2012 : What is Your Favorite Deep, Elegant, or Beautiful Explanation?

...walk alone .. if they answer not your call, walk alone ... if they are afraid and cower mutely facing the wall, open thy mind and speak alone...Gandhi...

Life is the highest value. Your life, the lives of all other people, and every living entity.

The Mandelbrot set

“Worry often gives a small thing a big shadow.”

Two words that go straight to the core of elegant scientific theory: Ockham's Razor... It is the original elegant theory that allows all other elegant theories to exist. It is usually the simplest answers that are correct.

Infinities are processes, not numbers.

I love both the playfulness and the humility herein:

--from Mark Pagel's "Infinite Stupidity" on Edge:

Karl Popper famously said the way we differ from other animals is that our hypotheses die in our stead; rather than going out and actually having to try out things, and maybe dying as a result, we can test out ideas in our minds. But what I want to suggest is that the generative process itself might be pretty close to random.

Putting these two things together has lots of implications for where we're going as societies. As I say, as our societies get bigger, and rely more and more on the Internet, fewer and fewer of us have to be very good at these creative and imaginative processes. And so, humanity might be moving towards becoming more docile, more oriented towards following, copying others, prone to fads, prone to going down blind alleys, because part of our evolutionary history that we could have never anticipated was leading us towards making use of the small number of other innovations that people come up with, rather than having to produce them ourselves.

The interesting thing with Facebook is that, with 500 to 800 million of us connected around the world, it sort of devalues information and devalues knowledge. And this isn't the comment of some reactionary who doesn't like Facebook, but it's rather the comment of someone who realizes that knowledge and new ideas are extraordinarily hard to come by. And as we're more and more connected to each other, there's more and more to copy. We realize the value in copying, and so that's what we do.

And we seek out that information in cheaper and cheaper ways. We go up on Google, we go up on Facebook, see who's doing what to whom. We go up on Google and find out the answers to things. And what that's telling us is that knowledge and new ideas are cheap. And it's playing into a set of predispositions that we have been selected to have anyway, to be copiers and to be followers. But at no time in history has it been easier to do that than now. And Facebook is encouraging that.

And then, as corporations grow … and we can see corporations as sort of microcosms of societies … as corporations grow and acquire the ability to acquire other corporations, a similar thing is happening, is that, rather than corporations wanting to spend the time and the energy to create new ideas, they want to simply acquire other companies, so that they can have their new ideas. And that just tells us again how precious these ideas are, and the lengths to which people will go to acquire those ideas.

A tiny number of ideas can go a long way, as we've seen. And the Internet makes that more and more likely. What's happening is that we might, in fact, be at a time in our history where we're being domesticated by these great big societal things, such as Facebook and the Internet. We're being domesticated by them, because fewer and fewer and fewer of us have to be innovators to get by. And so, in the cold calculus of evolution by natural selection, at no greater time in history than ever before, copiers are probably doing better than innovators. Because innovation is extraordinarily hard. My worry is that we could be moving in that direction, towards becoming more and more sort of docile copiers.

But, these ideas, I think, are received with incredulity, because humans like to think of themselves as highly shrewd and intelligent and innovative people. But I think what we have to realize is that it's even possible that, as I say, the generative mechanisms we have for coming up with new ideas are no better than random.

And a really fascinating idea itself is to consider that even the great people in history whom we associate with great ideas might be no more than we expect by chance. I'll explain that. Einstein was once asked about his intelligence and he said, "I'm no more intelligent than the next guy. I'm just more curious." Now, we can grant Einstein that little indulgence, because we think he was a pretty clever guy.

But let's take him at his word and say, what does curiosity mean? Well, maybe curiosity means trying out all sorts of ideas in your mind. Maybe curiosity is a passion for trying out ideas. Maybe Einstein's ideas were just as random as everybody else's, but he kept persisting at them.

And if we say that everybody has some tiny probability of being the next Einstein, and we look at a billion people, there will be somebody who just by chance is the next Einstein. And so, we might even wonder if the people in our history and in our lives that we say are the great innovators really are more innovative, or are just lucky.

Now, the evolutionary argument is that our populations have always supported a small number of truly innovative people, and they're somehow different from the rest of us. But it might even be the case that that small number of innovators just got lucky. And this is something that I think very few people will accept. They'll receive it with incredulity. But I like to think of it as what I call social learning and, maybe, the possibility that we are infinitely stupid.

http://www.edge.org/conversation/infinite-stupidity-edge-conversation-with-mark-...

That the scientific method is merely trial and error, and a hypothesis, at best, an educated guess.

That humans are very low on the evolutionary scale, unable to conceive of anything beyond three dimensions when in fact there are an infinite number of dimensions to the nth degree. As some species cannot discern color, we cannot perceive the infinite dimensions, reducing our logical thinking to a primitive cognition level: feelings are, after all, irrational to us yet; but there where is our experience lies.

"Accept me as I am - only then will we discover each other."
Federico Fellini (1920-1993), Italian director/screenwriter

My favorite quote for science/mathmatics is very much relavent to this post. I think it goes very nicely with Ziggy's posts below:

"For every problem, there exists a simple and elegant solution which is absolutely wrong." - J. Wagoner, U.C.B. Mathematics

This is my own. Prove me wrong if you like.

The dimensions are not polarized. Instead, Einstein's dimensions are rays that emanate in a tetrahedral fashion from a point. It takes four rays to define space, so the fourth dimension, which is thought to be time, is actually a vector which we are following through the universe. With rays there is no need for negative numbers. Negative numbers are imaginary throughout the universe.

This one requires a little thinking but it underscores simplicity as a key factor of it's elegant versus journeyman solution.

Here we go.

