Robert Boroch

« Abstraction in mathematics is the process of extracting the underlying essence of a mathematical concept, removing any dependence on real world objects with which it might originally have been connected, and generalizing it so that it has wider applications or matching among other abstract descriptions of equivalent phenomena.

« The advantages of abstraction in mathematics are:

« • It reveals deep connections between different areas of mathematics

« • Known results in one area can suggest conjectures in a related area

« • Techniques and methods from one area can be applied to prove results in a related area »

I submit that abstraction in modal metaphysics, mutatis mutandis, shares this description. Modal logic can be said to have developed from a concern to abstract the essence (so to speak) of the alethic modalites. It then went on to treat the formalisms more abstractly, removing them from, or taking them beyond, their initial alethic subject matter (similar to the way in which geometry was removed from its initial Euclidean subject matter), generalizing their formal relationships in various ways and applying them more widely in areas far removed from their initial subject matter, so that diverse areas of metaphysics can be seen to share similar abstract structure as represented by formal modal systems. For example, the modal system underlying laws of nature might be seen to share formal axioms etc. with the systems underlying human action or value (deontic) concepts.

The abstract systems of modal logic can reveal deep connections between different areas of metaphysics (and philosophy more generally), suggest conjectures in related areas; and methods and results etc. similar to those in one area can be used to generate results in another. This shouldn’t be surprising, since formal modal logic is now also an area of mathematics.

Granted, metaphysics is not pure mathematics, so the abstractive relationships will typically not be matters involving formal interdefinition. But the abstraction, whether explicit or implicit, formal or informal, is there.

And vogue or not, ontology has been a concern of metaphysics for a long time. As for Carnap – who is indeed a truly great philosopher – the problem with his early antimetaphysical stance is that the verifiability criterion or principle of verification, and similar principles of cognitive meaningfulness, on which he and the logical positivists (a.k.a. logical empiricists) relied to dismiss metaphysics as Scheinprobleme, are themselves unverifiable and hence cognitively meaningless by their own standard. But his 1950 article “Empiricism, Semantics, and Ontology” is certainly brilliant even if not the last word.

I think YES becouse the math, especially its basic ideas must emmanate from the state of illumination which is the same as the mathaphysical abstraction,

jerzy

PS. Itsc haracterized by persistance of 70 Hz gamma brain waves as prooved experimentally

Dear:

[1] Jerzy, (a) your is logically (at least, according to computer language) equivalent; do you mean: is identical to , or its reverse: is not identical to ; (b) please don't exchange 'illumination' (which belongs to creative thinking) with I describe; (c) reasoning (on which thought is founded) has to 'some' extent to do with brain; that is the 'hard problem' of consciouness (according to Chalmers).

[2] Roman, (a) do you know more about ? I would like to follow you very serously! (b) what about metaphysical Platonism of (say) Kurt Goedel (one of the famous mathematicians)?

Anyway, thanks for both answers! Marc.

I think it is true. metaphysics, more correctly "metamathematics" is beyond mathematics. Two remarks: 1. In English speaking countries the term "metaphysics" is frequently use also for mysticism, this presents unhappy homonymity. 2. Not all what lies in the foundation of mathematic is abstraction from aplied mathematics, from our world. Great part is human construction - e.g. the whole

Cantor set theory.

Dear Peter,

(i) You think that , that means that 'metaphysical abstraction' equals to 'mathematical abstraction'. If yes, then mathematics, must be a worldview (according to my description), q.e.d.

(ii) Your second sentence, beginning with is incomprehensible for everybody.

(iii) Upon your , the following: 1. No, "metaphysics" is not identical to . According to me 'mysticism' is scientificly higher in order than 'metaphysics'. 2. The whole Cantor's set-theory is approximately nearer to 'mysticism'.

Thanks anyway for the added answer. Marc.

