There seems to be certain disagreement as to whether Hilbert was
sufficiently acquainted with contemporary Italian mathematics before his
"Grundlagen der Geometrie" (GG) was published in 1899. It has been noticed
time and again that the great Peano is absent in that 'ars magna'.
In a recent article [1], dedicated to David Fowler on the occasion of his
60th birthday, Jeremy Gray remarks:
In his thorough study of Hilbert's route to his _Grundlagen_
Toepell [2] shows that although Hilbert did not read Italian
easily, he listed Peano's book on the Grassmann calculus in
its German translation [3] as one of the books on the axioms
of geometry. Toepell therefore disagrees with Morris Kline's
remark that Hilbert "did not know the work of the Italians"
(quoted from Kline [4], p. 1010). One might add that Italian
geometers visited Go"tingen, where Hilbert became a professor
in 1895. Significantly, Hilbert did not refer to Peano's much
more axiomatic work [5].
In a note of June 30th, Robert Tragesser referred to the following:
"(4c) When did -- and why -- this break with the Euclidean style
of rigor (good definitions first) occur? Clearly (?) Hilbert's
axiomatic thinking" is not Euclidean, but, rather, sets the
issue of definition aside in favor of special axioms."
The break with the Euclidean style seems to be due to internal pressures
within mathematics, basically the flowering of projective geometry and the
discovery of the non-Euclidean geometries -- pressures that Raymond L. Wilder
has characterised as the 'hereditary stress' [7, p.170]).
However, we should announce more regularly that we rarely know when, why
or by who a breakthrough takes place.
As Ken O. May once reminded us "Sometimes documents show that a mathematician
(usually one whose fame has preserved his papers) anticipated an idea that
was later published, but we have no way of knowing the unrecorded thoughts
of others. Such cases reveal much of interest. They indicate that an idea
was thinkable in the intellectual context prior to its publication, that
before an idea is recorded it is usually 'in the air' more or less as it
later appears, and so on. But they also establish the indeterminacy of the
concept of 'first person to think of it'. Hence, the most that we can hope
for is to find the first extant record of an idea, an apparently solvable
problem because of the finiteness of the literature".
Having recalled this, I would hazard a name: Moritz Pasch. Most historians
and mathematicians would concur in this name, I presume. Hans Freudenthal's
characterisation as "the father of rigor in Geometry is Pasch" is well-known.
And, anyone that states:
"In fact, provided the geometry is to be truly deductive, the
process of inference must be entirely independent of the
_meaning_ of geometrical terms, just as it must be independent
of the figures" (Pasch [6])
is somehow introducing the alpha version of "One must be able to say at all
times -- instead of points, straight lines, and planes -- tables, chairs,
and beer mugs".
I am grateful to Walter Felscher for bringing to my attention -- a few months
ago -- of an 18-page-dissertation (still unpublished?) of Walter S. Contro,
which was written under Matthias Schramm's supervision, thirty years ago!
An interesting issue noticed by Contro while researching for his thesis is
the fact that it was not only Fano 1892, as naively observed Freudenthal,
but already Peano 1889 [5] who 'cuts the umbilical chord between geometry
and reality' setting up geometry as an axiomatic system in the manner later
made popular by Hilbert. I cannot further comment on Contro's dissertation
since I have never read it.
I was aware of Freudenthal's oversight by my reading of a good article of
Huber C. Kennedy in the _Monthly_ [10]. In February 1972, Kennedy commented:
"Somewhat surprisingly Freudenthal overlooks Peano's monograph of
1889, even though it is cited in Fano's article, perhaps because
Fano says that Peano's work was based on that of Pasch. Peano's
work was indeed based on his reading of Pasch, but there are
important innovations, and one of them is the explicit statement
of the modern attitude toward the undefined terms of an axiomatic
mathematical system."
At all events, Hans Freudenthal ignores (?) Peano's _principii_, and
Kennedy's explanation does not sound convincing to me.
'Ignorance' seems to come and go, and, as Jeremy Gray notes in his novel
article [1], "A measure of the degree to which Italian developments were
not read may be Poincare's ignorance of them, to which Freudenthal drew
attention [11, pp. 620-621]".
It might be worth pointing out here and now that there is a fine essay
of Walter Contro [8] which traces the influence of Pasch's work. The title
of this paper, "From Pasch to Hilbert" (in German), is a bit misleading
since Contro by far gives a detailed analysis on Pasch's programme only.
References:
[1] Gray, Jeremy: "The Foundations of Geometry and the History of Geometry",
_The Mathematical Intelligencer_, vol. 20, no. 2, pp. 54-59, 1998.
[2] Toepell, M.-M: "Ueber die Entstehung von David Hilberts _Grundlagen
der Geometrie_", Go"tingen: Vandenhoeck and Ruprecht, 1986.