Given a flat earth, a homogeneous acceleration of gravity g, and neglecting air resistance, we are asked to determine at what distance a cannon ball hits the ground again, if it is shot away with an initial velocity v and angle phi with the horizontal direction. I have posed this problem to a number of grown-up professional mathematicians. A surprisingly large fraction could not really solve it --"It was too long ago that they had done problems like that."--, a few applied mathematicians solved it in a straightforward manner by writing down the differential equations of motion, solving them, adjusting the constants, adjusting the constants so as to satisfy the initial conditions, and finally setting y = 0 in the equation for the trajectory. (The story doesn't tell whether the eventual simplicity of the answer made them suspicious that they could have reached the answer in a simpler manner.) Those that could not solve it and those that solved it in the above manner, essentially reacted in the same way: they tried to solve the problem by means of reasoning techniques that were their daily routine; for the majority this routine pattern was insufficient, only the routine pattern of the applied mathematicians enabled those to solve the problem (but only after a fashion, as well shall see in a moment).

Two of them reacted quite differently: they gave the answer instantaneously because they knew it. And they knew it because they had solved the problem at the age of 14 --when both had posed it to themselves-- by a very simple argument, so simple that it is impossible to forget it once you have seen it. Their argument had to be simple because at that time 14-year old boys had not had analytical geometry --and so they did not know what a parabola was--, and they did not know what differential equations were, let alone that they knew how to solve them. They did not know much more than the definition of velocity --"the rate of change of position"-- and the definition of acceleration --"the rate of change of velocity"-- and a rather informal introduction to Newtonian mechanics. The boys' argument was as follows.

Because the horizontal speed is constant -- vx = v*cos(phi) -- the required distance is equal to t*vx, where t is the time of travel through the air. During the time of travel, the vertical speed has changed from vy = v*sin(phi) --for reasons of symmetry!-- to –vy, i.e the total decrement of the vertical speed equals 2*vy. Because the rate of change of vertical velocity is given to be g, we derive t = 2*vy/g, and the required answer is therefore

2*vx*vy
---------
g

(an answer of which all except the factor of 2 can be derived by dimensional analysis, a derivation which essentially repeats the boys' argument.)

The little experiment is interesting in two ways. Firstly we can try to appreciate why the boy's argument is so much more attractive than the applied mathematicians' derivation of the parabolic trajectory. It is, because in the boys's argument the reasoning consists of two different parts: a --very simple-- argument about the horizontal motion and an --also very simple-- argument about the vertical motion, the two being combined by the very "narrow" interface of t, the time of travel through the air. This simplicity is lost as soon as one writes down the equation of the trajectory. Secondly I don't think it very noteworthy that the little boys each found the argument: they used about the only tools they had. But it is worth observing that the grown-up mathematicians immediately tried to solve the problem with the familiar reasoning apparatus of their daily routines! The ones that did not solve the problem should have realized the unsuitability of their daily used equipment --and should, to my taste, have found the boys' argument--; the applied mathematicians that solved the differential equations should, to my taste, have realized that they were cracking an egg with a sledgehammer --an activity I don't like, no matter how effective a sledgehammer may be for cracking eggs--.

Source: http://www.cs.utexas.edu/~EWD/transcriptions/EWD06xx/EWD619.html

The beauty in this experiment is in the properties of phi. The elegance is in the simplicity of the equation.

added to "The 4000 Club",.....though apparently I was, temporarily , thrown out.

- - - CANT add it to "End Time Prophecy",............because I REMAIN,...uhm,....exiled. - sigh -

Thanks incredulous

Here is a simple one that is one of my favorites and doesn't require a degree to understand.

In the late 18th century a German schoolmaster gave --with the intention of keeping his pupils busy for another hour-- the task of adding a hundred terms of an arithmetic progression to a class of little boys who, of course, had never heard of arithmetic progressions. (To quote E.T.Bell: "The problem was of the following sort: 81297 + 81495 + ... + 100988, where the step from one number to the next is the same all along (here 198) and a given number of terms (here 100) are to be added.") The youngest pupil, however, wrote down the answer instantaneously and waited, gloriously, with his arms folded, for the next hour while his classmates toiled: at the end of the hour it turned out that little Johan Friedrich Carl Gauss had been the only one to hand in the correct answer. Young Gauss had seen instantaneously how to sum such a series analytically:

the sum equals the number of terms, multiplied by the average of the first and last term.

In two respects this is a classical example of elegance: firstly young Gauss produced the answer about a thousand times as fast as his classmates, secondly he was the only one to produce the correct answer. So much for the effective ordering of one's thoughts!

...I once asked an old man: which is more important, to Love or be Loved. He replied: Which is more important to a bird? The left wing or the right wing?"

We are all one .

What a great post. I have some reading to do. Much appreciated, Incredulous.

Cool, Thanks for this Post as well Incredulous, will have to view after work.... =))=

wow, bookmarked for detailed perusal, thank you

p.s. The theory of forms. ( and "The Myth of The Cave" )

Clemens. ( Twain )
or
H. L. Mencken
“On one issue, at least, men and women agree: they both distrust women.”
― H.L. Mencken

- - - kidding dudes,....kidding,....( not necessarily arguing,.... but ,....this -, for example . . .)

“Civilization, in fact, grows more maudlin and hysterical; especially under democracy it tends to degenerate into a mere combat of crazes; the whole aim of practical politics is to keep the populace alarmed (and hence clamorous to be led to safety) by an endless series of hobgoblins, most of them imaginary. Wars are no longer waged by the will of superior men, capable of judging dispassionately and intelligently the causes behind them and the effects flowing out of them. The are now begun by first throwing a mob into a panic; they are ended only when it has spent its ferine fury.”
― H.L. Mencken, In Defense Of Women

THIS is what "Culture" is about; added.

Archive of questions and answers from 1998 on at the link.