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A Mathematician’s Lament (2002) [pdf]

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Re: A Mathematician’s Lament (2002) [pdf]

#101

Post by sampo »

codyb wrote: post #18
sampo wrote: post #12

Earlier quoted context omitted.

"Are the ideas of math that inaccessible to the general population when compared to a work of Art?" I'd guess at least some fine arts (composing classical or modernist orchestral music, modern abstract painting and sculpting) would also be pretty inaccessible to general population.

Yes, but when a classical composer converses with you and says something along the lines of "There actually is, with all the creativity, a sort of formula I follow to produce my best pieces. You see I do this first, revise it three times, etc, etc." and no one will sit and shutter and have the PTSD style flashback the original comment talked about because they do not know and they do not have preconceived, strongly h…

But if the composer would use a more geeky language, and talk about e.g. E7#9 chords, mixolydian modes and chromatic passages? How long would a layperson bother to listen.

(Well I used rock/jazz terms, since I am not familiar with classical music.)

Re: A Mathematician’s Lament (2002) [pdf]

#102

Post by kalid »

stiff wrote: post #37

Please read the article with a critical eye, some of it is complete non-sense, for example: CALCULUS: This course will explore the mathematics of motion, and the best ways to bury it under a mountain of unnecessary formalism. Despite being an introduction to both the differential and integral calculus, the simple and profound ideas of Newton and Leibniz will be discarded in favor of the more sophisticated function-ba…

Hi, I hope you take my comment well. This need to avoid some theoretical logical contradiction far down the road is similar to explaining the perils of split infinitives before teaching a child to say "I want food."

99.9% of calculus students are there to expand their mind. They will never design a quantum mechanical reactor and think "Wow, I'm getting all these weird results, my calculus education must have held me back."

It's generally preferable to give students a first order approximation at first, the successively refine with additional terms down the road. The fact that most calculus students will have no idea I'm referring to a Taylor series explanatory strategy highlights the problems with an overly rigorous introduction. Math at this high level should primarily expand your thinking, and secondarily your storehouse of previously proven rigorous statements.

Re: A Mathematician’s Lament (2002) [pdf]

#103

Post by NoodleIncident »

stiff wrote: post #37

Please read the article with a critical eye, some of it is complete non-sense, for example: CALCULUS: This course will explore the mathematics of motion, and the best ways to bury it under a mountain of unnecessary formalism. Despite being an introduction to both the differential and integral calculus, the simple and profound ideas of Newton and Leibniz will be discarded in favor of the more sophisticated function-ba…

All of this misses the author's point.

How many students make it through calculus without learning that it is the mathematics of motion at all? I'd wager that it's more than half.

Re: A Mathematician’s Lament (2002) [pdf]

#104

Post by utxaa »

stiff wrote: post #41
vlasev wrote: post #39

Earlier quoted context omitted.

The point he's making is against unnecessary rigorization of introductory calculus and I think you are getting a bit too hung up on the "function-based approach". I repeat - introductory calculus. There's lot of time and space to make things more rigorous in a class like Analysis. When I help students with calculus most of them have no trouble with the ideas but the implementation that they are required to perform. I…

Introductory calculus classes are hardly ever rigorous. The only formalism you see is functions and limits, and that's a very useful one, far from unnecessary, it might just not always be motivated appropriately by poor teachers who themselves have little understanding of its usefulness. Your interpretation is also very far from what he has actually written. I edited my parent comment to make what I mean more clear.

it is true that rigour is eventually needed, but the principia, laplace's celestial mechanics and countless other works (heard of euler?) were all published before cauchy and weirstrass. all of the wonderful work in elliptic functions by gauss, abel and jacobi was done before rigour was en vogue. euclidean and non-euclidean geometries both flourished wonderfully w/o modern rigour (when was hilbert's book on geometry published?)

you're also wrong about your examples. it wasn't nowhere differentiable functions, but fourier series that motivated lebesgue. that's what the author is referring to regarding analytical traps ie, monotone convergence.

it is also quite a leap to assert the arithmetical definition of limits solves the zeno paradox!!! i few of my colleague's might disagree with you.

don't led your initial fascination with rigour (it can be addicting) get in the way of your intuition. rigour is necessary, but it comes after - sometimes to the chagrin of some. look at the teaching of modern algebra. pure abstraction and rigour, with complete detachment of all the wonderful ideas, and experiences that gave it rise.

up with triangles i say :)

Re: A Mathematician’s Lament (2002) [pdf]

#105

Post by dwc »

john_b wrote: post #40
nsphere wrote: post #5

"There is such breathtaking depth and heartbreaking beauty in this ancient art form. How ironic that people dismiss mathematics as the antithesis of creativity. They are missing out on an art form older than any book, more profound than any poem, and more abstract than any abstract. And it is school that has done this! What a sad endless cycle of innocent teachers inflicting damage upon innocent students. We could al…

I think that language (at least English) fails us here. The word "creative" can be used to mean both "relating to or involving the imagination or original ideas, especially in the production of an artistic work" [1] or "resulting from originality of thought, expression, etc.; imaginative" [2]. It seems that most people identify the word more strongly with the artistic sense of the word, in which creativity is a proxy…

> creativity is a proxy for a kind of self-expression that is not bound by any rules, logic, or structure

I do not find this to be the case. Art is full of structure, rules and logic. When artists "break" the rules, it usually means they been able to operate using underlying rules, and understand well the rules they break.

