----- Original Message -----
Sent: Monday, June 28, 2004 6:27 AM
Subject: Re: Mathematical Logic,
Thanks for your quotations from (or through)
Podnieks. Here are some comments.
"To the question "What is mathematics" - Podiek's (after Dave Rusin)
Mathematics is the part of science you could continue to do
if you woke up tomorrow and discovered the universe was gone."
BM: What a pretty quote!
It's a good description of what happened to me a long time ago. I woke up,
and realized the universe was gone. Only taxes remained ;)
JM: Remark: provided that YOUR mind is "out of this
world" and stays unchanged 'as is' after (the rest of) the universe was
point is "science" but I let it go now. (cf: Is math 'part of
BM:I really hope you don't doubt that. math
is certainly part of science. With comp and even with weakening of comp the
reverse is true: science is part of math.
(R-JM): Science in my
terms is the edifice of reductionist imaging (observations) of topically
selected models, as it developed over the past millennia: subject to the
continually (gradually) evolving (applied) math formalism. Will be back to
The JvNeumann quote:
In mathematics you don't
understand things. You just get used to them.
BM: I agree. But I think it is the same with loves,
cuisine and certainly physics. Children climb in trees before learning the
gravitation law ; and even that does not explain things.
JM: True. Once you want to understand them you have to
couple it with some sort of substrate, ie. apply it to "things" when the fix
on quantities turns the math idea into a (physical?) limited model preventing
a total understanding (some Godel?)
BM: It is your talk here. I am not sure I
understand. Of course we have a sort of build-in theory of our neighborhood,
as does cats and birds. But substrate and concreteness are illusion of
simplicity. Only many neurons and a long "biological" history make us
forgetting that nothing sensible can be obvious. And then with comp you can
have clues why it is so ....
(R-JM): (MY!) Simplicity is the 'cut-off' from the wholeness in our
models. Later you mention the causality: it is similarly a cut-off of all
possible (eo ipso 'active') influencings, pointing to the ONE which is the
most obvious within our topical cut. We make 'cause' SIMPLE.
JM: - Isn't this the way with
Einstein's "form": you first get used to it (in general)(?) then apply it to
substrates (shown later in the URL).
(My [_expression_]: Aspects of 'model' formation from different
agree with Podnieks, as you can guess.
For me, Goedel's results
are the crucial evidence that stable self-contained systems of
reasoning cannot be perfect (just because they are stable and
self-contained). Such systems are either very restricted in power (i.e. they
cannot express the notion of natural numbers with induction principle), or
they are powerful enough, yet then they lead inevitably either to
contradictions, or to undecidable propositions.
into my vocabulary it sais the same as the 1st sentence, (called) 'well
defined', topical and boundary enclosed and limited "models", - never
leading to a total (wholistic) result. I generalized it away from the math
thinking - eo ipso it became more vague.
But that's my
BM: I am not sure I understand what you ere saying here.
It is too much ambiguous.
Remember that comp entails the falsity of almost
all reductionist view of numbers, machines, etc.
(R-JM): Exactly. Comp (? I am not sure if I know what it is indeed) has
IMO brisk rules and definite qualia to handle by those rules. (I evaded:
'quantities'). Which means the omission of aspects OUTSIDE such qualia and
rules. The cut-off, ie. limitations, enable comp to become brisk, unequivocal,
well defined. Including unidentified and infinite variables, qualia, all sort
of influence (quality and strength) - meaning the wholeness-interconnection -
makes it more vague than any fuzziness could do (which still stays topical).
I don't expect this emryonic branch of thinking
(30-50years max?) even using the language of the millennia of reductionist
development, to compete in briskness with the conventional - what you and
others may call: - science. An embryo would recite Godel in a very vague
JM: Let us assume that PA is consistent. Then only
computable predicates are expressible in PA.
BM: This is ambiguous as it stands. All partial
computable predicates, including the total computable predicates are
expressible in PA. Incompleteness is linked to the fact that there is no
mechanical test to distinguish the total and partial predicates. See my
"diagonalization posts" to get the basic idea.
(R-JM): I pass on that. 'PA' I saw first ever in the
JM: ("3.2: In the first order
arithmetic (PA) the simplest way of mathematical reasoning is formalized,
where only natural numbers (i.e. discrete objects) are used..."
wholistic views an (unlimited, ie. non-model) complexity is non computable
(Turing that is) and impredicative (R.Rosen). In our (scientific!)
BM: I share with you that idea that the big whole is
vague and uncomputable, and that impredicativity is inescapable. Please note
that it is indeed provably the case concerning the experience of the universal
machine once you accept to define knowledge by true belief (proof) or other
theetetic definition of knowledge.
(R-JM): do we have ANY other knowledge-base? Proof (Popper's no-no) is within the belief system. "True" is a
1st pers. judgement. Even an 'accepted' 3rd p. truth is "1st p.
I haven't (yet?) included the universal mchine into
my vocabulary. It is not 'simple' (see above).
JM: No 'discrete objects': everything is interconnected
at some qualia and interactivity level.
BM: OK (except that "interactivity" like "causality")
has no clear meaning (for me).
(R-JM): My apologies: nor does 'interactivity make sense to me, it was a
lapse (did my cat walk over my keyboard?) Activity is something to be defined,
if, inter- does no job in that.
Causality I mentioned above, as a reductionist model
of influence(s?) within the chosen (topical) boundaries we observe in the
actual study. "A" cause never 'does' anything.
must be BM - or Podniek?): The end of the chapter:
"We do not know exactly, is PA consistent or not. Later in this section we
will prove (without any consistency conjectures!) that each computable
predicate can be expressed in PA." -
BM: Like Smullyan I believe we know that PA
is consistent. With comp that means (by Godel second theorem) that we are
"superior" than PA with respect to our ability to prove theorems in
What no machine can ever prove is its own consistency. But
machines can bet on it and change themselves. (The logic G and G* will still
apply at each step of such transformation, unless the machine becomes
my caution to combine wholistic thinking with mathematical (even "first
order arithmetic" only) language. It is,
I did not intend to raise
havoc, not even start a discussion, just sweeping throught the URL brought
up some ideas. Only FYI, if you find it interesting.
One remark to math vs science: I consider math a
human language, a mental activity (again this term!) on its own, (uninhibitied
by observational models - only by its intrinsic connotations).
Science, however, is a reductionist parcelling of observations -
according to the epistemic level of the age, the cognitive inventory and its
connectional capabilities of the by that time acquireds. Science applies math
in its formalizing of deductions, but such
math is quantitatively distorted -
adjusted to the models and the observations it pertains to. Which is also
subject to the actually achieved level of epistemic enrichment.
So it seems I deem math a higher level than
science. Maybe because I don't know it.