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A very nice fact that I coincidentally learned just yesterday, which is not in the linked answer but is touched on in one of the comments and in an edit on the accepted answer: there is an explicit bijection that maps a binary tree to seven binary trees (and vice-versa, by the inverse function).
Consider the definition of a binary tree: "a binary tree is either empty, or is a root with a left tree and a right tree". This we can write as
T = 1 + T²
T⁷
= T⁶ + T⁸
= T⁵ + T⁷ + T⁸
= T⁴ + T⁶ + T⁷ + T⁸
= T⁴ + 2T⁷
= T³ + T⁵ + 2T⁷
= T³ + T⁶ + T⁷
= T² + T⁴ + T⁶ + T⁷
= T² + T⁵ + T⁷
= T² + T⁶
= T + T³ + T⁶
= T + T² + T⁴ + T⁶
= T + T² + T⁵
= 2T + T³ + T⁵
= 2T + T⁴
= 1 + T + T² + T⁴
= 1 + T + T³
= 1 + T²
= T
"Bijection" -> https://en.wikipedia.org/wiki/Bijection
Monday was looking boring. I'm excited now about the new adventure I've set for myself to somehow inject this (new for me) term into a discussion appropriately.
As long as you surject it whenever you inject it you'll be fine.
This is fascinating, but I am still curious: would I ever have a good reason to repackage seven binary trees as one (or other way around)?
The particular example is not the point. The question you should be asking is
> would I ever have a need to understand what happens when you perform algebraic manipulations on data structures
and the answer is as usual that it depends on what you're working on. Parsers? Absolutely. CRUD apps? Yes, but only if you look for them. Numerical linear algebra? Probably not.
> CRUD apps? Yes, but only if you look for them.
I upvoted just for this breath of fresh air right here. CRUD apps are boring and crappy because people make them boring and crappy. I find uses for this type of thing all the time in "boring" CRUD apps, because I look for them.
This is like giving an artist a new shade of color they've never had before, like a new rich blue or something. It isn't that everything is improved by using that new color as much as possible everywhere. It's that it opens up new possibilities you may never have considered before. This insight may be useless, but there are other useful ones that can emerge that you wouldn't have thought of on your own in a long time.
I can't imagine any application. :-)
The fascination is more that of explanation: if you treat T = 1 + T^2 as an algebraic equation in the complex numbers, you can get a lot of expressions that are meaningless: it's hard to make sense of T = 1/2 ± i√3/2 or T^3 = -1 or even T^6 = 1 (there is certainly no bijection sending six trees to the empty tree). But then, T^7 = T is suddenly very much meaningful and we can turn it into an explicit description. What's going on? Which equations are "meaningful" and which are not?
Blass's paper gives an explanation: an equation is meaningful if you can derive it by working in (something like) N[T]/(T = 1 + T^2) (with some more conditions). Baez's discussion (linked above) puts it in terms of "categorification", and is also interesting.
Sharding a huge database index (or other binary tree structure) onto 7 different servers or hard drives? Or 49 (recursively apply it)? I don't know if that's actually useful here but it's the first thing that came to mind.
Wouldn't t be a cube root of -1?
t=t^2+1 => 0=t^2-t+1 => 0=(t+1)(t^2-t+1)=t^3+1
Same thing. A cube root of -1 is a sixth root of 1.
as someone that's self taught, i really wish math was taught via stronly typed programming languages.
Math is so full of loosey-goosey context dependent ambiguous notation. It makes it easy to have a very terse expression communicate a powerful idea, but it makes it impossible to just parse an arbitrary expression.
Just as an example, look at the maclauren series as defined in the link. It uses f-dot, f-double-dot, then f^(n), having to note that (n) is the nth derivative rather than exponentiation. You have syntax switching and an ambiguity in the same expression.
The issues you bring up aren't really like types, though. Expecting every domain of math to use the same notation and is more like expecting every programming language to use the same syntax.
