I like Nelson's explanation on symbolic logic as a basic of philosophy. Furthermore,"... the philosophy of mathematics can be regarded as a branch of the philosophy of science, next to disciplines such as the philosophy of physics and the philosophy of biology. However, because of its subject matter, the philosophy of mathematics occupies a special place in the philosophy of science. Whereas the natural sciences investigate entities that are located in space in time, it is not at all obvious that this also the case of the objects that are studied in mathematics..."

http://plato.stanford.edu/entries/philosophy-mathematics/

Mathematical or symbolic logic can serve as a basis of philosophy. It embraces mathematics, linguistics, cybernetics, natural and social sciences, jurisprudence, and philosophy itself. See Manuel Garrido Jiménez, Lógica simbólica, 3rd. ed. 1978, p. 7.

I like Nelson's explanation on symbolic logic as a basic of philosophy. Furthermore,"... the philosophy of mathematics can be regarded as a branch of the philosophy of science, next to disciplines such as the philosophy of physics and the philosophy of biology. However, because of its subject matter, the philosophy of mathematics occupies a special place in the philosophy of science. Whereas the natural sciences investigate entities that are located in space in time, it is not at all obvious that this also the case of the objects that are studied in mathematics..."

http://plato.stanford.edu/entries/philosophy-mathematics/

This is a good question with many possible answers.

In addition to what has already been incisively observed, it can be also observed that mathematics itself focuses of different forms of structures (mainly geometric structures such as points and lines either on plane or curved surfaces and algebraic structures such as  groups, rings and fields).   This suggests that a philosophy of mathematics would concern itself with questions and problems about geometric and algebraic structures, both their nature and paths to discovery.

Many of these questions about mathematical structures have their origins in antiquity, starting with Zeno's paradoxes about infinity and in more recent times such as Hadamard's view of seeing a structure all at once (flashes of genius) in arriving at mathematical discoveries.     See, for example,

P. Liljedahl, Mathematical discovery: Hadamard resurrected, 2004:

http://www.emis.de/proceedings/PME28/RR/RR116_Liljedahl.pdf

A good example of mathematical invention is Henri Poincare's story about his realization that the transformations he used to define Fuschian functions were identical to those in non-Euclidean geometry (p. 249).

I agree with all answers. I just want to add some small point of views.

What is currently called "philosophy of mathematics" is more or less philosophy inspired by mathematics, and its target is not really inside mathematics but in philosophy and in the rest of the world. It makes a good service for mathematics, because mathematicians do not do enough effort to explain themselves for the rest of the world. For them this philosophy is no more then their own method of thought and work.

Another good point is that younger mathematicians can learn a lot by reading some philosophy of mathematics, where big conclusions of the work of past mathematicians are well presented - because as a phd student you don't have the time to read all that has been done before. 

I remember some short stories showing how difficult is sometimes to separate mathematics from their own philosophy. We had a small group of students in Freiburg. We used to read some article on mathematical logic and then to meat and discuss about its philosophical relevancy. [It has been not a priori defined what it should be, we wanted to find out this also in discussions...] Every time I was discussing some aspect - and one or two colleagues answered "That is not of philosophical value, this is just [mathematical] technique." And sometimes there were also people finding the same aspect of philosophical value. Finally we thought that philosophy of mathematics is to analyze mathematics using philosophical methods of thought, in a larger scale containing also the real world or the rest of philosophy. However, this activity turned often to be pure mathematics, because most of philosophical methods are in fact mathematical. 

Or maybe, being ourselves mathematicians, could't do it better.

My article in Bulletin of Symbolic Logic

http://homepages.math.uic.edu/~jbaldwin/pub/catcomnovbib2013.pdf

addresses the issue of how formal methods (i.e symbolic logic- specifically modern  model theory) contributes to the philosophy of mathematics, the organiztion of traditional mathematics, and the solution of traditional progams.

http://homepages.math.uic.edu/~jbaldwin/pub/catcomnovbib2013.pdf

The philosophy of mathematics can only be created by mathematicians ("La philosophie des mathématiques ne peut être créée que par les mathématiciens" - Henri-Léon Lebesgue).

My respectful and dear friends and colleagues, I can not agree more with your ideas and commends on different aspects of philosophy of mathematics. Thank you very much.

According to my studies, if there are the fundamental defects in certain science branch disclosed by paradoxes; it would be the right time for the “philosophy of that science branch” to play a role. Not matter how beautiful and how complicate a “philosophy of mathematics” is, it should always remember its own form and one of the important missions: not only mathematics and not only philosophy, but the very integrating basic stone of mathematics and philosophy guaranteeing the clear nature and peaceful existence of mathematics without being disturbed by mathematical paradoxes.

The infinite related paradoxes have been troubling us for such a long time------the fundamental defects are not only mathematics and not only philosophy. Is it the faults of our not having been done enough on “philosophy of mathematics”?

What is “philosophy of mathematics” for? Do we need both theoretical and applied “philosophy of mathematics”?

I am sorry, is it too much requirements for “philosophy of mathematics”?

