The philosopher Anthony Quinton expressed an “anti-conventionalist” (or Platonist) position on the analytic when he wrote: “There is a non-conventional identity of concepts which would still exist even if no means of expressing the concepts had ever been devised.”
Anthony Quinton (1925 — 2010) was a British philosopher. He was educated at Christ Church, Oxford University, where he obtained a first-class honours in Philosophy, Politics and Economics. He was President of Trinity College, Oxford from 1978 to 1987; and a chairman of the board of the British Library from 1985 to 1990. Quinton was also the president of the Aristotelian Society from 1975 to 1976.
Quinton once presented the BBC Radio programme, Round Britain Quiz.
Axioms
It’s often argued that axioms don’t need to be true. That is, axioms can — and do — have a purely formal or syntactic role. What follows from axioms, however, must do so according to strict logical rules. This basically means that different geometrical and mathematical systems can be — and are — constructed on axioms which needn’t be taken as being true. ( As a result of various findings in geometry, this largely came to be seen to be the case in the late 19th century.)
Of course all this depends on how the word “true” is taken.
Anthony Quinton (in his 1963 paper ‘The A Priori and the Analytic’) saw axioms very differently. He argued that
“axioms only confer truth on theorems if they are true themselves”.
So how is the truth of axioms established?
Generally speaking, axioms are seen as “assumptions”. Alternatively, it has been argued that axioms can be “taken to be true” — but only in order to (as it were) get the ball rolling. (Axioms have also been seen as being evident or even self-evident.)
It’s worth stressing here that the term “axiom” is defined differently when used in mathematics, logic, and, indeed, philosophy. In philosophy, an axiom has traditionally been seen to be a statement that’s evident or simply well-established. In modern logic, axioms aren’t treated this grandly. They’re simply seen as being the starting points for reasoning. In mathematics, axioms are often taken to be statements which are often taken to be true — but only within the logical systems they define or generate.
So the word “axiom”, in this case at least, needs to be seen within the context of Anthony Quinton’s discussion of the a priori and the analytic — both terms hardly used outside philosophy. Indeed the term “analytic” only became widely discussed in analytic philosophy in the1930s. (For example, with the logical positivists and others; though such philosophers did work on Gottlob Frege’s ideas — see here.)
According to Quinton, a “formally sufficient axiom will be materially adequate only if it is intuitive”. This means that an axiom’s truth is discovered (or seen) intuitively.
This reliance on intuition (see also here) here must also mean that there’s no other way to see (or discover) the truth of axioms precisely because they’re so basic. In other words, they can’t be shown to be true by other axioms, the world, evidence, data and certainly not by theorems.
Remember also that axioms are used as the starting-points of systems or chains of reasoning. This means that axioms can’t rely on anything else. And that must explain why Quinton believed that their truth is seen intuitively. His position has the consequence that there’s no alternative to that position.
Still, what is meant by truth here? How are axioms true at all?
Philosophers of mathematics have often asked if it’s even meaningful to state (or believe) that axioms are true. (See Penelope Maddy’s paper ‘Believing the Axioms’.)
For example, the French mathematician Henri Poincaré expressed what has been called a conventionalist view. Poincaré’s use of non-Euclidean geometries showed him that the axioms used in geometry should be chosen not for their truth, but for what they can produce. What’s more, axioms shouldn’t be chosen (at least not exclusively) for their (supposed) coherence with human intuitions about the world.
Does all this mean that axioms aren’t about — or derived from — the world at all? And, presumably, do this also means that axioms aren’t about — or derived from — other axioms or theorems either?
So, again, why are axioms deemed to be true? Why can’t they simply be taken syntactically or even as simple scribbles on a page?
Analytic Statements
Anthony Quinton tied the intuitive truth of axioms to the intuitive truth of analytic statements. He wrote:
“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 oft-quoted analytic statements (such as “All bachelors are unmarried”) contain predicates about — and references to — worldly items. Axioms are usually nothing like that. (Quinton himself gave no examples in his paper.)
It can of course be argued that we (as it were) see the truth of the statement “All bachelors are unmarried” intuitively. However, that only means that we understand the concepts or words involved and see that they are synonyms. Surely that can’t be said of individual axioms.
Indeed it’s tempting to think that analytic truths — along with tautologies — are pretty pointless if they really are only about words (or symbols) and their synonymy. Moreover, if they’re all about conventionality, synonymity or analyticity (see W.V.O Quine’s own ‘Two Dogmas of Empiricism’), then the world (or reality) seems to completely drop out of the picture. (Of course analytic truths — as written down, spoken or even thought about — must still be part of the physical world.)
All that said, Quinton did make a distinction between analytic sentences and the (as he put it) “propositions they express”.
Propositions
Quinton argued that
“the conventionality principle fails to distinguish sentences from the propositions they express”.
This is a clear commitment to abstract propositions. Or, at least for now, a commitment to propositions.
In other words, Quinton must have believed that it’s the words (or sentences) used in analytic statements which abide by the rules of convention. However, the “propositions they express” have nothing at all to do with such rules.
