Is mathematics analytic or synthetic?

Question

This question is related to another question I posted but I think it requires its own treatment since of the already wide scope of the other question i.e. Is the classical theory of concepts compatible with logical positivism's view on analyticity of mathematics?

Consider the following definitions

Analytic propositions – propositions grounded in meanings, independent of matters of fact.
Synthetic propositions – propositions grounded in fact.

My view of analyticity of mathematics, which may be completely naive and wrong, is that all true mathematical statement are "true by definition"(a proof is manipulation of the defined objects as well as logic) and not because "facts about the world". It is just a matter of finding a proof, if the statement in question is in fact possible to determine (all statements do not have a proof of its validity or invalidity)

I think I might be misunderstanding the idea of "facts about the world" or "experience" because there seems to be serious researchers claiming that math is synthetic.

So what is the counterargument for my argument?

Answer 1

A possible counterargument is that the analytic-synthetic distinction you are using is inherently inadequate and outmoded language and thinking. For the first part, Quine in his Two Dogmas of Empiricism argues that the notion of analyticity is circular, and that culminates with the claim there is no method of reliable identity through synonymy, a notion he calls cognitive synonymy. From WP quoting Quine:

"It seems that the only way to assert the synonymy is by supposing that the terms 'bachelor' and 'unmarried man' are synonymous and that the sentence "All and only all bachelors are unmarried men" is analytic. But for salva veritate to hold as a definition of something more than extensional agreement, i.e., cognitive synonymy, we need a notion of necessity and thus of analyticity... So, from the above example, it can be seen that in order for us to distinguish between analytic and synthetic we must appeal to synonymy; at the same time, we should also understand synonymy with interchangeability salva veritate. However, such a condition to understand synonymy is not enough so we not only argue that the terms should be interchangeable, but necessarily so. And to explain this logical necessity we must appeal to analyticity once again. Thus, the argument is circular, and fails."

Secondly, the analytic/synthetic distinction was thought up by Kant over 200 years ago. That makes it a very dated notion of mind and language. The analytic-synthetic divide is so dated, in fact, that a modern book on the philosophy of language might not even consider it. In Szabo and Thomason's Philosophy of Language, the terms don't even appear in the table of contents. That's because in the last 200 years, some progress has been made on understanding language, particularly in linguistics. In fact, the linguistic turn has produced a number of ideas that have affected the philosophy of mind and language, such as the introduction of family-resemblances and the eventual prototype theory as a basis for conceptualization (SEP).

The analytic/synthetic distinction is not employed by modern philosophers of language, linguists, or cognitive scientists because it is has been replaced by a better understanding about the origin of language according to science. Relying on these concepts to describe language is the equivalent of using the vocabulary of 17th-century ship makers to describe nuclear-powered aircraft carriers.

But one need only go back to C.S. Pierce who questioned the divide more than 100 years ago:

Kant regarded mathematical propostions as synthetical judgements a priori; wherein there is this much truth, that they are not, for the most part, what he called analytical judgments; that is, the predicate is not, in the sense he intended, contained in the definition of the subject. But if the propositions of arithmetic, for example, are true cognitions, or even forms of condition, this circumstance is quite aside from their mathematical truth. For all modern mathematicians agree with Plato and Aristotle that mathematics deals exclusively with hypothetical states of things, and asserts no matter of fact whatever; and further, that it is thus alone that the necessity of its conclusions is to be explained. - Section 1: The Essence of Mathematics, The Simplest Mathematics

Thus, the analytic-synthetic distinction suffers from a number of flaws related to "subjects containing predicates" and "hypotheticals asserting fact" as well as other objections, and is a bad way to approach to building a philosophy of mathematics (SEP).

Answer 2

1. The two terms, analytic and synthetic, are two possible, mutual exclusive properties of statements. SEP introduces the following definition:

   “Analytic” sentences, such as “Pediatricians are doctors,” have historically been characterized as ones that are true by virtue of the meanings of their words alone and/or can be known to be so solely by knowing those meanings. They are contrasted with more usual “synthetic” sentences, such as “Pediatricians are rich,” (knowledge of) whose truth depends also upon (knowledge of) the worldly fortunes of pediatricians.

   I think that’s alike to the content of your definition.

2. According to this definition I consider true mathematical statements, i.e. mathematical theorems, to be analytic statements. Their truth is demonstrated by a proof. A mathematical proof derives the statement by logical conclusions from the axioms ot the theory, employing the definition of the terms in question. One can paraphase this procedure: Bringing to light by logical deduction what is contained in the axioms. That's much more and quite different from being "true by definition".

