How to Learn Advanced Mathematics Without Heading to University

Re: How to Learn Advanced Mathematics Without Heading to University – Part 3

#171

Post by fsloth » Tue, Sep 06, 2016, 7:36 PM UTC

kragen wrote: post #164

fsloth wrote: post #156

Earlier quoted context omitted.

Computers would be darn boring without calculus. Graphics, games, audio, animation - basically anything enabling creativity on a computer needs calculus tools. The interesting bits start to happen once one has built sufficient substrate out of the discrete parts. This is my personal opinion only, of course.

What those have in common is that they're numerical, not that they require (integral and differential) calculus. It's true that in a lot of cases, deeply understanding discrete numerical algorithms is a lot easier if you can analyze the continuous versions, which of course cannot be executed directly. But you can get really far with just the discrete versions, and you can understand useful things about the continuous…

I think we approach this from different ends. What one can achieve (your approach here) and into which boxes of science and mathematics are relevant to the said work. Yes, one can do lot of things by fumbling in the dark, so to speak, but that does not mean it's not isomorphic to the existing theory, rather, the experimenter lacks a map from the problem she is solving to the established theory. I'm all for experimentation! It's often better to first fumble a bit and then see what others have done. But it's often hard to map the relevant problem to existing theory without examples of application. Here comes the academic training part - it's a ridiculously well established training path to a set of tools forged by the greatest minds of humans.

A programmer equipped with a bit of calculus is so much more powerfull than a programmer without. It's like one is climbing from a canyon. Both the guy with the training and the utilities and the rookie with bare hands will probably reach the top, but it takes a shorter time for the better equipped person to reach the top, and he is already tackling other interesting problems when the other finally reaches the top.

Humans have a limited time on this planet. Really, learning calculus formallly is one of the most efficient and painless boosters for productivity when creating new bicycles of the mind. It's not the only one, and it's not necessary like you pointed out, but compared to the utility it's so cheap to aquire I can't really see no reason not to force it on people. This is still my opinion, I don't have sufficient practical didactic chops to even anecdotally prove this.

Re: How to Learn Advanced Mathematics Without Heading to University – Part 3

#172

Post by bpchaps » Tue, Sep 06, 2016, 7:50 PM UTC

throwaway729 wrote: post #162

clarry wrote: post #157

Earlier quoted context omitted.

I don't think it's a decent assumption for "almost everyone" in the west. I'd say it often is the case that you won't have enough free time for such studies unless you trade cash (income) for the free time. At which point you may not have much after paying the bills. Or you're unemployed and, well, don't have the cash. You suggest trading some of that free time for cash, as if unemployment were something people solve…

> You suggest trading some of that free time for cash, as if unemployment were something people solve by snapping their fingers and choosing to make a trade Not at all; I meant in the form of a loan, where free time is time you would otherwise spend retired. And I was also assuming that you already have a decent job that allows you to work 40 hours or so and make a comfortable enough living to consider spending your…

Such as Ramanujan. ;)

Re: How to Learn Advanced Mathematics Without Heading to University – Part 3

#173

Post by kragen » Tue, Sep 06, 2016, 8:57 PM UTC

fsloth wrote: post #171

kragen wrote: post #164

Earlier quoted context omitted.

What those have in common is that they're numerical, not that they require (integral and differential) calculus. It's true that in a lot of cases, deeply understanding discrete numerical algorithms is a lot easier if you can analyze the continuous versions, which of course cannot be executed directly. But you can get really far with just the discrete versions, and you can understand useful things about the continuous…

I think we approach this from different ends. What one can achieve (your approach here) and into which boxes of science and mathematics are relevant to the said work. Yes, one can do lot of things by fumbling in the dark, so to speak, but that does not mean it's not isomorphic to the existing theory, rather, the experimenter lacks a map from the problem she is solving to the established theory. I'm all for experiment…

I think you didn't understand what I wrote. I wasn't arguing for fumbling in the dark. My example of a ray-tracer is, I'm pretty sure, not something you can do by trial and error. I was arguing that the mathematical theory you need for the DSP things you mentioned isn't, mostly, the (integral and differential) calculus. There's a lot of mathematical theory you do need, but the calculus isn't it.

I totally agree that (integral and differential) calculus is a massive mental productivity booster. I'm not very convinced of the utility of schooling in acquiring that ability, because I've known far too many people who passed their calculus classes and then forgot everything, probably because they stopped using it. I've forgotten a substantial amount of calculus myself due to disuse. But I agree that schooling can work.

But I wasn't arguing against schooling, even though our current methods of schooling are clearly achieving very poor results, because they're clearly a lot better than nothing.

I was arguing that, for programming, the schooling should be directed at the things that increase your power the most. Two semesters of proving limits and finding closed-form integrals of algebraic expressions aren't it. Hopefully those classes will teach you about parametric functions, Newton's method, and Taylor series, but you can get through those classes without ever hearing about vectors (much less vector spaces and the meaning of linearity), Lambertian reflection, Nyquist frequencies, Fourier transforms, convolution, difference equations, recurrence relations, probability distributions, GF(2ⁿ) and GF(2)ⁿ, lattices (in the order-theory sense), numerical approximation with Chebyshev polynomials, coding theory, or even asymptotic notation.

