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Comment by hoosieree | Hacker News Reader

I love the "Notation and Nomenclature" section. So many academic papers assume readers will infer the type of a variable by its font styling, which is frustrating if you aren't already very knowledgeable in that particular field.

To clarify - using an actual programming language (any language, I'm not picky) would still be better than inventing an ad-hoc, unverifiable font-based type system every time you publish a new paper. But having a "Notation" section is still vastly better than the usual approach of "guess what I mean".

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To be fair, fields have conventions that help quite a bit. Matrices are almost always (I have seen exceptions) capital letters.

I used to be an advocate for some kind of universal notation, but honestly it is not practical. A useful notation for one paper can be incredibly cumbersome in another paper in the same field. This gets worse if you’re paper is at the border of two fields and their notational conventions are mutually inconsistent. It’s best to just have a notation section that sets forth your convention in the document and stick to it. Ultimately, if you’re reading a paper in an area in which you are knowledgeable and can’t parse the notation quickly, the author failed to write the paper well.

As Whitehead said, “By relieving the brain of all unnecessary work, a good notation sets it free to concentrate on more advanced problems, and in effect increases the mental power of the race.”

I agree with the Whitehead quote. Ken Iverson opened his 1979 Turing award lecture ("Notation as a Tool of Thought") with it. And APL would be a suitable replacement for 90% of the bespoke pseudo-notations I've seen in hundreds of academic papers in my domain (machine learning and reverse engineering).

Contrast this with 2 papers I've read recently which independently defined a "function" that returns 1 when its argument is true, and 0 when its argument is false. In both cases, the purpose of this function was to either keep or cancel part of an arithmetic expression by multiplying by either 1 or 0. Each paper defined this function with different notation (one used prefix notation: "Fn(condition)", the other used brackets: "[[condition]]"). Each paper spent several sentences defining the meaning of this notation and the associated function.

Meanwhile in APL, true and false are 1 and 0. If these authors had used APL, they could both have used the same notation, and neither would need to explain beyond saying something like "we use APL notation here". This behavior is not unique to APL - JavaScript, C, Python, and many others have "truthy" values which can be treated as 1 or 0 for the purpose of multiplication.

I don't care if authors use APL (or Python, or Idris, or whatever). I just would prefer that (when possible) they use an existing language which is actually learnable. If it's acceptable to not define your notation (relying instead on tacit conventions and context) why not use an existing language?

> So many academic papers assume readers will infer the type of a variable by its font styling

wait seriously??

Yes, have you never read a math paper before? Bold = vector is a common convention, for example.

Mathematical notation is a programming language.

The notation section is type definitions.

> Mathematical notation is a programming language.

It actually isn't - the notation has lots of ambiguities that don't confuse mathematicians because they are aware of the context.

On Proof and Progress in Mathematics[1][2] is a great essay by a Field's medalist. Part of the essay discusses this very topic, and why mathematicians choose not to use a formal notation for every day work. Some quotes from it:

> The standard of correctness and completeness necessary to get a computer program to work at all is a couple of orders of magnitude higher than the mathematical community’s standard of valid proofs. Nonetheless, large computer programs, even when they have been very carefully written and very carefully tested, always seem to have bugs.

> When one considers how hard it is to write a computer program even approaching the intellectual scope of a good mathematical paper, and how much greater time and effort have to be put into it to make it “almost” formally correct, it is preposterous to claim that mathematics as we practice it is anywhere near formally correct.

[1] https://arxiv.org/abs/math/9404236

[2] I discovered it via HN comments a while ago.

Yes, but I think this presents the core of the problem in modern pedagogical methods when it comes to mathematics. The Bourbaki attempted to reduce math to a highly axiomatic foundation, while disregarding the intuition and visualization that used to be a part of mathematics. The issue is that this sort of "code only" or "language only" approach really works when mathematics is a true "perfect language", the likes of which philosophers were attempting to construct, but is likely in fact impossible to create. Unfortunately, not only did the ideas of the Bourbaki fail, as modern research advances mostly still work with intuition instead of their ideas, but their approach polluted and ruined education. Many "textbooks" are terribly written reference books that have gaps and ambiguities that only people already knowledgeable in the field know about. Rudin's Analysis textbooks are probably the classical example of this. I would argue that any notation or abuse of notation is fine within insular fields and private practice, but there does need to be a leaning towards universal notation within all pedagogical works, at least up through all the core Algebra, Analysis, and Geometry and Topology work that you would see within a PHD qualifying exam.

There's a huge difference between "not being confused" and "understanding".

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