John, (01)
Lainaus "John F. Sowa" <sowa@xxxxxxxxxxx>:
> 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:
>
> Physicists don't do axioms.
>
> 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.
>
> 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. (02)
Physicists must have a very pragmatic, antifoundationalist
view to maths, and that's also the right way, I think.
If mathematics is useful in something, then it is, but
there are many things where using formulas does not
bring any extra clarity. Surely I was not claiming that
mathematics is bad. It is just that even water is bad
if one drinks it too much. As said before, formalization
doesn't give a better understanding of ''a man walks on
the street''. If this needs to be formalized, then NL
parsers are better than hardcoding it by hand, because
hardcoding usually faces an insuperable wall outside
laboratory conditions. But of course it is good to use
e.g. RDF schema as an information repository of a
smallscale application. (03)
> 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". (04)
So, would you want to state ''universe contains everything
that exists'' in mereology, in set theory, in CycL, or how?
If we are using Cyc, then yes. If we are writing a book
of metaphysics, then no. (05)
> But I will admit that any axiomatization requires an explanation
> and justification of each axiom in a natural language statement.
>
> AS> The act of axiomatization should help understanding the subject,
> > not to make it harder. Of course, pictures and graphs help too.
>
> 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. (06)
Yes, but math also scares many people away. Because of this,
I'd rather not use it unless it really brings extra benefit. (07)
> 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.
>
> 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. (08)
What good do you see in 3={{{}}}?? (09)
> 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. (010)
I think that accepting complete induction is the very same thing
as throwing away common sense. Horribile dictu: too horrible to
even say. After this, everything becomes possible. There are so
many ways to critisize it, that I won't even start with the
unnatural natural numbers. (011)
> And by the way, Goedel's theorem, despite its many "horrible"
> formulas, was an excellent example of the value of formalization. (012)
There are some things where Gödel, even though he was a great man,
could have used less formalization. For example, that it is
impossible to prove that 1+1=2, can be derived already from
the great of the greats, Aristotle: (013)
Evidently then such a principle is the most certain of all;
which principle this is, let us proceed to say. It is, that
the same attribute cannot at the same time belong and not
belong to the same subject and in the same respect;
Metaphysics book 4, chapter 3 (014)
$\ldots$ for not to have one meaning is to have no meaning,
and if words have no meaning, our reasoning with one another,
and indeed with ourselves, has been annihilated. For it is
impossible to think of anything if we do not think of one thing;
Metaphysics book 4, chapter 4 (015)
We do not need Gödel numbering to understand that 1+1=2
cannot be proved. It is so deeply tied with out cognitive
capabilities, that without understanding that 1+1=2, we could
not understand anything. If we try to prove that 1+1=2, we
have to use the same cognitive capabilities in the proof,
that we used when we understood that 1+1=2. This is the idea
of Gödel numbering: the things that are to be proved have to
be used in their own proof. (016)
Avril (017)
A little less formalization and a little bit of action (018)
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