspond to "what there is in the world". It is not simply about them being logically consistent with sets of rules we create. The diameter and circumference of every circle are really proportional to each other as 22/7, even though we cannot represent this relationship as a rational number. 

Godel certainly believed this, as did Plato, since both were realists about universals. Unfortunately, by also viewing math and geometry as formal systems, he and Plato failed to see how an ontology with 2 different kinds of interacting substances in just one world can explain what is most fundamentally true about "what there is in the world". By viewing space as a formal construct, instead of something ontologically ubiquitous that exists between all material particles, our epistemology-based philosophy of mathematics overlooks the ontology that would explain how math is objectively true of the world, not just a figurative or fictional convenience.

DCD

Daniel: I basically agree with you. Plato's fundamental error was the rejection of the material provided by the senses thus disconnecting concepts from reality-BTW: the term "ivory tower" really means intellectuals doing their abstractions divorced from the actual world--

Luis,

This process you describe, of trying out different hypotheses to see what works, sounds quite empirical to me. 

Luis and Ramon,

I appreciate your informed opinions and the arguments for them.

However, I think you may both be confusing the fact that we can use geometry and mathematics to derive and come to know true propositions from other true propositions, with the explanation of why they are true. The formal derivation process just shows that they are true. 

My suggestion here is that our science of geometry is true of (corresponds with) the three-dimensional nature of the ubiquitous spatial substance and how particular material substances coincide with parts of space.

DCD

Ramon, Superb summation of the state of mathematics in your post.  As to your "mathematical constraint," mathematics can only measure reality and to the extent the measurement is wrong or off, the mathematics as others have stated and never reality will be wrong.  Therefore, mathematical constraints can only derive from this reality, which to date has yet to be fully articulated by anybody.  Physics describes this reality piecemeal, but concerning 'space' as Daniel mentions, remains without a unified concept or cogent thought experiment.  Once that so-called 'theory of everything' is established, yes reality will be very predictive both mathematically and phenomonologically.  However, until empirically, physically and tangibly, independently objectively (not esoterically, but in plain view) a noncircular (not set theory) foundation of mathematics is firstly established outside of itself and I contend even outside of physics per se (pre-physics, pre-science; pre-all, not necessarily absolutely 'nothing,' possibly relatively 'nothing'), no testable/measurable unification between the proverbial Hatfield's and McCoy's (realism and construct; abstract and concrete; discrete and continuum; general and special relativity; cosmos and quanta; linear and curvature; certainty and uncertainty; absolute and relative) is forthcoming or can ever be expected to surface as is being played out here and for centuries preceding.

Cj Nev, also Ramon and Edwin Locke,

The "empirically, physically and tangibly, independently objectively (not esoterically, but in plain view) a noncircular (not set theory) foundation of mathematics is firstly established outside of itself and I contend even outside of physics per se (pre-physics, pre-science; pre-all, not necessarily absolutely 'nothing,' possibly relatively 'nothing')" that Nev is seeking could be found in an ontology that is thoroughly empirical.

Ontology can be explanatory in a non-trivial way if it identifies natural substances that constitute what exists in such a way that we can account for everything we find in the world, including conscious subjects who come to know how they are constituted.

Physical science uses laws of nature and efficient-causes to explain what happens in the world.

Empirical ontology uses naturalistic ontological causes to explain what exists as the interaction and grouping of the basic substances, including how these processes lead inevitably to rational subjects like us. 

DCD

In 2003 I published an article in the South African Journal of Philosophy in which this issue is investigated - "Frege's  Attack on “Abstraction” and his Defense of the “Applicability” of Arithmetic (as Part  of Logic)." I insert the whole article below. Special attention is given to the question: "What is ontic about number?" [The mathematical symbols for "element of" and "not an element of" are functional in this context - so I had to replace them with "e" and "-e". The footnotes are found at the bottom of the pages inside the text.]

Danie Strauss

Frege's  Attack on “Abstraction” and his Defense of the “Applicability” of Arithmetic (as Part  of Logic)

 Daniël F M Strauss

 E-mail:

Abstract:

The traditional understanding of abstraction operates on the basis of the as- sumption that only entities are subject to thought processes in which particulars are disregarded and commonalities are lifted out (the so-called method of genus proximum and differentia specifica). On this basis Frege criticized the notion of abstraction and convincingly argued that (this kind of) “entitary-directed” ab- straction can never provide us with any numbers. However, Frege did not consider the alternative of “property-abstraction.” In this article an argument for this alternative kind of abstraction is formulated by introducing a notion of the “modal universality” of the arithmetical and by developing it in terms of the distinction between type-laws (laws for entities – applicable to a limited class of entities) and modal laws (obtaining for every possible entity without any restriction). In order to substantiate this argument a case is made for the acceptance of an ontic foundation for the arithmetical (and other modes or functions of reality

– with special reference to Cassirer, Bernays, Gödel and Wang), which, in the final section, serves to give an ontological account of (i) the connections be- tween the arithmetical and other aspects of reality and (ii) the applicabality of arithmetic. In the course of the argument the impasse of logicism is briefly highlighted, while a few remarks are made with regard to the logical subject-object relation in connection with Frege's view that number attaches to a concept.

 The limitations  of typological abstraction

Originating in Greece, the employment of typological abstraction continued far be- yond this initial period of Western philosophizing. Its main concern is to account for concretely existing things or entities within reality. It may be surprising to us that Greek thought apparently found a point of rest in the delimitation provided to their world picture by the large world-sea, the Okeanos. The dominant position of Aristotle in Greek philosophy established the classificatory method which disregards distinctive particulars  in order to arrive at more abstract genera. Yet, according to their under- standing, the earth is a circular slice delimited and surrounded  by the Okeanos. It seems strange that they did not transcend the boundaries of the Okeanos. Our own modern acquaintance with the idea of infinity almost automatically forges this ques- tion.

However, in terms of the Greek mind this was impossible. The Okeanos was one of the primal forces that were subdued (and brought within delimitation) when the Olym- pic gods started their reign. The ordered cosmos owes its form, measure, harmony and determination (concept) to these gods. Whatever finds itself outside this limit does not display any form-delimitation and can therefore not be conceived. As a consequence Aristotle does not acknowledge an abstract  (or empty) space. He lacks our modern concept of space. According to the mature Greek understanding space does not exist, only place. Place is a property exclusively attributed to a concretely existing body. In the absence of a body there is no subject for the predicate place. From this it naturally follows that an empty place is the place of nothing – in other words, it is no place at all!

The possibility to understand (and encompass) the ordered cosmos flows from its fi- nite and limited nature – for that reason science is restricted to this finite, limited and ordered cosmos.

During the Middle Ages the Latin formulation of this (entitary-directed) method of classification became well-known in the form of the distinction between a genus proximum and differentia specifica. This method of concept-formation is well at home within domains where a typological classification is required, such as is the case within biology as a discipline.1

Although this kind of hierarchical progress does lead to higher levels of abstraction, it does not move away from entitary reality as such. Being a mammal is not less real than being this or that kind of mammal or even this individual mammal (except, once again, if one joins nominalism in rejecting all universals outside the human mind).

However, the higher (abstracted) levels reached in such a typological classification are still related to and focused upon entities (or at least their structures) and cannot serve as a foundation for those kinds of properties of things which come in sight when the typicality of entities is disregarded.2

For example, just think about the (supposed) abstract nature of mathematical think- ing. Some mathematicians tend to divorce their subject matter from every possible “application.” If some or other part of mathematics turns out to have found an applica- tion outside the domain of mathematics those mathematicians with a platonistic incli- nation find it “miraculous.” David Hilbert, the foremost mathematician of the 20th century, recalls the statement of Gauss, namely that pure number theory is the queen of mathematics and adds the remark that it did not find an application anywhere (Hilbert, 1935:386). Yet, Frege represents a different position in this regard by opposing the distinction  between  pure  and  applied  mathematics.  In  the  second  volume  of  his

1    The modern neo-Darwinian theory of evolution, given its nominalistic orientation (according to which “organisms are not types and do not have types” – Simpson, 1969:8-9), of course handles a different concept of structure altogether.

2    The (universal) conditions for being this or that type of thing must be distinguished from the (universal) way in which particular entities evince their conformity with these conditions. In being an atom or being human, this or that atom / human being shows that it meets the conditions for what it is. The term “struc- ture” is therefore ambiguous. It may refer to the order for (structural law or structural principle for) the existence of a specific type of entities, whereas the structures of these latter reveal what is correlated (and therefore distinct) from the said order for structures of entities. A structure for has the meaning of a law for, while a structure of represents the universal way in which individual entities reveal their con- formity with the given law for its existence (also known as its law-conformity).

 Grundgesetze (1903:§91)3  he writes: “It is applicability alone that raises arithmetic from the rank of a game to that of a science. Applicability therefore belongs to it of ne- cessity” (translation from Dummett, 1995:60).

Surely the difficulty with this distinction is not simply that different subdivisions of mathematics turned out to serve “applications” in other disciplines, but that this em- phasis constantly changed. In the foreword to Volume V of their work on the nature and application of mathematics (with reference to computers, algebra and analysis) Behnke et al. remark that what was abstract yesterday is now considered to be con- crete (for example matrices). What yesterday was seen as pure is today viewed as ap- plied (e.g., functional analysis). What yesterday was under suspicion is now seen as respectable (e.g., the theory of probability). Areas of mathematical reflection that had been considered beyond all possibilities of application later on turned out to be useful in different ways.

Dummett explains that it was Frege's objective to destroy the illusion that whenever mathematics finds an application, a miracle occurred:

The possibility of the applications was built into the theory from the outset; its foundations must be so constructed as to display the most general form of those applications, and then particular applications will not appear a miracle (Dum- mett, 1995:293, cf. 300).

Exploring the meaning of what is here called “the most general form” of a theory can help us to understand the ontic basis of seemingly purely conceptual clusters – but not without first considering the argument advanced by Frege to explain that numbers be- long to the class of “objects” which are both “non-actual” and “objective.” In order to achieve this aim Frege first of all launches a severe attack on the prevalent theories of abstraction as defended by some of his prominent contemporaries.

We have mentioned that the traditional (genus proximum and differentia specifica) method of concept-formation is focused upon concrete entities in reality. Pursuing its path allows for increasingly higher levels of generality. When Frege phrases his objections  against  the  idea  of  abstraction  his  arguments  are  restricted  to  this  kind  of entitary-directed abstraction.

He states that the properties through which entities distinguish themselves from each other are indifferent in respect of their number (1884:40 ff.). He explicitly asks the question from what one should abstract in order to arrive at the number “one” when one starts with the moon as an entity. By abstraction, he proceeds with his argument, one only arrives at concepts such as: “attendant of the earth,” “attendant of a planet,” “celestial body without its own light,” “celestial body,” “body,” “object” (Gegenstand) – and nowhere in this series the number “1” will occur (1884:57; §44).4 Dummett mentions the example where Frege refers to a white cat and a black cat in order to highlight the shortcomings of “abstraction”: “The concept ‘cat’, which has been at- tained by abstraction, indeed no longer includes peculiarities of either; but just for that reason, it is a single concept” (Frege, 1884:45-46; §34; translation by Dummett, 1995:84).

 3    In our subsequent analysis references to Frege (1884 and 1893) will simply be to the relevant para- graphs – thus following the practice employed by Dummett in his penetrating and encompassing work of 1995.

4    Note that this example fully fits the requirements of the genus proximum/differentia specifica method.

 Frege is first of all reacting against the prevailing view that number is to be seen as a set of units – the pure “ones” – obtained, via “abstraction” from the concrete “objects” we can experience. It is exactly at this point where Angelelli thinks that Frege's criti- cism becomes devastating:

by abstracting from the particular differences and natures of the given objects no plurality can be attained, but only one thing (the concept ‘cat’, for example) (Angelelli, 1984:467).

In his remark related to Cantor's definition of a subset (Cantor, 1962:282), Zermelo also refers to the attempt to introduce the notion of “cardinal number” with the aid of a process of abstraction, which would imply that a cardinal number is to be seen as a “set composed of pure ones.”5 If these “ones” are still mutually distinct then they sim- ply provide the elements of a newly introduced set equivalent to the first one, which means that the required abstraction did not help us at all (cf. his remark in Cantor, 1962:351).

Where Kant argues for the synthetic nature of mathematical judgments in his Cri- tique of Pure Reason (CPR), he clearly realizes that pure logical addition (a merely logical synthesis) cannot give rise to a new number (cf. CPR, 1787:15 where he considers the proposition that 7+5=12). In a different way Frege made the same point: entitary-directed abstraction can only proceed to more abstract entities, but can never yield any number as such. The logical addition of “ones” or “twos” cannot but end with the repeated identification of another number of the same kind: having identified a “two” and another “two” still result only in the “abstract” notion of “twoness.”

Number: the road to logical objectivity (Frege)

However, is it possible to distinguish different (modal) properties of one and the same entity? In terms of Frege's example of the moon, we may be more specific: does the moon have any numerical properties? Frege indeed realizes that “number” is an an- swer to the question “how many?” and explicitly discusses this question in connection with the moon (1884:57; §44). But in the absence of a theory of ontic functions (mo- dalities – to be discussed below), he cannot relate the numerical properties of entities to the (universal) ontic meaning of the quantitative aspect of reality and categorically denies that “number” is the “property of something” (Frege, 1884:63; §51). This ques- tion relates to the fundamental numerical question acknowledged by Frege, namely the question: ‘How many’?6 Is the moon “one” or “more than one”? he asks. Obviously, these questions point in another direction – the direction of what may be called “property-abstraction.” This kind of abstraction, surely, is fundamentally different from entitary-directed abstraction.

Yet, there is an even more fundamental issue at stake, because the question “How many?” requires a human response. Are there (universal) ontic features presupposed

5    The cardinality or power of a set disregards any order-relation between its elements. When such an or- der-relation is kept in mind, ordinal numbers are at stake. Counting the “first,” the ”second" and so on therefore employs ordinal numbers. Cantor holds that the concept of a cardinal number emerges when, with the aid of our active thought-capacity, we abstract from the character of the different elements of a given set M and also disregards the order in which they are given (1962:282).

6    In his Grundgesetze Frege distinguishes between cardinal numbers (Anzahlen) – an answer to the question “How many objects of a certain kind are there”? – and real numbers (numbers employed for measurements) (Cf. – Frege, 1903:§157 and Dummett, 1995:64).

 in our answer to this question which are quantitative in nature? Or, alternatively, do we have to revert to the position that number and all universals are creations of human thinking?7 In 1881 Frege wrote in an article on “Booles rechnende Logik und die Be- griffsschrift” (unsuccessfully submitted for publication): “individual things cannot be assumed to be given in their totality, since some of them, such as number for example, are first created by thinking” (quoted by Dummett, 1995:3).8

Frege's primary aim is to develop a logicist thesis according to which arithmetic ought to be reducible to logic. The beginning of the main text of Grundgesetze posits that the aim of Grundlagen was to show that arithmetic is a branch of logic. In the Introduction to Grundgesetze he specifies this aim by saying that “no ground of proof needs to be drawn either from experience or from intu- ition” (translation by Dummett, 1995:3). On the basis of his definition of an “ancestral” given in Begriffsschrift Frege defines natural numbers in Grundlagen (cf. §79, §83): natural numbers are “those objects for which finite mathe- matical induction holds good (Dummett, 1995:12).9

Within the realm of “objective ideas” Frege distinguishes between “objects” and “con- cepts,”  keeping  in  mind  that  the  correlate  of  a  concept  is  an  object  (Dummett,

1995:66). Frege's famous distinction between “Sinn” and “Bedeutung” (sense and ref- erence)  is  one  drawn  within  the  realm  of  the  “objective”  (Dummett,  1995:12). Whereas the Bedeutung (content or meaning) of an expression initially for Frege was at once its significance and what it signified, his new distinction between Sinn and Bedeutung subdivided the former notion of content into the said two components of sense and reference (cf. Frege, 1893:x and Dummett, 1995:67).

The proposal of John Stuart Mill that number could be obtained through inductive generalization  is  ultimately  rejected  by  Frege  because  he  places  natural  numbers within a domain which is not open to sensory perception. In his significant work on substance and function Cassirer explains the limitations of the psychological concep- tion of presentation (in contrast to the “logical meaning of the concept of number”) as follows: “The characteristic relations which prevail in the series of numbers are not thinkable as properties of the given contents of presentation. Of a presentation it is meaningless to say that one is larger or smaller than another, the double or triple of it, that one is divisible by another, etc.” (Cassirer, 1953:33).

Dummett points out that for Frege the concept natural  number comprises “all and only those objects attainable from 0 by reiterating the successor operation” (1995:

63).10

7    In his Principles  of Philosophy Descartes says “that number and all universals are only modes of thought” (Part I, LVIII).

8    In connection with the distinction to be drawn between modal and typical concepts we shall pay atten- tion to the peculiar meaning attached by Frege to the term “quantity.”

9    According to Freudenthal, Dedekind was perhaps the first one (cf. Dedekind, 1887, §59, §80) to call the conclusion from n to n+1 complete induction (“vollständige Induktion”). Neither Bernoulli nor Pascal is the founder of this principle. Its discovery must be credited to Francesco Maurolico (1494-1575) (cf. Freundenthal, 1940:17). In a mathematical context, where “bad induction” is supposed to be excluded (as Freundenthal remarks – 1940:37), no adjective is necessary to qualify the term induction.

10 Since “succession” finds its original “seat” within the numerical mode of reality this insight actually contradicts the logicistic aim of Frege – similar to the way in which, in general, the axiom of infinity ob- structed the attempt to reduce mathematics to logic.

 What is intrinsic to the concepts of arithmetic, according to Frege, preventing them from becoming synthetic – the mistake of Mill and Helmholtz – is the general princi- ple governing all possible applications  (Dummett, 1995:60). What is meant here is that when one answers the question “How many?” one needs to be “quite unspecific as to the type of objects concerning which the question ‘How many?’ could be answered by citing a natural number” (as explained by Dummett, 1995:61). Dummett adds the phrase: “for that reason, it involved no concept peculiar to any non-mathematical subject-matter” (1995:61).11

Where Cassirer introduces, in his rejection of “objects” either belonging to inner or outer reality, the alternative option of “acts of apperception” with which “the numeri- cal determination is connected,” he points out that even a person who wants to restrict knowledge to what is given in the senses “recognizes this universality” (i.e., in terms of our suggestion what we will call below modal universality): “but it understands it, according to its fundamental theory, as a thing-like ‘mark,’ which is uniformly found in a group of particular objects” (Cassirer, 1953:33-34). He then quotes Mill who says: “All numbers, must be numbers of something: there are no such things as numbers in the abstract. But though numbers must be numbers of something, they may be num- bers of anything. Propositions, therefore, concerning numbers, have the remarkable peculiarity that they are propositions concerning all things whatever; all objects, all existences of every kind, known to our experience” (Cassirrer, 1953;33-34 – reference to Mill, A System of Logic, Book II, Chapter 6, 2).

Moreover, the phrase which we have used, namely “the numerical aspect of reality,” implicitly refers to an alternative view of reality, which is foreign to Frege's under- standing of the nature of number, because, according to him (in the words of Dum- mett), “that to which, in general, a number is ascribed is a concept” (1995:74).

When something can act as a causal agent Frege calls it Wirklich. Dummett argues convincingly that it should not be taken to mean real as opposed to ideal, but rather be understood as his “manner of distinguishing between what present-day philosophers usually call ‘concrete’ and ‘abstract’ objects” (Dummett, 1995:80); and numbers be- long  to  the  class  of  “objects”  which  are  objective  but  not  Wirklich (Dummett,

1995:81).

Although Dummett claims that Frege “brilliantly, decisively and definitively” re- futed the abstractionist idea that numbers are to be seen as sets of featureless units (amongst others adhered to by Husserl and Cantor) (1995:82), the implicit assumption continues to be one which (albeit correct in itself!) argues that the inherent limitations of entitary-directed abstraction can never produce a concept of number:

His essential, and crucial, contention in Grundlagen was that abstraction is (at best) a means of coming to grasp certain general concepts: as a mental opera- tion, it has no power to create abstract objects or abstract structures (Dummett,

1995:85).

In all of this we have to appreciate the strength of Frege's argument, particularly evi- denced in the question whether or not the units of an abstract set of featureless units are identical with or distinct from one another. Frege argues that we “cannot succeed

11  Against the background of our suggested distinction between entitary-directed abstraction  and prop- erty-abstraction one may argue in this context that disregarding what is “peculiar to any non-mathemati- cal subject-matter” precisely amounts to what the idea of property-abstraction aims at.

in making different things identical simply by operations with concepts; but, if we did, we should no longer have things, but only a single thing” (Frege, 1884, §35; Dum- mett's translation, 1995:86).12

As abstract, non-actual logical objects (entities), numbers are determinate objects of scientific enquiry:

We speak of ‘the number one’, and indicate by means of the definite article a single, determinate object of scientific enquiry. There are not distinct number ones, but only a single one. In 1 we have a proper name, and, as such, it is as incapable of a plural as ‘Frederick the Great’ or ‘the chemical element gold’ ... Only concept-words can form a plural (Frege, 1884:§38; Dummett, 1995:87).

Frege finally gives the following answer to the question what is a number a number of, that is, what is a number ascribed to: “The content of an ascription of number consists in predicating something of a concept.”13 Dummett summarizes this result as follows: “what a number is ascribed to is a concept” (1995:88). Since the notion of a concept is prior to that of the extension of a concept, a class can be given only as the extension of a concept (Dummett, 1995:92).14 The question is whether the ascription of a number is not much rather an issue of – identifying it – which is different from ascribing  a number to a concept. However, we shall return to this question in connection with what will be called the logical subject-object relation below.

What is “ontic” about “number”?

Epistemology traditionally is well at home with sensory perception (“sense data” in the positivistic legacy) and with the ontic status of (at least: physical) entities (usually simply referred to as “objects”). Measured against this yardstick it is not surprising that the quantitative nature of number is transposed into the domain of thought prod- ucts as creations of the human mind. Particularly when it concerns the nature of infin- ity the temptation seems to be to transfer it to the domain of pure thinking. David Hil- bert (implicitly) still continues this orientation when he argues that after we have es- tablished the finiteness of reality in two directions (with regard to the infinitely small and in respect of the infinitely large), it may still be the case that the infinite does have a justified place within our thinking (I am italicizing – DS)!15

However, already the co-worker of Hilbert, Paul Bernays, holds that the treatment of number presupposes the representation of a totality of numbers as a system of things as well as the totality of sets of number – which is not an arbitrary construction: “One cannot justifiably object to this axiomatic procedure with the accusation that it is arbi-

12  Frege quotes Jevons who says: “It has often been said that units are units in respect of being perfectly similar to one another; but though they may be perfectly similar in some respects, they must be different in at least one point, otherwise they would be incapable of plurality” (cf. Dummett, 1995:86).

13  Supported  by  a  substantial and  convincing  argument  this  is  Dummett's  translational  equivalent  of

Frege's German statement: “dass die Zahlangabe eine Aussage von einem Begriffe enhalte.” (See Frege,

1884:59; §46 and Dummett, 1995:87, 88).

14  As “objects” classes are denoted by singular terms (Dummett, 1995:92), but although numbers attach to the concept of the objects being counted, they are not properties of concepts (cf. Dummett, 1995:96,

108).