Philosophy of Mathematics should do what the philosophy of anything else should do: make clear the concepts we use and the statements we make. Let me mention two examples. I have seen debates about whether numbers "are real" or not, which I did not appreciate. In my view, all that exist should be divided (classified) in three different spaces: (1) physical entities, (2) mental states, (3) abstract entities. A stone (physical), a love (mental), and a number (abstract) a very different entities, which belong to different realms of existence. We can call all them "real" or not: this is a matter of choice.

Second example: Bertrand Russell's notorious claim that "pure mathematics is the subject in which we do not know what we are talking about, or whether what we are saying is true", sounds nice, but it is actually misleading because it is not precise enough. Pure mathematics does not speak about anything, nor does it tell anything true or false. Pure mathematics is *a game with symbols and with symbolic expressions*; this game is played in accordance with certain adopted rules of shaping the expressions (formulas) and of producing (inference of) new expressions (formulas) from the existing ones. This game by itself contains no meanings and it does not claim anything, so that it cannot express either truth or falsity.

*Meaning and truth* enter into this story with the *interpretation* of a mathematical game (as a purely formal system or "pure mathematics"); the interpretation includes a projection of the symbols (variables and operators) to the phenomena (entities and relations) in a "world" (set of entities). With this, the meaning (of symbols and formulas) and truth (of formulas) enter the story; however, it is not pure mathematics that tells the truth or falsity, but its interpretation.

In my view, philosophy of mathematics should deal with such issues. The discovery (proof) of logical inconsistency in a mathematical theory is a technical task: this is a formal (which means, mathematical) issue, rather than philosophical one.

Dear Mario and Ramon，

The philosophy of mathematics deals with many fundamental things in our science (mathematics) system: makes clear the concepts we use and the statements we make. But those fundamental defects disclosed by paradoxes are most challenging: not only easy or leisure dialogues but for the impending health of science foundation. Just see following divergent proof of Harmonic Series Paradox example:

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）

Paradox is there whether or not we can produce infinite numbers each bigger than 1/2 or 1 or 100 or 100000 or 10000000000 or… from infinite infinitesimal 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.

What is infinite?

What is number?

What is limit theory?

What is mathematics?

Sincerely yours, Geng

I also believe that "most mathematicians are Platonist at heart".  Herewith is the presentation of some ideas of Albert Lautman - a brilliant French mathematician dead too young.

Dear Ms. Ruxandra Vintila, I really agree with you that "most mathematicians are Platonist at heart". But one thing is also agreeable that to remove those defects disclosed by paradoxes and have a health foundation for mathematics are the wish of most mathematicians not matter what school they belong to.

Sincerely yours, Geng

Maybe this article is not giving an answer to the question asked, but it is definitely related to this thread. "...The laws of nature have the final say in the functioning of his brain-computer model. The energy for this model comes from mathematics and Greek philosophy. And the intent is an educational system of virtue..." 

http://www.huffingtonpost.com/evaggelos-vallianatos/programming-our-computer-like-brains-with-greek-philosophy-and-mathematics-leads-to-virtue_b_6925434.html

Philosophy of mathematics deals with the cognitive logic i.e logical statements of building a reasoning and an analysis. Symbols are created to set the reasoning, the proofs, the theory etc..

I think in the natural world all that can be construed for contemplation are numbers and their combinations to form descriptions.  The rational mind knows only that 1+1=2; nature on the other hand has no name for the number =1 or 2 and a vast amount of diversity that can be  inferred rationally to be the consequent of amount and sequence.  As one might envision any observed pattern to be a numerical assembly of any complexity and elaborated via numerical relations,  the scientist then tries to establish paint-by-number formulae to account for observed structure. In the more shady realm is the use of like method for a mechanics of change; i.e. rate, in which logical reasoning, depending on the framing components, can ultimately refer theologically. Scientific tendency is to attempt to rarify motion, to extract description from a vast diversity of natural occurences to a logically isolatable entiity and to apply induced principles to all examples, i.e. to describe rationally how paint-by-number painting A becomes paint-by-numbers painting B.

  It is possible to evidence mathematics to exist naturally; though the mind can invent to capture change in terms of lines that do not exist on their own in nature, rather than volumes. Though all mathematical application entails form, often only a single view or face is described while  associated whole form/meaning may either not be known, accessible or referred to(i.e. conceived by an author but ommited from his/her description).  Einstein had put to question whether the rod and the clock are the elemental building blocks, I think nature has only design constructed of rules that ubiquitously have mathematical capability, though the clock is a matter of uniquenesses and their dispositions that only reflect in their property, i.e. category=change, those of  whole nature, cannot be rarified to mathematical terms beyond what Newton, or Einstein in the special theory, extracted in  terms of number capable shape, the triangle the simplest area/volume/shape posessing denomination...the constant speed of light, the general theory, matters of world view/philosophy  rather than mathematics alone-this view is evident in early Greek Math (e.g. Euclid) that reserves the triangle, human engineering, and nature to separate categories.

Our world views can effect how we apply mathematics, can be less than rigorously  rational.  

