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Comment by chongli | Hacker News Reader

for example: operating systems, programming languages, human computer interaction, etc.

Those topics might seem to have nothing to do with math but all of their components have mathematical underpinnings. Algorithms, data structures, complexity theory, and even the physics of end-to-end latency, colour perception, etc.

There probably are some people out there, working in these fields with only a high school math background, but I'd imagine they're exceedingly rare. Anyone who's completed a CS degree has done their fair share of math.

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Everyone who completed a physics degree has done at least as much math as a CS degree holder, is physics a subfield of math?

(I think there is a discussion to be had about making the distinction "CS" vs "theoretical CS" as the GP comment does, or if it should be "CS" vs some other term ("computing"? feels a bit general))

I tried reading "the road to reality" and came to the conclusion that math is physics minus all the boring parts.

(Mostly joking. Mostly!)

https://xkcd.com/435/

Physics is not a subfield of math. It has entirely non-mathematical origins (see Aristotelian physics [1]).

Computer science was developed by mathematicians as a study of algorithms, procedures for computing, and methods of abstraction. In the words of Hal Abelson, computer science is not a science and it's not really about computers in the same sense that geometry is not about surveying instruments [2].

[1] https://en.wikipedia.org/wiki/Aristotelian_physics

[2] https://www.youtube.com/watch?v=2Op3QLzMgSY

There is the Curry Howard Lambek Correspondence (or should I say: the Types, Logic, Cartesian Closed Category Correspondence). Curry Howard in particular, says the act of providing a term for a type is the same as providing the proof to a theorem (modulo a few details). Note that this isn't the same as saying writing a concrete computer program and proving a theorem in a type theory are the same type of activity.

Numerical methods and algorithms are fields of math as old as geometry, especially if we focus on the Babylonian or Chinese styles.

Hermann Grassmann sought to formalize arithmetic, not wishing to assume them as granted. In doing this, he also connects recursion, induction and the natural numbers (he would have known of recursion from its early application in the theory of combinatorics). Peano, Dedekind, Frege, Zermelo and many others would also work on the foundations and axiomatization of mathematics and deduction. Computing began as a side-effect of attempts to formalize just how far such an approach could be taken. The Turing Machine arose to tackle Hilbert's Entscheidungsproblem. The lambda calculus as an approach to the foundation of mathematics. Functional programming languages were originally part of tools meant to study formal mathematical objects while Logic programming sought to apply ideas from formal logic and the axiomatization of mathematics to automatically search for programs.

Dedekind said: "In speaking of arithmetic (algebra, analysis) as a part of logic I mean to imply that I consider the number-concept entirely independent of notions or intuitions of space and time, that I consider it an immediate result from the laws of thought."

What we find is computing reaches right to the foundations of mathematics. Whenever we try to systemize thought, we end up with ideas which seeming inevitably also lead to the foundation of computation.

It is like saying Chemistry is a subfield of math, because at some point they use numbers to describe things (mass and such)....

It is not. Applied Computer Science has as much common with math as Chemistry does with math.

I view theoretical computer science as mostly self-masturbatory, to the point that is very very divorced from real life applications and is benefiting very little to us.

Also the market has spoken as well. Someone with CS degree, and 5 years of experience can command a higher salary than someone that took 5 years to get his/her phd in CS. A phd degree is not seen as valuable, mostly because it is not seen as beneficial and it is very divorced to reality of applied computer engineering.

I wouldn’t use money as a measure of how useful something is. Useful for getting a job sure, but to society not so much.

The computer science that helps big companies get more control is the most useful by this metric. Oh look big data and AI are popular. Programming language theory to help create less buggy programs is less so.

You wouldn't have a compiler without someone having developed formal language theory. Or, at least, probably not one built on a solid theoretical foundation that actually happens to be helpful.

You wouldn't have complexity analysis of algorithms, with which most of us don't need to directly involve ourselves, but you do apply its results when choosing an algorithm based on the knowledge that was originally obtained through that analysis. Or if you're not choosing your algorithms, at the very least someone who chose them for your platform did.

You probably wouldn't have lossless data compression (and an understanding of it) at its present level without someone having done mathematically-based work on things like arithmetic coding [1] and range encoding [2]. Again, you probably don't write that code yourself (I haven't), but it's there.

The list goes on.

A PhD degree isn't really a good investment in terms of just salary in almost any field that I can think of. That just means work that's closer to (and directly applicable for) direct revenue streams tends to pay better than work that's further away from them. That doesn't directly mean work that's further away from revenue streams is less valuable down the road; it just means there's less certainty about its ability to help generate revenue, and that there are more steps, more interim work and a greater financial risk involved. While most businesses don't, and shouldn't, bother, that doesn't mean they might not benefit at some point if someone else does it. "The market has spoken" is a shortsighted way of looking at these kinds of things.

Sure, there are areas of theoretical computer science that are more similar to pure maths in terms of abstraction and applicability, and which are pretty much a pure intellectual pursuit. They are very far from engineering or applications. But theoretical work in CS is broader than that, and some of it underlies much of what we have in the practical world.

It's also true that most of software development and engineering work don't really require involvement with much of the theory, partially because someone else is already doing that work within the platform, and partially because most business software is actually theoretically more or less trivial.

Still doesn't mean the theoretical side is useless, because not all software is trivial.

[1] https://en.wikipedia.org/wiki/Arithmetic_coding

[2] https://en.wikipedia.org/wiki/Range_encoding

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