Derivatives are a bit of a weird case because some really intuitive and useful sub-domains only really need the 1-2 derivatives in time. So, hitting them with the whole (dy/dx)^(N) is overkill.
> it makes it impossible to just parse an arbitrary expression.
All language suffers from this. Without context (assumed by the reader, provided fully or at least referenced by the writer) all arbitrary expressions are impossible to parse. Tell me what this means:
map f xs
Math isn't any different here, don't try to parse arbitrary expressions until you've identified the context. Maybe a fool used notation foolishly, maybe an adversarial mathematician used notation to obfuscate and has, hidden away in their diary, a key that explains it all. But most of the time the context will be provided directly or indirectly. The notation will be explained in the text or the text will be in a field with notational conventions that the reader is expected to know. Same with code, same with every other human language.
Sure but I can load any program with that statement into my editor and mouse over each word to view the details or hit F12 to see the definition :p If any of those symbols are undefined, the editor or compiler will tell me.
Can you? What language is it?
As somebody with a PhD in math, I really wish that math was taught via strongly typed programming languages. There is so incredibly much low-lying work done through deep layers of abstraction that it can take hours or even days to digest a short paragraph in a field adjacent to one's expertise.
Of course, that's the dream. The reality is that people are making very strong efforts at computer proofs, and the languages I've worked with in that space are highly un-ergonomic. On the other side of the chasm, calculus of types is largely relegated to category theory which is its own onion of complexity. To my awareness, Haskell is currently the sweet spot. In the hopes of being proven wrong, I'd guess that we're at least a century away from a Grand Unified Notation that allows a fluid translation between what the mathematician is thinking and how the computer can compute.
Lean 4 seems promising, too.[1] Although it looks very much like Haskell.
[1] https://leanprover.github.io/functional_programming_in_lean/
> There is so incredibly much low-lying work done through deep layers of abstraction that it can take hours or even days to digest a short paragraph in a field adjacent to one's expertise.
Indeed, once you start to think about formalizing a research-level proof, it brings into focus the sheer number of trivial cases and natural transformations that a standard informal proof in a paper omits.
> I really wish math was taught via strongly typed programming languages
I'll use this opportunity to recommend Conrad Wolfram's "The Math(s) Fix" that outlines his Computer-based Math approach. In this book he asks (and answers) what would school look like if we taught mathematics as if computers exist.
Those are very familiar notations to anyone who has taken undergraduate math classes. There's some ambiguity but it's fairly easy to dispel with, like, a cheat sheet.
More importantly they are familiar intimately to those active in the field. And often reflect an underlying ideas that are useful in the field.
People often write equations out in full and marvel at the ludicrous clunkiness of it all. Usually because they expect that will help them understand a concept. That doesn't work.
The notation often suggests a manner of thinking about an idea that is more useful than seeing the idea spelled out.
Of course you probably should have spent some time peering into the blackbox of notation and physicists are probably most notorious for not caring to.
I'm unsure what your point is -- could you please clarify?
Not the guy you're replying to although I subcommented when I ought to have replied to you.
The notation often suggests a manner of thinking about an idea that most won't be able to see in the expanded notations.
As such those familiar with the field often abstract away the minutia.
I always heard a rather unfair terse statement about science. If you have to read the articles introduction section, you probably shouldn't be reading the article. That's wildly unfair but does reflect that academia is mostly concerned with academics in the same field.
Easy for you
And easy for you with a few hours of work. The notations are by faaar not the difficulty in understanding some like this.
Don't get me wrong, there are all sorts of disastrous notations in math that should be fixed. Those just aren't them.
Anyway, the computer science version with types is gonna be so obtuse that you're gonna wish you had this back.
Just as an example, look at the maclauren series as defined in the link. It uses f-dot, f-double-dot, then f^(n), having to note that (n) is the nth derivative rather than exponentiation. You have syntax switching and an ambiguity in the same expression.
