Why do mathematical platonists believe in the abstract when math clearly comes from FOL, a non-abstract?

Question

To assure ourselves first order logic is as free of paradox, errors, and impermanence, mathematicians and logicians "grounded" math in a language/system everyone can agree upon. Here is a quote I found helpful from mathoverflow, "you need to assume that we know how to deal with finite strings over a finite alphabet. This is enough to code the countably many variables we usually use in first order logic (and finitely or countably many constant, relation, and function symbols)...This is the hermeneutic circle that we have to go through since we cannot build something from nothing."

Finite strings and finite alphabets are clearly not abstract entities here, as I interpret this. That is the whole point, to have something so assuredly tangible no one can doubt it. Abstract objects are antithesis to tangibility, so abstractness can't have entered yet.

Platonists must think mathematics is only grasping at the "real" abstract realm. Math the language merely represents or approximates it.

I still don't get it though. In no other language have I begun to suspect in the abstract; natural language is just a tool to get by. I and it does not claim what is actually real. In claiming natural language is a tool, it merely represents (and serves a purpose), as does math the language. BUT, the rest of the world yet tangibly-experienced is categorically more accessible than the Platoninic realm. I can imagine traveling near c in a spaceship to nigh-in accessible locations in the cosmos.

I want to understand how Platonists like Penrose so strongly feel there is an abstract world. I clearly don't get it still. Am I too strongly a scientific realist and too deflationary about math? I did not come up with the above out of my own preferences though, they seem to be what each camp is saying the loudest, and they seem in-principle impossible to fit together. But to my eyes, the Platonists have the shakier footing.

Answer 1

You are correct that formal logic provides a link between mathematics, and tangible things we can operate on mechanically (formal proofs).

The concept of a "formal proof" is an abstract, mathematical one, which is no more or less real than perfect circles or the integers. However, the concept of a formal proof is also designed to be something we can physically write on paper, or physically store in the memory of a computer. This may be called "instantiating" the proof; creating a physical instance of the (abstract) proof. When instantiated in a computer or on paper, a formal proof is as real as a chair.

However, all of this is independent of whether mathematical objects exist in a Platonic realm. The instantiated proofs exist physically and concretely. But if a proof talks about Euclidean circles, the Euclidean circles are not instantiated; only symbols pertaining to them are physically present, not the circles themselves.

We have the option of saying the circles themselves exist Platonically, or do not exist. It really is just a question of how you want to define the word "exist." Perfect Euclidean circles apparently don't exist in our physical reality, because of the curvature of space. But we may just define the word "exist" in such a way that it applies to non-physical mathematical entities.

Answer 2

@jkusin

I empathize with you. I do not join sites like these because beliefs about mathematics and science are so incommensurable that almost anything one can say can be answered with a skeptical retort. I lost patience with it all long ago. I am grateful to the posters on this site like JR, conifold, Dcleve, and doubleknot. They, and a few others have given me words I had not had before.

My stance on mathematical foundations is semi-intuitionistic. My stance on physics is constructive empiricism. I am agnostic about the relationship of science to the truth of material reality . Ultimately, these views rest on my conclusion that the theory of evolution, if true, imposes severe epistemic limitations on knowledge claims. I certainly employ abstract objects to navigate within my sensible experience. I have never found any evidence to make me believe that the abstract objects of which I speak will persist when I perish. I expect to perish because that seems to be what happens to biological organisms.

What does it mean to be semi-intuitionistic? The mathematician with whom this is associated is Emile Borel. Michael Rathjen has written a paper on Solomon Feferman's attempt to understand semi-intuitionism. Using the metamathematics of formal systems and the arithmetical hierarchy of formulas, he attempts to characterize "definiteness" of sets to the Delta_0 class of formulas (formulas for which all quantifiers are bounded). One problem is that Feferman's analysis incudes Brouwerian intuitionism. There is no reason to think that a semi-intuitionist like Borel did not accept bivalent reasoning. Feferman is simply looking for a "dividing line" by which his mathematical paradigm can subsume Borel's views. (Do not misinterpret: it is good metamathematics.)

A different approach must come with the logic itself. Cornejo and Viglizzo have developed what they claim to be a propositional calculus for semi-intuitionistic reasoning. I could only find a partial description on arxiv because their papers are behind paywalls. However, that paper did include a statement for their rule of detachment.

I had to assume that their "primitive" logical connectives were the classical material connectives. I compared their rule of detachment to the rule of detachment for the propositional connective NAND as described on Wikipedia. For NAND, one has

( U | ( V | W ) ), U |- W

For Cornejo and Viglizzo, there is a reduction to

( U | ( W | W ) ), W |- W

relative to NAND.

I suspected this because of my personal researches. Thirty-five years ago I started studying the 16-set of truth tables because complete connectives made unary negation irrelevant. I began studying involutions on truth table representations because unary negation among such representations involves the exchange of valuation in the third column alone.

