Notation as a Tool of Thought

But that overhead is constant factor, more or less anything you can express well in matlab can be expressed straightforwardly in APL, too, if you have the right numerical routines. That's not true in the other direction though: there's a lot of stuff in APL you cannot express adequately in matlab at all. For example J (and these days Dyalog as well IIRC) have an operation called under which basically does this: u(f,g) = x => f^-1(g(f(x)). So you can write geometric_mean = u(mean, log).

It is completely impossible to implement something like "under" in matlab. Admittedly the J implementation at least of deriving a generalized inverse for an arbitrary function f is a somewhat ill-defined hack, but this is still something that is both conceptually and practically quite powerful. Also, whilst Matlab is really clunky for anything that is not a 2D array and hardcodes matrix multiplication as the one inner-product, APL has more powerful abstractions for manipulating arbitrary rank arrays and a more general concept of inner products.

Also, APL has some really dumb but cherished-by-the-community ideas that make the language less expressive and much more awkward to learn, e.g. the idea of replicating the terrible defect of normal mathematical notation where - is overloaded for negation and subtraction to every other function.

Tricky.

[0] https://www.numberphile.com/videos/juggling-by-numbers

[1] As far as we know. Without the notation we don't actually know what had been done, but when I took the new patterns to juggling conventions, no one knew them[2].

[2] Actually it's stronger than that. I showed people some of the new patterns at the British Juggling Convention in 1985 and no one knew them. Then at the European Juggling Convention just 4 months later, people from the USA were proclaiming them as the latest patterns that they had just learned, and were perplexed at how I not only knew them, but knew many, many more.

[3] And actually Paul Klimek had beaten us to it, but hadn't been able to get others interested in the notation. As far as we can tell, Paul was the first to get the notation.

Makes perfect sense. Matlab is for engineers, not for mathematicians. They use computer algebra systems, proof assistants, etc. Difference is that engineers (and physicists) want answers and don’t care about how they are obtained, while its the reverse for mathematicians.

I think APL, although it, too, is a language for computing numbers, spiritually is a bit closer to mathematics than Matlab.

In the outer expectation, there’s only one choice: X no longer exists as a potential random variable (as if it were a local variable in the inner expectation) so the outer expectation must be over Y.

I’m not saying that you’re wrong that those subscripts could be used (they often are) but the meaning of the expression is clear after a little while working with expectations.

I’m a little curious about this. Does J have a notion of the relationship between certain functions and their inverse? What is it that enables “under” in J which makes it impossible in Matlab?