That’s nothing. Study real analysis long enough to investigate sequence spaces and you’re looking at sequences of sequences. The notation for those will often use subscripts and superscripts as indices at the same time! But generally you’re not dealing with derivatives or exponents at the time so it’s not too bad.
I totally agree with the general idea.
Math always felt very random and arbitrary to me. Especially the fact that proofs are essentially just prosa and people believe in them or they don't. And that's why we ended up with proofs that essentially were correct but still had flaws and errors in them for a long time. Or proofs that actually turned out to be wrong in the end.
You can give a try formalizing some basic math with proof assistants. With such an attempt, you will discover that 90% of your time will be spent transforming types into other types and working through obvious transformations.
And when you are done, the least obvious steps (for your reader) will be hidden in a forest of obvious steps. Suppose that you want to communicate to someone how to get to B from A. You don't tell them "lift your foot in this manner, then lift your other foot in this other manner, go a little left when you see this small obstacle", etc... you assume that they have competence at navigating commonplace obstacles, and give just the directions that they should take at intersections.
But for math proofs, we shouldn't assume that people have competence of common basic parts. At least not to truly help them gain a good understanding of the constructive proofs.
To me, the example is rather to tell them get into the car then press the button and turn the wheel like this and you'll reach your destination. But they won't understand the car and I think they should, because at some point the need to, unless they never dig deeper - but then maybe we don't need to teach those theoretical things from the beginning.
Wow, you articulated something I've always been frustrated with, but was never able to put into words. I've always been convinced that most of (thought not all) the difficulty in learning math is because of sloppy and ambiguous explanations.
Unfortunately, even simple undergraduate proofs can be monumentally difficult or even impossible to translate into a type theory due to inherent limitations of the theory or the type checker.
SICM has a similar complaint about ambiguous notation. See https://groups.csail.mit.edu/mac/users/gjs/6946/sicm-html/bo.... Footnote 2 in particular is worth reading for a particularly good example of problems caused by ambiguity.
SICM as a textbook is intended to be worked out entirely with a computer.
The quote in the footnote[1] says that `f` means something different on the two sides of the equation ∂f/∂x = ∂f/∂u ∂u/∂x + ∂f/∂v ∂v/∂x.
Why is this true? I think about ∂/∂x, ∂/∂u, and ∂/∂v as higher-order functions that take the same `f` as their argument. What is wrong about this thinking?
[1] https://groups.csail.mit.edu/mac/users/gjs/6946/sicm-html/bo...
Edit: I'm guessing the footnote's reasoning is that on the left hand side, `f` is thought of as `f(x, y)`, and on the right as `f(u(x, y), v(x, y))`.
In software, it's very common to have someone with broad experience in a lot of fields need to jump in and understand these 20 lines of code that are causing some issue because of a change in business logic requirements. To have a temp or newbie be able to quickly close a Jira ticket, you absolutely want verbose, strongly typed, descriptive text. You want to optimize for comprehension by readers rather than convenience for writers.
Mathematics, though, is a language developed by academics for academics, who will spend a long time iterating through variations on the same pattern. When you're years into your post-grad, looking at the same tiny slice of mathematics over and over again, it's obvious what f^(n) means - you've probably written that and thought about it thousands of times - but yeah, it does suck for outsiders.
It's really annoying when you do the maths equivalent of Stack Overflow copy-pasting.
Often I'll see some maths equation in a paper or blog post or whatever that I want to use in software, but I don't understand the notation and so I can't use it without spending god knows how long learning all the context first.
> You want to optimize for comprehension by readers rather than convenience for writers.
Given the exorbitant cost of education (in the US), I think this is a reasonable desire or expectation for a student to have.
I'd love this. I'm in the camp where seeing an equation implemented in code helps my understanding much more than seeing the math notation.
As someone who has graduate level math education and about 25 years working as a professional programmer, I emphatically disagree.
