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Not a very interesting article imo, but the topic of self teaching mathematics is a pretty interesting one. There's a few challenges with self teaching math. Fi
by _hardwaregeek 7y ago
Not a very interesting article imo, but the topic of self teaching mathematics is a pretty interesting one. There's a few challenges with self teaching math. First, math is inherently about communication. If you cannot communicate your proof, you're not doing math–you're doing mysticism. For instance, the supposed proof of the ABC conjecture by Mochizuki is not verifiable because other mathematicians have not been able to read and vouch for it in its entirety. More mundanely, if you have a proof for the first Sylow theorem in your head with hand waves and loose intuition based arguments, that's extremely different from a proof with tight logic, well defined lemmas and carefully detailed phrasing. Basically if you're self teaching, you have to do problems and have their solutions verified by another mathematician with a degree of rigor.
Second, it's quite easy to convince yourself that you understand a topic. I don't know what it is about math, but there's so many people, myself included, who fall into the trap of thinking they understand a topic, when in fact they don't at all. For instance, there's a multitude of people who can recite verbatim the quadratic formula, or the power rule for differentiation. But ask them to explain why this is true, or give them variants on the power rule (what if we redefine derivatives to be based on multiplication? How does the power rule change?), and they won't be able to give you satisfactory answers. This goes back to my first point; the only way to truly understand math is to do problems and have them checked.
- theonemind 7y agoMath tends to bug me because "proving" theorems seems to come from convincing arguments, but we have enough computation power at this point that we should reduce them to mechanical/machine verification and then just run them and know. With enough rigor, you don't need to "convince" anyone--a machine can verify it.
- BigFish12 7y agoTo be rigorous you would need to prove that the machine verified it correctly. Good luck doing that. Also theorem proving is only good for basic rudimentary maths, and not abstract state of the art math research.
- artagnon 7y agoI agree that it's necessary to do exercises in order to understand the subject, but I disagree that you need to have them checked by someone else -- in many cases, you should be check it for yourself, honestly. Beyond the junior undergraduate level, computation-style problems go away, and you're just left with exercises that ask you to prove a certain something. It takes a long time to build up a sufficient degree of rigor, but it happens eventually -- I did go get some of my proofs checked by serious Mathematicians at some point, and it turned out that my intuition wasn't far behind.
- _hardwaregeek 7y agoThe main reason you'd want proofs to be checked by someone else, at least initially, is to get a feel for writing clear, rigorous and tight proofs. Much as a novice programmer should have their code reviewed, a novice mathematician needs to have their math reviewed for common mistakes and pitfalls.
- agumonkey 7y agoWeren't there people in history who managed a sound level of self testing ? I understand it's super hard to be non biased and not claiming victory when you think you did a few checks.
- jqgatsby 7y agoWait, "redefine derivatives to be based on multiplication"? That sounds really interesting, could you please expand on that?
- _hardwaregeek 7y agoIt's a pretty interesting topic: https://en.wikipedia.org/wiki/Multiplicative_calculus https://en.wikipedia.org/wiki/Multiplicative_calculus