Brent Meeker writes:
> Here's my $0.02. We can only base our knowledge on our experience
> and we don't experience *reality*, we just have certain
> experiences and we create a model that describes them and
> predicts them. Using this model to predict or describe usually
> involves some calculations and interpretation of the calculation
> in terms of the model. The relation of the model to reality, if
> it's a good one, is it gives us the right answer, i.e. it
> predicts accurately. Their are other criteria for a good model
> too, such as fitting in with other models we have; but prediction
> is the main standard.
This makes sense but you need another element as well. This shows up
most explicitly in Bayesian reasoning models, but it is implicit in
others as well. That is the assumption of priors.
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.
This implies that at some level, mathematics and logic has to come before
reality. That is the only way we can have prior beliefs about the models.
Whether it is the specific Universal Priori (1/2^n) that I have been
describing or some other one, you can't get away without having one.
> So in my view, mathematics and theorems
> about computer science are just models too, albeit more abstract
> ones. Persis Diaconsis says, "Statistics is just the physics of
> numbers." I have a similar view of all mathematics, e.g.
> arithmetic is just the physics of counting.
I don't think this works, for the reasons I have just explained.
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.