15 “Die Endlichkeit des Wirklichen haben wir nun in zwei Richtungen festgestellt: nach dem Unend- lichkleinen und dem Unendlichgroßen. Dennoch könnte es sehr wohl zutreffen, daß das Unendliche in unserem Denken einen wohlberechtigten Platz hat und die Rolle eines unentbehrlichen Begriffes einnimmt” (1925:165).

trary since in the case of the foundations of systematic arithmetic we are not concerned with an axiom system configured at will for the need of it, but with a systematic ex- trapolation of elementary number theory conforming to the nature of the matter (natur- gemäß).”16 Bernays explicitly questions the dominant conception that only one kind of factuality ought to be recognized, namely that of the ”concrete" (Bernays, 1976:122).17

Kattsoff, for example, also makes a plea for the acknowledgement of both physical and mathematical factuality, although “mathematical objects” are “quite different from physical objects”: “They are clearly not the sort of things that can be observed by means of the senses” (1973:30). Through intellectual involvement “mathematical ob- jects” come into sight: “In analogy to physical objects which are called sensory objects because they are observed by the senses, mathematical objects may also be called in- tellectual objects (or rational objects?) because they are observed by the intellect” (1973:33). Later on he calls his approach “quasi-empirical” (1973:40).

It is noticeable that some of the most prominent mathematicians of the 20th century increasingly struggled with the “objective” status of the subject matter of mathematics. Of course, the positivistic legacy, combined with the nature of experimental physics, emphasizing sense perception, constantly challenged mathematicians to come up with an account that will at least be similar (or: analogous) to the supposed way in which perception serve as a gateway to knowledge of reality. Yet its is clear to them that “mathematical objects” are not plainly open to sense perception. In other words, in or- der to account for the subject matter of mathematics there ought to be something “out there”  which  differs  from  the  “concrete”  entities  (“objects”)  we  can  experience through sense perception. The extreme platonist ought to clarify this issue, because the assumed “objective existence” of “mathematical objects” is in need of some other foundation if it is not merely postulated as constructions of the human mind.

The search for something “ontic” in the subject matter of mathematics, however, does not need to exclude a constructive role played by the mathematician. For, if we accept for a moment ontic quantitative properties which may serve as a starting-point for mathematical reflection, then understanding the meaning of these ontic quantitative properties requires human intervention; it needs the reflective and constructive disclo- sure and opening-up of this meaning through the formation of mathematical theories. If so, then what is the relationship between these ontic quantitative properties and the particulars of our (numerically specified) answer to the question: how many? To put it differently: in order to relate to a given unity or multiplicity humans have to grasp this quantitative meaning with the aid of numerals (numerical symbols). This entails that we can identify and designate ontic quantitative properties in assigning numbers to them through this act of identification and designation.

With an appeal to Gonseth we find an account of the subject matter of mathematics in the thinking of Bernays which does want to maintain a connection between experi- anceable  things  (“phänomenalen  Gegenständlichkeit”)  and  the  contents  acquired

16  “Gegen dieses axiomatische Vorgehen besteht auch nicht etwa der Vorwurf der Willkürlichkeit zu Recht, denn wir haben es bei den Grundlagen der systematische Arithmetik nicht mit einem beliebigen, nach Bedarf zusammengestellten Axiomensystem zu tun, sondern mit einer naturgemäßen systema- tischen Extrapolation der Elementare Zahlenlehre” (Bernays, 1976:45).

17  “Es scheint, daß nur eine vorgefasste philosophische Ansicht dieses Erfordernis bestimmt, die Ansicht nämlich, daß es nur eine Art von Tatsächlichkeit geben könne, diejenige der konktreten Wirklichkeit” (Bernays, 1976:122).

through processes of idealization and abstraction (this content has a structural charac- ter – Bernays, 1976:180).18  In particular mathematical idealization comes to expres- sion through axiomatization (Bernays, 1976:181).

The fact that Bernays, as we have mentioned above, questions the claim that there is only one kind of factuality, namely that of the “concrete,” urges us to investigate this issue in more detail, with particular reference to the conceptions of Gödel, Wang and Cassirer.

Gödel introduces the idea of “semiperceptions” when it concerns “mathematical ob- jects.” Next to a physical causal context within which something can be “given,” Gö- del refers to data of a second kind which are open to “semiperceptions.” The data of this second kind “cannot be associated with actions of certain things upon our sense organs” (quoted by Wang, 1988:304). Gödel says:

It by no means follows, however, [that they] are something purely subjective as Kant says. Rather they, too, may represent ‘an aspect of objective reality’ (my emphasis – DS), but, as opposed to the sensations, their presence in us may be due to another kind of relationship between ourselves and reality (quoted by Wang, 1988:304).

Wang is “inclined to agree with Gödel,” but he does “not know how to elaborate his assertions” (Wang, 1988:304). He says that he “used to be troubled by the association of objective existence with having a fixed ‘residence’ in spacetime,” but he now feels “that ‘an aspect of objective reality’ can exist (and be ‘perceived by semiperceptions’) without its occupying a location in spacetime in the way physical objects do” (Wang,

1988:304).

Clearly, Gödel and Wang contemplate the “reality” of “ontic” (designated by them as: “objective”) “aspects of reality” which are not like “concrete entities” occupying “a location in spacetime.”

If we connect this kind of non-entitary reality with the (above-mentioned) arithmeti- cal question formulated by Frege, namely the fundamental numerical question How many?, 19 then some significant insights of Cassirer may be helpful in this context.

His basic concern in 1910 is to make a distinction between entity (designated by him as: substance) and function. Although used in a slightly different context, his employ- ment of the terms “what” and “how” (1953:40) may be elucidating. An answer to the what-question results in the identification of an entity (or a certain kind of entity), whereas the answer to the how-question specifies a function (aspect) of reality. Cassirer approximates the position of Frege closely even though he uses the notion of “abstraction.” He does that, however, while having in mind not merely the entitary- directed kind of abstraction against which Frege argued. The act of abstraction has as its aim “to bring out the meaning of a certain relation independently of all particular cases of application” (Cassirer, 1953:39). Keeping in mind that “relational concepts” are for Cassirer the same as function concepts, his following statement is significant for the distinction between aspects (functions) and entities (things, so-called “ob- jects”):

18  “Die mathematische Gegenständlichkeit geht durch Idealisierungs- und Abstraktionsprozesse hervor aus der phänomenalen Gegenstandlichkeit des Strukturellen.”

19  As an answer to the question “How many objects of a certain kind are there” (Frege, 1903:§157 and

Dummett, 1995:64).

The fucntion of ‘number’ is, in its meaning, independent of the factual diversity of the objects which are enumerated. ... Here abstraction ... means logical con- centration on the relational connection as such (Cassirer, 1953:39).20

Surely, the mentioned considerations of Gödel, Wang and Cassirer call for an ac- knowledgment of a function or an aspect of reality which is ontically given but not given in the same way as physical (or other kinds of) entities. Understanding the meaning of the ontically given aspects or functions indeed requires a formative (con- structive) human activity, in its scholarly sense performed by mathematicians who are capable, through their articulation of mathematical theories, to make this given nume- rical meaning explicit and to disclose it in their erection of mathematical structures. This perspective is an argument in opposition to both platonism and pure constructivism. Against platonism it acknowledges that thinking about the meaning of number does not simply rest on the acceptance of an already existing transcendent (ideal) world of “mathematical objects” of the thinking human mind, since without the intervention of human thinking the ontically given meaning of quantity (plurality) can- not be disclosed and articulated  in mathematical structures. By contrast, it also op- poses constructivism in acknowledging that the subject matter of mathematics is not merely the product of the thought-activities of mathematicians, because such activities presuppose at least “an aspect of objective reality.”

So-called “evolutionary epistemology” runs into serious difficulties if it wants to ex- plain the ontic givenness of plurality (multiplicity), because long before the (paleonto- logical) appearance of human beings, entities (note the plural form: entities!) did func- tion in the quantitative mode of reality, did occupy space, did move, and so on. Furthermore, if this epistemology starts from sense-perception the only option open is that of entitary-directed abstraction – which is criticized by Frege as being incapable of justifying our concept of number. We have seen that another kind of abstraction is required, namely modal abstraction. The property argued for in the next paragraph, namely (modal) universality, is also inexplicable merely in terms of sense-perception.

The universality of functional  modes

We have noted that when Frege employs the term ‘logical’ (as in the phrase: “logical objects”), he always has the following distinctive feature of “the logical” in mind – in the words of Dummett: “its generality: it does not relate to any special domain of knowledge, for, just as objects of any kind can be numbered, so objects of any kind can belong to a class” (Dummett, 1995:224).

This view of Frege actually pertains to a common feature of each and every aspect of reality, not only to the logical aspect. It is part of the nature of an aspect or modus that its scope transcends the multiplicity of individual things merely functioning within it. This constitutes what ought to be designated as the modal universality to which we have alluded above – amongst others also holding for the quantitative aspect of real- ity. This modal universality of the numerical aspect clearly surfaces in Frege's account, as explained by Dummett:

A correct definition of the natural numbers must, on his view, show how such a number can be used to say how many matches there are in a box or books on a shelf. Yet number theory has nothing to do with matches or with books: its

20  Here Cassirer highlights what we have called property-abstraction (modal abstraction).

business in this regard is only to display what, in general, is involved in stating the cardinality of objects, of whatever sort, that fall under some concept, and how natural numbers can be used for this purpose (Dummett, 1955:272).

At this point Frege (and Dummett) were on the verge of the alternative theory which we have in mind when we speak about modal universality – if it was not the case that Frege identified the ontic meaning of the arithmetical aspect of reality with the nature of a concept. This theory first of all eludicates the uniqueness of functions as distinct from entities. In conceptual thought this difference surfaces in the distinction between the concept of number and type concepts.

Whenever (empirical) investigation, with the aid of entitary-directed abstraction as explained, arrives at the concept of this or that type of entity, there is an inherent re- striction to the number of entities involved, namely only to those belonging to that type.

Stated differently, i.e., in terms of a law-perspective: the type-law of a specific kind of entities only holds for that type of entity, i.e., for a limited class of entities. For ex- ample, the law for being an atom is limited to atoms and does not apply to molecules or planets. Similarly, the Coulomb law (applicable only to charged physical entities) and the Pauli principle (applicable to fermions) do not hold for all possible (physical) entities. Rather than delineating a limited class of entities, “property laws” (i.e., modal functional laws), however, have a generality (universality) which embraces all possi- ble entities because they “describe a mode of being, relatedness, experience, or expla- nation” (as phrased by Stafleu, 1980:11). The fact that modal laws – such as those of quantum physics – hold for all possible “objects,” is clearly observed by the German physicist Von Weizsäcker: “Quantum theory, formulated sufficiently abstract, is a universal theory for all Gegenstandklassen (classes of objects)” (1993:128). When he explains, on the next page, that one cannot deduce the kinds of entities of experience from the universal scope of quantum theory, he gives his own account of what we are designating as type laws.

This distinction shows a similarity with the way in which Frege employs the word

quantity. Dummett writes:

Frege so uses it that a phrase like ‘2.6 metres’ designates a specific quantity of one kind, ‘5.3 seconds’ a quantity of another kind, and so on. He thus takes quantities to be objects, distinct from numbers of any kind. There cannot be two equal quantities, on this use: if two bodies are equal in mass, they have the same mass. Quantities fall into many distinct types: masses form one type, lengths another, temperatures a third (Dummett, 1995:270).

Frege implicitly distinguishes between the general (modally universal) meaning of number and the specifications it receives when it is attached to different types of quan- tities – in which case he does not speak about number but about quantity.

Compare in connection with our notion of modal universality also the position of Russell and Gödel. From Russell's Introduction to Mathematical Philosophy Gödel quotes the second half of the following sentence: “Logic, I should maintain, must no more admit a unicorn than zoology; for logic is concerned with the real world (my emphasis – DS) just as truly as zoology, though with its more abstract and general fea- tures” (Wang, 1988:313). Whereas zoology has a foremost interest in living entities

(animals), logic, with its concern for the “more abstract and general,” operates on the level of modal universality.

In yet another fashion Dummett implicitly alludes to the nature of modal universal- ity in the following explanatory statement:

The contrast between arithmetical and empirical enquiry concerns not so much the discovery of individual objects as the delineation of the area of search. The astronomer need have no precise conception of the totality of celestial objects: he is concerned with detecting whatever is describable in physical terms and lies, or originates, outside the earth's atmosphere, and he need give no further specification of this ‘whatever’. In mathematics, by contrast, an existential con- jecture, to have any definite content, requires a prior circumscription of the do- main of quantification (Dummett, 1995:228).

Compare another instance of some remarks (the second one already mentioned in a similar context) by Dummett intended to explain Frege's concept of number, which also explicitly highlights this feature of modal universality (the universal scope of the numerical aspect of reality, in the thinking of Frege – mediated by the nature of “the logical / a concept”):

In view of the generality of number – the fact that there is no restriction on the type of objects of which we can say how many there are – ... (Dummett, 1995: 73); The term “logical”, in the phrase “logical objects”, refers to what Frege always picked out as the distinguishing mark of the logical, its generality: it does not relate to any specific domain of knowledge, for, just as objects of any kind can be numbered, so objects of any kind can belong to a class (Dummet, 1995:224).

Given his logicist prejudice Frege did not realize that his notion of the general appli- cability of arithmetic actually points in the direction of the modal or functional univer- sality of the arithmetical aspect of reality. This means that the logical and the arith- metical modes both share this feature of modal universality. However, as we shall see, the foundational position of the numerical aspect with respect to the logical aspect jus- tifies the view that the mathematical treatment of the arithmetical represents a higher level of abstraction. Bernays also defends this position: “However, in respect of the formal, the mathematical perspective, as opposed to the logical conceptual one, repre- sents the standpoint of a higher abstraction”.21

The implication of the stance taken by Bernays is that the arithmetical mode is foun- dational to the logical mode and that it is therefore not possible to reduce arithmetic to logic.

The cul de sac of logicism

Bernays noticed that whereas both Frege and Dedekind throughout their mathematical reflections and proofs evinced the most sincere and strict precision, they were wholly negligent in assuming the representation of a closed totality of all possible conceivable logical objects.22

21  “In Hinsicht auf das Formale stellt aber, ..., die mathematische Betrachtung gegenüber der begrifflich logischen den Standpunkt der höheren Abstraktion dar” – Bernays, 1976:27.

22  “So waren Frege und Dedekind, deren Beweisführungen und Überlegungen sonst überall durch äußerste

Präzision und Strenge ausgezeichnet sind, ganz unbedenklich in dem, was sie als vermeintlich selbst-

David Hilbert also points at the dilemma entailed in the logicist attempt to deduce the meaning of number from that of the logical-analytical mode. In his Gesammelte Abhandlungen Hilbert writes:

Only when we analyze attentively do we realize that in presenting the laws of logic we already had to employ certain arithmetical basic concepts, for example the concept of a set and partially also the concept of number, particularly as cardinal number [Anzahl]. Here we end up in a vicious circle and in order to avoid paradoxes it is necessary to come to a partially simultaneous develop- ment of the laws of logic and arithmetic (1970:199).

Cassirer first of all approaches this problem in terms of the numerical analogy within the logical-analytical aspect. His question is that it is not understandable why one only accepts logical identity and diversity, which enter the set concept as necessary ele- ments, as such basic functions, but that one does not do the same with regard to nu- merical unity and difference. He claims that a truly satisfactory deduction of the one from the other is also not achieved by set theory, which entails a persistent suspicion that all similar attempts will continue to harbor a concealed epistemological circle.23

That the logicist position taken by Frege is unsound was brought into the open when Russell made the absurdity of set theory known in 1900. Consider the set C with ele- ments A and the prescription that elements of set C may only be those sets A which do not contain themselves as elements.

Thus C = (A/A -e A). (The set of ten chairs is e.g. not itself a chair and does not con- tain itself as an element. On the other hand the set of thinkable thoughts is in itself thinkable and therefore does contain itself as an element.) Now suppose that C is an element of C (C e  C). Every element of C, however, does not contain itself as an ele- ment – this, after all, is the requirement for being an element of C. This implies that if C is an element of C, it must also meet this requirement – but then C e C implies C -e C! Suppose on the other hand that C -e C. Then C does meet the requirement for being an element of C, which means that C e C. In other words, C is an element of C if and only if C is not an element of C.24

Dummett mentions that Frege discovered by August 1906 that the flaw in his ac- count cannot be corrected within the framework of his theory.25

The assessment given by Dummett is straight-forward:

verständliche Voraussetzung dem Standpunkt der allgemeinen Logik zugrunde legten, nämlich in der Vorstellung von einer abgeschlossenen Gesamtheid aller überhaupt denkbaren logischen Objekte” (Bernays, 1976:47).

23  “In der Tat ist nicht einzusehen, warum man lediglich logische Identität und Verschiedenheit, die als notwendige Momente in den Mengenbegriff eingehen, als solche Urfunktionen gelten lassen und nicht auch die numerische Einheit und den numersichen Unterschied von Anfang an in diesen Kreis aufnehmen will. Eine wirklich befriedigende Herleitung des einen aus dem anderen ist auch der mengentheoretischen Auffassung nicht gelungen, und der Verdacht eines versteckten erkenntnis- theoretischen Zirkels blieb gegenüber allen Versuchen, die in dieser Richtung gemacht werden, immer bestehen” (Cassirer, 1957:73-74).

24  Of course this conclusion presupposes the application of the principle of the Excluded Middle (i.e., the acceptance of infinite totalities) – something rejected by intuitionistic mathematics.

25  That is, “with the abstraction operator as primitive and an axiom governing the condition for identity of value-ranges: but the underlying error lay much deeper than a misconception concerning the foundations of set theory. It was an error affecting his entire philosophy” (Dummett, 1995:223).

Frege had answers – by no means always the right answers – to all the philo- sophical  problems  concerning  the  branches  of  mathematics  with  which  he dealt. He had an account to offer of the applications of arithmetic; of the status of its objects; of the kind of necessity attaching to arithmetical truths; and of how to reconcile their a priori character with our attainment of new knowledge about arithmetic. His view of the status of the numbers, ontological and episte- mological, proved to be catastrophically wrong; for the last nineteen years of his life, he himself acknowledged it to have been wrong, and regarded that as bringing with it the collapse of his entire philosophy of arithmetic. In spite of efforts like those of Crispin Wright to defend it, we can clearly see that his view of this question was in error: but we have not supplied any very good al- ternative. In answering the remaining questions, we have not, save in one cru- cial respect, advanced very far beyond Frege at all (Dummett, 1955:292).

Logicism had to concede that the existence of an infinite totality cannot be proven. Consequently, infinity was introduced by an Axiom of Infinity.26

This concession had another consequence: no longer accepting the idea of the total- ity of all logical “objects” entails that also the acceptance of the totality of predicates became problematic.27

Once again: The “miracle” of applicability

In conclusion we have to return briefly to the issue of the applicability of mathematics.

In his work on “Warrant and Proper Function” Plantinga (1993:232, note 2) eluci- dates this “miracle” or “mystery” as follows:

This hasn't been lost on those who have thought about the matter. According to Erwin Schrödinger, the fact that we human beings can discover the laws of na- ture is “a miracle that may well be beyond human understanding” (What is Life? [Cambridge: University of Cambridge Press, 1945], p.31). According to Eugene Wigner, “The enormous usefulness of mathematics in the natural sci- ences is something bordering on the mysterious, and there is no rational expla- nation for it” (“The Unreasonable Effectiveness of Mathematics in the Natural Sciences,” in On Pure and Applied Mathematics, [13, p.2]) and “It is difficult to avoid the impression that a miracle confronts us here, quite comparable in its striking nature to the miracle that the human mind can string a thousand argu- ments together without getting itself into contradictions, or to the two miracles of the existence of laws of nature and of the human mind's capacity to derive them” (p.7). And Albert Einstein thought the intelligibility of the world a “mir- acle  or  an  eternal  mystery”  (Lettres  à  Maurice  Solouine  [Paris:  Gauthier- Villars, 1956], p.115).

26 Cf. Bernays: “Die Logistik verzichtet seither darauf, die Existenz einer unendlichen Gesamtheit zu beweisen und stellt vielmehr ausdrücklich ein Unendlichkeitsaxiom auf” (Bernays, 1976:47). Fraenkel et al remark: “It seems, then, that the only really serious drawback in the Frege-Russell thesis is the doubt- ful status of InfAx [InfAx = Axiom of Infinity – DS], according to the interpretation intended by them” (1973:186).

27 Cf. the remark of Bernays: “Mit der Preisgabe der Vorstellung von der Gesamtheit aller logischen Gegenstände wird aber auch die Vorstellung von der Gesamtheit aller Prädikate problematisch, und bei näherem Zusehen zeigt sich hierin eine besondere grundsätzliche Schwierigkeit” (Bernays, 1976:47).

We have quoted Dummett's account of Frege's position: “A correct definition of the natural numbers must, on his view, show how such a number can be used to say how many matches there are in a box or books on a shelf. Yet number theory has nothing to do with matches or with books: its business in this regard is only to display what, in general, is involved in stating the cardinality of objects, of whatever sort, that fall un- der some concept, and how natural numbers can be used for this purpose” (Dummett,

1955:272).

As soon as one acknowledges that “objective reality” displays an ontically given nu- merical (arithmetical) aspect,  the quoted phrase: “Yet number theory has nothing to do with matches or with books” has to be corrected. Material things – and whatever else belong to the dimension of entities within reality – invariably function within the numerical (and other) modes/aspects of reality. In so far as “matches” or “books on a shelf” therefore function within the numerical mode, this functioning has everything to do with the meaning of this aspect!

But if entities cannot avoid to have specific functions within the various modes of reality, it stands to reason that the universal modal laws obtaining within the numerical (and spatial) aspect(s) will hold for every kind of entity functioning within this (these) mode(s) – notwithstanding the fact that, for example, the application of arithmetical theories may acquire peculiar specifications within the context of different types of non-arithmetical relations or entities.28 This affirmation, however, presupposes the dis- tinctness and mutual coherence between the dimensions of modal aspects and that of concretely existing entities.

Because “(modal) abstraction” ultimately does not cut all ties with “reality,” mathe- matics will remain applicable – albeit for reasons different than those sustained by Frege and other platonists.29

In addition it should be pointed out that the applicability of, for example, arithmeti- cal insights, is not only dependent upon the fact that the arithmetical mode conditions whatever functions within it in a quantitative way, since other aspects of reality also stand in a relation of mutual coherence with the numerical aspect. This already sur- faced when we quoted Hilbert in respect of the fact that the development of logic is founded in a simultaneous development of arithmetic. To bring it closer to the “home” of arithmetic: topology and geometry, in their analysis of the meaning of space, are in- conceivable without the foundation of number, as is seen from terms such as (spatial) magnitudes and (spatial) dimensions – which undeniably reflect the foundational meaning of number. Although topology disregards metrical properties, it still employs the notion of open sets, which alludes to the spatial mode.

The interconnection between the arithmetical and the logical mode, furthermore, comes to expression in the logical subject-object relation. If the core meaning of the analytical mode (the logical aspect) is given in identification and distinguishing, then every act of identification objectifies within the logical mode whatever is identified and distinguished. Objectification is an act of an analytical subject. Surely, this activ-

28  Just recall Frege's distinciton between number and quantities (cf. Dummet 1995:270).

29  Keeping in mind the “non-logicist platonism” of Gödel (cf. Dummett, 1955:301), we may here refer to Dummett's words: “Logicism is not the most natural ally of platonism, because, on the most natural view of logic, there are no logical objects: it was a tour de force on Frege's part to combine a vehement advocacy of platonism with an unreserved logicism about number theory and analysis” (Dummett, 1955:

301).

ity is normed, which means that one can arrive at logically correct or incorrect (norm- conforming or antinormative) acts of objectification. Identification is actually nothing but the way in which we form a concept of something. A concept is constituted as a logical unity in the multiplicity of – (universal) features unified in it. In order to form a concept of number, logical acts of identification and distinction are involved.

What then about Frege's contention that there are not distinct numbers one, but only a single one, which is incapable of a plural?

