[FOM] Concerning Ultraformalism-to Slater&Ozkural
Mirco Mannucci
mmannucc at cs.gmu.edu
Tue Nov 7 10:51:12 EST 2006
---- Hartley Slater <slaterbh at cyllene.uwa.edu.au> wrote:
> Mirco Mannucci points me/us towards another paper by Nelson,
> 'Mathematics and Faith' in his 'Concerning Ultraformalism --to
> Slater', but I am afraid I find Nelson's dream of an 'overwhelming
> presence' in that paper less worthy of addressing.
I guess I have to apologize once again: my goal was not to endorse in any way the
"overwhelming presence": this is a (perhaps worthy) topic for Psychology,
Anthropology,or Theology, depending on one's interests, beliefs and biases. At any rate, I do not
think that it fits the present FOM thread (or even this list's scope).
Incidentally, I note in passing that your labelling it as a "dream" is also a bit biased,
though I respect your opinion. The only thing I wish to add here,
is that I entertain absolutely NO DOUBTS whatsoever about Nelson's integrity:
he simply related things as he deemed fit, without any hidden agenda, and I will
leave it at that.
The reason why I love that paragraph instead, and why I pointed you and the other FOM
fellows there, is that it hints at a view that I find fascinating, and, it seems to me,
(as yet) not fully explored in all its implications:
-----> from the (rigorous) formalist standpoint,
THERE ARE NO NUMBERS whatsoever.
What IS there is simply an "arithmetical game", and "numbers" are just the
"characters" of such a game.
Here is an important point,that goes against the grain, not only in the "platonic" camp,
but in the "constructivist" one as well:
-----> it is not true that numbers are constructed/known once and for all.
Quite to the contrary,
------> each "number" progressively unfolds as new "facts" become known about it.
Trivial (and a bit silly) example. I ask everyone here:
------> do you know the number 5?
You will probably answer: of course, ARE YOU KIDDING ME ??????
Answer: 5 = SSSSS0.
True, I say, but wait a minute:
Answer2: 5= SS0 + SSS0 as well
Answer3: 5 = S( SS0^SS0 )
Answer4: 5 = SSS(0 + 0 + S0) + SS0^SSS0 - SSSSSS0 -S0
........
Answer2^100000: ???
.......
You got my point (there is an indefinite number of answers, some even
beyond anyones' current imagination. It may turn out that 5 is the UNIQUE
number satisfying some incredibly sophisticated number theory conjecture,
or something along similar lines).
To say that 5 IS SSSSS0 would be exactly the same as saying that a vector
in R^2 is a list [x1, x2]. All right, what if I change coordinates? What
if I choose a completely different basis? Or even I choose to represent
it not in a basis, but using a over-complete frame?
SSSSS0 is just the CANONICAL representative term in the (temporary)
similarity classes of available terms denoting 5. Indeed is the simplest &
most rudimentary way of denoting 5, but also an extremely
expensive and clumsy one. Following the vector space analogy, one could say
that SSSSS0 is the representation of 5 in the standard basis (i.e. the standard denotation system).
The simple (and a bit puzzling) truth is that NOBODY knows 5 once and for all. As we further and
further play the "Peano Game", potentially meaningful new "facts" about 5, and its relations to
other "numbers", may unfold.
So, I ask again: what IS 5?
To me, 5 is a "character in the arithmetical game" that gets constantly re-constructed
and re-assessed, as our grasp of the arithmetical game itself increases.
To borrow a wondrous idea from quantum physicist David Bohm, I would say that PA (I mean
the formal rules and axioms) is the arithmetical world (completely) folded, whereas the bulk of
arithmetics that is based on PA is the same world unfolding.
----> BEGINS MINI-NOTE FOR DAVID ISLES
To my knowledge, thus far only David Isles hinted as something along similar lines (or,
to be more precise, and fair to Isles, that is the way I read him. I may be way off. I
hope he will comment on the above himself).
