In article <slrng14ggs.813.dbast0s@[EMAIL PROTECTED]
>,
"Daniel C. Bastos" <dbast0s@[EMAIL PROTECTED]
> wrote:
> 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.
You don't need to use mathematical induction. Consider your 10/2 + 12/3
example: Let your unit of accounting be 1/(2*3) of an orange. In terms
of these units, you're adding 10*(2*3)/2 + 12*(2*3)/3. The denominators
cancel in both, resulting in an integer addition (which I assume you
find non-problematic in its correspondence with the "real world").
Multiply the integer result by the unit's value of 1/(2*3), giving your
result in terms of oranges.
Note this generalises to the p/q + r/s = (ps + qr)/qs formula.
> 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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