When I was an undergraduate, I was on friendly terms (as I still am) with a retired professor in my department. This fellow was supportive of my mathematical endeavors (he wrote some of my recommendation letters for graduate school, for instance), but he was always encouraging me to explore my other interests (he himself had studied history before becoming a mathematician), and I always had the impression that he would have been more enthusiastic had I chosen to attend graduate school in another discipline, such as philosophy, where my undergraduate record was perhaps more indicative of promise (to a casual observer anyway). Now, since I already had enough trouble deciding which of my two declared major subjects (mathematics and music) to pursue "officially" at the graduate level, you might not think that adding a third option to the mix would have been particularly helpful. But there is at least one sense in which my friend's attempts to lure me away from mathematics were to prove not only helpful, but essential to my intellectual development (beyond merely testing my stubbornness): he was forcing me to work out in my mind the
reasons why I wanted to study mathematics. Just what, exactly, was it that was attracting a person like me, of such a "verbal" inclination, to a subject traditionally regarded as "non-verbal"? (Not that I would be the first such person so attracted:
Paul Halmos opened his memoir
I Want to Be A Mathematician with the sentence "I like words more than numbers, and I always did.") Or, how could the life of a scientist (of which a mathematician is a species,
in this context), subject as it is to the cold, hard facts of reality, possibly compete with the life of an artist (of which a musician is most definitely a species), who is in the business of satisfying his own fantastic desires? (Not that this alleged opposition actually makes any sense, of course...)
This process of "soul-searching" was (and is) a complex one, of course, and you will no doubt read more about it on this blog. But with regard to the relationship between mathematics and philosophy, and why someone with a philosophical turn of mind would want to study mathematics rather than philosophy itself, I can give something of an answer. I did so earlier today in an email to my retired mathematician friend, who, having recently run into a philosopher of our acquaintance, was, true to form, yet again raising the possibility of my switching fields. Here was my reply:
"It goes without saying that I am flattered by [the philosopher's] kind remarks. I am also amused by your persistence in trying to get me to study philosophy. However, you should realize that mathematics is philosophy: it's the branch of philosophy where philosophical problems actually get solved. Take, for example, the mystery of the nature of "space" and "continuity". Once upon a time, this was another perplexing conundrum, like consciousness or free will. But now we have the answer: "space" is when you have a collection of sets (which are called "open") that is closed under unions and finite intersections, and "continuity" means that the inverse image of every open set is open. Mystery solved! It is simple, unambiguous, definitive, and illuminating. The best part is, if you don't see how this resolves the problem, then you simply need to study some more mathematics and perhaps think a little harder. In particular, you don't publish a paper on how the 'Hausdorffian view' is 'mistaken' or 'problematic'. "
So there we have it:
mathematics is precise philosophy. In espousing this sort of view, I am (to my mind, at least) echoing
Scott Aaronson, who conceives of his field (theoretical computer science) as "quantitative theology". (Such a view yields the nice corollary that TCS is a subfield of mathematics!)
2 comments:
But there are in fact situations in which the so-called 'Haussdorfian' view is mistaken or problematic -- namely when you realize what a large class of spaces you're excluding when you impose the Haussdorf condition.
And the notion of a topological space hardly obviates philosophical speculation on the subject -- it's a purely mathematical structure invented precisely for its utility in formalizing the fundamental ideas behind continuity, compactness, etc. It was not motivated by some definitive physical understanding of the space we exist in -- which we still don't understand (we don't even know whether it's discrete or continuous at its most basic). That very recognition invites a hoard of philosophical inquiry: how do we develop mathematical intuitions about space and what are the implications of those intuitions for our ability to study the space we exist in?
All this is to say that mathematics is NOT the "precise" limit of philosophical investigation -- the analytics would certainly never admit that. There's nothing imprecise about the application of logic. Mathematics is merely concerned with different kinds of problems -- those dealing with quantity, shape, and structure.
If there's something that distinguishes mathematics from philosophy (besides this difference in the types of problems confronted), it isn't the rigor or precision by which it logically proceeds: it's the ability to work purely abstractly without the messier human details. Axioms in philosophy so often carry the burden of compatibility (or incompatibility) with someone's fundamental beliefs. An entire philosophy can change if one applies a different vocabulary or takes a different perspective, and it's hardly ever possible to agree on first principles. Mathematics on the other hand clearly lays out its first principles, so that whereas changes in point of view often undermine philosophical inquiry (say the belief that god is or is not omniscient), they are precisely what strengthen mathematics (Poincare's idea of giving different names to the same thing, e.g. functors).
But there are in fact situations in which the so-called 'Haussdorfian' view is mistaken or problematic -- namely when you realize what a large class of spaces you're excluding when you impose the Haussdorf condition.
To clarify: in this context, the "Hausdorffian view" refers to the abstract definition of topological space, not to the T2 separation axiom.
Of course we don't yet fully understand the nature of the space we live in. But notice that the question is now "merely" what kind of space the universe is; the intrinsic concept of "space" has been successfully analyzed. This, in my view, is where philosophy ends and science begins.
What I will concede to you is this:
If there's something that distinguishes mathematics from philosophy (besides this difference in the types of problems confronted), it isn't the rigor or precision by which it logically proceeds: it's the ability to work purely abstractly without the messier human details.
That, indeed, is the point. It's not that a philosopher couldn't have proposed the axioms of topology; philosophers are certainly capable of this type of precision and abstraction. The difference lies in what the response to the proposal by fellow philosophers would have been. Instead of developing the body of results that now constitutes the subject of topology, I suspect that they would have dismissed the axioms on the grounds that they admit objects as spaces that "shouldn't" be so considered. They would have complained, for example, about the fact that a general topological space need not be Hausdorff (i.e. T2), whereas the universe in which we live in "clearly" is. This would have been considered an "objection" to the theory itself; whereas for the mathematician, it just serves to locate the world within the theory (i.e. the universe would be modelled by a space that is T2, rather than one which isn't).
Mathematics isn't necessarily more precise than philosophy; it happens to be that way because philosophers are less willing than mathematicians to take a precise formulation and run with it, because they are afraid it might not match "intuition". The mathematical response would be to say: get over it--your intuition needs to broaden.
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