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Author Topic:   Mathematics and Nature
Son Goku
Inactive Member


Message 1 of 90 (268167)
12-12-2005 12:51 PM


Is any area of mathematics sacred?
That is to say are there any areas of mathematics which are pure abstractions.
(Or to put it more bluntly are the Pure Mathematicians really doing applied mathematics and they just don't know it yet?)
As a possible avenue of this discussion, what are we doing when we say we are doing mathematics.
addendum:
I am asking is there any area of current pure mathematical research which will escape being subsumed into physical theory.
Rather than asking is Maths a Science, I am asking is there any area of Pure Mathematics which is of no used to Science, especially physics.
Over the last century we have seen areas we thought useless to physics, such a noncommutative geometry actually come into play in the physical sciences.
Does Pure Mathematics mean no more than Mathematics with no current use.

Replies to this message:
 Message 2 by nwr, posted 12-12-2005 3:14 PM Son Goku has not replied
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nwr
Member
Posts: 6408
From: Geneva, Illinois
Joined: 08-08-2005
Member Rating: 5.1


Message 2 of 90 (268267)
12-12-2005 3:14 PM
Reply to: Message 1 by Son Goku
12-12-2005 12:51 PM


Wigner - unreasonable effectiveness
Wigner wrote a rather famous paper on the question of why mathematics is so useful to the sciences. The link I provided is to the wikipedia commentary, and that in turn has a link to the full article.
I might later post on why the observed effectiveness is reasonable, not unreasonable. But it may be a day or two before I can find time.

What shall it profit a nation if it gain the whole world, yet lose its own soul.
(paraphrasing Mark 8:36)

This message is a reply to:
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nwr
Member
Posts: 6408
From: Geneva, Illinois
Joined: 08-08-2005
Member Rating: 5.1


Message 3 of 90 (268271)
12-12-2005 3:18 PM
Reply to: Message 1 by Son Goku
12-12-2005 12:51 PM


Unlikely to be applicable
I am asking is there any area of current pure mathematical research which will escape being subsumed into physical theory.
The Goedel incompleteness theorem;
The set theory of large cardinals.
I'm sure there are others, but these two are the ones that immediately came to mind. Note that I am only marginally familiar with those areas.

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cavediver
Member (Idle past 3644 days)
Posts: 4129
From: UK
Joined: 06-16-2005


Message 4 of 90 (268384)
12-12-2005 7:11 PM
Reply to: Message 2 by nwr
12-12-2005 3:14 PM


Re: Wigner - unreasonable effectiveness
I might later post on why the observed effectiveness is reasonable
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

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RAZD
Member (Idle past 1405 days)
Posts: 20714
From: the other end of the sidewalk
Joined: 03-14-2004


Message 5 of 90 (268542)
12-12-2005 11:20 PM
Reply to: Message 1 by Son Goku
12-12-2005 12:51 PM


I am asking is there any area of current pure mathematical research which will escape being subsumed into physical theory.
Once something is known, then applications follow.

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Son Goku
Inactive Member


Message 6 of 90 (268688)
12-13-2005 6:28 AM
Reply to: Message 4 by cavediver
12-12-2005 7:11 PM


Re: Wigner - unreasonable effectiveness
This thread was actually inspired by work from John Baez and a few others.
They are currently working on Quantum Mathematics, which is basically the study of mathematics which is important to Quantum mechanics (Not yet field theory though).
Weird results have come up such as using measurements of quantum systems to study the Riemann zeta function and possibly solve the Riemann zeta hypothesis experimentally.
A very exciting field and I can't wait until they get to QFT.
My surprise isn't so much that Number theory has important applications to Quantum Mechanics, but that very obscure "curiosities" in number theory which only the number theorists knew about (Not even other Mathematicians) turn up all the time in Quantum Systems apparently and we might be seeing Quantum Number Theory as a research field in the next few years.
Bizarre!!

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cavediver
Member (Idle past 3644 days)
Posts: 4129
From: UK
Joined: 06-16-2005


Message 7 of 90 (268692)
12-13-2005 7:10 AM
Reply to: Message 6 by Son Goku
12-13-2005 6:28 AM


Re: Wigner - unreasonable effectiveness
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.
Weird results have come up such as using measurements of quantum systems to study the Riemann zeta function and possibly solve the Riemann zeta hypothesis experimentally.
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

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JustinC
Member (Idle past 4844 days)
Posts: 624
From: Pittsburgh, PA, USA
Joined: 07-21-2003


Message 8 of 90 (268700)
12-13-2005 8:09 AM


I found this quote by Andrew Pickering:
It is unproblematic that scientists produce accounts of the world that they find comprehensible: given their cultural resources, only a singular incompetence could have prevented members of the physics community from producing an understandable version of reality at any point in their history. 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
Thoughts? Are you guys (Son and Cavediver) getting an understanding of some deeper reality, or are you just creating a "myth for our times?"

Replies to this message:
 Message 9 by Dr Jack, posted 12-13-2005 8:28 AM JustinC has replied
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Dr Jack
Member
Posts: 3514
From: Immigrant in the land of Deutsch
Joined: 07-14-2003
Member Rating: 8.7


Message 9 of 90 (268706)
12-13-2005 8:28 AM
Reply to: Message 8 by JustinC
12-13-2005 8:09 AM


Try this easy experiment. Build a transistor. Model it according to QM. Compare and contrast the results.
Science is not a myth for our time. Interpretations of what the numbers mean are (I, incidently, think the idea of mathematics as the underlying reality is absurd).

