[HN Gopher] A list of advanced math tricks by Terence Tao
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A list of advanced math tricks by Terence Tao
Author : signa11
Score : 329 points
Date : 2022-12-12 09:03 UTC (13 hours ago)
(HTM) web link (mathstodon.xyz)
(TXT) w3m dump (mathstodon.xyz)
| jiggawatts wrote:
| To 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.
| ak_111 wrote:
| Tangential, 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.
| abdullahkhalids wrote:
| Actually a number of mathematicians I follow on Twitter have
| all partially or completely moved to this
| https://mathstodon.xyz server.
|
| I do follow them there as well, but its kind of difficult
| because my Mastodon timeline is all high-level math and
| requires a lot more mental engagement, whereas on Twitter there
| was always lighter stuff (from other accounts) in between the
| math.
| pohl wrote:
| Sounds like you just need to find some lighter folk to round
| out the set you follow a bit.
| ykonstant wrote:
| I 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 wrote:
| Mathematicians of all ages should sign up here,
| https://mathstodon.xyz/auth/sign_up
|
| I like how it has built in support LaTex!
| BeetleB wrote:
| If 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-
| formul...
| sitkack wrote:
| I could see many domain specific fediverse instances.
| Music, 3d modeling, knitting, etc.
|
| If you could I frame the content, then the origin could
| handle the proper rendering
| jvm___ wrote:
| mathstodon
| j2kun wrote:
| The 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.
| nmitchko wrote:
| As 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 wrote:
| Going 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 wrote:
| As 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.
| dysoco wrote:
| Yeah I know some of those words
| eachro wrote:
| It'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!
| S4M wrote:
| There 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-
| arbi...
| paulpauper wrote:
| Start with a simpler version of problem works well.
| vervez wrote:
| Equivariance 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 wrote:
| Nice to see Mastadon supports LaTeX
| ripvanwinkle wrote:
| This is the first Mastadon link I've seen on the front page of
| HN. Hope to see many more
| j2kun wrote:
| This is a feature of the math-focused instance, Mathstodon.xyz
| abdullahkhalids wrote:
| Not 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.
| dj_mc_merlin wrote:
| It 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.
| [deleted]
| scarmig wrote:
| For 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.
| AlexSW wrote:
| You may not be as far as you think.
|
| If you've done 'proof by induction', that's listed there
| (beginning of post 6), and should hopefully give you some
| grounding.
|
| Several of these are not as complicated as they may look.
| Though some of them I certainly have no ideas on at all.
| dj_mc_merlin wrote:
| For fun then, let's see just how far I am:
| > Reflection symmetry -> suffices to test odd and even
| functions separately.
|
| Maybe reflection means flipping a function? Or when it goes
| above or below an axis? > Translation
| invariance -> suffices to test individual plane waves (i.e.,
| to inspect the Fourier multiplier symbol).
|
| Fourier transform has something to do with frequencies. A
| plane wave sounds like part of a DnD spell.
| > Dilation invariance [if unitary] -> suffices to test
| homogeneous functions.
|
| If this didn't have "functions" in it I would think it's
| biology. > Rotation invariance -> suffices
| to test the case of spherical harmonic behavior in angular
| variable (separation of variables).
|
| Good thing the spherical harmonic behaviour sufficies, I do
| not know I would test the cubic dissonant one.
| > Invariance under linear combinations / convex combinations
| / algebraic operations -> suffices to check basis elements /
| extreme elements / generators
|
| Combinations has something to do with choosing things?
| Generators.. no, I give up. > Invariance
| under tensor products -> suffices to check one-
| dimensional/irreducible case
|
| Aha! Tensors! Like TensorFlow! I know one-dimensional too!
| This is the closest I've gotten to having a smart thought.
| > Invariance under limits -> suffices to check a dense
| subclass
|
| "Dense subclass" is exactly how I'm feeling right now.
| > Multiplicative structure (in analytic number theory) ->
| suffices to check prime powers
|
| -\\_(tsu)_/- > [Principle of induction]
| Preserved by successor -> suffices to test base case and/or
| limiting cases
|
| I do know this one, but I actually wouldn't have figured it
| out if he hadn't pointed out this is induction.
|
| I'm tired of showing how dumb I am for now, but you get me. I
| am as far as I think.
| AlexSW wrote:
| I think you're getting at the point with this though:
|
| > I actually wouldn't have figured it out if he hadn't
| pointed out this is induction.