Roman R Poznanski

Metaphysics has come a long way since the 19th-Century and in order to pursue many areas of metaphysics nowadays mathematical tools are used. For example, the foundations of modal logic was developed by philosophers/metaphysicians and modal logic has also become a branch of mathematics. Here is a webpage for a recent book on philosophical methodology; click on the table of contents button and you'll see what I mean:

• https://broadviewpress.com/product/more-precisely-the-math-you-need-to-do-philosophy-second-edition/?ph=d6f38e02c38be45d57993534#tab-table-of-contents.

Also, you might take a quick gander at this page describing what some of the metaphysicians at Stanford are up to:

• https://mally.stanford.edu/cm/

As for so-called "mystical philosophy", that kind of thing is more typically done by theologians or the folks in religious studies departments rather than by philosophers.

Dear Karl,

Thanks for the links! Marc.

Abstraction is a cognitive process of concept formation by identifying common properties of sets of objects or living beings, fixing attention and / or thinking about such concepts and ignoring other properties. The abstractions that exist must be represented or symbolized, either through oral language, written or simply graphic. Without language there could be no abstraction, so its origin must be sought in the cognitive coevolution that occurs in human ontogenesis.

All abstractions follow these basic principles, so in a psychobiological sense they respond to the same principle. However, the subject of what they are dealing with is always different or it will have its own characteristics and behavioral development differently, in this sense if they are different.

Dear Angel,

(i) Do you mean on the 1st and 2nd paragraph what scientists call today ?

(ii) What then is your answer upon what scientists call today ?

Thanks in advance for your answer upon my questions (i) and (ii). Marc.

Robert Boroch

I am a philosopher and I have taught many metaphysics courses, so let me tell you that ontology is a branch of metaphysics. Ontology specifically concerns the fundamental categories of being, namely the things to whose existence we must be committed. The traditional metaphysical question of whether universals exist, for example, is an ontological question: should universals be acknowledged as existing in their own right (Plato vs. Aristotle) or can they be reduced to something else or paraphrased away (nominalism)?

Whether we should be realists (platonists), nominalists, logicists, or formalists about mathematical entities such as numbers and sets is a metaphysical issue concerning the ontological status of such entities.

The logics deployed in doing modal metaphysics (initially concerning Aristotle's alethic categories of necessity, possibility, contingency, etc.) involve the same kind of abstraction as in mathematics, insofar as they have been generalized and applied to other problems in metaphysics (e.g. causality, laws of nature).

Have a look at the short sections on abstraction in mathematics and in philosophy:

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

Robert Boroch

« Abstraction in mathematics is the process of extracting the underlying essence of a mathematical concept, removing any dependence on real world objects with which it might originally have been connected, and generalizing it so that it has wider applications or matching among other abstract descriptions of equivalent phenomena.

« The advantages of abstraction in mathematics are:

« • It reveals deep connections between different areas of mathematics

« • Known results in one area can suggest conjectures in a related area

« • Techniques and methods from one area can be applied to prove results in a related area »

I submit that abstraction in modal metaphysics, mutatis mutandis, shares this description. Modal logic can be said to have developed from a concern to abstract the essence (so to speak) of the alethic modalites. It then went on to treat the formalisms more abstractly, removing them from, or taking them beyond, their initial alethic subject matter (similar to the way in which geometry was removed from its initial Euclidean subject matter), generalizing their formal relationships in various ways and applying them more widely in areas far removed from their initial subject matter, so that diverse areas of metaphysics can be seen to share similar abstract structure as represented by formal modal systems. For example, the modal system underlying laws of nature might be seen to share formal axioms etc. with the systems underlying human action or value (deontic) concepts.

The abstract systems of modal logic can reveal deep connections between different areas of metaphysics (and philosophy more generally), suggest conjectures in related areas; and methods and results etc. similar to those in one area can be used to generate results in another. This shouldn’t be surprising, since formal modal logic is now also an area of mathematics.

Granted, metaphysics is not pure mathematics, so the abstractive relationships will typically not be matters involving formal interdefinition. But the abstraction, whether explicit or implicit, formal or informal, is there.

