Register | Sign In


Understanding through Discussion


EvC Forum active members: 65 (9162 total)
3 online now:
Newest Member: popoi
Post Volume: Total: 915,806 Year: 3,063/9,624 Month: 908/1,588 Week: 91/223 Day: 2/17 Hour: 0/0


Thread  Details

Email This Thread
Newer Topic | Older Topic
  
Author Topic:   Mathematics and Nature
JustinC
Member (Idle past 4843 days)
Posts: 624
From: Pittsburgh, PA, USA
Joined: 07-21-2003


Message 16 of 90 (268746)
12-13-2005 10:33 AM
Reply to: Message 9 by Dr Jack
12-13-2005 8:28 AM


quote:
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).
Don't you need to interpret what the numbers mean in order to apply them to the world?

This message is a reply to:
 Message 9 by Dr Jack, posted 12-13-2005 8:28 AM Dr Jack has replied

Replies to this message:
 Message 17 by Dr Jack, posted 12-13-2005 10:54 AM JustinC has not replied

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


Message 17 of 90 (268756)
12-13-2005 10:54 AM
Reply to: Message 16 by JustinC
12-13-2005 10:33 AM


Yes, and no. The predictions of it's behaviour: yes, you do; the extrapolation from that to what is actually happening in the real world to make the results come out as they do: no, you don't.
These are different issues, no?

This message is a reply to:
 Message 16 by JustinC, posted 12-13-2005 10:33 AM JustinC has not replied

  
cavediver
Member (Idle past 3643 days)
Posts: 4129
From: UK
Joined: 06-16-2005


Message 18 of 90 (268771)
12-13-2005 11:42 AM
Reply to: Message 14 by Dr Jack
12-13-2005 10:11 AM


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.
I'm sorry, you're talking about the long history of mathematical discoveries?
By what means are your numbers structured
As a mathematician, I am surprised at you! What have numbers to do with anything? I'm not envisaging free floating arabic numerals
by which I mean how is one waveform separated from another, for example
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 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 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.
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).
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

This message is a reply to:
 Message 14 by Dr Jack, posted 12-13-2005 10:11 AM Dr Jack has replied

Replies to this message:
 Message 19 by Dr Jack, posted 12-13-2005 11:48 AM cavediver has replied
 Message 20 by nwr, posted 12-13-2005 12:14 PM cavediver has replied

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


Message 19 of 90 (268775)
12-13-2005 11:48 AM
Reply to: Message 18 by cavediver
12-13-2005 11:42 AM


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.
Oh, ok. Yeah, that all makes sense.

This message is a reply to:
 Message 18 by cavediver, posted 12-13-2005 11:42 AM cavediver has replied

Replies to this message:
 Message 22 by cavediver, posted 12-13-2005 1:16 PM Dr Jack has not replied

  
nwr
Member
Posts: 6408
From: Geneva, Illinois
Joined: 08-08-2005
Member Rating: 5.1


Message 20 of 90 (268786)
12-13-2005 12:14 PM
Reply to: Message 18 by cavediver
12-13-2005 11:42 AM


quote:
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.
I'm sorry, you're talking about the long history of mathematical discoveries?
LOL
I'll agree with Mr Jack on this.
Kronecker famously said "God gave us the natural numbers. All else is the work of man." Personally, I think Kronecker gave God too much credit.

This message is a reply to:
 Message 18 by cavediver, posted 12-13-2005 11:42 AM cavediver has replied

Replies to this message:
 Message 21 by cavediver, posted 12-13-2005 12:52 PM nwr has not replied

  
cavediver
Member (Idle past 3643 days)
Posts: 4129
From: UK
Joined: 06-16-2005


Message 21 of 90 (268801)
12-13-2005 12:52 PM
Reply to: Message 20 by nwr
12-13-2005 12:14 PM


"God gave us the natural numbers. All else is the work of man." Personally, I think Kronecker gave God too much credit.
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.

This message is a reply to:
 Message 20 by nwr, posted 12-13-2005 12:14 PM nwr has not replied

  
cavediver
Member (Idle past 3643 days)
Posts: 4129
From: UK
Joined: 06-16-2005


Message 22 of 90 (268810)
12-13-2005 1:16 PM
Reply to: Message 19 by Dr Jack
12-13-2005 11:48 AM


Oh, ok. Yeah, that all makes sense.
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.

This message is a reply to:
 Message 19 by Dr Jack, posted 12-13-2005 11:48 AM Dr Jack has not replied

  
nwr
Member
Posts: 6408
From: Geneva, Illinois
Joined: 08-08-2005
Member Rating: 5.1


Message 23 of 90 (268822)
12-13-2005 1:54 PM


Why mathematics is so useful in the sciences
What follows is my personal view. I welcome comments.

Science


Science is, to a large extent, the use of systematic methodology to study the world. It isn't always obvious how to systematize a study, so sometimes there is a period of trial and error while scientists experiment with various ways of systematizing.