   Of course there also are mathematical statements, which are false. Mathematics proves their falsehood by an indirect proof or by a counterexample.

   Sometimes one finds a mathematical statement which is independent from the axioms. It may happen then, that one is free to add either the statement or the opposite statement to the axioms without introducing a contradiction. The most prominent example is the continuum hypothesis (Cantor, Goedel, Cohen).

   Finally there are mathematical statements the truth of which is an open question, e.g. Riemann's hypothesis about the zeros of the zeta-function.

3. Different than Kant did, mathematics is no longer considered to deal with synthetic statements like the statements from theories in physics.

Answer 3

For the sake of the OP, I will assume that some version of the analytic/synthetic distinction is defensible. More specifically, I will assume that we can differentiate between analyzing a question from the inside to get an answer and synthesizing information from "outside" a question to get an answer. For example:

1. "Who is a bachelor?" has an analytic answer, "An unmarried man is a bachelor."
2. "Who is a bachelor?" has a synthetic answer, "Sam is a bachelor."

This doesn't need a difference between "relations of ideas" and "matters of fact," since at least on a thin theory of facts, there are facts about relations of ideas or, then, meanings. That is:

3. Bachelors are unmarried men.

... comes out to the same thing as:

4. The word "bachelors" refers to unmarried men.

But (4) would be synthetic inasmuch as we can "deny without intrinsic contradiction" that the word "bachelors" is being used in a certain way (i.e. the word can be used in other ways).

With that out of the way: go to various mathematical questions. I will assess the following:

5. 1 + 1 = 0 + 2?
6. How many prime numbers are there?
7. How many inaccessible cardinals are there?

(5) is, perhaps naively, a candidate for a question with an analytic answer. To see this, convert it into two questions:

8. What does 1 + 1 equal?
9. What does 0 + 2 equal?

Now 0 is the identity element for addition, since n + 0 = n for all n. Insofar as we correlate identity statements with triviality and hence analyticity, then, (9) seems to have an analytic answer, "2." Now assume that 1 + 1 = successor(successor(0)) or perhaps {{0}} or {0, 1}. In each case, we have two of something, over 0, so 1 + 1 = 2 looks analytic.

Oddly enough or not, if ultrafinitism is logically possible, then there aren't infinitely many numbers modulo ultrafinitism. So the answer to (6) would not be, "Infinitely many," without the background presupposition of infinitely many numbers whatsoever. Perhaps the conditional, "If there are infinitely many numbers at all, then there are infinitely many primes," can be determined analytically a priori, though.

Finally, see Hamkins[22] for examples of functions determining, "How many inaccessibles are there?" that answer the question possibly arbitrarily or "weirdly," but not, then, per the mere definition of an inaccessible, it seems. I.e. analyzing the definition of "inaccessible cardinal" will not tell you, from the inside of the query, what that "How many" should end up equaling, but it looks like you need some kind of outside information to get to the point.

Note: instead of the plainly erotetic account of the analytic/synthetic distinction, one might try out a functional one. E.g. take some function f(x) = x² and differentiate between replacing x with constants or variables. One might style the constant assignments as yielding synthetic outputs, whereas plugging x back into itself, here, yields analytic outputs. Then one might adapt this picture to (5), (6), and (7) again, to check to see if this picture lines up with the other one here painted.

Now, for all that, how useful does the analytic/synthetic distinction end up being? If analytic knowledge is uniformly trivial, I suspect that it does not end up being of much use. Of course, there is also the too-simple-to-be-simple phenomenon to consider, or the joke (another contributor here told me this one) about the two mathematicians, one of whom said a proof was trivial and spent two hours explaining so to the other, at the end of which explanation the other went, "Ah! I see now that it is trivial!" in all seriousness.

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Addendum 1: the cofinality of ℵ_ω

To illustrate further how tortured this issue can become, consider the expression cf(ℵ_ω). Offhand, it seems "true by definition" that this expression is to be evaluated to ω, and so that ℵ_ω is not regular, but is singular. However, if 0^♯ exists, then we say this curious thing, "Then ℵ_ω is regular in L." The seemingly full-and-real ℵ_ω is still singular, but we think then that L is off-key, etc. Yet do we say that this is all an analytic or synthetic matter? For we could ask the following questions:

• In L, is ℵ_ω singular?
• In V, is ℵ_ω regular?

