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John
Interjecting hopefully without disturbing the chain of thought, I want to learn from others perception on three concepts: Model Theory, Mathematics and Science.
From my recall of previous contributions you made to these topics individually, I recall that math can express a model theory precisely in the realm of validity of the parameters (for example linear, quadratic, second order tensor, etc.) Nature however defies closed form solutions (example: relativistic corrections to Newton's law for planetary spacecraft trajectories). How are models theorized, I think Logic and math interplay in the accurate realization of Models but are Model Theories described simply somewhere? Rick Murphy also addressed some aspects in past?
Regards.
Thanks.
Ravi
On Sun, Jan 24, 2010 at 12:13 AM, John F. Sowa <sowa@xxxxxxxxxxx> wrote:
Avril and Rob,
Your examples are consistent with what I said:
JFS>> First of all, it is essential to distinguish empirical sciences
>> from pure mathematics. In empirical sciences, the ultimate test
>> is agreement of predictions with observations. Mathematics,
>> however, is not an empirical science.
RF> Greg Chaitin might have issue with that statement.
JFS> No. He wouldn't object to that statement...
JFS> Before trying to prove a general theorem about a function,
> mathematicians usually start by performing "quasi-experiments"
> to "observe" how the function behaves on typical values.
>
> But those so-called observations are thought experiments,
> even when the thinking is carried out by a computer.
> They are not observations of the physical world.
AS> Hilary Putnam uses the term quasi-empirical with at least slightly
> different meaning in [1]. For example, calculus can be taken as
> quasi-empirical mathematics: it works perfectly with or without any
> mathematical proofs of it, although empirical proofs are available.
That is consistent with the above: Mathematicians perform thought
experiments. The use of informal reasoning instead of proofs is
irrelevant. People were using arithmetic and simple geometry for
thousands of years before they discovered formal methods of proof.
Mathematics is an aid to clear, precise thinking any subject,
but it does not depend on the nature of the world, and it does
not make any claims about the world. (Many mathematicians,
following Plato, talk as if the mathematical structures exist
in some Platonic heaven. But other mathematicians maintain
that such a heaven is a pure construction of their imagination.
In either case, that heaven is independent of the physical world.)
The test cases and thought experiments are the same kinds of things
that programmers do when they're writing and debugging their code.
They observe the effects of their own choice of rules and data.
They are not making observations or claims about the world.
RF> Here's a excerpt from Chaitin's "Randomness in Arithmetic and
> the Decline & Fall of Reductionism in Pure Mathematics" ...
GC> 5. Experimental mathematics
> Okay, let me say a little bit in the minutes I have left about
> what this all means.
>
> First of all, the connection with physics....
Those are good examples. Please continue reading and note:
1. Chaitin is comparing the techniques and methodologies of
physics to the techniques and methodologies of experimental
(or quasi-experimental) mathematics.
2. But in every case, the mathematical "experiments" are
thought experiments (which may be aided by computer, but
the simpler cases can be done mentally or with paper and
pencil).
3. In the "quasi-experiments" by Chaitin or Putnam, the results
do not depend in any way on the nature of the physical world,
nor do they make any predictions about the physical world.
Exactly the same results could be derived by aliens in a
universe with very different laws.
John
--
Thanks.
Ravi
(Dr. Ravi Sharma)
313 204 1740 Mobile
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