On math education, math as high risk
Veer Hitesh Patel, 29 Dec. 2024
It is my first year of university and I took two math courses this semester. An "introduction to proofs" and a "calculus" course. I have some feelings/opinions/observations regarding the state of math pedagogy, math textbooks. I had acquired these feelings/opinions/observations within the first three weeks of starting the degree and I write them down now.
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The thing I thought about math is that it’s very formalized and agreed upon in format, language, definitions so it is possible for one big wiki or encyclopedia which covered absolutely everything in math to exist. It seems like it’s not the case and different authors have different ways of presenting and reasoning.
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Why do some authors spend 5 sentences explaining how to do something trivial but then spend half a line STATING, not even EXPLAINING how to proceed from hard main body line of proof to other such line? Is there an actual reason behind this? Is the mathematician too full of himself?
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I had the understanding that in undergrad gatekeeping of information and reasoning wasn’t actually a problem but it seems you need to look through many different sources and then learn how to reason it by yourself using people to guide you (hence attending office hours is so encouraged), searching the web, use of large language models etc.
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Axiomatic method seems to be what is covered in undergrad. I’m not even sure if there’s anything in math which doesn’t follow axiomatic method. Such thing by definition may not even be math.
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Math has axioms, objects (which have properties and certain rules with how you may interact with them), logic which you use to deduce a statement from a previous statement etc.
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I'm surprised there’s no universally accepted interface which allows one to go and prove a statement from ground up, with alternative approaches etc. Like if you know all proof methods and all properties of objects, surely you can prove anything with said proof assistant engine. But no, have to refer to different textbooks. This seems like the sad absolute state of modern mathematics education, in the west at least.
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I feel sorry for the "20 year old proovers" who thought they were onto something but instead authors are still gate keeping solutions and explanations, making individual aspiring mathematicians waste their time on minute things which could have been presented very clearly and logically.
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Sometimes, the most beautiful things in math aren’t actually the results but understanding the intuitions and motivations from going to one place to another. Alternate viewpoints and approaches. It’s a breath of fresh air seeing the authors who wrote the most famous books incorporate these things.
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