Mathematical Beauty

[aeolium]

I thought this was an interesting article on the way the brain responds to mathematical theorems:

http://www.bbc.co.uk/news/science-environment-26151062

Bertrand Russell is very eloquent about this aspect of mathematical inquiry in his essay "The Study of Mathematics":

"Mathematics, rightly viewed, possesses not only truth, but supreme beauty--a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show. The true spirit of delight, the exaltation, the sense of being more than man, which is the touchstone of the highest excellence, is to be found in mathematics as surely as in poetry.....In the most beautiful work, a chain of argument is presented in which every link is important on its own account, in which there is an air of ease and lucidity throughout, and the premises achieve more than would have been thought possible, by means which appear natural and inevitable.....The discovery that all mathematics follows inevitably from a small collection of fundamental laws is one which immeasurably enhances the intellectual beauty of the whole; to those who have been oppressed by the fragmentary and incomplete nature of most existing chains of deduction this discovery comes with all the overwhelming force of a revelation; like a palace emerging from the autumn mist as the traveller ascends an Italian hill-side, the stately storeys of the mathematical edifice appear in their due order and proportion, with a new perfection in every part."

http://www.readbookonline.net/readOnLine/22896/

[Gordon]

Hm interesting. Seems as though the look of the thing [eg what the font was] may contribute to the perception of beauty as much as anything intrinsically mathematical. Russell wasn't talking about how maths "looks" to the eye, he's suggesting it has its own beauty that owes little or nothing to anything visible or audible or otherwise "sensible". On e test of this in conventional "beauty" terms might be to ask: Does an equation bring tears to the eyes?? Anyone who might dioubt that should see the Andrew Wiles Horizon programme by Simon Singh that celebrated Wiles' solution of Fermat's last theorem. Wiles describes his feelings on seeing the final element of his proof and is clearly moved.

http://www.youtube.com/watch?v=7FnXgprKgSE

In the case of Euler's formula many non mathematical people perhaps would not know what i, e or pi are from a mathematical point of view and why putting them all together in that way has a special resonance. Their significance is what enhances any other aspect of beauty to a mathematician.

One thing that does contribute to a sense of beauty in mathematics is how concisely a complex concept can be expressed in a formula or equation that belies its profundity - perhaps equivalent to Blake's grains of sand.

Whilst everyone knows E=mc^2 the depth of significance is Tardis like when one knows more about its context and origins. Maxwell's equations are concise and to anyone in electronic communications have a great density for what they tell us about radio waves, light, radiant heat etc. The great marvel is that radio waves travel apparently without any physical means - ie there is no "ether" as was once thought - just empty space itself. Pulls itself along by its own bootstraps, it has no choice. Magic!!



And then there is Mandelbrot - whatever happened to that?

[Honoured Guest]

Carol Vorderman

[french frank]

I managed a very poor pass in 'O' level Maths, and haven't progressed much beyond that. But there's something wondrous about an 'inexplicable truth': you can prove it's true but can't understand why it should be. And producing a formula which is so concise yet it contains hugely complex ideas.

Not sure I grasp what constitutes an 'ugly' formula, though.

[Gordon]

Well she is described mathematically by a set of functions called Elliptic Curves. They have properties that are not of this world!

[Gordon]

I managed a very poor pass in 'O' level Maths, and haven't progressed much beyond that.

You can still spot a problem with Roger Wright's reach statistics though!!

But there's something wondrous about an 'inexplicable truth': you can prove it's true but can't understand why it should be.

Betrand Russell got into a pickle with this in the late 1890s and rewrote maths texts because he wanted a rigorous proof that 1+1 = 2!! [I exaggerate a little, but not much].

Not sure I grasp what constitutes an 'ugly' formula, though.

You don't have to, there aren't any!!

[Frances_iom]

...But there's something wondrous about an 'inexplicable truth': you can prove it's true ...

but Goedel managed even better there being some truths that that not proveable

[Quarky]

I thought this was an interesting article on the way the brain responds to mathematical theorems:

http://www.bbc.co.uk/news/science-environment-26151062

Bertrand Russell is very eloquent about this aspect of mathematical inquiry in his essay "The Study of Mathematics":

"Mathematics, rightly viewed, possesses not only truth, but supreme beauty--a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show.

