A list of advanced math tricks by Terence Tao
mathstodon.xyz
mathstodon.xyz
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)
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.
> * 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!
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.
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.
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.
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.
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.
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.
The beauty of Tao's list is that he gives actual examples. And many of them too.
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.
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.
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.
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.
> Reflection symmetry -> suffices to test odd and even functions separately.
> Translation invariance -> suffices to test individual plane waves (i.e., to inspect the Fourier multiplier symbol).
> Dilation invariance [if unitary] -> suffices to test homogeneous functions.
> Rotation invariance -> suffices to test the case of spherical harmonic behavior in angular variable (separation of variables).
> Invariance under linear combinations / convex combinations / algebraic operations -> suffices to check basis elements / extreme elements / generators
> Invariance under tensor products -> suffices to check one-dimensional/irreducible case
> Invariance under limits -> suffices to check a dense subclass
> Multiplicative structure (in analytic number theory) -> suffices to check prime powers
> [Principle of induction] Preserved by successor -> suffices to test base case and/or limiting cases
I'm tired of showing how dumb I am for now, but you get me. I am as far as I think.
> 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.
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.
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.
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.
> Invariance under linear combinations / convex combinations / algebraic operations -> suffices to check basis elements / extreme elements / generators
> Multiplicative structure (in analytic number theory) -> suffices to check prime powers
For such functions with "multiplicative structure", if you want to know their value at any point, you only need to values at prime powers.
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.)
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.
So we all have Musk to thank if TT continues tweeting these high quality mastodon posts.
I like how it has built in support LaTex!
https://blog.nawaz.org/posts/2022/Dec/rendering-latex-formul...
If you could I frame the content, then the origin could handle the proper rendering
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.
It is annoying following Mathstodon from another server because the math does not render there.