In many cases, understanding the continuous case of a problem is easier than understanding the discrete case; but in other cases, the discrete case is easier, and trying to understand it as an approximation to the continuous case can be actively misleading. You may end up doing scale-space representation of signals with a sampled Gaussian, for example, or trying to use the Laplace transform instead of the Z-transform on discrete signals.

If you really want to get into arguing by way of stupid metaphors, I'd say that when you're climbing the wall of a canyon, a lightweight kayak will be of minimal help, though it may shield you from the occasional falling rock.

But I don't know, maybe you've had different experiences where itnegral and differential calculus were a lot more valuable than the stuff I mentioned above.

Re: How to Learn Advanced Mathematics Without Heading to University – Part 3

#174

Post by fsloth » Tue, Sep 06, 2016, 9:34 PM UTC

kragen wrote: post #173

fsloth wrote: post #171

Earlier quoted context omitted.

I think we approach this from different ends. What one can achieve (your approach here) and into which boxes of science and mathematics are relevant to the said work. Yes, one can do lot of things by fumbling in the dark, so to speak, but that does not mean it's not isomorphic to the existing theory, rather, the experimenter lacks a map from the problem she is solving to the established theory. I'm all for experiment…

I think you didn't understand what I wrote. I wasn't arguing for fumbling in the dark. My example of a ray-tracer is, I'm pretty sure, not something you can do by trial and error. I was arguing that the mathematical theory you need for the DSP things you mentioned isn't, mostly, the (integral and differential) calculus. There's a lot of mathematical theory you do need, but the calculus isn't it. I totally agree that…

Might be we have different chunking. In my preconceptions calculus is the first necessary stepping stone to the other stuff you mentioned. I have no idea how to approach Fourier transform conceptually for example than by the calculus route since the integral form is always introduced first. It's true linear algebra and calculus don't often meet at first - until one needs to do coordinate tranforms from e.g. spherical coordinates to cartesian.

It's true I don't need that suff in my daily work that much. But I recognise a lot of problems I might meet are trivial with some applied calculus. Like the newton iteration, which you mentioned.

Re: How to Learn Advanced Mathematics Without Heading to University – Part 3

#175

Post by ska » Tue, Sep 06, 2016, 9:51 PM UTC

Chris2048 wrote: post #153

ska wrote: post #151

Earlier quoted context omitted.

'versus "mathematicians" in the sense that they can use advanced math.' Those people are not "mathematicians", they have other useful names (most physicists, bio-statisticians, some engineers, etc.) Mathematicians are people who create new mathematics, not people who use mathematics.

Where does that definition come from? I feel the term is often used more broadly. http://c2.com/cgi/wiki?MathematicianDefinition In any case, the point stands - The need for persons who can construct mathematical proofs, versus those who simply need to derive the correct numerical result, is very different. As such, most classes that teach calculus are for practical, applied purposes - who don't need to "prove what t…

I don't find that discussion particularly insightful. Yes, the term is used more broadly, but doing so invites confusion.

Compare perhaps "composer" to "musician", they are both involved in music but operating on different axis. Most people would agree that there isn't a strict relationship superset, and there is overlap. There are skilled composers who are lousy musicians, and vice versa. There are a few people who are top rate at both. However, it is very useful to have the distinction between creating and performing.

It's much the same with mathematicians.

Re: How to Learn Advanced Mathematics Without Heading to University – Part 3

#176

Post by ska » Tue, Sep 06, 2016, 9:54 PM UTC

qrendel wrote: post #167

ska wrote: post #150

Earlier quoted context omitted.

If you are taking a degree program, it is the institutions responsibility to ensure that you have met all the requirements. If they don't have enough information on equivalence of a different institutions course, the easiest thing for them is to require you retake the sequence. The way to get around this isn't by taking pretests (which don't mean much) it's by writing the final exams. In some institutions you will be…

That's kind of my point - they could have tested me easily by giving me some problems from a previous final, or anything else, or even talking to me for five minutes, but instead they chose the path of petty legalism by assuming since the syllabus didn't agree with their's 100%, the only way to guarantee I knew the material was to make to pay to retake the entire sequence. I failed to mention I'd also already spent t…

It's not exactly petty legalism, they can lose their ability to grant degrees over stuff like this. This is one of the reasons that if you are transferring institutions, as a student it is your responsibility to check transfer credits. After all, it certainly isn't true that all undergraduate curriculum are equivalent.

It's a bit of a pain, but the point wasn't that they should give you some questions from the old exam but that you should actually sit the new one, under exam conditions. That resolves the problem for everyone without you having to spend the time repeating lectures etc. In the best case you don't pay full rate either.

