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Author Topic:   Mathematics and Nature
nwr
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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)

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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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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.

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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)

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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.

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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.

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


Message 39 of 90 (269465)
12-14-2005 9:01 PM
Reply to: Message 31 by RAZD
12-14-2005 8:21 PM


Re: Why mathematics is so useful in the sciences
Because math is a purely - and idealized - intellectual construction that does not exist outside the mind.
Take the counting of objects as an example: no two objects are really completely, purely 100% identical, ...
That's a strange thing to say. After all, our concept of "object" is an idealized intellectual construction that does not exist outside the mind.

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