If interested enclosed "Evolution at the Surface of Euclid:Elements of a Long Infinity in Motion Along Space

Article Evolution at the Surface of Euclid : Elements of a Long Infi...

Geng Ouyang puts forward difficult questions. I am not exactly a philosopher of mathematics: I spent a lot of time studying logic and logic programming, but this was almost thirty years ago. Regarding the questions about *paradoxes* and *infinity*, let me mention what I wrote in some my earlier texts.

Parmenides and Zeno were masters of conceptual games, but such games show the *limitations of human language*, rather than the "paradoxical nature" of reality. Zeno's paradoxes are "conceptual traps"; mathematics can "bypass" such traps, but it does not explain them at the *linguistic* (common sense) level. Language is a powerful means by which human mind created all sorts of gods and devils, incredibly complex theories, and even *infinitely many infinitely different infinities*. But language is a human creation, and it is imperfect. By skillful use of language, traps can be created from which language is *not capable of pooling itself out*.

Regarding infinity, we speak about something that we are not able to conceive of or to imagine, let alone to experience. It is not possible to conceive of or to imagine anything infinite. The attribute "infinite" expresses *the absence of a conceivable limit*, or simply the "inconceivable". We can say that God is infinite or that the series of natural numbers is infinite, but this does not mean that we know "how much" this is. "Infinite" is a "negative quality" which expresses the *absence* of limit, rather than the *presence* of a conceivable feature of an entity.

Philosophy of mathematics, the branch of philosophy that deals with the foundations of mathematics and mathematical discussions of epistemology.

At the beginning of the twentieth century, three schools of philosophy of mathematics to respond to some questions arose. The three schools intuitionism school and Mntqgrayy and formalism known names.

Mathematics is a human-invented language. Perhaps an even broader question therefore is: 'What is the philosophy of language' or perhaps 'What is the philosophy of communication accepting that language is one aspect of communication'?

the common point of phylosophy and mathematics is the reasoning.

Logic and reasoning --- common to philosophy or mathematics!

If mathematics is regarded as a science, then the philosophy of mathematics can be regarded as a branch of the philosophy of science, next to disciplines such as the philosophy of physics and the philosophy of biology. However, because of its subject matter, the philosophy of mathematics occupies a special place in the philosophy of science. Whereas the natural sciences investigate entities that are located in space in time, it is not at all obvious that this also the case of the objects that are studied in mathematics. In addition to that, the methods of investigation of mathematics differ markedly from the methods of investigation in the natural sciences. Whereas the latter acquire general knowledge using inductive methods, mathematical knowledge appears to be acquired in a different way: by deduction from basic principles. The status of mathematical knowledge also appears to differ from the status of knowledge in the natural sciences. The theories of the natural sciences appear to be less certain and more open to revision than mathematical theories. For these reasons mathematics poses problems of a quite distinctive kind for philosophy. Therefore philosophers have accorded special attention to ontological and epistemological questions concerning mathematics.

Philosophy of mathematics is concerned with problems that are closely related to central problems of metaphysics and epistemology. At first blush, mathematics appears to study abstract entities. This makes one wonder what the nature of mathematical entities consists in and how we can have knowledge of mathematical entities. If these problems are regarded as intractable, then one might try to see if mathematical objects can somehow belong to the concrete world after all.

On the other hand, it has turned out that to some extent it is possible to bring mathematical methods to bear on philosophical questions concerning mathematics. The setting in which this has been done is that of mathematical logic when it is broadly conceived as comprising proof theory, model theory, set theory, and computability theory as subfields. Thus the twentieth century has witnessed the mathematical investigation of the consequences of what are at bottom philosophical theories concerning the nature of mathematics.

When professional mathematicians are concerned with the foundations of their subject, they are said to be engaged in foundational research. When professional philosophers investigate philosophical questions concerning mathematics, they are said to contribute to the philosophy of mathematics. Of course the distinction between the philosophy of mathematics and the foundations of mathematics is vague, and the more interaction there is between philosophers and mathematicians working on questions pertaining to the nature of mathematics, the better.

There are four schools of philosophy of mathematics. The general philosophical and scientific outlook in the nineteenth century tended toward the empirical. Platonistic aspects of rationalistic theories of mathematics were rapidly losing support. Especially the once highly praised faculty of rational intuition of ideas was regarded with suspicion. Thus it became a challenge to formulate a philosophical theory of mathematics that was free of platonistic elements. In the first decades of the twentieth century, three non-platonistic accounts of mathematics were developed: logicism, formalism, and intuitionism. There emerged in the beginning of the twentieth century also a fourth program: predicativism. Due to contingent historical circumstances, its true potential was not brought out until the 1960s. However, it amply deserves a place beside the three traditional schools.

Dear friends and colleagues, we may not care for ancient Zeno’s Paradox, may not care for the philosophy of mathematics. But as a member working in mathematics, we have to answer following mathematical questions.

1, How many numbers each bigger than 1/2 or 1 or 100 or 100000 or 10000000000 or… can we produce from the infinite items in Harmonic Series by “brackets-placing rule" in following mathematical proof? Where are the logic and reasoning in such mathematical proof?

