lederhosen: (Default)
[personal profile] lederhosen
(Partly, I think, suggested by a passage in Oliver Sacks' Uncle Tungsten in which he discusses music and the periodic table.)

I know several of my artistic friends are tired of hearing lines like "You're so lucky to have that talent!" when a big part of their 'talent' comes not from luck but years of hard work.

For most media, that's the nature of art - people won't see the hard-earned technique unless they actively look for it, but the creative aspect is there for all to perceive. You can look at a Renoir and enjoy it without knowing a thing about Renoir's painting technique; you can listen to Beethoven's Ninth Symphony and feel its beauty and vibrancy without the faintest idea of how to read music, much less how to score for an entire orchestra.

Beethoven, of course, was deaf.

By the time the Ninth was finally performed, he was so utterly deaf that he was still attempting to conduct it several bars after the orchestra had finished, and he had to be turned around to see that the audience were applauding. For him, I presume, technique was the only medium through which he could know what he was writing, and hear it, if not with his ears, then at least in his mind.

Mathematics is a little like that. At first, the technique relates to things that we can perceive directly - we can lay out apples in a square, two by two, and see that two-squared is four; with a little bit of balancing, we can sit another layer of apples on top of the first four and see that two-cubed is eight. This is something comparable, perhaps, to playing a few notes on a scale and hearing what they sound like.

But take it a little further - what about two-to-the-fourth? - and we can't see it any more. We can still build physical analogies of one sort or another (imagine a second apple inside each apple, perhaps?) but they swiftly become so complicated or so bizarre that we can no longer visualise them in a way that lets us grasp the work we're trying to do; we have to build it inside our heads, a strange sort of shape beyond anything our physical senses can capture.

For the last six months, I've been working on a multi-stage sampling project, immersed in the mathematics that involves; I wish I could describe how that feels inside my head. It's a little like a telescope, with four or five stages that fold into one another; it's a little like an orrery, with a cloud of tiny satellites - each of them, viewed from a certain angle, represents a person - orbiting a swarm of invisible moonlets, which in turn orbit moons, each captive to a planet, each planet captive to the invisible sun at the heart of the system. But it's not really any of these things; trying to describe these structures in terms of things we can see is like trying to turn paintings into music.

So the only way I can share that with anybody else if if they've learned the relevant technique - and even then, they'll need to spend some time staring at the dry equations in order to go from 'notes' to 'music'. I suspect this is part of the problem: most people never get enough exposure to the technique that they're able to get past it to see what it represents. It's like staring at a Pointillist painting, and seeing only a sea of coloured dots.

To get very far with mathematics, you need to reach that level of comfort with the underlying technique, internalise it and practice to the point where you can forget it. I can learn how to press a key on a piano and produce a note of whatever pitch and volume I want, but that isn't enough to make me a musician**; knowing the rules of logic is vital as a first step in mathematics, but it doesn't take me very far on the road to doing what I want in the medium. A musician can listen to music they've never heard before, and guess what the next notes will be; there are similar instincts at work when I'm doing my job. Logic tells me what steps I can and can't take, but instinct guides me in choosing which of those countless possibilities will get me closer to where I want to be. It is a creative process in its own fashion, and knowing where that process will begin and end doesn't detract from that creativity, any more than painting loses its creativity when you're looking at a model.

Not, of course, that the distinction between 'technique' and 'creation' is sharply drawn. In mathematics especially, yesterday's creation is tomorrow's technique - and sometimes technique, picked apart and re-evaluated a la Godel etc, becomes creation again.


*Mind you, a good knowledge of technique almost always increases the ability to appreciate art - it's just that you can get a long way even without it.
**Sadly, I had almost ten years of piano lessons without ever really figuring out that there was more to music than pushing the keys in the right way at the right time.

Date: 2008-02-26 12:47 pm (UTC)
From: [identity profile] ruth-lawrence.livejournal.com
I have a couple of (minor) talents, but I didn't do the work and know it...wrong place, wrong time, not enough talent to be worth the arty scenes (I felt back then). It's too late now.

I've never thought of maths that way, though.

Interesting!

Date: 2008-02-26 02:39 pm (UTC)
From: [identity profile] silmaril.livejournal.com
Read, felt, liked, linked. Thanks.

