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Why there is no Hitchhiker’s Guide to Mathematics for Programmers

jeremykun.com

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Re: Why there is no Hitchhiker’s Guide to Mathematics for Programmers

#51

Post by j2kun »

bcoates wrote: post #16

If this is intended as a defense of Math culture, I think it falls short. He does a good job of pointing out the corresponding problems in the state of programming: Indeed, the opposite problems are familiar to a beginning programmer when they aren’t in a group of active programmers. Why is it that people give up or don’t enjoy programming? Is it because they have a hard time getting honest help from rudely abrupt mo…

I certainly agree that communication can be improved (and this is one of the main reasons I spend so much of my time improving my technical communication skills via blogging). But I stand by my defense of the need for flexibility. It's extremely hard to explain why without delving into technical details, but there are some times when abuses of notation are honestly much more helpful than they are hurtful. Usually they exist to decrease the amount of clutter in syntax, and they are often stated in words during presentation (and the abuses mostly go assumed only in the higher-end research papers, in fields where the fundamental book every practitioner is obligated to read spells out all of those assumptions in full).

Of course, the more of these you add on the harder it becomes to explain it to a newcomer. So I'm not trying to defend the math culture as being right in all ways. I'm just trying to explain why things are the way they are, and rationally see that there is a good reason to do so (or else why would it have been done?). When you learn the stuff in real life, each of these "abuses" are added on one small step at a time (maybe I learn about one new abuse of syntax every month or two) and so absorbing it all simultaneously is not such a big deal.

Re: Why there is no Hitchhiker’s Guide to Mathematics for Programmers

#52

Post by j2kun »

cschmidt wrote: post #36
tuxdev wrote: post #19

Earlier quoted context omitted.

Just because you don't explicitly write it down doesn't mean you didn't prove something. For optimization, you have to prove to yourself that the new code behaves equivalently to the old code. Anytime you use algebra you're doing a small proof to show that two expressions should be treated equivalently. Writing a proof down is the very last step, the hard part is thinking it through.

I do derivations all the time. I just do a series of algebraic transformations, starting from what I have, and getting to what I need. Some might say that I'm doing a proof. But I never use proof by contradiction, contrapositive, induction, or proper proof techniques.

Doing a derivation like that is sometimes called a proof by "chaining if and only ifs", though that's far from a standard name taught to students. It's basic, but most certainly a required method of proof to come up with convenient formulas.

Re: Why there is no Hitchhiker’s Guide to Mathematics for Programmers

#53

Post by j2kun »

Mgccl wrote: post #24
cschmidt wrote: post #7

I don't know about the OP's point on the value of proof to programmers. I do optimization and machine learning, and I use all kinds of math on a regular basis. But I don't prove things, and rarely need to follow someone else's proof either. You can get a lot out of applied math without touching a proof. And if it isn't applied, then you really don't need it ;-). I agree there is value in learning how to read math not…

You are restricted by the set of tools you already have right? I can't believe how many times a great advancement in theory in different field comes from pure math advancement (or applying pure mathematical results). Say in economics, if Nash didn't apply topology, how would one ever know Nash equilibrium exists. Stable matching, compressed sensing and others also comes to mind. I have a hypothesis that any field wou…

To be completely honest, the more common case is that the mathematicians follow behind physicists and engineers to verify that their applications are theoretically sound. But yes, great breakthroughs do come from pure theory.

Re: Why there is no Hitchhiker’s Guide to Mathematics for Programmers

#54

Post by j2kun »

thyrsus wrote: post #13

He's analogizing to the fictional "Hitchhiker's Guide to the Galaxy". There does exist an analog to the real world "Hitchhiker's Guide to the Galaxy": Carl E. Linderholm's "Mathematics Made Difficult". Utterly hilarious. Regrettably, it's out of print, and you have a choice of paying upwards of $138.50 (abebooks.com) for a used copy of a book of jokes, or finding the pirated pdf I once encountered in a search (I've n…

My upstairs neighbor has this book, and I've been meaning to steal it from him for a few months now :)

Re: Why there is no Hitchhiker’s Guide to Mathematics for Programmers

#55

Post by j2kun »

nborwankar wrote: post #42

Re: the original post IMHO, I find it has many logical flaws and while it is trying to draw attention to the fact that programmers need to learn math it may end up scaring people away. Background: I have a BS in EE from IIT Bombay, An MS in Applied Math from USC and an almost PhD(everything but thesis) in Pure Math. I also loved teaching math as a TA and was able to teach engineers multivariable calculus and differen…

I certainly don't think programmers need to learn math. The vast majority of programming requires no math, and there are tons of out of the box theorems and algorithms that can be used successfully by programmers to do wonderful things.

