Calculus with types

https://stackoverflow.com/questions/9190352/abusing-the-algebra-of-algebraic-data-types-why-does-this-work/41418365#41418365

by isomorph • 4 years ago

123 61 4 years ago

stackoverflow.com

chrsig5• 4 years ago

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.

ajkjk4• 4 years ago

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.

chrsig5• 4 years ago

I'm unsure what your point is -- could you please clarify?

psychphysic2• 4 years ago

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.

danielvaughn2• 4 years ago

Easy for you

valenterry2• 4 years ago

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.

jhanschoo3• 4 years ago

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.

falcor84• 4 years ago

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.

danielvaughn2• 4 years ago

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.

aimor• 4 years ago

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.

downboots• 4 years ago

Agree, but could that be like wishing that all video games are implemented using the same engine?

bee_rider• 4 years ago

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.

chongli• 4 years ago

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.

jshaqaw• 4 years ago

It’s not exactly easy reading but take a look at Homotopy Type Theory w Univalent Foundations which is free online.

tsss• 4 years ago

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.

LeifCarrotson• 4 years ago

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.

chrsig5• 4 years ago

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.

Asraelite• 4 years ago

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.

Jtsummers4• 4 years ago

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

If you guessed it was mapping the function f over the sequence xs, maybe you're right, maybe not. Maybe a fool wrote it who can't communicate and chose bad names. Maybe an adversarial programmer (wants job security, or is just a dick) uses `map` to mean `reduce` or `mapreverse` (so that it does most of what you expect, but produces a reversed output from what you expect).

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.

squeaky-clean2• 4 years ago

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.

btilly• 4 years ago

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.

acrefoot• 4 years ago

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.

nequo2• 4 years ago

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))`.

klyrs2• 4 years ago

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.

nequo2• 4 years ago

Lean 4 seems promising, too.[1] Although it looks very much like Haskell.

[1] https://leanprover.github.io/functional_programming_in_lean/

jhanschoo3• 4 years ago

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.

civilized• 4 years ago

Maybe not as the first introduction, but yes, this is worth doing.

vez-• 4 years ago

Check out Homotopy Type Theory / Univalent Type Theory and HoTTest summer school :)

6gvONxR4sf7o2• 4 years ago

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?

hither_shores2• 4 years ago

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...

svat2• 4 years ago

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

Solving the analogous equation z = 1 + z² in the complex numbers gives z to be a primitive sixth-root of unity (https://www.wolframalpha.com/input?i2d=true&i=z+%3D+1+%2B+Po...), i.e. z⁶ = 1, and so z⁷ = z. It turns out that we can similarly show T⁷ = T with some algebra (without subtractions) by repeatedly applying the transformations T→1+T² or 1+T²→T (a "game" version is at http://lebarde.alwaysdata.net/seventrees/):

    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

This can be used to give an explicit bijection: see Andreas Blass's paper "Seven trees in one": https://arxiv.org/pdf/math/9405205 (via https://twitter.com/CihanPostsThms/status/157901771352705024... — see also https://math.ucr.edu/home/baez/week202.html which has more.)

travisgriggs• 4 years ago

"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.

harveywi• 4 years ago

As long as you surject it whenever you inject it you'll be fine.

krastanov• 4 years ago

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)?

hither_shores2• 4 years ago

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.

svat2• 4 years ago

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.

jerf• 4 years ago

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.

feoren2• 4 years ago

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.

mjcohen• 4 years ago

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

6gvONxR4sf7o2• 4 years ago

Same thing. A cube root of -1 is a sixth root of 1.

keithalewis2• 4 years ago

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.

cvoss• 4 years ago

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.

nh23423fefe• 4 years ago

politest RTFA

keithalewis2• 4 years ago

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.

Calculus with types

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