Mathematics as a Natural Language

6 06 2007

My friend Luke Biewald pointed me to an interesting post suggesting that the language of mathematics is actually best viewed as a natural language rather than a formal language. I think some of the points the author makes about math don’t separate natural and formal languages (recursion, self-reference, alphabet and rules for combination, and so on). And I think that since he works in natural language processing, he may be thinking of being natural as a sort of simplicity rather than complexity (as I would think of it). But it’s an interesting point.

Mathematicians basically never write fully formal proofs in the sense that logicians like to talk about. They regularly “abuse notation” and overload symbols in order to simplify their way of speaking. Many of these changes are in fact quite historically contingent – if we hadn’t originally started abbreviating things one way, further developments that way would have looked incomprehensible to a community that had abbreviated things differently.

Given the fact that it is actually used by a relatively large community for certain essential (to those people) types of communication, it has most likely picked up a lot of the irregularities and “irrationalities” that plague natural languages – probably much more so than constructed languages like Esperanto and Klingon. I don’t know what all the relevant differences are between “real” natural languages, Klingon, and math, but they may help reveal something interesting about at least one of these languages.





Chmess

8 05 2007

Jason Stanley has a post on the Leiter Reports discussing the problem of “Chmess”. As he puts it,

Chmess, for those of you who haven’t heard, is just like chess – except the king can move two squares in either direction. As Daniel Dennett has pointed out, Chmess provides a rich source of a priori truths to explore. However, the a priori truths of chmess are not particularly worthy of exploration. Dennett’s challenge of Chmess is to explain the difference between describing the a priori truths of Chmess and practicing philosophy.

He says that the sciences don’t face the problem of Chmess because they are clearly investigating an independently existing body of truths. He suggests that for the humanities, the problem doesn’t even arise, because they don’t see themselves as investigating a body of truths, but rather as changing people’s lives. I think he suggests that something like the latter is the right way to avoid the problem for philosophy as well:

In philosophy, as much as any other discipline, one engages in the practice of investigating alternative conceptual structures, be they systems of value or systems of belief. Such investigations may seem like theorising about Chmess, if we conceive of philosophy as a science. But [if] we regard philosophy instead as an activity intended to help those who learn it acquire the resources to lead a dignified life, it quite clearly does not.

However, I’m not convinced that this is the right way for philosophers to avoid the problem. (Admittedly, I don’t know what Dennett’s preferred solution is – he must disagree with Stanley here because of his scientific picture of philosophy.) To see that another solution must clearly be available, consider the case of mathematics. Working on Chmess problems bears only a very distant resemblance to working on science or the humanities, and a somewhat closer relationship to philosophy (especially when characterized as the study of a priori truths). However, many mathematicians explicitly claim that mathematics is just the study of what can be proven in arbitrary abstract formal systems, and many further claim (echoing G. H. Hardy’s famous A Mathematician’s Apology) that there is absolutely no use to any of their work and they only do it for their own amusement.

If that were really the case about mathematics, then the problem of Chmess would be extreme. Why shouldn’t a top university hire someone who was an incredible genius and had made all sorts of deep and subtle discoveries about Chmess?

My thought is that all mathematics is applied – if not to the physical world, then at least to other mathematics. There is a deep sort of interconnectedness between all the disciplines of mathematics. Chmess is unlikely to have these connections, just like almost any other arbitrary formal system, including Chess (though Go may have helped lead Conway to the surreal numbers). Mathematicians only work on problems that are relevant to other problems that already have established interest. If a certain formal system arises in multiple contexts, then that system will become of interest in its own right.

In philosophy, if one wants to take a “scientific” sort of view, one can see the point similarly. Unlike the other sciences, philosophy encourages discussion of non-actual systems in order to uncover their structure. However, these systems will ideally have some connection to actual systems, and hopefully will illuminate some of the relevant issues. And although philosophy very often deals with non-actual circumstances (twin earth, Gettier, trolley problems, etc.) the only cases I can think of where people really deal with formal systems that are not intended to describe something actual are in analytic metaphysics, or when they are specifically trying to disprove someone else’s theories.

So I think Jason Stanley’s argument is far from conclusive in establishing that philosophy should not be thought of as a science, but rather as something like a therapeutic activity. (I know that’s not what he said, but I can’t resist comparing him to Wittgenstein.)





Mathematical Writing

15 04 2007

This recent post at The n-category Cafe reminded me of an idea for a post I had a while ago. It was partly inspired by these three posts at Adventures in Ethics and Science, but more by a comment a friend of mine made at the math department tea. He said something like the following:

Mathematicians should have some sort of publication that’s halfway between a talk and a paper. The paper is the authoritative source to go to for reference, but the talk is much more effective for actually learning what’s going on. In particular, rather than giving the standard sorts of unreadable definitions that mathematicians normally give, they should give the general motivation for a concept, and the two or three counterexamples from which any competent worker in the field should be able to construct the same unreadable definition for themself.

Since the focus of my research isn’t in mathematics itself, or in the history of mathematics, I haven’t often tried to read journal articles in math, but the couple times I have it’s been really tough going. I don’t know if math is worse off than other scientific disciplines in this way, since I’ve never tried to read anything in any other field (except a couple papers in theoretical physics that are really more about philosophy). But it seems to be connected to the same issues that John Baez brings up in the first link above.

Part of what seems to make it so hard is that, not having worked out the proof oneself, it’s hard to follow what the author is doing, because you have no idea where each step is leading. The individual steps don’t add up to a coherent story. Something about this style is useful – when you do understand what’s going on it’s often easier to refresh your memory by reading the proof in the forwards order, as produced in the “context of justification” rather than the “context of discovery”. It’s tough to train students to write things in this backwards way, and there is some value to it. But it sounds like it’s a habit that one has to moderate to some extent as well, so that others can follow what’s going on.





