[FOM] Fwd: Peano (1912) on generalized functions?

Martin Davis martin at eipye.com
Sat Dec 20 15:20:21 EST 2014

---------- Forwarded message ----------
From: Clark <terminal08 at verizon.net>
Date: Sat, Dec 20, 2014 at 10:27 AM
Subject: Peano (1912) on generalized functions?
To: davism at cs.nyu.edu

Note to moderator:

Dear Professor Davis,

I am not a subscriber to the FOM list and wish to post the information of
interest to the FOM list  which follows this note.

Best regards

Clark McGranery
I would like to thank Gabriele H. Greco for finding an answer and Michael
Barany for asking the question.

Professor Greco cites :
G. Peano. Resto nelle formule di quadratura espresso con un integrale
      Atti della Reale Accademia dei Lincei, 22 (1913) pp. 562-569

and in particular the text on the final page in which he introduces a
generalized impulsive function with special mention of the previous work of
Maxwell, Heaviside, and Giorgi, which many years later became known as
Dirac's delta function, and translates :

" -  As in algebra . . . the rational numbers
 are not numbers considered before, but they
 belong  to a wider category than the integers,
 so the “impulsive function” is not a function, as those which are
 defined in analysis, but it belongs to a wider category of entities. "

In the preceding paragraph, Peano discusses certain difficulties in

If no citation more closely corresponding to Schwartz's quotation is
discovered, it is plausible that this passage is the root of Schwartz
writing, perhaps slightly mistaken in the year and reals extending
rationals  in place of rationals extending integers:

The mathematician Peano wrote in 1912 on the difficulties of
differentiation: "I am sure that something must be found. There must exist a
notion of generalized functions which are to functions what the real numbers
are to the rationals." This was a marvelous intuition, and it arose long
1944. But the mathematical knowledge of the time did not make it possible
Peano to find the generalized functions, or even to conceive of them; at
time it would have been a superhuman task.

It made a deep impression on Schwartz's thought as he repeats this claim
multiple times on three further pages of the same chapter.
This contribution has not, so far as I am aware, been recognized elsewhere,
including by Schwartz's colleague at the College de France, Jean Dieudonné,
in his history of Functional Analysis.
In this respect, it can be added to a lengthy list of unrecognised results
and ideas of Peano, for which there is a  readily accessible article by
Gabriele H. Greco and  Szymon Dolecki.
One can hope that, at some point, the mathematical community will catch up
to Laurent Schwartz in appreciation of another of Peano's remarkable ideas.

Best regards

Clark McGranery
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