I think contest math is the best kind of math for a programmer since it sets out to solve problems and not to necessarily come up with theories like regular math.
Why there is no Hitchhiker’s Guide to Mathematics for Programmers
11–20 of 76 posts
Re: Why there is no Hitchhiker’s Guide to Mathematics for Programmers
#12I've come up against this recently and I think a lot of imperative programmers will. For me, the basic pattern, starting from a career based on perl/php/java was this series of realizations: - EECS circuit design is based on a conceit that gives chips a speed limit - as a result, chips aren't getting much faster and that multi-cores are happening instead - parallel programming will become more and more essential to m…
Message passing is an ancient and well-understood mechanism for producing parallel, scalable, mutable-state imperative programs. Feel free to continue to not learn math :)
Re: Why there is no Hitchhiker’s Guide to Mathematics for Programmers
#13He'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 no idea how complete that was).
Re: Why there is no Hitchhiker’s Guide to Mathematics for Programmers
#14"For the programmer who is truly interested in improving their mathematical skills, the first line of attack should now be obvious. Become an expert at applying the basic methods of proof"
This may be good advice if you want to get into pure mathematics, but huge parts of maths are not pure.
Applied mathematics is very relevant for programmers - especially game programmers. It's not a sin to sacrifice mathematical rigour for the sake of discovery.
It doesn't even have to be a direct map (e.g. actual implementations of calculus or such). Just understanding various bits and pieces in granular detail can help you get precise and testable quantitative ideas for problems.
For instance, a while ago I wanted to implement a kind of smooth up-and-down motion for an object in my game. Almost immediately I thought of equations of this form:
(y position) = (some amplitude)* sin((some frequency)*(time)) + (some value to denote the origin)
This is a first step for some fairly interesting up-and-down motion - the rest is all a matter of finding suitable constants.
Similar things come to mind when you want to move an object smoothly to another position along a line, with a decreasing speed:
position = ((position you want to move to) - (current position)) / (some constant) + (some minimum speed)
This stuff just flashes before your mind if you've done plenty of physics (or 'mechanics' in mathematics) before. What's more, you'll know countless other ways of increasing the complexity of your equations without having to do much trial & error hacking (except to find things like constants and such).
For games, I'm convinced that there's still an incredibly huge variety of mathematical behaviour that has yet to be harnessed for the implementation of actual game mechanics. (Games usually just use some game engine's implementation of only some parts of classical mechanics, for instance.)
Never underestimate the relevance of applied mathematics to real-world programming, even though it doesn't really occupy a spectacularly prominent position in CS-focused courses.
Re: Why there is no Hitchhiker’s Guide to Mathematics for Programmers
#15I think the biggest stumbling block for me was all of the stupid tricks involved in the proofs I had to do for school.
"If I add sin(x) to both sides of the equation, then I can use such-and-such identity which makes this equal foo."
If you don't make the correct mental leap, then you get completely stuck and it's really easy to give up.
Re: Why there is no Hitchhiker’s Guide to Mathematics for Programmers
#16If 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 moderators on help websites like stackoverflow? Is it because often when one wants to learn the basics, they are overloaded with the entirety of the documentation and the overwhelming resources of the internet and all its inhabitants? Is it because compiler errors are nonsensically exact, but very rarely helpful? Is it because when you learn it alone, you are bombarded with contradicting messages about what you should be doing and why (and often for the wrong reasons)?
The difference is that the CS community recognizes that these are problems; every single thing he's complaining about are open problems being taken seriously and attacked from multiple directions, and there is hope for serious improvement in the coming decades. Anyone who thinks rude snobs, bad documentation, or useless compiler errors are a beneficial is rightly ridiculed as a smug weenie or accused of having an ulterior motive.
