Does mathematics pose problems for physicalism?

For those who want to see the world as being comprised at a fundamental level entirely of such processes as physics and the other sciences might model or describe, mathematics seems to pose problems.

Some say that our moral sense, or love and emotion, or our perception of beauty somehow undermine a physicalist viewpoint. But they don't really. All these things can be understood as complex products of simpler physical processes (evolution, biology, social interaction, etc.).

And the realm of the mystic may be timeless, but is subjective – or at least cannot be shown to have an objective existence.

But mathematics seems to take us into a non-empirical but demonstrably objective realm.

Mathematics works in many ways like a branch of science (and of course is an inextricable part of science), but is essentially concerned with abstract patterns and relations rather than with the empirical world directly.

Nonetheless, as a human activity even pure mathematics is clearly a part of the empirical world.

In fact, as digital computers become more fully integrated into mathematical research and practice, and discrete mathematics continues (as I suspect it will) to replace continuous as the basis for our most plausible and accurate descriptions of reality (since physical reality at its most fundamental levels appears to be discontinuous), the view of mathematics as timeless and Platonistic will most likely fade.

Be that as it may, for various reasons many still adhere to a full-fledged mathematical platonism, and see the existence of mathematics and our access to mathematical truths (which on some views gives rise to what has become known as the 'access problem') as evidence of the inadequacy of physicalism; even sometimes as evidence for a religious or spiritual view of reality.

There are, of course, many competing philosophies of mathematics, some of them (like platonism) realist (in the sense of accepting the real existence of abstract mathematical objects), others anti-realist. In general, the former approaches seem more or less incompatible with physicalism, and the latter compatible.

As there is no scientific way of deciding between these approaches, the physicalist is really under no pressure. He or she can just point to one or other of those ways of seeing mathematics which do not entail accepting independently-existing abstract objects, etc.

I have alluded in the past to the anti-realist views of the mathematician Timothy Gowers. And I have just come across someone else whose views appeal to me.

Sharon Berry, who has recently completed work on her Ph.D. at Harvard*, is not an anti-realist or anti-platonist like Gowers, but her approach is basically empirical, and the platonism she countenances is sufficiently weak not to put me off too much.

Her dissertation is on the so-called access problem which she addresses in what seems like a refreshingly straightforward and down-to-earth way. She argues that mathematical knowledge can be reduced to 'knowledge of a kind of broadly logical possibility, that is possibility with regard to the most general principles about how any objects can be related by any relations.' She calls this notion 'combinatorial possibility', and claims we can account for our knowledge of combinatorial possibility 'by appealing to general constraints on relationships between concrete physical objects.'

What is particularly interesting about the notion of combinatorial possibility is that it is tightly tied to the empirical world: 'one can infer possibility from actuality.'

Our access to good (but incomplete) methods of reasoning about combinatorial possibility is explained by our experiences with concrete objects, and so – if indeed mathematical knowledge can be reduced to a knowledge of combinatorial possibility – our (partial) access to mathematical truth is also explained in terms of these ordinary experiences.

Berry believes that her approach to the access problem meshes neatly with a relatively robust approach to claims about mathematical objects. 'The key idea is that quantifiers can take on different senses in different contexts. These senses correspond to different standards that we might apply when assessing questions of existence.'

Her view is that lower standards operate in everyday contexts than are required in discussions of 'fundamental ontology'. And lower standards apply also in mathematical discussions.

So long as these higher and lower standards are seen in pragmatic terms (and not in terms of different kinds of objects actually having different degrees of being), this approach seems doubly attractive. It does justice to the subtleties of human communication as well as avoiding the implicit dogmatism of standard realist and anti-realist stances.

Berry herself, taking ontological and metaphysical discourse in general rather more seriously than I am inclined to, may not be entirely happy with my pragmatic interpretation. But her views do certainly reflect empirical and pragmatic tendencies.

As I said, there is probably no way to decide which, if any, of the available positions in the philosophy of mathematics are on the right track and which are not. Some look more plausible than others, it must be said, but such judgments are always going to be affected by prior metaphysical (or anti-metaphysical) tendencies and such like.

Which is not to say that all work in this area lacks significance. Arguably, both Gowers's and Berry's perspectives have significance and value.

As I suggested above, so long as there are plausible positions available which do not entail fully-fledged realism or platonism, the physicalist need not feel that his or her physicalist stance is under threat.

And, so, though I don't feel obliged to make (or capable of making) an unequivocal assessment either of Gowers's or of Berry's approach, I do value them both as possible (and, on the face of it at least, plausible) alternatives to full-blown mathematical platonism.


* Both a short and a long dissertation abstract are included in her CV which is available via her website.

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.

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.