[FOM] The unbearable ghastliness of 0/0

[katzmik at macs.biu.ac.il]

Dear Martin,

Thanks for this entertaining post.  An audience of logicians may not
find it baffling that a genius of Leibniz's caliber repeatedly wrote
in his work, both published and unpublished (response to Nieuwentijt,
Cum Prodiisset, and others) that he is working with a (binary)
relation more general than equality.

Thus Leibniz realized the difference between "equality on the nose"
and "equality up to".  Leibniz had several different pieces of
notation for equality, but he did not distinguish between different
types of equality by means of notation.

Nonetheless Leibniz's writings indicate that he had an answer to
Berkeley's logical criticism, in terms of Leibniz's generalized
relation of equality.  To follow up on your example, the replacement
of 2x+dx by 2x does not amount to "setting dx equal to zero" but
rather an application of Leibniz's transcendental law of homogeneity
which, in modern terminology, allows one to drop higher order terms.
The relation between 2x+dx and 2x is not that of equality on the nose
but rather a more general relation.  The principle is procedurally
somewhat similar (though not identical) to the standard part (shadow)
available in a hyperreal framework.

This is only one of the many aspects of Leibniz's theoretical strategy
for dealing with infinitesimals, which were analyzed in our 2012 study
in "The Notices of the AMS"
http://www.ams.org/notices/201211/rtx121101550p.pdf and the 2013 study
in the philosophy journal "Erkenntnis"
http://dx.doi.org/10.1007/s10670-012-9370-y and the 2014 study in the
specialized journal "Studia Leibnitiana"
http://arxiv.org/abs/1304.2137 and other articles (see
http://u.cs.biu.ac.il/~katzmik/infinitesimals.html)

If Hegel and Marx were confused about this, it is yet no reason for
the informed mathematical public to continue giving Berkeley's flawed
critique far more credit than it deserves.

MK

On Sun, September 6, 2015 03:19, Martin Davis wrote:
> 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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