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）

2, Can we really 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? Where are the logic and reasoning in above mathematical proof?

3, Can we really run away from the concepts of “infinite” or “limit” in our mathematical work?

Sincerely yours, Geng

Marcel, you cannot make a statement like "Mathematics is a human-invented language" without justifying it .....

Dear H. Chris,

If mathematics is not a language (e.g. a series of symbols which can be letters or numbers that are placed in a non-random order), please provide arguments.

@M. Lambrects, H. Chris

 ".... If mathematics is not a language (e.g. a series of symbols which can be letters or numbers that are placed in a non-random order), please provide arguments."

 If language is defined as the establishment of a series of connections that communicate meaning it is questionable to myself whether an arrangement of mathematical symbols constitutes language. If one thinks of syntax as the primary property of language, it is also inherent in language communications that meaning has also to do with who, where, and when it is spoken. Mathematics is similar in this respect, diferent solutions can allude to different things in the different minds and setting of mathematicians, but an arrangement of symbols in mathematic expressions does not entail qualifications to meaning as language does.   

    Emotion based reasoning can be full of redundancies, open loops that can be closed and understood with just exposure to mathematical method in laying out problems in a way that solutions are findable-in my view mathematics has to do with both language and life, but always remaining is potential contemplation about the nature of thought itself as well as that of the world-unexpressible with a connection of symbols alone but with syntax bearing language. 

    There is always something communicated when mathematicians agree on a solution to a problem, but it is not inherent to communications in the sense that one might ask of a persons words "what does that mean?" Per Socrates' saying "the unexamined life is not worth living", it is the examined life that brings about mathematical analysis, language emerging from thought, lingual communication that is individual preceeds all, though.

    Assuming this described nature of the language of mathematics, open to discussion is the existence of practical benefit of the philosophy of mathematics to life experience (see attachment "What is the philosophy of religion?") https://www.academia.edu/attachments/37105524/download_file?st=MTQyNzU4NDcxMyw3MS4xMzAuMjEwLjIwMSw1NzM0Nzgw&s=work_strip

Dear Marcel, it's  a subject and debate that has been visited very numerous times here on RG, in many different threads. No use revisiting what has been argued many times before.

I remember one of these RG debates on whether mathematics sprung up naturally from reality or whether it was a language even being quoted in an in-flight magazine in the US.

Dear Mr. H Chris Ransford, I agree with you that people have been talking about “philosophy of mathematics” in different situations and different times.

But I think what we are doing in this thread is different from previous, we are driven and powered by the newly discovered infinite related Harmonic Series Paradox and we have very clear target------we hope to solve some of the infinite related fundamental defects in our mathematics through the discussions here.

Sincerely yours, Geng   

Philosophy of mathematics has its root right from the birth time of arrival of child..All we know that the planetary chart also runs in 9 planets & we all are aware the magic root & working number of 9 .

In above line right from the counting of population covering importance of finance are the resulting fruits of number .I do not know it is the philosophy of number it is my personal opinion .

Dear Mr. Ramon Quintana and Mr. Rohit M Parikh,

1, Some people say both mathematics and philosophy are not sciences, because they deal with purely abstract entities (such as “limit”, “time”, …); others say both mathematics and philosophy are sciences, because the purely abstract entities they deal with are from objective world; still some others are worrying about the “perfectness” of some concepts (kinds of assumptions?) in our science.

2, Mathematics, philosophy and some concepts are right there in our science work, it doesn’t matter how we call them or how we like them in our present science mansion, we can not either neglect them or get rid of them, but have to improve them, tailor them as long as they are still alive in our present science created by us human and make them serve us human better.

3, Mathematicians have to tell “what there is, how mathematical things come and go” for some mathematical concepts because we must know these and we want to use them in our mathematical operations. So, philosophy plays a vital role in the foundation and development of our mathematics--------“philosophy of mathematics has its root right from its birth time”

Sincerely yours, Geng

Thank you very much for your ideas dear Ms. Brenda Jacono.

We are here in this thread seeking for a “good” philosophy of mathematics to solve the newly discovered infinite related Harmonic Series Paradox.

One of the most important natures for philosophy of mathematics is to get rid of mathematical fundamental defects found and insures the “healthy” of its foundation. It would be great if Logicism or Formalism or Intuitionism, or the integration of the three “schools” can solve the newly discovered infinite related Harmonic Series Paradox.

As our topic title says: what is philosophy of mathematics for?

Sincerely yours, Geng

Driven and powered by the newly discovered infinite related Harmonic Series Paradox, we have very clear target which is different from those of previous “just talks and writings”------hoping to solve some of the newly found infinite related fundamental defects in our mathematics (science, philosophy) through the discussions here.

Facing the newly discovered family member of 2500-year old Zeon’s Paradox------Harmonic Series Paradox, something should be done.

We have now more than a dozen of “philosophy of mathematics”; they are not being for “after dinner or tea time talks” but for the healthy foundation of mathematics.

