On Feb 18, 8:52 pm, Bruno Marchal <marc...@ulb.ac.be> wrote:
> On 18 Feb 2011, at 12:53, 1Z wrote:
> > On Feb 18, 9:48 am, Bruno Marchal <marc...@ulb.ac.be> wrote:
> >> Hi,
> >> What do you mean by Platonia?
> >> The kind of Platonia in Tegmark or in Peter's (1Z) post does not make
> >> sense for mathematicians. Even if you are using a theory like Quine's
> >> NF, which allows mathematical universes, you still have no
> >> mathematical description of the whole mathematical reality. Tegmark
> >> is
> >> naïve about this.
> >> *Arithmetical* platonia can be said to exist, at least in the sense
> >> that you can prove it to exist in models of acceptable set theories,
> >> like ZF. It is just the structure (N, +, x). It is used in all papers
> >> in physics, math and logic, including Pratt ...
> > Used as a formalism. It is not the case that everyone
> > who uses arithmetic is a Platonist
> I did not say that, even with platonism restricted to arithmetical
> realism, except for those using classical arithmetic or models of PA
> in ZF, etc. To believe in (N,+,x) you need a stronger realism than
> arithmetical realism, which says nothing about infinite sets.
To make use of in (N,+,x) you need no realism at all. Infinite
sets are irrelevant to the formalist
> And I am still waiting for you to explain me what *is* formalism
> without using arithmetical realism or equivalent.
"In foundations of mathematics, philosophy of mathematics, and
philosophy of logic, formalism is a theory that holds that statements
of mathematics and logic can be thought of as statements about the
consequences of certain string manipulation rules.
For example, Euclidean geometry can be seen as a "game" whose play
consists in moving around certain strings of symbols called axioms
according to a set of rules called "rules of inference" to generate
new strings. In playing this game one can "prove" that the Pythagorean
theorem is valid because the string representing the Pythagorean
theorem can be constructed using only the stated rules.
According to formalism, the truths expressed in logic and mathematics
are not about numbers, sets, or triangles or any other contensive
subject matter — in fact, they aren't "about" anything at all. They
are syntactic forms whose shapes and locations have no meaning unless
they are given an interpretation (or semantics)."
> Let me answer to you. To be able to use a formalism, you need to
> define what are the well-formed sentences;
I'e told you over and over that by fomalism I mean
mathematics as a game, not mechanisability
> for this you need to define
> them in the usual recursive way (or equivalent way) and this, together
> with simple rules (like finding the first and second in a couple of
> expressions) is ontologically as rich as sigma_1 realism.
SIgma_ 1 is just another formal game to formalists: to them,
it has no ontology.
> Formalism, and all form of finitism
Formalism has nothing at all to do with finitism
>which is a bit richer than
> ultrafinitism, is entirely constructed (implicitly or explicitly) on
> arithmetical realism.
How can anti realism be constructed on realism?
You are presumably indulging in your peculiarity
of using "realism" to mean "bivalence"
>Gödel showed the deep "bisimulation" of
> formalism and arithmetic.
They may well be structurally, formally, abstractly equivalent in some
That doesn't mean either is real
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