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Comment by zamalek | Hacker News Reader

> I wonder if it's even possible.

From a practical perspective it definitely is. I've picked up a fair amount of graph theory and with nothing but extreme persistence have grokked and used some fairly advanced stuff[1][2] (2nd-year dropout). It was, however, work-related. Just don't ask me to proof anything.

> the few lucrative jobs that make use of maths are in finance

There is also competency on the table here. Graph theory crops up day-to-day with the business software work I'm doing (three separate deliverables). Calculus is used to a point of absurdity in game development - e.g. the Fresnel term. Machine learning? Calculus, linear algebra, tensors. Profiling? A basic understanding of statistics. Compilers? Category theory, graph theory. Physics engines? ODE.

It's extremely valuable to know this stuff.

[1]: https://en.wikipedia.org/wiki/Tarjan%27s_strongly_connected_... [2]: https://eprint.iacr.org/2012/352.pdf

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>Just don't ask me to proof anything

I don't mean to offend, but being able to prove things is generally the main focus of advanced mathematics. If you can't prove what you know, or at least have a rough outline of a proof you could construct after referring to something, you haven't learned it in the same way those at a university have.

No offense taken, you are completely correct. I can, however, still explain it and use it.

And? You can use advanced math without having a deep understanding of it.

I just wanted to point it out. The thread title is "Learn Advanced Mathematics Without Heading to University", which could be read with the implication "Learn advanced mathematics, at the university level, without going to university". Under this context, someone replied that this is possible because s/he learned it, but can't prove what s/he's learned. Of course not being able to prove the useful theorems you've learned doesn't render them useless (although perhaps less useful as you're less likely to know precisely when they can be applied and how to extend them to new situations), but in the context of the discussion it seems like an important distinction to make.

What about calculus etc? The main focus there seems to be to get a result. There are plenty of fields that use advanced math, but leave extending the math to academia.

> If you can't prove what you know

Does the Bayesian vs. Frequentist debate hold back working statisticians?

Or debate wrt constructivism ( https://en.wikipedia.org/wiki/Constructivism_(mathematics) ) hold back math in general?

I admit I'm a bit out of my depth on this point though...

You won't do a calculus class for mathematicians without also heaps of real analysis and/or measure theory. It's a different story for a field like engineering, but that's not what the blog post is about. If you haven't studied the proofs, you haven't studied advanced mathematics.

As for Bayesian vs. Frequentist, it's another vim vs. emacs style debate most of the time - which is most appropriate to use, as opposed to which is right and which is wrong. Quite a lot of the time, it just doesn't matter.

Isn't there a very small demand for actual, proof-writing mathematicians, versus "mathematicians" in the sense that they can use advanced math.

I'd say being able to use advanced math, even for engineering, is a suitable definition for "having studied advanced mathematics".

   'versus "mathematicians" in the sense that they can use advanced math.'

Those people are not "mathematicians", they have other useful names (most physicists, bio-statisticians, some engineers, etc.)

Mathematicians are people who create new mathematics, not people who use mathematics.

Where does that definition come from?

I feel the term is often used more broadly.

http://c2.com/cgi/wiki?MathematicianDefinition

In any case, the point stands - The need for persons who can construct mathematical proofs, versus those who simply need to derive the correct numerical result, is very different.

As such, most classes that teach calculus are for practical, applied purposes - who don't need to "prove what they know" beyond demonstration procedural competence.

I don't find that discussion particularly insightful. Yes, the term is used more broadly, but doing so invites confusion.

Compare perhaps "composer" to "musician", they are both involved in music but operating on different axis. Most people would agree that there isn't a strict relationship superset, and there is overlap. There are skilled composers who are lousy musicians, and vice versa. There are a few people who are top rate at both. However, it is very useful to have the distinction between creating and performing.

It's much the same with mathematicians.

Excuse me, but you're the one who went off on that tangent.

If we're relating to how people use words, it's not "much the same with mathematicians"

>I don't mean to offend, but being able to prove things is generally the main focus of advanced mathematics.

That rather depends on the field. For engineers, the main focus of advanced mathematics is to be able to apply it to real world problems.

The kind of math we engineers study in typical BS/MS curriculum is quite basic.

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