Because of its subject matter, the philosophy of mathematics has a special place among the philosophy of different sciences. Mathematical knowledge has long been regarded as a model of human knowledge with truths that are both necessarily important and certain. Unlike the studies in Natural Sciences or Physical Sciences, the objects that are studied in mathematics need not be located in space in time. The methods of investigation of mathematics differ significantly from the methods of investigation in other sciences. Mathematical knowledge is acquired generally by deduction from basic principles, whereas in other science subjects are obtained using inductive methods.

How about abduction? 

http://plato.stanford.edu/entries/abduction/

http://mathforum.org/library/drmath/view/55620.html

Thank you dear Ljubomir for the documents, indeed inference is one basis and tool for the artificial intelligence science

Deduction, induction, abduction, or whatever ways in our scientific cognition process, the philosophy of mathematics we human build should help does some practical things to solve the defects being disclosed in the foundation of mathematics.

Our modest colleague @Marković G. Đoko has not cited his reference, but I am free to do it. THE NATURE OF TRUTH IN MATHEMATICS follows. My regards dear Đoko.

http://www.unite.edu.rs/vol/vol.1-no.1-Articles/Vol.1-No.1-Article10.pdf

Philosophy of mathematics -- gives logic and reasoning to way of life!

Dear Mr. Marković G. Đoko, Mr. Ljubomir Jacić and Mr. Subhash C. Kundu,

Thank you very much for your insightful ideas on the nature of mathematics and philosophy of mathematics.

My best regards, Geng

Science is our human’s, it is undoubtedly that we really can own (create) our common “scientific base” for our science through studies, discussions, …. This is one of the reasons that we are here and we cherish this destiny that ties many of us together.

I believe how to unify “objective world” and “subjective world”, how to cognize and translate “things in human science and things in natural world”, “sets in mathematics” and “sets in nature”, “infinitesimals in mathematics” and “infinitesimals in nature”,…, how to understand “approximate” and make it acceptable in our science…, has been the essential, important and toughest works for scientists.

We have two 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 which can be different kinds of real objective things (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,

Our science history tells us that in our present traditional infinite system, the abstract infinite concept is called “potential infinite” while the real objective carriers of the abstract infinite concept is called “actual infinite”. Now, the trouble is “the abstract infinite concept” and “the real objective carriers of the abstract infinite concept” have being confused and mixed up, “the infinite concept carriers’ theories” being neglected even unaware since at least Zeon’s time …This “half absence defect” in our present traditional infinite system failed many of our “actual infinite quantifying work (such as infinite sets comparing work and infinitesimal calculations)”

Our cognition defects to “infinite” and the confused--mixed up of the “potential infinite------ abstract infinite concept, and actual infinite ------real objective carriers of the abstract infinite concept” have been produced ill influences on our studies and cognitions to the foundations of limit theory and infinite related number theory, unable people’s qualitative and quantitative studies and cognitions to the “actual infinite ------real objective carriers of the abstract infinite concept (sets, infinitesimals, infinities, …)”.

Human intelligence is a physical object and a part of the objective reality. All concepts generated by its activities, including mathematics, respectively, too. It blurs the boundaries between abstract mathematics and physical reality. When think about it, everything looks a bit different.

 Vasiliy，I agree with you that Human intelligence is a kind of physical objects and a part of the objective reality.

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

So, cognizing and translating “things in human science and things in natural world”, “sets in mathematics and sets in natural world”, “infinitesimals in mathematics and infinitesimals in natural world”, “abstract mathematics and physical realities”, … are essential and important work for scientists though very difficult.

Do you think so, dear Mr. Vasiliy Fedorovich Komarov ?

My best regards, Geng

Geng Ouyang, I agree that the information field of intellectual activity is a separate layer however which intersects with a low-level (physical) reality. In mathematics (in contrast to many other branches of knowledge) self-organization is very clearly observed. This is an exact knowledge, all diversity of which is based on a simple set of natural rules. The fact of existence of such almost independent knowledge (which governs the behavior of the rest of the world) says a lot. Construction of mathematics followed the same laws of information ordering that govern self-organization at all levels in nature. In other words, mathematics is the product of a study of certain object (which is part of the nature), as well as any other science.

In the problem of countable and uncountable sets (as discussed in your thread) a simple analogy with reality can be an example if we consider quantum states as natural numbers and superposition as real numbers. Furthermore open physical systems provides a vast field for physical "study" of such mathematical objects.

 Vasiliy,

1, I agree with what you wrote that “mathematics is the product of a study of certain object (which is part of the nature), as well as any other science.” I think you mean that any branches of our science are objective based, am I right?

2, I proved that people used some mathematical techniques based on the traditional infinite theory system (present infinite relating number system and limit theory) in the uncountable proof of real number set----a small proof but disclosing some big fundamental defects.

3, I also agree with you that open physical systems are kinds of objective fields for mathematics.

My best regards, Geng

Geng,

1. Yes. I think, hypothetically, any problem which may be found in mathematics has a manifestation somewhere in nature.

Math is fairly isolated special branch of knowledge (remote from the natural sciences) due to our natural desire of formalize of thinking. Therefore system of mathematical knowledge (as a formal system) has now a non-obvious empirical evidence but integrity, which is clearly visible. If we divert it now from the nature and the history of mathematics, leave axiomatic geometries, concepts of algebra etc. - mathematics will remain the way it is. Despite this it remains objective based. Otherwise, mathematics would not have been invented and developed.

