I own this book and I'll tell you it's not easy learning math from this book. Most of it is just very light overview. There are much better rigorous textbooks that are simpler and more complete.
There are some great textbooks translated from Russian. Analysis by Kolmogorov, (rigorous) Linear Algebra by Shilov, Complex Analysis by Markushevich to name a few.
1) No such thing as universal mathematician in this day and age.
2) Engineer's PDEs(algorithms) are not the same as mathematician's PDEs(theory). Same as comparing a student in China who learned English to communicate with English speakers to English majors from English speaking countries.
Colleges are great in that they offer nice infrastructure that's hard to find elsewhere: amazing libraries, laboratories, possibly, access to people doing research in the area you are interested in...Ultimately, though, everyone has to teach their own selves.
I expect part of the problem is that people who are attracted by clear writing and thinking are turned off by the academic fields that suffer most from overly academic writing.
In math formal language is easier to read than the informal one. Mathematicians write proofs in informal, paragraph style while newcomers are taught to read/write proofs in the most formal way because the latter is much more structured and clear.
Could it be because academic writing is simpler than the literary one? For example, in math, 'for all yada, there exists bla' is different from 'there exists bla for all yada'. So, you literally memorize these two cliches because they greatly simplify math. There are many other such very precise cliches in math you keep reusing over and over. Even though, it makes your life simpler, to an outsider you come off as an academic bore. That's my theory anyway.
Human biology is an extreme hurdle on our way to achieve the unachievable: having to sleep, eat, excrete etc. Are there any tech and drugs being developed to deal with these things? For now I'd be contempt to at least control my sexuality :D
Watch - in a near future there'll pop up an article explaining how you can't afford to study a little of this and a little of that. It will tell you need to focus on one thing only to be real good at it so that you can sell that one skill.
In the olden days there used to be 3 channels on TV max. That's how I see Hollywood. It has maybe 2 and a half concept it keeps selling whereas TV, let alone Internet, wouldn't be able to get away with it these days.
> On the biology side, I got ridiculed for all the 'hand waving' that seems to happen with the math. Biologists want to see concrete experiments and results.
How interesting. Usually, it's the math folk who ridicule everyone else for being sloppy with their reasoning. For example, physics = hand-wavery :D
How to Think About Analysis by Lara Alcock.
Understanding Analysis by Stephen Abbot.
Mathematical Analysis and Proof by David Stirling.
Numbers and Functions: Steps into Analysis by Burn.
Analysis: With an Introduction to Proof by Steven Lay.
A First Course in Mathematical Analysis by David Brannan.