Mathematical rigor, I
Mathematical rigor.
This is an issue that has been at the front of my mind for a long time, and one which, despite first appearances, is surprisingly storied and even controversial.
Math is usually perceived as a completely rigorous field concerned with finding “correct” answers, and verifying the “correctness” of theorems. This is true in a very limited sense — math is indeed an axiomatic system under which certain conclusions follow from the premises, and others do not.
Of course, an axiomatic system can encompass any set of “truths” — after all, one can take any set of (non-contradictory) statements as a foundation. Thus, merely choosing axioms is an aesthetic decision not strictly based on any sort of reality.
While this may seem like a pedantic point, choosing axioms in math is actually a somewhat controversial issue. Apropos Godelian incompleteness, there is an infinite set of independent axioms to accept or reject. A few prominent examples are the axiom of choice and the continuum hypothesis in set theory, and a generalized Fubini’s theorem in analysis.
Historical axiomatization.
But has math always been a purely axiomatic system? Not really, as it turns out. Euclid‘s Elements is perhaps the greatest (or at least, most famous) specimen of axiomatization in mathematics. By taking several (reasonable) geometric truths as self-evident, Euclid was able to develop many non-obvious results in geometry.
(I must, of course, pause to note that one of his axioms is significantly less self-evident than the other. In fact, altering that axiom — the parallel postulate — yields a different kind of geometry, akin to conducting your business on the surface of a sphere instead of a plane.)
However, the lack of a concise shorthand for algebraic notions appears to have held back the rigorous treatment of non-geometric concepts. For instance, Newton’s second law, 
In fact, great mathematicians like Euler often stated results somewhat informally, and without proof. Even by the middle of the nineteenth century, Galois had to invent many terms in group theory to explain his highly-technical theory linking fields and groups.
Nicolas Bourbaki.
The true breakthrough in axiomatization began in the 1930s by a group of French mathematicians operating under the pseudonym Nicolas Bourbaki.
Bourbaki aimed to produce a coherent treatment of modern mathematics, publishing nine volumes covering a large portion of the field. While opinions on Bourbaki vary drastically, I think it’s evident that they accomplished their goal admirably, and in the process, very heavily influenced the way mathematics is performed.
Bourbaki took a very rigorous approach to mathematics, systematically building up concepts from set theory to algebra, topology, analysis, and beyond. The development is very rigorous and dry; no actual problems or applications are discussed, and virtually no diagrams are included.
At the time, Bourbaki’s books were undoubtedly the best references available; it is not surprising, then, that their new approach had a profound effect on mathematicians (particularly nascent ones). Even their vocabulary has stuck, such as the empty set symbol 
Since then, mathematicians have essentially agreed to conduct mathematics more or less in the manner of Bourbaki. (It may seem surprising that math was at one point much less rigorous, but this merely reflects the huge influence of Bourbaki.)
Rigor considered harmful?
In my view, the most important question at this point is how beneficial (or detrimental) rigor is to mathematics. For reasons I will explain in the next installment, it seems evident (though initially counterintuitive) that rigor often helps clarify the situation. At the same time, I believe there is an alarming dearth of non-rigorous treatments of math.
Please comment on this if you have anything to add or ask; I plan on writing at least several more posts about mathematics, and I would like to focus on whatever points everyone finds most interesting.
Very interesting! I'm reminded of Whitehead and Russel's Principia Mathematica, in which it takes almost 400 pages to prove and define "1+1=2". I can't wait to read what you write next!
August 17th, 2009 at 2:34 amExactly — PM was an attempt to be so rigorous that nothing possibly could slip through the cracks. Of course, Godel showed that such efforts were fruitless, because of inherent limitations in any sufficiently powerful logical system.So while PM was absurd, it really intended to use rigor to, in some sense, "clarify the system" (by removing all ambiguity). There are much more useful and currently-mainstream examples of ways in which rigor actually clarifies the system greatly, and I'll be touching on some of those (and on PM) in the next post, I think.
August 17th, 2009 at 9:19 amTagged some people I've discussed this with recently.
August 17th, 2009 at 9:51 amUnfortunately I'm a humanities major and have forgotten for too much of the math I knew in high school, but I'm nerdy enough to enjoy these things anyway.The question of rigor is a very interesting one to me because it seems related to a similar problem in philosophy- do we analyze arguments thoroughly and reduce all philosophy to linguistic analysis, or is there a better way?Issues I find interesting: how does the way we conceive of mathematics shape the way we think about wider metaphysical truths? http://www.frontporchrepublic.com/?p=4097 Is math education as awful as this man thinks?http://www.maa.org/devlin/LockhartsLament.pdfIs Chris Langan totally nuts or is he actually in the process of destroying the theoretical foundations of both math AND science? (Probably both, actually) http://www.ctmu.org/ (Click through to the "Theory of Theories" article- it's the only one I can really understand)
August 17th, 2009 at 10:36 amLockhart's essay is one thing I was thinking of discussing in the future. I think he generally has the right idea but carries it a bit too far.As to that Langan essay, he seems to have basic misunderstandings of the shortcomings of science and mathematics. By its very nature, there IS no way to "resolve the self-inclusion paradoxes" in set theory; this is precisely what PM attempted to do, and which Godel proved to not work. It's not that nobody has been clever enough yet; the system, by its very nature, cannot be patched up.Likewise, issues he points to like "quantum nonlocality" are not things which need fixing. The EPR paradox defies logic but is well-characterized by quantum mechanics, through Bell's inequalities. In fact, he seems to be hinting at things like hidden variable theories, which have been systematically proven to be incorrect.So basically, I think Langan demonstrably doesn't know what he's talking about, and is just waving his arms around.
August 17th, 2009 at 10:59 amYeah, I was a little doubtful about the idea of slapping "meta" in front of things and pretending like you've made a meaningful distinction that solves the problem. However, despite perhaps lacking knowledge of specific issues at stake, his negative critiques of the way science and math work seem pretty convincing to me. His positive solutions might be bunk, but if cared too much about that part I'd never listen to Nietszche either! I'd be interested to hear your thoughts about all the stuff that comes before Langan's "levels of proof" bits.
August 17th, 2009 at 11:20 amBasically, I think his paragraph beginning "Of the three kinds of theory" is completely wrong and I may address its ideas in full at some later point. Same with his paragraph beginning "In place of the scientific method" where he outlines the "axiomatic method." In fact, the latter was a set of ideas I was going to deal with (a misunderstanding of "aesthetics," as I'm going to call it) once I tied up this whole rigor thing.
August 17th, 2009 at 11:44 amGreat site, how do I subscribe?
October 3rd, 2009 at 1:51 pm