Avril, (01)
In a sense, any application of mathematics, could, in principle,
be axiomatized. But I recall a comment made by a physicist
about axiom systems such as von Neumann's: (02)
Physicists don't do axioms. (03)
He did not mean that physicists don't use mathematics. What
he meant is that physicists have a large number of physical
principles stated in mathematical form, and a large toolkit
of mathematical techniques. (04)
For any particular problem, they pick and choose from those
resources to assemble a mathematical model that characterizes
some aspect of the universe. Then they use math to compute
some prediction about how that model would evolve over time.
For any such model and computation, it is possible to list
every starting formula and call that list the "axioms" for
that problem. But it would be a different list for each case. (05)
AS> We have to separate two things:
>
> 1) formalization
> 2) axiomatization
>
> To axiomatize something, does not mean that we have to impose
> a horrible string of mathematical formulas upon it. In the
> case of philosophical ontology (the best part of metaphysics),
> plain text will do. (06)
Euclid did not use a formal notation to state his axioms
because at that time, the only known formalism, Aristotle's
syllogisms, was not adequate to do all the reasoning for
the problems he was trying to state. (07)
Euclid's approach with more formulas is still adequate for
many problems. But as you know, I have no sympathy for any
attempt to construct a "perfect" ontology. And I would have
an extremely negative attitude toward any claims of a "perfect"
ontology that was not stated in "a horrible string of formulas". (08)
But I will admit that any axiomatization requires an explanation
and justification of each axiom in a natural language statement. (09)
AS> The act of axiomatization should help understanding the subject,
> not to make it harder. Of course, pictures and graphs help too. (010)
I agree. But the use of math is an excellent way to make many
subjects easier to understand. On the other hand, there are
many poorly understood subjects for which math is merely a way
of disguising the lack of understanding. (011)
AS> The clearest cases of overformalization are the those where
> the thing that is to be formalized, is actually required to
> understand the formalization itself! Examples of these are
> e.g FregeRussell, von Neumann, and Zermelo definitions of
> natural numbers. (012)
I would not consider any of those cases to be examples of
over formalization. However, I agree that Frege's definition
of natural number was a very bad choice  the idea that the
number five, for example, *is* the set of all sets that have
five elements. (013)
Peano's axioms are much better because his successor function
relates the numbers to the very natural method of counting.
Furthermore, I agree with Kronecker, Brouwer, and many others
that arithmetic has a more solid foundation than any version
of set theory. (014)
In fact, Goedel's famous theorem involved a reformulation of
logic in terms of arithmetic. He thereby showed that higher
order logic could be reformulated in terms of arithmetic
functions. (015)
If Goedel had stated his theorem in logic, people would have
dismissed it as a proof of the inadequacy of his logic.
But by stating it as a theorem of arithmetic, it could not
be dismissed without dismissing arithmetic. (016)
And by the way, Goedel's theorem, despite its many "horrible"
formulas, was an excellent example of the value of formalization. (017)
John (018)
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