Dense Math Notation

Here is Jason Dyer's recent attempt to make a proof of the Fundamental Theorem of Arithmetic that is actually nice to read. He's looking for feedback! Dyer's proof

I do get the feeling that mathematics is made somewhat hard to approach for people who grew up programming computers by both the notation that's optimized for writing on blackboards and some unspoken cultural assumptions made by people who worked their way up to the stuff without the notion that a lot of the things might get programmed into computers or be analogous to computer programs.

There might also be a bias in math for excessively clever analytical solutions for things for which a more verbose and straightforward algorithmic approach would do as well. A recent blog post compared an inscrutable closed-form formula for calculating the weekday of a date into a much more readable programming language function.

I wouldn't take that as a very solid example though, in a mathematical sense, the calender system is a massively complex way to represent what is basically the number line. A lot of interesting math is about working with entirely new structures, instead of dealing with accidental complexity in established structures. The programming analogy starts to have trouble on two fronts here.

One problem is that new structures involve new formalisms, and programming languages are more about operating within an existing formalism. Diving into metamathematics by trying to keep things within one formal system when discussing another sounds much more headache-inducing than just doing the informal discussion of the new formal system that mathematicians do now.

The other problem is that the verbose variable names programming languages use, that are helpful when working with somewhat real-world phenomena, become much less useful once you dive sufficiently deep into abstraction. There might not be a better term for the operands of a suitably uncommon and abstract structure X than "things X operates on", so the single-letter names are as good as any. (You don't even have to go into strange theory for this, just see the higher-order functions from any functional language, like map or foldl, which operate on sequences of any values of the same type. Haskell calls the sequences x:xs for "current x" and "the rest of the exes", which is pretty much all you can say about them from the context of the higher-order function.)

If someone wants to experiment with this, I'd like to see how programmers who know little math get along with The Haskell Road to Logic, Maths and Programming and maybe what programming-savvy physics students with some advanced math skills get out of Structure and Interpretation of Classical Mechanics.

It probably depends a lot on the context, but in most situations I would find multi-character variable names distracting. It may work well for short formulae, but try to write down the Taylor series for the hypergeometric 2F1 function, and it may end up like

HypG (firstpar, secondpar, thirdpar, variable) = Sum (i=0, inf, i++) ( variable^i Pochhammer (firstpar, i) Pochhammer (secondpar, i) / ( Pochhammer (thirdpar, i) * Factorial (i) ))

Well, I rather prefer the standard way. Incidentally, once I have argued with a man who held even more drastic stance on this, saying not only that all mathematical formulae should be replaced by algorithms, but also that the raw TeX input is more readable than the processed output. And he even wasn't a programmer.

Equations can be crafted so that their layout on the sheet of paper resonates with our brains' pattern matchers. Concepts like factorization, additivity, conditional independence can be embedded in the spatial layout of equations in a way that makes them immediate without the need to go over the equation with a fine-tooth comb. Unfortunately, this requires brevity, which can decrease parseability overall if not used judiciously.

The main problem I have with mathematical notation right now is that I can't skim it. If I am reading a document with some math notation in, I tend to just skip past it and figure out what's going on from the surrounding text.

I can read it quickly and just see a bunch of apparently meaningless symbols. Or I can read it very, very slowly and carefully and figure out exactly what everything means and exactly what's going on. But there's nothing in between.

Computer code I find rather easier to skim, and natural language is much easier.

Is this a problem with the notation itself or is it just that I work with computer code on a day-to-day basis and don't with maths, so that I've learnt how to skim it and spot the relevant patterns much more easily?

Music theorists tend to use strings of capital letters as variables, which (at least to my non-programmer eye) gives the literature a computer-programming look.

For example, take this classic book by David Lewin, full of variables and operators with names like EQUIV, SAMETYPE, IFUNC, INJ, and so on.