Frege's and Bolzano's rationalist conceptions of arithmetic / Les conceptions rationalistes en arithmétique de Frege et Bolzano - article ; n°3 ; vol.52, pg 343-362

Charles Chihara

Description

Revue d'histoire des sciences - Année 1999 - Volume 52 - Numéro 3 - Pages 343-362
RÉSUMÉ. — L'essai présente une comparaison des conceptions rationalistes d'arithmétique de Gottlob Frege et Bernard Bolzano. Les deux philosophes élaborent une structure complexe de propositions posées comme vraies. Les axiomes, ou vérités fondamentales, représentent le socle sur lequel se constituent les fondations du sujet. Chaque élément de cette structure qui ne constitue pas un axiome est relié (objectivement) à sa base. Bien que cette relation à la base ne soit pas encore scientifiquement prouvée, parmi les propositions de cette structure se trouvent les vérités de la science. L'essai se conclut par une analyse de ces vues respectives, à la lumière du théorème d'incomplétude de Gödel.
SUMMARY. — In this article, I compare Gottlob Frege's and Bernard Bolzano's rationalist conceptions of arithmetic. Each philosopher worked out a complicated system of propositions, all of which were set forth as true. The axioms, or basic truths, make up the foundations of the subject of arithmetic. Each member of the system which is not an axiom is related (objectively) to the axioms at the base. Even though this relation to the base may not yet be scientifically proven, the propositions of the system include all of the truths of the science of arithmetic. I conclude the article by analyzing the respective views of Frege and Bolzano in the light of Gödel's first incompleteness theorem.
20 pages
Source : Persée ; Ministère de la jeunesse, de l’éducation nationale et de la recherche, Direction de l’enseignement supérieur, Sous-direction des bibliothèques et de la documentation.

Informations

• Publié par: REVUE_D-HISTOIRE_DES_SCIENCES
• Publié le: 01 janvier 1999
• Nombre de lectures: 8
• Langue: English
• Poids de l'ouvrage: 1 Mo

