[FOM] The unbearable ghastliness of 0/0

[jbell]

Concerning Martin Davis's most interesting post on the unbearable
ghastliness of 0/0:

The issue of the "value" of 0/0 arose in the 18th century with the use of
infinitesimals.

No less a mathematical authority than Euler held that infinitesimals were
formal  zeros so that a derivative, as a quotient of infinitesimals, could 
be
identified with the quantity 0/0.  Donning his formalist hat, as he so often
did, Euler asserted  that  the differential calculus was, at bottom, a
procedure for pinning  down the value of  0/0, which, without such
constraints, would be a totally arbitrary quantity.

As for Hegel, The Science of Logic (misnamed, certainly, in the traditional
sense, but then Hegel always had a quirky way with terminology) contains an
extensive discussion of the ideas underlying the calculus. Like Berkeley, d’Alembert
and Lagrange, Hegel was critical of the use mathematicians had made of
infinitesimals and differentials. But far from rejecting the infinitesimal,
Hegel was concerned to assign it a proper location within his philosophical
scheme, whose reigning principle was the division of reality into the triad
of Being, Nothing, and Becoming. For Cavalieri infinitesimals possessed
Being, and for Euler they were Nothing, but for Hegel they fell under the
category of Becoming. In Hegel’s subsequent review of how the infinitesimal
has been conceived by mathematicians of the past, those who regarded
infinitesimals as fixed quantities receive short shrift, while those who saw
infinitesimals in terms of the limit concept (which in Hegel’s eyes fell
under the appropriate category of Becoming) are praised. Thus, for example,
Newton is praised for his explanation of fluxions (in the Principia, see
....) not in terms of indivisibles, but in terms of “vanishing divisibilia”,
and, further, “not [in terms of] sums and ratios of determinate parts, but
[in terms of] the limits (limites) of the sums and ratios.”  Newton’s
conception of generative or variable magnitudes also receives Hegel’s
endorsement. The use of fixed infinitesimals, on the other hand, Hegel
deplores. Nor does Euler’s view of infinitesimals as formal zeros fare much
better.

Hegel goes on to discuss some of the methods that mathematicians have
employed to resolve the conceptual difficulties caused by the use of
infinitesimals. He pays particular attention to Lagrange’s attempt to
eliminate infinitesimals from the calculus through the use of Taylor
expansions. Hegel considers that the Taylor expansion of a function “must
not only be regarded as a sum, but as qualitative moments of a conceptual
whole.”  That being the case, he says, the basic calculus procedure of
omitting from the Taylor series terms with higher powers has “a significance
wholly different from that which belongs to their omission on the ground of
relative smallness”.

Hegel is often regarded a philosopher who did not take mathematics very
seriously. The fact that he devoted a substantial portion of The Science of
Logic to the infinitesimal calculus speaks to the contrary .

Given  Hegel's concern with the foundations of the calculus, it  is hardly
surprising that Marx, his greatest disciple, should devote considerable
attention to the issue.

The fact is that,  before Cantor succeeded in shifting  the attention of
philosophers to the infinite, it was the infinitesimal that captured their
imaginations. But then, Cantor idolized  the infinite, and execrated the
infinitesimal!


---  John Bell



Professor John L Bell, FRSC
Department of Philosophy
Western University
London, Ontario
Canada

http://publish.uwo.ca/~jbell/



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Message: 5
Date: Sat, 5 Sep 2015 17:19:27 -0700
From: Martin Davis <martin at eipye.com>
To: fom at cs.nyu.edu
Subject: [FOM] The unbearable ghastliness of 0/0
Message-ID:
<CA+cpueL7f1NsAV_vMK8BP=YNikdZDxeT+nRFHwHH5T9q1pVZ8g at mail.gmail.com>
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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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End of FOM Digest, Vol 153, Issue 21
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