Frege here emphasizes logical identity at the cost of the implied universal side of the property involved. Wherever the number “one” is applied to something, or wherever theoretical reflection enters into an analysis of the meaning of the number “one,” the universal orderliness of this number comes to expression. The conceptual “oneness” of the number “one” relates to its universality, its orderliness, i.e., to the way in which every instantiation of “oneness” testifies to the underlying condition for “being-one,” which makes it possible in the first place.30

The primary issue is therefore that of universality and specific instances and not that of one and many (plurality) as such.

Owing to his logicist bias, Frege inverted the order of foundation of the logical mode. Number does not in the first place attach to a concept – the primary vehicle of logical objectification – because the quantitative meaning of number can only be objectified in acts of identification, i.e., through the formation of number concepts. But the original domain to which the meaning of number “attaches” is the quantitative aspect of “objective reality” (in its ontic “givenness”). Only in a secondary (derived, analogical) sense does it relate to concepts.

In his attempt to develop number theory by eliminating mathematical primitives, Zalta mentions Frege's question: “How do we apprehend numbers given that we have no intuitions of them?” (1999:620). His theory concerns numbers which “are ab- stracted from the facts about concrete objects” and he adds that “the resulting numbers are even more closely tied to their application in counting the objects of the natural world than Frege anticipated” (1999:620). However, he has promised to discuss these philosophical issues more fully at another occasion. What is nonetheless clear from his current article is that he does not realize that the underlying logic employed by him presupposes the primitive meaning of all the pre-logical aspects of reality. (See the concluding remark of this article.)

Every logical unity and multiplicity presuppose a (foundational) arithmetical unity and multiplicity. In fact, within every aspect of reality distinct from the numerical one there are analogical specifications of the original quantatitive meaning of number. For example, within space we meet vector addition (where 3 + 4 equals 5 instead of 7); within the kinematical aspect Einstein employed the Lorentz transformations in terms of which 0.9c + 0.9c equals 1.80/1.81c (in stead of 1.8c); and so on.

The numerical aspect is therefore related to entitary reality through its modal univer- sality – evidenced in the fact that every entity whatsoever has a function within the arithmetical aspect – explaining the applicability of number theory, while it is connected to all other aspects of reality through its analogical appearances within each of them.

30  Similarly, wherever we encounter “gold” it is an instantiation of the the (universal) conditions for be- ing-gold.

Concluding  remark: A theory of modal aspects

Having reached this stage of our argument, the natural thing would be to explore more fully the basic elements of a constructive theory of modal aspects (functions). Up to this point the issues addressed were discussed in terms of the emerging problem of non-entitary ontic properties. In order to provide a first assessment it was necessary to introduce a minimal explanation of a theory of modal aspects.

However, since explaining such a theory in more detail will exceed the limits of the present article, the best option seems to be to treat such a theory in a follow-up article. As a point of departure this article will commence by investigating a well-known ex- pression within the English language: “Physical Objects?” The question mark suggests that the theory of modal aspects entails the argument that insofar as material things are physical they are not objects but subjects, and insofar as they are objects they are not physical for in the latter case they ought to be understood according to non-physical properties (so-called object-functions). Elucidating the grounds for this claim will then open the way to return to the outcome of our analysis of the position of Frege in this article. What will then be explained in more detail are the criteria underlying the asser- tion of non-entitary ontic aspects of reality.

Literature

Behnke, H., Bertram, G., Collatz, L., Sauer, R., & Unger, H. (eds.) (1968): Grundzüge der Mathematik, Volume V, Praktische Methoden und Anwendungen der Mathe- matik, Göttingen : (Publisher not known).

Cantor, G. (1962): Gesammelte Abhandlungen Mathematischen und Philosophischen Inhalts, Hildesheim (1932).

Cassirer, E. (1910): Substanzbegriff und Funktionsbegriff, Berlin (1910), Darmstadt

1969.

Cassirer, E. (1953): Substance and Function, New York 1953 (first edition of the Eng- lish translation of Substanzbegriff und Funktionsbegriff: 1923. First German edi- tion 1910).

Cassirer, E. (1957): Das Erkenntnisproblem in der Philosophie und Wissenschaft der neueren Zeit, Stuttgart: Kohlhammer Verlag.

Cassirer, E. (1958):The Library of Living Philosophers, The Philosophy of Ernst Cas- sirer, edited by P.A. Schilpp, New York: Tudor Publishing Company, First edition (second printing).

Dedekind, R. (1901): Essays on the Theory of Numbers, Chicago.

Dedekind, R. (1887): Was sind und was sollen die Zahlen, Braunschweig (10th ed.)

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Press, Cambridge, Second Printing.

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Theory, 2nd revised edition, Amsterdam; North Holland.

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(“Unveränderter Neudruck” – unaltered reprint, 1934).

Frege, G. (1893): Grundgesetze der Arithmetik, Vol.I. Jena. Frege, G. (1903): Grundgesetze der Arithmetik, Vol.II Jena.

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Hilbert, D. (1925): Über das Unendliche, Mathematische Annalen, Vol. 95, 1925 (pp.161-190).

Hilbert, D. (1930): Naturerkennen und Logik, in: Hilbert, 193 (pp.378-387).

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Husserl, E. (1970): Philosophie der Arithmetik (1891), Husserliana, The Hague: Martinus Nijhof, Vol. XII.

Kattsoff, L.O. (1973): On the Nature of Mathematical Entities, in: International Logic

Review, Number 7, 1973 (pp.29-45).

Meschkowski, H. (Editor) (1972): Grundlagen der modernen Mathematik, Darmstadt. Plantinga, A. (1993): Warrant and Proper Function, Oxford: Oxford University Press. Russell, B. (1956): The Principles of Mathematics, London (1903).

Schilpp, P.A. (Editor) (1958): The Library of Living Philosophers, The Philosophy of Ernst Cassirer, New York: Tudor Publishing Company, First edition (second printing).

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view of The Philosophy of Bertrand Russell, American Mathematical Monthly,

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Wang, Hao (1988): Reflections on Gödel, MIT Press, Cambridge, Massachusetts.

Weyl, H. (1966): Philosophie der Mathematik und Naturwissenschaft, 3rd revised and extended edition, Vienna.

Zalta, E.N. (1999): Natural numbers and natural cardinals as abstract objects: a partial reconstruction of Frege's Grundgesetze in object theory, in: Journal of philosophical logic, Volume / Issue 28, 6, pp.619-660.

I shall upload the two requested articles to ResearchGate.

Danie

Dear Mohammed, the notion of infinity is crucial for an understanding of modern mathematics and it deserves a discussion in its own right. Send to me your e-mail address and then I shall send to you a Book which I published in 2002 on the philosophical foundations of the modern natural sciences - Chapter 2 discusses maths with special attention to infinity. Intuitionism, for example, merely accepts the potential infinite (which I prefer to call the successive infinite), while axiomatic formalism also accepts the actual ifinite (my designation is the "at once infinite"). If the latter kind of infinity is rejected (as intuitionism does) Cantor's diagonal proof for the non-denumerability of the real numbers does not hold. On the basis of a new understanding of the aspects of number and space I have developed an alternative view on these two kinds of infinity that transcends the traditional opposition of these two views.

 Dear Daniel, does your proof disqualify the proof or the actual uncountability of the set of real numbers ? If it is the actual theorem, that means you have shown that the set of all reals is countable. This by it self disqualifies the continuum hypothesis and other long standing truths of mathematics.

In present traditional finite—infinite theory system, people have been creating many new “understandings” on “infinite”, “potential infinite” and “actual infinite” since Zeno’s time 2500 years ago. But it is difficult to solve those infinite related problems produced by the fundamental defects disclosed by the infinite related paradoxes since Zeno’s time, because within the present traditional finite—infinite theory system, “the infinite related problems” are strongly interlocked together with the foundation. So, though trying very hard willing to solve “some infinite related problems” with some new “understandings” within the present traditional finite—infinite theory system, but people finally discovered that nothing can be done because “everything is perfect” in present traditional finite—infinite theory system.

I know few people agree with me, but this is true.

Dear Dejenie, Ramon and Geng – I shall briefly respond to each of you.

Dejenie

Cantor's diagonal proof for the non-denumerability of the real numbers presupposes the acceptance of the at once infinite – which explains why intuitionistic mathematics, while rejecting the at once infinite, only arrives at denumerability. Send me your e-mail address and then I shall send a more extensive explanation of what is at stake!

Danie Strauss

[In 2006 I presented a paper on infinity and continuity:

Infinity and Continuity, The mutual dependence and distinctness of multiplicity and wholeness

Paper presented at the

Free University of Brussels

October 15, 2006

Infinity and Continuity

Table of Contents

Introduction [1]

Probing some foundational designs [2]

The impasse of logicism [2

]Contradiction and the meaning of analysis [4]

Tacit assumptions of axiomatic set theory[6]

The primacy of natural numbers and their succession (induction)  [7]

Ordinality versus cardinality  [ 8]

Wholeness and Totality – the irreducibility of the Whole-Parts Relation  [10]

Some crucial structural features.  [13]

The inevitability of employing analogical elementary basic concepts  [14]

The theory of modal aspects  [16]

Numerical and spatial terms  [18]

Deepening our understanding of infinity [21]

Mathematics and Logic [24]

The circularity entailed in set theoretical attempts to arithmetize continuity [25]

Aristotle and modern mathematics – infinity and continuity [27]

Aristotle and Cantor: an ‘intermodal’ solution  [30]

Concluding remark [32]

Literature [33]

Sketches [37]

Index of Subjects [39]

Index of Persons [41]

 Danie

Ramon

Thanks for your remarks! Perhaps you can proceed and read the section of my article on Frege that deals with “What is Ontic about Number?” and see if it does not shed some light on your concerns.

Note that my account rests on the acknowledgement of aspects (modes of being) referring the the how and not the what of reality. From Latin we inherited expressions such as modus operandi and modus vivendi in which the how is represented by the term “modus.” An aspect is therefore a specific (unique) mode of reality. In a general sense it is a modus quo or a mode of being. It provides a framework within which everything and all processes within reality function. As an equivalent for referring to facets, aspects or functions, one can therefore also speak about modalities, modal aspects or modal functions. Consider the first seven aspects in their succession: number, space, movement (the kinematic), the physical, the biotic, the sensitive and the logical-analytical modes.

Every successive aspect is founded on the previous aspects. Allow me just to give one illustration, formulated by the foremost mathematician of the 20th centry, David Hilbet concerning the foundational role of the meaning of number with respect to that of the meaning of analysis (the logical-analytical aspect). Every attempt to deduce the meaning of number from the meaning of analysis (or: logic) is faced with a vicious circle. Cassirer is also quite explicit in this regard. He claims that a critical analysis of knowledge, in order to side-step a regressus in infinitum, has to accept certain basic functions which are not capable of being “deduced” and which are not in need of a deduction. David Hilbert also points at this “catch 22” entailed in the logicist attempt to deduce the meaning of number from that of the logical-analytical mode. In his Gesammelte Abhandlungen Hilbert writes: “Only when we analyze attentively do we realize that in presenting the laws of logic we already had to employ certain arithmetical basic concepts, for example the concept of a set and partially also the concept of number, particularly as cardinal number [Anzahl]. Here we end up in a vicious circle and in order to avoid paradoxes it is necessary to come to a partially simultaneous development of the laws of logic and arithmetic” (GA – Vol.III 1970:199).

Danie

Geng

You say that “everything is perfect” in the “traditional finite—infinite theory system”. Are you acquainted with the initial aim of David Hilbert to prove the consistency of mathematics? And do you know what happened with his program? [At the young age of 25 Gödel astounded the mathematical world in 1931 by showing that no system of axioms is capable – merely by employing its own axioms – to demonstrating its own consistency (see Gödel, 1931).

Yourgrau remarks: “Not only was truth not fully representable in a formal theory, consistency, too, could not be formally represented.”

Or in the words of Grünfeld: “Gödel proved that if any formal theory T that is adequate to include the theory of whole numbers is consistent, then T is incomplete. This means that there is a meaningful statement of number theory S, such that neither S nor not-S is provable within the theory. Now either S or not-S is true; there is then a true statement of number theory which is not provable and so not decidable. The price of consistency is incompleteness.”

Weyl summed it up: “It must have been hard on Hilbert, the axiomatist, to acknowledge that the insight of consistency is rather to be attained by intuitive reasoning which is based on evidence and not on axioms.”

Danie

Dear Mr. Daniel F M Strauss, it is good to have your very insightful opinion, thank you.

Would you please tell me your frank opinion on following example?

Look at the widely accepted modern divergent proof of Harmonic Series.

1＋1/2 ＋1/3＋1/4＋．．．＋1/n ＋．．．                                 （1）

=１＋1/2 ＋（1/3＋1/4）＋（1/5＋1/6＋1/7＋1/8）＋．．．    （２）

>1+ 1/2 ＋( 1/4＋1/4 )+（1/8＋1/8＋1/8＋1/8）＋．．．           （3）

=1+ 1/2 + 1/2 + 1/2 + 1/2 + ．．．------>infinity                            (4）

Are the Un--->0 items in Harmonic Series “potential infinite” or “actual infinite”? How to treat them? No theories in the world now (including the theories by Gödel and David Hilbert)  can tell scientifically whether or not we can produce infinite numbers each bigger than 1/2 or 1 or 100 or 100000 or 10000000000 or… from infinite items (infinitesimal) in Harmonic Series by “brackets-placing rule" to change an infinitely decreasing Harmonic Series with the property of Un--->0 into any infinite constant series with the property of Un--->constant or any infinitely increasing series with the property of Un--->infinity.

Harmonious Series Paradox is a visible and touchable newly discovered “strict mathematically proven” modern ancient Zeno’s Paradox of Achilles--Turtle Race: the “brackets-placing rule" to get 1/2 or 1 or 100 or 100000 or 10000000000 or… from infinite items in Harmonic Series corresponds to different runners with different speed in Zeno’s Paradox while the items in Harmonic Series corresponds to those steps of the tortoise in Zeno’s Paradox.

Such cases (many) are the biggest reason for my statements about the defects in theoretical and practical cognitions of 'potential' and 'actual' infinite in present science theory system.

Thank you again.

Sincerer yours, Geng

Dear Geng

The issue involved in distinguishing between the successive infinite and the at once infinite does not hinge on the proof of the divergence of the Harmonic Series. What is at stake could be explained by simplifying the issue to the question: Is 0.999 . . . not equal to 1.000 . . . or is it equal to it?

Any succession of numbers [be it (0.90; (0.99); . . . or (1); (1/2); (1/2); . . . ] could be understood in terms of the most primitive meaning of the infinite: one, another one, yet another one, and so on, indefinitely, endlessly, infinitely. This is what I call the successive infinite.

Since number and space are both unique and mutually cohering, the question then arises if the original (primitive) meaning of the infinite – as the successive infinite – could be deepened (“enriched” or “disclosed”) by “borrowing” a key feature from space? The answer is: YES – one may borrow the whole-parts relation from space – wholeness or totality is original within the aspect of space – expressed in the idea of an infinite totality. In terms of the theory of modal aspects it could be phrased as follows.

When, under the guidance of our theoretical (i.e., modally abstracting) insight into the meaning of the spatial time order of simultaneity, the original modal meaning of the numerical time-order is disclosed (deepened), we encounter the regulatively deepened anticipatory idea of the actual infinite or of infinite totalities. Any sequence of numbers may then, directed in an anticipatory way by the spatial order of simultaneity, be considered as if its infinite number of elements are present as a whole (totality) all at once.

When only the successive infinite is employed 0.999 . . . is not equal to one; but if the at once infinite is employed, 0.999 . . . is equal to one!

Paul Bernays did see the essentially hypothetical character of the opened up meaning of number, without (due to the absence of an articulated analysis of the modal meaning coherence between number and space) being able to exploit it fully: “The position at which we have arrived in connection with the theory of the infinite may be seen as a kind of the philosophy of the 'as if'. Nevertheless, it distinguishes itself from the thus named philosophy of Vaihinger fundamentally by emphasizing the consistency and trustworthiness of this formation of ideas, where Vaihinger considered the demand for consistency as a prejudice ...” (Gesammelte Abhandlungen 1976:60).

Although the deepened meaning of infinity is sometimes designated by the phrase completed infinity, this habit may be misleading. If succession and simultaneity are irreducible, then the idea of an infinite totality cannot simply be seen as the completion of an infinite succession. When Dummett refers to the classical treatment of infinite structures “as if they could be completed and then surveyed in their totality” he equates this “infinite totality” with “the entire output of an infinite process” (Elements of Intuitionism 1978:56). The idea of an infinite totality transcends the concept of the successive infinite.

Although Lorenzen rejects the idea of the at once infinite, his description of it incorporates the legacy of combining infinity, time and eternity. According to him the at once infinite is incorporated in the modern concept of a real number, which allows for recognizing its descent from geometry [given at once – reflecting the spatial order of simultaneity]:

“Much rather all real numbers are imagined as really present at once . . . and thus every real number as an infinite decimal fraction is already represented as if the infinite multiplicity of numbers all exist at once (auf einmal) (Lorenzen, Das Aktual-Unendliche in der Mathematik, reprinted in Meschkiwski, H. Grundlagen der Mathematik, 1972:163).

Best wishes

Danie 

Dear Mr. Daniel F M Strauss, thank you.

From your post above, people may understand better my statement of “everything is perfect in the traditional finite—infinite theory system”; you prove nothing needs to be worry about in this system since Zeno’s time more than 2500 years ago. That is why Zeno created his Paradox of Achilles--Turtle Race.

Looking back into our history, so many new “ideas, understandings and terms, formal languages” have been created in order to solve or avoid the infinite related fundamental defects disclosed by Zeno’s Paradox of Achilles--Turtle Race, but invalided. The defects are still there unsolved or unavoidable, Zeno’s Paradoxes are still there unsolved or unavoidable: not matter how we change “terms and formal languages”, no theory is able to let us know how many numbers each bigger than 1/2 or 1 or 100 or 100000 or 10000000000 or… (0.4999 or 0.999 or 99.999 or 9999.999 or 9999999999.999 or ...) from infinite items in Harmonic Series by “brackets-placing rule"-------one of “strict mathematically proven” modern family members of ancient Zeno’s Paradox of Achilles--Turtle Race Zeno’s Paradoxes.

People have been focusing minds on how to create new “ideas, understandings and terms, formal languages” within the traditional finite—infinite theory system but not on understanding the nature of the fundamental defects disclosed by Zeno’s Paradox of Achilles--Turtle Race since Zeno’s time. So, in one of my articles I ask a question: in front of Zeno’s Paradox of Achilles--Turtle Race Zeno’s Paradoxes, do we still live in Zeno’s time more than 2500 years ago?

Thank you again for your frank opinion.

Sincerer yours, Geng

Dear Geng

Thanks for reaffirming some of the points you consider to be important. Your reference to Zeno's paradoxes prompted me to insert something below - Chapter 4 in a book on Metaphysics where I explicitly enter into a discussion of these paradoxes. The distinction between the aspects of number, space and movement is decisive. Whe continuity (space) is theoretically reduced to number the outcome is antinomic (I have argued this point elsewhere) and the same happens when a theoretical attempts is made to reduce motion to space (as Zeno aimed to do). These issues are explained in detail below.

Best wishes

Danie

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Preface  VII

Introductory Chapter   1

Mark Pestana

Chapter 1        The Metaphysical Character of Philosophy  9

Marie-Élise Zovko and Jure Zovko

Chapter 2        Appearance and Reality in Parmenides  45

Kenneth Dorter

Chapter 3        The Nature of Metaphysics and Science: The Problem of the One and the Many in Thomas Aquinas   65

Curtis L. Hancock

Chapter 4        Metaphysics Between Reductionism and a Non-Reductionist Ontology   83

D.F.M. Strauss

Chapter 5        Whiteheadian Structured Societies as

Open-Ended Systems and Open-Ended

Systems as Whiteheadian Structured Societies  111

Joseph A. Bracken

Chapter 6        The Meaning of Education in the Age of Technology  127

Bogomir Novak

The papers in this volume provide us with a good sample of contemporary metaphysical inquiries. The first essay, by Zovko and Zovko, develops an account of the  nature  of  metaphysical inquiry  in  general,  which account is  grounded in  the history of this discipline. Their essay illuminates the great divide between ancient and medieval thought, on the one hand and modern and post-modern metaphysical thinking, on the other. The authors finally argue for a species of anti-realist idealism and for the necessity of metaphysical thinking within the dominant contemporary focus on the subject of experience. They conclude with a powerful argument for the social embedded-ness of metaphysical reflection.

The next three essays, by Dorter, Hancock and Strauss, are inquiries into the work of classical metaphysicians. As noted above, not all metaphysical work survives the test of time. However, the works that have survived this brutal test are always worth revisiting because their original formulations of thought (or system of thoughts) are in many ways impossible to improve upon. Fresh insight can be gained from returning to the sources. The second essay in this volume is an investigation of a fundamental account of perhaps the fundamental metaphysical problem—the distinction between appearance and reality as first articulated by the (great grand) father of metaphysics, Parmenides. Dorter explores a fundamental topic addressed in human thought via problems that have arisen in the interpretation of Parmenides’ thoughts about that fundamental topic. And his essay is a perfect example of how reality can be illuminated for our minds by means of explicating a great and ancient thinker’s thoughts. The second historically oriented essay, by Hancock, addresses a comparably fundamental metaphysical problem—the problem of “the one and the many”. The issue here is that being is one, i.e., all things have being and just in so far as they have being or are beings, all beings are one being (or one in being) and being is many, i.e., all things are different from each other in their being and constitute a multitude of beings. So, the problem is to make sense of how being is both one, undivided, unitary, single, singular, the same and multiple, divided, plural, diverse, the different. Hancock explicates Thomas Aquinas’ Aristotelian answer to this question (and explains Thomas’ points of departure from Aristotle). In the course of this dense explication Hancock articulates a classical realist account of how the mind comes to know in general and how our minds “work” when we engage in inquiry into metaphysical realities. He includes, as needed in explaining the solution to the

in   modern   mathematics   to   explain   the   continuous   entirely   in   terms   of   the discontinuous and the attempt in ancient physics to explain or explain away motion. He ends by arguing that a non-antimonious ontology can only rest on a irreducible plurality of fundamental categories of being (instead of Quine’s desert, a tropical rainforest landscape).

The final two works in this collection bring us to the present. Joseph Bracken explains and champions the metaphysics, or more properly the theory of nature, developed by Alfred North Whitehead in the early part of the 20th  century (and in spite of the anti- metaphysical positivism of that time). Whitehead’s intriguing and powerful development of Aristotle’s basic analysis of becoming has always been of interest to scientists and scientifically oriented philosophers. In recent decades the study of “complex systems” has blossomed and Whitehead’s conceptions of the fundamental nature of nature perfectly extend such rational inquiry from the sensorily accessible realm of the real (studied by the empirical sciences) into the abstractly conceivable and foundational realm of the real (studied by metaphysics). Bracken introduces the reader to  and  explains  the  fundamental  concepts  that  Whitehead  developed  for understanding the fundamentals of reality, such as “prehension”, “societies of being”, “creative systems”. Work of this sort is relevant to contemporary attempts to explain living matter, conscious matter and other self-organizing, self-referencing realities. Bracken connects the Whiteheadian foundational analysis with analyses developed by Kaufmann, Sloan Wilson, Varela, Luhmann of the nature of such self-similar, fractal- like systems.

