There are some physical operations we perform on bags of objects --
well, on groups of whatever kind, but a "bag" is a nice tangible thing
to think about. One such is "bag-count" -- which is a procedure that
maps bags of objects onto the nonnegative integers; let me show you
with this bag of oranges, I just count them out, one... two... three...
Another is "bag-union" -- you see, here I have two bags of oranges,
and another bag which is empty of oranges (its count is zero), and
what I do is put each of the non-empty bags into the one that started
empty.
And now we can take data, and when we do, we find that the non-negative
integers appear to provide a useful mathematical description of the
properties of bags of objects, and in particular, that if A and B are
bags of objects, we have:
bag-count( bag-union( A, B ) ) = bag-count( A ) + bag-count( B ).
And incidentally, it doesn't matter what order you take the bags in:
bag-count( bag-union( A, B ) ) = bag-count( bag-union( B, A ) )
The way I think about it is, that these empirical properties of
bag-count and bag-union -- long since elevated to the status of
physical laws (except for things like raindrops and bunny rabbits),
are what make the counting numbers, and perhaps most of analysis as we
know it, of interest to persons who deal with the physical world. The
experiments with bag-count and bag-union are perhaps the fundamental
experiments in mathematical physics. If the count of the union were
not the sum of the counts, but some other function thereof, then
although mathematicians might have thought up the Peano postulates and
developed analysis in the abstract, "one, two, three..." might not be
something that every toddler learns.
One approach to a "1+1=3" world might be substituting some different
function of bag_count( A ) and bag_count( B ), than their sum, in the
above.
It is interesting that folklore and fantasy provide several examples
of instances of failure of this law, notably the never-ending penny.
-- Jay Freeman, First Extropian Squirrel