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.
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.
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.
Bruce
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