I only recently got around to counting up the results, and of those 48, I grouped them into six essentially different responses:

• pound (or pound sign, or “pound that shit bro”),
• hash (or hatch),
• number (or number sign),
• sharp,
• octothorpe,
• and a few other responses (tic-tac-toe board, and 0×23).

Many respondents seemed to have a primary answer, followed by caveats (e.g., “Pound, except in music, where it’s a sharp.”) so I differentiated between the primary and subsequent responses for each person. The following graph shows the results, where dark blue indicates primary responses, and light blue indicates all subsequent responses.

If you want the raw responses for some reason, and you don’t want to cull it yourself from my Twitter and Facebook, I can send you a spreadsheet. Graph is courtesy of the Google Charts API, which is a pretty sweet way to make easily-modified charts. (Check out the image URL if you’re curious how it works. I wrote a trivial Perl script to make it easier to incrementally edit the parameters and view the results.)

In mathematics, a proof is no more than a convincing argument of a statement’s validity. But as justification for a theorem, a proof alone is wanting. Why state one theorem over another?

The metric of real-world utility seems not to apply. After all, the very objects of study are, in some sense, artificial.

But the allure of mathematics–to me, at least–is how well those objects fit together: the degree to which mathematics unifies apparently different concepts is staggering. Perhaps nobody said it better than G.H. Hardy in A Mathematician’s Apology: “The ‘seriousness’ of a mathematical theorem lies in…the significance of the mathematical ideas it connects.”

If research mathematics is about connecting ideas, surely math education ought to convey that interconnectedness. On the contrary; until reaching senior-level courses, mathematics appears an amalgam of essentially disparate concepts.

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, , was expressed by him as “The alteration of motion is ever proportional to the motive force impress’d.” While it sounds pompous now, that’s simply how equations were discussed back in the day. Imagine solving a set of equations written that verbosely!

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 and the words injective, surjective, and bijective.

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.

I’ve touched on this subject before — what is beauty (or perhaps, elegance) in mathematics? Last time I addressed it, I concluded that elegance, to me, is a high results to complexity ratio. I gave the example of Euler’s formula, which I find incredibly simple but deep. Beauty is about the insightfulness and depth of results, not just the usefulness.

(Example from group theory: the classification of finite simple groups, as a single corpus, is neither insightful nor deep nor useful! And it’s certainly one of the ugliest — and longest — proofs out there! Not that there aren’t good moments. It’s just kind of silly to represent the whole thing as a “proof.”)

But I’m mostly commenting again on mathematical beauty because of a comment Hofstadter made about beauty in general — what is found beautiful and by whom? (After all, most people don’t find math beautiful at all!)

He noted that, while people claim “beauty is in the eye of the beholder,” there’s clearly some sort of standard for beauty. After all, if there weren’t, why would we have art museums filled with the most “beautiful” art? It does, however, take a special sort of person to enjoy the sort of beauty that math offers. Though, in that regard, math isn’t unlike most other activities — after all, why do marathoners want to run 26 miles for pleasure? It’s just that, as Hofstadter points out, enjoying mathematics requires somebody who wants instead to… “run a matharon,” so to speak.

Have I mentioned that I love puns?

Wow, there are some great acts here. In particular, I think the basis we just heard was great. His music spanned our three-space quite nicely. Anyway, I was going to bring some predatory birds here, but then I realized it wasn’t a talon show.

Okay, I’d like to make a request of you before I start my act. Please laugh very loudly at everything I say, because nobody might actually find it funny.

So, math comedy. When I told my friends I was going to do a math stand-up act, one of them replied, “Chuck Norris knows the tangent of pi over two!” Well…okay. I’m not sure how to respond to that.

Math comedy is certainly a niche audience, though. Even among mathematicians. If you ask a statistician if they’ve heard a joke before, they say “Probably.”

Anyway, there are a lot of oldies-but-goodies. There’s the joke about the mathematician who gives a talk about 13-dimensional space. Afterwards, an engineer comes up to him and says, “Wow, how could you possibly visualize 13-dimensional space?” The mathematician responds simply, “That’s easy, I just visualize n-dimensional space, and set n equal to 13.”

Of course, many sub-disciplines come with their own occupational hazards. They say topologists can’t tell the difference between a doughnut and a coffee mug, them being homeomorphic and all. I’m not sure if that’s true; I’ll ask Kent after the show.

And then physicists get their own brand of flak from mathematicians. Physicists, you see, use a special brand of mathematics. The really fuzzy type…that’s usually wrong, but somehow comes up with the right answers all the time. I think one thing in particular illustrates physicist math. Those of you who know some physics may know that electric and magnetic waves propagate as orthogonal sinusoidal waves. The direction in which they are pointing, the vector representing the energy flux of the wave, that’s called the Poynting vector. I don’t know about you, but I never make distinctions about which of my vectors are pointing. They all are!

Anyway, the other day I was going to a geometry conference, and I was speaking on constructible diagrams. I was flying out of the airport, but I was stopped at security because of my straightedge and compass. They found my weapons of math construction. I ended up missing my plane. But it’s okay; luckily I had three points in my pocket, so I defined my own plane and got there on time.