I am just pointing out why math can scare programmers away. The first step in overcoming that fear is to identify it.

And the fact is that programmers do need to tweak the algorithms they use. Your analogy with a car works perfectly: can you expect someone who only knows how to drive a car be able to convert a sedan into something that can go off road? Then how can you expect someone who only knows how to call a function to alter its contents to suit their particular application?

I also think that I haven't set a high bar. The basic four methods of proof is quite a low bar (this could, and should be taught in high school mathematics, and it baffles me why it's not). Learning common syntax will come as you learn.

Re: Why there is no Hitchhiker’s Guide to Mathematics for Programmers

#56

Post by j2kun »

j2kun wrote: post #51
bcoates wrote: post #16

If this is intended as a defense of Math culture, I think it falls short. He does a good job of pointing out the corresponding problems in the state of programming: Indeed, the opposite problems are familiar to a beginning programmer when they aren’t in a group of active programmers. Why is it that people give up or don’t enjoy programming? Is it because they have a hard time getting honest help from rudely abrupt mo…

I certainly agree that communication can be improved (and this is one of the main reasons I spend so much of my time improving my technical communication skills via blogging). But I stand by my defense of the need for flexibility. It's extremely hard to explain why without delving into technical details, but there are some times when abuses of notation are honestly much more helpful than they are hurtful. Usually the…

It's kind of like list comprehensions in python. For loops do just fine, but they're such a pain to write all the time and for some cases we just want a clear one-line way to say "add 1 to everything in this list."

Re: Why there is no Hitchhiker’s Guide to Mathematics for Programmers

#57

Post by alok-g »

+1 for the book recommendation alone [1]. I finally see a book on set theory that explicitly states that the term "set" by itself is undefined.

Reading the first chapter preview at Amazon, I already see the need for a human to explain something to me:

Two sentences from the book [1]: "A possible relation between sets, more elementary than belonging, is equality." and "Two sets are equal if and only if they have the same elements". These already seem to be contradicting each other -- how can equality be more basic than belonging if the former is defined in terms of the latter.

While I agree with OP on the difficulty I face when learning advanced mathematics (notation, generally tacitly assumed by the authors), one thing that also bugs me is use of human language (!) as a part of the proofs. How do I know that the proof is correct and is not impacted by something fundamental about the language itself? Merely via inclusion of a human language in the proof, a lot more axioms "may" have been included than what meets the eye. Probably not, but I often find it hard to convince myself of this.

[1] www.amazon.com/Naive-Theory-Undergraduate-Texts-Mathematics/dp/0387900926

Re: Why there is no Hitchhiker’s Guide to Mathematics for Programmers

#58

Post by cschmidt »

coolSCV wrote: post #46
cschmidt wrote: post #7

I don't know about the OP's point on the value of proof to programmers. I do optimization and machine learning, and I use all kinds of math on a regular basis. But I don't prove things, and rarely need to follow someone else's proof either. You can get a lot out of applied math without touching a proof. And if it isn't applied, then you really don't need it ;-). I agree there is value in learning how to read math not…

An analogous case is the programmer who relies on libraries and abstraction and "knows just enough to be dangerous." Statistics and optimization mathematics in particular are vulnerable to this. If you haven't read the proofs or you cannot prove it yourself, then you truly don't understand it. I'm not claiming this is you since your post doesn't imply that. However as an example, if someone is solving linear or non-l…

I've benefited from lots of theoretical classes, and you're absolutely right that you need to understand the theory of what you're doing. For example, I took a lovely class from Prof. John Hooker (at CMU) that derived Linear Programming duality as a natural consequence of Farkas' Lemma, rather than just mechanically grinding through LP tableau. I certainly followed the proofs discussed in many classes.

However, I can honestly say I got that understanding without doing hardly any proofs of my own. Graph theory was the only class where I "did proofs", and I didn't find it my favorite. (Later I did algorithmic graph theory and network flows - and that was fantastic, with no proofs.)

I guess my point is just that an engineer can understand and use a lot of pretty advanced math without spending much effort learning about proof techniques. That certainly wouldn't be how I'd spend my time. (Of course, if you enjoy doing that then knock yourself out.)

Re: Why there is no Hitchhiker’s Guide to Mathematics for Programmers

#59

Post by compee »

I work with mathematics applied to various applied computer science problems with a couple startups.

If anyone is interested in learning a bit of mathematics to structure programs (for real-world usage, even!) feel free to let me know what you'd be looking for. I'd love to help!

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