The final paper returns to issues concerning the very nature of metaphysical inquiry. The overarching theme of this closing chapter concerns the relationship between technology  and  education  and  the  basic  claim  that  Nocak  advances  is  that technology has brought about a significant loss of meaning for human beings. The reality we now encounter, so Novak argues, is completely mediated to us through technologies and, thus, “our” reality is now essentially artificial or artifactual. The consequent impact of this fact on our metaphysical understanding of reality, on our most fundamental grasp of the most fundamental realities is tremendous—the ultimate meaning of everything is also artificial. Thus, metaphysics as the science of

Preface   IX

 being as being has now become the science of artificial being as being! This is catastrophic for the human psyche since to the extent that we are what we know, we ourselves have become artificial in a fundamental sense. Accordingly, Novak issues a clarion call to our educators to explicitly inform their charges of these facts and thereby pre-empt our descent into un-reality. There is much to ponder in this closing essay.

It  is  our  hope  that  this  collection  will  give  readers  a  sense  of  the  type  of metaphysical  investigations  that  are  now  being  carried  out  by  thinkers  in  the Western nations. We also hope that the reader’s curiosity will be peaked so that further inquiry will follow.

Professor Mark Pestana, Grand Valley State University, Allendale, Michigan, USA

4

Metaphysics Between Reductionism and a

Non-Reductionist Ontology

D.F.M. Strauss (NWU)

1. Introduction

Philosophy and all the academic disciplines are sensitive to the aim of sound reasoning – except   for   the   dialectical   tradition   which   sanctions   contradictions   and   antinomies (Heraclitus, Nicolas of Cusa, Hegel, Marx, Vaihinger, Simmel, Rex, and Dahrendorf). A brief overview is presented of conflicting theoretical stances within the various academic disciplines  before  an  assessment  is  given  of  the  positive  and  negative  meaning  of

‘reductionism’ and the implications of a non-reductionist ontology. These implications are

explained by distinguishing between different fundamental irreducible modes of being, such as the numerical, spatial, kinematic, physical, biotic, sensitive, logical-analytical, cultural- historical, lingual, social, economic, aesthetic, jural, moral and certitudinal aspects of reality. When these original modal functions are not acknowledged theoretical thought entangles itself  in  insurmountable  antinomies.  Every  single  academic  discipline  therefore  has  to employ such basic (and irreducible) concepts. Precisely because these concepts are basic they cannot be defined. Various disciplines acknowledge this state of affairs by explicitly introducing “primitive terms.” Furthermore, when it is attempted to reduce what is irreducible the antinomy involved at once expresses itself as a logical contradiction. We shall argue that an antinomy as such is inter-modal (such as when Zeno attempts to reduce motion to static positions in space), while a contradiction is intra-modal (for example when a triangle and a circle is confused). A clear example of the irony of an ismic ortientation will be discussend in sub-paragraph 10 when the impasse of historicism is discussed.

Against the background of historical lines of development the multiple terms employed in this context are mentioned and eventually positioned within the context of the normativity holding for logical thinking. It is argued that the logical contrary between logical and illogical serves as the foundation of other normative contraries, such as polite – impolite, legal – illegal and moral – immoral.

It will be shown that through the discovery of irrational numbers the initial Pythagorean conviction that everything is number reverted to a geometrical perspective that generated a static metaphysics of being which challenged the ideas of plurality and motion. This development uncovered the well-known problem of primitive terms in scientific discourse as an alternative for those metaphysical attempts aimed at reducing whatever there is to one single  mode  of  explanation.  Zeno's  paradoxes  are  used  to  demonstrate  an  alternative

understanding of the difference between the potential and the actual infinite as well as the nature of (theoretical) antinomies. It is argued that genuine antinomies are inter-modal in nature (such as is found in the attempt to reduce movement to static positions in space) and therefore differ from a logical contradiction (such as a ‘square circle’ which merely confuses two figures within one modal aspect). Although every antinomy does entail logical contradictions, the latter do not necessarily presuppose an antinomy. The implication is that logic itself has an ontic foundation – as is seen from the nature of the principle of sufficient reason (ground) and the principle of the excluded antinomy – and therefore only acquires meaning on the basis of a non-reductionist ontology. When the method of immanent critique unveils genuine antinomies, the way is opened for meaningful intellectual interaction between different philosophical stances. In distinguishing between contradiction and antinomy philosophers are actually challenged to contemplate the implications of a non- reductionist ontology as an alternative to all metaphysical attempts to over-emphasize one or another aspect of empirical reality.1

2. Setting the stage: Unity and diversity

Our human experience of reality is embedded in an awareness of unity and diversity. For that reason we have to discern, that is, we have to identify and distinguish. Whereas it is quite natural and meaningful to articulate differences between distinct (kinds of) entities in our everyday life, it is equally natural that we are (analytically) sensitive to a confusion of what is distinct. It is a standard practice amongst philosophers and logicians to designate this kind of confusion by using the terms contradiction and antinomy – which are normally taken to be synonymous.

Concurrent with the rise of philosophy and some of the disciplines in ancient Greece (such as mathematics, astronomy and the medical sciences) an awareness of the logical-analytical capacities of human beings surfaced. Early Greek philosophy also witnessed the emergence of dialectical conceptions in which it was attempted to accommodate (and sanction) contradictions. A later disciple of Heraclitus said:

For all things are alike in that they differ, all harmonize with one another in that they conflict with one another, all converse in that they do not converse, all are rational in being irrational; individual things are by nature contrary, because they mutually agree. For rational world-order [nomos] and nature [physis], by means of which we accomplish all things, do not agree in that they agree.2

In the course of the intellectual tradition of the West academic disciplines more and more acquired the status of independent special sciences, since they also increasingly adhered to principles for logical reasoning. Yet a first glance at the history of the various disciplines

1  I want to thank Dr. Darrell Patrick Rowbottom, University of Durham, England, for many valuable comments and suggestions that are incorporated in the final version of this article. The same applies to Prof. Hubertus Bargenda (a mathematician from the University of the Free State). The underlying perspective of this contribution draws upon my work Philosophy: Discipline of Disciplines (2009) as well as Strauss 2006.

2  These words, expressed by a later disciple of Heraclitus, were erroneously ascribed to Hippocrates' writing, Peri diatès, I, xi, 6.

shows that within each of them alternative, often conflicting orientations developed – a state of affairs that cannot be explained on purely logical grounds. This predicament rather suggests that theoretical (i.e., scientific) thought cannot escape from considerations exceeding the boundaries of logicality.3

In  order  to  substantiate  this  suggestion,  a  distinction  will  be  introduced  between  an antinomy and a contradiction. It may turn out that this distinction is closely connected to the above-mentioned issue of unity and diversity. The problem of the one and the many, alongside others, such as the relationship between universality and individuality, constancy and dynamics, the finite and the infinite, what is necessary and contingent and what is considered to be knowable and unknowable, co-determined the development of philosophy and the disciplines.

From the history of these disciplines we learn that the foundational problems within the (natural and social) sciences are indeed philosophical in nature. This explains the mentioned historical fact that in their development all the academic disciplines reflect divergent philosophical schools of thought and it urges us to ask how this situation ought to be assessed in terms of the requirements for logical thinking and sound reasoning. The following (incomplete) overview may help to portray the background picture of our subsequent discussion of the distinction between contradiction and antinomy. This succinct overview merely provides a glimpse and not a detailed exposition – for that would require more than an article on each mentioned discipline. The rationale for providing this glimpse is to allude to the widespread reality of opposing ismic positions – with a view to the fact that such positions mostly entail antinomies since they are normally reductionist (the connection between reductionism and antinomies will be explained below). The brief overview below first mentions the name of prominent trends (schools of thought) within various academic disciplines and then mention key scholars who adhered to this theoretical stance within these special sciences.

·      Mathematics: Axiomatic formalism (Hilbert), logicism (Russell, Frege) and intuitionism

(Brouwer, Heyting, Troelstra, Dummett) in modern mathematics;4

3 I do not intend to view the relationship between “purely logical” and what exceeds the logical aspect in terms of the distinction between statements within an object language and statements belonging to a meta-language (utens/docens). The original distinction – probably going back to the 13th century – continued an old question regarding logic as a scientific part of philosophy or merely as an instrument of philosophy. The dialectica utens was viewed as treating arguments within all disciplines, whereas the dialectica docens was seen as a special science (scientia specialis) focused upon the dialectical syllogism or with the secondary intentions connected to dialectical conclusions.

4 Salmon merely refers to “intuitionistic philosophers of mathematics” – without acknowledging the truly mathematical character of this trend in 20th century mathematics (see Salmon, 2001:23 – he refers to Körner's work The Philosophy of Mathematics, 1968). By contrast, Stegmüller remarks: “The special character of intuitionistic mathematics is expressed in a series of theorems that contradict the classical results. For instance, while in classical mathematics only a small part of the real functions are uniformly continuous, in intuitionistic mathematics the principle holds that any function that is definable at all is uniformly continuous” (1970:331). Beth also highlights this point: “It is clear that intuitionistic mathematics is not merely that part of classical mathematics which would remain if one removed certain methods not acceptable to the intuitionists. On the contrary, intuitionistic mathematics replaces those methods by other ones that lead to results which find no counterpart in classical mathematics” (1965:89).

·      Physics:  Classical  determinism  (Einstein, Schrödinger,  Bohm  and  the  school  of  De Broglie) and the mechanistic main tendency of classical physics (last representative Heinrich Hertz)5  versus the Kopenhagen interpretation of quantum mechanics (Bohr and Heisenberg); the contemporary ideal to develop “a theory of everything” (Hawking and super string theory: Greene).

·      Biology: Mechanistic orientation (Eisenstein), physicalistic approach (neo-Darwinism), neo-vitalism (Driesch, Sinnott, Rainer-Schubert Soldern, Haas, Heitler), holism (Adolf Meyer-Abich), emergence evolutionism (Lloyd-Morgan, Woltereck, Bavinck, Polanyi) and pan-psychism (Teilhard de Chardin, Bernard Rensch);

·      Psychology:  The  initial  atomistic  association  psychology  (Herbart),  the  stimulus- response approach, Gestalt-psychology [the Leipzig school (Krüger and Volkelt) and the Berlin school (Koffka and Köhler)], depth psychology (Freud, Adler, Jung), the logo- therapy of Frankl, phenomenological psychology, contemporary system theoretical approaches (under the influence of von Bertalanffy).

·      The science of history: Compare the conflict between linear and cyclical conceptions of history, the Enlightenment ideal of linear accumulative growth, the recurrence of the Greek conviction that history is eternally recurrent in the thought of Vico, Herder, Hegel, Goethe, Daniliwski, Nietzsche, Spengler and to a certain degree also Toynbee;

·      Linguistics:  Two  lines  of  thought  dominated  the  19th  century  –  Rousseau,  Herder, Romanticism, von Humboldt and the rationalistic trend running from Bopp, Schleicher, and ‘Jung-Grammatici’ to Paul (with his historicistic conception of language-in- development). Cassirer, by contrast, developed his neo-Kantian theory of language (in which language is a thought-form imprinted upon reality), Bühler pursued the stimulus of behaviorism in his theory of signs, at the beginning of the 20th century Wundt dominated the scene, De Saussure contributed to the development of a structuralist understanding (followed by Geckeler, Coseriu and others), Reichling explored elements of Gestalt- psychology in his emphasis on the word as the core unit of language, Chomsky revived the doctrine of the a priori within the context of his transformative generative grammar, more recently the manifestation of systems theory within general and applied linguistics;

·      Sociology: The initial organicistic orientation (Comte, Spencer) was continually opposed by mechanistic and physicalistic approaches (cf. L.F. Ward – late 19th  century – and in the second half of the 20th century W.R. Catton), the dialectical heritage of Hegel permeated Georg Simmel's formalistic sociology with its individualistic neo-Kantian focus (Park and Burgess explored this direction in the USA), Max Weber developed the sociological and economic implications of the neo-Kantian Baden school of thought, Talcott Parsons made the systems model (based upon von Bertalanffy's generalization of the second main law of thermodynamics) fruitful for sociological thinking, opposed by conflict sociology (Dahrendorf, C. Wright Mills and Rex and by the Frankfurt school of  neo-Marxism),  a  systems  theoretical  approach  was  recently  revived  by  J.C. Alexander, A. Giddens developed his structuration theory – and during the past two decades J. Habermas elaborated his theory of communicative actions;

5 Mario Bunge says: “It is now generally understood that mechanics is only a part of physics, whence it is impossible to reduce everything to mechanics, even to quantum mechanics.” Although he holds that the physicalism of the Vienna Circle and the Encyclopedia of Unified Science is dead, “the sharp decline of physicalism has not been the end of reductionism” (see his “The Power and Limits of Reduction” in Agazzi, 1991:33).

·      Economics:  The  classical  school  of  Adam  Smith,  the  neoclassical  approach  (from Cournot and Dupuit to Menger, Jevons, Walras and Pareto), the marginalism of Marshall,  Keynes's  ‘General  Theory,’  alternative  approaches  to  competition (Chamberlin and Robinson);

·      The science of law:  The historicistic orientation  of  von  Savigny  –  followed by  the

Romanist  (von  Jhering)  and  Germanistic  (von  Gierke)  schools,  neo-Hegelianism (Binder), neo-Kantianism (Stammler, Radbruch, Kelsen), the revival of natural law theories after the second world war, and legal positivism (which seems to remain alive amongst legal scholars);

·      Theology: Dialectical theology (Barth, Gogarten, Brunner) in its dependence upon

Kierkegaard and Jaspers, Bultmann (dependent on Heidegger), theology of hope (Moltmann  –  dependent  upon  the  neo-Marxism  of  Ernst  Bloch),  the  historicistic design  of  Pannenberg  (dependent  upon  Dilthey  and  Troeltsch),  the  ‘atheistic’ theology of Altizer and Cox (influenced by neo-positivism), existentialist- hermeneutical trends (Fuchs, Ebeling, Steiger), theology of liberation (influenced by neo-Marxism).

What is particularly striking regarding these (philosophically founded) schools of thought within the disciplines is that many of them are entangled in what should be labeled reductionism in a pejorative sense.6 Surely there are also positive and largely unrelated connotations attached to the term reduction in different special sciences. For example, mathematicians may speak about the construction of numbers from sets and then designate it as “reduction”. Separating chemical compounds into their simpler constituents is also known as “reduction”, and so on.

By designating more problematic situations the term reductionism emerged by the middle of the 20th century. In 1953 Quine used it in his discussion of “The Verification Theory and Reductionism” (see Quine, 1953:37 ff.) and in the early seventies the work “Beyond Reductionism” appeared (see Smythies & Koestler, 1972). Smith considers Polanyi to be “perhaps the severest and most comprehensive critic of reductionism” because he “was a major scientist of this century and was drawn into philosophical debate primarily because of the threat to scientific freedom, political democracy, and to humane values that he saw in reductionism”. To this he adds the remark: “His works The Contempt of Freedom, The Logic of Liberty, Science Faith and Society, Personal Knowledge, and The Tacit Dimension have as a common theme the criticism of reductionism in all its scientific, cultural and moral forms.”7

6   Popper  states:  “As  a  philosophy,  reductionism  is  a  failure”  (Popper,  1974:269).  And  Goodfield remarks: “Reductionist methodology may have been extremely successful, but the history of science abounds with examples where forms of explanation, successful in one field, have turned out to be disastrous when imported into another” (Goodfield, 1974:86). A positive appreciation of reductionism is, for example, found in the thought of Dawkins and Dennett (see Dennet, 1995:80 ff.).

7    See   Smith,   G.L.,   1994.   On   Reductionism.  Sewanee,  Tennessee  –   available  on   the   WEB   at:

http://smith2.sewanee.edu/texts/Ecology/OnReductionism.html (accessed  on  22-01-2005).  Putnam holds  that  scientism  and  relativism  are  reductionist  theories  (Putnam,  1982:126).  In  respect  of

‘phenomenalism’ he remarks: “the idea that the statements of science are translatable one by one into

statements about what experiences we will have if we perform certain actions has now been given up as an unacceptable kind of reductionism” (Putnam, 1982:187).

Our approach in what follows will be to investigate the limits of logical discernment (identification and distinguishing) in order to account for the real antinomies arising from the attempt to reduce what is truly irreducible. This approach is similar to the strategy defended by the physicist Henry Margenau (in following some ideas of Mario Bunge). He takes this to be “the strategy consisting of reducing whatever can be reduced without however  ignoring  emergence  or  persisting  in  reducing  the  irreducible”  (cf.  Margenau,

1982:187, 196-197). Once we have assessed the systematic distinction between antinomy and contradiction  its   implication   for   understanding   the   nature   of   the   various   “ismic” orientations within the disciplines will briefly be highlighted.

3. Brief historical contours

Since both academics and non-academics enjoy the fun of wrestling with “logical” problems we proceed by mentioning the liar-paradox attributed by Diels and Kranz to Epimenides (5th Century B.C.) – where it is asserted that one of the Cretans, their own prophet, said all Cretans are liars. In the account of Titus 1:12-13 it is reported that the Apostle Paul holds that this testimony is true. How can such a statement, made by a liar, be true without being false at the same time?8

The  mere  statement  of  this  contradiction  shows  that  ancient  Greek  thought  already wrestled with the above-mentioned basic logical ability of humans to identify and to distinguish.9 The school of Parmenides postulated the primordial nature of being and even identified it with thought.10 But in Plato's dialogue Parmenides one finds a negative argument concerning the mutuality (relatedness) of identification and distinguishing, ultimately also highlighting the limits of concept formation, for conceiving the One (and the Many) in an absolute sense, escapes the grip of logical concept formation.11 In the Sophist it is consequently acknowledged that trying to know what being and non-being in themselves are present thought with an aporia, i.e., an unresolved problem.12  Yet, whenever being is thought  non-being  is  thought  as  well.13  In  other  words, identification refers  to  what  is distinct from it.

8   In this formulation an escape route is given by observing that normally a liar is a person who

sometimes tells a lie, but not always.

9  Derrida applies the mutuality of identity and difference to language: “The identity of a language can only affirm itself as identity to itself by opening itself to the hospitality of a difference from itself or of a difference with itself” (Derrida, 1993:10).

10 Diels-Kranz I, 231; Parmenides, B. Fr. 3: “For thinking and being are the same.”

11  The final conclusion to the four paths of the dialectical argument regarding the One and the Many reads: “Therefore, if the One is, it is everything and nothing, in relation to itself and to the many” (Parmenides 160b1-3).

12 Logic eventually used the term “aporia” in connection with the theoretical truth of a statement where

there are grounds for and against it. In Latin aporia turned into dubitation and question (see Waldenfels,

1971:448).

13  Spivak explains Derrida's view of deconstruction in similar terms: “Deconstruction, as it emerged in Derrida's early writings, examined how texts of philosophy, when they established definitions as starting points, did not attend to the fact that all such gestures involved setting each defined item off from all that it was not” (Spivak, 1999:426).

Whereas Plato therefore already had a clear understanding of the meaning of the logical principles of identity and contradiction,14 Aristotle, in addition, already understood the meaning of the principle of the excluded middle (see Metaph. 1057a).

During the middle ages, alongside the continuation of Aristotle's predicate logic, a notable dialectical tradition, proceeding from Heraclitus and the dialectical logic of Plato, remained in force. This so-called via negativa of neo-Platonism (Pseudo-Dionysius, Plotinus) eventually brought Nicholas of Cusa to his notion of the coincidence of opposites (coincidentia oppositorum). Nicholas of Cusa explored the so-called actual infinite in terms of which he claimed that God, as the actual infinite, is at once the largest and the smallest (De Docta Ignorantia, I,5), i.e. the coincidentia oppositorum (see De Docta Ignorantia, I,22). A remarkable analysis eventually came from Georg Cantor, the founder of modern set theory and the theory of transfinite numbers, who demonstrated something similar about the smallest transfinite ordinal number omega (ω), because this number is both even and uneven and at the same time neither even nor uneven.

Perhaps the most significant elaboration of the dialectical tradition on the one hand is found in the thought of Hegel, Marx and those sociologists of the 20th century who are known as conflict  theorists  (Simmel,  Rex  and  Dahrendorf),  and  in  the  philosophy  of  “As  If”  of Vaihinger on the other. The significance of the latter is linked to its relevance for various academic disciplines (such as mathematics, physics, linguistics, economics, and the science of law – to name some of them). Vaihinger claims that the use of inherently antinomic constructions (designated as fictions) may serve human (scientific) thought in surprisingly efficient ways. For example, he characterizes mathematical constructs such as negative numbers, fractions, irrational and imaginary numbers as “fictional constructs” (that are not hypotheses) with a “great value for the advancement of science and the generalization of its results in  spite of the  crass contradictions which they  contain” (Vaihinger,  1949:57).  In general Vaihinger aims at providing an explanation of “the riddle that by means of such illogical, indeed senseless concepts, correct results are obtained” (Vaihinger, 1949:240). His answer is given in what he terms to be “the general law of fictions,” i.e., in the “correction of the errors that have been committed” or in a procedure called “the method of antithetic error” (Vaihinger, 1949:109). However, since this method holds that thought “progresses by means of  antithetic  operations,”  and  since  including  under  one  concept  antithetic  operations creates fictions viewed as merely the symbol of “such antithetic operations and antithetic errors” (see Vaihinger, 1949:119-120), it is clear that his “method of antithetic error” simply duplicates the initial problematic construction of internally antinomic or illogical fictions – as if two logical errors in practice will generate what is right. The coherence of what is irreducible within reality requires an alternative approach, namely that what we have called a non-reductionist ontology.

At this point the above-mentioned problem of unity and diversity comes to mind again – not only with respect to the various ismic positions within the disciplines, but also regarding the opposing and oftentimes contradicting philosophical schools of thought – including the

14 The following phrase highlights both principles: “No objection of that sort, then, will disconcert us or make us believe that the same thing can ever act or be acted upon in two opposite ways, or be two opposite things, at the same time, in respect of the same part of itself, and in relation to the same object” (Politeia, Book IV, Ch.XIII, 436 – translation by Cornford 1966:130).

dialectical tradition that affirms contradictions. But what is meant when different terms are employed alongside the term ‘contradiction’? This question calls for a clarification of the terminological problem regarding multiple terms and for an investigation of the question whether or not there is more at stake than purely logical distinctions. We therefore proceed first of all by looking at the multiple terms that are employed in this context.

4. Multiple terms

Both in scholarly and within everyday contexts we hear about contradictions, antinomies, paradoxes, riddles, dilemmas and even puzzles. Particularly since Immanuel Kant explained apparently stringent proofs, in the Transcendental Dialectics (second Book, second Chapter) of his Critique of Pure Reason (CPR), for a set of four theses and antitheses, under the category of antinomies,  the  latter  term  became  common  knowledge  in  subsequent  philosophical literature and reflection.

The different terms used in this context are seen in the following arbitrarily selected references. In 1849 a posthumously published book from the German philosopher- mathematician,  Bernard  Bolzano,  appeared  with  the  title:  Pardoxien  des  Unendlichen (Paradoxes of the Infinite). Russell speaks about contradiction, paradox and antinomy (Russell,

1956:144, 190 ff.). E. Teensma published a book with the title: The Paradoxes (1969), while E.P.

Northrop wrote a work on Riddles in Mathematics (1944 – Penguin version 1964). In a work with the title “Dilemmas” Gilbert Ryle discusses Achilles and the tortoise as an example (Ryle, 1953:50-69). Sometimes the word “puzzles” is used – Martin Gardner employed it in a (Pelican) book on Mathematical Puzzles and Diversions (1968). Recently Michael Clark published another work: Paradoxes from A to Z (2002).

By and large the legacy of classical and modern logic as well as philosophy in general did not develop a systematic analysis of the differences between these diverse designations. In general contradictions, antinomies and paradoxes are used interchangeably. For example, when Fraenkel et al discuss the known contradictions and paradoxes they are called antinomies.15

We commence by considering the nature of normative contraries in order to highlight the

normativity of logicality.