Re: A Mathematician’s Lament (2002) [pdf]

#106

Post by dsego »

dsego wrote: post #97

Still haven't finished reading this and all the painful memories came back from middle school. Not math (although it too wasn't a fun experience) but art. For four years we had a teacher that thought music and visual arts by dictating. For 45 minutes we would write down everything word for word. Then after a few weeks she would ask us broad questions and we'd have to recite everything back to her. I still remember (I…

oops, meant taught instead of thought.

Re: A Mathematician’s Lament (2002) [pdf]

#108

Post by abecedarius »

westoncb wrote: post #80
lmartel wrote: post #52

I've read this article several times at this point (it does tend to pop up everywhere) and it resonates with me but I'm not sure what to do about it. I really want to experience the kind of math the author writes about; can anyone recommend a place to start as someone who has only ever done "fake" high school math? I'm in college now and I'm halfway through a computer science degree; I've tried a few times to break i…

Hey Imartel -- I've been in a similar situation. I think my first starting point was "Mathematics and the Imagination" or "Gödel's Proof." Mathematics and the Imagination is a good high level overview, and would provide some foundational notions that'll reappear repeatedly -- but, it won't given you any practice in mathematical methods. For that I would recommend "What is Mathematics?" by Courant and Robbins. Can be…

Courant and Robbins is really great. I'd recommend checking it out after the Lockhart book (or together with it), as it's more textbookish; there's more danger of bouncing off.

Re: A Mathematician’s Lament (2002) [pdf]

#109

Post by stiff »

utxaa wrote: post #104
stiff wrote: post #41

Earlier quoted context omitted.

Introductory calculus classes are hardly ever rigorous. The only formalism you see is functions and limits, and that's a very useful one, far from unnecessary, it might just not always be motivated appropriately by poor teachers who themselves have little understanding of its usefulness. Your interpretation is also very far from what he has actually written. I edited my parent comment to make what I mean more clear.

it is true that rigour is eventually needed, but the principia, laplace's celestial mechanics and countless other works (heard of euler?) were all published before cauchy and weirstrass. all of the wonderful work in elliptic functions by gauss, abel and jacobi was done before rigour was en vogue. euclidean and non-euclidean geometries both flourished wonderfully w/o modern rigour (when was hilbert's book on geometry…

it is true that rigour is eventually needed, but the principia, laplace's celestial mechanics and countless other works (heard of euler?) were all published before cauchy and weirstrass. all of the wonderful work in elliptic functions by gauss, abel and jacobi was done before rigour was en vogue. euclidean and non-euclidean geometries both flourished wonderfully w/o modern rigour (when was hilbert's book on geometry published?)

Now any hard-working student of university calculus can without problems understand and reproduce the results of those treatises. I think this would not be possible without the modern systematic methods, including the epsilon-delta approach.

you're also wrong about your examples. it wasn't nowhere differentiable functions, but fourier series that motivated lebesgue. that's what the author is referring to regarding analytical traps ie, monotone convergence.

I mentioned fourier in another comments. Those weird functions were postulated in the discussion that arised somewhere in the same time period. I don't know what the author is referring to, because he is irritatingly vague, especially for a mathematician.

it is also quite a leap to assert the arithmetical definition of limits solves the zeno paradox!!! i few of my colleague's might disagree with you.

If you are familiar with the concept of a limit, you can notice that Zeno considers a limiting process of two related quantities, time and the difference in the position of Achilles and the Tortoise. Since in this limiting process time get arbitrarly close to some definite value (the meeting time of Achilles and Tortoise), but never gets equal to it, it stops being so surprising that the distance between them never reaches zero, though it gets arbitrarly close. The formalism clarifies what seems parodoxical when described in natural language. I find this quite convincing, and I never found a better explanation, altough I know philosophers still dispute this.

I never said rigour takes precedence over intuition. I just think it's the inherent difficulty of calculus that stops students from understanding it, and not the epsilon-delta stuff.

Re: A Mathematician’s Lament (2002) [pdf]

#110

Post by saraid216 »

marcusf wrote: post #92
saraid216 wrote: post #24

Earlier quoted context omitted.

One of the key mistakes of educational strategy is dividing up knowledge into a bunch of subjects like "Language", "Math", and so on. You can find linkages in just about everything; this siloing really only serves to make people think they're particularly good at one thing and not another, when really it's a preference for one perspective over another.

So the pragmatists question is then naturally, how do you teach it? How do you shard the curriculum if not by subject? It seems a bit overbearing to ask each teacher to have the proper renaissance man's education of knowing a bit of everything.

To me, a teacher's proper role isn't having knowledge and dumping it into someone's brain. It's knowing where to find that knowledge. My three Rs are "reading, research, and reflection": a teacher's job is to (1) provide useful material to consume, via lecture or homework or whatnot, (2) point towards larger resources for further exploration, and (3) guide thought processes to make useful conclusions.

A teacher's job is not to teach. It's to provide a space in which a student can learn. A focus is useful for this, but the focus doesn't need to be an abstract subject. It's a MacGuffin; it can be anything.

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