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Dictionary of the History of Ideas

Studies of Selected Pivotal Ideas
170 occurrences of ideology
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170 occurrences of ideology
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The only general history of infinity is the book of Jonas
Cohen; a supplement to it, heavily oriented toward theol-
ogy, is the essay of Anton Antweiler. Of considerable inter-
est is a collection of articles in the 1954 volume of the Revue
de Synthèse.

A comprehensive study on infinity in Greek antiquity is
the work of Mondolfo. The author is a staunch defender
of the thesis that Greek thought had fully the same attitude
towards infinity as modern thought. About infinity in the
Old Testament see the books of C. von Orelli, Thorleif
Boman, and James Barr. Occasionally one encounters the
view that, in a true sense, infinity was originally as much
a Hebraic intuition as a Greek one, and perhaps even more
so. Such a view is implied in the books just cited, and it
was expressly stated in Revue de Synthèse, p. 53 (remark
by M. Serouya).

Infinity in the pre-Socratics is competently dealt with in
the recent work of Guthrie. Infinity in all of Greek philoso-
phy, Hellenic and Hellenistic, is also fully dealt with in
the great Victorian standard work of Eduard Zeller. It is
still very good on infinity in Plotinus, and also in Philo,
in spite of recent special studies on the two, especially on
Plotinus. In the case of Philo, it is not easy to locate infinity
specifically in his work, and even in Wolfson's detailed study
of Philo there is very little direct reference to it.

About infinity in medieval philosophy, European and
Arabic, and in subsequent philosophy up to and including
Spinoza, there is a wealth of material in Wolfson's two-
volume work on Spinoza. All of volume I is very pertinent,
and not only the parts dealing expressly with infinity, like
Chapter V, part III (Definition of the term “Infinite”), and
Chapter VIII (Infinity of Extension). The latter chapter is
of special interest for the genesis of Descartes' view on the
nature of infinity of his extension (or étendue); see section
II above.

About infinity in scientist-philosophers, or cosmologists,
or astronomers from Nicholas of Cusa to Newton and Leib-
niz there is the informative work of Koyré, which features
a judicious selection of verbatim excerpts, all in English.
There are also recent books about the relevance of infinity
to nonscientific general philosophy, such as the books of
Bernardete, Welte, and Heimsoeth.

Infinity in mathematics is accounted for in any general
history of mathematics, but especially in Boyer's The History
of the Calculus.
For the history of Zeno's paradoxes the


main account, with full references, is the article in nine
parts, commencing in 1915, by F. Cajori in the American
Mathematical Monthly.
The references are carried to 1936
in the lengthy introduction to Ross's edition, with commen-
tary, of Aristotle's Physics. To judge by an incidental remark
in Cajori's account, the first outright association of the
paradoxes with mathematics is documented only from the
seventeenth century A.D., in the work of Gregory of St.

For the roots and rise of Georg Cantor's set theory there
is much material in Cantor's Collected Works which have
been edited by Ernst Zermelo. The principal memoirs of
Cantor were translated into English, with introduction and
notes, by P. E. B. Jourdain. There is a lack of studies on
how the emergence of Cantor's set theory fits into the
history of ideas; there is, for instance, no special study on
how it reflects itself in the philosophical system of Charles
S. Peirce (cf. Collected Papers of Charles S. Peirce, ed. C.
Hartshorne and Paul Weiss, Cambridge, Mass. [1933], Vol.

