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.