Following on from my previous post, here are a few more thoughts on the philosophy of mathematics and on Timothy Gowers's views.
On rereading his talk, I thought the section on what it could mean for 2+2 to equal 5 in an alien world to be too clever by half and ultimately unconvincing. (Gowers virtually concedes this himself, so why include the material in a short talk?) It reminded me of Wittgenstein's equally unconvincing (to me) arguments about deviant forms of counting.
And then there is Gowers's politics, including his role in the campaign against the scientific publisher, Elsevier. (Wittgenstein too had strong moral and social convictions - including a conviction that ideas should not be 'owned' - but, unlike Gowers, he was, so far as I know, never an activist.) I am not aware of Gowers speaking anywhere of his general ideological or political position. I would guess that his views are left-leaning, but I don't know for sure, and I don't know what effect (if any) his general political and moral views might have had on his views on the philosophy of mathematics.
Which brings me back to the main point of the previous post: there seems to be no objective way of deciding on the truth or otherwise of the many and various options within the philosophy of mathematics. So when someone shows a strong commitment to a particular view, I wonder whether extraneous factors - such as ideology - might be playing a role.
I should also say something about neo-Meinongianism, having raised the topic in my previous post. Needless to say, it's complicated, but the general gist of it is something like this. Neo-Meinongians think that you can make statements about mathematical objects (like numbers) without being committed to believing they exist. They would claim that the statement that there are infinitely many prime numbers is literally true even though numbers may not exist as such. This view is based on the (I think) plausible idea that expressions like 'there is' or 'there are' are used in various ways and do not necessarily entail any ontological commitment.
As I understand it, (non-neo) Meinongianism is supposed to countenance gradations or different kinds of existence or being, whereas neo-Meinongians claim that certain uses of expressions such as 'there is' involve no ontological commitment.
In fact, Gowers makes a somewhat neo-Meinongian point in his talk, endorsing Rudolf Carnap's distinction between internal and external questions, only the latter (possibly) involving ontological commitments. So, on this view 'there are two senses of the phrase "there exists". One is the sense in which it is used in ordinary mathematical discourse - if I say that there are infinitely many primes I merely mean that the normal rules for proving mathematical statements license me to use appropriate quantifiers. The other is the more philosophical sense, the idea that those infinitely many primes "actually exist out there". These are the internal and external uses respectively.'
Or again, Gowers writes: 'One view, which I do not share, is that at least some ontological commitment is implicit in mathematical language.'
I'm not sure where I stand on all this. I have doubts about the worthwhileness of much of what goes on under the designation of 'philosophy of mathematics', but see some issues which seem real and important. Though I am certainly drawn to the position outlined by Gowers, I am acutely aware that this general orientation may be, like general religious or political predispositions, largely a function of inherited or early-environmental factors.
It seems to me that platonistic or anti-platonistic intuitions lie behind most philosophical work in the area. But what are these intuitions worth if they are in large part the result of arbitrary genetic and developmental factors?
The fact that little progress has been made in answering apparently real and interesting questions in the philosophy of mathematics and related areas suggests to me that the standard, traditional philosophical approaches are somehow flawed. Or perhaps the questions are ill-conceived, based on an inadequate understanding of the intellectual disciplines in question.
In fact, the advent of digital computers and new ways of conceptualizing information and information processing is changing the way we see mathematics (and much else besides). As new ways of doing and looking at mathematics and science emerge, questions that once seemed meaningful and important may no longer seem so.