Mathematics and Nature

Hmmm, can't wait Actually, these days it's almost the other way round: why is physics so effective at mathematics? E.g. how the hell does Topological Quantum Field Theory (TQFT) manage to russel up the Donaldson Invariants in a tiny fraction of the time it takes the mathematicians?BTW, I wouldn't consider Incompleteness an area of mathematics in SG's sense; rather a property of mathematics.And large cardinals? Don't hold your breath. You have to remember that fundemental physics pushes back to categories and even topoi. Once you realise that, you understand that nothing is sacred An old colleague was devastated when I showed him Connes' work on p-adics. He really thought that he had an area of maths in which no physicist would ever show an interest.Admittedly, fundemental physics is in reality an area of mathematics, but try convincing the pure mathematicians...This message has been edited by cavediver, 12-12-2005 07:13 PM

Exactly. John's enthusiasm for this stuff has inspired me a great deal. He has a very similar outlook and has made the transition to deeply knowledgable mathematician with annoying ease, something that he will strenuously deny He more than anyone else makes me wish I was still in the field. I do a great lecture on this (well, I think it is!)Anyway, this together with the TQFT stuff I mentioned all fits into my own metaphytsics - reality is simply mathematics... we are an Irrep (Irreducable representation) of SU(Everything) if you get my vague meaning...I suppose what hinted this to me was the point where I discovered that an electron is almost a pure rep of the Lorentz Group - simply specified by the Casimir invariants of the LG (mass and spin). Add on a U(1) element and you've got an electron. Then remember that electrons have no size, no strict locality, and at this point you realise that there are no "things", no "objects" anymore with which to play physics.Counter-ideas that suggest the mathematics is simply a derived concept inspired by the physical universe will always be left with the big question: what is the physical stuff, the stuff of existence?
In my mind, I have greatly simplified this. But of course, I could be completely wrong

Yeah, you would from Pickering. If he wants to talk authoratatively on this, he should have spent a bit longer on his original studies (when he was still in physics).

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And, given their extensive training and sophisticated mathematical techniques, the preponderance of mathematics in particle physicists' account of reality is no more hard to explain than the fondness of ethnic groups for their native language

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Ha ha, I would expect so much better from him. It is not the "preponderance of mathematics in particle physicists' account of reality " that is surprising. It is the utter simplicity of the fundemental physics when couched in mathematical terms: both particle physics and GR. No-one would have been too surprised if these areas turned out to be tractable using existing mathematics, given the history of mathematics and physics, but the simplicity is astounding and unprecedented in scientific enquiry.

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is no more hard to explain than the fondness of ethnic groups for their native language

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Thinking that mathematics is the language of choice for physics is so far off the mark, I don't know where to begin. Mathematics is not one of numerous possible systems or languages. It is a natural extension of formal logic. Would alternate languages of physics be based upon illogic?

Fortunately, concepts consistently regarded as absurd certainly seem to form core parts of reality.But in your own case, may I ask why?

I remember your background and philosophical take on this. Just to be clear - this is your philosophical take on mathematics, one take of several (and the interested reader can look here for more details Wikipedia Link)And I appreciate your views. As you're aware, my views have been shaped by exposure to fundemental physics. I would not describe them as Platonist, as I view our reality as the Platonic ideal. There is no separate reality. True, but this basis of mathematics causes most of the conceptual problems with the world: if we are talking matter, 1 + 1 = 0, not 2, and hence the exclusion principle for example (fermionic statistics coming from Grassmannian algebra).The fact that we can get 1 + 1 = 2 from apples is exceptionally complex in itself. I think the problem is that everyday experience is the biggest distraction from "reality".And you are still left with "what is stuff?". You are stuck with new-age concepts of energy and vibrations and "things" and a naive separation of foreground object and background space. This is fine if you are happy with that, but that appears far more absurd to me than my ideas. It is interesting that everyone I have worked with or met that has grappled with existence at the (current) core level has some Platonistc type leaning. I don't mean this as evidence that my ideas are correct, just that it is interesting that there is something at this level that is suggestive, compared with other levels of scientific enquiry. There is also the observation that we have no real physical description of the last 100yrs of theoretical physics work - it is all mathematical, other than analogies constructed for laymen and teaching. What are quarks, electrons, gluons? What is gravity? As does physics. At the moment, we only have laws of GR and QFT. Everything is part of one or other of these. These are very much bottom-end.This message has been edited by cavediver, 12-13-2005 09:45 AM

I'm sorry, you're talking about the long history of mathematical discoveries? As a mathematician, I am surprised at you! What have numbers to do with anything? I'm not envisaging free floating arabic numerals Ahhh, ok, let you off. But what do you mean by separation? A wave-fn exists over all space - it's just a functional, a point in a Hilbert Space. But position for example is simply a case of an n-tple of coords.In string theory, these n-tples are just values in n fields defined on a 2d space. The n-d space (4d space-time in our case) is purely derived (or emerges) from these 2d-fields.This is on place where we are getting away from the everyday foreground/background divide. I think it is this divide that primarily makes my idea look absurd (possibly hence your comment on separation). I know I struggle here and it's something of which I am fully aware! It is just such an ingrained concept. I disagree. They do not describe "things" at all. They are mathematics. There are mathematical entities: fields, groups, etc, but these are understood. The "things" come into play when we try to match these theories to different length-scales, where we still talk about "things". For instance, GR is not about motion of planets. It has no knowledge of matter. But fortunately we can use it for such work because of the simplicity of reality. It does not tell us how planets "work" becasue it is not a theory of planets. But it certainly tells us how space-time works to give rise to what we call gravity. This is what sets GR apart from any other theory. It actually tells us what is going on. That is why I describe GR as bottom-up. Interesting comment about simpler players. GR plays with one exceptionally simple player: the universe. It may seem odd given the length scale involved (big) but GR is unbelievably fundemental. It is so incredibly simple. Forget that there's lots of interacting matter inside the universe. GR doesn't care about that. This is why we can have a theory of quantum cosmology where we model the universe with one degree of freedom... ONE!!!! We have two simple fundemental theories, GR and QFT, that exist at opposite ends of the length scale. Both I would describe as bootom-up (though not as bottom as I would like ) The top-down stuff is all the horrible mess in the middle. You don't get fundemental just by going small...Rambling now so will stop