And vogue or not, ontology has been a concern of metaphysics for a long time. As for Carnap – who is indeed a truly great philosopher – the problem with his early antimetaphysical stance is that the verifiability criterion or principle of verification, and similar principles of cognitive meaningfulness, on which he and the logical positivists (a.k.a. logical empiricists) relied to dismiss metaphysics as Scheinprobleme, are themselves unverifiable and hence cognitively meaningless by their own standard. But his 1950 article “Empiricism, Semantics, and Ontology” is certainly brilliant even if not the last word.

Marc Carvallo. What I want to expose in the first paragraphs of my answer, is what can currently be understood by conceptualism. Abstract ideas, abstractions and concepts have no real entity in the physical world, but with their mental creation cognitive tools are achieved that can change behavior a lot. For me, non-conceptualism does not exist.

Mathematics begins with the achievement of numerical abstraction, once done, they serve to better know the physical reality. Metaphysical abstractions follow the same cognitive process, but its foundation and use is different.

Ángel

Abstraction is a way of generalization quantifying observations, using logic deeply thinking connections among various elements. Quantum physics & the general theory of relativity are examples with mathematical abstraction to physics. Ontology in general as mentioned above may connect mathematics with metaphysics, which try to understand the very mind that thinks to produce this effect!!!!!

I have modeled electrochemistry & physics with the process of abstraction. The particular technique several decades ago I have presented at prestigious situations universities. The IPZ model of electrochemistry that I then have developed appear at Stanford Review. Hence I have proved abstractions quite successfuly applied to have quantification of sciences. These are extendable to understand thought processes with metaphysics, that I call it metrix protocol a pure logic mathematical quantification connecting logic with language........

A beautifully subtle question.

We can abstract the physical by

-expressing general truths about physical reality, or

-by describing actual systems we would not normally see as physical but from which the physical are made.

In both cases these, in principle, can be non-hypothetical and non-mathematical.

Mathematical abstractions that exist exist on their own whether applicable to anything physical or not, and involve mathematical concepts.

These can be artificially brought close together by stretching to define that "a" metaphysics describes or underlies a "real or unreal physical world," in which case all that remains different is that a metaphysics might possibly not be mathematical in nature.

Dear Angel,

Please explain me (and other readers) your

Thanks for the further explanation. Marc.

Dear Angel,

Please explain my question in plain English! (or translate what you have written). Thanks for the link. Marc.

Sorry I was wrong to pass the translated text

I think that all cognitive expression (concepts, ideas, abstractions, sensations, etc.) have a psychobiological foundation (evolutionary, neurological, psychological and social) that produces it. That we do not know with absolute precision how it works, does not mean that it exists and sooner or later it will be known. The most abstract concepts or ideas have had in the course of their creation and development a material base that could be learned sensory through our senses. Their learning and relationship with other sensations, more or less related, would lead to more complex ones, forming a complex cognitive network where the end seems to be disconnected from the beginning (causal cognition). The materiality of the world, using our attention, perception and other cognitive capacities evolutionarily acquired, is what will produce our entire culture. Thus, to think of a "non-conceptualism" seems to me little credible.

We should not think about these concepts and ideas without knowing how (even briefly) our brain works and how it evolved. For those who are interested in these last subjects I have published a work that deals with it, I leave the address in this same portal.

Article MECHANISMS OF COGNITIVE EVOLUTION OF GENUS HOMO

Dear Angel,

Do you really mean that 'non-conceptualism' is not worth discussing? Please explain me (and other readers) that. Thanks for your answer. Marc

I think I already explained in my previous answer. I reaffirm in what has already been comment

Dear Angel,

It seems worthwhile to explain what (according to you) non-conceptualism precisely is. Thanks for your answer hereupon. Marc.

Marc,

• Conceptualism says abstract entities are mind dependent, i.e. they are conceived or "invented" by the mind. It is also called an antirealist position regarding abstract entities.
• Nonconceptualism says abstract entities exist independently of mind; i.e. they are discovered by the mind. It is a realist or platonist position regarding abstract entities.

Abstraction can be viewed as either a means by which abstract entities are invented or a means by which abstract entities are discovered.