Mathematicians are chronically lost and confused (2014)

I did not know that about MacDonald. Thanks for pointing it out. Do you agree, though, that such examples are rare? The original premise is that in math it appears self taught is not really a viable route for all but a very small few.

This is also largely a consequence of the fact that most linear algebra courses are computational by nature. They'll ask you to compute lots of things as a means of assessment. Work out this determinant. Find the eigenvalues of that matrix. "Matrices are grids of numbers". Take this matrix and write it in terms of this other basis. I only really perceived the impact of linear transformations being vector-space-structure preserving mappings long after I'd seen their analogues in the form of homomorphisms, homeomorphisms, diffeomorphisms etc. Nobody told us that determinants are just the alternating k-tensor on real k-space (up to a constant factor). At some level, this is because most students taking linear algebra don't have the mathematical maturity to stomach a course in finite dimensional vector spaces. I was indignant when I found out that a matrix was not just a grid of numbers, but rather a manifestation of a linear transformation, with respect to a particular basis.

I only really started to understand linear algebra when I was forced to in a differential geometry class. The opening chapters were a review intended to fix my university's notoriously broken linear algebra training. As I said, it comes with the territory. You can't design a course based on Halmos' FDVS and expect students coming in, that is, students who've scraped through calculus 1, to manage. So, the recipe book / cookbook style abounds. Granted, there were inklings of mathematics in my linear algebra course. I don't think anyone really appreciated it however. It's hard to grok "vector space over a field" when you've never been introduced to the abstract concept of a field.

When my second year stats lecturer told me that a determinant of a 2x2 matrix was an area, I almost didn't believe him.

Fun exercise I was told about just the other day. Every invertible matrix with integer coefficients has determinant +-1. I would never have known how to solve that after my linear algebra course.

for me your comment is so sad to read

mathematics is unconstrained by any desired artificial 'engagement' requirement

engagement is as easy as acting on and developing an interest

and i encourage everyone to engage with mathematics

your attempts to create an artificial toll or make a case for one, especially in lieu of your argument being unsolicited from the contents of the linked article, is suspicious at best, and wholly detrimental at worst

> very hard to engage with outside of formal education - or at least nobody has really found a great model for doing so yet

this seems like contradictory logic.. you appear to be denying 'informal' students from using the same model you advocate from these 'formal' sources

also i think you need to flesh out your definitions a bit..

what precisely do you mean by: formal education, outperform, general programming, scientific programming?

you also seem to be setting up a logical fallacy in your attempt to define your thesis of 'mathematical maturity'

> my colleagues who entered industry straight after their Masters or even Bachelors, and those who completed Doctorates or even held postdoctoral positions

are you comparing a ~20 year old at their first job to a ~30 year old who spent thaer twenties in academia? have you tested your hypothesis by comparing others of similar time spent on the subject but lack receipts for the money they paid into the academic institution? are there anomalies present in your investigations?

your argument seems to lack any substantial scrutiny, and this would seem to me to be the defining element of some such concept of an interest in mathematics 'maturing': devotion to rigor

The Second Edition of Courant & Robin's 'What is Mathematics?', revised by Ian Stewart is good. This is from the preface:

" In short, it wanted to put the meaning back into mathematics. But it was meaning of a very dif- ferent kind from physical reality, for the meaning of mathematical ob- jects states "only the relationships between mathematically 'nndefined objects' and the rules governing operations with them." It doesn't matter what mathematical things are: it's what they do that counts. "

https://www.amazon.com/Mathematics-Elementary-Approach-Ideas...

I recently wrote two articles I think you would enjoy :)

The first is high level perspective, the second is nitty gritty math, proof, and implementation.

[1]: https://jeremykun.com/2016/04/18/singular-value-decompositio...

[2]: https://jeremykun.com/2016/05/16/singular-value-decompositio...