For example. For algebraic calculation is not important how the empirical evidence comes: in electrical circuits of CPU or in the human brain. Background formal rules are objective in itself.

2. I unfortunately not a mathematician. So, for the paradox I see no contradiction since as a physicist I can give a silly explanation:

Look for example at http://math.stackexchange.com/questions/229266/harmonic-series-paradox

The paint (which have limitations due to the physical structure of matter at elementary particles level) could create an infinitely thin layer (in the case of Gabriel's horn) or infinitely thin line (in case of the harmonic series paradox) only in case if we adjust the geometrical size of objects, which will be painted. They will match to physical paint (which consist from particles) only in the case of infinite size and volume of horn and infinite size and area for harmonic series paradox respectively.

Unfortunately Observable Universe at the moment has a limited (countable and not infinite) amount of materials for paint, as well as size and entropy.

3. Open physical system is infinite in every sense and free of such paradoxes (contradictions) in the limiting case. It is mover of evolution, scientific knowledge (with its principles) and endless process of cognition.

Here it is necessary remember something like Hilbert's second problem. Everything has a beginning (axioms) and the rest including you and me and what everyone is doing there are fundamental defects :)

I think that all problems of math from the impossibility to finally build (axiomatic free) consistent formal system that can cover whole knowledge base.

With best regards, Vasiliy.

Sorry for my English.

PS: Perhaps my explanation in discrete world does not help for the problem of countable and uncountable sets too. As far discrete world?

Dear Mr. Vasiliy Fedorovich Komarov, thank you.

1, I agree with your understandings on “things in human science and things in natural world” in our scientific work------ We see flour and rice on the storage rack in grocery stores but we understand very well that the grocery stores never grow flour and rice.

2, I agree with your ideas on the relationship between math and physical world, physics seems closer to real world than math. Human science is ours so we are working hard to remove fundamental defects and trying our best to build a “healthier” human science.

English is my second language too, your English is better than mine. I am sure that we can benefic from any discussions in RG on scientific topics, thanks again. 

My best regards, Geng

Geng, a few thoughts in addition...

I've been thinking a lot on this topic last few years too. Mathematics of course can be considered as an area of virtual (ie higher layer) reality which generated inside our level of intelligence (like microcosmos for cells inside a multicellular organism). In such (abstract) reality possible objects/processes that are unavailable abroad. But this reality should be considered only as an extension of the physical world. This is just another layer of information, which would not exist without the physical world (which acts as the carrier for knowledge and this virtual reality). From the point of view of the theory of chaos, mathematics at least must also fit into the system integrity of our Universe. Perhaps it owes its existence to some larger system. Maybe somewhere there is a fundamental defect.

Returning to the problem of countable and uncountable sets. All this is connected with the dichotomy in my humble opinion too. It can be traced throughout the history of mankind. In simple words, to something born in the infinite world it must be separated from the outside world. As it turns 1. Next step of cognition should break up 1 into parts (on the one hand) and identify 1 as some part of something bigger. For example, we moved the center of the Universe as we expand our outlook up to the center of Big Bang for General Relativity step by step. Before now we are constantly engaged to identify The Yet Another External Absolute Reference Frame aligned to something. The internal coordinate system is constructed from the shell/1. Look there is difficult. Usually self-knowledge is mediated through the study of their own kind (sometimes fatal).

The whole infinite world can be transformed into a system of relations inside/outside in a completely relative space with a simple finite set of rules? Such world is irrational! Maybe somewhere there is a fundamental defect. Probably so looks turbulence in the space-time from inside.

Besides those already mentioned paradoxes there are objects that give the ground for thought too:

various golden sections,

Minkowski's question mark function,

limit (1+1/n)^n as n->0 & infinity,

limit (1+n)^1/n as n->0 & infinity,

...

At least there is connection some irrational numbers with rational and thus with the natural. Consider that 1 with golden section Pi in the Gabriel's horn (in addition to infinity) also plays an important role.

The truth is somewhere nearby.

With best regards, Vasiliy.

Dear Mr. Vasiliy Fedorovich Komarov,

It is convinced that people can solve all the problems you arise with the ideas of “potential infinite” and “actual infinite”, but the pity is we can not because no one really knows what “potential infinite” and “actual infinite” are and these two concepts are confused and used arbitrarily------endless debates and unsolvable paradoxes in present traditional infinite relating science system by “potential infinite” and “actual infinite”.  

Yours, Geng

Geng, from the position of these two concepts these questions I had not considered (I did not pay attention to the terms). It will be necessary to have a closer look.

One thing I can say immediately. What is contained within the 1 refers to the actual infinity and rest refers to the potential infinity. Can these two infinity overturn each other? This is the same question.

