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Sure.  And I tried to teach my students to guess the answers before going ahead and solving properly.  

Without intuition you get nowhere.  

But none of my colleagues would ever admit that in public.  

Why do you think is that?  

The Fates are kind.

by Gaianne on Wed Jan 2nd, 2008 at 04:39:28 AM EST
[ Parent ]
I have absolutely no idea: the idea that you can work without intuition is bizarre to me. I mean, intuition -> experiment/proof and peer review -> new intuition(s) is how it works. Where and when was this, in general terms?
by Colman (colman at eurotrib.com) on Wed Jan 2nd, 2008 at 04:43:17 AM EST
[ Parent ]
There is the ongoing pedagogical problem where the process is very seldom taught, at least in mathematics, where everything is generally presented in a linear fashion - because that's the easiest way to read it once you're done - and people start to imaging that's how the results were arrived at, which is almost never the case.
by Colman (colman at eurotrib.com) on Wed Jan 2nd, 2008 at 04:50:50 AM EST
[ Parent ]
That's exactly the problem I had when I was teaching "finite mathematics" which was the first undergraduate course where proof techniques were introduced.

I had to explain to my students that the proofs in their textbooks made no sense because they were discovered inductively in the opposite direction than they were written, and then sharpened for "elegance" in order to impress other professors so they will select the textbook. Take, for example, a typical epsilon-delta exercise from calculus. The official answer will start with "let epsilon equal blah blah bla" and the student will stop right there and scratch their head "how did they come up with that?". Well, that was the last thing they came up with.

We have met the enemy, and he is us — Pogo

by Migeru (migeru at eurotrib dot com) on Wed Jan 2nd, 2008 at 07:00:03 AM EST
[ Parent ]
I had similar troubles with proof techniques during secondary schooling... I could follow the logic of the proof, but not how anyone could come up with it out of thin air, which even then seemed slightly more important to me.
by Nomad on Wed Jan 2nd, 2008 at 07:25:30 AM EST
[ Parent ]
It's the pedagogy and sociology of mathematics that are broken, not the subject matter.

We have met the enemy, and he is us — Pogo
by Migeru (migeru at eurotrib dot com) on Wed Jan 2nd, 2008 at 07:28:33 AM EST
[ Parent ]
Ahhhh, I wish I could have been your student, my relationship with math and some of science would probably be very different. :-)
by Fran (fran at eurotrib dot com) on Wed Jan 2nd, 2008 at 05:07:38 AM EST
[ Parent ]
Because they're mediocre?

Dirac admitted it in public. The way he put is is that what people call "physical intuition" is knowing what the solution looks like without solving the equation.

Seeing how some of my classmates who were better than I was at symbolic algebra and functional analysis and never got caught by a trivial counterexample, but were mystified by even slightly nontivial PDEs, I came to the conclusion that I did the right thing studying physics before mathematics. Physical intuition allow the mathematics to come alive.

We have met the enemy, and he is us — Pogo

by Migeru (migeru at eurotrib dot com) on Wed Jan 2nd, 2008 at 06:57:03 AM EST
[ Parent ]
against abstract mathematics on much these grounds.  

(It wasn't the abstraction he objected to, but the disconnect.)

The Fates are kind.

by Gaianne on Wed Jan 2nd, 2008 at 07:18:47 AM EST
[ Parent ]

Gaianne: ... Without intuition you get nowhere.  

But none of my colleagues would ever admit that in public.  

Why do you think is that?  

Because they were prisoners of a false image of what real maths or science was like - from their own education and the way papers are written - tidily, after the messy process of creative thinking (cf Colman below).

Koestler's "The Act of Creation" was published back in 1964 and gave many examples from science (as well as the arts) of how the creative process worked; bascially:

felt need (arts) existing problem (sciences)

initial conscious work

unconscious activity (so important to leave time for this - a very important lesson that B. Russell learnt the hard way after years of struggling to do it all consciously)

Insight/inspiration

The latter often occurred in dreams or half-awake state and often involved imagery - including in sciences - Kekule snake with tail in mouth I seem to recall - yes:

 "While researching benzene, the German chemist dreamed of a snake with its tail in its mouth. Kekulé interpreted the snake as a representation of the closed-carbon ring of benzene"

http://en.wikipedia.org/wiki/Cryptomnesia

 Einstein even said his creative "thinking" involved muscular sensations.