You know, we mathematicians are always trying to prove to everyone that there’s math everywhere. In particular, there’s a lot of math in the Bible; did you know that? For example, a lost story from the gospels. One day, Jesus said, “The kingdom of heaven is like x squared plus 3x plus 5!” Somebody went up to Matthew and asked him, “What is Jesus talking about?” “Don’t worry,” responded Matthew, “that’s just another one of his parabolas.”

Then there’s also the story in Genesis with Noah’s Ark. After the ark landed, Noah told all the animals to go forth and repopulate the world. Two snakes stayed behind, and told him, “We can’t do that until you build us a wooden desk.” So, whatever, he built it, and lo and behold, they started to reproduce. He asked them what the problem was, and they said, “Well, we’re adders. We need log tables to multiply.”

The other day I was proving a theorem. It was a long theorem, with a lot of significant intermediate stages. I got to one of those stages, and I said to myself, “Do I have to finish? Lemma stop here.”

Medicine has made great strides recently. When right triangles get old, they sometimes start to sag, their right angle turns into 89 degrees, 88 degrees… Anyway, they made this injection, you can just apply it to the triangle, and the angle will snap back up to a right angle. It’s called Pythagorean serum.

The other day I was at the concession stand. I wanted a medium order of Fibonachos, and my friend wanted a small order. But then I realized that a small plus a medium cost the same as a large.

I usually eat more healthily. I found a grape that could commute, it’s called an abelian grape.

I thought up a great anagram for Banach-Tarski. Ready? It’s… “Banach-Tarski Banach-Tarski.”

Some people have wondered why Newton didn’t contribute to group theory. It’s because he wasn’t Abel.

Have you heard? A former vice president recently released some rap tapes to teach computer science. It’s called “Al Gore Rhythms.”

Even mermaids like math. They wear algae bras.

Okay, just one more and I’ll leave you guys alone. So, as you know, lately, the military has been having issues with how its officers are perceived. Some kernels have expressed concern at their rather zero-dimensional images.

However, I had a question for everyone. Right now, the Writings section of my website is kind of bare. I have written some papers I am pretty proud of, and I was wondering whether people would actually be interested in reading them. Of course, I would only post interesting papers on interesting subjects. For example, I have written some pretty nifty papers this year on luminiferous aether and on special relativity with which I am rather pleased. If you are somewhat interested in the history of science, I think you would find them good discussions of fascinating topics. I could also dig into the Carlo vault and post older papers, for example, my McCarthyism paper that demonstrated I am in fact competent with history, just not at remembering all its minutiae.

Are people interested? I know proposing that some people might like to read papers in their spare time is a pretty crazy thing to say, but some weirdos actually like that sort of thing.

A show of brute computation doesn’t interests me nearly as much as an elegant result. If I had to define elegance in this context, I think I would say that a subject with a high results to complexity ratio is very elegant, especially for math. I think first-year calculus (I and II) is very elegant, as it makes many very difficult problems easy, and makes insoluble problems possible.

To me, the most elegant result obtained by first-year calculus is Euler’s formula, or in its most famous special case, that e^(i*pi) = -1. The fact that this can be conclusively proved to or even discovered by high-schoolers with knowledge of Taylor series is, I think, a testament of how elegant calculus is. Of course, there are other ways to prove it, but I think Taylor series are the most elegant way–the clearest, most straightforward, least contrived; no clever logic is required at all.

Of course, Euler’s many formulae remain the gold standard of elegant results in mathematics. While he didn’t always convincingly prove his results, I think it was more important that he simply generated them. Sometimes simpler logic goes further than more bulletproof logic. I certainly don’t believe I should be mentioned in the same paragraph as Euler, but as to myself, I was quite happy with my intuitive proof using Gram-Schmidt that, for vector space V and subspace W, that dim(W) + dim(W perp) = dim(V). The idea is too conceptual to forge into any sort of formal proof, but it makes perfect sense; I would be interested in seeing a more rigorous proof, but I haven’t gone out of my way to look for one. I am, for the most part, satisfied with rationality.

Andrew Wiles’s proof of Fermat’s last theorem was certainly a tour de force in the mathematical world, and certainly, as I understand, contains many interesting ideas (which no, I am not myself qualified to appreciate). It’s certainly nice that the theorem–and subsequently, the full Shimura-Taniyama theorem–have been proven, but the proofs are long, complex, involved, and *gasp* even, in some places, significantly case-oriented. It’s a true achievement, but to me isn’t as awesome (in the pre-slang sense of the word) as much shorter results. The epsilon conjecture hinted at great possibilities, but to me, V – E + F = 2 has for centuries been quite profound enough.

I don’t mean to say that I don’t consider the above to be true accomplishments; they certainly are. I suppose it’s simply that I am more interested in the concept than the execution itself. This is why I doubt I will be entering math itself as a field, though I fully expect to be using math in whatever I end up doing.

And it’s also, I suppose, why I have always been so happy to help somebody with math who seems actually interested in the ideas themselves, not just how to do well on the next test. It’s easy to tell the students who truly want to hear the concepts and understand them for themselves apart from the students who simply want an algorithm to memorize. And it’s perfectly understandable why math could be so hated by students who view it as simply a set of algorithms. Because then, as John once sagely pointed out, math is no more than a “demented mind-game.”