>It's been rumoured that Brian D Harper said:
>>
>> Rick: feel free to send this to the fractal group if you
>> think its appropriate. BTW, how would I go about
>> subscribing to this group? I have an interest in chaos
>> and complexity and might want to participate.
>
>The fractal group is actualy "fractal-art" and they don't
>really like people to talk technical.
>
Hmmm, well, this doesn't sound like its for me. :)
>> >The real theological question is:
>> >"Is mathematics an accident, or was math cleverly designed
>> >by an omnipotent God?"
>>
>> This reminds me of some questions I raised on the Evolution
>> list awhile back:
>
>Well, I have an opion about everything, so ...
>
>> 1) Was math invented or discovered?
>
>Contrary to popular opinion, its discovered, not invented.
>Mathematicians, nor anyone else, have the power to change
>2+2=4. (although they can discover other algebras)
>Niether, for that matter, does it seem possible for even
>God to change that fact. Numbers and algebras existed
>long before life evolved on planet earth.
>
I tend to agree on this, despite the inherent danger
of appearing Platonic. :)
John Barrow (refs at end) constructs an interesting
conversation between two mathematicians taking
opposing sides on this issue. A good source for
understanding the various arguments and counter
arguments for both sides.
[...]
>
>> 2) Were natural laws invented or discovered?
>
>Contrary to popular wisdom, they are invented, not discovered.
>I base this on the fact that no natural law ever exactly
>reflects nature: Newtons' laws don't really describe gravity,
>nor do Einsteins, except as an approximation. And when
>quantum gravity is finnaly worked out, it will prove to be
>an approximation as well. Newton, Einstein, etc. are heros
>because they invented a law that more accurately describes
>nature than any of thier contemporaris did. (Yes, there
>are alternaitves to Einsteins formulation, and not all
>of them have been ruled out yet.) That's the very nature
>of science.
>
I tend to waffle somewhat (alternate tuesdays) on the
first question but not on the second. It is true that
mathematical expressions of laws are often approximations
but the approximations tend to get better and better
and, IMHO, are approaching some real law of nature that
exists independently of our representations of it.
In other words, there is a real law that is discovered
despite the imperfection of our representation of the
law.
Let me try explaining it this way. Some time ago I read
something by a FemiNazi :) stating that Newton's <Principia>
was a manual on rape and that science would be completely
different if it had been created by women instead of
european white males. Well, one wonders then if women
scientists would have invented gravitational forces that
varied in proportion to surface area and inversely as
the distance cubed or something. Or perhaps if a woman
dropped heavy balls from a tower they would reach the
ground at different times depending on their color. etc.
>> I'm kicking myself for not throwing in a related, and
>> probably more significant question:
>>
>> 3) Why are natural laws mathematical?
>
>Because they cannot be anything else. Mathematics is a kind of a
>language, a shorthand for the english language that allows you to say
>more with fewer marks on the page. Thus, if you can say it
>in english, you can always recast it as a formula. Conversly,
>if it cannot be said, then it is not truly understood, and therefore,
>not expressible mathematically.
>
Well, I think you should have inserted "Contrary to popular
wisdom ..." for this one as well :), since the peculiarity
that nature is described by mathematics has been a
puzzlement for many great physicists and philosophers.
One of the classic papers on this subject is "The Unreasonable
Effectiveness of Mathematics in the Natural Sciences" by
Eugene Wigner. The paper begins humorously as follows:
==========begin quote===============
There is a story about two friends, who were classmates
in high school, talking about their jobs. One of them
became a statistician and was working on population
trends. He showed a reprint to his former classmate.
The reprint started, as usual, with the Gaussian
distribution and the statistician explained to his
former classmate the menaing for the symbols for the
actual population, for the average population, and
so on. His classmate was a bit incredulous and was not
quite sure whether the statistician was pulling his
leg. "How can you know that?" was his query. "And what
is this symbol here?" "Oh," said the statistician,
"this is Pi." "What is that?" "The ratio of the
circumference of the circle to its diameter." "Well,
now you are pushing your joke too far," said the
classmate, "surely the population has nothing to do
with the circumference of the circle."
Naturally, we are inclined to smile about the simplicity
of the classmate's approach. Nevertheless, when I heard
this story, I had to admit to an eerie feeling because,
surely, the reaction of the classmate betrayed only
plain common sense. ...
