One bonobo's view of the world...and stuff.

Wednesday, December 15, 2010

Was God a Mathematician?

Last night I watched a BBC4 tellything called 'Beautiful Equations' in which the presenter, an artist with no scientific background, struggled to get to grips with the idea that some scientists have talked about the aesthetic quality of equations. I don't recommend the programme to anyone with a basic understanding of science - it was one of those that was more travelogue than science1 - but the basic idea's interesting enough.


Einstein famously said:

"The only physical theories that we are willing to accept are the beautiful ones."


and Paul Dirac:
“God used beautiful mathematics in creating the
world."
There are implications here that the universe is 'constructed' with an underlying, elegant pattern. Neither was necessarily saying that the universe was created by God. Einstein was certainly an atheist who was at pains to make it clear that he only ever used 'God' as a metaphor. Dirac perhaps took the idea more literally - although in point of fact he probably didn't give it much thought. Certainly neither saw any connection between the ordering of the universe and the conduct of our daily lives: they weren't theologians or philosophers.

(Interestingly, there is a school of thought within Islam that wheras only God can fully understand the universe, we have a duty to practice science to gain insight into the Oneness of God. Sometimes this is described as scraping back the surface of the universe to reveal glimpses of the underlying 'greeness' - green being associated with God, life, etc.)

Even so, the idea that science and mathematics reveal the inherent beauty of the universe is arsey-versy, isn't it? It's an anthropocentric notion. The universe is complicated. We are evolved to grub for roots, spear antelope and/or gather shellfish. We're on a par with other beasties in our ability to Comprehend Nature. Granted, we're extraordinarily adaptive by virtue of our faculties for problem solving. Nevertheless, when we bump our heads against the difficulty of understanding the inner workings of the universe, there's no reason to suppose our capacities are any more limitless than, say, a bonobo. We're undoubtedly better at it...but even our best minds can find it awfully hard.

The reason we're better is that, especially over the last 400-ish years, we've come up with some little tricks to simplify the picture. It goes without saying that equations are useful if they allow us to predict the way the universe behaves. But that wouldn't necessarily make them beautiful. A beautiful equation is something like:


E = mc2
...which has the additional advantage of simplicity. It's not only simple in that it only has three terms, but the way it falls out of Special Relativity is elegant. Plus it tells us a lot about the the universe and has various practical applications.


Or take the Dirac equation:







Now, OK, I'm not going to bullshit that I understand the first thing about this, but my understanding is that its a simple, clean expression which, by manipulating its variables, predicts the existence of various particles (e.g. anti-matter) which are experimentally verifiable.

So what these 'beautiful' equations have in common is that they're neat little bundles with the power to tell us a lot about the universe. Einstein's probably wouldn't have caught on if it went 'E=mc2 except in February minus the number you first thought of...' and on for twenty pages. For an example of an inelegant equation, see the Computus (origin of the word 'computer') by which the date of Easter is calculated. Its main predictive power is to explain why nobody ever knows what date it's going to fall on in any given year. The Dirac equation is slightly different. Wheras most people can grap the bones of Relativity after a bit of thinking about trains, watches and flashlights, even particle physicists struggle with Dirac. My understanding is that in deriving it he 'boiled down' some concrete stuff into abstract variables. E, m and c we can get to grips with, but nobody can quite explain the real-world concept represented by Ψ.

Note, incidentally, that not all mathematical descriptions of the world are considered elegant. I well recall my A-Level in Pure Mathematics with Mechanics2. The Pure, I liked. The Mechanics...sheesh!....all those long, long expressions representing the forces acting on a ladder leaning against a wall on a rough surface. The underlying maths was simple (and repetitive) - basically variants on Newton - but it was pure handle turning, without elegant shortcuts. Subsequently, throughout my so-called career, I've worked with people doing various forms of mathematical modelling. While in no way denegrating them, the type of maths they're dealing with is getting computers to spit out anwers using more data and doing morecalculations than humans can get their heads around. It's a matter of brute force rather than elegance.

But wait a minute. In contrasts to the messy maths, there are the beautiful equations which show themselves to be powerful tools for understanding and manipulating the real world. Doesn't the very fact of their existence demonstrate a beautiful order?

It's Douglas Adams' Intelligent Puddle once again, surely?:

"Imagine a puddle waking up one morning and thinking, 'This is an interesting world I find myself in, an interesting hole I find myself in, fits me rather neatly, doesn't it? In fact it fits me staggeringly well, must have been made to have me in it!"3

What we've done is to go out looking for ways to simplfy the world, either for good, practical purposes or simple curiosity. Some parts of it we find can be described in nice, neat equations. But these will only be tractable if they'll fit within a human head or can be worked out on not too many sheets of paper or, more recently, in MATLAB.