This message is a reply to:
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Replies to this message:
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 Message 16 by JustinC, posted 12-13-2005 10:33 AM Dr Jack has replied

  
cavediver
Member (Idle past 3644 days)
Posts: 4129
From: UK
Joined: 06-16-2005


Message 10 of 90 (268713)
12-13-2005 8:45 AM
Reply to: Message 8 by JustinC
12-13-2005 8:09 AM


I found this quote by Andrew Pickering:
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).
quote:
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
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.
quote:
is no more hard to explain than the fondness of ethnic groups for their native language
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?

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cavediver
Member (Idle past 3644 days)
Posts: 4129
From: UK
Joined: 06-16-2005


Message 11 of 90 (268716)
12-13-2005 8:49 AM
Reply to: Message 9 by Dr Jack
12-13-2005 8:28 AM


I, incidently, think the idea of mathematics as the underlying reality is absurd
Fortunately, concepts consistently regarded as absurd certainly seem to form core parts of reality.
But in your own case, may I ask why?

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 Message 9 by Dr Jack, posted 12-13-2005 8:28 AM Dr Jack has replied

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 Message 12 by Dr Jack, posted 12-13-2005 9:04 AM cavediver has replied

  
Dr Jack
Member
Posts: 3514
From: Immigrant in the land of Deutsch
Joined: 07-14-2003
Member Rating: 8.7


Message 12 of 90 (268719)
12-13-2005 9:04 AM
Reply to: Message 11 by cavediver
12-13-2005 8:49 AM


Probably because I'm a Mathematician (by education anyway). Maths is an entirely human construct; a useful invention in the same way that the spanner, the computer and logic are. Maths exists within our heads, and has reality only were we find correlations; taking an apple and putting it next to another apple doesn't make two apples because of mathematics. I've also studied a reasonable amount of Philosophy, and I find striking parallels between the Platonic notion of ideals and the notion of mathematics-as-reality - and I have very little credance for Plato's works.
I find nothing surprising about the success of mathematics as a description of physical reality. Maths can essentially embrace anything that operates according to rules; which physics does (although I don't mean to imply that the rules exist either; I think the behaviour of physical systems comes bottom-up not top-down). While mathematics roots lie in macrosopic physics (take an apple and another apple get two apples; draw a circle of diameter 1, measure its circumfrence, etc.) and these macroscopic physics themselves emerge from the behaviour of the microscopic physics in a reversal of the way mathematical constructs have emerged in generalisation and derivation from basic maths. It does not surprise me that the two intermesh.
But I freely accept your point about absurdity; what I find absurd is in no way a reasonable guide to what actually is.

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Replies to this message:
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cavediver
Member (Idle past 3644 days)
Posts: 4129
From: UK
Joined: 06-16-2005


Message 13 of 90 (268723)
12-13-2005 9:43 AM
Reply to: Message 12 by Dr Jack
12-13-2005 9:04 AM


I remember your background and philosophical take on this.
Maths is an entirely human construct; a useful invention in the same way that the spanner, the computer and logic are. Maths exists within our heads
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.
While mathematics roots lie in macrosopic physics (take an apple and another apple get two apples; draw a circle of diameter 1, measure its circumfrence, etc.)
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?
although I don't mean to imply that the rules exist either; I think the behaviour of physical systems comes bottom-up not top-down
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

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Dr Jack
Member
Posts: 3514
From: Immigrant in the land of Deutsch
Joined: 07-14-2003
Member Rating: 8.7


Message 14 of 90 (268736)
12-13-2005 10:11 AM
Reply to: Message 13 by cavediver
12-13-2005 9:43 AM


Just to be clear - this is your philosophical take on mathematics, one take of several
This is true. However, if I make the lesser claim "Mathematics is a human construct" then that is factually verifiable by studying the history of mathematics whether that construct reflects a deeper reality or not.
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.
I agree; I don't believe any modern interpretations of physical reality are true. I'm also far from convinced that this is an answerable questions. But I don't see how taking mathematics as your physical reality helps you avoid this; how do you go from your numbers to the physical reality we experience in day-to-day life? By what means are your numbers structured (by which I mean how is one waveform separated from another, for example)? And, of course, since I consider mathematics to be an entirely human construct I cannot also consider it to be a (the?) fundamental part of reality; one of these positions must be wrong and given the one I have more knowledge of and am in a better position to judge is my position on mathematics I'm inclined to take that as the one I'm correct on.
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.
I don't think I made myself clear on this. I would consider GR and QFT top-down, in that they describe the behaviour of objects from a lofty looking down kind of perspective. They describe how things behave rather than how they work. I think that the behaviour of reality works the other way round and GR and GFT emerge from the interaction of simpler players each of whom has no knowledge of the target rules (in the same way that the gas laws emerge from the properties of the particles of that gas).
And, again, I accept this is complete speculation on my part. I wouldn't wish anyone to get the idea that I'm expounding on the actual reality of the universe from a position of privileged knowledge.

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JustinC
Member (Idle past 4844 days)
Posts: 624
From: Pittsburgh, PA, USA
Joined: 07-21-2003


Message 15 of 90 (268745)
12-13-2005 10:25 AM
Reply to: Message 10 by cavediver
12-13-2005 8:45 AM


quote:
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.
Is the fact that it is modeled simplistically simply a result that science tries to model things as simply as possible?
But if your argument is that simplicity of fundamental physics is unreasonable, doesn't it suffer from the ambiguities of "simple"? I certaintly don't find theoretical physics simple, but then again I have never studied it beyond college level physics II. I just don't think an argument the hinges on the notion of simple is very effective.
On the other hand, are you referring to the fact that it is unreasonable that the universe is so "simple" that humans can come to have a good understanding of it in a lifetime?
And earlier, when I said "myth of our time" that was just a nod to a book I read by Gribbon. I know science makes good predictions about the world, but does that mean it is a more "real" depiction of reality than any other one? Can there be empirically equivalent theories?
That may be for the "Theory Evolution" thread though.

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