Saturday, 5 July 2014
Roger Scruton argues that the platonic position on the true nature of mathematics is basically based on four fundamental points. He lists them thus:
i) We know many arithmetical truths, and know them without a shadow of a doubt.
ii) Arithmetical truths are about numbers.
iii) Numbers are the subject-matter of identities, and indeed identity of number is one of the primary mathematical concepts.
iv) Truth means correspondence to the facts.
I suppose that I can provisionally agree with 1) above. We do seem to know arithmetical truths without a shadow of a doubt. We also intuitively believe that arithmetical truths are about numbers. I suppose if arithmetic is about numbers, then this aboutness implies that such numbers are not our own invention.
As for 3), numbers are essentially about identities. 1+ 1 = 2; thus 2 = 2. Or, more complexly, 2 + 2 = 4 itself equals 4 = 4.
Poincare said that the whole of mathematics boils down to a gigantic A = A – though only if one thinks that mathematics deals with tautologies!
Now 4). We say that the statement ‘snow is white’ is made true by the fact of snow’s being white. Thus the inscription ‘2 + 2 = 4’ is made true by the mathematical fact that 2 + 2 = 4. Mathematical statements correspond with mathematical facts. Indeed, each number has its own reference. The inscription ‘4’ refers to the number 4,just as ‘snow’ or ‘Tony Blair’ has a reference when embedded in a true sentence. (Though does ‘4’ also have a Fregean ‘sense’? Indeed does ‘2 + 2 = 4’ have a Fregean ‘sense’ or does it express a ‘Thought’?)
The upshot of all this is simple. We can now say that it is hard to accept 1) to 4) and still "deny that numbers are objects" (383). Of course many philosophers do reject 1) to 4) – especially 2) and 4). Wittgenstein, primarily, rejected the view that mathematics is about objects, correspondence and mathematical facts. This is a mistake. It is to conflate what is true about world-directed or empirical facts with what is true about mathematical propositions. They are not, in fact, the same, and for many reasons which Wittgenstein forcibly gives.
There is one final and often noted problem with the platonic position on numbers or mathematics generally. If numbers really are as Plato thought they are, then numbers "take no part in any change or process; they are causally inert" (383). That is why Plato put them in a transcendent realm. Though if they are causally inert, how do we gain access to them in the first place? Of course Plato had an answer to this. We ‘intuit’ them with our intellectual faculty. Is this really an answer or a bunch of bullshit?