The rationals form a ring with addition defined as
p/q + r/s = (ps + qr)/qs.
I'm used to the fact that this works for the addition of fractions as we
want to; for example, a basket with half of 10 oranges if dumped on a
basket with a third of 12 gives 9 oranges. So I know that
10/2 + 12/3 = (30+24)/6 = 54/6 = 9.
So if we think of adding oranges in baskets, we want the addition of the
rationals not to give us any surprise. It seems to me then that what I
want is to prove that the abstract system of the rationals (with respect
to addition) reflects what we want them to reflect.
So if I ask for a proof that all fractions of oranges in one basket, if
added to another basket with another fraction of oranges, will give me
what my adding in the real world would give, how would I go about
proving this?
I have a feeling that this is not possible because there is no guarantee
that the real world will ever work as some abstract system does.
On the other hand, I have mathematical induction in mind which would be
perhaps the strongest argument for proving that the definition of
addition will work as we expect.
Suppose I confirm that some base case does work such as the example
above. Then I could suppose that there is another quantity (let's
represent it by the quantity k) when added to another basket also works,
and then try to show that some quantity k+1 works as well.
I have never written any proof using mathematical induction for the
rational numbers, but I always thought it possible due to the fact that
the rationals are countable. If anyone has any thoughts on this matter,
I'd appreciate hearing about it. Thank you.
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