The following works are additional references for the study
of infinity. Paul Alexandroff, “Über die Metrisation der im
kleinen kompakten topologischen Räume,” Mathematische
99 (1924), 294-307. Anton Antweiler, Unendlich,
Eine Untersuchung zur metaphysischen Weisheit Gottes auf
Grund der Mathematik, Philosophie, Theologie
(Freiburg im
Breisgau, 1935). Saint Thomas Aquinas, Summa theologiae,
Latin text and English trans. by Blackfriars (London and
New York, 1962), Vol. II. Aristotle, Physica, trans. R. P.
Hardie and R. K. Gaye in the Oxford translation of Aris-
totle's works under the general editorship of W. D. Ross,
Vol. 2 (Oxford, 1930). See also W. D. Ross, below. A. H.
Armstrong, Plotinus (New York, 1962). James Barr, Biblical
Words for Time
(London, 1961). José A. Bernardete, Infinity,
an Essay in Metaphysics
(Oxford, 1964). Salomon Bochner,
The Role of Mathematics in the Rise of Science (Princeton,
1966); idem, “The Size of the Universe in Greek Thought,”
Scientia, 103 (1968), 510-30. Hermann Bondi, Cosmology,
2nd ed. (Cambridge, 1960). Thorleif Boman, Das hebräische
Denken im Vergleich mit dem Griechischen,
4th ed. (Göttin-
gen, 1965); 3rd ed. trans. as Hebrew Thought Compared With
Greek Thought
(Philadelphia, 1961). Carl B. Boyer, The
History of the Calculus
(New York, 1959). F. Cajori, “The
History of Zeno's Arguments on Motion,” American Mathe-
matical Monthly,
12 (1915), 1-6, 39-47, 77-82, 109-15,
143-49, 179-86, 215-20, 253-58, 292-97; idem, Sir Isaac
Newton's Mathematical Principles of Natural Philosophy and
His System of the World,
trans. Andrew Motte (1729), re-
vised by F. Cajori (Berkeley, 1934; many reprints); cited
as Principia. Georg Cantor, Gesammelte Abhandlungen
mathematischen und philosophischen Inhalts,
ed. Ernst
Zermelo (Berlin, 1932). C. Carathéodory, “Über die Beg-
renzung einfach zusammenhängender Gebiete,” Mathe-
matische Annalen,
73 (1913), 343-70. Morris R. Cohen and
I. E. Drabkin, A Source Book in Greek Science (New York,
1948). Jonas Cohen, Geschichte der Unendlichkeitsproblems
im abendländischen Denken bis Kant
(Leipzig, 1869). James
H. Coleman, Modern Theories of the Universe (New York,
1963). H. S. M. Coxeter, Non-Euclidean Geometry (Toronto,
1957). E. J. Dijksterhuis, Archimedes (New York, 1957).
Diogenes Laërtius, Lives of Eminent Philosophers, 2 vols.
(London and Cambridge, Mass., 1925). Abraham Edel, Aris-
totle's Theory of the Infinite
(New York, 1934). George
Gamow, The Creation of the Universe (New York, 1952).
Marvin T. Greenberg, Lectures on Algebraic Topology (New
York, 1967). Gregory of St. Vincent, Opus geometricum
quadratura circuli et sectionum coní
(Antwerp, 1647).
W. K. C. Guthrie, A History of Greek Philosophy, Vols. 1 and
2 (Cambridge, 1962 and 1965). T. L. Heath, The Thirteen
Books of Euclid's Elements
(Cambridge, 1908); idem, History
of Greek Mathematics,
2 vols. (Oxford, 1921). Heinz Heim-
soeth, Diesechs grossen themen der abendländischen Meta-
physik und der Ausgang des Mittelalters,
3rd ed. (Stuttgart,
1954). Werner Heisenberg, Physics and Philosophy, the
Revolution in Modern Science
(New York, 1958). David
Hilbert, Grundlagen der Geometrie (Leipzig, 1899), many
editions and translations. P. E. B. Jourdain, Contributions
to the Founding of the Theory of Transfinite Numbers