There ought to be nothing objectionable about this to modern metamathematical studies because they claim the sensible impression of symbols to be the ground of their "foundation." It is, however, highly objectionable because of folklore, presupposition, and the belief that have arisen with respect to these things. The folklore is the "arithmetization of mathematics." The presupposition is that mathematics is reducible to algebraic formal systems. And the belief is the underlying platonism behind Goedel's uses of natural numbers. I am not at all certain that a semi-intuitionist requires all of the machinery of formal systems and their transcendant hierarchies just to understand epistemic limitation.

You have been reminded that mathematics is an old subject. Respecting that view, the lingustic analysis through which formal systems arise is posterior to the language practices of ordinary mathematicians. To some extent, analytical philosophy arose in response to Kant's attempt to answer Humean skepticism. To say that "mathematics is analytic" is to assert a belief. It does not matter how many people believe you. Few people who recite statements coming from analytical philosophy have actually read Section 74 of Kleene's "Introduction to Metamathematics" where the difference between ordinary practices and the practices involved with formal systems is acknowledged.

The significance of my decision to study 16-sets lies with a paper by William Kantor on 2-transitive symmetric designs and a paper by Edward Assmus and Chester Salwach on 2-(16,6,2) designs. Lemmas 6.5, 6.6, and 6.7 in Kantor's paper explain the ubiquity of these designs. The paper by Assmus and Salwach provide a means of visualizing these designs with respect to a 4x4 array. The "sensible impressions" underlying my investigation of involutions between truth table representations now has a format related to the symmetries of dihedral groups. Moreover, Assmus and Salwach explain how the double transitivity of the 16-element group can be shown with the complete graph K_6 and the symmetric group S_6.

Curiously, S_6 is unique among finite symmetric groups because it has distinct inner and outer automorphisms. So, it is associated with a fundamental notion of "twoness" (Brouwer) understood with respect to "inner" and "outer" (Kant).

When you start looking for "convergence" to 16-sets, you will find it to be ubiquitous: free Boolean lattice on two generators, 3-dimensional projection of a tesseract, trilattice logic, skew lattices over rings, Coxeter's collection of 4-dimensional regular polytopes as given by his group-theoretic definition of "regular," the quaternionic decomposition of Williamson matrices, and, of course, the finite basis for the Kummer surfaces of algebraic geometry (important to string theorists and quantum cryptographers).

Historically, the natural numbers are a system of aliquot and aliquant parts. When you declare that "number theory" is based upon "counting, " you break group theory except as an interpreted language signature.

The admonitions against circularity that are used to enforce the philosophical reduction of mathematics to foundationalism in the sense of objectual ontologies is based on legitimate concerns about infinities. Mathematicians do not seem overly concerned about thinking in terms of relations and implicit definitions. So, the idea that mathematicians are unable to coherently reason about a downward infinite regress of relations (Bradley) is an element of modern folklore because of the influence of logicists (Russell).

However, objectual ontology is assumed by physicists with their "configuration space." I certainly do not know if "points" exist. But, I do know that they are implicit to the basic pedagogy of physics when the dichotomy between potential energy and kinetic energy is explained. A potential field is asserted to coincide with the elements of the objectual ontology (witnessable "space"). Arc connectedness is asserted as a relation between the elements of the objectual ontology (witnessable "trajectories"). Of course, arc connectedness presupposes that "time" has the form of an algebraic dimension with an "unwitnessable measure."

Usually, though. "time" is measured in a "part of space" separated from where "evidential measurements" are being taken.

Here is a simple observation: Draw a circle on a piece of paper and denote it with 'A'. Draw a point inside the circle and a point outside the circle. Denote the former with 'x' and the latter with 'y'. Then the formal expression,

( x in A ) AND ( NOT ( y in A ) )

is a lingustically competent expression for "distinctness."

Its negation would be

IF ( x in A ) THEN ( y in A )

Conditionals are reflexive, transitive relations. There is nothing "singular" here. There is nothing symmetric here.

What has been imposed upon this is "obvious" to some and not to others.

I hope you can take something of value from these remarks. Since I will not join sites like these, I will make no further comments.

Answer 3

correction:

( U | ( W | W ) ) , U |- W

for Cornejo and Viglizzo.

Given your comment, look up "differential ontology" and consider this as a "branch point" relative to the objectual ontologies of domains of discourse.

The first-order paradigm is not exactly metaphysics. If one is justifying the necessary truth of reflexive equality statements with the law of identity, it is metaphysical. But, the necessary truth of reflexive equality also follows from declaring mathematics to be analytical. A word ought to carry the same meaning (warrant for substitutivity in a logical calculus) with every occurrence.