Do Mathematical Concepts Have Essences?

24 01 2006

In John MacFarlane’s seminar today on Brandom’s Making it Explicit, the distinction was discussed between necessary and sufficient conditions for a concept and the essence of a concept. The distinction is roughly that necessary and sufficient conditions for the application of a concept doesn’t necessarily tell you in virtue of what the concept is satisfied. For instance, we have two extensionally equivalent notions – that of being a pawn in chess; and that of being permitted to move forwards one square, capture diagonally forwards one square, move two squares forwards in certain contexts, and so on. At first it might seem correct to say that the piece can move forwards one square because it is a pawn, but further reflection suggests that this would leave the notion of being a pawn unanalyzed. After all, a piece is not a pawn in virtue of its shape (we could be playing chess with an elaborately carved ivory set, or labeled checkers rather than standard chess pieces, or even patterns of light on a computer screen) nor by its position on the board (any piece could be in most of the positions a pawn could be in), nor most anything else. It seems that in fact, the reason it is appropriate to call this piece a pawn is because we give it the normative status of being able to move in certain ways (along with giving other pieces related permissions and obligations). Thus, it seems that it is a pawn in virtue of its normative statuses, and it has these statuses in virtue of our agreement to treat it thus (or something like that).

Now, whether this “in virtue of” makes sense or not is a contentious debate I’m sure. But if it does, then it seems to motivate various projects, both philosophical and otherwise. For instance, the physicalist program seeks to find physical facts in virtue of which all other facts hold (whether about consciousness, laws of nature, normativity, life, etc.) and in general any reductionist program seeks to show that a certain set of facts holds in virtue of some other set (though they may argue that even the distinction between these two sets of facts is merely illusory).

Another example that was used in seminar to motivate this distinction was that we know that necessarily, any straight-edged figure whose external angles are all equal to the sum of the other internal angles is a triangle, and vice versa. However, there is a sense in which it is a triangle in virtue of its having three sides, rather than in virtue of this fact about the external angles. So I wondered how much this idea can be extended in mathematics.

At first I thought it would quickly fail – after all, it’s extremely common not to have a single definition for a mathematical object. For instance, an ordinal can be defined as a transitive set linearly ordered by the membership relation, a transitive set of transitive sets, the Mostowski collapse of a well-ordering, and probably in many other ways. In different presentations of set theory, a different one of these is taken to be the definition, and all the others are proven as theorems. Similarly, a normal subgroup of a group G can be defined either as the kernel of a homomorphism from G to some group H, or as a subgroup of G that is closed under conjugation by elements of G, or probably in many other ways as well.

However, I’m starting to think that maybe there still is a notion of essence here. For most of the uses of the normal subgroup concept, the fact that it is the kernel of a homomorphism is really the fundamental fact. This explains why you can take quotients by normal subgroups, and more easily generalizes to the notion of an ideal in a ring. With the ordinal concept, it’s a bit harder to see what the fundamental fact is, but it’s clear that well-ordering is at the bottom of it – after all, when generalizing to models of set theory without the Axiom of Foundation, we normally restrict the notion of ordinal to well-founded sets unlike the first two definitions.

If this is right, then I think that a lot of the history of the development of mathematics can be seen as a search for the essences of our concepts (and for the important concepts to find the essences of). Although we often think that theorems are the main product of mathematics, it seems that a lot of the time just identifying the “right” structures to be talking about is really the goal.

Something like this can be seen in the history of algebraic geometry. At first, it was the study of curves in the real plane defined by polynomials. Eventually, it was realized that setting it in the complex plane (or the plane over any algebraically closed field) makes a lot of things make more sense. (For instance – Bezout’s theorem is true in this setting, that a curve of degree m and a curve of degree n always intersect in exactly mn points counting multiplicity.) Then it was generalized to n dimensional spaces, and projective spaces as well to take care of a few troubling instances of Bezout’s theorem, and to make sure that every pair of algebraic curves (now called varieties) intersect. After noticing the connection between algebraic curves and ideals in the ring of polynomials on the space (there is a natural pairing between algebraic subsets of a space and ideals closed under radicals in the ring of polynomials), it became natural to define a ring of polynomial-like functions on the algebraic curves themselves. With this definition, it was clear that projective spaces are somehow the same as algebraic curves in higher-dimensional spaces, and affine spaces are their complements. Thus, instead of restricting attention to affine and projective n-spaces over algebraically closed fields, the spaces of study became “quasiprojective varieties” – intersections of algebraic subsets of these spaces and their complements. In the ’50s and ’60s, this notion was generalized even further to consider any topological space with an associated ring satisfying certain conditions – that is, the objects of study became sheaves of rings over a topological space satisfying certain gluing conditions. Finally (I think it was finally), Grothendieck consolidated all of this with the notion of a scheme.

At various points in the development of algebraic geometry, the spaces under study changed in various ways. At first, extra restrictions were imposed by requiring the field to be algebraically closed. But then other restrictions were removed by allowing the dimension to be higher. Moving to projective spaces was another restriction (in a sense), but then moving to quasiprojective varieties was a generalization. Moving to locally ringed spaces, and then sheaves, and finally schemes were greater generalizations that (I think) incorporated the spaces that were originally removed by requiring algebraic closure. However, the ones that were excluded by that first move could now be understood in much better ways using the tools of ideal theory, the Zariski topology, and much more that was naturally developed in the more restricted setting. I am told that the notion of a scheme helped tie everything together and allow algebraic geometers to finally reach a concept that had all the power and interest they wanted to give good explanations for facts like the analogs of Bezout’s theorem, and also to start dealing with problems of number theory in geometric terms.





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.