Dear Bruno,
I did some browsing in the Podieks website and
found interesting statements.
Without connotation and order:
*
To the question "What is mathematics" - Podiek's
(after Dave Rusin) answer:
Mathematics is the part of science you could continue to do if you
woke up tomorrow and discovered the universe was gone.
Remark: provided that YOUR mind is "out of this
world" and stays unchanged 'as is' after (the rest of) the universe was
gone.
Another point is "science" but I let it go now.
(cf: Is math 'part of science'?)
*
The JvNeumann quote:
In mathematics you don't understand things. You just get used to
them.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?) - 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: Aspects of 'model' formation from different
directions).
*
Podnieks:
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.
Translated 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 problem.
*
Let us assume that PA is consistent. Then only computable
predicates are expressible in PA.
("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..."
In (my) wholistic views an (unlimited, ie.
non-model) complexity is non computable (Turing that is) and
impredicative (R.Rosen). In our (scientific!) parlance:
vague.
No 'discrete objects': everything is
interconnected at some qualia and interactivity level.
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." -
underlines my caution to combine wholistic
thinking with mathematical (even "first order arithmetic" only) language.
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.
John Mikes
- Original Message -
From:
Bruno Marchal
To: [EMAIL PROTECTED]
Sent: Saturday, June 26, 2004 11:30
AM
Subject: Mathematical Logic,
Podnieks'page ...
Hi George, Stephen, Kory, All.I am thinking hard finding to
find a reasonable way to explain thetechnical part of the thesis, without
being ... too much technical.The field of logic is rather hard to
explain, without beinga little bit long and boring in the beginning
:(At least I found a very good Mathematical Logic Web page:http://www.ltn.lv/~podnieks/index.htmlThe page
contains also a test to see if you are platonist (actually it testsonly if
you are an arithmetical realist!). Try it!From that page I will be
able to mention easily set of axioms, and rules.For example below are
the non logical axioms of Peano Arithmetic.Does it makes intuitive sense
?I suggest you try to find the logical axioms and the inference rules
inPodnieks page.
SKIPAny comments ?BrunoPS I have finished my french
paper, and I will write the paper forAmsterdam. The goal is always the
same: how to be clear, short andunderstandable (given the apparent
"enormity" of the result!)
http://iridia.ulb.ac.be/~marchal/