As this review points out, unfortunately it got a very bad review from Medawar (and probably other scientists) so not too many young scientists may have read this - a pity:

"The Act of Creation" offers a theory to account for the "Ah Ha" reaction of scientific discovery, the "Ha Ha" reaction to jokes and the "Ah" reaction of mystical or religious insight. In each case the result is produced by a "bisociation of matrices" or the intersection of lines of thought which brings together hitherto unconnected ideas and fuses them into a creative synthesis. When the lines of thought are scientic the result is a scientific discovery, when they are concerned with devotional matters the result is mystical insight and when they are on a more homely plane the result can be a joke.

The model is fleshed out with a great deal of information ranging from the religions of the world to a theory about the nervous system to account for the build-up of tension and its discharge at the puchline of a joke. Peter Medawar's review was scathing in his comments on Koestler's science, which is a shame because the book can have the desirable effect of encouraging young scientists to read far beyond the usual range of their literature.

http://www.amazon.ca/Act-Creation-Arthur-Koestler/dp/0140191917



Maybe it's because I'm a Londoner - that I moved to Nice.
by Ted Welch (tedwelch-at-mac-dot-com) on Wed Jan 2nd, 2008 at 11:43:24 AM EST
[ Parent ]
In my top 10 of 'books that have changed me'

You can't be me, I'm taken
by Sven Triloqvist on Wed Jan 2nd, 2008 at 11:56:48 AM EST
[ Parent ]

We probably agree on LOTS of things :-) By the way I went on courses run by Tom Hudson and Terry Setch too.

Maybe it's because I'm a Londoner - that I moved to Nice.
by Ted Welch (tedwelch-at-mac-dot-com) on Wed Jan 2nd, 2008 at 12:06:13 PM EST
[ Parent ]
Wow, that is a neat bit of serendipity. Studying under Tom Hudson certainly shook me out of my middle-class complacency! The other who was a great influence at that time in Leicester was Mike Sandle, who to a certain extent took me under his wing.

You can't be me, I'm taken
by Sven Triloqvist on Wed Jan 2nd, 2008 at 12:17:04 PM EST
[ Parent ]

I only did short courses with them at Barry summer school after finishing art school (Camberwell). I'm afraid I found it a bit superficial - I generated quite a lot of visual ideas, as they encouraged us to do, and I think it was Setch who said I had enough ideas already for 10 years work and a show. I thought what I had done was all a bit trivial - no real aha moments :-) But they could be very encouraging and liberating if you were stuck in some limited mode. I see Setch was still working quite recently with some stuff supposedly having an ecological orientation - though it wasn't really clear in the work itself on his site.

Maybe it's because I'm a Londoner - that I moved to Nice.
by Ted Welch (tedwelch-at-mac-dot-com) on Wed Jan 2nd, 2008 at 12:40:35 PM EST
[ Parent ]
This on the workings of the conscious and the subconscious in problem solving...

Then Poincaré illustrated how a fact is discovered. He had described generally how scientists arrive at facts and theories but now he penetrated narrowly into his own personal experience with the mathematical functions that established his early fame.

For fifteen days, he said, he strove to prove that there couldn't be any such functions. Every day he seated himself at his work-table, stayed an hour or two, tried a great number of combinations and reached no results.

Then one evening, contrary to his custom, he drank black coffee and couldn't sleep. Ideas arose in crowds. He felt them collide until pairs interlocked, so to speak, making a stable combination.

The next morning he had only to write out the results. A wave of crystallization had taken place.

He described how a second wave of crystallization, guided by analogies to established mathematics, produced what he later named the "Theta-Fuchsian Series." He left Caen, where he was living, to go on a geologic excursion. The changes of travel made him forget mathematics. He was about to enter a bus, and at the moment when he put his foot on the step, the idea came to him, without anything in his former thoughts having paved the way for it, that the transformations he had used to define the Fuchsian functions were identical with those of non-Euclidian geometry. He didn't verify the idea, he said, he just went on with a conversation on the bus; but he felt a perfect certainty. Later he verified the result at his leisure.

A later discovery occurred while he was walking by a seaside bluff. It came to him with just the same characteristics of brevity, suddenness and immediate certainty. Another major discovery occurred while he was walking down a street. Others eulogized this process as the mysterious workings of genius, but Poincaré was not content with such a shallow explanation. He tried to fathom more deeply what had happened.

