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.