Ask HN: Learn Math the Hard Way

I think there is a point of diminishing returns.

Baby Rudin I'm dubious about. But Royden and big Rudin (both of which you recommended) I have certainty about. There are good reasons that I never saw CS students in my real analysis classes. I don't think it is particularly valuable for CS either then or now, to acquire a deep understanding of real analysis.

And yes, I know about measure theory. I know how it applies to probability. But I went the other way. I learned measure theory. Then I learned probability. Then I began having to do probability stuff in the real world. And not once has my measure theory background been particularly relevant.

As for Hilbert and Banach spaces, they are key pieces of mathematics. In fields from wavelets to optimization theory, they come up over and over again. But I would wager that most computer science professors do not need to know what Hilbert and Banach spaces are. I'd even bet that most have not heard of the Hahn-Banach theorem. Again, if you find yourself going that way, learn it later.

On combinatorics and graph theory, you claim that people will learn enough of that material elsewhere. Maybe, maybe not. But it is clear that programming problems routinely get turned into graph theory problems, many of the most important programming algorithms are about graph theory (start with the traveling salesman problem and work your way through the list of NP-complete problems), and at its heart, analyzing an algorithm's run-time is a combinatorics problem. Acquiring the necessary concepts and vocabulary for those is necessary, whether you classify the book you're learning from as a math text or a CS text.

One thing I'd like to point out is that measure theory is not the only and probably the least interesting way to study probability. There is the more elegant (IMO) approach via Nonstandard Analysis. And the fun more practical approach via Game Theory and Markets (which also support "imprecise probabilities").

I also think theres room for different approaches to the same thing, each offering their own unique insight. Many differential equation modelling problems, especially those involving populations could be fruitfully replaced by agent modelling.

Yes, there is a big question about what to learn, about how much to invest in such things.

> Baby Rudin I'm dubious about. But Royden and big Rudin (both of which you recommended) I have certainty about.

But Baby Rudin is a prerequisite to Royden and big Rudin.

I'm sorry, but probability, stochastic processes, and mathematical statistics were junk for me until I went at them via measure theory.

I floundered terribly with random variables until I saw the measure theory definition; it's terrific: Go take 10,000 measurements. Now have the values of 10,000 random variables. Any 10,000 measurements at all. So far, no concept of 'randomness' at all. So, random variables are very general things and, e.g., handle even deterministic processes as a special case.

E.g., sufficient statistics is just an application of the Radon-Nikodym theorem, and a total train wreck to do otherwise. Yes, order statistics are always sufficient, maybe nice to know in 'data mining'. That sample mean and sample variance are sufficient in the Gaussian case is mind blowing; nice opportunity for 'data compression'!

E.g., measure theory and the Radon-Nikodym theorem define conditional expectation, that is, under mild assumptions, E[Y|X] = f(X) for some measurable f. Then easily f(X) is the best non-linear least squares approximation of Y. Nice.

Further, if 'cross tabulate' Y on X, then have a discrete approximation to E[Y|X] which shows that cross tabulation is a discrete version of the best non-linear approximation of Y given X.

Measure theory permits working with all the forms of convergence of random variables, especially strong convergence, at least awkward to do otherwise.

Then martingale theory makes little sense without measure theory.

Measure theory, the Kolmogorov extension, shows that we really can have a collection of random variables with desired properties.

Measure theory is crucial in even defining E[Y|U(t), t Constructions such as

E[Y|U(t), t are crucial in the nice qualitative, axiomatic definition of the Poisson process.

Similarly for independence of two collections of random variables where each collection has uncountably infinitely many random variables.

Measure theory was crucial in the standard results of ergodic theory.

I wrote a paper on anomaly detection in server farms and networks, and the key idea in the paper was a finite group of measure preserving transformations lifted roughly from ergodic theory.

Via measure theory we can show that the space of real valued L^2 random variables is complete, and, thus, a Hilbert space, which continues to blow my mind that any such thing could be true. I'd also like to have locally compact, but that's a bit much to hope for!

The Doob decomposition shows that every stochastic process is the sum of a martingale and a predictable process, all measure theory!

It's tough enough to believe in probability with the measure theory foundations; otherwise, I couldn't swallow the stuff!

There is a broad point: Maybe OP wants to know what to learn for the applications of the future. Then what current CS profs know is not necessarily very relevant!