Reply to Hersh: On Platonism

Reuben Hersh is a well-known contributor to the debate on the philosophy of mathematics. He wrote a follow-up to Davies' article, also published in the EMS Newsletter (June 08). He seems to be in general agreement with Davies, but focuses on a different aspect of criticism of platonism, namely the social-construction view of mathematics.
Already in the first paragraph, Hersh declares that most practicing mathematicians have a "rough-and-ready, naive" view of the philosophical underpinnings of mathematics. This is a very typical comment which mathematicians have heard from philosophers so many times that they finally stopped listening. It is a rather arrogant attitude to claim that people who spend a lot of time reflecting on various arguments for or against ideas, have a more developed and mature view of things than people who deal with the actual objects in question on a day to day basis. Since Hersh has written much about the importance of founding philosophy on the experience of mathematics, one might have hoped that he would pay specific attention to the experience of mathematicians who de facto experience mathematics closer and deeper than anybody else. Unfortunately, Hersh has chosen the view that most mathematicians do not understand what it is they are working with, and are philosophically naive and illusioned.
This type of attitude of philosophers towards mathematicians has been very damaging for the relations between philosophy and mathematics in the past, and one may hope that the future of the philosophy of mathematics will be more interested in a dialogue with mathematicians, rather than continuing the tradition of dismissing mathematicians as philosophically naive.

Hersh chooses to reject Platonism on the grounds that it is incompatible with the materialist outlook of reality, a view whose correctness he takes as self-evident and unquestionable. This type of view is a relic from the positivist era, and as such has lost much of its power since the days when it was at its most popular. Nowadays there are many rational scientifically-minded people who are not materialists, and we now know that materialism/mechanism/reductionism and so forth are not proved by science, but in fact just assumptions that one may or may not hold in connection with a scientific view of the world. One simply can't base an argument against platonism on a premise that lacks any proof or self-evident qualities.
One problem with Hersh's social-construction view of mathematical reality is that it could just as well be applied to, for example, physical science. If mathematical objects are social constructions, then why are not the objects of physics, such as planets, forces, energy, also social constructions? Despite the social aspects of science, its 'corporeal ground', its postmodern critique, deconstruction, and whatever, almost no sane person will deny that planets are really 'out there'. In the same way, a social-construction view of mathematics cannot necessarily imply that the platonist 'out there' view must be rejected. Of course there are cultural aspects to any human activity, but most people still hold that there are matters of fact beyond the cultural sphere. These facts are relations between objects existing independently of human culture. What this comes down to is the much repeated observation that a social-construction view of mathematics can't be used as an argument against platonism, it can only be used as an alternative after one has chosen to reject platonism.
The article also mentions the problem of mathematical 'intuition' and how, if the platonic realm exists, humans can obtain knowledge of it. Here Hersh mentions the curiously irrelevant observation that blood flow through various parts of the brain can be correlated to mathematical thought. As I have written before in the post on Davies' article, this is completely irrelevant as to the existence of a platonic realm, but again, it can be used to fill the vacuum after one has chosen to reject platonism. Hersh completely agrees with this when he writes:

From this point of view, the facts presented by Davies are relevant and interesting. They do not refute Platonism, they are part of the scientific program that one focuses on after rejecting Platonism.

In spite of Hersh being (rightfully) sceptical of Davies' line of argument in rejecting platonism, Hersh agrees with Davies in dismissing the possibility of a mathematical intuition interacting with a platonic realm:

It seems that Davies regards the evidence that thinking takes place in the brain as proof that there is no such “intuition”, in the sense of a special mental faculty for connecting to “out there.” But with or without neurophysiological evidence, it is pretty clear that the posited “intuition” is an ad hoc artifact, lacking any specificity or clear description, let alone empirical evidence.

The lack of specificity or clear description of the interaction between the mind and the abstract objects of platonism is no argument for or against the existence of such a function of the brain.
As I described in the post on Davies' article, we have no specific or clear description of how memory is stored, or how consciousness interacts with it or with other faculties of the mind. Does this provide a basis for rejecting the existence of stored memories in the mind? Obviously not. It is probably a risky and possibly not very useful strategy to limit one's philosophical positions to what science can tell us at the moment. Philosophy is at its most fruitful when it is not pretending to be science.
Hersh formulates his materialistically based conclusion despite admitting the lack of the proper empirical evidence asked for above:

The mental, social and cultural, including the mathematical, are grounded in the
physical – the flesh and blood of past and present humans, especially mathematicians. We can recognize this, even while the detailed nature of this grounding – just how our thoughts are carried out by our brains – may
never be completely understood.

If one has chosen ontological materialism, then Hersh's arguments are a natural and rational way to proceed from there. However, this position has some major disadvantages. Firstly, as we have already observed, it demands that one ignores the overwhelming consensus and experience of practicing mathematicians, and therefore alienates a whole group whose opinion is most relevant for the matter. Secondly, one looses the possibility of a good explanation of how mathematics can be so universal, stable in time, and above all, utterly useful for understanding the physical universe. Hersh dismissively calls this a puzzle (no doubt, he chose this word consciously, rather than the word 'problem'), while in fact it is a major problem and a great mystery (to use Wigner's words), a mystery whose only satisfactory explanation to date is platonism. It is nowhere nearly enough to say that all human cultures can observe and count pebbles or fingers, and that therefore they develop the same mathematics, and as long as we can see pebbles and count our fingers, mathematics will be universal. The thing is that the universality and timelessness of mathematics is manifest on a much higher level which cannot simply be reduced to counting pebbles. Examples of this include the remarkably congruent and simultaneous but independent discoveries of infinitesimal calculus (Leibniz and Newton), and non-Euclidean geometry (Lobachevski, Gauss, Bolyai), respectively. An example of the 'mysterious' effectiveness of mathematics is the application of group theory (which started as an area of pure mathematics) to physics.
If one adheres to platonism one must explain how our minds interact with the platonic realm. If one rejects platonism one must explain why mathematics is culturally and geographically universal, tolerant to the test of time, and why it is that mathematics is so efficient in describing our physical universe, even when it was not originally conceived for this purpose! These two are both very difficult problems, but given that we go beyond scientism and materialism, we have a chance of finding good solutions to them. I claim that reality does not simply consist of things that are empirical on the one hand, and things which are socially constructed on the other. This type of ontology (empirical objects + social constructions) is of course what remains after the two big trends in Western thought, exemplified by positivism and social constructivism, respectively. The problem is that both of these two trends ignored, or did not provide an adequate place for, a fundamental domain of intellectual pursuit: mathematics.