-- Eugene Wigner
====end==========
But the closing paragraph is more appropriate to our
present discussion:
====begin quote==============
Let me end on a more cheerful note. The miracle of the
appropriateness of mathematics for the formulation of
the laws of physics is a wonderful gift which we
neither understand nor deserve. We should be grateful
for it and hope it will remain valid in future research
and that it will extend, for better or for worse, to
our pleasure even though perhaps also to our bafflement,
to wide branches of learning .... [ellipses in original]
-- Eugene Wigner
======end=====================
This is also discussed in chapter 2 "The Relation of
Mathematics to Physics" in Richard Feynman's excellent
book <The Character of Physical Law>.
"The strange thing about physics is that for the
fundamental laws we still need mathematics" --Feynman
It would be difficult to do justice to Feynman's ideas
in a brief discussion, but I'll do my best. Feynman
discusses at some length your point above that mathematics
is just a language. The general context is that of
explaining to a lay person illiterate in mathematics
why it is that he cannot just translate the ideas
into English, why it is that a person must go through
the torment of learning mathematics before they can
truly appreciate the physical world.
===begin quote===
You might say, 'All right, then if there is no explanation
of the law, at least tell me what the law *is*, why not
tell me in words instead of in symbols? Mathematics is
just a language, and I want to be able to translate the
language'. In fact I can, with patience, and I think I
partly did. [...]
But I do not think it is possible, because mathematics is
*not* just another language. Mathematics is a language
plus reasoning; it is like a language plus logic. Mathematics
is a tool for reasoning. [...] The apparent enormous
complexities of nature, with all its funny laws and rules,
each of which has been carefully explained to you, are
really very closely interwoven. However, if you do not
appreciate the mathematics, you cannot see, among the
great variety of facts, that logic permits you to go from
one to the other. -- Feynman
=========end======
Let's consider an example (borrowed from Feynman). Kepler
analyzed literally a mountain of data to find his three
laws of planetary motion. These physical observations
can, of course, be expressed in terms of mathematical
formulas. Similarly, Galileo *discovered* :) the laws of
motion for a projectile by analyzing the results of many
experiments. Again, Galileo's results can be expressed
in terms of mathematical formulas.
Now, along comes Newton, who finds the universal law
of gravity which once again can be expressed in terms
of a mathematical formula. That one could express all
these different physical phenomena in terms of mathematical
formulas is not at all surprising. What is surprising is
that they are all connected to each other purely by
logic. From Newton's inverse square law one can deduce,
purely by logic, that planets follow elliptical orbits
and sweep out equal areas in equal times (Kepler's
first two laws).
I personally find this amazing to contemplate. Also
amazing to me is some folks merely reacting to
something like this with a shrug of the shoulders.
"So?". Surely its possible to imagine a universe in
which there were a great many laws and facts for
many situations but with those laws and facts being
disjoint and unconnected. Just having laws in themselves
is somewhat amazing, but having them all connected
by logic is startling, for me anyway. Perhaps I'm
simple minded ;-).
Oh, I forgot to mention another peculiarity discussed
by Feynman. One could take the equal area law of gravity
say and from it deduce the conservation of angular
momentum. It seems perfectly reasonable to assume that
this conservation law would apply then only for the
law of gravity, but it turns out to much more general
than the context in which it was derived. As Feynman
notes:
"Now we have a problem. We can deduce often from one
part of physics, like the Law of Gravitation, a
principle which turns out to be much more valid than
the derivation. This does not happen in mathematics;
theorems do not come out in places where they are not
supposed to be." -- Feynman
A similar idea was expressed by Hertz (in discussing
Maxwell's equations:
"One cannot escape the feeling that these equations have an
existence and intelligence of their own; that they are wiser
than we are, wiser even than their discoverers; that we get
more out of them than was originally put into them."
references:
John D. Barrow, (1988). <The World Within the World>,
Clarendon Press. especially chpt 5 "Why are the laws
of nature mathematical".
Richard P. Feynman, (1965). <The Character of Physical
Law>, The MIT Press.
Eugene P. Wigner, (1960) "The Unreasonable Effectiveness
of Mathematics in the Natural Sciences," <Communications
on Pure and Applied Mathematics> 13:1. [also in the
collection <The World Treasury of Physics, Astronomy,
and Mathematics>, Timothy Ferris Ed., Backbay Books, 1991]
Brian Harper
Associate Professor
Applied Mechanics
The Ohio State University
"... we have learned from much experience that all
philosophical intuitions about what nature is going
to do fail." -- Richard Feynman