(Chicago and London, 1915). Immanuel Kant, Critique of
Pure Reason,
trans. Norman Kemp Smith (London, 1929).
G. S. Kirk and J. E. Raven, The Presocratic Philosophers
(Cambridge, 1957). Alexandre Koyré, From the Closed World
to the Infinite Universe
(Baltimore, 1957). P. Kucharski,
“L'idée de l'infini en Grèce,” Revue de Synthèse, 34 (1954),
5-20. Earle Loran, Cézanne's Composition, 2nd ed. (Berke-
ley, 1944). Anneliese Maier, DieVorläufer Galileis im 14.
(Rome, 1949); idem, Zwei Grundprobleme der
Scholastischen Naturphilosophie
(Rome, 1951); idem,
Zwischen Philosophie und Mechanik (Rome, 1958); idem,
Metaphysische Hintergründe der Spätscholastischen Natur-
(Rome, 1955). Rodolfo Mondolfo, L'infinito nel
pensiero dell'Antiquità classica
(Florence, 1965). Isaac
Newton, see Cajori, above. C. von Orelli, Diehebräischen
Synonyma der Zeit und Ewigkeit, genetisch und sprachver-
gleichlich dargestellt
(Leipzig, 1871). Erwin Panofsky,
Albrecht Dürer (Princeton, 1945); idem, “Die Perspective
als 'Symbolische Form,'” in Vorträge der Bibliothek Warburg
(1924-25); the latter is reprinted in Panofsky's Aufsätze zu
Grundfragen der Kunstwissenschaft
(Berlin, 1964). W. Pauli,
ed., Niels Bohr and the Development of Physics (New York,
1955). Charles S. Peirce, Collected Papers of Charles S.
ed. C. Hartshorne and Paul Weiss, 6 vols. (Cam-
bridge, Mass., 1933), Vol. IV. Revue de Synthèse (Centre
international de Synthèse), 34, New Series (1954). J. M. Rist,
Plotinus: The Road to Reality (Cambridge, 1967). B. Rochot,
“L'infini Cartésien,” Revue de Synthèse, 34 (1954), 35-54.
Vasco Ronchi, The Science of Vision (New York, 1957).
W. D. Ross, ed., Aristotle's Physics, A revised text with intro-
duction and commentary
(Oxford, 1936); idem, Aristotle, a
complete exposition of his works and thought
1959). Meyer Schapiro, Paul Cézanne (New York, 1952). A.
Schoenfliess, “Projective Geometrie,” Encyclopädie der
mathematischen Wissenschaften,
Vol. III, Leipzig, 1898-),
Abt. 5. Basil Schonland, The Atomists (1830-1933) (Oxford,
1968). Oswald Spengler, The Decline of the West, trans. C. F.
Atkinson, 2 vols. (New York, 1926-28). Norman Steenrod,
Topology of Fibre Bundles (Princeton, 1965). Leonardo
Tarán, Parmenides, A Text with Translation, Commentary,


and Critical Essays (Princeton, 1965). Mario Untersteiner,
Parmenide, Testimonianze e Frammenti (Florence, 1958). G.
Veronese, Grundzüge der Geometrie (Berlin, 1894); the orig-
inal edition in Italian is almost never quoted. Richard
Walzer, Greek into Arabic; Essays in Islamic Philosophy
(Oxford, 1962). Bernhard Welte, Im Spielfeld von Endlich-
keit und Unendlichkeit. Gedanken zur Deutung der men-
schlichen Daseins
(Frankfurt-am-Main, 1967). Christian
Wiener, Lehrbuch der darstellenden Geometrie, 2 vols.
(Leipzig, 1884). Harry Austryn Wolfson, Crescas' Critique
of Aristotle, Problems of Aristotle's Physics in Jewish and
Arabic Philosophy
(Cambridge, Mass., 1929); idem, The
Philosophy of Spinoza, Unfolding the Latent Processes of
His Reasoning,
2 vols. (Cambridge, Mass., 1934); idem, Philo:
Foundations of Religious Philosophy in Judaism, Christi-
anity, and Islam,
2 vols. (Cambridge, Mass., 1947). Eduard
Zeller, DiePhilosophie der Griechen in ihrer geschichtlichen
3 vols. (1844-52); the English translation ap-
peared in segments.


[See also Abstraction; Axiomatization; Continuity; Cos-
mology; Mathematical Rigor; Newton on Method; Number;
Rationality; Space; Time and Measurement.]