Bruno Marchal wrote:
On 13 Jun 2015, at 03:29, Bruce Kellett wrote:
LizR wrote:
On 12 June 2015 at 17:40, Bruce Kellett <[email protected]
<mailto:[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)).
That is the logicians' understanding of 'tautology': "a compound
proposition that is true for all truth-possibilities of its components
by virtue of its logical form." A more common understanding of
'tautology' is: "a proposition that is true because of the meaning of
the terms involved." For example, "a brother is a male sibling".
In the case of '2+2=4', this is not a compound proposition that is true
for all truth-possibilities of its components, because '2' and '4' do
not stand for propositions that can take on truth values -- we can not
say that '2' is 'true', or that '2' is false. So this cannot be a
logical tautology. However, we have to assign a meaning to the symbol
'2', or to the word 'two', and so on for '4', '+', and '='. Once we have
assigned these meanings, then the proposition is true by virtue of those
assigned meanings. The tautological nature of '2+2=4' has nothing to do
with the logical structure of the proposition -- it is entirely due to
the meanings of the terms involved.
Thus, while what you say about logic, and the axiomatic basis of
arithmetic, may be all very well, it has absolutely nothing to do with
my assertion that '2+2=4' is a tautology by virtue of the meanings of
the terms involved.
Bruce
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
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