Mathematics, he said, isn't merely a question of applying rules, any more than science. It doesn't merely make the most combinations possible according to certain fixed laws. The combinations so obtained would he exceedingly numerous, useless and cumbersome. The true work of the inventor consists in choosing among these combinations so as to eliminate the useless ones, or rather, to avoid the trouble of making them, and the rules that must guide the choice are extremely fine and delicate. It's almost impossible to state them precisely; they must be felt rather than formulated.

Poincaré then hypothesized that this selection is made by what he called the "subliminal self," an entity that corresponds exactly with what Phadrus called preintellectual awareness. The subliminal self, Poincaré said, looks at a large number of solutions to a problem, but only the interesting ones break into the domain of consciousness. Mathematical solutions are selected by the subliminal self on the basis of "mathematical beauty," of the harmony of numbers and forms, of geometric elegance.

"This is a true aesthetic feeling which all mathematicians know," Poincaré said, "but of which the profane are so ignorant as often to be tempted to smile."

But it is this harmony, this beauty, that is at the center of it all.

Problem solving is what I have always enjoyed more than anything else. I think that "pattern recognition" is a good way of describing the process.

My motivation is to find the simple answer that everybody knows, but which doesn't yet exist: Naoto Fukasawa


"The future is already here -- it's just not very evenly distributed" William Gibson
by ChrisCook (cojockathotmaildotcom) on Wed Jan 2nd, 2008 at 12:14:03 PM EST
[ Parent ]
Yes, it's remarkable how important beauty can be to those in maths and physics - there are a number of examples in Kaku's "Parallel Worlds".

Maybe it's because I'm a Londoner - that I moved to Nice.
by Ted Welch (tedwelch-at-mac-dot-com) on Wed Jan 2nd, 2008 at 12:44:04 PM EST
[ Parent ]
Why is it remarkable? Isn't aesthetic pleasure important to engineers, architects, sculptors, painters, photographers, filmmakers, writers...?

We have met the enemy, and he is us — Pogo
by Migeru (migeru at eurotrib dot com) on Wed Jan 2nd, 2008 at 12:47:53 PM EST
[ Parent ]

Because it is so different from the general image the general public has of the nature of science and maths (I don't recall it being mentioned during my education :-)), and, has been noted here, even many scientists do not like to acknowledge the creative/intuitive nature of their work.

Maybe it's because I'm a Londoner - that I moved to Nice.
by Ted Welch (tedwelch-at-mac-dot-com) on Wed Jan 2nd, 2008 at 12:56:20 PM EST
[ Parent ]
As Sam has mentioned perviously, we come from a college background where drinking in the pursuit of theorems was a semi-serious  strategy. I just find this whole part of the discussion culturally weird.
by Colman (colman at eurotrib.com) on Wed Jan 2nd, 2008 at 12:58:49 PM EST
[ Parent ]
It is remarkable that the general public has the wrong idea about the nature of science and maths?

We have met the enemy, and he is us — Pogo
by Migeru (migeru at eurotrib dot com) on Wed Jan 2nd, 2008 at 12:59:33 PM EST
[ Parent ]
A big factor is that a vast majority of people in the written press have arts degrees, and are activly disinterested in science. Unless you are reading more specific science press, the quality of science reporting is poor.

On top of this you have a vast amount of celebrity reporting that compresses the space that's available for other reporting. The vast increase in the number of TV channels also allows people to avoid any exposure to science.

why is Science that important to the average person when Brittneys got no underwear on?

Any idiot can face a crisis - it's day to day living that wears you out.

by ceebs (ceebs (at) eurotrib (dot) com) on Wed Jan 2nd, 2008 at 01:15:25 PM EST
[ Parent ]
And elegance as a criteria for suggesting truth in mathematics is a widely known heuristic, surely? Not that it always works, but people are extra suspicious of inelegant hacked together proofs.
by Colman (colman at eurotrib.com) on Wed Jan 2nd, 2008 at 01:00:09 PM EST
[ Parent ]
This book, edited by Graham Farmello - subtitled 'Great Equations of Modern Science' goes into great detail on this subject.

You can't be me, I'm taken
by Sven Triloqvist on Wed Jan 2nd, 2008 at 01:08:50 PM EST
[ Parent ]

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