|
| A lot of these other ones are not that complicated either,
| it's just hard to parse and sometimes requires a small bit
| of math knowledge, which I think is within reach if you're
| familiar with induction and would only take e.g. a minute
| for you to understand if someone explained it (ideally with
| a blackboard than in text).
|
| Some of those are definitely nonsense to me. But, to put my
| money where my mouth is and to give some examples (as you
| spent the time to type that out), 'invariance under linear
| combinations' just means that, if something holds for f(x),
| then it holds for 2f(x), f(x) + 10, etc. (these are all
| linear combinations). So then saying 'suffices to check
| basis elements' means 'just check f(x), and the others all
| fall out for free'.
|
| As another example, convex combinations: Google 'convex
| hull' for images, but basically this one is just saying
| just check the 'extreme elements', i.e. the 'corners' at
| the boundary, and everything in the middle falls out for
| free because we have 'invariance under convex
| combinations'. A convex combination here is just a some
| point in the middle of these extremes.
|
| While these may not be immediately obvious when reading
| them, the pictures or ideas are actually sometimes quite
| simple.
|
| I hope that helps.
| dj_mc_merlin wrote:
| You're right there. Or at least you definitely have a
| point.
|
| I would question whether being able to see this pinhole
| perspective of the maths he's talking about really means
| I "understand" it, or if that's just a semantic game. I
| don't feel like I am any more able to do something with
| the information than a second ago, even though I
| "understand" it more.
| AlexSW wrote:
| Glad it made a bit of sense! But that's a fair point
| about whether it really constitutes 'understanding'.
|
| I suppose it's equivalent here to whether knowing the
| meaning of a particular sentence in the alien language
| qualifies as understanding it, vs being able to
| meaningfully speak about it in this alien language, and
| use the sentence in a conversation. Seems like just
| semantics to me.
| BeetleB wrote:
| > 'invariance under linear combinations' just means that,
| if something holds for f(x), then it holds for 2f(x),
| f(x) + 10, etc. (these are all linear combinations).
|
| I'm not sure how Terence meant it, but for some people,
| it actually means that some property that holds for f(x)
| will also hold for f(ax + b).
|
| > So then saying 'suffices to check basis elements' means
| 'just check f(x), and the others all fall out for free'.
|
| Yes, but your statement confuses me more than Terence's
| :-)
|
| A given vector space has basis elements (e.g. x, y and z
| unit vectors for 3-D Cartesian space). It means that if
| you can show the property is true for the basis elements,
| you've now shown it's true for any vector in that space.
| One needs to show linearity holds to assume this.
|
| > As another example, convex combinations: Google 'convex
| hull' for images, but basically this one is just saying
| just check the 'extreme elements', i.e. the 'corners' at
| the boundary, and everything in the middle falls out for
| free because we have 'invariance under convex
| combinations'. A convex combination here is just a some
| point in the middle of these extremes.
|
| That actually helped - thanks.
| btilly wrote:
| No, you're closer than it feels. Here are what they mean in
| more every day terms.)
|
| Reflection symmetry - the laws of physics should be the
| same for an observer who is a mirror image of ourselves.
| (In classical mechanics this is so. In quantum mechanics
| you also have to reverse the sign of all charges.)
|
| Translation symmetry - The laws of physics should be the
| same for an observer in a different position in space.
| (This one is true.)
|
| Rotational symmetry - the laws of physics should be the
| same for an observer who is rotated from ourselves. (Again
| true.)
|
| Combinations - just refers to combining things by some
| rule. A linear combination of vectors is just adding
| scalars times vectors. So the set of linear combinations of
| 2 vectors is the plane spanned by those vectors. A convex
| combination of points is any average of them. So the convex
| combinations of 3 points is a triangle. And if you allow
| some set of algebraic operations, from a fixed set of
| values you can generate a whole system. Those fixed values
| are generators for that system. So, for example, (1, 0) and
| (0, 1) with the algebraic operations of addition and
| subtraction generate the whole grid of points (n, m) with
| n, m integers. In all of these cases you can often work
| with just the few things from which the
| plane/triangle/grid/whatever was created, without having to
| look at the rest.
|
| Tensors - not going through that topic here. Start with
| https://en.wikipedia.org/wiki/Tensor_product if you're
| curious.
|
| Dense subclass - for any point you choose, for any distance
| you choose, the dense subclass includes something at least
| that close. For example the rational numbers are dense
| within the reals. Pi is not a rational number, but we have
| no trouble finding rational numbers within 1 billionth of
| pi. So you can sometimes prove something for all rational
| numbers, and then find you've proven it for all real ones.
