Naturalism in Mathematics?

3 06 2005

Warning: the views expressed here are caricatured representations of papers that I’ve read through somewhat, but not carefully enough yet. They’re probably not actually the views of the philosophers mentioned.

Penelope Maddy makes a distinction in the philosophy of mathematics between questions that are “methodological” and ones that are “philosophical”. On her preferred view (which she calls “naturalism”), it is primarily the former that are actually interesting questions. The idea is that questions that tell us which sets of axioms to pursue are useful to actual mathematical practice, whereas questions about how we can know that 2+2=4 don’t seem to be. (That is, if they tell us that we don’t know this fact, then we can’t really do any math, and if they tell us that we do, then we’re right back where we started.) However, these methodological questions are principally the domain of mathematicians and “naturalistic philosophers of math”, who have basically become real mathematicians without the occasional philosophical worries. In a sense, this would suggest that philosophical questions properly stated can’t be relevant to mathematical practice.

However, during FEW I came up with what seems to be a reductio of this position:

1. If mathematicians care about question Q, then the answer should be relevant to their practice. (This seems clear if we’re talking about mathematicians caring about the question qua mathematician, and if they’re doing their job right.)

2. Philosophical questions can’t be relevant to mathematical practice. (This is my caricature of Maddy’s naturalism.)

Therefore,

3. No question mathematicians care about is philosophical.

However, this seems clearly false. I recently stumbled upon an article in the 1994 Mathematical Intelligencer by Bonnie Gold that basically just lists dozens of philosophical questions about mathematics that are interesting to mathematicians (and in many cases, much more interesting to mathematicians to philosophers). For example, “What is common to those subjects (e.g., algebra, analysis, topology, geometry, combinatorics, category theory) that are classified as mathematics which causes us to classify them, but not other subjects, as mathematics?” Or “Is mathematical knowledge more sure than other forms of knowledge? It seems so …, but is that simply an illusion?” Many of the other questions in the article seem more directly sociological than philosophical, but these ones seem philosophical to me. And I don’t think she’s being derelict in her duties as a mathematician to wonder about these questions.

I think premise 1 seems pretty clear. If we were to come up with a good classification of some subjects as mathematics and some as not, it seems that this would provide some insight into useful methods for mathematical work, and ideas for new fields of mathematics that haven’t been studied yet. And if mathematical knowledge can be shown to be no more sure than other forms of knowledge, this would increase the importance of alternate proofs and double-checking in mathematics. So the fault must lie with premise 2, which I’m guessing is therefore a misreading of Maddy’s position. Perhaps it’s because she’s using “philosophical” as almost a technical term, separating off all the “methodological” questions from it. If philosophical questions are just defined to be the ones that have no relevance to the working mathematician, then there’s no problem. But the questions mentioned above do strike most people as at least somewhat philosophical.

I think the real answer is that I’ll have to re-read some of Maddy’s more recent papers, and actually read her book “Naturalism in Mathematics”. I think she’s right to reject at least some seeming questions in the philosophy of mathematics, but it does generally seem to me that she goes a bit farther than I’d like.

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4 responses

4 06 2005
Peter McB.

I’ve not read Gold’s paper, so this response may be redundant: But surely what connects the subjects you list (algebra, analysis, etc) is that they are all topics which mathematicians explore. In other words, mathematics is what mathematicians do, no more and no less.

Such a socially-constructed definition of mathematics means that the subject will change over time. A typical 19th-century British mathematician reincarnated in 1980 would be very surprised to see the emphasis given to non-numerical and non-quantitative reasoning, eg category theory, and the almost complete lack of attention paid to studying specific infinite sums and sequences of numbers. And he (such a 19th century mathematician would be unlikely to be female) would be very surprised to find all the attention paid to topics considered on the lunatic fringe of the discipline in 1880 — formal logic, for example. Babbage, Boole, de Morgan, Keynes pere, et al, were not working in the mainstream of 19th-century British mathematics.

4 06 2005
Kenny

Well, that certainly seems to be one thing linking these disciplines. But it seems plausible that there is something more. After all, many mathematicians juggle, solve Rubik’s cubes, and play violin, but none of those activities is generally considered to be part of math. (There are mathematical studies related to the study of each of these activities, but the actually activity of juggling, solving a Rubik’s cube, or playing violin is not mathematics.)

And formal logic, from what I can tell, is still almost considered a lunatic fringe of the discipline. There don’t seem to be many set theorists outside of California, except at Wisconsin, CMU, Rutgers, and in Israel. I suppose the difference now is that these places are highly regarded, while Queens College and University College London (where Boole and de Morgan were) were not then.

5 06 2005
Peter McB.

I think you are right, Kenny, that most mathematicians still regard logic as being on the lunatic fringe of the subject. So is it mathematics? It has an AMS (American Mathematics Society) classification, so some powerful mathematicians clearly think so. However, it is still not taught as standard fare to mathematics undergraduates in most Universities.

The three things you mention (juggling, solving rubik’s cube, playing the violin) are all physical activities, not the elaboration of formal theories (mathematical or otherwise) transmitted in written form. While some branches of mathematics are usually taught through physical actions (eg, Euclidean geometry, category theory, game theory), the overwhelming bias of the discipline is for written transmission of theories, as it is indeed for our entire culture.

So let us look at the written transmission of formal theories of juggling, playing Rubik’s cube, and violin playing. On juggling, we have a book, “The Mathematics of Juggling” by Burkard
Polster (2003), published by Springer in their Mathematics series. The AMS classification would be: Combinatorics (05-xx). On Rubik, there are several books, for example, “The Mathematics of Rubik’s Cube”, by W. D. Joyner, or, “Mathematics of the Rubik’s Cube Design”, by H. M. Bizek. The AMS classification would be: Combinatorial choice problems (subsets, representatives, permutations) (05Axx:05A05). And the study of the mathematics of music and sound has a history that goes back at least to Pythagoras. The relevant AMS classification would be Hydro- and aero-acoustics (76Q05). Elaborating formal theories about each of these three activities is therefore considered mathematics.

Lest you think this is fanciful, note that there is serious research on the mathematics of origami — see here, for example. As yet, there appears to be no specific AMS classification for this work, although it would fall into one or more sub-parts of Geometry (51-xx). That there is yet no specific AMS classification for the mathematics of origami just reflects the fact that the mathematicians doing this work are not yet sufficiently numerous or powerful enough to revise the AMS classification. They may be insufficiently numerous because origami problems are not yet considered important by our society — although, as it happens, the work on origami turns out to have application for the optimal design and packing of spacecraft. If the AMS classification had been devised by 18th-century Japanese mathematicians — when, the study of similar geometric problems was popular well beyond the intelligentsia — the classification would no doubt be different.

5 06 2005
Peter McB.

My embedded link to the “Mathematics of Origami” page did not work. Here is the

http://www.merrimack.edu/~thull/OrigamiMath.html

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