Long time ago I read all six volumes of the Swedish edition (called Sigma) of The World of Mathematics. Last spring, when I dug out volume 6 from the bookshelf it fell open on Oswald Spengler’s Meaning of Numbers. A few weeks later I found the American edition in a nearby antiquarian bookshop (Röde Orm on Husargatan 18D in Haga, Goteborg, for locals or if you happen to pass by. At the time of writing the present text, there is one more copy to be found there.).
Since I now have started to write a report on the project I’ve also decided to do some reading. I’ll start with Leslie A. White’s article The Locus of Mathematical Reality: An Anthropological Footnote vol. 4 pages 2348-2364 (vol. 6 pages 2441-2458 in the Swedish edition).
White (1900-1975) who was an American anthropologist, addresses the question of where the mathematical concepts and truths reside. Do they belong to the external physical world or are they human mental constructions? His text is interesting in many ways. He refers back to earlier discussions about this issue and he is very eloquent about his own point of view: mathematics is purely a cultural phenomenon.
Another way to phrase the question is to ask whether mathematics is discovered or invented. White reviews how mathematicians have the feeling that they are discovering something which is external to themselves, citing G.H. Hardy as an example (among others). On the other hand, it seems just as clear that mathematical concepts are human inventions. So the answer would be that mathematics is both discovered and invented. (Perhaps one could say that the concepts are invented whereas the truths are discovered?) He clearly renounces any notion of a Platonic abstract realm where mathematics reside.
White’s answer to the dichotomy of discovery vs. invention is to claim that mathematics is a cultural phenomena. This sounds reasonable, but he does it in an anthropological framework which to me takes it a bit too far. When thinking of mathematics as a mental phenomenon there are of course two senses to the concept of a mental: mental concepts in the individual human being and shared mental concepts of the species. Here are some quotes from White (where the italics are mine in the following quotes):
“What we propose to do is to present the phenomenon of mathematical behavior in such a way as to make clear, on the one hand, why the belief in the independent existence of mathematical truths has seemed so plausible and convincing for so many centuries, and, on the other, to show that all of mathematics is nothing more than a particular kind of primate behavior.”
Clearly it is too simple-minded to consider “external physical reality” and “internal mental reality” as the only possibilities for where mathematics could reside. The human culture is another possibility. White writes:
“Mathematical truths exist in the cultural tradition into which the individual is born, and so enter his mind from the outside. But apart from cultural tradition, mathematical concepts have neither existence nor meaning, and of course, cultural tradition has no existence apart from the human species. Mathematical realities thus have an existence independent of the individual mind, but are wholly dependent upon the mind of the species.”
This makes sense to me, but then he continues with:
“Or, to put the matter in anthropological terminology: mathematics in its entirety, its “truths” and its “realities,” is a part of human culture, nothing more. ”
It’s the nothing more part I don’t agree with. These quotes are from the beginning of the text. White then goes on to argue his case and much of it makes perfect sense, but he takes it too far. Even physical theories like Maxwell Electrodynamics and Einstein General Relativity becomes purely cultural in White’s view. Certainly, the detailed formulation of the theories of fundamental physics are dependent upon culture, but surely there is a basis in physical reality here – independent of human culture – or any culture anywhere. But to White, culture was a “super-human” entity that could only be explained in terms of itself. The role of the individual is reduced to null: discoveries are not made by individuals – it is something that happens to them. In this way, culture becomes almost meta-physical.
In more recent times, Reuben Hersh has argued for mathematics being a cultural phenomena. The argument is popularized in his book What is Mathematics, really? Another reference is his article Some Proposals for Reviving the Philosophy of Mathematics in New Directions in the philosophy of Mathematics (ed. T. Tymoczko). Hersh does not take the argument as far as White.
The arguments for mathematical concepts residing in the common human culture are very convincing, but does it explain everything? Still we have the Wigner problem of the “The unreasonable effectiveness of mathematics in the natural sciences” (Comm. Pure Appl. Math, 13:1, 1960). I cannot escape the feeling that there must be something in physical reality that serves as a basis for mathematics. A point of view very different from the “cultural basis view” is put forward by Roger Penrose in his The Road to Reality. Penrose seems to be a Mathematical Platonist and his discussion on the interactions between three worlds: the Platonic mathematical world, the Physical world and the Mental world in the first chapter is very intriguing. Where is culture in that picture?
Come to think of it, these questions also reminds me of Gottlob Frege’s struggle (The Foundations of Arithmetic) with defining numbers independently of individual mental states in order that mathematics not become a part of psychology.
Much more here to think about, obviously, and to write about in the report!