Of course. God only gave us the empty set to play with. Why do you think Russell was an atheist? He was just so pissed off with God for making him do all the work As I was explaining to MrJack, natural numbers are only really natural at this length scale. Natural number counting is not as fundemental in physics as you would think. Counting requires distinguishable objects, and such things do not always exist below the nuclear scale.

re-reading it, maybe not too much? My point is concerning the perceived separation of objects and the arena in which these objects sit (space). What I am saying is that this separation is probably false. Furthermore, I think it is this separation that makes my suggestion regarding the role of mathematics in reality particularly hard to swallow.

Where would you see this? Just out of interest, why? Always is a very strong word. Have you any examples of when this has not been the case?

I think it is a very sound exposition of the role of mathematics and science through the ages, but it omits the shift in focus in fundemental physics that began around the end of 19c. I agree with this, but when I first read it I took away the impression that you meant that the independence is a relatively recent phenomenon, where-as I would say we easily have a 6-700 year solid history of pure mathematics. I think this is a slightly naive/misleading view with regard to fundemental physics. Mathematical models stopped being simple models of observed behaviour, and became descriptors of the previously unknown underlying principles. The mathematics itself started to perform the role of the physicist.This would appear ludicrous if it wasn't for the fact that the multitude of predictions that dropped out of these descriptions of reality went on to be confirmed to astounding accuracy. So much so that particle physics was accused of new-age mysticism - apparently we were only observing the predictions to be true (finding the right particle in the right place) because we were willing it to be there! We could predict anything and we would still find it! Nonsense of course The above is no better demonstrated than the SU(3) non-Abelian gauge theory of QCD, and its prediction of quarks and gluons. Of course, GR itself is just as good an example. In both of these cases we see a vastly greater output (in terms of predictions) than obvious inputs (initial observations). This appears to be based upon consistency. If a mathematical theory of reality is to be consistent, it is immensely constrained.The mathematics that is GR was built upon a requirement of consistency. It has one free parameter. Let me repeat: it has one free parameter... and is the most accurately tested theory ever considered in human history (if you want to argue for QED, I won't complain) The mathematics seems to have done this for us. There was no trial and error in GR. It arrived wholesale once the consistency was realised (and this wouldn't have been such a chore if Einstein had had Hilbert's insight). Similarly with QCD (albeit with more parameters).These theories demonstrate that mathematics is playing a new role in the physics/mathematics relationship.

Interesting. I wasn't aware of this, but it is easy to see how this can happen. Most excellently stated. And I certainly agree that there is an "appearance" to the outsider...

This is a statement of one's philosophy of course; a philosophy with which many mathematicians would not agree. At your length scale, you are approximately correct. But you are made up of fermions and bosons. These are identical... completely identical. This is why we have the Pauli exclusion principle, which is a good thing because we couln't exist without it! I'm not sure what you mean here. I have made hundreds of mobius strips.A mathematical space can be realised by far more than just a surface. Rotations of objects are better realisations, and rotations of fermions and bosons are even better. And these spaces are far far more bizarre than the good old mobius band.Mathematics is certainly idealised at the length scales with which you are familiar... I have no problem with that. But I am not referring to those length scales.

You are confused in your concept of mathematical space. A mobius band is a statement about the topology of a space. My cellotaped piece of paper has the correct topology and is hence a mobius band. The thickness is immaterial. The point is that a mobius band need have no thickness to be a mobius band, in the same way that a 2-sphere has no interior. The surface of the earth is a 2-sphere, despite the earth having an interior. How would I tell? They are identical. This is a pivotal point of quantum/particle physics. It is not an idealisation. The quantum theory of electrons falls apart if there is the slightest hint of even a theoretical possibility of being able to distinguish two electrons.

Following on from my discussion with RAZD...You have two electrons. Shake them up so they are equally likely to be spin up or spin down.What is the prob they are both spin up?
What is the prob they are both spin down?
What is the prob one is spin up and the other spin down?

Again, statement of philosophy, not statement of fact. "Dancing"? "Changing partners"? Sorry, I don't speak Layman-ese as I told Randman recently. What are you talking about? Of what relevance is this to my point? No confusion. What I mean is that it is not a point regarding difficulty of measurement. Like the uncertainty principle, this is a fundemental facet of reality. Not at all. I am saying that the theory has been verified by observation. More verified, I should add, than any theory other than GR. Nothing of course. We would have some work to do... all of the mathematics of QED would be wrong. The mystery would be how an amazingly complex set of calculations based on totally incorrect mathematics arrives at precisely (10+ decimal places of accuracy) the observed values. This is one piece of observational evidence that electrons are identical.Another is based upon my quiz. You didn't answer the questions. When this quiz is carried out in experiment, what probabilities do we observe? Or what probabilities should we observe if electrons are potentially distinguishable? Are these probabilities then observed in experiment? Answer: no.You seem to be under the impression that my outlook is

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the mathematics is so wonderful it couldn't possibly be wrong, and who cares about any evidence

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If so, you are greatly mistaken.This message has been edited by cavediver, 12-15-2005 04:47 AM