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Comment by jondubois | Hacker News Reader

I enjoy the pure logical aspect of maths, but really dislike the notation/abstraction aspect.

I think that if we could rewind time and humanity had to reinvent math from scratch, it would probably end up very different from what we have today; most of the underlying logic and ideas would be the same, but the abstractions, notations and representations of those ideas would be completely different (and possibly better/simpler and more logical).

When it comes to code; whenever something is kept a certain way because of historical reasons, that's called technical debt and it's a bad thing.

In maths, basically all of the abstractions and notations are the way they are because of complex historical reasons. Math never gets refactored.

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I disagree, there's been quite a few attempts at refactoring math. Whether it's the Hilbert Program, which attempted to formalize the foundation of mathematics, the Cartesian plane as a way to describe functions in space or the modern reformulation under homotopy type theory, mathematics has gone through some serious refactoring. Even something as simple as algebra has gone through extreme changes in notation and in phrasing. Try comparing a copy of Euclid's elements to your standard geometry textbook. Or a proof of the Galois Correspondence Theorem in an Algebra textbook to the original in French.

Also, I highly doubt that math would significantly simpler or "logical" (a rather peculiar word to use, as math is inherently about logic, so what would more logical logic look like?). Math is inherently about communication and is therefore inherently subjective. People have a million different ways of writing math because they have a million different ways of speaking. And depending on who you ask, one way may be more simple or more logical than another.

Do you have any examples of things due for a refactor? As another commenter mentioned, mathematical notation and language have evolved heavily over time.

When you talk about abstractions, what in particular do you mean? Math is in many ways about describing objects through abstraction.

Actually, the view that Mathematics was "discovered" and not "invented" is quite popular among Mathematicians. There is an area of research called the Foundations of Mathematics, which investigates this question.

It's untrue that Mathematics does not undergo any refactoring -- in fact, it does: even in a field as new as Algebraic Geometry, the "classical" textbooks which build up the subject using ideals and varieties are very different from the modern ones which start from Scheme Theory.

Moreover, Category Theory shows that there is a beautiful underlying structure in all fields that can be extracted out into a new field. For example, the concept of adjunctions existed much before Category Theory was invented -- CT came along and showed that similar structures exist in Algebraic Topology, Algebraic Geometry, Differential Geometry, and other fields.

Definitions and notations are not discovered, they're invented. I think that's what OP was talking about.

The dy / dx notation has a very deep meaning from differential geometry. With some other notational things, there is a deep reason behind it.

Other times, notation is bad and it is refactored. The thing is, mathematics by its very nature is a lot of highly coupled 'code bases'. So any time you try to refactor, you end up with sprawling rewrites or a boundary with some translation method.

This is different in coding because there is less coupling, and the translation methods (Foreign function interfaces / shims) can be automated.

Math isn't context free. Neither are most/all human languages. Yet we communicate succesfeully (within some margin of error).

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