A type system is itself a programming language. If it is going to implement complex enough logic, it will be Turing complete. It does not magically become clearer simply because it ideas have been expressed in a type system. For an amusing demonstration, read through https://aphyr.com/posts/342-typing-the-technical-interview. That is a working program that solves the 8 queens problem entirely in Haskell's type system.
And the type systems that you are used to are simpler in part because they are NOT Turing complete. This limits what you can say in them. In this case, for example, the author was unable to express higher derivatives as types.
Now perhaps your desire is for unambiguous machine-checkable logic? If so then I highly recommend learning something like https://coq.inria.fr/. It is a worthwhile enterprise, and there are serious mathematicians who say that it, or something like it, should be implemented for all of math. However one of the objections to it is that it is harder to express and understand yourself in such systems. So unambiguous machine-checkability is a win, but not necessarily the one you thought it was.
And then we come to the real issue. Are you optimizing for use, or for the person who doesn't understand the subject? It is literally impossible to think about complex things without a concise notation. The actual notation that we have has a lot of history and rough spots. But verbosity impedes comprehension. So what makes life easy for the novice and expert are exactly opposite. Also most of the use of the material is by experts, who are also who the notation is designed to be most helpful for. However even for students it helps. So much so that learning the notation generally pays off within a course or two. That is, learning the same ideas WITHOUT the notation would have been harder than learning both notation and ideas. So notation merely creates an obvious barrier, but then helps with the real one.
I know you don't believe me. You've told others that it is about what works best for you. But with all due respect, you can't have any idea what actually works best for you until you've actually mastered a sufficient amount of the subject. Which you apparently haven't. Until then you are best off believing people who HAVE mastered the subject, and have spent sufficient time teaching and tutoring to know where the real barriers to understanding are for students. And everyone I know in that boat, including me, says that the notation helps more than it hurts.
Agree, but could that be like wishing that all video games are implemented using the same engine?
Check out Homotopy Type Theory / Univalent Type Theory and HoTTest summer school :)
It’s not exactly easy reading but take a look at Homotopy Type Theory w Univalent Foundations which is free online.
Maybe not as the first introduction, but yes, this is worth doing.
Stuff like this is so cool. It reminds me of umbral calculus, another place where identities we’re used to from single variable calc apply correctly in other surprising places.
It’s especially amazing how we can invent “illegal operations” like the square root of a datatype (or the square root of minus one) and prove that we’ll get the right answer in the original domain if it exists. I’d bet I miss lots of opportunities to take advantage of inventing nonsensical operations to solve hard problems.
It seems like a lot of the logic we use for reasoning about complex systems has the same combinatorial underpinnings (e.g. continuous systems in calc, recurrence relations and polynomials elsewhere, and algebraic datatypes in OP). I’d love to see an intro to all these variations of “calculus” written up from those combinatorial foundations that applies to all of the different applications, rather than via epsilon-delta derivations that only apply to continuous systems.
Anybody seen a write-up like that anywhere?
I think you're looking for "combinatorial species". This paper gives a decent introduction in a programming context: https://www.cas.mcmaster.ca/~carette/publications/labelled-s...
Calculus already has types. The type of a function is f:R -> R.
If X and Y are normed linear spaces and f:X -> Y the type of the (Frechet) derivative is Df:X -> L(X,Y), where L(X,Y) is the space of linear operators from X to Y.
The linked discussion is actually about performing calculus-like operations on type-theory types. (A natural next step after observing the similarities between algebraic types and algebra.) It's not about imposing a type system onto traditional calculus.
I had a different read on that but I am familiar with the link to Haskell zippers. People seem to assume if someone wrote something about Haskell they must be smart.
Their "derivative" of trees implicitly assumes the cartesian product A x B is isomorphic to B x A.
Just like the Frechet derivative for non-commutative Banach spaces, the derivative of T^2 is not 2 T dT, it is T dT + dT T.
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