| (For example, with powers, roots, and division we can
| define 2^x for all rational numbers x and prove properties
| about it. From that we can actually define 2^x for all real
| numbers.)
| jordigh wrote:
| > Invariance under linear combinations / convex
| combinations / algebraic operations -> suffices to check
| basis elements / extreme elements / generators
|
| Linear combinations of a basis (b1, b2, ..., bn) are sums
| of the form a1*b1 + a1*b2 + ... an*bn using some
| coefficients (a1, a2, ..., an). Convex combinations add the
| requirement that a1 + a2 + ... an = 1. Algebraic operations
| adds other things besides adding and multiplying by
| coefficients. Anyway, he's saying that if you have a
| function for which f(a1*b1 + a2*b2) = a1*f(b1) + a2*f(b2),
| then you only need to know what f does to b1 and b2 and the
| other basis elements in order to know what it does to
| anything. > Multiplicative structure (in
| analytic number theory) -> suffices to check prime powers
|
| Here he's talking about how (for example) some functions
| f(ab) = f(a)f(b) (sometimes with the extra condition that
| `a` and `b` have no factors in common).
|
| For such functions with "multiplicative structure", if you
| want to know their value at any point, you only need to
| values at prime powers.
| 082349872349872 wrote:
| Are 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]
| mrmuagi wrote:
| Induction 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.
| commandlinefan wrote:
| I came to the comments to see if anybody did understand it. (I
| know I didn't).
| bryanrasmussen wrote:
| I often upvote things that I don't have time to say anything
| about, not that I've upvoted this, but there can be lots of
| reasons to upvote something and not comment on it.
| dj_mc_merlin wrote:
| Sure.. but you've been around this site for at least as long
| as I have, so you must've seen that you can directly tell how
| technical a discussion is by the points/comments ratio. The
| ranking goes like this:
|
| 1. Political/current events stories have a million comments
| -- sometimes more comments than points if the OP is a
| controversial opinion
|
| 2. Software engineering opinion pieces have the 2nd most
| comments (everyone has an opinion), followed by
|
| 3. Software engineering technical pieces. These usually have
| a few comments by some smartypants that has worked with the
| software, and a few other people hallucinating technical
| challenges around them for some reason.
|
| 4. The least commented on are these kinds of pieces that
| require at least a master's in some other non-SWE field.
| Mathematics gets quite popular since most of HN was "good at
| maths", and thus will upvote it, even though the most maths
| they have understood in the last decade was when they copied
| some formulas from a random paper.
| johnfn wrote:
| I really enjoy this comment because I feel like it
| articulates a pattern which we are all kind of familiar
| with.
|
| Perhaps you also need a 5th category, for the rare post
| which is so well-written (or remarkable along some other
| axis) that people are too stunned to leave comments. E.g.
| [1] which arguably fits into 2) but is so on-point that the
| normal opinionated argumentative comments didn't even break
| out for the most part.
|
| [1]: https://news.ycombinator.com/item?id=31840331
| stocknoob wrote:
| It's great. The infamous HN "well actually" contrarian is
| left speechless.
| sdwr wrote:
| Well, being contrary on HN is actually the majority
| viewpoint. Piggybacking on other people's thoughts and
| poking holes in them is the easiest way to maintain a
| healthy defensive wall of superiority while
| communicating.
| Workaccount2 wrote:
| We all know this post is going over 95% of our heads.
| hardware2win wrote:
| Ah, good old mathematics (business model: uber for abstract
| concepts)
| antman wrote:
| I upvoted because the sentences look beautiful and it is pretty
| sure they are right. I could not not upvote.
| jordigh wrote:
| He's talking about the general principle that if something only
| depends on composable atoms, then in order to know how that
| thing changes, you only need to know how those composable atoms
| change.
|
| Probably a simple example is a rotation in the plane about the
| origin. If you know it's a rotation, then in order to know how
| any point in the plane is moved by this rotation, you only need
| to know how one point is moved.
| nerdponx wrote:
| > 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.
| 082349872349872 wrote:
| Is there a reason @JadeMasterMath's comment doesn't generalise
| the list?
| red_trumpet wrote:
| What 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 wrote:
| I 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 wrote:
| Ha, 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 wrote:
| mcphage wrote:
| It 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.
| hgsgm wrote:
| It generalizes even further: If you need to prove B, and you
| know A implies B, all you need to prove is A.
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