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.
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!
It’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.
> 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.
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.
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.
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.
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.
Well 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.
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.
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.
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.
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.
ChatGPT 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.
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.
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.
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.
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.
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.
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.
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.
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.
> 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
¯\_(ツ)_/¯
> [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.
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.
> '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.
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.
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.
> 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.
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.
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.)
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.
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.
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.
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).
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.
