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I think most people commenting in this thread are reading the headline and/or reading the article as outside the context of mathematics. This is Terry Tao's blo
by hervature 3y ago
I think most people commenting in this thread are reading the headline and/or reading the article as outside the context of mathematics. This is Terry Tao's blog and not a VC's microblog. The very first block of text is a quote from "I want to be a mathematician". He is purely talking about how to approach mathematics at a level sufficiently close to the frontier. What he is saying is that when you are taught something like optimization theory, the definition of convexity is presented to you in the first lecture. A "stupid" question to ask is what makes convex functions special. After thinking a lot about this question, I've come to the conclusion that convex functions are functions that are trivial to optimize but the converse is not true. Not all function that are trivial to optimize are convex functions. There are then plenty of questions like which other classes are easy to optimize. In Terry's words, "the answers to these questions will occasionally lead to a surprising conclusion, but more often will simply tell you why the conventional wisdom is there in the first place, which is well worth knowing.". Another example I ran into was the notion of metric spaces and completeness. It is "stupid" to ask what happens if you get rid of completeness. Stupid because you were just taught completeness for a reason. Trying to understand gives you a whole new appreciation than being presented the formal definition refined over many years.
- alberth 3y agoSounds like “First Principles” thinking for mathematics.
- kragen 3y agoall of mathematics is first principles thinking, and that's not a brand name now get your damn car off my lawn
- kdwikzncba 3y agoAll maths is first principles, without the pretentiousness of using the phrase "first principles".
- fivestones 3y agoI asked gpt4 about this first and it didn’t really know. What is it about “First Principles“ that’s cringeworthy or pretentious? I think I’m out of the loop on this one.
- hcks 3y agoHoly cringe
- creer 3y agoAnd it would be great if a lot of this (why these first principles) was at least mentioned in passing in the teaching. This is not just a question of being close to the frontier but about understanding why and where. Why some precept and where we are on the map. Necessarily in teaching this would be "just in passing" but still useful. Useful in all subjects but all the more so in highly abstract ones.
- kdwikzncba 3y ago> I've come to the conclusion that convex functions are functions that are trivial to optimize I think a lot of maths is created like this. I'm trying to solve problem P in domain D. I couldn't do that, but I've noticed that I could do it in subset D1 of the domain. If problem P was important enough and D1 large enough, I become very famous within mathematics and people start studying D1 by itself, without connection to P. After a generation or more a lot of experts in D1 might not know the original reason why someone decided to study it. Eventually some historians of science re-discover the original motivation and publish it as a curiosity.
- xanderlewis 3y agoVery true, combined with the fact that deep understanding of D1 turns out, centuries later, to be of practical use in building some sort of physical machinery.
- borg16 3y ago> Trying to understand gives you a whole new appreciation than being presented the formal definition refined over many years. This is so true, and which is why makes learning from certain professors way more enjoyable than others. What you have explained is in some sense an explanation of 'thinking from the first principles' or 'intuition behind the concept', at least imo.