Benjamin.J.Tilly
What about PA? A collection of simple statements, all of which are
manifestly true about the positive integers.
Jan Willem Nienhuys
These statements look simple. But they can do things for you only
if you have a vast mathematical apparatus that includes:
- conventions regarding names of variables;
- conventions regarding scopes of thoses names;
- rules about what kind of substitutions are permittable;
- conventions regarding definitions;
- conventions regarding what is considered: sets, functions, products.
- rules of inference (i.e. logic).
I just mentioned in my previous post 1 (variable names).
If one proposes to name variables x, x', x'' and so on, one certainly
needs to know that all these names are different. Of course, in my
heart I know they are different. But if the formal system is to be absolutely
watertight, I must prove that they are different. In other words, I
must apply the Peano axioms to the language used to talk about them.
Benjamin.J.Tilly
I think that you are going overboard. For instance at first all that
you need to use are PA with a finite number of variables. The other
conventions are taken care of by first-order logic. (A series of rules
that are again very reasonable. If I was to learn that first-order
logic was hopelessly flawed then I would be severely disturbed.)
If you are really concerned then you can use a "boot-strap" technique
so that statements which involve an arbitrary number of variables are
talked about in a system that uses only finitely many of them. Then you
can talk about several groups of variables, each group of which could
have any number of variables in it, and so on ad nauseum. The principle
would be like the distinction between first-order logic, second-order
logic, ect. In principle this could get rid of the application of the
same version of Peano's axioms to the version that you are using at any
point.
Personally I do not see that there is any point in bothering.
Foundations, after all, at some point come down to belief. So what do
you believe in? Personally I accept PA completely, and ZF seems
reasonable. Beyond that I do not worry about it since I do not
personally use it...
Jan Willem Nienhuys
The fact that my experience with nonnegative integers makes me believe
that the Peano axioms are manifestly true is irrelevant. My experience
certainly doesn't include having often verified that adding 1
consecutively 10^80 times gives each time a different result.
The basic trouble with the Peano axioms is that they implicitly
assert that any set of positive integers has a smallest element.
If you think of simple Ramsey numbers, then you'll realise that
even in simple cases this is more a metaphysical assertion than a
matter of "manifestly true". How can I have confidence that any
set of positive integers, no matter how unwieldy its definition
has a smallest element? The unwieldyness of definitions is made
possible by having an unlimited amount of positive integers (and
variable names) at your disposal.
Benjamin.J.Tilly
Well, if you want to get nasty about it, there is an infinite set of
positive integers such that no positive integer can be proven to be in
it in your (consistent) formal system. (The set's definition depends on
what formal system you are interested in, but that can be rectified in
that I can define it in such a way that, for instance, no formal system
stateable in under a hundred pages of writing can verify that any
number is in the set.)
If things like this bother you, then become a constructivist.
And as for the Ramsey numbers? I have no problem with those. I know
that there is an answer. I know how to find that answer. That this is
impractical is a matter of practice, not principle. (Then again, I am a
mathematician. :-)
Jan Willem Nienhuys
The Peano axioms plus a formal system in which they are embedded
must be free of contradictions, i.e. for no statement A both
A and not-A may be provable. (Note that I apparently want to allow
variables like A to denote statements; the formal system should
have watertight descriptions - no handwaving or illustration by
example allowed - of how to go about making variables represent
statements.)
Only such a system as a whole can conceivably contain a contradiction.
Without the formal framework a contradiction doesn't make much sense.
Benjamin.J.Tilly
Ummm...I think not. There are (as has become apparent in some
discussions) many ideas that make a great deal of sense, whether or not
they have been formalized. For instance things like Russell's paradox,
the exam paradox, and the barber paradox all make sense to people and
can generate discussion even without a formalization.
Personally I think that mathematicians are sometimes too hung up on the
details of formalizations and lose track of the actual math they are
talking about. As the old saying goes, mathematicians are platonists on
weekdays and formalists on Sunday. I personally think that this
tendancy is a good and healthy one!
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