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Comment by floatboth | Hacker News Reader

> [Foundations and logic] Why: This is the assembly language of mathematics—the stuff at the bottom that everything else compiles to.

> [Basic mathematics on the real numbers] Why: You need to be able to understand, write, and prove equations and inequalities involving real numbers

If logic is the assembly, real numbers must be the Enterprise Java frameworks :)

But really, aren't they from non-discrete (continuous?) mathematics? How are they useful for computer science? Like, sure we have approximations of them on computers (floats, rationals) but aren't they mostly used for those pesky boring real-world-ish calculations? Isn't CS mostly about integers?

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You can't have calculus without the real numbers, and you can't have asymptotic analysis and probability theory without calculus. There goes the whole field of the analysis of algorithms.

You can do discrete math (including analysis of algorithms) without reals.

Reals are only needed for calculus on infintesimals, which are not relevant to much of CS. Everything you can do with formal Taylor expansions doesn't need reals. In fact knowing calculus on the reals often causes confusion in students learning formal expansions for CS.

All of optimization, stats and theoretical ML involves solid grasp of calculus over the reals

Doing numerical computation is definitely something that comes up in CS, as well as in applied maths. For example, deep learning is all about basic calculus, and algorithms to do this. Many discrete problems can be posed as a continuous problem, which is usually called a relaxation, and more easily (approximately) solved in the continuous domain.

You only need reals for exact solutions. Once you are approximating with smooth functions, you can approximate with rationals/decimal paramters on polynomials.

How do you work with smooth functions without using reals?

Yeah, but you got to get that PhD somehow and there are only so many unique topics for your dissertation. Eventually you gotta start reaching!

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