By contrast, mathematicians are defensive and complacent about their arcane, non-inclusive notation and communication: "At this point you might see all of this as my complaining, but in truth I’m saying this notational flexibility and ambiguity is a benefit." Look at the litany of problems he just presented. Consider the fact that mathematics is not the only complicated subject that requires complicated, flexible, and rigorous notation. It just isn't credible that the shitty state of mathematical notation is either necessary or unavoidable. The occasional counterexample, where someone with a good understanding of a subject presents it in full rigor without resorting to the usual obfuscation, is a hint of what could be.
If your publications cannot be read without an expert interpreter, they are defective. Hypertext has been around for decades, if you're going to invent your own ad-hoc (or even standardized!) syntax to solve a problem your readers have a right that you document the meaning of your notation.
Re: Why there is no Hitchhiker’s Guide to Mathematics for Programmers
#17> “The proof is trivial.” This is the mathematician’s version of piping output to /dev/null.
Heh heh. Never thought about it like that. Amazing write-up, btw. As somebody who dabbled in both CS & Math, I'd say the cultures are vastly different. You can spend years, decades even, just teaching undergrad calc courses while having barely 1-2 papers to your name, and yet you'd be considered a legitimate mathematician & get paid & all that. I know scores of math professors who are in that category. With CS, if you don't have productive output in a week, you are just idling & companies will fire you. CS academia isn't a whole lot different...the paper output is a lot more, though much of it is backed by programatic machinery.
As a consequence, a standardized notation/syntax has developed out of sheer pragmatic necessity. Everybody knows what you mean when you say its a hash, or a bst, or an lfsr, or a trie or a monad ( you wish :) Well atleast some of these are standard concepts across all pgmming languages.
With math, you can labor for years in some obscure field ( heh heh a mathematician will kill for that pun!) in which less than ten people know what you mean. At a conference, Hilbert was supposed to have gotten up and asked a speaker "What do you mean by a Hilbert space?". That should tell you something. The syntax is remarkably nonstandard & even simple things like edges, nodes, edge weight, graph correspondingly become arcs, vertices, payload, network depending on where the literature originates from. When I studied a few math papers in grad school, I had a lurking suspicion the author was going out of his way to obscure his thought process & result. With CS papers, you atleast get straightforward pseudocode & you can run off to your favorite language whether php or haskell & give it a shot. There are math texts out there where you pick up one of them, you feel like a complete fool, you pick up the other, you instantly get the point, and they are both talking about the same exact thing !! artin vs dummit & foote vs herstein comes to mind...i got 1000% more out of herstein than the other two.
I agree with the author that in the absence of standard math terminology, I'm afraid programers struggling to learn advanced math just learn bits & pieces necessary to get their job done & move back to fighting their daily battles with git rebase & jira tickets.
Re: Why there is no Hitchhiker’s Guide to Mathematics for Programmers
#18I 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…
I think that theorems are just like libraries you can pull in for your purposes, and the proofs are the unit tests that you can dive into if needed ;) !
Re: Why there is no Hitchhiker’s Guide to Mathematics for Programmers
#19I 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…
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.
Re: Why there is no Hitchhiker’s Guide to Mathematics for Programmers
#20If 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…
This isn't entirely true. Numerous mathematicians have publicly bemoaned the difficulty of communication between mathematicians in different specialties. So it's definitely not something that is ignored.
The question is what to do about it. We're not just talking about confusion arising from different notation between mathematical specialties (resolving that would be as easy as defining your notation in an appendix), but different, equally valid ways of mathematical thinking.
In some ways, it may be better to think of different mathematical specialties as different programming languages. Proficiency in one will help you, but won't guarantee that you can interpret another. Except that in mathematics, the differences are more extreme. If you have two Turing-complete programming languages, then you have two different tools that can solve the same class of problems. But different fields in mathematics deal with entirely different mathematical objects which require a conceptual instead of notational leap on the part of the reader. It's not simply a matter of figuring out how to write for loops or manipulate strings in the new language. You actually have entirely different ideas in each, and trying to impose some common notational standard among them is fraught with problems.