[FOM] Peano (1912) on generalized functions?

[Michael Barany]

Vladik,

Naturally, the 1912 article you mention was the first place I looked.
I am certainly not an infallible speed-reader of Italian, so it is
possible I missed the phrase in question, but I did not see anything
looking like the quote in Schwartz's autobiography in that article or
in the handful on related themes Peano published over the few years in
either direction of 1912. The 1912 article does treat many themes one
would retrospectively associate with Schwartz's quote.

Since a few have asked about this off-list, I can spell out what I
currently suspect are the most likely explanations for the quote:
1) Schwartz was better at reading Italian than I am (true, as far as I
know) and the quote is hidden in a usage I didn't pick up (less
likely, I hope).
2) The quote is contained in some correspondence from around the time
of that 1912 article, possibly related to the article itself.
3) Peano said or wrote something like that quote elsewhere and because
of the thematic relevance to the 1912 article Schwartz attributed it
to that.
4) The quote is from someone else, and Schwartz mis-remembered it as
coming from Peano.
5) The quote is the product of a rumor mill, without an identifiable
origin but leaving a strong enough impression that Schwartz could
assume his rumor-inflected recollection was accurate without checking
it up for his manuscript.

As part of my broader project, which has required me to fact-check a
fair number of claims from Schwartz's autobiography, I've encountered
situations resembling each of those 5 (with #1 here being a case of a
suspicious memory of an obscure document turning out to have been
accurate).

At the start my best guess was #2 or #3, which are both cases where I
think an FOM query would have had a high probability of success (as
with #1). Since it doesn't seem like those on the list recognize the
quote however, that points me more toward #4 and #5.

Thanks to everyone for your help on this, and I'll try not to clog
your inboxes further on this bit of philosophy-of-math-themed
philology.

Sincerely,

Michael
Princeton University
Program in History of Science
http://mbarany.com

> Date: Mon, 24 Nov 2014 23:16:05 +0000
> From: "Kreinovich, Vladik" <vladik at utep.edu>
> To: Foundations of Mathematics <fom at cs.nyu.edu>
> Subject: Re: [FOM] Peano (1912) on generalized functions?
> Message-ID:
>         <49A1861CB3A6E84A95F43B695D2B5C381A2A796D at ITDSRVMAIL011.utep.edu>
> Content-Type: text/plain; charset="utf-8"
>
> Most probably, it is from Peano, G. 1912 "Derivata e differenziale," Atti Accad. sci. Torino 48: 47?69 since this is a paper he wrote in 1912 about difficulties of diffentiation
>
> From: fom-bounces at cs.nyu.edu [mailto:fom-bounces at cs.nyu.edu] On Behalf Of Michael Barany
> Sent: Monday, November 24, 2014 8:12 AM
> To: Foundations of Mathematics
> Subject: [FOM] Peano (1912) on generalized functions?
>
> Dear FOMers,
> In Laurent Schwartz's autobiography, he writes the following:
> 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."
> I haven't been able to find the source of this quote in articles circa 1912 in Peano's collected works, and Schwartz does occasionally take liberties with these sorts of attributions, but I was hoping someone on this list might recognize the quote or have a quick way of knowing where it's from, if it is indeed genuine.
> Many thanks,
>
> Michael
> Princeton University
> Program in History of Science
> http://mbarany.com