Content-free mathematics

All of mathematics is content-free.

That’s a claim that would probably get me lynched in any self-respecting math department, and anyone who has taken a real math class would feel that it isn’t true. How could anything that causes so much pain and confusion be content-free?

In some sense, however, every statement in mathematics is trivial. After all, a proof is no more than a series of logical implications, and every such implication is logically trivial (i.e. correct). The difficulty in understanding a proof is simply a parsing problem. For a technically challenging proof, there may be many definitions and lemmas to parse, and the writing may not be very clear, but all of the logical steps are there. In mathematics, proving is hard, but once a proof has been written, checking the proof requires much less inspiration and genius. In fact, given a proof, it could be written in computer-checkable form, and we all (hopefully) believe that computers aren’t yet sentient.

This isn’t to say that reading and understanding proofs is entirely easy; if this were true, learning math would take no time at all and checking Perelman’s proof of the Poincare Conjecture would not have taken many months and years. Even though I can try to justify a theoretical claim that mathematics is trivial, it is nonetheless true that I struggle to understand things. My brain isn’t a computer; though it’s probably better as being creative, it’s certainly worse at parsing. An argument about the triviality of math because purely philosophical and is itself content-free.

However, there are some parts of math that I do believe to be content-free, not just because all proofs are logically correct, but because they have been generalized to the point of being eaten up by other broader fields. These domains of mathematics become special cases of general statements, and though they may maintain their clever tricks and elegant arguments, work in these fields can contribute little that is not trivially known.

Consider Euclidean geometry as an example. In the time of Euclid and the Greeks, geometry was considered the essence of mathematics; they even used it to prove facts about the primes. Now, however, Euclidean geometry as moved into the realm of the content-free. Every statement in pure Euclidean geometry can be restated in terms of basic building blocks like collinearity and concurrency, and these can be translated into systems of polynomial equations that can be solved by standard techniques (e.g. Gröbner basis). So the ability to solve problems in Euclidean geometry is a corollary of an understanding of systems of equations, and the work of Euclid can now be automated by a computer.

Of course, plenty of people still care about Euclidean geometry. Even if the mathematical statements themselves are no longer important for the development of mathematics as a whole, Euclidean geometry still provides a good place to learn how to think about math, and that is a skill that transcends mere problems or mere fields. In addition, Euclidean geometry is still a great source of problems for contests because of its abundance of elegant results. For me, an argument in pure Euclidean geometry still carries an aesthetic appeal that a system of equations will never be able to match.

Will every mathematical field end up as a subfield of a more general theory? As I see more math, I believe that the answer is yes: There is always a more general framework in which old ideas are simplified and trivialized. There’s no reason that anyone would care about such a general framework, however, unless it can serve to unify seemingly unrelated ideas. The pursuit of generality for generality’s sake feels to me as rather silly; adding needless abstraction to a problem does not make it any more important or more interesting. Instead, I believe in abstraction when mathematics is ready — and not before.

In the end, there will always be problems and puzzles for mathematicians to ponder, and that is what is important. Regardless of whether mathematics is actually content-free (and throughout this discussion, we never defined “content-free”, so this is an subjective opinion), mathematics will always be interesting and worth studying.

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5 responses to “Content-free mathematics

  1. Elliott December 28, 2010 at 2:16 pm

    If P != NP, then finding a proof could be much harder than checking a proof! There would be problems for which finding a proof would require brilliance and creativity that a computer could never match (because, for example, it would take longer than the age of the known universe), even though that same computer could easily check if a given proof is correct.

    But P = NP would mean finding proofs is “just as easy” as verifying proofs. In this case, mathematicians might as well go find other day jobs (or become philosophers).

    • Moor Xu December 29, 2010 at 12:00 am

      That’s an interesting thought. Simply doing a search through all proofs and checking each one would take far too long. Finding a proof is like factoring numbers — it’s relatively easy to check that a given answer is correct, but it is much harder to actually come up with an answer or a proof. That’s what mathematicians do: They come up with heuristics and intuition to do the hard work of discovering proofs. In that sense, mathematics is sort of like an NP problem, and mathematicians are needed to solve the problems that computers cannot do.

      However, I’m not convinced that mathematics actually falls into the class NP. Even if P=NP, there are classes of problems beyond P and NP, and those problems would still be intractable. Actually, I’m not sure if a computer could ever do mathematics, as math is subjective. Yes, a computer could produce proofs (even do a depth first search over the space of all proofs) and hence theorems, but that’s not enough for it to be considered mathematics. In fact, what is (subjectively) considered good mathematics consists of theorems that seem particularly elegant, have useful applications, are sufficiently general, etc. In addition, mathematicians also value intuitions and connections between results. Instead of having a laundry list of results, people like to have context of how certain results fit in a broader framework; this context is what a computer would not be able to provide.

      So I suppose that despite the logical nature of mathematics, it still needs a human element regardless of whether computers can do math. At least, that’s if we want to preserve mathematics in its current form. Whether purely automated mathematics is still worth considering is something that I can’t answer, however. I think of mathematics as a complicated game with endlessly evolving rules that people play for fun, with the side effect of occasionally having useful applications. In that sense, a computer would produce practical applications without keeping mathematicians happy. The rest of the world might consider that a good thing, but I’d prefer not to be rendered obsolete.

      • Elliott December 29, 2010 at 9:25 am

        Mathematics as a whole probably isn’t “in NP”, but I’m pretty sure finding proofs is; I remember Luca Trevisan gave a talk where he explained (1) you can encode a proof as some sort of a graph, (2) you can check the correctness of that proof by checking some property of that graph (something to do with 3-colorability; I’ve forgotten the details), and (3) even a probabilistic check will suffice, and the time it takes to check is constant. Plus, he definitely used Riemann Hypothesis as the example :-P

        Otherwise, I agree that humans won’t be obsolete for a while. The Robot Overlords still need to keep humans around to come up with the right axioms in the first place (at least until the Singularity, at which point They can do it Themselves)!

        Maybe an ideal situation would be one in which each individual mathematician relegates whatever busywork he or she doesn’t want to do to a computer, whether that’s checking proofs, “proving” routine consequences of definitions, or doing drawn out computations. The important point would be that each mathematician would have a different “definition” of busywork.

  2. zhai2nan2 March 13, 2011 at 12:16 am

    All of mathematics is content-free.

    That’s a claim that would probably get me lynched in any self-respecting math department, and anyone who has taken a real math class would feel that it isn’t true.

    They wouldn’t lynch you at Carnegie-Mellon. They would just nod and say, “Yeah, I think Bertrand Russell said something like that. No one cares what math is about, just so long as it’s logically true.”

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