Date: 2008-02-26 03:00 pm (UTC)
From: [identity profile] nefaria.livejournal.com
I'm afraid I got to be good at mathematics for a while without ever becoming an artist at it. I learned the processes and formulas and how to solve problems described in the book, but I never developed a flair for venturing off into unchartered territory. When I did end up there, for one reason or another, I got hopelessly lost. Calculus was my high point, after that I started to get very confused. Topology hurt my brain, and Galois theory (at least as taught by my crazy instructor) marked the end of my math career.

I used to be very good at computer programming, back when it was an elegant mix of math and logic, but lately it has become neither, and I'm feeling mediocre at best.

Date: 2008-02-26 08:47 pm (UTC)
From: [identity profile] lederhosen.livejournal.com
I never got to grips with Galois theory either. (Which is annoying, because I'm pretty sure I'd like it if it clicked, but it never did. IIRC I was dealing with Angsty Breakup at the time, which didn't help matters.)

My topology lecturer was an intimidating Russian fellow whose philosophy of marking assignments was to give you something approximating zero regardless of what you'd got, and then mark you up again if you could justify it to him - turned everything into a de facto oral exam. The subject matter was interesting, but it was a very stressful way to learn. Algebraic topology, I just didn't get (again, the Angsty Breakup didn't help).

Date: 2008-02-26 10:49 pm (UTC)
From: [identity profile] nefaria.livejournal.com
Funny you should mention, my topology teacher had a recommended book for us. It was in French. o_O

Seems like whenever I have a good book to follow, I learn what I need. Without the book, I'm doomed.

Date: 2008-02-26 05:45 pm (UTC)
michiexile: (Default)
From: [personal profile] michiexile
Read it. Loved it. Enough that I just got you submitted to the next Carnival of Mathematics?

Date: 2008-02-26 08:47 pm (UTC)

Date: 2008-02-26 10:14 pm (UTC)
From: [identity profile] psychowoof.livejournal.com
This reminds me: I have a statistical type question to ask you.

I'm advising a doctoral student who is way in over her head in terms of analyses. Basically, she's examining the question of whether type of medication (e.g., antipsychotic v. other v. none v. combo) influences choking (yes, physical airway obstruction) among the mentally retarded (level of mental retardation may be a moderating variable). However, she's playing in some state generated data base of choking incidents and other info on mentally retarded individuals. So I suggested a logistic regression using dummy coding for the IV(s).

She started off so over her head that she simply provided a break down in terms of meds for all the patients who had choked. Um, big problem if the DV is now a constant! So I sent her off to get a sample of people who have not choked.

Now, she's at a point that is slightly over my head. In gathering a final sample for analysis, how big should the group of non-chokers be. For instance, if choking generally occurs in 1% of the patient population, should the sample also be 99% non-chokers and 1% chokers regardless of the IVs?

Yeah, I'm befuddled because I've never done research from existing data sources, so I've never known the outcome on my DV until I've completed my study.

Date: 2008-02-27 08:57 am (UTC)
From: [identity profile] lederhosen.livejournal.com
For instance, if choking generally occurs in 1% of the patient population, should the sample also be 99% non-chokers and 1% chokers regardless of the IVs?

I'm not familiar enough with logistic regression to give you a reliable answer to that, but I suspect that doing it that way might break some of the assumptions of the method. (In a regular experiment, the ratio of chokers to non-chokers would be a random variable, and part of logistic regression is dealing with that variation; done this way, it would be a constant.)

If you have good data about the overall percentages of the population who are chokers and who are in each of those medication categories, and the things that ought to be independent are, you can probably get an answer that way (and it should be slightly more accurate than grabbing a sample of non-chokers); I'll think about this a bit and see if I can come up with anything useful.

Edit: Partial response here.
Edited Date: 2008-02-27 10:23 am (UTC)

Date: 2008-02-27 12:27 am (UTC)
From: [identity profile] mr-figgy.livejournal.com
Been listening to a bit of the Ludwig Van as well?

Date: 2008-02-27 09:34 am (UTC)
From: [identity profile] culfinriel.livejournal.com
That was beautifully written.

Date: 2008-02-28 05:27 pm (UTC)
From: [identity profile] harliquinn.livejournal.com
I'm taking this post, printing it, framing it and keeping it forever.

Date: 2008-02-28 08:28 pm (UTC)

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