Yes interesting. I have some experience of this, having been inspired by equations and theorems to read physics at University, when I would actually have been better off taking a vocational course at a polytechnic.

But I am a bit sceptical. Does Eulers Theorem viewed numerically possess any more intrinsic beauty than a Brian Ferneyhough score? Actually I greatly prefer the graphical representation, an arc within a circle.

Admittedly the equation is of fundamental importance to physics and electronics, but it is not an ultimate truth. I have seen papers which have dug around the edges of this representation, and have produced an even more fundamental "truth".

Although maths and science aim for "truth", history has shown they are no more than approximations to truth. Newton relied on the new differential calculus to formulate his laws, Einstein relied on Riemannian Geometry, and no doubt the string theorists have some recently formulated fancy theorems to assist in their researches. So new scientific discoveries frequently rely on newly formulated mathemmatical techniques.

My view is that the proof of the pudding is in the eating, not gazing at it on the dinner table. What do we think of Ferneyhough's latest composition, when performed? How did we get on working out that tricky RF circuit, using Eulers representation, rather than rows of sines and cosines?

[David-G]

Although maths and science aim for "truth", history has shown they are no more than approximations to truth.

A mathematical theorem is a mathematical theorem, it is not an approximation of any kind. Science is the best current interpretation, often mathematically expressed, of our understanding of the physical world. Science always welcomes disproof and replacement by a new and better theory, on the basis of experimental data.

[ferneyhoughgeliebte]

What do we think of Ferneyhough's latest composition, when performed?

We think it's even better than expected, don't we?

(Hope that doesn't count as a "recently formulated fancy theorem". Is that like a mathematician's "fancy man"?)

[aeolium]

Although maths and science aim for "truth", history has shown they are no more than approximations to truth.

But while mathematical theorems which have not been proved are not necessarily "true", is it not the case that theorems which have been proved - as was recently done with Fermat's Last Theorem - are permanently, unalterably true?

[speaking as a non-mathematician ]

[Gordon]

But I am a bit sceptical. Does Eulers Theorem viewed numerically possess any more intrinsic beauty than a Brian Ferneyhough score?

No, as an object to be looked at, it doesn't. But that is what I suggested in #2, the look of the thing has its own beauty but that need not have any connection with any other property of the object eg this post!!

I agree with David, the maths as applied to science need not be rigorously precise to be useful. Whilst Dirac's electron equation was mathematically sound unfortunately the electron seemed to be made of more elusive stuff and needed Schwinger, Feynman, Dyson [BTW he was the son of composer George] et al to improve predictions of the behaviour of said electron. Whilst Schwinger's original model was complex, Feynman's was shorter and more elegant [beautiful?] and Dyson showed they were actually the same.

I doubt the electron takes a blind bit of notice of the theory of QED which is only a model that uses mathematics. That electron and that mathematics have no fundamentally necessary connection.

The fact that the difference between the squares of two consecutive integers is always an odd integer is a mathematical truth. However, remember that this truth is expressed as an equation: it shows that two apparently different things are in fact the same or that one is equivalent to the other. That is the basis of mathematical discovery - finding things that are precisely the same [yes there are mathematical approximations too] and there is a beauty in that, ie maths is a set of tautologies. How they apply to the behaviour of natural phenomena is another thing.

is it not the case that theorems which have been proved - as was recently done with Fermat's Last Theorem - are permanently, unalterably true?

Quite so! Unless Wiles did make a mistake [he did at one stage after announcing his first proof look at the video link in #2] and nobody can see it!! That proof was a series of very intricate steps taking in many other proofs supplied by others - the room for making mistakes was significant. Mathematicians are human beings and our idea of "mathematics" is what we say it is. What constitutes a mathematical "proof"? Using something you already believe to be true and then showing that it is the same as something you didn't know before or something you suspect is true but don't know yet. How rigorous is "true"? That is the conundrum that Russell addressed that I mentioned in #2 and there is the start of the road that leads to Goedel.

