Chris, (01)
I'll have to explain the background a little. The overall mathematical
ideology that I'm holding is finitism, and I argue that finitism is not just
an alternative to platonism and transfinitism, no more than slavery is an
alternative to freedom. Slavery is wrong, and so is uncontrolled platonism.
The reason for this is that it is better for the human kind to throw away
the abracadabraic things from mathematics. And that's a good enough
justification for finitism. (02)
It is not enough that axiomatic systems are 1) consistent internally, but
they have to be 2) applicable too, or there should at leat be some sort of a
chance tp applicability. If fact, 2) should be given an equal priority than
1), but this seems not to be the case with transfinitism/Cantorism. (03)
more comments below. (04)
Lainaus Christopher Menzel <cmenzel@xxxxxxxx>:
> > None of these makes arithmetic any more secure, better, or more
> > usable in any way.
>
> Well, while I tend to agree with you, there are some serious issues
> here. For instance, the point of the logicist program was to show
> that mathematics could be derived from purely logical principles. If
> the program had been successful, that would have put mathematical
> knowledge generally on a very secure epistemological foundation. The
> program failed, of course, at least as initially envisioned. But
> there are still lots of good reasons to formalize arithmetic over and
> above any epistemological worries, e.g., for the sake of clarity, to
> investigate properties of the theories, connections to other theories,
> for use in automated reasoning and automated theorem proving, etc.
>
> > Still they talk about an incompleteness and completeness of
> > arithmetics.
>
> Point being? (05)
The point is that self-evident things are being proved. If axiomatizing
arithmetic is of a real use in automated reasoning, or in anything else,
then of course it is good to axiomatize it. But for a person to feel more
secure about addition and multiplication after the axiomatization, is not
good. One feature of finitism, as a protestant attitude, is to emphasize
that formalization does not make common sense any better. Another is to
prune off unnecessary features from mathematics. (06)
A certain fundamental feature in Peano's axioms deserves to be pruned off,
namely, the interpretation of axiom 9 as something where complete induction
is applied. Complete induction is not only useless, but also harmful. E.g.
the proof you gave did not need complete induction. In Wikipedia Peano's
axioms were numbered as: (07)
5. 0 is a natural number.
6. For every natural number n, S(n) is a natural number.
9. If K is a set such that:
* 0 is in K, and
* for every natural number n, if n is in K, then S(n) is in K, then K
contains every natural number. (08)
Axiom 9. can be maintained, but the meaning of 'every' has to be interpreted
to denote a totality of something around 10^120, or Ackermann(5 5), or
something finite that can be written down or understood in some way. (09)
I'll just type three of the unbeareable features of the set K with the
complete induction -interpretation:
1) K contains something neverending as a totality.
2) K contains infinitely many finite things that grow on each step, when it
is clear that if something grows or has grown infinitely, it has to be
infinitely big.
3) K's proper part has a magnity as big as the whole K. To justify this,
Cantorists turn things around and claim that this is the very definition of
transfinity. The same political twist is made with Burali-Forti paradox:
just flip flop and fly, and an evident paradox becomes a definition of the
'truly' infinite. (010)
> > It is clear even without any proof about it. One way to prove that
> > all cannot be proved is the simple fact 1+1=2 cannot be proved.
>
> In one sense, yes. It is quite reasonable to think that one cannot
> expect to find a proof of 1+1=2 that uses principles that are somehow
> epistemically more basic or certain than 1+1=2 itself. However, you
> can most certainly prove it in a system of formal arithmetic -- Peano
> Arithmetic. 1+1=2 is not an axiom of PA. It has to be proved
> formally from the axioms. Here's the proof; "S" indicates the
> successor function:
>
> 1. (x)(y)(x + Sy = S(x + y)) Axiom
> 2. S0 + S0 = S(S0 + 0) From 1
> 3. (x)(x + 0 = x) Axiom
> 4. S0 + 0 = S0 From 3
> 5. S0 + S0 = SS0 From 2 and 4
>
> > Or, if you prove it in some formal system, then the axioms of that
> > formal system cannot be proven true.
>
> From within the system (if the axioms are independent). Yes. What's
> the point? Everyone knows this. (011)
The point was to emphasize that proving 1+1=2 does not make 1+1=2 any
more secure. But, as said above, if proving it helps in automated reasoning
then the axiomatic system on which the proof is based on is useful, but the
complete induction should be erased from the (interpretation of the) system
because it is useless and contradictory. (012)
> > The theorem was a clear blow to some people who thought that
> > everything can be governed, or put under some mathematical axioms.
> > There's no denying that.
>
> Certainly not. It pretty much completely torpedoed "Hilbert's
> Program" of finding a complete and consistent axiomatization of all of
> classical mathematics. But again, there are still lots of good
> reasons to axiomatize various branches of mathematics. (013)
Yes, and I don't have much (anything) to say about or against proof analysis
and consistency proofs of axiomatic systems, but surely there is also space
for being more down-to-earth in mathematics, and finitism is the
enlightenment and protestant movement that aims to inform people about this. (014)
Many mathematicians take the axioms of ZFC as the true axioms of
mathematics. I interpret the axioms as equivalent to "you can do anything
you can imagine". We can distinguish two ways (among others) to look at ZFC
axioms: 1) as the God-given licence that a mathematician can do whatever at
all (like contraction out of contradiction) and forget all the controversy
of the axioms, and 2) as a measure of craziness, of just how crazy things
are required, in order to make a very bad idea intelligible. The bad idea
that is 'being' made intelligible is that continuous magnitudes consist of
points. No transfinities would be needed if Aristotelian-mereological
continuum was adapted, but there's actually a much better alternative
available: mathematical continuum, or a line consists of nothing. It does
not consists of anything, we just use geometry if we need it for some
purposes. It is wrong to give 'foundations' for things that do not need
them. Still, mathematicians are holding on to ZFC, and teaching that idelogy
to young students. This is wrong, but it also gives the finitist a reason to
fight against it. Of course, anti-foundationalism is not a new idea, but
people don't understand well enough that point-continuum can be seen as the
source of all transfinitism. (015)
Avril (016)
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