Mathematical Knowledge: A Dilemma

[Philosophy Now]

Rui Vieira thinks of a number.

http://philosophynow.org/issues/81/Math ... _A_Dilemma

[Jonathan.s]

If mathematical objects such as numbers, perfect circles exist as abstract entities beyond space and time, neither affected by nor causing anything in our everyday world, as Platonism asserts, then how is it that we can come to have knowledge of them?

The problem that many people have is in understanding "where" the perfect realm in which such things as numbers and forms are to be found might be. Heaven, perhaps? Of course such notions are all rejected by modern philosophy; the article notes that empiricism has made great efforts to ‘to bring mathematics back down to earth from its heavenly perch’.

But perhaps this is because the word 'realm' is really a metaphorical description. After all in talking about the nature of number, we are talking about something comparable to language itself, or to thinking itself. So this makes it a very hard thing to talk about. Discussing the nature of number is completely different to discussing the composition of stars, or the migratory patterns of birds, or the causes of an epidemic. It is a discussion of a completely different kind. So we struggle to find metaphors and analogies for facts of this kind; and 'realm' is one such metaphor, because 'the world of number' does indeed appear to be a whole world unto itself.

A well-known platonist was Frege:

Frege believed that number is real in the sense that it is quite independent of thought: 'thought content exists independently of thinking "in the same way", he says "that a pencil exists independently of grasping it. Thought contents are true and bear their relations to one another (and presumably to what they are about) independently of anyone's thinking these thought contents - "just as a planet, even before anyone saw it, was in interaction with other planets." '

Furthermore in The Basic Laws of Arithmetic he says that 'the laws of truth are authoritative because of their timelessness: "[the laws of truth] are boundary stones set in an eternal foundation, which our thought can overflow, but never displace. It is because of this, that they authority for our thought if it would attain to truth."

(Frege on Knowing the Third Realm Tyler Burge, Mind, New Series, Vol 101, issue 404, Oct 1992)

So this is typical of the platonist approach - also mentioned in the article on Gödel - to see numbers as real, but not real in the same way as material objects. Of course this poses problems for materialism which insists that only matter is real. But it is indubitable that many mathematicians and mathematically-inclined philosophers tend towards some variety of platonism.

That said, it is easy enough to use number, to discover many extraordinary things that we would otherwise have no way of knowing; the discoveries of mathematical physics are replete with them. It is also demonstrably the case that many of these discoveries have been made because of the qualities of number, such as symmetry, which in some cases have led to very counter-intuitive discoveries about the nature of matter (Eugene Wigner won a Nobel for discoveries in the nature of atomic structure that were due in part to the exploration of mathematical symmetries.)

Here is another way of conceiving of Frege's 'Third Realm': that the reality expressed by number is constitutive of the world. That is, such things as perfect shapes, real numbers, and logical laws, are inherent or implicit in the structure of reality itself. They are 'the way the world works'. They are the tendencies within nature for things to exist or to develop in a particular way. They may not be causal in any direct sense, but they act as constraining factors in the way things develop.

Furthermore, they are also constitutive of the way that thought itself operates. They reflect the nature of reality - but at a deeper level than can be understood purely in terms of sensory experience. The Greeks believed this underlying order was the 'logos', which was the origin of the very idea of logic. (Perhaps also we can ideas like this in the notion of the 'universal grammar' of Chomsky.)

So this ‘realm’ that Platonism refers to might actually be ‘the formal realm’. And the ‘formal realm’ is not anywhere. It is not something that exists in its own right but which precedes existence. It is, if you like, the realm of the possible - that from which the actual is condensed, as it were.

Not for nothing is it called 'the realm of ideas'.

For Kitcher, mathematics consists not of abstract Platonic entities, but of generalized human empirical operations performed on physical objects: such as the collecting, correlating or segregating of, for example, pebbles. For instance, the mathematical statement 2 + 3 = 5 would, according to Kitcher, first refer to the collecting operation performed on objects called ‘making two’, then to the collecting operation of ‘making three’, and then to the final combining operation of ‘making five’. Higher mathematics would then be developed from these primitive perceptual beginnings, eventually allowing us to perceive the world’s empirical structures through mathematics

But it is nevertheless the case that only human beings demonstrate the ability to grasp anything like ‘higher mathematics’. You could sit in front of your African Gray Parrot for centuries, showing him flash cards and patiently talking to him, and he still would never get the concept of ‘prime number’. This is why the Greeks called man the Rational Animal; and rationality is a difference in kind to the abilities possessed by non-rational intelligences.

Anyway, Kant demonstrated that the structures of logic which organize and interpret abstract observations were built into the human mind and organized the nature of experience a priori. The empiricist might counter that we believe certain things to be true because we have enough individual instances of their truth to generalize about them. But there is no way around Kant's a priori logic. You might say that the very principles of logical deduction are true because we observe that using them leads to true conclusions - but this depends on our ability to recognise a true proposition when we see one. Again, it goes back to the fact that ultimately, there must be such a thing as an ‘innate truth’, something which the rational ability in man is able to comprehend.

Just as Plato said.