[3] Peano, Giuseppe: "Die Grundzu"ge des geometrischen Calculs", Autorisirte
deutsche Ausgabe von Adolf Schepp, Leipzig: Teubner, 1891.
[4] Kline, Morris: "Mathematical Thought from Ancient to Modern Times",
Oxford: Oxford University Press, 1972.
[5] Peano, Giuseppe: "I principii di Geometria logicamente esposti",
Torino: Fratelli Bocca, 1889.
[6] Pasch, M.: "Vorlesungen u"ber neuere Geometrie", Leipzig: Teubner, 1882.
[7] Wilder, Raymond L.: "Evolution of Mathematical Concepts", New York:
Wiley, 1968.
[8] Contro, Walter S.: "Von Pasch zu Hilbert", Archive for History of Exact
Sciences, vol. 15, no. 3, 283-295, 1976.
[9] Contro, Walter S.: "Die Entwicklung der Geometrie zum hypothetisch-
deduktiven System", Frankfurt a. M.: Hochschulschrift, Univ., Diss., 1968.
[10] Kennedy, H.C.: "The Origins of Modern Axiomatics: Pasch to Peano",
_The American Mathematical Monthly_, vol. 79, pp. 133-136, 1972.
[11] Freudenthal, Hans: "The Main Trends in the Foundations of Geometry
in the 19th Century" (pp. 613-621), in "Logic, Methodology and Philosophy
of Science", Proceedings of the 1960 International Congress (edited by
E. Nagel, P. Suppes, and A. Tarski), Stanford, CA: Stanford University
Press, 1962.
PS: In 1985, Walter S. Contro wrote "Eine schwedische Axiomatik der
Geometrie vor Hilbert: Torsten Brodens Om geometriens principer von 1890"
["A Swedish Axiomatics of Geometry before Hilbert: Torsten Broden's Om
geometriens principer from 1890"] (pp. 625-636) in "Mathemata", Festschrift
fu"r Helmuth Gericke, edited by Menso Folkerts and Uta Lindgren, Boethius -
Texte und Abhandlungen zur Geschichte der Exakten Wissenschaften [Texts and
Essays on the History of Exact Sciences], XII. Wiesbaden: Franz Steiner
Verlag GmbH, 1985. ISBN: 3-515-04324-1.
Comments, criticisms, and further discussion would be most welcome, and,
I believe, the *raison d'e^tre* of this forum!
Greetings from Montevideo,
Julio Gonzalez Cabillon
On Mon, 13 Jul 1998, Walter Felscher <walter.felscher@uni-tuebingen.de> wrote:
| ...
| It seems that implicit definitions were first considered by J.D.Gergonne in
| his "Essai sur la the/orie des de/finitions", Ann. de Math. 9 (1818) 1-35,
| where he started from the example that a system of equations may uniquely
| determine its solution. In this spirit, however, Gergonne then restricted
| himself throughout to the case of objects uniquely determined by an
| implicit definition.
|
| As noticed in the 1968 Frankfurt dissertation of W.Contro, the first
| author to admit implicit definitions with non-unique solutions seems to
| have been Peano with his axiomatization of geometry in "I Principii di
| Geometria Logicamente Esposti", Torino 1889, where he writes
|
| Si ha cosi una categorie di enti, chiamati punti. Questi enti non
| sono definiti. Inoltre, dati tre punti, si considera una relazione
| fra essi, indicata colla scrittura c(e)ab , la quale relazione
| non e parimenti definita. Il lettore puo intendere col segno 1 una
| categoria qualunque di enti, e con c(e)ab una relazione qualunque
| fra tre enti di quella categoria; avranno sempre valore tutte le
| definizioni che seguono (para 2), e sussisteranno tutte le
| proposizioni del para 3. Dipendentemente dal significato attribuito
| ai segni non definiti 1 e c(e)ab , potranno essere saddisfatti,
| oppure no, gli assiomi. Si un certo gruppo di assiomi a verificato,
| saranno pure vere tutte le proposizioni che si deducono, non essendo
| queste proposizioni che trasformazioni di quegli assiomi e delle
| definizioni.
|
| As noticed already by the late Professor Freudenthal, similar
| formulations were used by G.Fano in "Sui postulati fondamentali della
| geometria projettiva in uno spazio lineare a un numero qualunque di
| dimensioni", Gior. Mat. (Battaglini) 30 (1892) 106-132 :
|
| A base del nostro studio noi mettano una varieta qualsiasi di enti
| di qualunque natura; enti che chiameremo, per brevita, punti,
| independemente pero, ben intesto, dalla lora stessa natura.
|
| Peano wrote ten years, Fano seven years before Hilbert.