How to be successful as a research mathematician? Follow your gut

I've talked about this in a replay to another comment, but I don't think she's making any value judgment in the article, or (at least in the article) actually articulating any claim about colonialism and mathematics. In fact, I don't even think she's saying "Europeans bad":

  We might marvel at, say, Amish people raising a barn or Inuit people building a kayak without any European formal maths training. We might declare that this is maths — and I think it is. But it’s problematic if we think we’re doing them some great honour by bestowing the label of maths on what they do.

I've just finished reading Feyerabend's Tyranny of Science which goes down a very similar path to OP's interview:

>In this wide-ranging and accessible book Feyerabend challenges some modern myths about science, including the myth that ‘science is successful’. He argues that some very basic assumptions about science are simply false and that substantial parts of scientific ideology were created on the basis of superficial generalizations that led to absurd misconceptions about the nature of human life. Far from solving the pressing problems of our age, such as war and poverty, scientific theorizing glorifies ephemeral generalities, at the cost of confronting the real particulars that make life meaningful. Objectivity and generality are based on abstraction, and as such, they come at a high price. For abstraction drives a wedge between our thoughts and our experience, resulting in the degeneration of both. Theoreticians, as opposed to practitioners, tend to impose a tyranny on the concepts they use, abstracting away from the subjective experience that makes life meaningful. Feyerabend concludes by arguing that practical experience is a better guide to reality than any theory, by itself, ever could be, and he stresses that there is no tyranny that cannot be resisted, even if it is exerted with the best possible intentions.

In other words: there's a 'tacit knowledge' that we ignore by just writing down what we can measure in numbers. Feyerabend died before it became en vogue to talk about colonialism's impact, but his points are similar. (This ties in with her gut feeling: the scientific literature pretends the gut feeling does not exist, we invent a logical story of how we've deduced an insight after we've stumbled over the insight)

I'm curious: what field did you work in? One of my reasons for not pursuing math was going to a summer undergrad research mentoring thing by a well-known-enough American algebraic topologist (the field that Eugenia works in), and finding out that even his students were having problem getting positions in higher ed in the US that weren't temporary – that was simply the state in general academia, with an ageing professoriate.

Maybe I should add a 5th point, but I need to phrase it very cautiously.

5. If you're interested in doing mathematics in an institution of higher ed in the US, focus your work on teaching, and be prepared to live in the middle of nowhere.

The author teaches maths to art students. The paragraph on maths research:

  But now I realize that, in maths research, you don’t just follow logical steps. If you do, you’ll never get anywhere interesting. You have to use your gut instinct and feel your way through something first, and then back it up with logic afterwards.

Opposite to TFA, in school, I gave up on maths because explanations were not provided; you had to take things on faith (e.g. algebra of limits, epsilon-delta proofs were not given), and the questions were carefully manufactured to be solvable by the given methods. i.e. puzzle-solving, not general powerful methods. Also, no proof of the uniqueness of prime factorization.

TBF, it's hard to load proofs on to school students, and since Euclid was dropped, the focus is on technicians and results - not truth. But it can be done: spend the time on building foundations; or omit subjects where you can't. I'd like an axiomatic approach.

Yes: "Eureka!", benzene ring, https://wikipedia.org/wiki/Eureka_effect

BTW TFA uses "feelings" in two different senses: hunch, inkling, intuition, suspicion, gut instinct vs. emotion (as in music).

In mathematics, there is even the formal term "conjecture" for an unproven theorem.