[...]
>> >> 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.
>
>Hmm, well, being a much dimmer thinker than Wigner or Feynman,
>I can only suggest that they failed to propose "what else it
>could have been, if not mathematics".
>
Well, the short answer would be non-mathematical :). Surely
a mathematical universe would have to be a special case.
"The words of Galileo ... recognize a profound fact of
life: the laws of nature are mathematical in character.
They need not have been expressed in the language of
mathematics, but somehow the nuances of that language
are infinitely adaptable to the facts of experience."
-- John Barrow
But I'll try to do better than that obvious reply.
Another response might be that of Bertrand Russell:
"Physics is mathematical not because we know so much
about the physical world, but because we know so little:
it is only its mathematical properties that we can
discover."
Now, I don't really agree with Russell here, but he
does have a good point. There may be a kind of
selection effect imposed by our own mental abilities
which allows us to find only the simple laws of
nature amenable to mathematical description.
Nahhhhh ..... ;-)
Another point is that not all physical theories are
mathematical. Feynman gives an example of such but
as I left his book at home, I'll have to think up
my own :). What comes to mind first is the plasticity
of metals. First, let's review a basic feature of
the theory of gravitation discussed previously.
In the beginning, things were extremely complicated,
but as people looked closer and closer, things
became simpler and simpler. This is due to the
deep underlying "structure" wherein a wide variety
of seemingly disparate observations can be unified
by mathematical logic. In the case of plasticity,
the situation is the opposite. The deeper one goes,
the messier it gets. One starts with simple formulas
but finds they have to get more and more complex
to account for various phenomena. For example, one
can take a particular set of experimental data and
develop a theory that describes this data very well.
This theory is expressed, of course, in mathematical
language. But one finds that when one applies mathematical
reasoning to these formulas in an attempt to predict
something new, the theory fails and one has to add
some new parameters to the theory to describe the
new data. If one has to do this too many times then
one scraps this theory and tries to find another.
To date, the history of plasticity theory is one
of scrapping, with new theories becoming ever more
complex rather than simpler.
My last point on this will be that, historically, the
mathematical nature of physics took people by
surprise. When Newton published his explanation
of physical phenomena in the form of what was
essentially a mathematical treatise people were
shocked and dismayed. Newton is often given credit
(blame!) for having introduced the mechanical world
view with all its inherent ugliness and pessimism :).
But nothing could be further from the truth. Here
we have to recognize that an equation is not a mechanism.
Newton offered no mechanisms, only equations, and this
at a time when only mechanical explanations were
thought to be scientific. In fact, Leibnitz accused
Newton of introducing occult practices into science
for daring to give an equation for a phenomena with
no mechanism.
Unfortunately, I'm kinda out of time now and won't be
able to respond to the rest of your post at this time.
It might be a couple days before I can get back into
this.
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