... and so on. No doubt, such questions might be accommodated by the aforementioned descriptions of the analytic/synthetic distinction, but one wonders about the point of going through all the trouble to adapt such inquiries to such a metatheory (why not just see where L and so on go on their own terms?).

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Addendum 2: diagram-chasing in category theory

I don't have much to say about this, mostly, except to ask: is checking whether a diagram commutes an analytic, synthetic, categorical, or hypothetical procedure? I don't know how to check that kind of thing, nor do I fully (if at all!) understand the significance of diagram-chasing, but I appreciate that category theorists have something important in mind by this practice. I will leave off this addendum with a link to Eugenia Cheng's "Mathematics, morally" for reflection on the relations and differences between mathematical proofs, truths, and "moral" aesthetics in a category-theoretic context.

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Further reading (on the PhilosophySE):

1. An answer to a question about the analytic/synthetic in algebra.
2. An answer to a question about whether all logics have tautologies as well-formed.

Answer 4

There are various ways to define "analytic" and "synthetic". Those word are generally thought to apply to propositions, but there are different ideas of what a proposition is. Depending on the author, a proposition can be a propositional sentence, a propositional mental state, or an abstract propositional concept. What you consider a proposition is critical because these things are all very different. If you are talking about sentences, then issues such as meaning, definition, and convention come into play. If you are talking about mental states, then the nature of mental representations may be relevant and there may be questions as to the objectivity of the distinction or whether it applies to all minds. If you are talking about abstract propositions, then you have to talk about the relationships between propositions and between propositions and objects and you may need to discuss abstract objects.

There have been various attempts to prove that the very distinction between analytic and synthetic propositions is misconstrued, but in my experience, all of these arguments treat propositions as either sentences or mental states, while the people who make use of the distinction, who find it valuable, generally view propositions as abstract concepts. I won't say any more about the people who argue against the distinction. Instead I'll assume we are talking about a distinction between abstract propositional concepts.

Kant's definition of an analytic proposition is one in which the predicate is already contained in the subject. He was thinking of the logic of his day where the canonical form of proposition was considered a proposition of the form "[all/some] X [is/is not] Y" where X and Y are class concepts. Kant's test for analyticity would have been something like this: "if it is analytic that all X is Y and you don't immediately understand that all X is Y, then you don't understand how I am using the words".

This may look like a definition in terms of propositional sentences rather than concepts, but the rigorous distinction between language and concept is something that postdates Kant. Earlier logicians tended to view the sentence as just a representation of the concept, and Kant was pretty clearly concerned with the concept rather than the language because he didn't discuss linguistic issues, only structural issues which arguably apply to the concept expressed by a sentence more than to the sentence itself. The test I suggested is not the definition of "analytic"; it is, rather a way of expressing the nature of the conceptual relations. It's the same as the law of thought, "X is X". If someone disagrees with "X is X", how am I to characterize their disagreement? Assuming they are acting in good faith, I can only assume that our communication is failing in some way. That doesn't mean that I consider them wrong for failing to assent to the sentence token "X is X". I would have exactly the same reaction if they failed to assent to any of the language tokens "X=X", "X is identical to X", "X and X are the same thing", or "X es X" (from high-school Spanish). These are all different language tokens that express the same concept, and it is the concept that I am concerned with, not any specific language token.

Given this original distinction by Kant, clearly "37*16 is 592" is not analytic in this same sense because someone can see that sentence, clearly understand it, and not know whether it is true or not. Now along comes Frege and the logicists who took this position:

The proposition "37*16 is 592" is analytic but not directly analytic. It takes some thought to see that it is analytic. Anything that can be logically inferred from an analytic proposition is also analytic.

This is something of an extension of the notion of "analytic", but it makes some sense. So the next step for the logicists was to show how you can use logic to prove that "37*16 is 592". Neo-Fregeans are still working on this. Their efforts are frustrated by a dilemma. If they define a number as a concept, then some equivalences that ought to hold do not, but if they define a number as an object, they have to justify how you can prove from logic alone that an object exists.

Answer 5

all true mathematical statement are "true by defintion"

Define "true" and "false".

It is simply not possible to define words only using more words without circularity between definitions and ultimately self-reference, which only results in nonsense.

This applies to natural languages as well as to mathematics or any formal language.

Understanding mathematics always involves a human being.

A language makes sense if and only if someone can make sense of it, and this requires ostensive definitions. And then all definitions end up ostensive by proxy.