From: *Bruno Marchal* <marc...@ulb.ac.be <mailto:marc...@ulb.ac.be>>
On 18 Dec 2018, at 01:14, Bruce Kellett <bhkellet...@gmail.com
<mailto:bhkellet...@gmail.com>> wrote:
On Tue, Dec 18, 2018 at 6:15 AM Bruno Marchal <marc...@ulb.ac.be
<mailto:marc...@ulb.ac.be>> wrote:
On 17 Dec 2018, at 08:50, Bruce Kellett <bhkellet...@gmail.com
<mailto:bhkellet...@gmail.com>> wrote:
What I am curious to know is how how many of these
statements you agree with:
"2+2 = 4" was true:
1. Before I was born
2. Before humans formalized axioms and found a proof of it
3. Before there were humans
4. Before there was any conscious life in this universe
5. As soon as there were 4 physical things to count
6. Before the big bang / before there were 4 physical things
"2+2=4" is a tautology, true because of the meanings of the
terms involved. So its truth is not independent of the
formulation of the question and the definition of the terms
involved.
What about ExEyEz (x^3 + y^3 +z^3 = 33) ?
What about it? Not all syntactically correct formulae are either true
or false; some are undecidable.
A is undecidable means ~[]A & ~[]~A.
It does not mean ~A & ~~A.
That A (= ExEyEx(x^3 + y^3 + z^3)) is either true or false is a direct
consequence of the excluded middle. Either such numbers exists or they
don’t. If it is undecidable in PA, it means that PA cannot decide it,
but it does not mean that ZF or ZF+kappa could not decide it. And even
if they couldn’t, it is still does not mean that A is not true, or
false, which it is, certainly.
So you have no idea whether it is true or not, because you do not know
in which system the Turing program designed to search all possible
combinations of integers to find an (x,y,z) combination with the
required properties would halt, or in which systems the similar Turing
machine would not halt, and you could not prove halting or not in any
system. Then the problem becomes totally trivial, we could define a
system in which your A above is an axiom, or one in which ~A is an
axiom. In either case the question is trivially decidable, but not in
any useful sense -- such systems still do not produce the required
triple of integers
Unless and until you find some x,y,z that satisfy this relationship,
the statement is neither true nor false.
In intuitionist logic. I have made clear that I use classical logic
(indeed, in mathematical theology, we can expect many statement to be
undecidable by the finite creatures/theories). Same already in
theoretical computer science. The notion of totality is non
constructive, “halting” is non constructive, in fact all attribute of
programs can be shown to be necessarily non constructive. I recursion
theory, constructive is equivalent to limiting research in the
security zone of some RE subset of TOT. Machines have no right to
search for a number which might not exist.
Who gets to say what rights machine may or may not have? I can write a
program to search all possible integer combinations for a solution to
(x^3 + y^3 +z^3 = 33). But I cannot determine whether this program will
halt or not. You cannot claim that I do not have a right to write such a
program.
Unperformed experiments have no results!
Even in physics, this is doubtful and indeed contradict Einstein’s
physical realism.
Einsteinian realism is already contradicted by experiment. The rejection
of counterfactual definiteness is intrinsic to quantum mechanics, even
in MWI.
No problem, as we know it has to be tampered with the Mechanist
assumption.
You might assume this, but you cannot know it!
Bruce
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