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The Mathematical Hacker (2012)

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Re: The Mathematical Hacker (2012)

#131

Post by jhanschoo »

kaashif wrote: post #125
BeetleB wrote: post #58

Earlier quoted context omitted.

For most mathematicians, calculus as is often taught in typical undergrads is not "true" mathematics. It's just a tool for computation. For them, calculus is analysis (theorems/proofs that are used to build up calculus). So my question is: Did you study analysis and would you credit programming in helping you get good at it?

I don't understand this comment. Calculus as taught to mathematics undergrads is analysis, right? Where is calculus "often taught" without "theorems/proofs that are used to build up calculus"?

Not US, but for GCSE A's further math equivalent, calculus primarily involved applying intermediate rules (chain rules, trig identities, etc.) to evaluate differentiation/integrals on elementary functions, and a bit on geometric and arithmetic series.

Analysis discusses less well behaved functions and spaces than these.

Re: The Mathematical Hacker (2012)

#132

Post by JBiserkov »

hayley-patton wrote: post #109

Yes, the numeric language does numeric things. But these examples of closed form solutions ignore that numbers on computers are not the usual mathematical objects, rather having finite size and precision, and so it is hard to say immediately if these solutions using floating-point arithmetic provide the same precision. Of course the precision can be calculated and improved with analysis, still, but this falls under "…

Cool story!

Only way I = I + 1 is if I is infinity.

Re: The Mathematical Hacker (2012)

#133

Post by zozbot234 »

jhanschoo wrote: post #131
kaashif wrote: post #125

Earlier quoted context omitted.

I don't understand this comment. Calculus as taught to mathematics undergrads is analysis, right? Where is calculus "often taught" without "theorems/proofs that are used to build up calculus"?

Not US, but for GCSE A's further math equivalent, calculus primarily involved applying intermediate rules (chain rules, trig identities, etc.) to evaluate differentiation/integrals on elementary functions, and a bit on geometric and arithmetic series. Analysis discusses less well behaved functions and spaces than these.

You can do the first part without even relying on analysis in a mathematical sense. You simply define a differential algebra, by introducing a derivation function that just happens to respect the correct rules. Then "calculus" is the topic of how to perform computations in such an algebra. Note however that you do need analysis to rigorously address other parts of a typical "calculus course", especially those dealing with infinities, sequences and etc.

Re: The Mathematical Hacker (2012)

#134

Post by kthielen »

kovac wrote: post #124
dysoco wrote: post #53

Earlier quoted context omitted.

> mathematics is orders of magnitude more intensive and difficult than most programming But what level of programming and mathematics are you comparing here though? because college-level algebra and calculus is really not that hard imho (once it "clicks" for you, but it's the same for programming), and if we are comparing math as in what you see in a BSc/Msc of Mathematics (or research-level) then I agree it's hard b…

College-level means undergraduate-level? If so, how is algebra/calculus not that hard? Abstract algebra is one of the hardest stuff I've come across. Calculus? Do you think it's not that hard to prove convergence/bounds/limits of random series and sequences... I agree though that calculus is not that hard, compared to the rest. Programming is child's play compared to undergraduate mathematics taught in math departmen…

> Programming is child's play compared to undergraduate mathematics taught in math departments.

One thing you might learn in math is to avoid making overgeneralized statements that you can’t support.

A valid substitution in your statement for “programming” is writing a compiler. And for “undergraduate mathematics taught in math departments”, basic differential calculus.

Yet we regularly teach smart high school students and first-year undergraduates calculus, and almost never try to teach them to write a compiler, contradicting your proposition.

But what do I know? I’m just a dumb programmer. I can’t read your mind, so maybe you had something a little more specific you wanted to say.

Re: The Mathematical Hacker (2012)

#135

Post by hayley-patton »

JBiserkov wrote: post #132
hayley-patton wrote: post #109

Yes, the numeric language does numeric things. But these examples of closed form solutions ignore that numbers on computers are not the usual mathematical objects, rather having finite size and precision, and so it is hard to say immediately if these solutions using floating-point arithmetic provide the same precision. Of course the precision can be calculated and improved with analysis, still, but this falls under "…

Cool story! Only way I = I + 1 is if I is infinity.

There is some irony in juxtaposing a statement on floating point arithmetic before; I = I + 1 if I is a sufficiently large float.

Re: The Mathematical Hacker (2012)

#136

Post by subradios »

zozbot234 wrote: post #44
subradios wrote: post #39

Earlier quoted context omitted.

