On 13 Jun 2015, at 03:29, Bruce Kellett wrote:
LizR wrote:
On 12 June 2015 at 17:40, Bruce Kellett
<[email protected] Arithmetic is, after all, only an
axiomatic system. We can make up
an indefinite number of axiomatic systems whose theorems are every
bit as 'independent of us' as those of arithmetic. Are these
also to
be accepted as 'really real!'? Standard arithmetic is only
important
to us because it is useful in the physical world. It is invented,
not fundamental.
So you say, and you may be right. Or you may not. The question is
whether 2+2=4 independently of human beings (and aliens who may
have invented, or discovered as the case may be, arithmetic).
It may well be independent of humans or other (alien) beings, but it
has no meaning until you have defined what the symbols '2','4','+',
and '=' mean. Then it is a tautology.
I can be OK, but logicians (pure or applied) reserve tautology for the
purely logical formula (like:
(p & q)->q, or A(t) -> ExA(x)).
In propositional logic, worlds can be defined by the assignment or
truth value (true, false), and a tautology is something true in all
worlds.
Then we can add non-logical axioms, introducing some functional
constant, like 0 + x = x..
We cannot define what are numbers, but we can agree on some axioms.
In mathematics the word number has obviously many different, yet
related, meaning. In high school we learn that there the natural
numbers, and that from them, by (computable) equivalence class we get
the integers, and the rational numbers. Using topology (limit) we get
the real number, and that all this extends in the plane (complex
numbers), then in the fourth dimension (the quaternion, so useful to
handle relative 3d rotations, and then in the eight dimension: the
octonion).
Set theorist have then axioms leading to the transfinite numbers, and
then the logicians (but in fact everyone including nature) have used
the intensional properties of natural number, where not only 17 is
prime, but 17 get properties like being the code for some other numbers.
Depending on which numbers we want to talk about, we use this or that
theory. I use the natural numbers, and it is only asked if you agree
with the following axioms. For all numbers x and y we assume
0 ≠ s(x)
s(x) = s(y) -> x = y
x = 0 v Ey(x = s(y))
x+0 = x
x+s(y) = s(x+y)
x*0=0
x*s(y)=(x*y)+x
That is Robinson Arithmetic. It is basically Peano Arithmetic without
the induction axioms.
Then, the "easy", but still quite tedious thing consists in defining,
in that theory the observers.
I define the observers, roughly, by Peano arithmetic. That is, a
believer in the axiom above, who believes also the infinitely many
induction axioms:
(F(0) & Ax(F(x) -> F(s(x))) -> AxF(x),
with F(x) being a formula in the arithmetical language (with "0, s, +,
*), and the logical symbols as said above.
This can be done by the Gödel technic of arithmetization of meta-
arithmetic.
First order logic have rather clear mathematical semantic, and they
inherit from calssical propositional calculus the notion of
completeness, so a theorem is true in all models (mathematical
structure satisfying formula) and what is true in all models is a
theorem in the theory.
Now, it is the PA (emulated by the "ontogical RA") that I "interview
about how they see and make sense of what is there.
Since Gödel 1931 a lot of progress have been made, so that it is
relatively easy to get the formulation of the problem, notably in the
form of intensional variants of Gödel beweisbar predicate, which
incarnate the explanation of the functioning of PA in the language
that PA can understand.
By a theorem of Solovay, the propositional logic of correct platonists
machine is axiomatized by a modal logic G, for the part provable by
the machine, and by G*, for the true part, which by incompleteness
extends properly the provable part. Incompleteness also provides sense
to the distinction between provable("2+2=5") and "provable("2+2=5") &
2 + 2 = 5, and other nuances making us able to ask the main question,
and to isolate the "proximity spaces" and the "orthogonal realities"
to see if we got eventually the measure needed for computationalism
making sense. Not unlike some parts of physics we are confronted to
infinities, perhaps too many, but that remains to be seen, and the
first simple discovery shows some sign of the existence of a measure,
in the form of three quantizations of the sigma_1 arithmetical formula.
Bruno
Bruce
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