From the point of view of Zeno's paradox inside can always add another segment, this is no different from the potential infinity. Each segment again can be splitted in the same way. But the external +1 element can also be split in the same way. And this is no different from the actual infinity.

This is probably one of the first motives why I am inclined to believe that the physical space is curved up to the Lobachevsky geometry within a single universe despite the limitations of real space.

When recording a numerical value the same division occurs with appending every decimal place. The concept of actual infinity is abstract. It is more like a cascade process, not a static condition for everything that does not touch the irrational values, such as golden sections for example. Actual infinity disappears as soon as calculations are stopped. But about a potential infinity can say exactly the same thing. On the other hand, abstract infinity sign recorded to potential infinity automatically convert it into actual infinity.

With regards, Vasiliy.

Vasiliy, in present traditional infinite relating science, we have been in fact facing 2 troubles at least since Zeno's time:

1, unable to define what “potential infinite” and “actual infinite” conceptually (theoretically) are, so people have been arguing and debating at least since Zeno's and will continue endless in present defected infinite relating science system------both parties are impossible to know what they against and what they are for.

2, unable to identify those things being treated “potential infinite” or “actual infinite” operationally (practically), so members of infinite relating paradox families have been born one generation after another. For example, we have Zeno’s Paradox of “Achilles--Turtle Race” 2500 years and now we have a “strict proven” modern version of ancient Zeno’s Paradox------ Harmonious Series Paradox.

With regards, Geng

If we really know what “potential infinite” and “actual infinite” conceptually (theoretically) and operationally (practically) are, we would not have the “strict mathematically proven” modern version of ancient Zeno’s Paradox------ Harmonious Series Paradox.

Geng, I think the debate on this issue will also be continued indefinitely. One can endlessly normalize 1 into infinity. Comparison of actual and potential infinity, as already noted above, is equivalent to comparison of the state and the process. Even abstract mathematics in actual fact has the irreparable dualism (discrete and continuous).

You can infinitely compare the speed of convergence or divergence of various processes. The only thing you can say about them is the fact that converging objects corresponds to the actual infinity, while divergent relate to potential infinity.

It is noteworthy that the Poincaré recurrence theorem (in classical thermodynamics) has the same explanation of what I proposed above: "for typical thermodynamical systems the recurrence time is so large (many many times longer than the lifetime of the universe) that, for all practical purposes, one can not observe the recurrence".

Physics unlike this all its history doing renormalization of infinity.

Dear Mr. Vasiliy Fedorovich Komarov, thank you for your frank opinion.

I think the debate on this issue will not be continued indefinitely because we are not just discussing on “what potential infinity is and what actual infinity is”. We are discussing on “how many 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".

This “strict mathematically proven” paradox (paradox of finite or infinite) exposes the fundamental defects in the traditional finite-infinite theory system.

Mathematical proof can have only one result. Vasiliy, could you give your frank opinion on the above “how many” question------- finite or infinite?

Sincerer yours, Geng

Geng. Actually, I wanted to point out that mathematics as the algorithmically precise activity is not without decision problem and halting problem. There is a danger to become an unwitting participant in Zeno's paradox trying to eliminate it in another area. Is it possible to become aware of real position in this situation? Perhaps, this question can replace all previous thoughts.

Yours sincerely, Vasiliy.

https://en.wikipedia.org/wiki/Decision_problem

https://en.wikipedia.org/wiki/Halting_problem

Dear Mr. Vasiliy Fedorovich Komarov, thank you. It is true sometimes we meet decision problem and can not help by some reasons.

Looking back into our science history, the infinite related fundamental defects disclosed by Zeno’s Paradox of Achilles--Turtle Race are really difficult to be solved, but people have been trying hard and so many new “ideas, understandings and terms, formal languages” have been created in order to solve or avoid Zeno’s Paradoxes, but invalided. The fact is: the defects are still there unsolved or unavoidable, all the family members of Zeno’s Paradox are still there unsolved or unavoidable challenging us human intellectual ability. Something should be done whenever there is still human science in our human life, it is impossible to compromise or run away from those Zeno’s Paradox.

Thank you again for your frank opinion.

Sincerer yours, Geng

That's a complicated one, but the answer is more or less the same for all "x" vs "philosophy of x" type questions. Mathematics is the study of logical consequence. We take a collection of axioms, a logical system, and find results from those two. Philosophy, as a whole, questions our thought process.

We might as questions like What is mathematics? Is it reasonable to think this way? Could we be thinking about this in a different way? What is it reasonable to do with mathematics? What is the limit of mathematics?

Dear Mr. Mushtak T. S. Al-Ouqaili, thank you for your ideas.

The infinite related mathematical fields have a closed relationship with philosophy of mathematics, but the fundamental defects in both modern philosophy of mathematics and infinite related mathematic unable us to dispel the thousands years suspended black cloud of “infinite related paradox families” over the sky of present mathematics and science--------the existing fundamental defects (syndrome of "potential infinite--actual infinite") disclosed by the paradox families in analysis and set theory are not only mathematical and not only philosophical.

It is the time for philosophy of mathematics to work and the philosophy of mathematics should not only be furnishings or “something leisure” after meals and tea time.

