<<written in the symbolic notation invented by Peano; only the introduction
and a few comments (such as the one I quoted in my last note) appear in
ordinary language. To read them, therefore, amounts to first decode long
strings of symbols - only to find out that their mathematical content does
not go much beyond what was said earlier, say by Pasch.>>
Secondly, with my plodding German, I wonder if the widely read works of
Baumann (Julius Baumann, "Die Lehren von Zeit, Raum und Mathematik," Berlin,
1868, -- the encyclopedia used by Frege in 1884), and of Pl"ucker (Julius
Pl"ucker, "Neue Geometrie des Raumes gegrundet auf die Betrachtung der geraden
Linie als Raumelemente," vI, ed., Felix Klein, Leipzig, 1868) contain anything
that points toward Pasch's argument. They fit between Reye (1868) and Pasch
(1882). 1868 was also the year that Helmholtz published his major article on
the foundations of geometry, very much the contrary of Pasch's ("Uber die
Thatsachen, die der Geometrie zum Grunde liegen" in "Nachr. KGW zu G"ottingen"
#15, June, 1868).
And third, I think a historical argument can be made with Felscher's statement
that:
<<Peano invented a good notation, but what he lacked was the conceptual
apparatus to put it to use with: the formal analysis of logical arguments by
their syntactical form, the setup of logical rules, expressing by purely
syntactical descriptions that one notational string be a consequence of one or
several other notational strings. Such apparatus, however, was present already
in form of Frege's "Begriffsschrift" of 1879/1884 which used a much less
convenient, two-dimensional form of notation.>>
I'd like to make a place in this story for Charles S. Peirce, who introduced
several symbols, including the crucial "-<" for set inclusion, together with
what Russell would call "the logic of relations" in his "Description of a
Notation for the Logic of Relatives from an Amplification of the Conceptions
of Boole's Calculus of Logic" published in Memoirs of the American Academy of
Arts & Sciences (n.s. 9, 1873) and for Ernst Schr"oder, who seems to have
borrowed Peirce's notation and improved on it, beginning in 1877 ("Der
Operationskreis des Logikkalkuls,"). Peano read both and profited from their
work when he was inventing his own notation; but he did not pick up on Frege
until afterwards, which is why the history of notation leaves "Begriffschrift"
in a backwater and why the history of symbolic logic has to double back to
Frege via Russell. In Felscher's words:
<<Still, it took until Russell's article in the American Journal of
Mathematics of 1908 and Whithead-Russell's monolithic treatise of 1910 , that
Frege's machine was married with Peano's notation, giving rise to mathematical
logic as we know it. Curiously enough, it was Hilbert, not having even
mentioned Peano in 1899 , who in his articles after 1920 introduced the
simplifications of the Peano-Russell notation which mathematical logic uses
today.>>
-Bill Everdell, Brooklyn