On 09 Jun 2015, at 02:37, LizR wrote:
On 9 June 2015 at 11:26, Bruce Kellett <bhkell...@optusnet.com.au>
wrote:
LizR wrote:
Reality isn't defined by what everyone agrees on. What makes ZFC (or
whatever) real, or not, is whether it kicks back. Is it something
that was invented, and could equally well have been invented
differently, or was it discovered as a result of following a chain
of logical reasoning from certain axioms?
Why do not those same arguments apply equally to arithmetic? What
axioms led to arithmetic? Could one have chosen different axioms?
The arguments do apply. The point is that once the axioms are
chosen, the results that follow are not a matter of choice.
Arithmetical truths appear to take the form "if A, then
(necessarily) B".
However, some of the elementary axioms (or even perhaps axions! :-)
do appear to be demonstrated by nature - certain numerical
quantities are (apparently) conserved in fundamental particle
interactions, quantum fluctuations can only occur in ways that
balance energy budgets, etc. So one could say that for anyone of a
materialist persuasion, the assumptions of elementary arithmetic
aren't unreasonable, at least (Bruno often mentions that comp only
assumes some very simple arithmetical axioms - the existence of
numbers and the correctness of addition and multiplication, I think)
So if you choose Peano arithmetic, then such-and-such follows, while
if you choose modular arithmetic, something else follows. The
"kicking back" part is simply the fact that the same result always
follows from a given set of assumptions. To put it a bit more
dramatically, an alien being in a different galaxy, or even in
another universe, would still get the same results. Nature is
telling us that given A, we always get B.
The difference is that for arithmetic (non modular arithmetic of the
natural numbers), although there are many different axioms systems
possible, either they have all the same theorems, or they are included
in each other (one theory being just more powerful than another), but
they all get the same theorems, when they get them. That is not true
for set theory, where the theories can overlap, but also have
different incompatible theorems.
For the comp TOE, we need only to assume a (Turing) universal theory:
we get the same physics, the same consciousness, etc. The kicking back
is done at the elementary finite combinatorial level. For set theory,
you need transfinite induction, which is philosophically much more
demanding.
Bruno
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