Re: How to Learn Advanced Mathematics Without Heading to University – Part 3

#177

Post by kragen » Tue, Sep 06, 2016, 10:29 PM UTC

fsloth wrote: post #174

kragen wrote: post #173

Earlier quoted context omitted.

I think you didn't understand what I wrote. I wasn't arguing for fumbling in the dark. My example of a ray-tracer is, I'm pretty sure, not something you can do by trial and error. I was arguing that the mathematical theory you need for the DSP things you mentioned isn't, mostly, the (integral and differential) calculus. There's a lot of mathematical theory you do need, but the calculus isn't it. I totally agree that…

Might be we have different chunking. In my preconceptions calculus is the first necessary stepping stone to the other stuff you mentioned. I have no idea how to approach Fourier transform conceptually for example than by the calculus route since the integral form is always introduced first. It's true linear algebra and calculus don't often meet at first - until one needs to do coordinate tranforms from e.g. spherical…

http://www.dspguide.com/ch8/1.htm talks about the discrete Fourier transform, which decomposes a discrete (periodic) signal into a sum of a discrete set of sinusoids. The Fourier transform is actually a case where the continuous case is misleading — in the continuous case, you unavoidably have the Gibbs phenomenon, a complication which disappears completely in the discrete case, and the argument for this is a great deal simpler than the analogous reasoning for analytic signals. And even if you show that, for example, sinusoids of different frequencies are orthogonal in the continuous case, it doesn't immediately follow that this is true of the sampled versions of those same signals — and in fact it isn't true in general, only in some special cases. You can show by a simple counting argument that no other sampled sinusoids are orthogonal to the basis functions of a DFT, for example. Showing that the DFT basis is orthogonal is more difficult!

You definitely don't need calculus to transform between spherical and Cartesian coordinates. I mean I'm pretty sure Descartes did that half a century before calculus was invented. You do need trigonometry, which is about a thousand years older.

Newton iteration is a bit dangerous; it can give you an arbitrary answer, and it may not converge. In cases where you think you might need Newton iteration, I'd like to suggest that you try interval arithmetic (see http://canonical.org/~kragen/sw/aspmisc/intervalgraph), which is guaranteed to converge and will give you all the answers but is too slow in high dimensionality, or gradient descent, which does kind of require that you know calculus to understand and works in more cases than Newton iteration, although more slowly.

Re: How to Learn Advanced Mathematics Without Heading to University – Part 3

#178

Post by bitL » Tue, Sep 06, 2016, 11:47 PM UTC

FabHK wrote: post #84

bitL wrote: post #62

Earlier quoted context omitted.

The main problem of contemporary mathematics is that it is unbelievably obfuscated to most people, unnecessarily so. So even if you a have super-simple thing, mathematicians invented ways how to completely obfuscate meaning (often unfortunately in order to achieve prestige and being considered elite as a form of intellectual pride). Imagine Dirichlet's box principle, a thing that a 5-year old should understand; now l…

Just because something seems obvious does not mean that it is. Famously, for example, Bertrand Russell and Alfred Whitehead prove in Volume II of their Principia Mathematica, using theorem 54.43 from page 379, Volume I, that 1+1=2 (adding that "the above proposition is occasionally useful.") Now, that is clearly obvious to everyone, and yet what Russell and Whitehead achieved in the intervening 400+ pages was more th…

FYI, this might shed light on some problems introduced by Russel and Whitehead in Principia Mathematica: http://www.academia.edu/13159243/2015_Pragmatism_the_A_Prior...

Re: How to Learn Advanced Mathematics Without Heading to University – Part 3

#179

Post by Chris2048 » Wed, Sep 07, 2016, 9:22 AM UTC

ska wrote: post #175

Chris2048 wrote: post #153

Earlier quoted context omitted.

Where does that definition come from? I feel the term is often used more broadly. http://c2.com/cgi/wiki?MathematicianDefinition In any case, the point stands - The need for persons who can construct mathematical proofs, versus those who simply need to derive the correct numerical result, is very different. As such, most classes that teach calculus are for practical, applied purposes - who don't need to "prove what t…

I don't find that discussion particularly insightful. Yes, the term is used more broadly, but doing so invites confusion. Compare perhaps "composer" to "musician", they are both involved in music but operating on different axis. Most people would agree that there isn't a strict relationship superset, and there is overlap. There are skilled composers who are lousy musicians, and vice versa. There are a few people who…

Excuse me, but you're the one who went off on that tangent.

If we're relating to how people use words, it's not "much the same with mathematicians"

Re: How to Learn Advanced Mathematics Without Heading to University – Part 3

#180

Post by jamez1 » Fri, Sep 09, 2016, 3:44 AM UTC

chris_wot wrote: post #155

Retra wrote: post #138

Earlier quoted context omitted.

I'm not sure there are many places where Trigonometry is considered advanced mathematics.

Rather missing the point I fear.

She's on point, you're trivializing something you don't understand