Re: what relation do mathematical models have with reality?

From: Hal Finney <hal.domain.name.hidden>
Date: Mon, 25 Jul 2005 14:08:49 -0700 (PDT)

Brent Meeker wrote:
> [Hal Finney wrote:]

> > When you observe evidence and construct your models, you need some
> > basis for choosing one model over another. In general, you can create
> > an infinite number of possible models to match any finite amount of
> > evidence. It's even worse when you consider that the evidence is noisy
> > and ambiguous. This choice requires prior assumptions, independent of the
> > evidence, about which models are inherently more likely to be true or not.

> In practice we use coherence with other theories to guide out choice. With
> that kind of constraint we may have trouble finding even one candidate
> theory.

Well, in principle there still should be an infinite number of theories,
starting with "the data is completely random and just happens to
look lawful by sheer coincidence". I think the difficulty we have in
finding new ones is that we are implicitly looking for small ones, which
means that we implicitly believe in Occam's Razor, which means that we
implicitly adopt something like the Universal Distribution, a priori.

> We begin with an intuitive physics that is hardwired into us by
> evolution. And that includes mathematics and logic. Ther's an excellent
> little book on this, "The Evolution of Reason" by Cooper.

No doubt this is true. But there are still two somewhat-related problems.
One is, you can go back in time to the first replicator on earth, and
think of its evolution over the ages as a learning process. During this
time it learned this "intuitive physics", i.e. mathematics and logic.
But how did it learn it? Was it a Bayesian-style process? And if so,
what were the priors? Can a string of RNA have priors?

And more abstractly, if you wanted to design a perfect learning machine,
one that makes observations and optimally produces theories based on
them, do you have to give it prior beliefs and expectations, including
math and logic? Or could you somehow expect it to learn those? But to
learn them, what would be the minimum you would have to give it?

I'm trying to ask the same question in both of these formulations.
On the one hand, we know that life did it, it created a very good (if
perhaps not optimal) learning machine. On the other hand, it seems like
it ought to be impossible to do that, because there is no foundation.

> > Mathematics and logic are more than models of reality. They are
> > pre-existent and guide us in evaluating the many possible models of
> > reality which exist.

> I'd say they are *less* than models of reality. They are just consistency
> conditions on our models of reality. They are attempts to avoid talking
> nonsense. But note that not too long ago all the weirdness of quantum
> mechanics and relativity would have been regarded as contrary to logic.

I guess we could agree that they are "other" than models of reality?
It still strikes me as paradoxical: ultimately we have learned our
intuitions about mathematics and logic from reality, via the mechanisms
of evolution and also our own individual learning experiences. And yet
it seems that at some level a degree of logic, and certain mathematical
assumptions, are necessary to get learning off the ground in the first
place, and that they should not depend on reality.

I'm pretty confused about this right now.

Hal Finney
Received on Mon Jul 25 2005 - 18:04:22 PDT