Tuesday, August 28, 2012

Science, philosophy, ideology

Previously I have discussed* the implications of studies which indicate that a person's basic political (and religious) orientation is influenced greatly by genetic and early developmental factors. We generally only engage in political or religious debate because we have strong ideological or religious convictions, and those convictions set the general tone and direction of our contributions. Rationality comes in only later - to help us elaborate and defend that general position which feels so true to us (but strangely not to our antagonists).

These facts (as I take them to be) are rather inconvenient. It takes all the fun out of argument if one feels obliged to be skeptical towards one's own deeply felt convictions!

But on the plus side, it allows one (I believe) better to understand what is really going in much ideological, religious and philosophical debate.

In my previous post on this site, I touched on these issues, suggesting that platonists and anti-platonists in the philosophy of mathematics may be caught up in a debate which is superficially rational but ultimately driven by non-rational factors - deep convictions similar to religious or political convictions.

If progress is to be made in any of these areas, I think there has to be an acceptance that we are less rational than we would like to think; and so we need to depend more on scientific methods (which incorporate mechanisms to counter individual biases etc.), and less on convictions (or the elaborate arguments which we have built upon them).

A boring conclusion, I know. Especially for those of us who have strong convictions and a taste for argument and debate about the big questions.

Within the (rather ill-defined) area of philosophy, history certainly seems to indicate that arguments and debates are most fruitful (albeit somewhat constrained) when the dividing line between philosophy and science is blurred or non-existent, and most pointless and futile whenever philosophy is disengaged from science.

I recognize, however, that we are inveterately ideological creatures**, and there will always be a role for those who can identify, articulate and criticize the ideological frameworks we inevitably create and seek to live by.

Could this be what will replace the bits of philosophy which are not swallowed up by the various sciences: the scientifically-informed critique of ideologies?

* For example, here and here.

** There are problems with the term 'ideology', I know. I am using it in a very broad sense to mean something like a system of beliefs involving values and prompting certain forms of action, often in concert with others who share the ideology and sometimes in opposition to those who don't. It may be that I would do better to speak of us being inveterately tribal. 'Ideology' may be just the intellectual's (way of rationalizing) tribalism.

Sunday, August 19, 2012

Timothy Gowers and the philosophy of mathematics

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.

Tuesday, August 14, 2012

A strange form of amusement

If you look up an encyclopedia entry on the philosophy of mathematics you will usually find yourself presented with a list of competing approaches dating back to Plato. For someone of my temperament this is unsatisfactory. Well, it's fine having competing views, but which one (if any of them) is true?

With areas like ethics or art, the fact that there is no consensus may be explained by the very real possibility that these areas are largely subjective. But mathematics? Surely there is something that mathematics is, some broad understanding at any rate that we can agree on?

In the philosophy of mathematics there is a basic division between realists (or platonists) who believe that mathematical objects exist (but not in time and space); and anti-realists who don't believe this (seeing mathematics simply as a human activity, for instance).

Most mathematicians are thought to embrace some form of realism (mathematical truths are 'out there' to be discovered); but not all do. The distinguished mathematician Timothy Gowers is an anti-realist (aligning himself with Wittgenstein in this matter).

This basic division is just the beginning, however, as there is (as in just about any area in which philosophers are involved) a proliferation of arguments and counter-arguments resulting in an ever-proliferating list of divisions and subdivisions and so of positions to attack or defend.

Amongst which is the gloriously named neo-Meinongianism. (Could anything so called actually be true?)

A part of me says: 'Steer clear of all this, my dear fellow. Life is too short. And it's not about what it seems to be about. In part it's merely a perverse, self-perpetuating amusement for philosophers, in part an attempt by serious, religiously-inclined thinkers to defend a metaphysico-religious (or should that be religio-metaphysical?) view of the world.'

Unfair, no doubt. What of the serious, non-religious and/or anti-platonistic participants? Like Timothy Gowers.

In fact Gowers has himself asked the question of whether mathematics needs a philosophy, and I find his thoughts on the matter very persuasive.

I will probably not be spending a lot of time researching this area, but I would like to follow up on Gowers's views.*

And also, I intend to have a closer look at neo-Meinongianism. Why not?

* Gowers is a very political (and influential) figure within the mathematical and broader intellectual community and seems to have some fairly radical views about the ownership and distribution of ideas.