Much recent work in the philosophy of mathematics has focused on the Philosophy of Mathematical Practice, that is, a broad outward-looking approach to the philosophy of

mathematics which engages with mathematics in practice including issues in history of mathematics, the applications of mathematics, cognitive science, etc.).

Taken from: http://www.philmathpractice.org/about/

See my recent book, published by Cambridge University Press:

Model Theory and the Philosophy of Mathematical Practice

Formalization without Foundationalism

Major shifts in the field of model theory in the twentieth century have seen the

development of new tools, methods, and motivations for mathematicians and

philosophers. In this book, John T. Baldwin places the revolution in its historical

context from the ancient Greeks to the last century, argues for local rather than

global foundations for mathematics, and provides philosophical viewpoints on the

importance of modern model theory for both understanding and undertaking

mathematical practice. The volume also addresses the impact of model theory on

contemporary algebraic geometry, number theory, combinatorics, and differential

equations. This comprehensive and detailed book will interest logicians and

mathematicians as well as those working on the history and philosophy of

mathematics.

Part I. Refining the Notion of Categoricity: 1. Formalization; 2. The context of formalization;

3. Categoricity; Part II. The Paradigm Shift: 4. What was model theory about?; 5. What is

contemporary model theory about?; 6. Isolating tame mathematics; 7. Infinitary logic; 8.

Model theory and set theory; Part III. Geometry: 9. Axiomatization of geometry; 10. π, area,

and circumference of circles; 11. Complete: the word for all seasons; Part IV. Methodology:

12. Formalization and purity in geometry; 13. On the nature of definition: model theory; 14.

Formalism-freeness; 15. Summation.

Dear Mr. John T. Baldwin, thank you for your ideas and your offer.

Can modern philosophy of mathematics solve following challenge from Zeno's Paradox family：

In present mathematics, it is “scientific and reasonable” to use the well known Oresme's “bracket computing operation” of creating infinite items each greater than 1/2 or 1 or 100 or 1000000000000000 or 1000000000000000000000000000000 or ... from the Harmonic Series to change the Un--->0 Infinite Harmonic Series into a “Vn ＞any positive constant” infinite series (such as the positive constant of 1000000000000000000000000000000):

1 + 1/2 + 1/3 + 1/4 + ... + 1/n + ... ＞ 1/2 + 1/2 + 1/2 + 1/2 + 1/2 + …--->∞

1 + 1/2 + 1/3 + 1/4 + ... + 1/n + ... ＞ 1 + 1+ 1 + 1 + 1 + 1 + 1 + …--->∞

1 + 1/2 + 1/3 + 1/4 + ... + 1/n + ... ＞ 100 + 100 + 100 + 100 + 100 + …--->∞

1 + 1/2 + 1/3 + 1/4 + ... + 1/n + ... ＞ 1000000000000000 + 1000000000000000 +1000000000000000 + 1000000000000000 + …--->∞

…

The philosophy of mathematics can never tell us logically what "potential infinite, actual infinite" are and never know how to have quantitative cognitions to them since Zeno's time.

According to my studies, the infinite related fundamental defects disclosed by Zeno's Paradox family in mathematics and science are very philosophical.

What is philosophy of mathematics and where is philosophy of mathematics, what is it for?

"A good introduction to the philosophy of mathematics by Ray Monk. He considers the issue of the nature of mathematical truth, what mathematics is actually about, and discusses the views of Plato, Aristotle, Immanuel Kant, Frege and Bertrand Russell... What are numbers? What is mathematics actually about? Is it something discovered or is it something constructed by the mind? From the time of Plato onward, people have regarded mathematical truth as an ideal. Unlike ordinary, empirical truth, mathematical truth seems necessary, eternal, incorrigible, and absolutely certain. This talk considers some of the ways in which philosophers have tried to account for the special nature of mathematical truth..."

https://www.youtube.com/watch?v=bqGXdh6zb2k

Dear Mr. Ljubomir Jacić, thank you for your ideas.

It is true that the philosophy of mathematics should help to solve some fundamental defects in mathematics and should not only be “something leisure” after meals and tea time.

Five of my published papers have been up loaded onto RG to answer such questions:

1，On the Quantitative Cognitions to “Infinite Things” (I)

https://www.researchgate.net/publication/295912318_On_the_Quantitative_Cognitions_to_Infinite_Things_I

2，On the Quantitative Cognitions to “Infinite Things” (II)

https://www.researchgate.net/publication/305537578_On_the_Quantitative_Cognitions_to_Infinite_Things_II

3，On the Quantitative Cognitions to “Infinite Things” (III)

https://www.researchgate.net/publication/313121403_On_the_Quantitative_Cognitions_to_Infinite_Things_III

4 On the Quantitative Cognitions to “Infinite Things” (IV)

https://www.researchgate.net/publication/319135528_On_the_Quantitative_Cognitions_to_Infinite_Things_IV

5 On the Quantitative Cognitions to “Infinite Things” (V)

https://www.researchgate.net/publication/323994921_On_the_Quantitative_Cognitions_to_Infinite_Things_V