Extrait

M CHARLES CHIHARA
Frege's and Bolzano's rationalist conceptions of arithmetic / Les
conceptions rationalistes en arithmétique de Frege et Bolzano
In: Revue d'histoire des sciences. 1999, Tome 52 n°3-4. pp. 343-362.
Résumé
RÉSUMÉ. — L'essai présente une comparaison des conceptions rationalistes d'arithmétique de Gottlob Frege et Bernard
Bolzano. Les deux philosophes élaborent une structure complexe de propositions posées comme vraies. Les axiomes, ou vérités
fondamentales, représentent le socle sur lequel se constituent les fondations du sujet. Chaque élément de cette structure qui ne
constitue pas un axiome est relié (objectivement) à sa base. Bien que cette relation à la base ne soit pas encore scientifiquement
prouvée, parmi les propositions de cette structure se trouvent les vérités de la science. L'essai se conclut par une analyse de ces
vues respectives, à la lumière du théorème d'incomplétude de Gödel.
Abstract
SUMMARY. — In this article, I compare Gottlob Frege's and Bernard Bolzano's rationalist conceptions of arithmetic. Each
philosopher worked out a complicated system of propositions, all of which were set forth as true. The axioms, or basic truths,
make up the foundations of the subject of arithmetic. Each member of the system which is not an axiom is related (objectively) to
the axioms at the base. Even though this relation to the base may not yet be scientifically proven, the propositions of the system
include all of the truths of the science of arithmetic. I conclude the article by analyzing the respective views of Frege and Bolzano
in the light of Gödel's first incompleteness theorem.
Citer ce document / Cite this document :
CHIHARA CHARLES. Frege's and Bolzano's rationalist conceptions of arithmetic / Les conceptions rationalistes en arithmétique
de Frege et Bolzano. In: Revue d'histoire des sciences. 1999, Tome 52 n°3-4. pp. 343-362.
doi : 10.3406/rhs.1999.1360
http://www.persee.fr/web/revues/home/prescript/article/rhs_0151-4105_1999_num_52_3_1360Frege's
and Bolzano's rationalist conceptions
of arithmetic
Charles Chihara (♦)
RÉSUMÉ. — L'essai présente une comparaison des conceptions rationalistes
d'arithmétique de Gottlob Frege et Bernard Bolzano. Les deux philosophes élabo
rent une structure complexe de propositions posées comme vraies. Les axiomes,
ou vérités fondamentales, représentent le socle sur lequel se constituent les fonda
tions du sujet. Chaque élément de cette structure qui ne constitue pas un axiome
est relié (objectivement) à sa base. Bien que cette relation à la base ne soit pas
encore scientifiquement prouvée, parmi les propositions de cette structure se trou
vent les vérités de la science. L'essai se conclut par une analyse de ces vues respect
ives, à la lumière du théorème d'incomplétude de Gôdel.
MOTS-CLÉS. — Arithmétique ; Bolzano ; Frege ; le théorème d'incomplétude
de Gôdel ; base ; rationaliste.
SUMMARY. — In this article, I compare Gottlob Frege's and Bernard Bolza
no 's rationalist conceptions of arithmetic. Each philosopher worked out a complica
ted system of propositions, all of which were set forth as true. The axioms, or basic
truths, make up the foundations of the subject of arithmetic. Each member of the
system which is not an axiom is related (objectively) to the axioms at the base. Even
though this relation to the base may not yet be scientifically proven, the propositions
of the system include all of the truths of the science of arithmetic. I conclude the
article by analyzing the respective views of Frege and Bolzano in the light of Gôdel "s
first incompleteness theorem.
KEYWORDS. — Arithmetic ; Bolzano ; Frege ; Godel's incompleteness theo
rem ; base ; rationalist.
(*) Charles Chihara, Dept of Philosophy, Univ. of California, Berkeley, California
94720, USA.
Rev. Hist. ScL, 1999, 52/3-4, 343-361 344 Charles Chihara
Part I
As any student of The Foundations of arithmetic knows, Frege
thought he could show that the truths of are all analytic.
To accomplish this task, he had to show that all arithmetical truths
could be proved from definitions and primitive logical laws - logi
cal laws which neither admit nor require any proof to be
known (1). The aim of constructing such proofs would be not
merely to establish the certainty of the propositions proved, but
also to show us the objective dependency of the truths upon one
another (2). By pursuing these researches, it was hoped that one
could uncover « the ultimate ground upon which rests the justifica
tion for holding » arithmetical propositions to be true. If that could
be done, we would discover the very foundations of arithmetic.
The philosophical and mathematical significance of uncovering
this objective dependency is, in one respect, obvious : all doubts
about the truth of the arithmetical laws would be thereby removed.
But it can be argued that Frege had another, more subtle, goal in
mind : such an uncovering would make it clear that arithmetical
reasoning did not depend upon intuitions, as Kant had argued. To
this extent, as William Demopoulos puts it,
« Frege's intellectual motivations echo those of the nineteenth century
analysts who sought to free the calculus and the theory of the reals from
any dependence on the sciences of geometry and kinematics (3) ».
Paul Benacerraf suggests that Frege's conception of the analyti-
city of arithmetic « derives from the rationalist conception [...] of a
hierarchy of propositions, some of which are absolutely basic and
(1) Although Frege does not say explicitly that the primitive logical laws must be « self-
evident » in the above way, he implies as much by declaring that all analytic propositions are
a priori and characterizing an a priori proposition as one that is provable from general laws
« which themselves neither need nor admit of proof » (Frege, 1959, 4). (Les références comp
lètes se trouvent en bibliographie placée en fin d'article.) Cf. also, Frege's response to the
discovery by Russell of the inconsistency in his system : « I have never concealed from
myself its [Basic Law V] lack of the self-evidence which the others possess, and which must
properly be demanded of a law of logic » (Frege, 1964, 127).
(2) Frege, 1959, 2.
(3) Demopoulos, 1995, 76. Frege's and Bolzano's rationalist conceptions of arithmetic 345
form the foundations on which all the others "rest" (4) ». The
rationalist conception described above is not one that contempor
ary foundationalists find congenial. Few contemporary logicians
would accept, for example, Frege's idea that, for every true arith
metical proposition / there is some ultimate ground for holding /
to be true - a ground which can be known to be true even though
it cannot be proved. Despite the importance of this remarkable
presupposition of Frege's central project in The Foundations of
arithmetic, there is surprisingly little in that work which justifies or
even explains the above rationalist conception. However, it may be
illuminating to compare Frege's rationalistic conception of arith
metic with that of another mathematician who has written a great
deal about just such a conception.
Bernard Bolzano, whose name is well-known to calculus stu
dents because of the Bolzano- Weierstrass Theorem, sketched a
view of mathematics that has many of the salient features of
Frege's conception of arithmetic. This is not to say that Bolzano's
conception is essentially identical to Frege's : as I shall point out
later, there are some enormous differences. However, from our
perspective, the similarities are striking.
Why did Bolzano feel the need to prove the Intermediate
Value Theorem when many proofs of the theorem already exis
ted? Consider, for example, those proofs of the theorem that
made use of the geometric fact that every continuous line of simple
curvature of which the ordinales are first positive and then negative
(or conversely) must necessarily intersect the x-axis somewhere at a
point that lies between those ordinates. Bolzano did not think that
this geometric theorem was incorrect or questionable in any way.
He did hold, however, that an acceptable proof (a « scientific
proof ») of a theorem in mathematics should provide « the objec
tive reason » for the truth of the proposition (5). Since the Inte
rmediate Value Theorem was a truth that «holds equally for all
quantities, whether in space or not », he was convinced that the
objective reason for its truth could not possibly lie in a truth
which holds only for quantities which are in space. Evidently, Bol
zano wished to find a proof of the theorem that was not depen
dent on the sciences of geometry and kinematics. In this respect,
(4) Benacerraf, 1981, 32.
(5) Bolzano, 19966, 228. 346 Charles Chihara
Bolzano pursued a goa