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.

And I am still waiting for you to explain me what *is* formalism without using arithmetical realism or equivalent.

Let me answer to you. To be able to use a formalism, you need to define what are the well-formed sentences; 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.

Formalism, and all form of finitism which is a bit richer than ultrafinitism, is entirely constructed (implicitly or explicitly) on arithmetical realism. Gödel showed the deep "bisimulation" of formalism and arithmetic.

With your use of the term Platonia, the theory I am working on, is usually called finitism, and is usually considered as anti platonism. This use is misleading because it is platonist, and even pythagorean, in the sense of the neoplatonist.

I think you are confusing people on the genuine issues, here.

Bruno



Now, with computationalism, we don't even need such a mathematical
arithmetical Platonia. We need only the idea that arithmetical truth
(even a tiny effective part of it) is independent of you and me. Like
in Plotinus, the ultimate being (arithmetical platonia) is not a being
itself (nor is matter!).

So neither Platonia, nor even arithmetical Platonia needs to exist.
Numbers needs to exist in some sense, and do exist in theories like RA
or PA, in the sense that such theories formally proves that Ex(x =
sssssss0) for example.

Just to be a bit precise.

Bruno

On 18 Feb 2011, at 02:49, Stephen Paul King wrote:



Hi All,

    Question: Why must Platonia exist?

Onward!

Stephen

“It is amazing what can be accomplished when nobody cares about who
gets the credit.”
Robert Yates

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