ABSTRACT
For me, and I suppose for most mathematicians, there is another reality, which I will call ‘mathematical reality’; and there is no sort of agreement about the nature of mathematical reality amongst either mathematicians or philosophers. Some hold that it is ‘mental’ and that in some sense we construct it, others that it is outside and independent of us…I will state my own position dogmatically…I believe that mathematical reality lies outside us, that our function is to discover or observe it, and that the theorems which we prove, and which we describe grandiloquently as our creations, are simply our notes of our observations. This view has been held in one form or another by many philosophers…from Plato onwards…(Hardy, 1941, pp. 63-4)
Although Hardy, in the above passage, is careful to allow the possibility that mathematical reality is ‘mental’ in that we ‘in some sense construct it’ his very usage of the term ‘mathematical reality’ implies a Platonic view; and when later he says ‘A mathematician…is working with his own mathematical reality’ (ibid., pp. 69-70) he is careful to maintain what he calls a ‘realistic’ view of mathematical reality: mathematical reality is independent of how our minds are made.
