please, permit me to add a few more remarks to our topic.
Moritz Pasch's "Vorlesungen "uber neuere Geometrie", Leipzig
1882, appeared after the admirable geometrical treatises by von
Staudt 1856/57 and by Reye 1866/68 . From the inner-mathematical
point of view, one of the essential new discoveries presented in
Pasch's "Vorlesungen" was the axiomatization of the betweenness
relation (or order, acknowledged in Hilbert's axiom group II) to
which he succeeded to reduce notions such as rays, half-planes
etc. which previously has been employed only intuitively. In so
far, Pasch obtained the tools which permitted him to develop
geometry purely deductively from his axioms.
But with the last remark we leave technical mathematics and come
to questions of methodology. There actually are two of them:
(1) mathematics, and in particular geometry, presented as a
deductive science, and
(2) the basic notions of a deductive science viewed as being
defined implicitly through their axioms.
Pasch, in his "Vorlesungen", commented extensively on (1), so on
p.98 (|* italics *|)
Es muss in der That, wenn anders die Geometrie wirklich
deductiv sein soll, der Process des Folgerns "uberall
unabh"angig sein vom |*Sinn*| der geometrischen Begriffe, wie
er auch unabh"angig sein muss von den Figuren; nur die in den
benutzten S"atzen, beziehungsweise Definitionen
niedergelegten |*Beziehungen*| zwischen den geometrischen
Begriffen d"urfen in Betracht kommen. W"ahrend der Deduction
ist es zwar statthaft und n"utzlich, aber |* keineswegs
n"othig *| , an die Bedeutung der auftretenden geometrischen
Begriffe zu denken; so dass geradezu, wenn dies n"othig wird,
daraus die L"uckenhaftigkeit der Deduction und (wenn sich die
L"ucke nicht durch Ab"anderung des Raisonnements beseitigen
l"asst) die Unzul"anglichkeit der als Beweismittel
vorangeschickten S"atze hervorgeht. Hat man aber ein Theorem
aus einer Gruppe von S"atzen - wir wollen sie |*Stamms"atze*|
nennen - in voller Strenge deducirt, so besitzt die
Herleitung einen "uber den urspr"unglichen Zweck
hinausgehenden Werth. Denn wenn aus den Stamms"atzen dadurch,
dass man die darin verkn"upften geometrischen Begriffe mit
gewissen anderen vertauscht, wieder richtige S"atze
hervorgehen, so ist in dem Theorem die entsprechende
Vertauschung zul"assig; man erh"alt so, ohne die Deduction zu
wiederholen, einen (im Allgemeinen) neuen Satz, eine
Folgerung aus den ver"anderten Stamms"atzen.
[As a matter of fact, should geometry be really deductive, then
the process of inference should everywhere be independent of
the |* meaning *| of the geometrical notions, as it also
should be independent of the drawings; only the |* relationships*|
between the geometrical notions may be considered, as they
are laid down in the sentences or definitions made use of.
During a deduction it is permitted and useful to keep in
mind the meaning of the geometrical notions appearing there,
but it is |* by no means necessary *|; actually, should
consideration of the meaning become necessary, this will
show the deduction to be defective and (should it not be
possible to remove the gap by a change of argument) show
the insufficiency, for this proof, of the statements
established previously. However, once a theorem has been
deduced rigorously from a group of statements - let them be
called the |* root-statements *| - then the deduction
obtains a value going beyond its original purpose. Because
if the root-statements result in correct statements again if
the geometric notions connected in them are exchanged by
certain others, then the corresponding exchange in the
theorem is admissible as well; without repeating the
deduction, we so obtain a (usually new) theorem, a
consequence of the transformed root-sentences. ]
as quoted already by M. Cabillon. Clearly, in the last sentence
Pasch thinks of the duality principle in projective geometry.
