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A list of advanced math tricks by Terence Tao
- 082349872349872 4y agoIs there a reason @JadeMasterMath's comment doesn't generalise the list?
- red_trumpet 4y agoWhat makes you think it doesn't? It certainly generalises some of Tao's points, but I lack the expertise to judge all of them. Also, Tao's list provides lots of helpful examples, not everyone might be easily able to see their problem through the lense of JadeMasterMath's comment.
- 082349872349872 4y agoI have no reason (also lacking expertise to judge all); I found it a good summary and that's why I ask if anyone else has a reason. Good point about the examples. There's an ancient injunction to not multiply examples unnecessarily, but in this day and age of general literacy and nearly free storage, we probably do well to offer a grand variety of examples and let readers pick and choose the ones they wish to contemplate.
- isaacfrond 4y agoHa, I snagged on that one to. Obviously what he is saying is true, but is add nothing. It like saying, to prove A, find a structure B, and exploit it. It is 100% true but helps you nothing. The beauty of Tao's list is that he gives actual examples. And many of them too.
- hgsgm 4y ago
- hgsgm 4y agoIt generalizes even further: If you need to prove B, and you know A implies B, all you need to prove is A.
- mcphage 4y agoIt does, but it really isn't actionable in the same way. Given a specific structure, you need to know what the generators are for that structure, and so that ends up being a list of "if you've got this structure... then you need to check ..."—which is exactly what Tao provided.
- S4M 4y agoThere is also this blog post from 2007 by the same author with some math tricks, very neatly explained (it is Terence Tao after all...): https://terrytao.wordpress.com/2007/09/05/amplification-arbitrage-and-the-tensor-power-trick/ https://terrytao.wordpress.com/2007/09/05/amplification-arbi...
- dj_mc_merlin 4y agoIt pleases me to see the classic "high votes, no comments" on this post, as people upvote it since they can tell its smart, but have literally no clue what the words mean. Neither do I for that matter, he might as well be speaking an alien language.
- scarmig 4y agoFor something more accessible of his, check out his Real Analysis textbook! Most people can understand it without difficulty; he starts by teaching you how to count and add the natural numbers. It's far more readable than other introductory analysis texts (e.g. Rudin) without sacrificing rigor.
- 082349872349872 4y agoAre you familiar with the idea that to prove a loop correct only takes two things: A/ proving (o->o), that it makes progress with each iteration, and B/ proving (->o), that it terminates? If so, he's speaking a bunch of different alien languages, showing how the same basic pattern (prove something for an infinite number of things by looking at their structure and proving something for the finite number of ways each of those things can be put together) works in each of the alien languages. Or at least that's what I got from the few of those languages which I speak — all the rest are, as they say, greek to me.
- deleted 4y ago[deleted]
- mrmuagi 4y agoInduction proofs are indeed useful, the strongest argument for learning these sort of formal tricks, is you can convince another human that your algorithm correct with pure math. It's one thing to say, look, my program's test suite (that I wrote) passes, or it passes a few criteria, it is another thing entirely to prove without a shadow of a doubt the algorithm is logically sound and correct. Induction proofs are useful with loops, but there are a lot of tricks useful for programmers.
- hardware2win 4y ago
- nmitchko 4y agoAs I read through this, I remind myself, there are some people that are so smart and in-depth within their field, that I have zero idea what any of the concepts they discuss are.
- Workaccount2 4y agoGoing from know-nothing to "know-at-least-enough" in a few dense fields, I've found that the concepts, once you get them, are usually pretty plain and straightforward. It's the custom terminology, unique vernacular, and commonly used abstractions that can make it so hard to internalize initially.
- rrrrrrrrrrrr2 4y agoAs I read through this, I just feel this great sense of relief that I switched from math into CS. I am just so glad I wasn't stupid enough to lock myself into a 6-10 year PhD/postdoc process, and I don't have to care about any of these tricks or existence proofs about technical constructions that will always exist only in theory. That being said, Tao works on twin primes; I don't really care about the means because I don't care about the end. There are other areas of math (geometry, etc.) which seem a lot more fun. But honestly why does anyone care enough about the twin prime conjecture to go through this pain, and what possible practical application could there be compared to algebra (cryptography) and geometry/topology (physics). And I am saying this as someone who took some graduate-level mathematics. I know some of what he's talking about. I'm just so overwhelmingly glad I don't have to care about any of it. Maybe Adderall could help me care, like it did for Erdos. Tao is obviously a genius, but to me he's an alien.
- ak_111 4y agoTangential, but it is interesting to note that it seems TT has been triggered enough by Elon's twitter drama to make a commitment in using mastodon. He has been very active in the blogsphere (and even google circles) for over more than a decade but I don't think he ever used twitter, somehow the recent drama has finally pushed him to start tweeting on mastodon. So we all have Musk to thank if TT continues tweeting these high quality mastodon posts.
