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The Mathematical Hacker (2012)

evanmiller.org

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Re: The Mathematical Hacker (2012)

#161

Post by kaashif »

BeetleB wrote: post #147
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"?

In most US universities, the courses called "calculus" are mostly about computing integrals and derivatives. Yes, they'll have some theorems (Fundamental Theorem, Mean Value Theorem, etc), but most of the problems are related to computation than proving. Unfortunately, since engineering students outnumber math majors by a large margin, the departments cater to them and not the math majors. The latter study analysis i…

> They begin taking analysis in their 4th year, and it's offered only one semester a year!

This is insane to me. In the UK about ten years ago, on literally the first day of my degree, my first class was real analysis. Yes, it was the easy stuff like proving sequences and series converge, various things about continuous functions, but we learnt how to prove it all and the exam was all about proving various things. And we built up to harder stuff as the year went on.

What is even happening if math majors aren't studying analysis until their fourth year?

Sorry I'm so incredulous, it's just that I literally don't know what I would've been studying if analysis had been delayed so much.

Re: The Mathematical Hacker (2012)

#162

Post by __MatrixMan__ »

eternityforest wrote: post #156
__MatrixMan__ wrote: post #140

Earlier quoted context omitted.

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 t…

If you're learning math for career reasons rather than just pure curiosity, engineering is the main/possibly only place you'd use it besides statistical analysis.

Yeah that sounds about right. I was more musing about the philosophical boundary between math and engineering (math being a creative pursuit and engineering being about outcomes).

I can't quite pin it down, like with a definition, but I'm tempted to say that if it has a linter it's not math anymore, even if it once was.

Re: The Mathematical Hacker (2012)

#163

Post by noobermin »

We're here (computational scientists) and we're quietly working doing our thing. The only reason for the disconnect we don't really swim in the same circles as the lispy programmer types, we tend to be relegated to academic circles which by their nature are rather insular. There is also a very different approach to computation in general, and so there is "laughing" perhaps, but such is how it is I guess.

Re: The Mathematical Hacker (2012)

#164

Post by dinkumthinkum »

zozbot234 wrote: post #115
dinkumthinkum wrote: post #85

Earlier quoted context omitted.

Self teach Stokes Theorem by inputting it into a proof assistant? Are you serious? That is very inefficient; the OP was talking about learning the kind of vector calculus taught in first calculus sequence. I think just watching a short YouTube video and doing a few exercises will work and is a proven method. Proofs of theirebs are very often much more complicated than applying them (understatement intended).

If you don't understand the proof of any theorem, you haven't really "learned" it in any real sense. Wrt. doing computational exercises in vector calculus, that requires knowing the "rules of the game" which is also something that you can test precisely in a proof assistant.

Many proofs of theorems in calculus would require topology to understand. You’re suggesting that you cannot be competent in vector calculus without knowing the proofs at a professional. I think I, along with probably everyone, will have to disagree with that.

Re: The Mathematical Hacker (2012)

#165

Post by subradios »

bee_rider wrote: post #137
subradios wrote: post #120

Earlier quoted context omitted.

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 l…

Yeah sorry the issue that I run into frequently on this topic is "US bad because those poorer, more conservative states are terrible and if we just governed them like the utopia of New York City, we'd do much better in rankings", and while I think red states have their own failure modes - the stats show that at least some of them are doing just fine.

Sorry for being a bit reactionary there, sometimes a certain opinion is common enough that you respond like it's being stated due to similar comments being a prelude to it.

Re: The Mathematical Hacker (2012)

#166

Post by strangattractor »

bmitc wrote: post #46
strangattractor wrote: post #18

Earlier quoted context omitted.

Implementing numerical solutions for math concepts can give you insight into the math ex. Newton-Raphson Method or FFT. I would not sell yourself short on being able to understand. You may not be skilled at doing all the symbol manipulation required for what people consider typical "math" but that does not mean you cannot understand it.

I do agree that programming is very useful as a medium of exploration, a la what you said. I really like how the book Turtle Geometry approaches this. But that is really something different than what the quoted text is attempting to get at. > I would not sell yourself short on being able to understand. I have a master's in mathematics and have continued to take courses while working full-time. It's just that I'm more…

I am more a recreational mathematician. There seems to be a lot of unnecessary math phobia lurking about. Simply like to encourage people that sound like they fall into that category - obviously you are not one of them:)

Re: The Mathematical Hacker (2012)

#167

Post by ratorx »

eternityforest wrote: post #155
geysersam wrote: post #114

Earlier quoted context omitted.

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…

Is it the learning process, or the subject itself? Programming works with manmade abstractions, carefully designed to have very few interactions, keep mutable state contained, and to have all the parts structured in a hierarchy without recursion. In math you have systems of equations that all reference each other. And they all happen at the same time because there's no steps and time or lines of code just 5 equations…

Isn’t math just manmade abstractions as well? I’m not saying they’re equally hard (I think maths is harder too), but you seem to be overly simplifying programming, under the assumption that the system you are working with and building on is actually well designed, rather than what is actually more likely (a munge of SOME clean design with many layers of hacks on top).

I think Maths is harder because the abstractions are higher level, have less intuitive bases, the feedback loop is longer and doesn’t have robust testing. I don’t think the abstractions themselves are that much harder in general, but getting an intuition and doing anything useful (correctly) at a really high level of abstraction is quite difficult.

Re: The Mathematical Hacker (2012)

#168

Post by ThrowawayR2 »

kthielen wrote: post #134
kovac wrote: post #124

Earlier quoted context omitted.

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 studen…

> "Yet we regularly teach smart high school students and first-year undergraduates calculus..."

High school students are taught plug-and-chug calculus where one uses rules and formulae without any real understanding of the underlying subtleties that make calculus work.

Re: The Mathematical Hacker (2012)

#169

Post by kthielen »

ThrowawayR2 wrote: post #168
kthielen wrote: post #134

Earlier quoted context omitted.

> 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 studen…

> " Yet we regularly teach smart high school students and first-year undergraduates calculus... " High school students are taught plug-and-chug calculus where one uses rules and formulae without any real understanding of the underlying subtleties that make calculus work.

Bulletproof counter argument, you sure showed me.

Re: The Mathematical Hacker (2012)

#170

Post by anthk »

mschoenert wrote: post #157
bibanez wrote: post #149

Earlier quoted context omitted.

There are many ways to obtain the formula mentioned in the article. One of them is by diagonalizing this matrix and then applying exponentiation. How can another algorithm be better than constant time in the most general case?

No - there isn't really an O(1) solution. And the reason is that the Fibonacci numbers grow without limit. So there can't be an O(1) algorithm - even just writing down the answer takes O(N) - because the answer has O(N) bits. Concretely Fibonacci(1480) is the largest Fibonacci number that fits into a double. So for higher Fibonacci numbers you need to compute this with arbitrary size integers (or floats). And then lo…

Well, with Scheme, the difference on computing fib(n) recursively vs the iterative way shows up really fast.
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