We haven't discovered an underlying pattern, pleasing in its beauty. It's more that we've found we cen get our heads around parts of it and have been pleased with our ability.

(Btw, when I say 'we', I mean 'they'...those cleverer people than I who've made scientific discoveries.)

Finally: As a further test of Snow's 'two cultures'...can any scientists amongst us tell me why I have a picture of a vase in this post? See last two lines here. No cheating, now!




1 Actually, it was a good illustration of CP Snow's 'Two Cultures'. At the outset, the presenter seemed to have little notion of the idea of manipulating variables in equations, finding limiting values, etc.
2 My school's assumption was that if you did science and weren't clever enough to be a doctor, then you'd be an engineer, so you needed Mechanics. Alternately, if you did Arts and weren't posh enough to be a solicitor, you'd be an accountant and would need Statistics. It was only at university that I encountered - and was good at - statistics, which comprised a large part of my Experimental Design and Analysis.
3 Completing the quote, to illustrate the potentially malign consequences of anthropocentrism (or puddlecentrism):
"...This is such a powerful idea that as the sun rises in the sky and the air heats up and as, gradually, the puddle gets smaller and smaller, it's still frantically hanging on to the notion that everything's going to be alright, because this world was meant to have him in it, was built to have him in it; so the moment he disappears catches him rather by surprise. I think this may be something we need to be on the watch out for."

19 comments:

Dan | thesamovar said...

As you know, I'm a poetry ignoramus so I won't even attempt to respond to that last bit... ;-)

Do you know about Wigner's essay on "The Unreasonable Effectiveness of Mathematics in the Natural Sciences"? If not, take a look at wiki on it and read the essay:

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

Myself I'm of pretty much the same view as you on the anthropic principle way of looking at it. If mathematics weren't effective, we wouldn't be interested in it. More specifically, there's an infinite amount of possible mathematics that could exist, and we are only interested in a tiny fraction of it. That fraction is chosen because of its effectiveness.

The question is really: why is there some structure rather than none? But I'm not sure this metaphysical sort of question has any reasonable answer, other than the anthropic principle one. If there weren't, we wouldn't be talking about it.

Incidentally, dunno if I've talked to you about this before, but you might be surprised that as a mathematician having moved into the field of neuroscience, my view is that mathematics is not a very appropriate instrument for understanding the brain. More precisely, we use very basic mathematics like differential equations, but there's an awful lot of effort put in by mathematical neuroscientists into trying to understand the brain using dynamical systems and high powered mathematics that I feel is very wrong headed. The one somewhat mathematical thing that might be applied to the brain is the idea of optimality: for certain tasks we'd expect to be close to optimal because that's what evolution would predict. Not sure if optimality is exactly mathematical though, certainly not in the pure mathematical sense although maybe in the applied sense.

Edward the Bonobo said...

Ta for the link. I'll take a look.

'An infinite amount of possible mathematics...' - Now that makes a lot of sense of the relationship between mathematics and reality. It seems to me that the notion that Mathematics is the fundamental building blocks of nature is an over-fetishisation. I don't think it's going too far to say that it's a work of the human imagination - albeit a staggeringly clever and impressive one - only a tiny part of which, such as the square root of minus one, turns out to have any value in describing orpredicting reality. (and i was thought to be useless for a couple of hundred years, until folk cottoned on to vectors).

Another possible example is the enormous interest in Primes (and patterns therein). Book recommendation: 'Uncle Petros and Goldbach's Conjecture' by Apostolos Doxiadis http://www.amazon.com/Petros-Goldbachs-Conjecture-Apostolos-Doxiadis/dp/1582340676 (have I recommended this before?). A novella which includes the implication that primes are only interesting to us because sometimes we find we have remainders when trying to divide things up into equal piles. Granted in the search for proofs for (eg) Golbach's Conjecture or Fermat's Last Theorem, people have come up with tools - including computational mathematics - which have turned out to have practical application.

On 'Why is there some structure rather than none?' Are you talking about something more metaphysical than Turing, Chaos Theory and why the universe is clump-y? I'll have to think about whether the metaphysical issue is the same as the concrete one.

Mathematics and Neuroscience: Perhaps there are two issues here. The first one is the classic differention in cognitive science between experience and explanation. Explaining how consciousness works doesn't tell us how it feels, nor even what it is. The second one...if an explanation is to add value to our understanding of consciousness, we have to be able to (ahem) get our head around it. High-falutin', roccoco theories may describe without explaining. There's also the approach of (is this the kind of thing you do?) sticking lots of simple-ish maths in a great big computational engine and seeing what pops out. This can have predictive power. But in the middle, as far as the observer is concerned there's, effectively a black box (if a person could follow all the number crunching, there'd be no need for the computer, right?). It's not clear that this is anymore comprehensible than the pinkish, squidgy thing it models.