So take the following sentence:
“All bachelors are unmarried men.”
Clearly, that’s a sentence because it occurs within quotation marks (among other things).
But we also have this:
All bachelors are unmarried men.
Is that a proposition?
It’s of course the case that the the (?) about bachelors directly above is again expressed by a sentence. However, when we take away the inverted commas, it (to put it ironically) suddenly becomes a proposition… Or does it? Isn’t proposition still expressed by a sentence even if the quotation marks have been taken away! And, without sentences (as well as minds), what purchase can we have on these supposedly abstract propositions?
Moreover, even if I write
The state of affairs of bachelors being unmarried men.
or more simply
Bachelors being unmarried.
they are still two sentences.
In any case, these linguistic expressions simply seem to be about facts or states of affairs (or the philosophers’ truth conditions). Yet propositions aren’t deemed to be worldly or concrete things. That is, propositions have often been seen (from Frege to Quinton) as being abstract entities which don’t belong to time and space and which have no causal relations with the world.
So how do human minds gain access to these abstract propositions?
Concepts
Quinton moved on from using propositions as an argument against mere conventionality (or mere analyticity) to arguing exactly the same thing about concepts.
(Wikipedia defines concepts as “abstract ideas”. Yet an abstract idea is an abstraction that needn’t itself be taken to be an abstract object.)
So now it’s concepts which are seen as being non-conventional and, therefore, abstract entities. (To be more accurate, Quinton was putting the position of what he called the “anti-conventionalist”.) Quinton wrote:
“[T]he 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 (or Platonists) should move from propositions to concepts because, on Frege’s picture, senses are the non-spatial and non-temporal parts of non-spatial and non-temporal propositions. In Frege’s own words:
“[T]houghts are neither things of the outer world nor ideas. A third realm must be recognized.”
To use Frege’s own terms, senses are (non-spatial) parts of what he called Thoughts. And both senses and Thoughts are “eternal” (see here). In that case, again, a sense (though not identical to a concept in Frege’s philosophy) is simply an abstract part of a larger abstract entity — a proposition.
[See Frege’s own ‘The Thought: A Logical Inquiry’, from 1918. Also bear in mind here that propositions and Thoughts aren’t deemed to be identical in Frege’s philosophy.]
Yet problems remain.
If we can’t make sense of abstract propositions, then perhaps we can’t make sense of abstract concepts (or even senses) either.
For one, it seems utterly bizarre that Quinton should have argued that the concepts (to use Quinton’s spatial metaphor) “behind” the terms “bachelor” and “unmarried man” are actually “timeless and objective”. Really? Perhaps it can provisionally be accepted that an abstract, timeless and objective number lies behind the symbol, say, “2”. However, can we really say the same about the concepts behind the words “bachelor” and “unmarried man”? Surely not.
For one, the words “bachelor” and “unmarried man” have only existed since the institution of marriage began. And therefore the concepts themselves (if expressed by different words in different languages) must have only begun to exist when the first words about (or even institutions of) marriage existed. So surely none of these things are timeless, atemporal or objective.
[It must be assumed here that Quinton used the word “objective” to refer to those words which had abstract objects — in this case, abstract concepts — as their referents.)
Perhaps it can be accepted that the concept [bachelor] is both objective and abstract. However, is it also timeless? That’s unless it’s believed — and it often is — that timelessness must necessarily always come along with being abstract and also with being objective. Yet if abstractness and objectivity do come along with timelessness, then the concept [bachelor] may not be abstract or objective either…
Conventionalism
In the philosophy of mathematics, and even in mathematics itself, a link was made — roughly from the 17th century to the 19th century — between the conventionality of mathematics and the fact that mathematical statements are true simply because they assert identities. (See here for the basic mathematical position on identities or equalities.) In other words, the identities of mathematical statements (according to Quinton’s conventionalists) follow from the fact that mathematics is about symbolic conventions — not truth. This was — very broadly speaking — the position advanced by David Hume (see here), Henri Poincaré (see here) and then Ludwig Wittgenstein in his Tractatus (see here). (See also ‘Tautology’.)
However, perhaps mathematical conventionality and identities don’t go together. Alternatively, perhaps conventionality and the importance of identities can both 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, according to Quinton, Leibniz believed that mathematical statements — indeed truths — were (mere?) identities. That is, he believed that both sides of the equality sign state the same thing — though in different ways.
This can be seen as a deflationary view of mathematics and it has been almost universally rejected. (The rejection largely set in the early 20th century; though perhaps before.)
Quinton even touches on Kant’s “synthetic a priori truths”.
If mathematical statements aren’t identities, then does that mean that they’re synthetic a priori truths? That is, truths that are neither mere identities nor determined by the meanings of the symbols on both sides of the equality symbol.
It is still argued that such truths too can be known a priori. However, that’s not because they’re identities or simply because both sides of the equation state the same thing. Yet such truths can still be known — according to Kant — without (further) experience.
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