On 20 Jun 2016, at 04:32, Bruce Kellett wrote:
On 20/06/2016 3:10 am, Bruno Marchal wrote:
On 18 Jun 2016, at 02:25, Bruce Kellett wrote:
All these problem dissolve if you reject the notion of a platonic
realm for arithmetic and accept physicalism.
If you succeed in making me doubting that 2+2=4, I might doubt even
more on Hphi = Ephi. If you reject elementary arithmetic, you can
invoke directly the God of the Gap. (and I don't believe in a
platonic realm for arithmetic, I have no real clue what you mean by
that).
I think you really do know what is meant by a 'platonic realm' --
you know about Plato's theory of Forms that exist in a separate
sphere (world). Platonism in mathematics is just an formalization of
the feeling that mathematician's report that they seem to be
exploring an existing landscape rather than inventing things.
Given that at some moment I compare the machine's discourse with
Plato's Platonism, I prefer to use the term "realism" for that feeling
of the mathematicians that the sum of initial odd number give squares,
independently of them or of anything else.
The formal counterpart is the acceptance of the classical tautology A
v ~A, for A sigma_1 or pi_1. In english, you need to understand that
either a running digital machine will stop or will not stop (in
arithmetic).
Although this feeling may be understandable, it is not a proof that
mathematics and mathematical objects are objective in any sense
other than that they are implied by the axioms and the rules of
inference, and are thus intersubjectively valid.
That is formalism philosophy or logicism. It is debunked by
mathematical logic.
Even if valid (which it is not) the same could be said (as much
wrongly) for physical formalism and theories.
But nowhere I claim anythings like that.
Knowing that 2+2=4 is just a matter of knowing the definitions or
rules.
So take Riemann hypothesis. Since Turing we know that it is a Pi_1
arithmetical proposition.
You are confusing theories and models.
And "2+2=4" is also different from "knowing that 2+2=4". For a human,
knowing that 2+2=4 might need more than definition and rule, also many
examples and some intuition.
You know that in chess the King can only move at most one square a
time in any direction. The knowledge is just knowledge of rules and
their consequences -- no separate existence is implied.
You are the one coming and telling me that some God exist, and I am
not satisfied by that explanation of the appearances.
Don't talk like if you knew what exists. Nobody does. Just present
your theory, that is your assumption.
In Smolin's phrase, mathematics is "evoked" rather than pre-existent
in another realm.
I don't do metaphysics. Existence is managed by the existential
quantifier in the formal base theory. If you have a physical proof
that one plus one is two, which does not use this at any point in the
formal physical proof, then you might exploited, but that does not
exist, and untill now, all physical theories assumes numbers as much
as I assume them.
A non realist is someone who does not let its kids follow a course
where they are taught that 1+1=2.
The rest is attributing me metaphysical supplementary axiom which are
not there.
It is really quite annoying that you keep suggesting that doubting
that 2+2=4 involves the rejection of arithmetic
? If you reject 2+2=4, you do reject arithmetic.
and and an appeal to a "God-of-the-Gaps".
Then explain how assuming a physical universe can select a mind from
the infinitely many computation running that mind in arithmetic,
without throwing out the computationalist assumption.
There is no suggestion of this in anything I have said.
What I say concerns all people believing/assuming both
computationalism and weak materialism.
Bruno
Bruce
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