5. Normative contraries

The scope of the logical principles of identity and (non-)contradiction applies to the human ability to conceive and to argue. Copi states a generally accepted conviction when he says that the “principle of contradiction asserts that no statement can be both true and false” (Copi,

1994:372).

The classical example of an illogical concept stems from Immanuel Kant and concerns a “square circle” (see Kant 1783:341; § 52b). Establishing that this concept is illogical entails that a normative standard has been applied and that the said concept does not conform to the requirement of ought to be inherent in this normative standard. It is contradictory not to distinguish between a square and a circle, or, to put it differently, confusing two spatial

15 He distinguishes between logical antinomies (those of Russell, Cantor and Burali-Forty) and semantic antinomies (those of Richard, Grelling and The Liar – 1973:5-12).

figures violates the demands for identifying and distinguishing properly: a square is a square (logically correct identification) and a square is not a non-square (such as a circle – logically correct distinguishing).

Thinking in a logically antinormative way, i.e., thinking illogically, remains bound to the structure of logicality and does not turn into something a-logical (non-logical), such as the economic, the moral or the jural. These (non-logical) facets of our experience are said to be a- logical – but they are not illogical. In fact they also have room for contraries similar (or analogous) to the contrary between logical and illogical, namely economic and uneconomic, moral and immoral and legal and illegal. Although the history of humankind tells the story of different assessments of what may count as economically, morally or legally proper behaviour, one can hardly deny the normativity inherent in these dimensions as it is manifested in the mentioned contraries. The logical contrary actually lies at the foundation of all these other instances of normative contraries – the latter analogically reflect within their own domains the meaning of logical analysis (identification and distinguishing).

Yet the phenomenon of contradictions does not tell the full story. Let us return to the confusion of spatial figures present in the illogical concept of a square circle and compare it with something more drastic, namely the attempt to explain whatever there is purely in spatial terms. This happened in Greek philosophy after the discovery of irrational numbers – an event that led to the geometrization of Greek mathematics (after its initial Pythagorean arithmetization).  This   alteration   within   mathematics   inspired   the   development   of   a speculative metaphysics in which material entities were exclusively characterized in terms of their spatiality. The result was that the Greeks did not contemplate an empty space. According to their mature understanding space does not exist, only place. Place is a property exclusively attributed to a concrete body. In the absence of a body, there is no subject for the predicate place. From this it naturally follows that an “empty place” is the place of nothing – in other words, it is no place at all! But what then do we have to say about the movement of a material body? Will it be possible to assert that motion is a “change of place”? Surely, given the identification of a body with its place, motion would then be an impossibility – at least when a body is supposed to be the subject of motion – for a change of place will amount to a change of “essence”! In terms of such a metaphysics of space the introduction (or: “definition”) of motion yields to the contradiction that a body can move if and only if it cannot move – which actually approximates the arguments of Zeno against multiplicity and movement alluded to above.

The attempt to explain whatever there is exclusively in spatial terms, is nothing but pursuing the aim of reducing everything to space (similar to the Pythagorean assertion “everything is number”). But, as we have noted, the first “victim” of such a spatially oriented reductionism is found in the function of motion (the school of Parmenides). In order to acquire a better handle on this problem we first have to pay attention to the underlying problem of the “coherence of irreducibles” – which is just a different formulation of the basic philosophical problem of unity and diversity. Russell refers to Hegel in respect of the difference between a continuous magnitude (wholeness) and a discrete magnitude – as “different” instances of the “class-concept” and then proceeds by saying that he “strongly” holds “that this opposition of  identity  and  diversity  in  a  collection  constitutes a  fundamental problem  of  Logic  – perhaps even the fundamental problem of philosophy” (Russell, 1956:346).

6. Uniqueness and coherence

The claim that a reduction is unwarranted implicitly presupposes the conviction that there exist “irreducibles” (and: primitives).16  Typical (reductionsistic) all-claims, such as the mentioned Pythagorean conviction that everything is number, the statement that everything is physical (materialism) or that everything is interpretation (postmodernism), challenge the idea of uniqueness and irreducibility.17 All-claims like these are mainly monistic in nature – in the sense that they want to find one single, all-encompassing perspective or principle of explanation capable of accounting for the entire diversity manifest in our experience of the universe.18 An argument in favour of the acknowledgement of irreducibility – as one side of the coin (with the mutual coherence of what is unique as the other side) – ought to show that an unwarranted reductionism gets entangled in unsolved problems (normally referred to as contradictions, paradoxes or antinomies).19

Ernst Cassirer, the philosopher from the neo-Kantian Marburg school (perhaps best known for his Philosophy of Symbolic Forms), is also quite explicit in this regard when he claims that a critical analysis of knowledge, in order to side-step a regressus in infinitum, has to accept certain basic functions which are not capable of being “deduced” and which are not in need of a deduction (Cassirer, 1957:73).

Every single academic discipline has to employ such basic (and irreducible) concepts. Precisely because these concepts are basic they cannot be defined. Various disciplines acknowledge this state of affairs by explicitly introducing “primitive terms.”

For example, in Zermelo-Fraenkel set theory first order predicate calculus is assumed and on that basis it introduces as an undefined term the specific set-theoretical primitive binary predicate e which  is  called the  membership  relation  (Fraenkel  et  al.,  1973:23).20   Bertrand Russell states: “The relation of whole and part is, it would seem, an indefinable and ultimate relation” (Russell, 1956:138). For the sake of an economy of primitive terms even the term

16 Salmon refers to “primitive terms” in “pure mathematics” (Salmon, 2001:32).

17  P. Hoyningen-Huene writes about irreducibility in the context of complementarity: “But this property

is just identical with the epistemological non-reducibility of these features. In other words: in order to establish that in a certain situation complementarity prevails, it has to be shown that the features involved are irreducible to each other” (see his Theory of Antireductionist Arguments: The Bohr Case Study, in: Agazzi, 1991:67). Weingartner refers to primitive terms: “Term (concept, idea) t is scientifically analyzable iff it is reducible to primitive terms. t is reducible to primitive terms iff t is itself a primitive term or it can be traced back to primitive terms by a chain of definitions” (see his article on Reductionism and Reduction in Logic and in Mathematics, in: Agazzi, 1991:124).

18  With reference to Einstein's thirty year search for a unified field theory, Brian Greene, a specialist in the theory of super strings, believes that physicists will find a framework fitting their insights into a “seamless whole,” a “single theory that, in principle, is capable of describing all phenomena” (Greene,

2003:viii). He introduces Super String theory as the “Unified Theory of Everything” (Greene, 2003:15;

see also pp.364-370, 385-386).

19  Compare, for example, the remark of Weingartner regarding the failure of logicism: “Logicism is an

example of reduction which was as a whole unsuccessful” (Weingartner, 1991:130).

20 This approach follows a general pattern: an axiomatic theory (axiomatic theories of logic excluded) “is constructed by adding to a certain basic discipline – usually some system of logic (with or without a set theory) but sometimes also a system of arithmetic – new terms and axioms, the specific undefined terms and axioms under consideration” (Fraenkel et al., 1973:18).

identity need not to be taken as primitive, since in the approach to axiomatic set theory explained by Lemmon it could be defined by the use of the axiom of extensionality (see Lemmon, 1968:124). In general linguistics the term “meaning” is primitive; in kinematics the term constancy (“invariance” – normally associated with a uniform movement) is primitive; in the discipline of law the term “retribution” is primitive. When Russell discusses the mathematical meaning of constants and variables he says that “constancy of form must be taken as a primitive idea” (Russell, 1956:89) – and so on.

The upshot of  this  is  that  the  acquisition of  concepts  and the  formulation of  definitions

ultimately rest upon primitive terms – they are not defined and they cannot be defined.

The question how does one know these indefinable (primitive) terms? is an epistemological issue which is rooted in philosophical assumptions about the world in which we live and therefore it involves ontological commitments.21 However, since a discussion of this issue exceeds our present context, we return to the space metaphysics of the school of Parmenides.

7. Zeno's paradoxes – A different understanding of antinomies

In the school of Parmenides Zeno argued against multiplicity and movement by assuming an absolutely static being. The well-known reasoning regarding the flying arrow, Achilles and the tortoise as well as what is known as the dichotomy paradox is reported by Aristotle in his Physics (239 b 5 ff.). Aristotle's own approach proceeds from the assumption that “it is impossible for anything continuous to be composed of indivisible parts” (Phys. 232 a 23 ff.) and that “everything continuous is divisible into an infinite number of parts” (Phys. 238 a

22).  The  basis  of  the  first  paradox  is  found  in  divisibility and  that  of  the  third  in  the successive addition of two distinct series of diminishing magnitudes both converging to the same limit – but given the different points of departure the first one is nested within the second one. It looks as if the Aristotelian account of paradoxes one and three collapsed movement ab initio into an issue of spatial divisibility and the addition of diminishing magnitudes (therefore both cases are related to what modern mathematics calls the density of spatial continuity), whereas the account of the paradox of the flying arrow seems to allow for movement to begin with and then “freezes” it into distinct “moments” of time – as if something moving from “moment” to “moment” has a definitive place in space.22

It may be worthwhile to mention the fourth Fragment of Zeno known to us, for it explicitly starts by granting the reality of movement and then it proceeds with an argument launched from the perspective of the static nature of space in order to rule out the possibility of movement: “That which moves neither moves in the space it occupies, nor in the space it does not occupy” (Diels-Kranz B Fr.4). This certainly explains why Grünbaum distinguishes between the “paradoxes of extension” and the “paradoxes of motion” (Grünbaum, 1967:3) –

21 Contemporary formal ontologies intersect with certain basic ideas of a non-reductionist ontology but unfortunately did not develop a theory inter-modal coherences between the various modal aspects of reality, causing this trend also to by-pass the importance of modal universality. However these issues cannot be treated in this context.

22 If “being at one place” means “being at rest,” and if this is “every moment” the case with the “flying arrow,” then the arrow is actually only “at rest” – i.e., it is not moving at all. Of course, modern kinematics holds that “rest” is a (relative) state of motion. But without reference to some or other system one cannot speak about the motion of a specific kinematical subject (see Stafleu, 1980:81, 83-84).

but  he  explicitly  distances  himself  from  the  authenticity  of  the  historical  sources  by restricting himself to the legacy of Zeno in “the present-day philosophy of science” (Grünbaum, 1967:4).23

Unfortunately, in his encompassing treatment of Zeno's paradoxes, Grünbaum does not anywhere in his analysis pay attention to the problem of uniqueness and irreducibility (with the accompanying problem of primitive terms and indefinability). Yet this does not mean that in his mode of argumentation there is not an implicit acceptance of the uniqueness of the core meaning of motion. In his work on space and time, for example, he discusses Einstein's “principle of the constancy of the speed of light” (Grünbaum, 1974:376) and points out that it concerns an upper limit that is only realized in a vacuum (Grünbaum, 1974:377). Einstein's theory  of  relativity  proceeds  from  the  hypothesis  that  one  singular  light  signal  has  a constant velocity (in respect of all possible moving systems) without necessarily claiming that such a signal actually exists. Stafleu remarks: “The empirically established fact that the velocity of light satisfies the hypothesis is comparatively irrelevant” (Stafleu, 1980:89).

The postulate concerning the constancy of the velocity of light explores an insight already advanced by Galileo and his predecessors in the meaning of inertia. Galileo reversed the Aristotelian view that whatever moves requires a causing force in order to continue its movement. He did that with the aid of a thought-experiment concerning a body that is in motion on a plane which is extended into the infinite – and from this experiment he derives the law of inertia. The question is whether or not the meaning of uniform motion is primitive and unique in the sense that it ought to be distinguished both from the static meaning of space and the dynamic meaning of physical energy-operation (causes and effects)? Since physics always deals with dynamic forces operative in the interplay of energy transformation, and since a (constant) uniform motion can indeed be envisaged without making an appeal to a cause (causal force), it is clear that something unique and irreducible is here at stake. It entails that in a functional sense movement is something original. Whatever moves will continue its (uniform) motion endlessly. Motion is not in need of a cause – only a change of motion needs a cause. Both acceleration and deceleration require an energy-input (i.e., a physical cause).

Although modern physics was dominated by a mechanistic inclination until the end of the

19th  century, it eventually realized that a purely kinematical explanation of physical phenomena is untenable. A consistent mechanistic approach, such as that still found in the posthumously published work by Heinrich Hertz (the German physicist who did experimental work about electromagnetic waves more than a hundred years ago) on “The Principles of Mechanics developed in a New Context,” demonstrates the dilemma of reductionism, for his aim to restrict physics to number, space and movement only (re- presented by the concepts mass, space and time), led him to reject the (physical) concept force. He claimed that the concept of force is something inherently antinomic (cf. Katscher,

1970:329).

23  This position closely imitates a similar disclaimer found in Russell's treatment: “Not being a Greek scholar, I pretend to no first-hand authority as to what Zeno really did say or mean. The form of his four arguments which I shall employ is derived from the interesting article of M. Noël, ‘Le mouvemont et  les  arguments  de  Zénon  d’Elée,’  Revue  de  Métaphisique  et  de  Morale,  Vol.I,  pp.107-125.  These arguments are in any case well worthy of consideration, and as they are, to me, merely a text for discussion, their historical correctness is of little importance” (Russell, 1956:348, note).

But as soon as the dynamic physical sense of force is acknowledged, as it has been done by

20th century physics, what Hertz deemed to be an antinomy turns out to require an acknowledgement of another unique and irreducible functional mode24 of reality (in addition

to number, space and movement), namely the physical.25

8. The inter-modal meaning of an “antinomy”

But the kinematic function of uniform motion also differs from the functional modes of number and space. The above-mentioned B Fragment 4 of Zeno actually demonstrates that the unique and irreducible meaning of uniform flow (motion) cannot be captured purely in spatial terms, except in an antinomic way. In order to explain what this means we first have to alter the meaning of the term “antinomy.” The most obvious way to accomplish this is to allude to its literal sense, which is intended to designate a clash of laws: anti = against, and nomos = law. The attempt to explain movement in terms of space results in a (theoretical) conflict between kinematic laws of motion and spatial laws.

Such a conflict or clash between distinct functional (modal) laws indeed demonstrates the nature of a theoretical antinomy. After all, in the actual world these two modes of being are unique and are mutually cohering.26 Yet the attempt to reduce one unique mode to another one invariably results in genuine (theoretical) antinomies.

In  this  sense  antinomies  therefore  concern  an  inter-modal  confusion,  i.e.,  a  lack  of distinguishing properly between different modes, functions or aspects of reality.

Furthermore, an antinomy always entails a logical contradiction, whereas a contradiction does not necessarily presuppose an antinomy. The above-mentioned illogical concept of a “square circle” exemplifies an instance where two spatial figures are not properly identified and distinguished. In other words, a contradiction such as this one has an intra-modal character since its confusion relates to givens within the modal-functional boundaries of one aspect or function only.

24 The aspects of reality are not brought to sight by asking questions about the concrete what of entities and processes, for these aspects represent the way (manner) in which such entities and processes function

– i.e., they relate to the how of concrete entities and processes. From Latin we have inherited expressions

such as modus operandi and modus vivendi in which the how is represented by the term ‘modus’. An aspect is therefore to be seen as a specific (unique) mode which, in a general sense, is a modus quo, a mode of being. As an equivalent for referring to facets, aspects or functions, one can therefore also speak about modalities or modal aspects or modal functions. Already in 1910 Cassirer highlighted the importance of this distinction between entity (‘substance’) and function (see Cassirer, 1953). When entities and processes are resolved into functions we meet functionalism; and when modal functions are treated as entities they are reified. An in-depth analysis of the decisive role of functionalism in the development of the modern natural sciences is found in an important work of Rombach (see Rombach, 1965-66).

25  It is therefore understandable that Janich distinguishes between phoronomic and dynamic statements (Janich, 1975:68-69). Also Einstein highlights the difference between the mechanical point of view (where all processes are reversible) and thermodynamics (the most general physical discipline where courses of events are irreversible) (Einstein, 1959:42-43).

26 The “path” of a movement highlights the undeniable interconnection between motion and space. The

serial order of events reveals a connection with the numerical meaning of succession. We shall briefly return to this perspective below.

Within the dialectical tradition of Marxism this difference between an (inter-modal) antinomy   and   an   (intra-modal)  contradiction  surfaced   strikingly.   In   following   the dialectical-materialistic conception of Engels the Marxist physicist Hörz talks about a “dialectical Widerspruch (antinomy).” One can say that a moving body (i.e., a body involved in a change of place) at the same time is and is not at a specific place. According to him this is the “dialectical antinomy (Widerspruch)” of change of place. But a formulation precluding every logical contradiction runs as follows: “as the result of movement a body finds itself at a specific place and with regard to the movement itself the body does not find itself at a specific place” (Hörz, 1967:58). Hörz's designation of this situation as the “dialectical antinomy (Widerspruch)” of change of place is also found in the thought of Hegel:

When we speak about movement as such, we say: a body is at a specific place and then moves to a different place. While it is moving, it is no longer at the first place, but also not yet at the second. When it is at any one of the two it is at rest. When it is said that it is between both, this is not said for between both there is also a place and therefore the same problem occurs. But movement means: to be at this place and at the same time not be there; this is the continuity of space and time and this is it that makes possible motion (Hegel, 1833:337 ff.).27

Hegel and Hörz distinguish the aspect of movement (when a body does not find itself at a specific place) and the spatial aspect (the position of a body, when it has a definite place). In other words, they are actually making an appeal to two different aspects in order to side-step the accusation of a logical contradiction but at once they appreciate the (inter-modal) distinction that they make as a “dialectical Widerspruch.”

9. Different aspects involved in Zeno's ‘paradoxes’

In the case of Zeno's arguments different modal aspects are at stake. The theoretical attempt to reduce the meaning of movement to that of space is antinomic – but this antinomy shows itself in an implied logical contradiction: For example, as we have seen, in his fourth Fragment Zeno grants movement to begin with, but then concludes that movement is impossible. Something therefore can move if and only if it cannot move. This logical contradiction is the outcome of his antinomic attempt to reduce the original (primitive and indefinable meaning of) motion to static spatial extension. In other words, since an antinomy results from an attempt to reduce what is irreducible, it is always inter-modal in nature and simultaneously it expresses itself (intra-modally) within the logical mode as a logical contradiction.28

Of course this perspective does not eliminate a meaningful analysis of the numerical and spatial aspects of a (concretely) moving body (compare many of the arguments found in Grünbaum,  1967).  Material  (physical)  entities  and  processes  display  various  functional

27 “Wenn wir von der Bewegung überhaupt sprechen, so sagen wir: Der Körper ist an einem Orte, und dann geht er an einen anderen Ort. Indem er sich bewegt, ist er nicht mehr am ersten, aber auch noch nicht am zweiten; ist er an einem von beiden, so ruht er. Sagt man, er sey zwischen beiden, so ist dieß nicht gesagt; denn zwischen beiden ist er auch an einem Orte, es ist also diesselbe Schwierigkeit hier vorhanden. Bewegen heist aber: an diesem Orte seyn, und zugliech nicht; dies ist die Kontinuität des Raums und der Zeit, und diese ist es, welche die Bewegung erst möglich macht.”

28  Stafleu is correct in suggesting that one can interpret Zeno's arguments “against” motion as a de- monstration that motion cannot be explained by numerical and spatial relations (see Stafleu, 1987:61).

properties without being exhausted in their concrete many-sided existence by any single one of these modes of being (which are at once modes of explanation).

The solution of Zeno's problem of Achilles and the tortoise is certainly not given by the view that Zeno understood the metaphor of the  “moving observer” in a literal way, as it is claimed by Lakoff and Johnson (see Lakoff and Johnson, 1999:157-158), since what is ultimately shown by this ‘antinomy’ is that it is impossible to define uniform motion (exhaustively or exclusively) in spatial terms.

The acknowledgement of irreducible modes (functions or aspects) of reality is intimately connected with the idea of the identity of an entity which has a concrete and many-sided functioning within each one of these aspects without ever totally being absorbed by any one of them. Consider the four most basic functions of an atom. Besides the arithmetic function which an atom has (think about the atomic number), it also clearly has a spatial function since it is characterized by a particular spatial configuration – the nucleus of an atom with peripheral electron  systems.  According  to  wave  mechanics,  we  find  quantified  wave  movements around the nucleus of the atom – the kinematic function of the atom. Already in 1911, in Rutherford's atomic theory, the hypothesis was posed that atoms consist of a positively charged nucleus and negatively charged particles which move around it (a view which was inspired by the nature of a planetary system). In the following year (1912), Niels Bohr set up a new theory which contained two important ideas: (i) the electrons move only in a limited number of discrete orbits around the nucleus and (ii) when an electron moves from an orbit with a high energy content to one with a low energy content, electromagnetic radiation occurs. Therefore an atom is a micro-totality is qualified by its physical function of energy- operation.

The (relative) motion of a material entity concerns the ontic functioning of such an entity within the kinematical aspect of reality. But the motion of a physical entity pre-supposes the spatial function of physical entities – just think about the path of movement – as well as the numerical function – normally evinced when the measure of motion acquires a numerical specification (designated by establishing its speed).29 Although Salmon is correct in stating that Zeno, “[I]n his attempt to demonstrate the impossibility of plurality, motion, and change” points at “problems lying at the very heart of our concepts of space, time, motion, continuity, and infinity” (2001:5), none of the selections contained in the work edited by him on Zeno's Paradoxes enters into a discussion of the mutual ireducibility of these functions of reality (namely the numerical, spatial, kinematical and the physical). Neither does any one of them

29  The notion of ‘speed’ in phoronomy is similar to the notion of ‘magnitude’ in metric spaces. The classical ‘definition’ of a line as the shortest distance between two points is mistaken. Hilbert rather speaks about the straight line as the shortest connection between two points (Hilbert, 1970:302 – problem

4 of his classical 23 mathematical problems presented at the International Congress held in Paris in

1900). In this work Grundlagen der Geometrie (1899), Hilbert abstracts from the contents of his axioms and proceeds upon the basis of three undefined terms: “point,” “lies on,” and “line.” Within the functional structure of (a metrical) space distance (i.e., one dimensional extension) is the (numerical) measure of the extension of a line, the continuous extension of the line itself is primitive, just as specifying the speed of a moving body requires a measure of movement while movement itself remains a primitive. In both cases we may speak of the fact that the quantitative meaning of number is analogically reflected within the aspects of space and movement. Physics designates the numerical analogy within the function of energy-operation with the term mass.

consider the scope and limitations of different modes of explanation in respect of the uniqueness (primitive meaning) of motion. However, it does sometimes happen that an author highlights different aspects of an event. For example, when Max Black summarizes his argument by saying: “But Achilles is not called upon to do the logically impossible; the illusion that he must do so is created by our failure to hold separate the finite number of real things that the runner has to accomplish and the infinite series of numbers by which we describe what he actually does” (Black, in Salmon, 2001:80), then he actually distinguishes between different aspects through which we can approach such an event – namely the numerical and the physical.