Is 1 + 1 = 2? It is only because we say it is axiomatic, "by definition" and only if our notation is assumed to be decimal. If that notation was say ternary ie 3 = 0, it is still true that 1 + 2 = 3 in decimal but 1 + 2 = 10* in ternary; it looks different but it is still "true", the notation does not change the "truth".

[french frank]

But I am a bit sceptical. Does Eulers Theorem viewed numerically possess any more intrinsic beauty than a Brian Ferneyhough score?

I don't think there was any comparative claim that a mathematical theorem is, more or less 'beautiful' than a painting or a piece of music (although apparently within that particular field, formulae are more or less beautiful, which was a concept I didn't understand - I suspect it to be subjective ). It was that boring, dry old Maths has a 'beauty' of its own at certain levels, in that neurological reactions can be very similar to the reaction in the brain to more obvious entities that are thought of as 'beautiful'.

It also seems a nice illustration of: 'Beauty is truth, truth beauty'!

[Honoured Guest]

But while mathematical theorems which have not been proved are not necessarily "true", is it not the case that theorems which have been proved - as was recently done with Fermat's Last Theorem - are permanently, unalterably true?

[speaking as a non-mathematician ]

Axioms are the fundamental "building blocks" of Mathematics. The proof of theorems is based on these axioms. So, the theorems are "permanently, unalterably true" in a mathematical world with those axioms.

These axioms are understood to be "true" in our real universe, which is why Mathematics is useful in Science.

When a scientific observation or experiment is currently inexplicable, Science strives to explain it, and I suppose that some future event might be shown to be impossible with these axioms of Mathematics. So, then mathematicians would need to come up with a revised set of mathematical axioms for the universe. When they succeeded, every theorem would need a new proof, on the basis of the new axioms. Some of our current proofs might turn out to be Special proofs (true in certain situations) and not General proofs (universally true).

[Gordon]

I don't think there was any comparative claim that a mathematical theorem is, more or less 'beautiful' than a painting or a piece of music (although apparently within that particular field, formulae are more or less beautiful, which was a concept I didn't understand - I suspect it to be subjective ). It was that boring, dry old Maths has a 'beauty' of its own at certain levels, in that neurological reactions can be very similar to the reaction in the brain to more obvious entities that are thought of as 'beautiful'.

It also seems a nice illustration of: 'Beauty is truth, truth beauty'!

One could postulate that there is, neurologically, some part of the brain that houses the experience of "beauty". A pleasurable or satisfying [are these part of beauty?] subjective sensation that we could associate with the word "beauty" results from the objective stimulus of the appropriate neurons?

There could be many paths to that centre channeled via the senses of sight, hearing, touch etc but perhaps mediated by other brain regions that house experiences and/or training. The phenomenon of synaesthesia is interesting because it suggest links between disparate senses eg sounds induce colours and vv [viz Sir A Bliss] which then suggests that some kind of "beauty centre" is in contact with other pleasure/beauty inducing processes. Some kind of resonance occurs when one sensation/idea causes a jump to another. One of those processes need not involve the "external" senses at all but result from internal "thinking" about concepts that idealise from those external sensations and actually build on them - resonate - to make "beautiful" ideas that have no real world connections. Perhaps mathematics is part of that? Maybe the search for beauty is what drives mathematicians just like everyone else.

An artist sees a curve on paper and thinks it beautiful to look at; a mathematican sees the same curve but sees the equation that might generate it and thinks that is beautiful. The curve means different things to each of them. They may be neurologicaly the same sort of experience and the two may describe their experiences with similar words and even the emotions and feelings will be similar but for different reasons and I suspect that we will never know what really goes on inside someone else's head.

Oh well, back to the washing up; beautiful coloured bubbles you get with soap - or are they just thin walled spheriods that preferentially diffract light of certain wavelegths?