A small anecdote in relation to a recent coffee conversation I had with @TaliaRinger (which they relate over at https://twitter.com/TaliaRinger/status/1681410191278080000 ): Yesterday I spoke with a children's book author who was interviewing me as part of a series she was writing on contemporary scientists. She freely admitted that she did not have great experiences with her math education at an under-resourced school and chose very early on to focus on writing instead. Nevertheless we had an excellent conversation about many mathematical topics that she was not previously familiar with, such as proof by contradiction, Cartesian coordinates, Mobius strips, or compressed sensing, all of which she found fascinating (and said she would read up on more of these topics herself after our interview). I posed to her the isoperimetric problem (using the classic story of Queen Dido from the Aeneid as the intro) and she correctly guessed the correct shape to maximize area enclosed by a loop (a circle), and instantly grasped the analogy between this problem and the familiar fact that inflated balloons are roughly spherical in shape. I am certain that had her path turned out differently, she could have attained far greater levels of mathematical education than she ended up receiving.

This is not to say that all humans have an identical capability for understanding mathematics, but I do strongly believe that that capability is often far higher than is actually manifested through one's education and development. Sometimes the key thing that is missing is a suitable cognitive framework that a given person needs to align mathematical concepts to their own particular mental strengths.

Talia Ringer on Twitter

“Terry Tao and I spoke over coffee for like two hours yesterday, in part about diversity in how people think about math. We both agreed that people who hit these walls early mostly don't learn the way of thinking about math that works for them. It's an educational failure”

Twitter

@tao @TaliaRinger I think of the preface to Penrose’s Road to Reality where he tries to explain to a mathematics-phobe what a fraction is and in doing so comes to realize there really is some essential subtlety to the notion of equivalence classes and perhaps it is he who hadn’t appreciated the concept in its fullness.

One thing that leads us astray (as Stephen Jay Gould repeatedly said in his writings against IQ determinism) is the thought that there is some unitary notion of intelligence both inside and outside mathematics. Imagine one is confused about: what is allowed in Euclidean geometry proofs, what is an inertial frame, what it means to assign probability to an event that happens once, why can we use Fourier series to find the solution to an equation etc. One can easily imagine someone prescribing respectively: a course in formal logic, a course in the philosophy of physics/probability, a course in rigorous real analysis etc.

Such a prescription would not necessarily allow everyone to get past their confusions, but at present we seem to assume that such confusions, up to (and sometimes including) undergraduate level education, point to a fundamental flaw in the individual and a sign that a subject as vast and multifaceted as mathematics is “not for them”.

It is a great tragedy that the free and full development of our capacities and personalities is left to the mercy of such pernicious notions as the inherent hierarchy of people.