One day I'll retire and go back to school. The idea of learning Math - really learning & understanding Math - as a fun pastime is so appealing.<p>What's stopping me now? That sweet overpaid SDE salary and the endless obligations that come from being an adult. I suspect I am not alone...
I love Terry Tao’s writing on math. One thing that strikes me about him is that, despite being an absolute technical powerhouse, he writes in a very down to earth style that connects disparate areas of math - e.g. his article on “what is a gauge” <a href="https://terrytao.wordpress.com/2008/09/27/what-is-a-gauge/amp/" rel="nofollow">https://terrytao.wordpress.com/2008/09/27/what-is-a-gauge/am...</a> where he explains how dimensional analysis might be viewed as a change of coordinates. Too often exposition in math is myopic and fails to impart a unique perspective on the subject, but Tao imbues his writing with a wisdom that I consider the sign of a true genius.
Related:<p><i>There’s more to mathematics than rigour and proofs</i> - <a href="https://news.ycombinator.com/item?id=9517619" rel="nofollow">https://news.ycombinator.com/item?id=9517619</a> - May 2015 (32 comments)<p><i>There’s more to mathematics than rigour and proofs</i> - <a href="https://news.ycombinator.com/item?id=4769216" rel="nofollow">https://news.ycombinator.com/item?id=4769216</a> - Nov 2012 (36 comments)
I understood math much better after following the Software Foundations tutorials:<p><a href="https://softwarefoundations.cis.upenn.edu/" rel="nofollow">https://softwarefoundations.cis.upenn.edu/</a><p>Another good starting point (for mathematicians) would be the Lean community group and the Math lib project:<p><a href="https://arxiv.org/pdf/1910.09336.pdf" rel="nofollow">https://arxiv.org/pdf/1910.09336.pdf</a><p><a href="https://github.com/leanprover-community/mathlib" rel="nofollow">https://github.com/leanprover-community/mathlib</a>
I feel like this is how all domain expertise works, no? Start with intuition which helps you solidify some of the foundation. Flush out the foundation and start building complicated structures. Now that you've built up the experience, go back and use your intuition to figure out new types of buildings to build.
This is remarkably accurate and resonates with me a lot. I did mathematical olympiads in high-school, where intuition to crack problems plays a major role. Then went on to college to study an undergraduate degree in maths (concentration in analysis). Analysis requires, at least in its rigorous foundations, to be careful and have a skilled knowledge of logic/quantifiers (more than elementary abstract algebra in my humble and biased opinion), often very scrupulously. Then in my graduate studies intuition along with the maturity of rigor work to produce new theorems. I'm impressed that several times I look at a paper or series of results and can read them "diagonally" to get the motivation without scanning all the text (of course, if the aim is to cite/build on top of/generalize/apply it then close attention to reasoning should be paid).
You can infer from this how Artificial Intelligence and Logic should be combined: AI enables a "post-rigorous" mode, and Logic is how you know you are still doing something sensible, and how you expand your sure footing.
#2, #3 means being at the stage where you can look at something and be like "no this cannot work" or "maybe this can work" without having to do all the steps.
Which are the newest developments in math? Someone told me Geometric Algebra has been around for a long time but wasn't really useful until some recent theorems.