It's often said that axioms themselves don't need to be true. What follows from axioms, however, must do so according to strict logical laws. Thus different geometrical and mathematical systems are constructed on axioms which needn't be taken as true. (This largely came to be seen to be the case in the late 19th century.)
However, Anthony Quinton (in his 1963 paper 'The A Priori and the Analytic') sees axioms very differently. He says that “axioms only confer truth on theorems if they are true themselves”.
How is the truth of axioms established?
According to Quinton, a “formally sufficient axiom will be materially adequate only if it is intuitive”. Thus truth is discovered or seen intuitively (which is a common idea in mathematics).
The idea of intuition here must mean that there's no other way to see (or discover) the truth of axioms precisely because they are so basic. In other words, they can't be shown to be true by other axioms and certainly not by theorems.
Remember that axioms are the starting-points of systems or chains of reasoning. Thus they can't rely on anything. And that must explain why their truth is seen intuitively – there's no alternative to that.
Still, what is meant by truth here? How are axioms true?
Axioms aren't about the world and presumably they aren't about other axioms or theorems. So why are they true at all? Why can't they simply be taken syntactically or even as simple marks on a page?
Quinton ties in the intuitive truth of axioms to the intuitive truth of analytic statements. He writes:
“According to the analytic thesis, an a priori truth is intuitive if its acceptance as true is a condition of understanding the terms it contains.”
To state the obvious, analytic statements are very unlike simple axioms. Most of the quoted analytic statements (such as “All bachelors and unmarried”) contain predicates and references to worldly items. Axioms are nothing like that. (Although Quinton gives no examples.)
It can of course be said that we see the truth of “All bachelors are unmarried” intuitively. Though that only means that we understand the concepts or words involved and see them as synonyms. Surely that can't be said of individual axioms.
It's tempting to think that analytic truths are pretty pointless if they really are only about words. If it's all about conventionality, synonymity or analyticity, then the world seems to drop out of the picture. (Except of course that analytic truths - as written down or thought about - must be part of the world.)
However, Quinton makes a good distinction between the analytic sentence and the proposition its expresses.
He says that “the conventionality principle fails to distinguish sentences from the propositions they express”.
In other words, it is the words or sentences which abide by conventional rules. However, the propositions they express have nothing directly to do with rules.
Thus we effectively have the following:
“All bachelors are unmarried men.”
Clearly, that is a sentence because it occurs within inverted commas.
But we also have:
All bachelors are unmarried men.
Sure, the proposition is expressed with a sentence again; though when you take away the inverted commas, it becomes a.... proposition. Or does it? The proposition is still expressed by a sentence even if I've taken away the quotation marks!
Even if I write:
The states of affair of bachelors being unmarried men.
Bachelors' being unmarried.
that's still a sentence.
In any case, a proposition on these expressions simply seems to be a fact or a state of affairs (or the philosophers' “truth conditions”). Yet propositions aren't deemed to be such things. Propositions are seen as abstract entities which don't belong to time and space and have no causal relations with the world. (All that is extremely problematic; though I'll leave that there for now.)
In any case, Quinton himself moves on from using propositions as an argument against mere conventionality (or mere analyticity), to saying exactly the same thing about concepts. Here it's concepts which are seen as non-conventional and therefore, possibly, also abstract entities. (Or, to be more accurate, Quinton is putting the position of what he calls the “anti-conventionalist”.) Quinton writes:
“... the anti-conventionalist maintains that there is a non-conventional identity of concepts, lying behind the conventional synonymy of terms, which would still exist even if no means of expressing the concepts had ever been devised.”
It's not really a surprise that “anti-conventionalists” should move from propositions to concepts because, on Frege's picture, concepts are (non-spatial) parts of propositions. (Or, to use Frege's own terms, concepts are parts of Thoughts.) In that sense, a concept is simply an abstract part of a larger (as it were) abstract entity – a proposition.
The problem remains: if we can't make sense of propositions, then we can't make sense of concepts either. Or at least we can't make sense of concepts if they're seen as abstract entities of some kind.
For one, it seems utterly bizarre that Quinton should say that the concepts behind (to use a spatial metaphor) the terms 'bachelor' and 'unmarried man' are “timeless and objective”. Really? I can happily accept the the concept or symbol '2' lies behind a concept or abstract entity which is timeless and objective. Though can we say the same about the concepts behind the words 'bachelor' and 'unmarried man'? Surely not. For one, the words have only existed since the institution of marriage began. And therefore the concepts (expressed, yes, by different words in different language) must have only begun to exist when the first words or even institutions of marriage existed. None of these things are timeless.
Are they “objective”?
Well, that depends. In fact I can accept that the concept [bachelor] is both objective and abstract; though not that it's timeless. That's unless you believe that timelessness must necessarily come along with abstractness and objectiveness. If it does, then the concept [bachelor] isn't abstract or objective either.... and that's all I can say.
In the philosophy of mathematics, and even in mathematics itself, a link was made between the conventionality of mathematics and the fact that mathematical statements are true because they assert identities. In other words, the identities of mathematical statement follows from the fact that mathematics is about symbolic conventions – not really about truth as such. This was the position from David Hume through to Jules Henri Poincaré and Ludwig Wittgenstein's Tractatus.
However, perhaps mathematical conventionality and identities don't go together. Or perhaps both can be rejected.
According to Quinton: “Leibniz knew to much about mathematics to regard it as conventional.” However, it seems that “he did not know enough about it to realise that its propositions were not identities”. So Leibniz did believe that mathematical statements – indeed truths – were mere identities. That is, both sides of the equality sign states the same thing; though in different ways. This can be seen as a very deflationary views of mathematics and it's been almost universally rejected since Leibniz's time. (The rejection largely set in, I think, in the early 20th century; though perhaps before.)
If mathematical statements aren't identities, does that mean that they are Kantian synthetic a priori truths? (I.e., truths that aren't mere identities or determined by the meanings of the symbols on both sides of the equality symbol.) They can of course be known a priori; though that's not because they are identities or simply because both sides of the equation state the same thing. Yes, they can be known without (further) experience; though that knowledge of their truth isn't simply a result of symbolic identities.