**Axioms**

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 19

^{th}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?

**Analytic Statements**

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.

or
simply:

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.**Conventionalism**

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 20

^{th}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.
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