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).
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 -----
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 the
technical part of the thesis, without being  ... too much technical.
The field of logic is rather hard to explain, without being
a little bit long and boring in the beginning :(

At least I found a very good Mathematical Logic Web page:

The page contains also a test to see if you are platonist (actually it tests
only 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 in
Podnieks page.
Any comments ?


PS I have finished my french paper, and I will write the paper for
Amsterdam. The goal is always the same: how to be clear, short and
understandable .... (given the apparent "enormity" of the result!)

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