On 06 Aug 2011, at 22:00, Stephen P. King wrote to Craig Weinberg:
Natural numbers are an invention
of an entity that thinks,
Bruno: The existence of numbers, with the laws of addition and
multiplication, entails the existence of universal numbers. They can
introspect themselves and discover, for themselves, the numbers and
their laws. They can even discover themselves in there, and this
variety of levels.
Craig: I don't think that you can say that they do that without a
mathematician being there to watch and understand, or a silicon chip
to prove it. What numbers help you discover is the logic behind sense
and the sense behind logic, but they don't necessarily reveal a logic
independent of sense. (That may be my main point right there).
Stephen: I think that you are both wrong! Numbers as independent
primitives can do nothing without the schemata of ordering and
relations that even allows the notion of "introspection" and
"discovery" to be meaningful.
I said numbers *with the laws of addition and multiplication*. You
can define the order x < y by Ez(not(z = 0) & (x + z = y)).
In fact with addition and multiplication, you have the dreams and
their coherence property leading to physics.
OTOH, requiring the physical presence of a mathematician is missing
the point that the relationships upon which 'introspection' and
'discovery' supervene are not limited some just some particular
kinds of things. You are missing the true part of functionalism.
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