So, you arrive at a second "branch point" by which the substitutivity warrant becomes the denial of distinctness. To make sense of this, you must distinguish between equality, discernibility, and distinctness. This requires an observation from first-order model theory regarding the persistence of truth for existential formulas with respect to model extensions (upward persistence) and the persistence of truth for universal formulas with respect to submodels (downward persistence). If one is not convinced by logicism and rejects the influence it has had on formalist foundations, this actually corresponds with the characterization of the consistency problem as described by Hilbert and Bernays. They interpreted intuitionistic criticisms in terms of a paradigm difference in which a "given" domain of discourse with well-construed (decidable) equality relations could not be assumed. Moreover, they explicitly understood the consistency question as obfuscating any role for existential import.

The paradigm difference expresses itself through description theory and the use of Tarski's transitivity axiom to embed existential import into equality statements.

It is possible to write a formal sentence "introducing" a discernibility relation that is irreflexive and symnetric with no quantifiers other than the two universal quantifiers with global scope.

It is possible to write a formal sentence "introducing" a distinctness relation whose two nested universal quantifiers carry no existential import. Related by a disjunction, the denied distinctness relation is expressed by the conjunction of existential subformulas.

The correlate to first-order formalism is to warrant substitutivity with a denied discernibility relation. Following the terminology in description theory (outside of received studies in mathematical logic) call this the attributive paradigm (Russellian).

Among the logics which manage existential import, one has free logics. In particular, there is a close connection between recursion theory and negative free logic because recursion theory admits a relation by which arbitrary expressions are comparable. Defined expressions are discernible from undefined expressions, and, all undefined expressions are indiscernible from one another. This corresponds with the principle of indiscernibility of non-existents in negative free logic.

Thoralf Skolem's paper on arithmetic and "the recursive mode of thought" has generally been interpreted as primitive recursive arithmetic relative to Goedel's effective (and platonistic) characterization of recursive arithmetic. Such an analysis is based on the comparison of "quantifier-free" methods. If, however, you read Skolem's paper, he is clearly critical of Russell's description theory. Much like Kant's call for the development of geometries different from Euclidean geometry (see Ewald's anthology), Skolem's criticism of Russellian description theory is lost in folklore.

It is well known that one can prove reflexiveness from transitivity and symmetry. If one takes Tarski's transitvity axiom for equality as the "symbol introduction" for the sign of equality, then one need only formulate a symmetry axiom to support a logic supportive of a non-Russellian description theory.

Implementing this, however. requires both discernibility and distinctness.

One criticism of the use of descriptions is that one cannot assure "singularity." So, if one can use a formula to introduce a constant using a description, one must first declare or otherwise prove that satisfying instances are indiscernible from one another. This is a different problem from how existential priority cannot be expressed within the first-order object languages.

What this also means is that a logic supporting non-Russellian descriptions does not even work like Russellian description.

A basic existential axiom will have two existential quantifiers

ExEy Phi(x,y)

So that the first quantifier can become a universal quantifier,

Ax( x = a IFF Ey Phi(x,y) )

The quantifiers here are restricted with respect to reflexive equality statements,

Ax Phi(x)

IF a=a THEN Phi(a)

Ex Phi(x)

a=a AND Phi(a)

With regard to descrpition theory as discussed outside of received views for mathematical logic, these would more properly be called demonstrative descriptions. The kind of problematic description theory associated with Frege's idea that the word "the" is sufficient for "singular" reference is properly called referential description.

Consequently, this approach should be called the demonstrative paradigm as a contrast to the attributive paradigm of Russellian description.

As a "branch point" this is all motivated by the fact that the necessary truth of reflexive equality statements does not follow from Tarski's semantic conception of truth.

Analytical philosophy makes a knowledge claim--- "mathematics is analytic"--- before applying Tarski's truth analysis.

This is why a semi-intuitionist of Borel's persuasion would take issue with characterizing semi-intuitionism using formal systems. It is not a domain of discourse with decidable equality relations which is "given." It is the experiential continuum which is given. For Borel, at the time that he lived, this would have been equivalent to accepting real spaces as given through witnessability. This is why he is associated with problems in "descriptive set theory."

The development of these ideas on non-Russellian description can be traced to the inquiries into "slingshot arguments." I received some clarity on "demonstrative description" through Stephen Neale's work in fact theory.

Tarski had used the liar paradox to demonstrate that his conception of truth had correspondence with naive notions of how truth related to language. Frege had used a square of opposition to demonstrate correspondence with analyses of traditional logic such as Port Royal. Naturally, I have the proof schemes corresponding with Skolem's "descriptive function" explanation of numeric succession. So, while I am well aware that there may be fatal errors in my work because of the intrinsic difficulty, I am not making these remarks flippantly. Of course, my work will never see publication. I have no credentials and no one is interested in mathematical foundations that question analytical philosophy and its apparent determination to use mathematics to ground reductionism to physics as "truth."

Being the ever-competent dialectician, Russell actually discussed something similar to this paradigm in Chapter 19 of "Principles of Mathematics" under the classification "the relative view of quantity." It is similar with respect to the fact that the necessary truth of reflexive equality statements is not assumed.

Good luck in finding answers to your questions. Hopefully, you will not have to work as hard as I have.