[FOM] The unbearable ghastliness of 0/0

[Martin Davis]

Consider the calculation:

                                (x^2-a^2)/(x-a) = (x-a)(x+a)/(x-a) = x+a

As in the case of what I wrote about EFQ, we can consider the case x=a. On
the left we get the meaningless 0/0. On the right 2a. Nevertheless, this is
useful: it is an example of a basic operation in calculus, calculation of
derivatives. In this case, the calculation shows that the derivative of x^2
is 2x.

The contemporary textbooks say that we never set x=a; we compute the limit
as x -> a. But the savvy student still sees that what you are doing amounts
to setting x=a.

This situation gave rise to philosophical discussions over the centuries.
Bishop Berkeley's famous tract "The Analyst" pointed out, in effect, that
the calculation was valid only for x different from a, so by setting x=a
after the computation, one was making an illegitimate shift in hypothesis.
The good bishop went on to conclude that mathematicians using this kind of
fallacious reasoning had no standing to criticize theologians for their
reasoning.

Hegel, in his misnamed "Science of Logic" devoted over 100 pages to
discussing this matter. Karl Marx in his "Capital" noting that his economic
analysis led to conclusions contradicting "all experience based on
appearance" went on to say: "For the solution of this apparent
contradiction, many intermediate terms are as yet wanted, as from the
standpoint of elementary algebra many intermediate terms are wanted to
understand that
0/0 may represent an actual quantity."

The point of this story (in addition to the fun I have telling it) is that
dealing with the case where two items are identified can be both useful and
anomalous

Martin.
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