Mathematics


Mathematicians often see their discipline as the study of pattern, or of regularity or symmetry. But we could equally consider it the study of systems and systematicity.
There we see the connection. Science uses systematic methods, and mathematics studies the principles of systematicity. That makes mathematics a study of some of the underlying principles of science, albeit abstracted and idealized from what happens in reality.

Examples


Example 1: Counting is probably one of the earliest systematic methodologies used by man. The study of natural numbers is mainly a study of the principles and consequences of counting. The natural numbers are a kind of fictitious objects to which we can apply our idealized system of counting.
Example 2: Measuring of distance, length, etc depends on the systematics use of a portable measuring rod. Euclidean geometry is little more than the theoretical analysis of the consequences of measuring. In a ruler and compasses construction, the line between the tips of the compasses are, in effect, the measuring rod.

Usefulness


If the systematic method happens to work perfectly, then the mathematical properties of that system can be directly applied to what is studied, and we can expect perfect fit.
Even if the systematic method does not work perfectly, the mathematics is useful. For the mathematics tells us how the system would behave, purely on account of its systematicity, if reality did not intrude. That make it easier for us to see interesting features of reality in the failure of the mathematics to exactly match the world.

Comments


For a long time, mathematics advanced along with science, by studying the systems used within science. However, as mathematics became more independent it spent increasing energy in studying systematic methods in their own right, without depending on the origin of those systems in physics. As a consequence, mathematicians have been inventive in discovering many additional systems worthy of study.
More recently science, and most particularly physics, has been looking at the systems studied by mathematics, to see if some of those systems can be adopted for use in systematic empirical methodology.

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

Replies to this message:
 Message 24 by Ben!, posted 12-13-2005 9:07 PM nwr has replied
 Message 27 by RAZD, posted 12-14-2005 7:02 PM nwr has replied
 Message 30 by cavediver, posted 12-14-2005 8:21 PM nwr has not replied

  
Ben!
Member (Idle past 1398 days)
Posts: 1161
From: Hayward, CA
Joined: 10-14-2004


Message 24 of 90 (269007)
12-13-2005 9:07 PM
Reply to: Message 23 by nwr
12-13-2005 1:54 PM


Re: Why mathematics is so useful in the sciences
I think this is a clear, concise, correct summary.
Math is a tool that fit the bill.
I hope we can find appropriate math tools to describe cognition and neuroscience. Looks like some signal analysis might be doing the trick. But looks like some of those tools (such as wavelet analsyis, independent component analysis, and "Parallel Factor Analysis" [which I'll start reading about tomorrow] ) are being driven by the science.
So it's not even just a one way street; further evidence of the relationship you described in your post.
At least, that's how I see it.
Ben

This message is a reply to:
 Message 23 by nwr, posted 12-13-2005 1:54 PM nwr has replied

Replies to this message:
 Message 25 by nwr, posted 12-13-2005 9:27 PM Ben! has replied

  
nwr
Member
Posts: 6408
From: Geneva, Illinois
Joined: 08-08-2005
Member Rating: 5.1


Message 25 of 90 (269019)
12-13-2005 9:27 PM
Reply to: Message 24 by Ben!
12-13-2005 9:07 PM


Re: Why mathematics is so useful in the sciences
I think this is a clear, concise, correct summary.
Thanks.
I hope we can find appropriate math tools to describe cognition and neuroscience.
When I finish writing some stuff up, there will be a reference to Gillman and Jerison, "Rings of Continuous Functions" (1960, van Nostrand). But, in all honesty, the connection with cognition is not going to be obvious, so I suggest that you don't spend a lot of time looking it up.

This message is a reply to:
 Message 24 by Ben!, posted 12-13-2005 9:07 PM Ben! has replied

Replies to this message:
 Message 26 by Ben!, posted 12-13-2005 9:34 PM nwr has not replied

  
Ben!
Member (Idle past 1398 days)
Posts: 1161
From: Hayward, CA
Joined: 10-14-2004


Message 26 of 90 (269021)
12-13-2005 9:34 PM
Reply to: Message 25 by nwr
12-13-2005 9:27 PM


Re: Why mathematics is so useful in the sciences
Well I'm IN the library right now, so...
(looking it up)
found it.
(Running upstairs to get it... )

This message is a reply to:
 Message 25 by nwr, posted 12-13-2005 9:27 PM nwr has not replied

  
RAZD
Member (Idle past 1404 days)
Posts: 20714
From: the other end of the sidewalk
Joined: 03-14-2004


Message 27 of 90 (269373)
12-14-2005 7:02 PM
Reply to: Message 23 by nwr
12-13-2005 1:54 PM


