Frege's argument for the existence of abstract mathematical objects

Question

I have some trouble understanding Freges argument in particular as presented here, https://stanford.library.sydney.edu.au/entries/platonism-mathematics/#FreArgForExi

In particular the first premise i.e

"The singular terms of the language of mathematics purport to refer to mathematical objects, and its first-order quantifiers purport to range over such objects"

If we refer to our language how is then the object that we introduce independent of us as the "intelligent" agents, which Platonism suggests it should be?

Answer 1

First comment : here "platonic" must be understood in a broad sense, because Plato asserted the "real" existence of non-sensible objects : the Forms (or Ideas).

Frege has been defined a "platonist" because he asserted that numbers are real non-sensible objects, i.e. abstract objects.

Having said that, "singular terms" are "names" for numbers : zero, one, two, etc. According to Frege, they are real names, denoting objects : the natural numbers.

Thus, the quantifier of a formula like $∀n(n ≥ 0)$ must be interpreted - using the now standard semantics for first-order languge - as ranging over a domain of objects. These objects are the numbers.

But Plato's Forms are more than "abstract objects" :

These Forms are the essences of various objects: they are that without which a thing would not be the kind of thing it is.

A Form is aspatial (transcendent to space) and atemporal (transcendent to time). They are non-physical, but they are not in the mind. Forms are extra-mental (i.e. real in the strictest sense of the word).

The Forms are perfect and unchanging representations of objects and qualities. For example the Form of beauty or the Form of a triangle.

Some of these characteristics of platonic Forms can be predicated also of numbers, but not all of them.

Answer 2

You would do better to actually read Frege's writing and then decide if an explanation that admits to misrepresentation should even be given credence.

First of all, Frege's definition of zero is obtained by an application of ths principle of indiscernibility for non-existents from negative free logic. So, it is not even correct to compare it with the semantics of first-order logic. If, in Frege's account, a singular term does not denote, the formula containing it is simply false. In particular, reflexive identity statements can be false.

Frege's reasoning is in no way reflected in the claims of the link. He contrasts his extensional view of logic with the intensional logicians of his day. Here, extension and intension refer to the converse orders of the Aristotelian term hierarchy. Logic as inherited from Leibniz inverted this order, and, a focus on syntactic structure with the rise of formalism meant that logic had been moving away from Aristotelian notions of substance. Frege objected to formalism. Moreover, when he retracted his logicism, he chose to advocate for a geometric foundation for all of mathematics.

Frege's notion of truth depended upon the distinction between a concept (intensional) and the extension of a concept (extensional)(importantly, note the definite article in "the extension of a concept"). The importance of singular terms is grounded in Aristotle's distinction between "primary substance" and "secondary substance". Only individuals are primary substances. Species and genera are secondary substances.

With exception for his zero object, Frege's account of number was metaphysical. Truth was to be substantiated by the existents -- that is, primary substances -- which would "fall under a concept". Defined as a class whose only possible members could be fictional or self-contradictory, the zero object could be declared meaningful even if it was not substantive. In turn, it could be proven to be unique. Consequently, it became the ground for showing that the class of individuals is not vacuous. From that point, Frege used the inclusive disjunction to implement the logicist successor function.

This last item, namely the use of inclusive disjunction, had been one of Russell's criticsms of formalism. Other than claiming the "obviousness" of succession, as Skolem had done, formalism has no justification for its account.

It is important to acknowledge the role of definite description in Frege's account. Imparting singularity, the definite article conveys that extensions of concepts are well-construed individuals. Russell offered an alternate interpretation because Frege's semantics would identify non-existents with one another. Yet, the principle of indiscernibility of non-existents is widely accepted as a well-formedness condition today.

Look at how Moschovakis deals with undefined terms in his paper on formal recursion theory. Less explicitly, Shoenfield describes a relation admitting undefined terms ss arguments in his account of recursion theory. And the purpose of that relation is to identify non-existents with one another just as Frege's semantics had done.

On the other hand, the interpretation of Russell's paradox, as a problem to be addressed through the limitation of size, undermined the idea that definite description could actually govern the nature of sigularity. One cannot form Fregean number classes in Zermelo-Fraenkel set theory.

Meanwhile, definite descriptions rely upon a grammatical form of the principle of identity of indiscernibles. Historical views of the intensional logicians holding that singularity ought be understood in relation to proper names began to be reasserted. In addition, Hilbert and Bernays showed how definite descriptions could be eliminated under a common presupposition concerning the syntactic structure of definitions (to my knowledge, only Padoa correctly portrays the "non-creativity" as an assumption that can be rejected). So, the Fregean tactic has suffered a great deal of disrepute.

Then again, the homotopy type theory advocates claim to be rediscovering Frege.

