fool, the point is that abstract mathematics, invented by humans, has been found to describe the physical world with little "forcing," like a puzzle which just fits together (without the need for reshaping the basic pieces themselves) and results in a coherent picture.
Although I think frozen-tears view is correct for certain aspects or time periods in mathematics, I don't believe it is applicable to pure math today. For the last 200 years or so, mathematicians have been attempting to solidify the foundations of math(as a branch of logic), mainly by restricting the number of axioms(or starting assumptions). Sure, there was a bit of trial and error, a little testing of the waters to find the most simple axioms that keep the desired structure, but that is mostly done with- the axioms are set(ok so if you bring up the axiom of choice, we can talk).
At this point, we have a self-consistent logical structure, where theorems are not tweaked to fit reality, but are either true or false(or both-stupid Godel). This means if a mathematical theorem doesn't jive with reality, we don't just throw it out or modify it as frozen-tears suggested.
So the question is, restated, why does a logical system based on such a simple starting point have describe reality so well? Why do the theorems about groups apply to molecules? Why do physicists always find such convenient mathematical structures like Calub-Yau shapes to hang their theories on? (I've got the feeling most of the posters aren't mathematicians, come on Ben where are you?)
Why do the theorems about groups apply to molecules?
I asked this question of the most eminent group theorist now alive, Michael Aschbacher. His answer was remarkably simple and exactly on target. "Because things like molecules have automorphism/symmetry groups in quite natural way."
At the foundation, some aspects of mathematics existed before anyone could define them in terms of axioms. The concept of numbers might have been an invention, but cardinality certainly existed. Suppose you have a molecule with six binding sites, and six ligands. Whether or not anyone knew it, you now have a one-to-one mapping between ligands and sites. From something as basic as this, we can extrapolate the tendencies of molecular self-assembly in crystals, which even now is a hot topic in physical chem. (Hehe, I guess I can't talk about the axiom of choice, as I'm still puzzling over the proof to the Banach-Tarski paradox. Then again, I'm not a mathematician. Talk to me about chemistry.)
And then, of course, there is the group of people who believe that our simple logical system does not work everywhere in reality, but only works from our point of reference. I don't know enough about the basis of this belief to say anything else about it. I also don't see how Max Tegmark's theory of an infinite multiverse is holding up, if he's illustrating its existence simply by saying that it's "very probable." Soooo if anyone wants to direct some light into my hole here, I'd be grateful!
I hope the lack of posts is not a general consensus to let the thread die.
Anyways, here are my latest musings on the subject.
Starting with the central role of the ideas of conservation and symmetry in science, I hope to show the influence of scientific concepts on human thought.
The idea of conservation is such a bastion of scientific reasoning, I think it has become a philosophical idea, something so beautiful it must be true. Strikingly elegant, the image of a universe with a set amount of energy, energy which merely changes form not quantity has become almost axiomatic. I’m not trying to argue that conservation is not a physical law, I’m merely pointing out the influence such a powerful concept has on one’s ideology.
On the other hand, the idea of symmetry seems purely philosophical- why should symmetry be favored over asymmetry except for matters of taste. But symmetry is definitely a prized ideal for scientific theory: just look at the uproar about CP violation. Accepted a priory, symmetry in nature is a principle which has been used to great effect by many scientists(see Maxwell’s equations), but why?
The wonderful connection comes by Emily Noether’s proof that conservation and symmetry are inescapably intertwined- in fact, they are practically the same thing. This still invokes a sense of wonder in me, that something ingrained in the fabric of the cosmos, and something entirely aesthetic should be related so strongly. My conclusion is that the influence of the natural world on our sense of beauty is so strong that those theories which seem aesthetically pleasing are likely to be true.
So how does this relate to mathematics? Perhaps the influence of what we see around us and how the world fits together so colors our thinking that when a mathematician says, “I am thinking a beautiful thought”, he is really saying “I am thinking a thought applicable to the real world”. In this way by pursuing a completely abstract thought for its own sake, for its complexity or simplicity or symmetry or whatever attribute makes it interesting, mathematicians are fulfilling their conditioned sense of beauty, and consequently creating structures completely in tune with the real world.
Wow, looking back this post seems mostly obvious and totally convoluted. Oh well, if anybody has an idea, lets hear it. I notice Ben(post 3) and nephilim(post 10) hint at strong opinions on the topic, you two care to expostulate?
In the end, it's because human beings are awesome.
If there's anything that humans are absolutely fantastic at, it's pattern recognition. From what I can see, mathematics, as well as science, are all about pattern recognition. It's then natural that humans can observe and identify similar patterns across fields. Just as math that was developed 50 years ago as part of some pure field is now finding its way into physics, there are physicists who have recently been classifying the riemann zeta function as a partition function (an important concept in thermodynamics).
But isn't thermodynamics fundamentally asymmetric? I guess don't really see the idea of asymmetry as being different in type from the idea of conservation.
And I too am interested in nephilim's post #10...
This is a great discussion. I am a current Caltech parent with nothing useful to contribute but as I was reading the posts (not really understanding half of what is being said) I could not help but think that anybody wanting to know why is Caltech different from other schools (and amazing to some of us on the outside) should be pointed to this thread.
I think the answer is this: certain notions -- linearity, symmetry, exponentiation (of operators), eigenvalues, are just quite recurrent. They show up again and again because much that happens in the world produces manifestations of them. One example: we discover eigenvectors by thinking of which vectors are nearly unchanged by a simple skewing-stretching-rotating of the plane, and then we notice the same notion applies to much more complicated linear operators (in quantum mechanics, or whatever).
The fact that this happens is an interesting collusion of our adeptness at modeling (something to which Mike alluded) and the recurrence of similar kinds of phenomena in nature. For example, we like to model things linearly, and nature gives us lots of things that we can model reasonably well in a linear way.
Of course, the underlying consistency of nature is still a bit of a mystery. There are lots of functions out there --- why does nature seem to like (approximate) linearity and exponentiality so much? But that mystery is much more fundamental, closely related to why nature is "lawlike" at all --- i.e., why any principles hold consistently over time and space.
The two-pronged quality of this account also lets you tweak it to your taste, placing most of the emphasis on the part of the explanation you find most plausible. If you think nature really isn't all that close to the nice math we keep using, you can attribute most of this "miraculousness" to our insistence on modeling the world in ways we understand well. After all, it's not like physics jumps on you like a giddy toddler and says "I have eigenvalues inside!". Our decisions about how to model phenomena bear very heavily on what we think the world is like under the hood.