Re: Why mathematics is so useful in the sciences
most excellent breakdown. some quibbles:
so sometimes there is a period of trial and error while scientists experiment with various ways of systematizing.
and {making\using\accumulating} predictions and subsequent evidence to validate or invalidate certain systematizations, to separate the wheat from the chaff. Math may be involved in the predictions and in the analysis of the data, but the data is not based on math.
In a ruler and compasses construction, the line between the tips of the compasses are, in effect, the measuring rod.
A unitless measuring rod, generalized to all measurment conditions.
Even if the systematic method does not work perfectly, the mathematics is useful. For the mathematics tells us how the system would behave, purely on account of its systematicity, if reality did not intrude. That make it easier for us to see interesting features of reality in the failure of the mathematics to exactly match the world.
Fully concur. Add that both the math needs to be revisited (and have nuances proposed to 'correct' it), AND the investigation to possibly validate the math should be done.
More recently science, and most particularly physics, has been looking at the systems studied by mathematics, to see if some of those systems can be adopted for use in systematic empirical methodology.
The only problem I see here is a tendency to depend on the math and not involve the real world. Math should always be regarded as a purely theoretical model with no foundation in reality, so mathematically precise predictions do absolutely need to be validated on real observations.
Don't mistake the clothes for the emperor.
This message has been edited by RAZD, 12*14*2005 07:03 PM

Join the effort to unravel {AIDS\HIV} with Team EvC! (click)

we are limited in our ability to understand
by our ability to understand
RebelAAmerican.Zen[Deist
... to learn ... to think ... to live ... to laugh ...
to share.

This message is a reply to:
 Message 23 by nwr, posted 12-13-2005 1:54 PM nwr has replied

Replies to this message:
 Message 28 by cavediver, posted 12-14-2005 7:28 PM RAZD has replied
 Message 29 by nwr, posted 12-14-2005 8:13 PM RAZD has not replied

  
cavediver
Member (Idle past 3643 days)
Posts: 4129
From: UK
Joined: 06-16-2005


Message 28 of 90 (269380)
12-14-2005 7:28 PM
Reply to: Message 27 by RAZD
12-14-2005 7:02 PM


Re: Why mathematics is so useful in the sciences
The only problem I see here is a tendency to depend on the math and not involve the real world.
Where would you see this?
Math should always be regarded as a purely theoretical model with no foundation in reality
Just out of interest, why? Always is a very strong word.
so mathematically precise predictions do absolutely need to be validated on real observations.
Have you any examples of when this has not been the case?

This message is a reply to:
 Message 27 by RAZD, posted 12-14-2005 7:02 PM RAZD has replied

Replies to this message:
 Message 31 by RAZD, posted 12-14-2005 8:21 PM cavediver has replied

  
nwr
Member
Posts: 6408
From: Geneva, Illinois
Joined: 08-08-2005
Member Rating: 5.1


Message 29 of 90 (269415)
12-14-2005 8:13 PM
Reply to: Message 27 by RAZD
12-14-2005 7:02 PM


Re: Why mathematics is so useful in the sciences
and {making\using\accumulating} predictions and subsequent evidence to validate or invalidate certain systematizations, to separate the wheat from the chaff. Math may be involved in the predictions and in the analysis of the data, but the data is not based on math.
I wouldn't call that a quibble. You were just filling in the details that I glossed over.
A unitless measuring rod, generalized to all measurment conditions.
Mathematicians like to idealize and generalize.
The only problem I see here is a tendency to depend on the math and not involve the real world.
This is always a potential problem. I see it a lot in discussions of cognitive science/artificial intelligence. However, reality has a habit of eventually intruding on such theorizing.
There is perhaps an appearance of this sort of problem in cosmology, but I suspect that most physicists and cosmologists well understand the need to involve reality.

This message is a reply to:
 Message 27 by RAZD, posted 12-14-2005 7:02 PM RAZD has not replied

Replies to this message:
 Message 33 by cavediver, posted 12-14-2005 8:28 PM nwr has not replied

  
cavediver
Member (Idle past 3643 days)
Posts: 4129
From: UK
Joined: 06-16-2005


Message 30 of 90 (269424)
12-14-2005 8:21 PM
Reply to: Message 23 by nwr
12-13-2005 1:54 PM


Re: Why mathematics is so useful in the sciences
What follows is my personal view. I welcome comments.
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.
For a long time, mathematics advanced along with science, by studying the systems used within science. However, as mathematics became more independent it spent increasing energy in studying systematic methods in their own right, without depending on the origin of those systems in physics
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.
More recently science, and most particularly physics, has been looking at the systems studied by mathematics, to see if some of those systems can be adopted for use in systematic empirical methodology.
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)
systematic empirical methodology.
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.

This message is a reply to:
 Message 23 by nwr, posted 12-13-2005 1:54 PM nwr has not replied

  
Newer Topic | Older Topic
Jump to:


Copyright 2001-2023 by EvC Forum, All Rights Reserved

™ Version 4.2
Innovative software from Qwixotic © 2024