If you look into these matters carefully enough, you will find that these people have been doing little more than arguing in circles for centuries. Max Black wrote a paper in which the antagonist against the principle of identity of indiscernibles uses spatial intuition to describe a symmetric universe. Kant used spatial intuition as the basis for his criticism in "Critique of Pure Reason". And, Strawson uses spatial intuition to differentiate between the qualitative identity one obtains from logic and numerical identity one obtains from geometric arrangements.

Good luck with your studies.

Answer 3

You are welcome.

Let me try to give you something more immediate to your question.

When we begin our studies of "mathematical logic" we have a naive appreciation for the nature of truth that is conflated with our personal beliefs. We talk of "truth" and "falsity", we represent "truth" and "falsity", we formulate a logical algebra with which we investigate proof structure, and we find that we must differentiate between material consequence and logical consequence precisely because of our interest in a logical algebra to be used in a logical calculus.

Where have we actually discussed what we mean by truth?

There are a multiplicity of theories of truth. The two which are most relevant here are the correspondence theory and the coherence theory. A majority of philosophers will discount the coherence theory because it can only have a relationship with material reality by sheer luck.

There are also conceptions of truth. Tarski proposed the semantic conception of truth. The analytic conception of truth probably originates with Kant. It holds that the truth of a statement depends upon the meaning of its words. There is also an epistemic conception of truth. This is closely related to the typical mathematical practice of proving that a definition is non-vacuous. When it involves singular terms, it also involves a uniqueness proof.

You are questioning how one surmises existence of an object by merely using a singular term purporting to denote it. By virtue of its quantifier rules, first-order logic is a logic with existential import. If you accept the logic, you are bound to its presupposition. Free logics are different in this regard, and, Frege's use of the indiscernibility of non-existents had been intended as a well-formedness condition providing for the substantive denotation of his terms.

Now, all instances of reflexive identity are true in an analytic conception of truth. It is assumed that the two occurrences of syntactically identical symbols have uniform denotation (if they denote at all). This is why formalists find the possibility of a false reflexive identity statement confusing. Yet, a semantic conception of truth insists upon the substantive denotation of singular terms in true statements.

Tyler Burge, for one, has held that Tarski's semantic conception of truth clearly justifies the idea of false reflexive identity statements under a correspondence theory of truth. Within mathematical contexts, examine the transitivity axiom from Tarski's cylindric algebras. It has the effect of binding identity statements with the existential quantifier. In particular, reflexive identity statements only hold for existents.

Philosophers and symbolic logicians will have a problem with Tarski's axiom from cylindric algebra. It would, in effect, be an existence predicate and philosophers are taught to reject such things. Symbolic logicians will object to the fact that it has an infinite expansion relative to self substitutions into the definiens. But, again, in mathematical contexts, this seems closely related to the model-theoretic investigation of identity in the presence of apartness done by Statman and van Dalen and the proof-theoretic investigation of the same by Borretti and Negri.

Writing before the more modern developments of free logics and cylindric algebras, Abraham Robinson wrote the paper, "On Constrained Denotation." Its principal motivation had been the description of a language given through the formulation of definite descriptions. One simply cannot use a model-theoretic identity relation because the identity relation is essentially constructed as terms are introduced. From this, one may surmise that an epistemic conception of truth is a constructive methodology. Using ideas from free logic makes the epistemic conception more understandable. But the first-order logician who insists upon an extra-logical stipulation of the domain will find it unpalatable.

Now we can compare the correspondence theory and the coherence theory.

Our naive intuition is correspondence. In a sealed room with one chair we say "the chair", using the definite article referentially. In a restaurant we might invoke the ostensive language act of pointing to a table while uttering "that table." But if we apply a correspondence theory of truth to mathematical objects, we seem committed to the existence of mathematical objects without any ability to produce such an existent. To a large extent, we reach this point because of commonly held beliefs concerning a relationship between mathematics and material reality.

This is why I began by pointing out how we begin our studies with the conflation of truth with belief.

Kant had probably been the first to propose a coherence theory of truth, although there had been no vocabulary to express it as such at the time. He had not been trying to delineate "mathematics". He had been responding to Hume's modern account of skepticism. The argument applies to any correspondence theory of truth intended to make claims about material reality. By relegating mathematics to "intuition" Kant had been trying to provide a basis for "objective knowledge" in contrast with claims of "absolute knowledge" (correspondence with material reality).

Not surprisingly, philosophers -- especially analytic philosophers -- took issue with this. Frege's logical studies arise in this context. Hence one sees the motivation behind his emphasis on truth and his emphasis on extensionality.

Ultimately, these are waters you must navigate for yourself. I find the argument that mathematical objects exist because science is useful to be analogous with the argument that God exists because prayer has efficacy. To be skeptical of the latter without being skeptical of the former is incoherent in my view.