A Tribute to Euler (2008) [video] | Hacker News Reader

A Tribute to Euler (2008) [video]

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40 pointsrfreytag13 comments

Euler was a master of generating functions. Here is a little taste. Start with a Haskell data type:

   data Tree x = Leaf | Node x (Tree x) (Tree x)

The question is how many different Trees of size n are there, where n is the number of x values in the tree.

  Size 0: Leaf
  Size 1: Node x Leaf Leaf
  Size 2: Node x (Node x Leaf Leaf) Leaf, Node Leaf (Node x Leaf Leaf)

It turns out this sequence goes 1,1,2,5,14,42,...

Now we do black magic: we take the Tree x = Leaf | Node x (Tree x) (Tree x) equation and replace Tree x with a function T(x), replace each constructor (Leaf or Node) with the number 1, and replace | with +. We get:

  T(x) = 1 + 1*x*T(x)*T(x)

Simplifying:

  T(x) = 1 + xT(x)^2

We can solve that:

  T(x) = (1 - sqrt(1 - 4x))/2x

Now we do a series expansion of T:

  T(x) = 1 + 1x + 2x^2 + 5x^3 + 14x^4 + 42x^5 + ...

https://www.wolframalpha.com/input/?i=series+(1+-+sqrt(1+-+4...

Magic!

You can try it yourself with:

    data List x = Leaf | Node x (List x)
    data Tree2 x = Leaf | Single x | Node x (Tree2 x) (Tree2 x)

And if you're really adventurous, try:

    data Tree3 x = Node x (List (Tree3 x))
    data Tree4 x y = Leaf y | Node x (Tree4 x y) (Tree4 x y)
    data Tree5 x = Tree2 (Tree2 x)
    data Foo x = Leaf x | Node (Foo x)

More on this, for the curious: https://codewords.recurse.com/issues/three/algebra-and-calcu...

If you are interested in learning more about this: https://www.math.upenn.edu/~wilf/DownldGF.html

There's also http://algo.inria.fr/flajolet/Publications/book.pdf. The first chapter of this book is more or less equivalent to generatingfunctionology, but has a more modern flavour.

Wow, nice!

Euler was a genius, and he wrote a bunch of calculus textbooks. I wonder if they are still worth studying - other than out of historical interest, that is.

Historical interest is fine, but if you just want to learn math I wouldn’t bother.

You say this, but I personally tried many approaches to math, and historical approach is the only one that seems to keep my attention. It feels much more like an adventure in time, space and mind like this. Currently looking forward to Liber Abaci sometime in 2019.

It definitely is a roundabout and non-pragmatic way, I agree, but I find it rewarding enough to keep me going. Unrelated, I think children's math education could benefit a lot from this kind of approach.

There are quite a few books that take this approach actually. One of my favourites: https://www.springer.com/gp/book/9781441960528

What makes you so sure? Euler's approach to problem solving is legendary for still being good pedagogical material today: http://eulerarchive.maa.org/hedi/index.html

In Euler's time they still didn't have a correct understanding of higher order differentials, so the work from this time period has genuine errors that you would need to be aware of. Might I suggest an alternative? There's a wonderful little book by Nathanial Grossman called 'The Sheer Joy of Celestial Mechanics' [1]. It assumes vector calculus of course but otherwise might be just what you're looking for. From a prepublication review:

> Don't look for axioms to memorize. Too many courses are consecrated to teaching students to play chords on a set of axioms. This book celebrates the heroic age of calculus, the time of Euler, Maclaurin, Clairault, Lagrange, and Laplace, a time before delta and epsilon. [...] mathematics was invented to do things, not just to be talked about, and today - still - its greatest triumphs are what it can do.

[1] https://www.amazon.com/Sheer-Joy-Celestial-Mechanics-dp-0817...

There might be some genuine errors, but the fact that you could manipulate differentials algebraically (a la Leibniz or Euler) is a big deal.

A big deal that was later made concrete: https://en.wikipedia.org/wiki/Non-standard_analysis

You don't need nonstandard analysis to make differentials concrete - Cauchy showed us how to model them with ordinary variables, and this method is taught in standard undergrad textbooks like Stewart.

The issue i was bringing up is that the early use of differentials does not correspond to this. The algebraic models of differentials presented in Cauchy or Robinson's nonstandard analysis is not the same thing as the algebraic models used in Euler. The book i suggested uses differentials from beginning to end, but uses the modern, correct form that doesn't lead one to wrong answers when manipulated with standard algebraic rules. There is absolutely no reason to go back in time and inflict this confusion on yourself by intentionally unlearning the hard-won right answer.

Although I should point out that Archimedes correctly understood how to compare different orders of infinitesimals in his book 'On Spirals' (via tangents), circa 200 BCE! Wow!!!

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