Dividing Zeno's arguments into (i) the paradox of plurality and (ii) the paradoxes of motion, may seem to juxtapose two disconnected areas of reflection, but, as Salmon correctly remarks, they are not unrelated (Salmon, 2001:vi). He is also justified to hold at the same time that (i) is more basic than (ii). He writes: “we shall see that the paradox of plurality is logically more basic than the paradoxes of motion” (Salmon, 2001:7). However, we may want to expand his qualifier “logically” to read “onto-logically” – because numerical considerations in an ontic sense are foundational to an understanding of the meaning of space and motion.30

Even in respect of (ii) most philosophers and mathematicians focus on the problem of an infinity  of  points  or  intervals  that  ought  to  be  passed  /  traversed  in  a  finite  time.  A constantly recurring consideration concerns the “logical impossibility” to “complete” an endless (infinite) series. The “mathematical solution” of this problem is apparently found in the observation that, in terms of an arithmetical perspective (mode of explanation), that the successive partial sums of the series 1 + 1/2 + 1/22 + 1/23 + … 1 – 1/2n (n = 1, 2, 3, …) do not grow beyond all limits but converge  towards 1 (see Weyl, 1966:61). But Weyl immediately adds the remark that if the stretch of length 1 really consists of infinitely many partial stretches of length 1/2, 1/4, 1/8, … as separated wholes, then it would contradict the essence of infinity as the “Unvollendbaren” (what cannot be completed) that Achilles finally had to pass through (Weyl, 1966:61).31  In other words, as soon as the idea of a completed totality is combined with the infinite – for instance in speaking about an infinite whole or an infinite totality – then the true nature of infinity is contradicted (in this case in the claim that Achilles in the end completely passed through that which cannot be completed).

Interestingly Max Black argues from an understanding of infinity as “uncompleted” when he says that what is meant “by the assertion that the sum of the infinite series 100 + 10 + 1 + 1/10 + 1/100 + is 111 1/9” and that this “does not mean, as the naive might suppose, that mathematicians have succeeded in adding together an infinite number of terms” (Black, in Salmon, 2001:70). Black and Wisdom both criticize the “mathematical solution” (see Wisdom

30  In passing it should be mentioned that prominent 20th  century mathematicians, such as Gödel and

Bernays, argued for the ontic status of the numerical aspect of reality (see Wang, 1988:304 and Bernays,

1976:45, 122).

31  “Die Unmöglichkeit, das Kontinuum als ein starres Sein zu fassen, kann nicht prägnanter formuliert werden als durch das bekannte Paradoxon des Zenon von dem Wettlauf zwischen Achilleus und der Schildkröte. Der Hinweis darauf, daß die sukzessiven Partialsummen der Reihe 1/2 + 1/22   + 1/23 + …,

1 – 1/2n (n = 1, 2, 3, ...) nicht über alle Grenzen wachsen, sondern gegen 1 konvergieren, durch den man heute das Paradoxon zu erledigen meint, ist gewiß eine wichtige, zur Sache gehörige und aufklärende Bemerkung. Wenn aber die Strecke von der Länge 1 wirklich aus unendlich vielen Teilstrecken von der Länge 1/2, 1/4, 1/8, … als ‘abgehackten’ Ganzen besteht, so widerstreitet es dem Wesen des Unendlichen, des ‘Unvollendbaren’, daß Achilleus sie alle schließlich durchlaufen hat.”

in Salmon, 2001:83). Wisdom concludes: “The idea that the limit of an infinite series is attainable is a mistake. If a physical action is interpreted by means of an infinite series, then the completion of the action is self-contradictory” (Wisdom in Salmon, 2001:87).32

Owen  points  out  that  one  “beneficial  result”  of  Zeno's  “arguments  (on  this  familiar account),” was “to compel mathematicians to distinguish arithmetic from geometry” (Owen in Salmon, 2001:139). But Owen questions the idea of an infinite divisibility by posing the question whether or not such a division ever could be (could have been) completed?33

Likewise, the arguments found in Ryle's Dilemmas are based on the same assumption of the “uncompleted infinite,” although in addition he does introduce into his discussion the whole- parts relation (with reference to the classical slogan that the “the whole is more than the sum of its parts”). He says that the question “how many parts have been cut off from an object?” must be distinguished from the question “in how many parts did you divide it?” (Ryle, 1977:61). The first point proceeds from a notion of wholeness containing all its (finite) parts, whereas the second point reverts to the perspective and explores an on-going process of division.

This distinction actually imitates B Fragment 3 of Zeno in which he argues as follows (in the translation of Guthrie): “if there is a plurality, it must contain both a finite and an infinite number of components: finite, because they must be neither more nor less than they are; infinite, because if they are separate at all, then however close together they are, there will always be  others  between  them,  and  yet  others  between  those, ad  infinitum” (Guthrie,

1980:90-91). Therefore assuming a plurality leads to the contradictory conclusion that it

contains  “a  finite  and  an  infinite  number  of  components.”  But  given  the  fact  that Parmenides and his school, as an effect of the discovery of irrational numbers, switched from an arithmetical mode of explanation to a spatial one, we may look at the spatial whole- parts relation in order to understand what is here at stake.34 If the plurality of the first argument refers to a perspective from the parts to the whole, then the number of these parts must be limited while at once they constitute the world as a whole (the universe). By contrast, if the argument proceeds from the whole to the parts, the infinite divisibility operative in this move entails that “there will always be others between them” and so on indefinitely. Fränkel explicitly employs the whole-parts relation to explain the meaning of B Fragment 3 of Zeno (see Fränkel, 1968:430).35 Perhaps Zeno's B Fragment 3 could be seen as the first ‘two-directional’ discussion of the spatial whole-parts relation.36

32  The standard mathematical formulation of the nature of a limit does not hold “that the limit of an infinite series is attainable” (see note 36 below).

33 “For suppose we ask whether such a division could be (theoretically, at least) continued indefinitely:

whether any division can be followed by a sub-division, and so on, through an infinite number of steps. Let us say, to begin with, (A) that it does have an infinite number of steps. Then could such a division nevertheless ever be (or ever have been) completed?” (Owen, in Salmon, 2001:142).

34  Whatever is continuously extended is a coherent whole in the sense that all its parts are connected – therefore the terms coherence and connectedness are mere synonyms for the terms wholeness, totality and continuity. A whole (or totality) contains all its parts. Paul Bernays affirms that ‘wholeness’, i.e., the totality- character of spatial continuity, will resist a “perfect arithmetization of the continuum” (see Bernays, 1976:74).

35 Guthrie has a positive appreciation of this article of Fränkel. He refers to an English translation of it:

Zeno of Elea's Attacks on Plurality (see Guthrie, 1980:88 ff., 512).

36 In response to the phrase “If they are just as many as they are, they will be finite in number” Russell states: “This phrase is not very clear, but it is plain that it assumes the impossibility of definite infinite

10. Confusing the nature of infinity

The mere fact that so many scholars involved in discussing Zeno's paradoxes argue in terms of the ‘uncompleted’ nature of infinity demonstrates that they probably do not have a clear understanding of the difference between what traditionally is called the potential infinite and the actual infinite. Even when a competent scholar like Russell – who certainly has a sound understanding of the difference between these two kinds of infinity – sets out to present a historical presentation of the problem of infinity (see Salmon, 2001:45-68), one does not encounter an exploration of these two kinds of infinity.

Before Gregor of Rimini lectured on the Sentences at Paris in 1344, the objections to the actual infinite formulated by Fitzralph resulted in a rejection of both the potential and the actual infinite. In his argumentation he speaks about the simultaneous infinite (simultanes infinitum). In their discussion of the problem of infinity (during the 14th century) Henry of Harclay and others contemplated the difference between what was labeled the infinitum successivum and infinitum simultaneum (cf. Maier, 1964:77-79). These designations make and appeal to our most basic intuitions of number (succession) and space (at once). Therefore, it is recommendable rather to distinguish between the successive infinite and the at once infinite than between the potential infinite and the actual infinite.

The successive infinite brings to expression the most basic meaning of infinity – in the literal sense of without an end.37 It is determined by the primitive quantitative meaning of succession: one, another one, and so on indefinitely. The reason why the phrase actual infinity ought to be replaced by the expression the at once infinite is found in the inter- connection  between  number  and  space.  The  awareness  of  simultaneity  (at  once)  is  the correlate of any spatially extended figure (subject) for if the different parts of a spatial figure (as a whole) are not given at once the spatial figure itself is not given. Any understanding of infinity in terms of the idea of infinite wholes or infinite totalities is therefore dependent upon

‘borrowing’ a crucial element of space. In fact it seems to be impossible to develop set theory without “borrowing” key-elements from our basic intuition of space, in particular the mentioned spatial order of at once and its correlate: wholeness or totality. Hao Wang mentions that Gödel speaks about sets as being ‘quasi-spatial’ and then says: “I am not sure whether he would say the same thing about numbers” (Wang, 1988:202).

There is no constructive transition between the successive infinite and the at once infinite (see Wolff, 1971). But a full-blown treatment of the real numbers, transcending the ap- proximative approach found in intuitionistic mathematics based upon the denseness of the rational numbers, does require the employment of the at once infinite. Even though it seems as if the limit concept can be formulated merely in terms of the “endlessness” of the successive infinite, the requirement of the numerical nature of an arbitrary limit is dependent upon the use of the real numbers while an account of the real numbers, in turn, requires the use of the at once infinite. It is not possible to define (irrational) real numbers with the aid of converging sequences of rational numbers, because the classical definition of a limit (stemming from Cantor and Heine) presupposes that whatever functions as a

numbers” (Russell, in Salmon, 2001:47). Clearly Russell did not explore a “two-directional” use of the spatial whole-parts relation suggested above.

37 Russell holds that an endless series neither has a beginning nor an end (Russell, 1956:297).

limit in advance already must be a number – explaining why it cannot be ‘created’ through a ‘convergence process.’ In 1883 Cantor expressly rejected this circle in the definition of irrational real numbers (going mainly back to Cauchy in 1921 – see Cantor, 1962:187). The eventual description of a limit still found in textbooks today was only given in 1872 by E. Heine, who was a student of Karl Weierstrass.38 In 1887 Cantor pointed out that the core of the ideas in Heine's article was borrowed from him (Cantor, 1962:385).

What is important to realize is that as soon as the successive infinite meaning of the numerical order of succession is deepened through the spatial order of at once, the meaning of successive infinity is enriched through its connection with the meaning of simultaneity. Any successive sequence of numbers could then, under the guidance of this deepened hypothesis, be viewed as if its elements are given all at once. The deepened and disclosed meaning of the infinite (under the guidance of an insight into the spatial meaning of simultaneity) encountered here, justifies our choice to designate it as the at once infinite. Under the guidance of this hypothesis the initial successive infinite sequences of natural numbers, integers and rational numbers could be viewed as actual infinities, i.e., as infinite totalities given at once.39

Both Salmon and Grünbaum are sufficiently acquainted with (Cantor's) Set Theory (see the Appendix in Salmon, 2001:251-268; and Grünbaum, 1967)40  and they are justified in employing set theory in an understanding of the numerical side of physical movement (even though they do not realize that set theory is not a purely numerical theory but a spatially deepened arithmetical theory).41

Similar to those who argue against the “completed” infinite by applying the standard of the successive infinite, already Immanuel Kant thought that he can bring his first antinomy42 to a ‘solution’ by taking recourse merely to ‘endlessness’ (i.e., the successive infinite). In his remarks about  the first thesis of the second antinomy43  Kant states that space is not a

38  In general a number l is called the limit of the sequence (xn), when for an arbitrary 0 < e a natural number n0 exists such that |xn – l| < e for all n ≥ n0. (See Heine, 1872:178,182).

39  Lorenzen, although rejecting the at once infinite in his constructive logic and mathematics, does

provide a lucid description of the classical theory of real numbers in its dependence upon the use of the

at once infinite: “One imagines much rather the real numbers as all at once actually present – even every real number is thus represented as an infinite decimal fraction, as if the infinitely many figures (Ziffern) existed all at once (alle auf einmal existierten)” (Lorenzen, 1972:163).

40  That they did not discern the circularity present in Cantor's attempt to develop a supposedly purely

arithmetical understanding of a continuum of points cannot be analyzed in the present context.

41 Grünbaum's sympathetic quotation from Weyl (in Salmon, 2001:175) raises the suspicion that he is not

aware of the fact that Weyl, in rejecting the at once infinite, also rejects the Cantorean view employed by him  in  his  attempt  to  develop  a  “consistent conception of  the  extended  linear  continuum as  an aggregate of unextended elements” (see Grünbaum, 1952). Just compare the following words of Weyl: “In agreement with intuition Brouwer sees the essence of the continuum not in the relation of the element to the set, but in that of the part to the whole” (Weyl, 1966:74).

42 “The world has a beginning in time, and is limited also with regard to space” versus “The world has no beginning and no limits in space, but is infinite, in respect both to time and space” (cf. Kant, 1787:454).

43  The two sides of this second ‘antinomy’ corresponds to the “two-directional” nature of the spatial

whole-parts relation referred to above – Zeno B Fragment 3 and Ryle (in connection with the antithesis Kant raises the issue of infinite divisibility): “Every compound substance in the world consists of simple parts, and nothing exists anywhere but the simple, or what is composed of it” versus “No compound thing in the world consists of simple parts, and there exists nowhere in the world anything simple” (Kant, 1787:462 ff., 467).

compositum (in reaction to atomistic views of space), since in determining its parts space is a totum.44 He therefore does have an eye for the totality character of spatial continuity.

Clearly those who attempt to liberate themselves from the impasse in Zeno's arguments by pointing at the ‘impossibility’ of the ‘completion’ of endlessness (including the untenability of “infinity machines”),45 have to account for the difference between the potential and actual infinite (the successive and the at once infinite). The assumption in Zeno's bisection paradox seems to be that it is logically absurd to argue that all of an infinite number of tasks have been (can be) completed (in a finite time). Many authors did not come to terms with the endlessness of the successive infinite since they constantly alluded to a “last element.” Likewise it is self-contradictory to speak of an infinite division of a continuum if this entails a “last division” – though it is meaningful to speak about infinite divisibility.46 Yet, when the distinction  between  the  successive  and  the  at  once  infinite  is  applied,  a  deepened arithmetical perspective does allow for the acknowledgement of an infinite totality where any given succession can be viewed as being given at once, as an infinite whole. The fact that the at once infinite is irreducible to the successive infinite makes it meaningless to argue against the former by using the latter as yardstick.

But all these considerations still side-step the basic fact that most of the treated positions enter into a combat that takes place on the wrong battlefield! The real issue is not whether it is possible to develop a sound mathematical (or: set theoretical) analysis of the meaning of the successive infinite or of the at once infinite and then apply it to the numerical and spatial side of an actual physical movement. The issue is whether or not spatial continuity could be arithmetized fully47 and whether or not it is the task of a mathematical theory of number (or space) to explain the core meaning of motion in the sense of defining it in quantitative terms or in terms of the static meaning of space.

The employment of the at once infinite in the mathematics of Weierstrass, in fact misguided him to an understanding devoid of phoronomic and physical connotations, i.e., stripped of the idea of constancy and change. Boyer remarks:

In making the basis of the calculus more rigorously formal, Weierstrass also attacked the appeal to intuition of continuous motion which is implied in Cauchy's expression that a variable approaches a limit. Previous writers generally had defined a variable as a quantity or magnitude which is not constant; but since the time of Weierstrass it has been recognized that the ideas of variable and limit are not essentially phoronomic, but  involve  purely  static  considerations.  Weierstrass  interpreted  a  variable  x  as simply a letter designating any one of a collection of numerical values. A continuous variable was likewise defined in terms of static considerations: If for any value x0  of the set and for any sequence of positive numbers d1, d2, ..., dn’ however small, there

44  “We ought not to call space a compositum, but a totum, because in it its parts are possible only in the whole, and not the whole by its parts” (Kant, 1787:467).

45  This idea was initiated by Weyl (see Weyl, 1966:61). See also Salmon, 2001:26 ff. and Thomson (in Salmon, 2001:89 ff.) and Benecerraf (in Salmon, 2001:103 ff.).

46  Grünbaum distinguishes between infinite divisibility and an actual infinite dividedness (Grünbaum, 1952:300).

47 This problem is related to Zeno's paradoxes of plurality.

are in the intervals x0  – di, x0  + di  others of the set, this is called continuous (Boyer, 1959:286).48

This position even caused Russell to settle for the idea that Zeno's arrow is truly at rest at every moment of its flight!

After two thousand years of continual refutation, these sophisms were reinstated, and made the foundation of a mathematical renaissance, by a German professor, who probably never dreamed of any connection between himself and Zeno. Weierstrass, by strictly banishing all infinitesimals,49  has at last shown that we live in an unchanging world, and  that  the  arrow, at  every  moment of  its  flight,  is  truly  at  rest  (Russell, 1956:347).

The alternative approach advanced in this article holds that only when the uniqueness and mutual coherence of number, space and movement are observed is it possible to avoid the threat of antinomies inherent in Zeno's arguments against plurality and motion.

The spatial metaphysics of Parmenides, for that matter, inspired Zeno to defend a view of unitary wholeness that excludes plurality. In other words, Zeno wants to deny the ‘part’- element of the spatial whole-parts relationship while at the same time holding on to the ‘wholeness’ which entails it.50 His position is that reality is both one and indivisible. Yet, in order to argue for his position, he explored the whole-parts relation in his argument that is aimed at the denial of plurality! The reason why Zeno considers plurality to be self- contradictory is that plurality requires a number of (indivisible) units and because it also implies  that  reality  is  divisible  (see  Guthrie,  1980:88).  But  divisibility  threatens  the wholeness of a unit, since anything divisible has to be a magnitude which must be infi- nitely divisible. The supposed indivisibility of a unit clashes with its infinite divisibility. “Hence, since plurality is a plurality of units, there can be no plurality either” (Guthrie, 1980:89).

48  However, in the course of their development during the 20th  century both logic and mathematics realized that it is impossible to side-step the idea of constants and variables – thus showing that an analysis even of the meaning of number cannot be separated from the interconnections between various modes of explanation.

49  Of course the new introduction of infinitesimals in the non-standard analysis of Abraham Robinson outdates this remark of Russell. Robinson developed his new theory on the basis of a fertile use of Cantor's theory of actually infinite sets (transfinite cardinalities). A number a is called infinitesimal (or infinitely small) if its absolute value is less than m for all positive numbers m in      (   being the set of real numbers). According to this definition 0 is infinitesimal. The fact that the infinitesimal is merely the correlate of Cantor's transfinite numbers, is apparent in that r (not equal to 0) is infinitesimal if and only if r to the power of minus 1 (r-1) is infinite (cf. Robinson, 1966:55ff).

50  This is a quasi-Wittgensteinean position. Whereas Wittgenstein had to throw away the ladder after climbing up it (Tractatus, 6.54), Zeno started on top with wholeness and then discarded the ladder of infinite divisibility supporting it. The reverse took place in intuitionistic mathematics, which started with the original spatial whole-parts relation but then distorted it by accentuating the part-element (with its implied infinite divisibility) at the cost of the whole-element (with its givenness all at once). The intuitionistic theory of the real numbers and the continuum followed a similar kind of Wittgensteinean approach – it used the “spatial ladder of wholeness” but immediately afterwards discarded it while holding on to the infinite divisibility implied by it.

Therefore, by denying the foundational meaning of multiplicity, Zeno not only distorts the meaning of number (plurality) but also misrepresents the meaning of space. The infinite divisibility of a spatial whole analogically reflects the original and primitive numerical meaning of the successive infinite. Through spatial continuity the endlessness of the numerical infinite is “turned inwards.” But divorced from its connections with number the meaning of space collapses.51 The original numerical meaning of the unitary one is non- original within space, for within space the magnitude of an extended spatial figure (as a whole) provides a different context for unity – a unity (totality) that is infinitely divisible. The speculative (metaphysical) notion of a unitary whole excluding plurality robs both number and space of their meaning and mutual connections.52

From this subsection it is clear that the issues involved – regarding the uniqueness and mutual coherence of number, space and motion – transcend the confines of logic as such, although they certainly also cohere with the meaning of logical analysis. This consideration prompts us to look at the self-insufficiency of logic.

11. The limitations of logic

As long as one merely considers the logical principles of identity and non-contradiction (whether or not amended by the principle of the excluded middle), no material criterion of truth is available, for in terms of these principles one can at most affirm that two contradictory statements cannot both be true at the same time and within the same context.53

What refers thought irrevocably beyond logic is first of all the principium rationis sufficientis (also known as principium rationis determinantis and principium reddendae rationis) – in English formulated as the “principle of sufficient reason.” Since one can at most affirm that two contradictory statements cannot both be true at the same time and within the same context” it is clear that logic alone cannot resolve the contradiction. What is needed is to ask for non- logical (extra-logical) grounds, that is to say, a reference to states of affairs “in reality” (“outside” logic) is required.

This principle, originally formulated by Leibniz, was subjected to an extensive investigation by A. Schopenhauer in 1813. He called it the principle of sufficient ground of knowledge (principium rationis sufficientis cognoscendi) (Schopenhauer, 1974:156).

51 Mathematical dimension theory explores the notion of dimension as an order of extension – captured by the natural numbers 0, 1, 2, 3, … – thus analogically reflecting the foundational meaning of number. In metrical spaces magnitude (such as length – one-dimensional extension; surface – two dimensional extension; and volume – three dimensional extension) in a correlated way analogically reflects the foundational meaning of number. Exploring the suggestions of Poincaré, Brouwer in 1913 introduced a precise (topologically invariant) definition of dimension, which was independently recreated and improved in 1922 by Menger and Urysohn. Menger's formulation (still adopted by Hurewicz and Wallman) simply reads: “a) the empty set has dimension -1, b) the dimension of a space is the least integer n for which every point has arbitrarily small neighborhoods whose boundaries have dimension less than n” (Hurewicz, 1959:4, cf. 24; cf. also Alexandroff, 1956:165, 167 note 12 regarding the intuitive meaning of dimension as it is present in the principle of invariance of Brouwer).

52 Just recall our earlier remarks about speed and the length of a line where it was argued that within the domains of space, movement and the physical we can discern numerical analogies.

53 This was already clearly understood by Kant (see Kant, 1787:84-85).

The general legacy of Leibniz is captured in the phrase: there is nothing without a sufficient ground (nihil est sine ratione sufficiente). Of course already Plato affirmed that assertions require a foundation (Timaeus 28a), whereas Aristotle distinguished four causes: material, formal, effective and final ones.

In his Monadology Leibniz formulates his view as follows:

… and the second the principle of sufficient reason, by virtue of which we observe that there can be found no fact that is true or existent, or any true proposition, without there being a sufficient reason for its being so and not otherwise, although we cannot know these reasons in most cases (Leibniz, 1976:646 – see Sections 44 and 196).

Ultimately the combined perspective of the principle of sufficient reason and the ontic requirement to avoid antinomies by acknowledging what is unique and irreducible opens up the plea for a non-reductionist ontology. In terms of this remark one can view each and every monistic ism as an argument against the position defended in this article.54 Our conjecture is that the untenability of such a truly monistic orientation is shown by the antinomies that it entails. An interesting feature of an antinomic position is that it always reaches the opposite for what it aims for.

In addition to what has been said about the antinomies entailed in the space metaphysics of the school of Parmenides, we briefly consider another example by looking at the intentions of historicism. Under the heading: Change and Permanence: On the Possibility of Understanding History Hans Jonas examines the impasse of historicism. He argues that radical historical skepticism is self-defeating (Jonas, 1974:241). The problem is that change can only be detected on the basis of an enduring or persistent element, of something constant.55  For example, if law is intrinsically historical it is supposed to have ‘happened’ somewhere in the past – which is not at all the case, for jurisprudence knows much about legal history. If the jural itself was history, it could not have had a history. The irony of radical historicism is therefore that the opposite of what is aimed for is achieved – if everything is history, there is nothing left that can have a history. Jonas refers to the said element of constancy as something transhistoric in his assessment that runs parallel with the irony just mentioned: “Actually, there is no paradox in this. For history itself no less than historiography is possible only in conjunction with a transhistoric element. To deny the transhistorical is to deny the historical as well” (Jonas, 1974:242).

