Intuition and Rigor
I was recently chatting with an acquaintance and (don’t judge me) the topic of mathematics came up. K ranted that his mathematics classes had always seemed irrelevant to his work or anything real, and that mathematicians were an annoyingly pedantic lot, rigor-nazis who voluntarily blinded themselves to the bleeding obvious.
This got me thinking about how I’d myself felt nearly the same way once. When I took MA101, my first math class as an engineering undergrad, I struggled to understand why it was important to prove such seemingly obvious things as . I’d loved high school math, but I’d always had an intuition for geometry and numbers and had always relied on insights from my intuition to solve problems. Rigor was boring and stuffy and useless. As a result, in my college math class, the more “obvious” a result seemed, the more I struggled to prove it from first principles. Needless to say, my professors had a diabolical taste for just these sorts of results.
In hindsight, the trouble with high school math was that it was never made clear to us that math existed independently of all notions of “reality”. Common sense and intuition could only ever be used in high school math because it so happened that the kind of math taught in high school was that which was readily useful in loads of real world situations, and which we had therefore developed an intuition for. This is an infinitesimally small subset of all possible mathematical systems. I suppose I must not blame my high school math teachers for not telling me this – enough kids shun math already without having to be told that it is mostly useless in daily life.
Rigor in mathematics is a system of rules, a structure if you will, that ensures that the mathematician stays true to his/her assumptions i.e axioms. Mathematics serves as a torch with which to explore the rest of the Platonic universe created by a set of arbitrary axioms. First, immediate outcomes of the axioms are visible. Then immediate results of those outcomes together with the axioms are visible. A whole universe is mapped inch by boring inch. It can seem restrictive and cumbersome in cases where the universe is a familiar one (read, built carefully with “common-sense” axioms) to have to wade through a dozen lemmas to get to an unremarkable result. The fruits of rigor aren’t always apparent in such cases.
Mathematics does not concern itself with the axioms themselves – the most outlandish of axioms you can think of are still permitted. Mathematicians do not shy away from creating universes that bear no apparent relation to the one we live in. All that is required of the mathematician is that he/she stick to the axioms. On the other hand, intuition and common sense are honed only on domains that are relatively simple and familiar to us. If we had to rely only on intuition to understand the world around us, we would be largely crippled on unfamiliar territory (which, incidentally, is exactly what we usually wish to explore). This is why it is important to liberate our means of acquiring knowledge from intuition. Rigor does just this – it enables the discovery of truth in unfamiliar universes, the consequences of non-common-sense axioms. Even seemingly common sense assumptions can lead to outlandish-looking outcomes.
Scientists attempt to craft mathematical models of the observed world that rely on progressively more compact sets (pardon the pun) of axioms to explain all that is observed (Occam’s razor). All of science is about progressively refining these models. If something is predicted by the model that is contradicted by observations, it points to missing or incorrect axioms. Similarly, if something is observed, yet not an outcome of the model, the model needs revision. If something is predicted by the model that is yet to be observed, experiments are then conducted to examine the veracity of predictions and thus help validate the model. Science thus differs from pure mathematics in that it is tied down by an almost annoying insistence on fidelity to empirical observations.
By contrast, mathematics has this fantastic, untamed magnificence that gives most people goosebumps when they first experience it.
PS 1: Just found Poincaré’s Intuition and Logic in Mathematics . Recommended reading!
PS 2: I offered my sympathies to K and moved on – I did not know this person well enough to get away with this rant without making it awkward, whence this post.