On 'Optimality': is it perhaps backwards thinking to say that organisms are optimised to their environments? All we can say is that those that exist are those fit enough to survive. That's all we measure optimality against. (Another recommendation: 'Candide' by Voltaire - Dr Pangloss's 'Best of all possible worlds.' It's an easy read, even in French.)

Jaysus! I'm way out of my bullshit zone here talking about a) mathematics (you know how little I know about that!) and b) computational neuroscience. But then...you've failed the Two Cultures test, so I'll now allow you to click the link, scroll down and read the last two lines. The poem's quite famous. ;-)

(I'm neither fish nor fowl. Not a proper scientist; not artsy-fartsy enough.)

the quiet one said...

Another reason mathematics can be seen as beautiful is how, with few initial axioms, one can derive huge theories and unveil hidden structures and symmetries in them.

This could be seen as a measurement of how 'natural' axioms are: I can invent any theory by just inventing random axioms, but there is no reason that it would give anything interesting, either because no non-contradictory mathematical object could possibly satisfy the axioms, so you'd end up basically studying the empty set, or because the axioms would be too loose to be able to say anything meaningful about what you define.

But usually, yes, any application to something somewhere is appreciated.

Another beauty is how sometimes two relatively unrelated situations appear to be connected when you scratch the surface. I'm thinking of Dynkin diagrams here if you really need to know.

All this to say I'm not a huge fan of equations. I like Einstein's one because something so fundamental in our physical world turns out to be ruled by a (particularly simple) polynomial equation.

A non-equation result which is quite cool is Wedderburn's theorem (finite fields are commutative). I'm still wondering what strange property of nature inheritently forbides non-commutative finite fields.

@Dan: Oh, I have read (too long ago) Wigner's essay, I liked it! Maybe you had linked to it somewhere else.

David McLaughlin said...

I see a beautiful symmetry in Maxwell's field equations (I prefer the differential form.)

The first one just says that the "divergence" of an electric field is proportional to the charge density. That's fairly intuitive. If you have a single point charge, you can imagine the field rays all diverging radially from it. If you generalise that to a distributed charge, you can still imagine that the rays diverge most sharply from the most concentrated charges.

The second one is to magnetism as the first is to electricity. It says that the divergence of a magnetic field is proportional to the concentration of magnetic monopoles - but they don't exist, so the concentration must be zero.

The third one is Faraday's law of induction. Faraday observed that if you move a magnet through a coil of wire, it will induce a current in the coil. In other words, a moving magnet creates an electric field that curls around it. The equation says that the "curl" of the electric field is proportional to the rate that the magnetic field changes over time.

The fourth one is Ampere's law. A current in a wire produces a magnetic field that curls around the wire. The "curl" of the magnetic field strength is proportional to the current. Like the first equation, this can be generalised from a current in a wire to a more distributed current density. Maxwell added another crucial component to the fourth equation to deal with the situation where the electric field changes with time.

If you look at the equations in pairs, the first two show a beautiful symmetry, its perfection only marred by the absence of magnetic monopoles. Likewise the last two (this time, the "missing" term in the third is the current density of passing magnetic monopoles).

With a bit of manipulation, and with no electric charges to complicate matters, you can turn these into a wave equation (actually two identical wave equations, one in the electric field and one in the magnetic field). And of course the wave speed is the speed of light.

Is that not close to magic?

Haven't a clue about the Keats unless it's something about the shape of the urn (though I can't see any simple mathematical form). [Clicks link, penny drops - it's symmetry again!]

What's a Grecian urn?

David McLaughlin said...
This comment has been removed by the author.
Edward the Bonobo said...

In answer to your question:

'A couple of hundred Euros less per month, if they're a civil servant.'

Dan | thesamovar said...

On primes, I've always found the idea strange that if you wanted to communicate with aliens, and show that you're an intelligent form of life, you should send out sequences of primes as if it were some sort of universal thing that any intelligent life would be interested in. Primes are great, but I can easily imagine a different mathematics which isn't interested in them.

For the question about why there is some structure rather than none, I think I'm talking about something more metaphysical. I'm saying: why is that it's EVER possible to predict anything? Why is that there appear to be stable 'rules' in the universe at all?