Every scientific methodology (and epistemology) is founded in an ontology. This insight is captured in the title of an article of Neemann: “Das Primat der Ontologie vor dem der Metholodologie” (“The Primacy of Ontology before that of Methodology” – see Neemann, 1986). The idea of a non-reductionist ontology carries it to its ultimate epistemological consequences by making a plea to side-step the logical contradictions entailed in underlying antinomies.

12. The foundational role of the principium exclusae antinomiae

If the principium rationis sufficientis refers thinking beyond the limits of pure logicality, the logical principle of non-contradiction is further enriched by an underlying ontical principle,

54  Monistic isms are for example arithmeticism, holism, physicalism, vitalism, psychologism, logicism, historicism, and so on.

55 The term ‘constancy’ is preferable to ‘permanence’.

namely   the   principle   forbidding  inter-modal   reductions  which  invariably   result  in antinomies. This principle is ontical in nature and should be called the ontical principle of the excluded antinomy (principium exclusae antinomiae).56

The perennial philosophical problem of explaining the coherence of what is unique and irreducible (the “coherence of irreducibles”) therefore opens the way to an acknowledgment of the foundational position of the principium exclusae antinomiae in respect of the logical principle of non-contradiction – and at once it explains why the distinction between antinomy and contradiction is not a purely logical distinction. The principium exclusae antinomiae not only depicts the limits of logic but at once also underscores the significance of a commitment to transcend the one-sidedness of reductionnsitic isms within philosophy and the various disciplines. For this reason we understand our argument as being supportive of a non- reductionistic ontology. Laying bare theoretical antinomies in an ‘ismic’ stance serves as a strengthened form of immanent criticism – no thinker can turn away when it has been shown that her position is internally antinomic. In fact, once an antinomy has been articulated, the challenge is reverted, for then an alternative must be presented not subject to the same immanent criticism. If it is successful, i.e., if the alternative view put forward does not harbour a similar antinomy, then progress has been made in the critical intellectual encounter.

13. Concluding remark

The logical principle of sufficient reason refers us to those (extra-logical) grounds on the basis of which a valid argument can be pursued. However, if something truly basic (primitive) and indefinable becomes the victim of an attempt to reduce what actually is irreducible, theoretical thought gets entangled in genuine antinomies. Since the latter bring to light an attempt to reduce different functional modes of reality to each other (such as the Eleatic attempt to reduce motion to space), antinomies demonstrate that they are inter-modal in nature – thus differing from the intra-modal reference of a mere logical contradiction – such as the confusion of two spatial figures in the (illogical) concept of a “square circle.” In order to explain some of the intricacies of this distinction between contradiction and antinomy – a distinction that exceeds the confines of pure logic – we had to take into account related problems, such as the meaning of unity and diversity, the problem of reductionism and the issue regarding the urge of monistic isms to find one all-embracing mode of explanation. In the final analysis it turned out that the need to distinguish between contradiction and antinomy amounts to a plea for the acknowledgement of a non-reductionist ontology. The aim of such an ontology is to avoid the dead alleys accompanying all instances of a metaphysical reductionism because the latter always terminate in the antinomic position of one or another monistic orientation.

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Dear Mr. Daniel F M Strauss, it is really a good way to develop our science through discussions; you may have more information about the topic from my uploaded papers. 

Sincerely hope everything goes well with your book of METAPHYSICS.

Best wishes, Geng

I'm not sure that mathematical propositions are about the world at all! Their truth or falsity does not depend on their having any obvious application to it (e.g. alternative geometries, developed before an application was found for them).

Incidentally, paradoxes such as the Turtle are (sorry!) a red herring - false in their application to the real world for reasons to do with how they are commonly expressed, which involves a little trickery. There are two sequences involved in the Turtle - a progression of distance covered and a progression of time taken. Neither of them reaches the end of the race!

Dear Mr. Roger John Sapsford, thank you for the insightful idea. But how do you think of following normal and “pure” mathematical case?

Look at the widely accepted modern divergent proof of Harmonic Series.

1＋1/2 ＋1/3＋1/4＋．．．＋1/n ＋．．．                                       （1）

=１＋1/2 ＋（1/3＋1/4）＋（1/5＋1/6＋1/7＋1/8）＋．．．        （２）

>1+ 1/2 ＋( 1/4＋1/4 )+（1/8＋1/8＋1/8＋1/8）＋．．．                （3）

=1+ 1/2 + 1/2 + 1/2 + 1/2 + ．．．------>infinity                                 （4）

Can we produce infinite numbers each bigger than 1/2 or 1 or 100 or 100000 or 10000000000 or… from infinite items in Harmonic Series by “brackets-placing rule" to change an infinitely decreasing Harmonic Series with the property of Un--->0 into any infinite constant series with the property of Un--->constant or any infinitely increasing series with the property of Un--->infinity?

Sincerer yours, Geng

Dear Geng

I think we are starting to spin our wheels - so I am dropping out of this discussion now!

Dear Roger

Regarding what you remarked "not [being] sure that mathematical propositions are about the world at all!" I want to make just one observation: without the ontic quantitative aspect of "the world" mathematics would be unable to formulate theories without any "obvious application"! What is primitive in axiomatic set theory reveals the irreducibly unique meaning of number.

Best wishes

Danie

To Daniel Strauss; in reference to Ramon Quintana 

Danie,

I agree with what you say above in your reply to Geng: "without the ontic quantitative aspect of "the world" mathematics would be unable to formulate theories without any "obvious application"!"

By "ontic" I am assuming you mean to distinguish it from purely phenomenal or conceptual appearance. You refer to the being or reality of what is experienced or known, apart from any "ontological" or "metaphysical" analysis of it. 

If so, then the ontic quantitative aspect of "the world"  you refer to would be the aspect(s) of being that explain(s) (ontologically) how the quantification or counting of particular real things is possible. 

So far, so good. Then you move on to add. "What is primitive in axiomatic set theory reveals the irreducibly unique meaning of number." Again, I am in full agreement provided that the primitives in set theory can be shown to refer to correspondingly primitive ontics in "the world" which are NOT phenomenal or conceptual in nature. 

By referring to ontic rather than phenomenal aspects of the world and what we find in it, we are attempting to describe exactly what Kant says we cannot: the thing in itself, or what I would prefer to call the intrinsic "nature of substance". We are trying to do what Ramon might call in his recent post, "...  to coax ontological truths out of mathematics, ...." As long as we allow the ontological argument to be reduced to the epistemological argument about how we know about mathematical truths in the apparently non-empirical way that we do. These are distinct arguments and which one we take to be the most basic makes a big difference in how the conclusions work out.

But I'm still not sure in your view, Danie, which you consider to me more basic. My point is that, pretty much since Plato prevailed in the pre-Socratic debate, Western philosophy has pursued an epistemological approach to solving problems in philosophy generally. The epistemological project produced

all intended to replace the ontic with the epistemic. But I argue that the epistemic has to be less basic than the ontic because we have a chance to explain how non-intelligent, non-knowing, non-organic being can result in knowing, conscious beings like us.  Whereas, 2000 years of epistemological acrobatics have universally failed to show how we can "coax ontological truths out of mathematics, logic or any other formal methodology. Godel's incompleteness theorem proves this by showing there is no mathematical formalism that can "complete" the set of true statements the define a set of epistemologically necessary propositions.

So instead, I propose just leaving the epistemology aside at first, and trying to develop an approach that begins with propositions that refer to what exists in the world, rather than to what we can know about what exists.  It's a choice and the history of philosophy (everywhere, really) is largely the history of one of these 2 choices.

The irony is that Western philosophy begins with the iconoclasitc notion that we know best about the world when we use naturalistic descriptions of the observed world rather then rationalistic descriptions of the motives and reasons of super-natural, anthropomorphic beings whose actions, invisible to us, brought about and sustain the natural world we experience only imperfectly through our senses.

We have Thales and the other Milesian Greeks to thank for this radical shift that makes the beginnings of the scientific worldview which was largely lost to culture for over 1000 years until it was "reborn", leading to the rise of modern science and technology which have reshaped the landscape of the planet and of culture. But, philosophically, we are still just heirs to Plato and this is nowhere more clear than in the philosophy of mathematics. For it, after all, is the model of "perfect" knowledge that inspired rationalism and remains the domain of philosophy, which now has ceded all of its ontological might to the scientists... along with the rest of the wisdom it used to love. 

So it the very refreshing to me to see your thoroughly detailed analysis of the ontic quantitative aspect of the world. You seem to be doing ontology, for a change,  but I am still not sure how you want to connect the need for an ontic correlate of formal axiomatic systems like mathematics without some "obvious application". I hope you will enlighten me on this "killer app". :)

Thanks,

Daniel C. Davis

I think I agree with much of Dr. Davis's views though not all. Yes, math has to tie to the real world in the end or it is useless. The question is how do we know it? Kant said we can't. Plato said we can by divorcing concepts from sensory experience (rationalism). Both disasters. But western civilization, especially science and politics, derives from Aristotle. I don't mean his cosmology which was wrong; he did not have a telescope. But he grounded concepts in sensory experience, including his biology.

Edwin,

Thanks for the reply. It's always nice to find someone local on here. Go Terps!

But Aristotle was still a rationalist. His teleological cosmology is not incidental to his fundamental understanding of the nature of substance. Remove the final cause of substances and we are left with the problem of explaining how things are naturally grouped into kinds by their functions... that is, by what they WILL do in the future or by the future consequences of their behaviors. Nature seems to Aristotle to have intelligent intentions built into it at its core. We moderns find this belief to be anathema to scientific methodology but have really found nothing to replace it as an efficient cause. Even Darwin's natural selection of species is seen as accidental or else as trying to re-introduce final causation. 

This is particularly problematic when we consider that in order to explain how we know something in the "material world" post-Aristotle, How are there universal kinds of things? What, actually, are the units that arithmetic counts, each as "one", and reasons about? What, indeed, is left of "substance" at all, other than bundles of sensations with no ontic properties other than in our habits or structures of mind?

Please see my recent comments to Daniel Strauss and how they align with these to you.

Thanks,

Daniel C. Davis

Daniel D: I agree that Aristotle ended up with some Platonic elements--the key problem with to figure out the nature of concepts--from what I am told he made a Platonic mistake here- (Ayn Rand solved the problem)-but I would not agree that he was a rationalist--he was tied to reality thru the senses--his logic was mostly deductive but not wrong, I think--he did not have a real theory of induction but was on the right track with his biology (though did make mistakes)--it was unfortunate that the Catholics turned his views (e.g., cosmology) into dogma (A,. would have revised his views if he had a telescope) but nevertheless his legacy, not Plato's, paved the way for modern science--I recall he was a hero to Galileo-- Have you read: The Cave and the Light by Herman? Basic theme:  most of the bad things that happened in history were due to Plato and most of the good things to Aristotle- (I am not an historian)

Wow, Ramon, you seem to have touched a nerve.  I think that is because your question goes to the heart of philosophy and the many differences that we observe between philosophers, past and present.  On the question of the reality of numbers and related questions I have been heavily influenced by Wittgenstein, both early and late.  Since very little has been said about Wittgenstein's approach to this question, I will add a few thoughts to this discussion. 

Wittgenstein famously made the point that there is something very odd in the claim that a fundamental philosophical foundation is established when philosophers say things like, "There is a table here."  Wittgenstein thought it odd because the statement, as it's intended by the metaphysical realist, plays no such role in ordinary language.  Thus, when you speak of a natural language that can refer directly to objects in the world, I think that Wittgenstein would emphasize that there are no necessary metaphysical commitments in our ordinary language about objects in the world.  Of course, the metaphysical or philosophical realist finds Wittgenstein's position to be a ludicrous one.

The question about the reality of numbers, however, can provide some clarity to Wittgenstein's approach to this sort of question.  Number systems are taught to children who then use numbers in their ordinary discourse with others, whether learning to count change at the store or developing engineering software.  What can be said about those numbers is shown in the life that we share together. 

What is the ontological status of numbers?  Well, what are you looking for?  What do you think an ontological status would tell you about numbers?  Would this status be essential for their use in human discourse?  Would it be more important than the use itself?  The common problem with a search for an ontology is that it often takes you away from the meaning and use of the terms in question while tempting you to think that you are saying something fundamental to the nature of those terms.

So, when you ask a question about the ontological status of numbers what follows is a lot of positioning and arguing between philosophers of different traditions.  I don't mean to suggest that the question of the nature of the reality of numbers is an unimportant one; I would only suggest that the discussion of that question would bear much more fruit if it were more closely tied to the common discourses of humanity rather than a search for some ontological status.

Dear Daniel Davis

Thanks for your constructive remarks - and for pointing at the priority of ontology above epistemology. From what you (and Edwin) wrote I am not sure that your have read the relevant sections of my article on Frege - where I discuss the question what is ontic about number. Please send me your e-mail addresses so that I can send you this article in PDF format (as well as some other material).

Danie

Danie,

Thanks for sending your publications to me. I'll need some time to read and digest them but will reply soon as I can. Briefly, I think we both see an need for an ontic foundation for numbers but our ontological approaches are pretty different. 

The way to understand the applicability of abstract math to concrete reality is to understand first what the number "one" is. What is it to be one? How is it that there are "ones" or "units" out there that we can count as "1"? Without objects to count, there would be no counting.

Thanks,

DCD

Here is why ontology is necessary: 1) is there is real world out there? 2) does the universe obey causal laws? 3) does consciousness exist? These involve axioms, starting points. Without these you cannot go anywhere--they cannot be proved because they are the basis for proof of anything--if anyone is interested I can discuss how you validate something as an axiom--the problem of the many is: the problem of how you form valid concepts--I can talk about that too if there is any interest--

Edwin,

There seem to be more fundamental questions at stake: Must we make sense in order to be understood on an Internet blog?  Does it make sense to ask, on a blog with other philosophers, whether there is a real world out there?  What is asking?  What is a blog? What is the Internet?  What does it mean to ask whether there is a real world out there if the sense of all of these questions entails a real world?  What are questions or speaking or Internet blogs if there is no real world out there?  I don't mean that these establish a real world; I don't see how we can even make sense of the question.  How can we speak of determining whether there is a real world out there before we have determined what it means to say something, to ask questions, to determine something? 

Further: Is the universe a thing like a house or a watch that we could observe it?  Could there be questions without consciousness? 

"Without these you cannot go anywhere."  But that is precisely what we do.  Whole cultures and peoples have come to be and passed away without answers to these questions.  But not because they didn't ask them or they couldn't get the right answers; they didn't have answers to these questions because the questions don't make sense.

The clearest statement of our situation comes from Rush Rhees, one of Wittgenstein's literary executors: "Language makes sense if living makes sense."  I recommend his book, Wittgenstein and the Possibility of Discourse and his two volume collection: In Dialogue with the Greeks.

All cultures answer those questions, if not explicitly then implicitly-- and the implications of the premises are enormous--let's say you are a mystic and believe the world is created in your head--how would that affect your life? it would stop all  progress and not just science--want to "cure" disease? attribute it to the devil and that's the end of medicine--reject the law of identity and everything occurs by magic--can there be questions a/o consciousness? No. Who would ask them? I have to say that I reject Witt if his view is to study language use divorced from reality--

Neither Wittgenstein nor Rhees ever suggested that we could study language divorced from reality.  What they suggested is that the very idea of a separation between language and reality is a confusion, brought on mostly by philosophers.  The confusion leads philosophers to absurd conclusions such as suggestions that mystics believe the world is created in your head and that attributing disease to the devil eliminates medicinal cures.  When philosophers assume that a bridge between language and reality needs to be established in order for us to find sense and truth, they tend to make statements such as "All cultures . . ." before they have even looked at cultures. 

Language is part of the reality that we share so it cannot makes sense to speak of connecting language to reality in any general sense since that would require that language is something independent of reality.  It does not follow that there are no general questions about the nature of reality; it only follows that such general questions will not involve connecting a label (language) to an object (reality).  Such questions will begin with what it means to say something in a shared life (Can we see or understand what is real in what is said?).  The Sophists seemed to think that what it means to say something is to get people to agree with you.  So, reality just is what your peers will let you get away with.  Plato recognized that there is a half-truth in this that has a corrupting influence on our common discourses, our understanding of ourselves and of the reality that we share.  He struggled to understand how and why this is the case.  We cannot understand what Plato understood without having to struggle just as much as he did.

Very important questions are raised in our discussions of metaphysics, ontology, and epistemology but surely we should be more careful when we claim, "This is what a culture(or a people or a group) is doing, whether they realize it or not." 

Well  I guess we disagree- I believe that words are symbols that stand for concepts and that valid concepts are tied to reality starting with the material provided by the senses- concepts are mental integrations---for example, in forming the concept of triangle, you integrate the common element (3 sided plane figure bounded by straight lines) and ignore the differences (e.g., angles, size, etc.)--this ties the concept to reality and prevents the Platonic error of concepts divorced from reality--if you think reality is what people let you get away with rather than what is, would you cross the street blindfolded? jump off a building? write a check for more than your account has in it? To me philosophy is for the purpose of living in the real world--

Well, I hope that that we agree that reality includes things like words, symbols, and concepts.  Reality is a concept, too, right?  And, on your view, if it is a valid concept, then the concept of reality is tied to reality.  How do we make sense of that?  Can we tie a concept of reality to reality?  Does this mean that the second reality is not a concept?  How is that possible?  I know that we climb mountains and not concepts of mountains, but I know that in and through the concept of mountains.  I can't know it independently of the concept of mountains.  So how do I know that reality is tied to reality without reliance upon a concept of reality, and have we said anything at all when we say that reality is tied to reality?

You seem to be putting a heavy burden on the term "reality," a weight that I fear it cannot bear.  Are concepts real?  Your answer is that some are valid and some are not.  But I'm asking about the concept of concept and the concept of reality?  Are these valid concepts?  Are we using terms that must be assumed and accepted without a validation procedure?  If so, why should I consent to such assumptions?  So I can be like other philosophers who let their peers speak this way?

These are reasonable questions that can be asked of any account that you might give of efforts to tie concepts to reality.  These questions illuminate the difficulties in speaking this way, i.e., as if concepts and reality were in need of being tied together.  People speak and communicate, and cultures have bequeathed whole literatures without having to have philosophers insure that concepts are tied to reality. The point of these questions, if they cannot be clearly answered, is to consider whether we should find other ways of talking about the nature of reality.

I am not suggesting that reality is what people let you get away with.  But sometimes it looks and feels as if reality is what people let you get away with.  I'm interested in the question of why reality is not simply what people will let you get away with but I'm interested in discussing that question without falling into the logical confusion of trying to tie concepts to reality.  That road is not even a road; it's an illusion.  I would like for us to have discussions of the nature of reality that begin with living in the real world.  An ontology that assumes that every culture is implicitly answering these same questions whether they realize it or not seems to me to be a great distance from the real world.  Wouldn't an account of the real world be one in which people would recognize the reality being described?

Phil starts with axioms without which you cannot go anywhere- so the Q is how do you validate an axiom: 1) self evident to one's first perceptions; 2) precedes any further knowledge; 3) can't be contradicted without using them in the process. The three key axioms are: existence, identity and consciousness. One example from your post:: the concept of illusion can't be formed without first knowing what is real.

I do not agree that math is tautological--if it did not connect to reality, then modern civilization would have been impossible--you could not make progress in a single field of endeavor--and there can be no randomness in nature--the law of identity: everything is some particular (Aristotle) and thus has specific attributes and characteristics--randomness is an epistemological concept: it refers to lack of advance knowledge-

Ontological, fictional or in the literal sense?

1,  Science is our human’s, it is undoubtedly that we really can own (create) our common “literal scientific concepts and terms” for our science through studies, discussions…. But how to unify “objective world” and “subjective world”, how to cognize and translate “things in human science and things in natural world”, “ontological and formal”, “sets in mathematics” and “sets in nature”, “infinitesimals in mathematics” and “infinitesimals in nature”,…through “literal scientific concepts and terms” have been the essential, important and toughest works for scientists.

2,  The cognitions of “literal concepts and terms” are essential, important and toughest works for scientists because I think these basic stones exert great impact on logics and fundamental theories-------if the concepts and terms are unscientific, the relationships (logics) between and among them are unscientific.

I shall soon respond to the issue of an ontology!

Dear Bernard, Ramon and others who may be interested in the status of ontology, let us look at what Putnam says: “ ‘Objects’ do not exist independently of conceptual schemes. We cut up the world into objects when we introduce one or another scheme of description.”

When Putnam claims that our “conceptual schemes” “cut up the world into objects” he proceeds from an uninterrupted (not yet “cut up”) world. Without being aware of it, Putnam in fact employs two opposing “schemes of description” – the one “cutting up” the world in multiple distinct (discrete) “objects” and the other one assuming a continuous whole preceding any “cutting up.” In short, Putnam's point of departure is the DISCRETE and the CONTINUOUS – about which Fraenkel et. al., Foundations of Set Theory (2nd revised edition 1973) states: “Bridging the gap between the domains of discreteness and of continuity, or between arithmetic and geometry, is a central, presumably even the central problem of the foundation of mathematics” (page 211).

In terms of his own approach Putnam should have recognized the alternative “conceptual scheme” behind his claim – namely that of a continuous whole not yet divided [cut up] into “objects.” Implicitly the idea of continuity does function as an “unconceptualized reality”!

Putnam also holds that ultimately “we have no access to unconceptualized reality” – contradicting his “non-contemplated” access to the continuity of the pre-divided world!

Putnam could have benefited from the way in which Kant accounted for the unknowable “Thing-in-Itself” (Ding an sich): there must be a thought-form in which we THINK what is UNKNOWABLE. This form of thought is the transcendental idea.

What is known as the opposition between a correspondence theory of truth and a coherence theory of truth actually one-sidedly emphasizes either the factual object side or factual subject side of the logical analytical aspect. In its modal universality, this aspect embraces whatever there is and therefore underlies the ability we have to logically objectify whatever is identifiable and distinguishable. Although the conceptual framework within which knowledge is embedded co-determines our knowledge acquisition, it is always at once related to what is logically objectified (similar to the difference between the connotation and denotation of a word or sentence occurring within the lingual subject-object relation).

If it was not the case that the entire universe had either a subject-function or an object-function within every aspect, human beings would have been opposed to things with which nothing is shared. Subject-object relations are ontic – they are embedded in the ontic modal structures of the various aspects of reality.

However, what lies behind Putnam's denial of a non-conceptualized reality is modern nominalism.

Two of my articles may be considered in this context (I shall attach them):

1)         Strauss, D.F.M. 2013. Understanding the Linguistic Turn and the Quest for Meaning: Historical Perspectives and Systematic Considerations. In: South African Journal of Philosophy, 32:1, 90-108. To link to this article: http://dx.doi.org/10.1080/02580136.2013.810419

2)         Strauss, D.F.M. 2009. The (social) construction of the world – at the crossroads of Christianity and Humanism. 28(2):120-131.

Objects do not exist except in our conceptual schemes? Oh, really--do you get out of the way of cars when you cross the street? Do you dress in real clothes or concepts? Do you eat real food or conceptual food? How can you take seriously a philosophy which detaches you from the real world???

Daniel's assessment of Putnam is quite right.  We can rightly ask of Putnam, "From where did you get this concept of 'the world' (which is then cut into pieces) if not from a conceptual scheme?" 

However, the difficulties that Daniel raises reveal not a problem in need of a solution but a puzzle created by our own language when philosophers go in search of an ontology.  Daniel's "answer," like much of Kant's ingenious analysis, is an elucidation of our grammar and is nothing more than a tautological observation: Physical reality is characterized by the subject-object relation.  In creating technical terms to express these tautologies we fool ourselves into thinking that we have provided a profound answer to one of philosophy's great traditional problems.  And yet, the best of these "answers" turn out to be tautologies dressed in sophisticated terms. 

