My attempt at answer:

In Mathematics, if one were to ask a question such as what a "set" is, then the formal answer would be around the lines "an object which obeys a list of axioms". The obeying of axioms is captured by certain logical formula involving the set is valid.

Also in the context of mathematics, the question like "where does the set exist in the physical world" is non sense. A set existing, to my understanding, simply means a logical formula involving the definition of that set is well formed.

I conclude that the most formal answers for questions regarding sets is simply questions of syntax.

This also leads me to the following: the semantic aspect of mathematics is only important when figuring out what is the right definition. Once the theory is made, there is no aspects of semantics in the theory itself. That is a mathematical definition says no more than the logical formula.

Question in more words

Is my inference about the nature of relation between semantics and syntax valid? What are considered to be the most important prespectives on this matter by mainstream ph£losophers ?

  • Wouldn't model theory collapse into proof theory if this were so? OTOH, perhaps one could construe model theory as higher-order proof theory, or even that a theory can sustain multiple syntaxes. Nov 14, 2022 at 20:11
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    "Similarly now that most of the Mathematics community is past the platonicism phase". Is that really true?
    – cody
    Nov 14, 2022 at 20:29
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    OK I take that back @cody google.com/url?sa=t&source=web&rct=j&url=https://… Nov 14, 2022 at 20:33
  • What do you mean by "formal answer"? If you mean "rigorously exact", as people often mean, then that's not the formal answer. What that is is the answer within a formal system. However, sets existed before formal systems and are not defined by them. Nov 14, 2022 at 22:28
  • I take it for granted that axioms are used in context of a foundational formal system. Also, while I do agree there is a primitive conception of set which exists outside formal system, I think we have developed way past that Nov 14, 2022 at 22:38

1 Answer 1


I am not sure what modern philosophers have written specifically about this, but I can offer a few comments.

One can understand a concept like 'set' as an attempt to formalise, or at least make rigorous, a pre-theoretic or naive concept. We start with the naive idea of a collection of things. Then we wish to make this concept of a collection rigorous by writing down some rules, so we can say with certainty whether a given inference about it is correct or not. At this point our naive understanding informs our choice of rules, and the rules may in turn reveal interesting features of the concept that we hadn't previously considered. Hence the interplay you asked about.

But we can easily run into problems. An early attempt by Frege to write down the rules for sets ran into Russell's paradox. This was a sharp wake-up call for mathematicians, because previously it had been assumed that mathematics was just obviously correct and yet here was an 'obvious' piece of mathematics that entailed a contradiction. This was one of the motivations for the logicist project of attempting to reduce all of mathematics to formal logical systems.

Rigorous definitions and systems also commonly give rise to edge cases, and these can be unintuitive. Most mathematicians consider that the empty set qualifies as a set. It is certainly very convenient to allow it. If we didn't, all of our rules concerning sets would become much more complicated because the empty case would need to be treated as an exception. But we don't ordinarily think of emptiness as being a collection. Come to that, we don't ordinarily think of zero as being a number. Suppose I say, "I have a number of coins in my pocket," and you ask me, "How many?" and I reply, "I don't have any coins, but zero is a number". You would rightly think I was messing with you. But it is highly convenient to treat zero as a number and the vast majority of mathematicians do so, though actually not all.

The resulting rigorous account may only approximate our original naive ideas, and it may include unintuitive edge cases, but as long as it is successful and useful, we may eventually progress to the point where we regard it as having completely replaced our naive conception. Although even then, it may still make sense to ask whether our formalisation is the best one possible, and whether some alternative might have desirable epistemological properties. ZFC was not the end of set theory, though it is the most commonly used. We also have NBG, New Foundations and many others.

At the end of the day, if a theory, mathematical or otherwise, is to be useful then it will have an interpretation that will allow us to say of some propositions that they are true or false. Many theories have an intended interpretation and this constrains the choice of axioms and rules. So the syntax is not arbitrary in that it must ultimately answer to some practical use, while the semantics does not float free of the syntax because of the desirability of having rigorous formalisations that are amenable to formal proofs. Each continues to inform the other.

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