-----> ENDS MINI-NOTE FOR DAVID ISLES
************************************************
Meanwhile, here is another question/proposal:
------> Can we develop formal ways of seeing basic arithmetics from an "invariant standpoint"?
In other words, can we develop a general framework of different arithmetical denotation systems,
and mappings of one denotation system into another (coordinate transformations), together with
measures of their computational advantages-disadvantages?
Note: this GENERAL THEORY OF DENOTATION SYSTEMS is needed to develop and
rigorously formalize a notion I introduced on this list a few months ago:
utterable and unatterable numbers (see postings on Utterable Numbers).
Meta-Conjecture: given ANY reasonable definition of utterability, MOST
numbers will be unutterable (in palin words, most numbers are not only
unfeasible, but they cannot even be NAMED!!!!)
.
***************************************************
Before I leave the section dedicated to you (Hartley Slater), I would like
to say that I found your sentence
>if one does not incorporate into one's formal language the ability to say
>what it means.
very intriguing and a bit mysterious. I assume that you would
like a language that is both ground-language and meta-language at the
same time.
But PA does that already, via Godelization. What am I missing???
Can you either elaborate on that one, or point me somewhere for refs? Thanks
**************************************************************
---- Eray Ozkural <examachine at gmail.com> wrote:
>
> I think that a computationalist approach may fare better at
> this ambitious goal. Here is why. Abstract concepts help us predict
> physical events better. However, all abstract concepts are computational.
> There is no use for any abstract concept that does not lend to
> computation.
I basically agree, but... how can you develop a computational approach
OUTSIDE formalism? I hope you do not mean PHYSICAL machines. Physical
machines were built, as you know, because people like Turing had FORMALIZED
an abstract notion of computability.
>
> However, I think the (ultra) formalist position does not necessitate getting
> rid of potential infinity, because in my view the concept of potential
> infinity has a crystal clear formalization which is a non-terminating
> program, that everybody can objectively examine.
I am afraid you are wrong. What does it MEAN to say that a computation is not
terminating?
Answer: that its length can get arbitrarily large without ending up in
a terminating state.
And what does it mean arbitrarily large?
Potential infinity (and circularity) again...
Not to mention that there are programs for which you do NOT know a
priori whether they will terminate.
By the way, I do NOT advocate killing potential infinity. My own view is quite more
subtle than that.
------> I claim that the very distinction of finite-infinite is not ABSOLUTE,
but CONTEXTUAL.
In order to substantiate my claim, I intend to build mathematical structures, let us
call them *ultrafinitistic universes*, such that what looks like infinity from inside
is very small from outside...
> Is not this already achieved by Kolmogorov complexity theory?
It Would be nice indeed, but the claims made by Chaitin that his "complexity approach"
EXPLAINS Godel are so far (alas!) unsubstantiated (which does not mean there
is nothing in his view). What Chaitin did was to CREATE ANOTHER undecidable sentence,
NOT showing how all incompleteness phenomena are in fact derived from unmanageable
complexity issue.
I personally think that Chaitin's overarching project is meaningful and promising;
however, not there yet...
> What do you have in your mind when you say that? Do you want to
> state Godel's theorem in another way?
No. I am asking the following question: is it possible to interpret Godel's
incompleteness as saying something about the arithmetical game ITSELF (i.e. the way
it works, the way it is played, etc.), as opposed to some (perhaps fictional) "intended "
standard model of arithmetics?
My answer to my own question is yes (it will be posted under the header "The Grand Peano Game").
But, as I said, I would like first to see if someone else has a view to
put on the table...
>
> PS: Also, what do you think of Godel's statements when he said that
> the second incompleteness theorem is also valid for finite systems. This
> seems to weaken the finitist position a little, if we listen to Godel.
>
I am not sure I understand you here. Please elaborate this point.
Best Wishes
Mirco Mannucci
P.S. Apparently, a subset of strictly positive Lebesgue measure on this
list has *crystal clear* ideas on a number of issues... well,
I do not. The only crystal clear idea I have is: ---I AM--- and
sometimes I feel like doubting that one too...
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