The problems and "answers" created by our use of language when seeking an ontology ought to reveal to us that there is something wrong in the way that we are approaching the question of the nature of mathematical statements.  Perhaps the discussion should begin with mathematical statements. What is a mathematical statement?  Then, what is one looking for when asking whether mathematical statements are figurative or literal?  I would think that a discussion of figurative and literal would follow and what that comparison may or may not have to do with mathematical statements. 

So, for example, a mathematical statement is sometimes used to describe relations of logic in logical symbolism and sometimes used only of mathematical relations.  Already, there is need for work.  And "literal" simply means common usage and "figurative" means a departure from common usage.  So, as long as a mathematical statement is made in common parlance and according to common usage, it is a literal statement.  Is it thus possible for a statement to be mathematical and figurative?  Perhaps new observations in higher mathematics could be understood as figurative. But there are problems here.

It seems to me that this line of discussion would help us to become much clearer about the nature of a mathematical statement than a search for an ontology.

Dear Edwin and Patrick

I like Edwin's immanent critical questions to an over-estimation of conceptual schemes: “Oh, really--do you get out of the way of cars when you cross the street? Do you dress in real clothes or concepts? Do you eat real food or conceptual food? How can you take seriously a philosophy which detaches you from the real world?”

Then Edwin asks: “Are mathematical statements ontological in a figurative or fictional sense, or in the literal sense of ‘what there is in the world’?” – to which Patrick adds the question “of the nature of mathematical statements.”

Patrick also claims that we fool ourselves because we are merely formulating tautologies.

However, shifting the question of ontology to “the nature of mathematical statements” does not avoid the problem of ontological commitments in mathematics. The issue is straight-forward: how does one define mathematics?

I one of the articles attachments to this response I argue that defining mathematics exceeds the confines of mathematics! It is philosophical in nature and cannot avoid (implicitly or explicitly) to reflect upon the ontic status of the aspects of number and space.

Skolem (1979:70) summarised his assessment regarding these issues in 1922 as follows: “Those engaged in doing set theory are normally convinced that the concept of an integer ought to be defined and that complete induction must be proved. Yet it is clear that one cannot define or provide an endless foundation; sooner or later one encounters what is indefinable or non-provable. Then the only option is to ensure that the first starting points are immediately clear, natural and beyond doubt. The concept of an integer and the inferences by induction meet this condition, but it is definitely not met by the set theoretic axioms such as those of Zermelo or similar ones” [Quoted in my article on Defining Mathematics]

Regarding tautologies just a brief remark (I attach an article on Primitive terms and the limits of conceptual understanding). The core meaning of each of the various ontic modal aspects is indefinable and therefore needs to be designated by a term which succeeds in calling forth what conforms to our immediate, intuitive insight. Suppose we characterize what is primitive about number by using the phrase discrete quantity, what is primitive about space by designating it with the phrase continuous extension, what is primitive about the biotic as life, and so on. Then tautologies like “what is discrete is discrete” or “life is life” are meaningful primitive intuitions. Yet there is more to it, for the meaning of whatever is (tautologically) unique only comes to expression in its coherence with other unique aspects – as understood by Gödel in his own way.

Yourgrau explains that Gödel “insisted that to know the primitive concepts, one must not only understand their relationships to the other primitives but must grasp them on their own, by a kind of ‘intuition’ ” (Yourgrau, 2005:169). [Yourgrau, P. 2005. A World Without Time. The forgotten Legacy of Gödel and Einstein. London: Penguin Books.]

Asking “whether mathematical statements are figurative or literal?” enter into a different but related field of investigation, namely that of the basic concepts of a discipline, embracing also an account of different kinds of analogies and how they relate to metaphors or figurative statements. [see: George Lakoff & Rafael E. Nunez, 2000. Where Mathematics Comes From. How the Embodied Mind brings Mathematics into Being. New York: Basic Books.]

I attach also the following article: Discerning similarities: ‘concept’ and ‘word’ at the intersection of ‘analogies’ and ‘metaphors’ Acta Academica, 2005 37(2):1-20.

Danie Strauss

I repeat what I said before:if math were not tied to reality, then civilization as it is today could never have arisen--could you build a bridge or a building or an airplane without your math being tied to the real world--there are thousands of examples one could find in every field--

Of course, Daniel, defining mathematics is not a mathematical problem.  The question is whether it must then be an ontological problem.  Seeking a definition is first and foremost a language problem.  What is a mathematical statement?  It does not follow from this, Edwin, that I am trying to say that math has no reality.  Clearly it does (though I don't think we have to show that it's tied to reality precisely because it is so central to civilization).  As you say, civilization would not be what it is without mathematics.  But the question is; What kind of reality does math have?  I have tried to argue that this question can be pursued without having to pursue talk of ontology.

I understand the history of the use of the term "ontology" in philosophy.  The fact that we philosophers, as a consequence of that history, have given such great weight to the notion of "implicit reflection" ought to raise enormous red flags for us.  How did we get into this position?  How is it that we are claiming that people must see reality in a certain way, whether they realize it or not?  The arrogance of our discipline is astounding and embarrassing.  This is the result of seeking an ontology.

I don't have a problem with tautologies.  They are extremely valuable, especially in teaching children how to speak.  And they can be powerful reminders to philosophers about how we speak.  What I find offensive is the practice of passing off tautologies as if they were profound truths about the nature of reality when they are simply (admittedly, sometimes profoundly so) elucidations of how we speak.  This seems to me to be the domain of ontology seeking.

In order to avoid that temptation, I have suggested that we could pursue the question of what a mathematical statement is by discussing the role of mathematical statements in the life and lives that we share together.  That question is neither a mathematical question nor does it seek an ontology.  it is a philosophical question that seeks the definition of a mathematical statement.

Dear Bernhard

I enjoyed your reflections and often felt that I can relate to many of your brief remarks – even though it is not easy to discern a systematic threat in what you have said. But allow me to “test” your appetite, as a physicist, for systematic distinctions! What I have in mind is what I have called the Achilles' heel of positivism.

Positivism did not realize that sensory perception relates to things and events, to the concrete WHAT of experience, but that it does not give access to the terms employed in describing the HOW of what has been observed, for these terms actually stem from the various modal aspects of reality. And these modal functions as such are never open to sensory perception. Yet, these aspects provide theoretical thinking with modal terms (aspectual terms) that are indispensable for the formation of scientific concepts and theories. As soon as the inevitability of employing modal terms is acknowledged, the Achilles' heel of positivism is laid bare.

Let us explore this issue in some more detail (involving some of the distinctions discussed in some of the articles which I attached earlier). In order to highlight the limitations of the senses in the acquisition of knowledge, we may consider the concept of MATTER in terms of some of its main conceptual transformations.

The Pythagoreans adhered to one statement above all else: everything is number. After the discovery of irrational numbers – revealing within the seemingly form-giving and delimiting function of number something formless – Greek mathematics as a whole was transformed into a spatial mode (the geometrization after the initial arithmetization). As a consequence, material entities were no longer described purely in arithmetical terms. The aspect of space now provided the necessary terms required to characterize material entities. This spatial angle of approach remained in force until the rise of modern philosophy, since philosophers like Descartes and Kant still saw the ‘essence’ of material things in their extension. Particularly through the work of Galileo and Newton, the main tendency of classical physics eventually underwent a shift in perspective by attempting to describe all physical phenomena exclusively in terms of (kinematic) motion.

Since the introduction of the atom theory of Niels Bohr in 1913, and actually already from the discovery of radio-activity in 1896 and the discovery of the energy quantum h, modern physics realized that matter is indeed characterized by physical energy-operation.

From this brief explanation it is clear that different aspects served to characterize matter – starting with the perspective of number and then proceeding to the aspect of space, the kinematic aspect and eventually the physical aspect of reality.

The key question is whether these modal aspects could be observed in a sensory way. Can they be weighed, touched, heard, seen or smelled? The answer must be negative, for they are not things but aspects of things (or rather ontic aspects within which concretely existing things function). The first step positivism had to take in order to digest “sense data” theoretically has already eliminated the restriction of reliable knowledge to sense data!

The only way out of the impasse of positivism is to acknowledge the ontic status of the various aspects of reality.

Danie Strauss

I Attach an Article: Popper and the Achilles’ heel of positivism

I believe the flaw in positivism was that ONLY sense experience was part of science--(this is generally known as empiricism)-one example was the fight against the atomic theory of matter--until fairly recently you could not see atoms--but based on many types of evidence they could be inferred by building conceptually on what was known from the senses and experiments--neither the senses alone nor reason alone but reason applied to sensory material--

Ramon, to your post on Page 5:  "If these folks are correct, then mathematics should possess within its framework, the logical powers to express the fundamental nature of reality? So why do formal languages appear ontological agnostic? What will it take to coax ontological truths out of mathematics, or the proposed ontologese?"

The logical powers mathematics possesses within its framework (thus right away violating Gödel's incompleteness theorem) to coax out any ontological truths are limited inductively, that is, derived from infinite specifics, e.g., repeated experiments (probabilities), to arrive at still not an entirely 100% positive conclusion or generalization or ontological truth.  So coaxing out any ontological truths from mathematics will never definitively happen.  Whereas, inversely or deductively extracting mathematics (specifics) from a (general) presumed ontological truth might prove considerably more productive.  Indeed, that presumed ontological truth would have to have a property in common with mathematics (which no languages have yet articulated, pending forthcoming paper), while remaining unique to itself (self-referential).  In other words, not all mathematics implies ontological truth, but  ontological truth always implies mathematics.

I disagree. Math is a conceptual achievement based on perception--how you you form the concept of say, 2? You abstract out number from the objects in question--2 can be 2 of anything,2 is the label you put on that many-- That is the basic tie between math and reality--that is why (rational) math works in reality: to enable civilization to progress and flourish--

Dear Edwin

You wrote:

I disagree. Math is a conceptual achievement based on perception--how do you form the concept of say, 2? You abstract out number from the objects in question--2 can be 2 of anything, 2 is the label you put on that many-- That is the basic tie between math and reality--that is why (rational) math works in reality: to enable civilization to progress and flourish--

Are mathematical statements ontological in a figurative/fictional sense, or in the literal sense of “what there is in the world”?.

Let us focus on: “how do you form the concept of say, 2? You abstract out number from the objects in question--2 can be 2 of anything, 2 is the label you put on that many”.

If “you abstract out number” then it is “already out there” but if “2 is the label you put on that many” then the “2” is not out there (to be abstracted) but derived from the counting (and labelling) subject!

There is a difference between “quantity” and “counted quantity” – “multiplicity” and “numerals”.

This reminds us of Frege who argues against the appropriateness of (entitary-directed) abstraction, given in his example that higher levels of abstraction starting with the ‘moon’ does not yield the number one, but only more general categories – such as attendant of the earth, attendant of a planet, celestial body without its own light, celestial body, body, object (entity). This capacity is inherent in all the everyday (non-scientific) categorizations and classifications we are used to.

Therefore it must be clear that abstraction in the context of entities cannot produce the kind of functional concepts within a scientific universe of discourse. Those of us who had to use an abacus in order to be taught how to go about with the most basic arithmetical operations, such as addition and subtraction, will understand this straightaway.

When we learn to calculate with the help of the abacus, we begin by involving different aspects of reality at once: we take into account the colour, the movement, the shape and the quantity of blocks on the abacus, that is, we initially include the many-sidedness of these blocks by leaving their physical, kinematical, spatial and numerical properties intact. However, gradually we had to disregard the colour, movement and shape, and concentrate on the quantitative side only, i.e. we had to elevate the numerical aspect in order to simultaneously ignore the non-numerical aspects (namely the spatial, the kinematic, the physical and the other aspects not yet mentioned). Keep in mind that the abacus is a cultural artefact (its formative aspect), that it has a name (its function within the sign mode), that it belongs to someone (the jural aspect evinced in the accompanying property right), and so on.

Edwin, your mode of assessing the issue appears to suffer from the same shortcoming which Tait claims that Frege tends to confuse, namely the following two questions: “What are the things to which number applies? And, what are numbers?” (Tait, 2005:241). [Tait, W. 2005. The Provenance of Pure Reason, Essays in the Philosophy of Mathematics and Its History. Oxford: University Press.]

Augustine already realized that every number is unique and that the meaning of number involves succession. He states: “Moreover, each number is so defined by its own properties, that no two numbers are equal. They are therefore both unequal and different from one another; and while they are simply finite, collectively they are infinite” (Augustine 1890:383). [Augustine, A. 1890. (Ed. by Phili Schaff). St. Augustin's City of God and Christian Doctrine. New York: The Christian Literature Publishing Co.]

Danie

What are the things to which numbers apply? Anything that exists if you mean counting 2 can be 2 of anything--)-

what are numbers? like all concepts they are cognitive abstractions (concepts)-- concepts are mental existents- they are man's way of grasping reality above the level of the lower animals--to prevent rationalism (Platonism) valid concepts have to be tied to reality starting with the senses- and you build from there: you can form abstractions from abstractions (thus getting for example to higher level math-

I must state that I reject Augustine, Kant, Plato and Wittgenstein--historically I would be closest to Aristotle, though he did make an error in his theory of concepts but he did view sensory knowledge as the starting point which i think was right-

I do not see any big problem about infinity. I think Aristotle said infinity refers to a potential: what ever number you have, there is the potential to add one more--(the same would go for subdividing)--what's the big deal? let's take an inch and admit that if you have a big enough microscope, you can see that you can always find a deviation of a zillionth of a millimeter--this does not invalidate measurement-- you use the degree of exactitude that you need for your particular purpose--for a machine tool you might need a more exact measurement than for a house--none of this invalidates measurement or numbers--

Dear Bernard

You wrote: “One problem is that there could be as many sorts and kinds of numbers as there are numbers... It is very difficult to reason on numbers in general. The properties of 'numbers in general', taken as one category, are very unclear to me, besides the fact they are numerable.”

I think the idea of “numbers in general” derives from Mill:

“All numbers, must be numbers of something: there are no such things as numbers in the abstract. But though numbers must be numbers of something, they may be numbers of anything. Propositions, therefore, concerning numbers, have the remarkable peculiarity that they are propositions concerning all things whatever; all objects, all existences of every kind, known to our experience” (Cassirrer, 1953;33-34 – reference to Mill, A System of Logic, Book II, Chapter 6, 2). [Cassirer, E. (1953): Substance and Function, New York 1953 (first edition of the English translation of Substanzbegriff und Funktionsbegriff: 1923. First German edition 1910).

I agree with you that more precision is needed concerning “numbers in general”. Normally the sequence commences with natural numbers, integers, fractions, real numbers, transfinite numbers, complex numbers and ends with imaginary numbers.

Danie

Danie Strauss responds Roger John Sapsford who wrote: “Stating that the two progressions do not reach the end of the race.” It means that two converging sequences do not “reach” the limit-value they increasingly approximate, thus merely showing that one operates with the successive infinite. When this numerical meaning of the successive infinite is deepened by pointing towards (anticipating) the spatial time-order of simultaneity (at once) and the spatial whole-parts relation, we meet the idea of the at once infinite enabling the regulative hypothesis of infinite totalities (where all the elements are present at once). And the latter cannot be seen as the completion of an endless sequence of numbers, showing that the at once infinite is irreducible to the successive infinite. Recall the example whether or not 0.999... is equal to 1.000... When the successive infinite is employed it is not equal to 1 (similar to “the two progressions [that] do not reach the end of the race”) and when the at once infinite is employed the infinite totality of .999s is equal to 1. Weierstrass and Cantor defined real numbers by using the idea of infinite totalities.

To Danie, et al,

This question has become lost in formalist conundrums because it became a conversation among mathematicians trying to do some philosophy, instead of a philosophical discussion about the nature of mathematics. Partly it is because the question is framed using philosophically imprecise terms. "Fictional" contrasts with "factual" , while "figurative" contrasts with "literal". Figurative language like, "It's raining cats and dogs", conveys the factual information that a hard rain is falling but the language it uses to do so is not literally a reference to real or fictional dogs and cats falling from the sky. Math could calculate quite literally how many pets would be falling if we knew it was raining 25 cats plus 75 dogs; or the product, difference and division of these ... or any number of dogs and cats.  If one unicorn leaves New York City travelling at 10 miles and hour and another one leaves Chicago .... I don't need to finish the word problem to illustrate that it's not a matter of what natural language statements refer to, versus mathematical statements. They both refer to their respective objects literally, not figuratively, whether the references are factual or fictional.

To determine the ontological status of anything requires doing ontology, not just mathematics or epistemology. Decisions about what to include in one's ontology are not decisions about which metaphor to use or whether the ontology is factual or just a sufficiently convenient fiction. Ontology is literally descriptive of what constitutes the same world we all find ourselves in. Its questions, if they mean anything at all, are more basic than other philosophical questions, scientific  or mathematical questions.

The conversation here is not ontological at all, anymore. These are debates within mathematical theory and not philosophical debates about mathematics.  Math is not a theory of being. Plato's idealism is a theory that the apparent certainty with which we know mathematical statements to be true implies that these true statement must correspond to and be caused by "formal" entities ... not properties ... that have the property of existence; they are the real constituents and causes of the world we find ourselves in. This is not figurative language, even though the Allegory of the Cave is used to illustrate the literal claim about what really exists beyond what appears to us through our imperfect senses. Descartes' arguments are not that different; just more narcissistic.

It is the apparent certainty and usefulness of math and geometry that makes them such compelling candidates for ontological models. But we still need to ask the questions about why they seem so certain in the context of  our theory about the nature of being, without assuming our sense of certainty implies the existence of entities unlike the tables and chairs described by natural language.

A wide variety of diverse but related natural languages have occurred here on Earth. Many are now extinct; some are still extant. They all included systems of counting and numbering whatever distinct objects and geometrical structures of objects that could be found in those cultures. More than anything else, these natural languages constitute the cultures we identify them with. Languages are patterns of behavior that organize the individual people into functional units that enable their ability to keep reproducing their kind over many generations.

All people perceive objects in 3 spatial dimensions that are always spatially related according to the laws of geometry, described first by the ancient Greeks, Pythagoras and Euclid. The science of geometry came long after natural languages used geometry and mathematics to organize and quantify things and people. So math is more like a part of natural language than something opposed to it, as Ramon's original question suggests.

The assumptions of the question assume a kind of naive realism about mathematical statements. I am suggesting that we need the kind of critical thinking that philosophical reflection alone can provide to get us beyond the naive realism and rationalistic ontology mentioned here. Unfortunately, this discussion has veered off in directions that miss and dismiss my recommended line of inquiry.

Thanks,

DCD

Descartes before the horse -- very narcissistic!

Hahaha, CJ.

I'll be ordering a la Descartes. Or maybe the Blue Plato Special.

Ramon,

Straying is part  of discovery, I think. It can lead to unexpected insights sometimes.

I think you tried to ask the question in an ontological way but it strayed back into epistemology and purely formal considerations. I liked the question as an ontological one and was just trying to re-state it in a way that clarified it as such. I don't know if that is your intention but I'd be interested to pursue the ontological basis of  mathematical reasoning further.

Thanks,

DCD

Dear Daniel Davis

Gödel's proof has shown that any axiomatic system is based upon intuitive evidence exceeding the formalism of that system – therefore your remark that “math is more like a part of natural language than something opposed to it” is quite right. [Weyl aptly remarked: “It must have been hard on Hilbert, the axiomatist, to acknowledge that the insight of consistency is rather to be attained by intuitive reasoning which is based on evidence and not on axioms” (Weyl, 1970:269); see Weyl, H. 1970. David Hilbert and His Mathematical Work. In: Reid, 1970 (pp. 243-285).]

You then say that this “question has become lost in formalist conundrums because it became a conversation among mathematicians trying to do some philosophy, instead of a philosophical discussion about the nature of mathematics.”

From this I deduce that you did not have a chance to read my article on how to define mathematics, showing that it is not a mathematical question but one which is philosophical in nature. I attach it to this communication, merely here lifting some of the issues dealt with in this article. [I shall attach it.]

Any definition of mathematics falls outside its field of investigation. When mathematics is set theory, the history of mathematics prior to the investing of set theory is eliminated. Arguing that the aspects of number and space delimit mathematics makes it possible to avoid both Platonism and constructivism in mathematics. Every philosophy of mathematics should be able to account for the nature and status of the infinite. That set theory is a spatially deepened theory of numbers cannot be accounted for by what Lakoff and Núñez call the Basic Metaphor of Infinity. Gödel’s 1931 results point to an immediate, evident, intuitive insight.

Moreover, you say: “natural languages used geometry and mathematics to organize and quantify things and people.” Perhaps you meant: mathematicians proceeded from a natural language when they developed arithmetic and geometry to deepen our understanding of quantitative and spatial relationships.

You state categorically: “Math is not a theory of being.” Implicit in your statement is the identification of “being” with concrete being (“entitary being”), i.e., that the term “being” solely refers to one kind of factuality, namely “concretely existing things.” However, the co-worker of David Hilbert, Paul Bernays, is more articulate in his discussion of Wittgenstein where he rejects the view of those who merely acknowledge one kind of factuality, that which is concrete: “It appears that only a pre-conceived philosophical view determines this requirement, that view namely, according to which there can solely exist one kind of factuality, that of concrete reality.” [Bernays, P. 1976. Abhandlungen zur Philosophie der Mathematik. Darmstadt: Wissenschaftliche Buchgesellschaft (page 122).] The other kind alludes to the various aspects of reality (according to Bernays accessible through “idealized abstraction”).

What, according to Bernays, is characteristic of our distinction between the “arithmetic” and the “geometric” intuitions? He rejects the widespread view that this distinction concerns time and space, for according to him, the proper distinction needed is that between “the discrete and the continuous” [“Es empfiehlt sich, die Unterscheidung von ‘arithmetischer’ und ‘geometrischer’ Anschauung nicht nach den Momenten des Räumlichen und Zeitlichen, sondern im Hinblick auf den Unterschied des Diskreten und Kontinuierlichen vorzunehmen” (Bernays, 1976:81).]

Paul Bernays is also convinced that mathematical axiom systems are not created out of thin air because they are not in their entirety an arbitrary construction. “One cannot justifiably object to this axiomatic procedure with the accusation that it is arbitrary since in the case of the foundations of systematic arithmetic we are not concerned with an axiom system configured at will for the need of it, but with a systematic extrapolation of elementary number theory conforming to the nature of the matter (naturgemäß)” (Bernays 1976: The “nature of the matter” contains an implicit reference to the ontic status of the “multiplicity-aspect” of reality. [“Gegen diese axiomatische Vorgehen besteht auch nicht etwa der Vorwurf der Willkürlichkeit zu Recht, denn wir haben es bei den Grundlagen der systematische Arithmetik nicht mit einem beliebigen, nach Bedarf zusammengestellten Axiomensystem zu tun, sondern mit einer naturgemäßen systematischen Extrapolation der Elementare Zahlenlehre” (Bernays,1976:45).]

The statement “Math is not a theory of being” should be reformulated: “Math is not a theory of entitary being, since its theoretical attention is directed to the aspectual being of the numerical and spatial modes of being of reality.”

Finally, allow me to say something about your observation: “The science of geometry came long after natural languages used geometry and mathematics to organize and quantify things and people.” This statement distinguishes between “geometry” and “mathematics” – perhaps you had in mind to distinguish between geometry and arithmetic. Furthermore, multiple things do not owe their function in the quantitative aspect of reality through being “quantified” by mathematics, because they are multiple prior to being counted. Numerals (number-words used in counting) presuppose the existence (“being”) of the numerical aspect and the quantitative properties entities may acquire by functioning within this aspect.

I shall attach another article, originally posted in a different context, on Frege – in which I discuss the question: What is ontic about number.

Best wishes

Danie Strauss

[23-05-2015]