[jfv]

I think if you're a programmer wondering what math will help you in your career, which is the question this article is trying to answer, this article is misleading.

Most working programmers won't have to know any math beyond arithmetic.

For instance,

"Also, how does one distribute a large computation across globally distributed data centers? You have to understand some physics to do this well: at Internet scale, the speed of light starts to be a bottleneck. Heat dissipation, density of electrical current draw per unit area, etc, are all real world considerations that go into what programmers do"

Come on, let's be real. In grade school we learned that light goes around the earth 8 times a second. That means 1/16 of a second (63ms) to go around to the other side of the world. That doesn't set a useful lower bound since this is the ballpark of latency within a country anyway. You're just going to ping between the two servers and measure the latency.

That being said, programming could in theory require any kind of math. If you're doing machine learning, you'll need to understand statistics. If you're doing graphics programming, you'll need to understand trigonometry, linear algebra, etc.

If you're writing software for the medical industry you'll presumably have to understand a little bit about how that industry works, but I wouldn't suggest programmers go learn that just for the sake of improving their "general" programming skills.

Here's a more realistic take:

1) Learn math as necessary for a field you're interested in

2) Know binary and hexadecimal representations of numbers.

3) If you're going to use a lower-level language, understand how signed/unsigned arithmetic works, boolean algebra, and some basics of the floating point representation (in a nutshell, don't assume that computations are exact, so don't rely on equality (==)).

4) Understand time and space complexity. For most programmers, this is going to be simple: don't nest too many for-loops. I'm guessing 90% of programmers will never have to construct an algorithm whose complexity involves a logarithmic function, except for maybe binary search. Even then, most programmers will rely on libraries to do those things.

5) Be able to do back of the envelope calculations, and understand that order of magnitude is more important than exact numbers. Try to do them in your head so you can quickly evaluate possibilities.

6) Increasingly, statistics and machine learning are becoming important for programmers to know. Learn some basic statistics and it will make you much more desirable. Play around with R or Octave or numpy and don't make the mistake of applying too much statistics without thinking, unless you're in the marketing department.

[jerf]

Programmers use tons of math, they just don't recognize it. Partially because you can still come out of even a bachelor's program in computer science and still think "math is just about numbers", but some of it is also just excessive familiarity and not realizing the easy things that you're doing that are actually mathematical because school trained you that Math must be Hard and you have to be Sitting There and Thinking Very Hard or you're not Doing Math.

Every day I'm using mathematical reasoning such as "I reached this line of code, and since this is a structured program I am guaranteed that this variable was initialized by the time I got here", "this function can only return either a non-empty string XOR an error and has nowhere it can throw an exception so if I handle those two cases I have fully handled everything this function can do", "the error string has the username in it, and values like that must come from somewhere, so even though I've been looking for an hour and can't find it, mathematically the thing generating the error message must have access to the username", "if I construct this as a pure function that does no IO, on the flip side, no users of this function will have to worry about what errors could arise during IO, reducing the mathematical cognitive load of using this function for users".

Then you start stringing together those steps by the dozens and hundreds and thousands and millions (and you get qualitative differences there too, but those don't fit into a little post like this anywhere near as well), and you can build software.

You don't have to realize you're reasoning mathematically (proof: look around), but you're leaving useful tools in the toolchest if you don't. Even if you won't believe me about building systems this way, I can assure you those magical stories about people who can debug systems at a glance that have defeated others for hours or days comes back to someone who either formally or by experience has a fairly mathematical understanding of the system in question and was able to apply some sort of logic, even if only intuitively, to arrive at that conclusion.

[oreganoz]

But that's mostly logic, at best mathematical logic. I don't think anyone formally reasons when writing code anyway, we mostly rely on intuition. Maths isn't about numbers but it's not about intuition neither.

Programmers doing actual math would be modeling problems and creating proofs for their solutions. Usually we clasify those problems based on the field of math we use to model them.

Saying we do tons of math implies we use a lot of knowledge from a lot of those fields. But most of the time, programmers use only the introductory notions of those fields. We are usually guided by a domain expert anyway so that the math part is correct.

Programmers don't use a ton of math. They use some math.

[tzs]

Some of your points remind me of Underwood Dudley's 2010 article "What is Mathematics For?" [1]. Briefly, he argues that we greatly exaggerate how much math you actually need in most jobs that people usually think of as needing math, but nevertheless learning math is worthwhile and important because of what it teaches you about how to think.

[1] http://www.ams.org/notices/201005/rtx100500608p.pdf

[jfv]

I agree. It's (usually intuitively learned) reasoning that's very similar to the logical reasoning of mathematics. I think if you were to study, say, calculus in a manner typical of a first course in college, you'd be missing out on that sort of problem solving, but you'd definitely get that if you took a proofs-based class in analysis ("This function is continuous on this interval, so it must have a maximum. How can I use that to prove what's being asked?").

[jordigh]

> Come on, let's be real. In grade school we learned that light goes around the earth 8 times a second. That means 1/16 of a second (63ms) to go around to the other side of the world. That doesn't set a useful lower bound since this is the ballpark of latency within a country anyway. You're just going to ping between the two servers and measure the latency.

So there's some funny folklore regarding this...

http://www.ibiblio.org/harris/500milemail.html

[ben509]

> 1) Learn math as necessary for a field you're interested in

I think this a Rumsfeldian problem: you can address the known unknowns, but what about the unknown unknowns?

There's no great answer to this, and it's why I think the original post was advocating the broad fields of math.

Just knowing math exists, though, doesn't address the depth to which you study it. I've gone at monads about five times now and I think I get it, to where I see why a promises library works the way it does. Same with databases, learned SQL, then stumbled across the wonderfully cranky dbdebunk.com, read Date's book, finally spent some months working on an implementation of the relational algebra and finally grokked it.

The problem with some of these deeper subjects is you'll have many "aha" moments only to look back at your early "ahas" and realize you had no idea what you were doing.

[lorddoig]

I'm not so sure. Programming is isomorphic to mathematics (via logic) as proven by the Curry-Howard Correspondance, and so is not just applicable to ones problem/industry domain—it's applicable in a much more general sense. In my experience, the learnings on both sides of the coin filter back to the other in surprising and often highly insightful ways.

[troupe]

> Most working programmers won't have to know any math beyond arithmetic.

If your job is to write code that moves information back and forth between a database and a web page, this may be somewhat true. However, once you get beyond that to the point you are actually solving problems in code (not just moving information) you are going to need math beyond arithmetic. Algebra, boolean logic, and statistics are just a few things programmers use all the time--almost to the point that they may not think of them as "using math" because it is just built into the way they think.