- jvm___ 4y agomathstodon
- j2kun 4y agoThe math-focused Mastodon instance has a pleasant focus on math topics in a way that Twitter (before or after Musk) tends to interrupt to grab your attention.
- ykonstant 4y agoI would be extremely happy if Terry Tao could pull young mathematicians into services like mastodon. It is hard to overcome inertia, and brand name helps a lot; especially when it is accompanied by extremely high quality exposition and a measured, polite attitude (because it sets the standards and atmosphere of the space).
- sitkack 4y agoMathematicians of all ages should sign up here, https://mathstodon.xyz/auth/sign_up https://mathstodon.xyz/auth/sign_up I like how it has built in support LaTex!
- BeetleB 4y agoIf you are on another instance and want to view the equations rendered, you'll need some sort of browser addon. Or if you're using Emacs: https://blog.nawaz.org/posts/2022/Dec/rendering-latex-formulae-in-mastodonel/ https://blog.nawaz.org/posts/2022/Dec/rendering-latex-formul...
- nerdponx 4y ago> If one is trying to prove a Hilbert space identity or inequality which is invariant under a unitary group action, one can often reduce "for free" to the irreducible components of that group action. This is really, really cool. I love a good unifying principle. Also this sort of "bag of tricks" comes in very handy. It's a lot like the analogous tricks and idioms in programming. You don't have to rediscover it from first principles, if you remember that exists while solving a problem.
- dysoco 4y agoYeah I know some of those words
- eachro 4y agoIt's interesting that so many of these tricks are specifically for testing invariance properties. I wonder if it's because those sort of questions pop up a lot, or maybe those are the type of problems that TT enjoys, or that those problems are particularly amenable to coming up with tricks to solve. > * Invariance under linear combinations / convex combinations / algebraic operations -> suffices to check basis elements / extreme elements / generators This is the insight that led me to really start to understand linear algebra while working through Axler's Linear Algebra Done Right. > * Invariance under tensor products -> suffices to check one-dimensional/irreducible case I suppose this is case dependent, but I can imagine some groups irreducible representations are way messier to handle than others!
- tel 4y agoIt’s because it’s all the same trick applied in many different specific domains as more approachable (for domain-relevant experts) examples. It’s all the same trick. That’s sort of the point of what he’s saying. Read until you find one that makes sense and then realize the rest are just variations.
- vervez 4y agoEquivariance is growing in popularity in machine learning, so these tricks will be helpful if one wants to study, or publish, in that area (I'm thinking about it). I'd recommend for any ML-related folks to look into this area and save this thread whenever they're trying to implement an equivariant neural network.
- kidme5 4y agoNice to see Mastadon supports LaTeX
- j2kun 4y agoThis is a feature of the math-focused instance, Mathstodon.xyz
- abdullahkhalids 4y agoNot so much of a feature. They just added Mathjax, which is great and all that is needed. It is annoying following Mathstodon from another server because the math does not render there.
- ripvanwinkle 4y agoThis is the first Mastadon link I've seen on the front page of HN. Hope to see many more
- jiggawatts 4y agoTo try and explain some of the bullet points, because the common sentiment here seems to be that he's speaking an alien language. Terrance is providing a list of things where if "you already know a fact" about a problem or function, then you can take shortcuts based on that fact. So for example, he mentions: > Reflection symmetry -> suffices to test odd and even functions separately. This means that if you know that there is a "mirror image" in the function around the origin, then you can use this fact. The definition of mirror symmetry is: A function f(x) is even if f(-x) = f(x). The function is odd if f(-x) = -f(x) That's quite a strong statement! You can use this to simplify proofs, eliminate cases, etc... Even if you don't know if the mirror symmetry is odd or even, this doesn't really matter because it's just two scenarios! Just check both and you're done.
- paulpauper 4y agoStart with a simpler version of problem works well.
- swframe2 4y agoChatGPT is overloaded at the moment but it would be interesting ask it to explain the intuition of each trick. I suspect it won't be prefect but it will shed a little more light with relatively little effort.
- gerdesj 4y agoWell thanks a lot. It's going to take me roughly 30 bloody (fucking?, yeah fucking, who says bloody these days?) ... fucking years to be able to get to the point when I can possibly comment usefully on this. OK so we have something close to a paper (OK a discussion) in the form of toots. Not sure that's the best way to publish but the world is changing rather quickly and this is surely a rather obvious starter for a chat about media for publication ie media for formal dissemination of ideas. Maths. Shmaths. lol etc.