For the computational neuroscience stuff, it's true that there is a danger of coming up with black box models, and there is some sense in which even if you could do it this would be a waste of time. I'll temper that with a few observations. Firstly, if we could have a virtual brain, that would make exploring it much easier than exploring a real brain (we'd have access to all the variables without having to messily poke electrodes into brains), which would have enormous experimental value. Secondly, there would be practical value to something that could solve tasks intelligently like people can, or handle perceptual tasks like understanding speech or reading text. Thirdly, knowing what the minimal set of rules that could generate human-like behaviour may or may not be interesting to know and might lead to further insight. So that's what I'll say in favour of the black box stuff. But I agree with you that the black box isn't understanding on its own, and it's important to work with more conceptual models as well. For my part, I like to work at both levels. At the moment I mostly work with readily understandable conceptual models (albeit ones with vast numbers of computational elements in them, my latest paper featured simulations with around 1M simulated neurons). However, I'm not against playing around with stuff where you wouldn't necessarily be immediately be able to understand how it's doing what it's doing.

Incidentally, the next task I'm going to tackle is the binding problem! Never let it be said that I set my sights too low. ;-)

Dan | thesamovar said...

Organisms are not always entirely optimal to their environments, but they're very often close to optimal in quite specific senses. For example, the sand scorpion has eight legs organised roughly in a circle around its body that can detect the motion of the sand it is standing on via 'mechanoreceptors'. A prey moving on the sand near the scorpion causes waves in the sand which can be detected by these receptors. By combining this information, it can work out where the prey is and attack in that direction. This is very simply represented as a mathematical problem, which once you introduce noise (in the environment and in the sensors) is probabilistic. You can solve the mathematical / probability problem optimally, and it turns out that this almost precisely predicts what actual scorpions will do.

Or another example closer to the work I'm doing is the human ear. Sounds we hear come into our ears and cause vibrations of our ear drum and various connected bones. These in turn cause motion in a liquid. Attached to our 'basilar membrane' and suspended in this liquid are very fine hairs which move when the liquid moves (i.e. when the drum vibrates). The motion of these hairs causes mechanically sensitive ion channels to open, causing an electrical effect in the hair cells, which travels down the hair cell, causes voltage sensitive channels to open, release neurotransmitters which then stimulate auditory nerve fibres (the first neurons that receive the auditory signal). It turns out that given there is brownian motion in the liquid, this system is physically optimal. If it were more sensitive it would just respond to the inherent brownian motion induced noise. The system can detect motion of the hairs of distances comparable to the size of an atom.

OK, so that's two examples where evolved systems behave close to optimally (which I described in detail mostly because they're incredibly cool and I thought you'd like to hear about them). In general we would expect this sort of behaviour because if a small mutation can improve the performance of an organism, or reduce its energy costs (which will usually lead to improved performance because it can spend more time doing more useful things than gathering energy), then if that mutation happens the organism will have a relative advantage over others in the same species, and it will tend to become more frequent until they all have it. Interestingly, you can mathematically formalise evolution and it turns out to be the optimal search strategy for optimal genes! I have to say though that I tried to read the book that explains that and I found it too hard going! ;-)

the quiet one - yes the surprising connections between seemingly unrelated mathematical fields is extraordinary, and maybe even a little difficult to explain on the anthropic principle point of view.

I like the Wigner essay and often post it, so it's quite possible you came across it that way. Do I know you from H2G2? The name doesn't immediately ring a bell...

Edward the Bonobo said...

@DMcL:

So what exactly do we mean by the 'curl' of a field. Or, indeed, a 'field'. Surely they're imaginative abstractions devised to explain real-world phenomena?

Similarly when we say (e.g.) that an electron is a particle that also behaves like a wave. Surely it's neither? It simply helps us (slightly) to get our heads around them if we imagine them that way.

Edward the Bonobo said...

btw...

>>Haven't a clue about the Keats unless it's something about the shape of the urn (though I can't see any simple mathematical form). [Clicks link, penny drops - it's symmetry again!]

Tut! No, no, no! CLICK THE LINK!!! http://www.eecs.harvard.edu/~keith/poems/urn.html Scroll down. Last two lines. Very famous.

I'm reminded of the story of how they used to send Education Officers around to lecture to the conscripts in WWII. One time, a sergeant major got his men together and told them:

"Gather round, you 'orrible shower! We've got a very h'educated man 'ere to talk to us about Keats. Now I bet none of you h'ignoramuses even knows what a Keat is..."

Edward the Bonobo said...

@Dan re. optimality:

You know the old joke about the hunter and guide out in the savannah? Suddenly they see a cheetah. The guide takes his backpack off, pulls out a pair of running shoes and starts putting them on.