Then why do all New England states outcompete most European countries? Why is most STEM research produced in Eastern Europe, the US, or China? This really depends on what metrics you are using and has infinite room for gaming. US undergrads (both native and foreign born) absolutely crush all of these nondescript places you are suggesting are better, and graduate programs make the gap even wider. A non trivial fractio…

U.S. undergrads have to complete "general educational requirements" that are taken care of in high school in practically every other developed country. Why does that happen? Because U.S. colleges don't trust K-12 to provide a satisfactory education.

We do not trust the average student, however access to universities in the US is less gated on ability or achievement than most of the nations you're going to be comparing it to.

Re: The Mathematical Hacker (2012)

#137

Post by bee_rider »

subradios wrote: post #120
bee_rider wrote: post #57

Earlier quoted context omitted.

New England is mostly sanely governed, it shouldn’t be used as a stand in for the rest of the country. The US was a neat idea we had but it went a little off-kilter around the second half of Pennsylvania.

Texas A&M is ranked 67, Urbana Champaign 41. (USNWR) After you pull out HPYSM, these are not ridiculous rankings either. Georgia Institute of Technology is 38th in the world by research impact, with the top ten or twenty being your standard prestige universities. (Elsevier) If the US is so bad, why is a poorer southern state competitive with far more prestigious institutions from other countries you would perceive as…

I should have been more explicit; I was being a bit flip, trying to inject a little lightheartedness.

To specifically respond to just that one point, New England is in many ways an unusual region so I don’t think we should use the scores from here to come to any particular conclusions about the country in general.

This shouldn’t be taken as a response to your other points, most of which seem reasonable enough or at least I don’t know anything in particular about them.

Re: The Mathematical Hacker (2012)

#138

Post by j2kun »

bmitc wrote: post #9

In Steve Yegge’s linked post: > Math is a lot easier to pick up after you know how to program. In fact, if you're a halfway decent programmer, you'll find it's almost a snap. This couldn’t be more wrong. Mathematics is the hardest thing I have ever done. I’m sorry, but mathematics is orders of magnitude more intensive and difficult than most programming. A simple fact that shows this is the amount of programmers who…

I wrote a book based on this premise: www.pimbook.org.

The ebook is pay what you want.

Re: The Mathematical Hacker (2012)

#139

Post by logarhythmic »

constantcrying wrote: post #96
zozbot234 wrote: post #94

Earlier quoted context omitted.

A constructive proof of existence is exactly a program. (It might not always be a program for a Turing machine, because "constructive" and "computable" are not exactly the same - but that's beside the point.) Even a non-constructive proof of existence for x can be significant in a programmatic context; it tells you that you can posit an oracle for x (e.g. asking for it to be input by the user, introducing further ass…

>but that's beside the point It seems to me that it is an important point though. > Even a non-constructive proof of existence for x can be significant in a programmatic context Which has no relation whatsover to proofs and programms being the same thing.

Programs and proofs are the same thing though according to the Curry Howard Correspondence, if I'm not wrong.

Re: The Mathematical Hacker (2012)

#140

Post by __MatrixMan__ »

geysersam wrote: post #114
nickelpro wrote: post #69

Earlier quoted context omitted.

I would hard disagree that undergrad level Analysis or even just the trickier corners of vector calculus are within the bounds of what programmers can easily pick up without dedicated and guided study. Everybody's gangster until they have to parameterize some bullshit helical structure in R3. Comparable levels of programming, what we expect of CS juniors, are regularly picked up by "the guy who is good with Excel" in…

I think you're hitting the nail on the head here. Something about the learning process makes programming much easier to pick up. What if we had something similar for mathematics? Rapid feedback, error messages, maybe even linters and highlighting for the "mathematical syntax". I've though about this before and I think tools like this could unlock math for a lot of people, and also increase the effectiveness of profes…

I'd love to play around with such tools, but I think they'd only get you so far before they'd start to become a hinderance.

The linter in mathematics is whether the other mathematician (whoever you're proving to) knows what you mean. If you're locked into a rigidly defined syntax, an obvious line of questioning is: what's not expressible in this syntax?

I fear that by the time the tooling was agreed on, built, and taught in schools, you'd have something like APL, which might be cool to code in, but from which the mathematical conversion would have moved on a while ago. Efforts like that, after all, are how math becomes engineering.

Consider, for instance, Russel's theory of types, which was interesting math at the time and now strikes the student with an engineering background as "pretty much just Java" (or any "normal" statically typed language).

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