On 09 Jun 2015, at 07:40, Bruce Kellett wrote:
LizR wrote:
On 9 June 2015 at 11:26, Bruce Kellett <bhkell...@optusnet.com.au <mailto: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.
Yes, exactly. That is why I would say that arithmetic is invented as
a codification of our experience of the physical world. If we had
chosen a set of axioms that did not reproduce the results of simple
addition -- add two pebbles to the two already there, to give four
in total -- then we would have abandoned that set of axioms long
ago. Axiom systems are evaluated in terms of their utility, nothing
else. In more advanced mathematics, utility might be measured in
terms of simplicity and fruitfulness for further applications. But
in the beginning, as with arithmetic and simple geometry/
trigonometry and so on, utility is measured entirely in terms of the
applicability to the experienced physical world, and of the utility
of the system in helping us live in that world.
But that concerns the way human discovered arithmetic, not its
fundamental or not status.
Anyway, comp makes no sense if we have doubt about 2+2=4, or about the
less trivial fact that there are universal diophantine polynomials, or
that all natural numbers can be written as the sum of four squared
integers, 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.
Given a set of axioms and some agreed rules of inference, the same
results always follow, regardless of by whom or at what time the
application is made. This is not what is usually referred to as
"kicking back". Johnson did not apply some axioms and rules of
inference in answer to the idealists, he kicked a stone.
But people can kicked stone in dreams too.
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.
Nature doesn't particularly tell us that. Rigorous application of
the rules of inference to certain axioms tells us that. The physics
might, after all, be different in a different universe, but using
the same rules of inference on the same axioms will give the same
result, regardless of the local physical laws.
Yes. Then with comp, physics is the same for all universal machine,
and this can be proved in all (Turing complete) theories. Physics is
made theory independent, except for assuming at least one universal
system. Physics is very well grounded in arithmetic or Turing
equivalent. It is made more solid that the extraoplation that we can
do from observation. Of course comp might be wrong, and that is why it
is nice that it becomes testable.
Bruno
Bruce
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