"Surely you don't think you can outrun a cheetah?" says the hunter.

"I don't have to," says the guide, "I only have to outrun you."

Edward the Bonobo said...

It would be interesting to speculate what a universe without stable rules would be like. There wouldn't be Things in it, because they would (to quote Chinua Achebe - 2 cultures again? ;-) ) Fall Apart.

This is why I'm thinking that the Metaphysical question is really the same as the Physical one. (But I haven't quite thought it through yet).

While I'm teasing you with the 2 cultures thing, I've suddenly remembered an apt line for you from the poem that Achebe took his title from. (The Second Coming by Yeats):

"Things fall apart; the centre cannot hold/ Mere anarchy is loosed upon the world."

Anonymous said...

I've actually read the Chinua Achebe book! ;-) (Although not the poem, of course...) You see, I'm not a general cultural ignoramus, but rather a specific poetry ignoramus.

Potentially agree on the 'world without rules' stuff, but still not on the optimality stuff. Yes, optimality is defined with respect to the environment and the environment to a significant extent is defined by the other organisms living in it, but the point you maybe miss is the extent to which the requirement to gather and expend energy is a constraint. I don't think it's a coincidence that we so often see near optimality in animal behaviour. At the very least, it makes a very good working hypothesis! :)

Dan | thesamovar said...

Oops, that last one was me.

the quiet one said...

Dan: You may know me indeed, I'm toybox on h2g2.

The thing about primes, besides their universality, might be that they are simple enough to be discovered quickly even without really wanting to. So if an alien civilization did develop some 'mathematics', it seems likely that they might have popped up somehow, even if their study wasn't pursued any more.

I mean, they do arise in different situations (weird things keep happening at prime numbers) so even when you are not interested they do tend to pop up.

Now I am curious as to what you would use instead of primes, even disregarding the fact whether it's easy to send the notion as a signal in space.

Dan | thesamovar said...

Ah, toybox, got it! :)

It's difficult to say for sure one way or the other if any civilisation that did something like mathematics would end up being interested in primes, but I feel like I can imagine that they might not. Our interest in primes comes from the ancient Greek interest in ratios, I guess. That could have been otherwise. They found irrational numbers quite upsetting, but imagine if mathematics had developed from a culture that took the idea of a real number as somehow more primary than integers and rational numbers. Perhaps primes would never have been looked at.

The most obvious place where they come up I guess is finite groups and fields, but maybe these aren't essential to mathematics? Maybe we see these structures in so many places because we know about and like primes and integers. There might be all sorts of structures that crop up all over the place that we don't know about, but... we don't know about them. A culture that approached mathematics from a very different starting point might find our way of looking at things equally unimaginable.

the quiet one said...

I'm not sure... I'd say that primes is something that an alien mathematician would eventually run into even if not looking for them (and groups too, but that may be just me). Even if not interested, they would, I think, recognise that primes have a property which makes them 'remarkable' to some extent. Maybe for them, Catalan numbers are the most fascinating, and primes something anecdotal. If aliens sent us Catalan numbers as a pattern, would we recognise it as an intelligent signal? Well, probably.

Anyway, yes, obviously I implicitly believe that integers would be something a priori studied by mathematicians universewide. Because counting stuff is a very basic activity. Then again, if we had let the Romans instead of the Greek....

Ah, basically it's hard to imagine the unimaginable ;-)

David McLaughlin said...

Ed,

You're right, a field is just an abstraction, but it's one that can model the behaviour of real stuff in a way that is remarkably consistent with observations.

As I'm sure you know, the curl of a field has a precise meaning, but the name "curl" is pretty descriptive.

I didn't know anyone else had read Uncle Petros!

As for optimality, you're right to allude to "survival of the fittest". Just as the auction price of an item is a little bit higher than the second-highest bidder is willing to pay, so the "fittest" organism need only be little better than the competition. (BTW, I first heard the "savannah" story in the context of a briefing to submariners on the "buddy system" and polar bears.)

That said, over a few million years, iterative refinement can lead to some very well designed features. Dan's description of the human ear glossed over the ossicles. These were originally part of the reptilian jaw, but have evolved into a very efficient impedance converter that allows acoustic energy to pass from air into the water in the cochlea.

I assume you've come across the anthropic principle in cosmology. But are you familiar with Boltzmann's anthropic brain?

And yes, I had clicked the link and read the Keats while I was writing my comment, that's what the bit in [] was about.

Tan Morgan said...

Hey there!

Re. your "gather shellfish" link, this is just in case you want to know it's broken and the new url for Elaine Morgan's website is: http://www.elainemorgan.org/

:)