Footnote 1 : Reading Pasch's above explanations, today's
mathematician will find their content to be rather obvious:
mathematical proofs as logical deductions from axioms and
postulates. This, after all, had been already the aim of
Eudoxos' miraculous theory of proportions (the overlooked
axioms of Archimedes and of the fourth proportional granted)
to which later generations paid lip service, recanting their
assurances that things always could be straightened "in the
manner of the Ancients". But then recall the situation of
1882 : it was in this year that there also appeared the great
Paul du Bois-Reymond's "Allgemeine Functionentheorie", still
debating a 'geometrical' versus an 'arithmetical' foundation
of the continuum. And look at today's professors teaching
elementary analysis: each of them will readily profess his
belief that mathematics can be based on set theory - and most
of them will be quite unaware that they use AC already when
they study limits with help of sequences. [End of Footnote 1 ]
Footnote 2 : Pasch's observation on the value of a deduction
beyond its original purpose, provided an exchange of basic
notions preserves the validity of the axioms ruling their
relationships, was formulated similarly already by George
Boole in the opening sentences of his "The mathematical
Analysis of Logic", London 1847 :
Those who are acquainted with the present state of the
theory of Symbolical Algebra, are aware that the validity
of the process of analysis does not depend upon the
interpretation of the symbols which are employed, but
soleley upon the laws of their combination. Every system
of interpretation which does not affect the truth of the
relations supposed, is equally admissible, and it is thus
that the same process may, under one scheme of
interpretation, represent the solution of a question on
the properties of number, under another, that of a
geometrical problem, and under a thrid, that of a problem
of dynamics or optics. This principle is indeed of
fundamental importance ... [End of Footnote 2 ]
But Pasch did not desire to study a geometrically inspired
deductive machinery as a formal structure; he studied geometry
based on a semantics of geometrical truths (as expressed in
"correct" statements):
Because if the root-statements result in correct statements again ...
Consequently, Pasch did not take up the methodological question
(2), but rather wrote about basic notions and axioms on pp. 16-17
|* Die Grundbegriffe sind nicht definirt worden *|; keine
Erkl"arung ist im Stande, dasjenige Mittel zu ersetzen,
welches allein das Verst"andnis jener einfachen, auf andere
nicht zur"uckf"uhrbaren Begriffe erschliesst, n"amlich den
Hinweis auf geeignete Naturobjekte. ... Die Mathematik stellt
Relationen zwischen den mathematischen Begriffen auf ... ;
die (ausser den Definitionen der abgeleiteten Begriffe) zur
Beweisf"uhrung nothwendigen Erkenntnisse bilden selbst einen
Theil der aufzustellenden Relationen. Nach Ausscheidung der
auf Beweise gest"utzten S"atze, der Lehrs"atze, bleibt eine
Gruppe von S"atzen zur"uck, aus denen allen "ubrigen sich
folgern lassen, die Grunds"atze; diese sind unmittelbar auf
Beobachtungen gegr"undet ...
[|* The basic notions have not been defined *|; no
explanation is able to replace the one tool which alone
opens the understanding of those simple notions, not
reducible to others: the reference to appropriate objects
of nature. ... Mathematics sets up relationships between
mathematical notions ... ; the insights, required to carry
out a deduction (except the definitions of derived concepts),
themselves form a part of the relationsships to be
established. Once the statements secured by proofs, the
theorems, have been removed, there remains a group of
statements, the basic statements, from which all others can
be deduced; these are based directly on observations. ]
and on p.43 :
Die Grunds"atze kann man ohne entsprechende Figuren nicht
einsehen; sie sagen aus, was an gewissen sehr einfachen
Figuren beobachtet worden ist. Die Lehrs"atze werden nicht
durch Beobachtungen begr"undet, sondern bewiesen; jeder
Schluss, der im Verlaufe des Beweises vorkommt, muss in der
Figur seine Best"atigung finden, aber er wird nicht aus der
Figur, sondern aus einem bestimmten vorhergegangenen Satze
(oder aus einer Definition) gerechtfertigt.
[The basic statements cannot be understood without corresponding
drawings; they express what has been observed from certain,
very simple drawings. The theorems are not founded on
observations, but are proven; every inference, occurring
during a deduction, must find confirmation in a drawing, yet
it is not justified by a drawing, but from a certain
preceding statement (or a definition). ]
So there is a world of a difference between Pasch's explanation
of basic notions through reference to objects of nature - and
Fano's and Peano's choice of basic notions as "enti di qualunque
natura" or "enti non sono definiti".