On 13 Jun 2015, at 06:40, meekerdb wrote:
On 6/12/2015 6:29 PM, Bruce Kellett wrote:
LizR wrote:
On 12 June 2015 at 17:40, Bruce Kellett <[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.
Bruce
It is commonly thought to be discovered and so to be "ought there"
independent of human beings or any cognition. But when considered
more carefully what was discovered is that one can group pairs to
things together (at least in imagination) and have four things. So
two fathers grouped with two sons is four people. Except when it's
three people. So we said OK we'll *define* units to be things that
obey the rules that 2+2=4. Then we discovered that these rules
implied a lot of things we hadn't thought of. But they aren't "out
there", they're in our language.
An expression in a language is grammatically correct, or not.
Here we do have a semantic, a notion of truth. That the arithmetical
truth is not tautological is reflected in the fact that we need non
logical axiom (like x + 0 = x) and this is amplified by the fact that
most arithmetical truth are not provable by any theory, despite we do
have the intuition that it is either true or false (an intuition that
we lack for richer theory having axiom of infinity).
Bruno
Brent
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