To what extent was Cantor motivated by Galileo's paradox? More generally, to what extent were late 19th century mathematicians motivated by, or even aware of, Galileo's paradox?

This is an issue I've been wondering about for many years but I have never gotten around to looking into it until this morning.

Yesterday, I read Omen's answer [later deleted] to the History of Science and Math StackExchange question What motivated Cantor to invent set theory and this led me to make the following comment: It was my understanding that Cantor did not know about Galileo's writing on infinity. In fact, I've been under the impression (for maybe 20 years) that Galileo's thoughts on infinity were mostly unknown to mathematicians until Edward Kasner called attention to the connection in 1904, but I have not (yet) looked into this matter.

(NEXT DAY) mweiss suggested in a comment that I isolate my question from my partial attempts at an answer, which I think is a good structural idea for this site.

  • $\begingroup$ A little contribute : the 14th Century philosopher Nicole Oresme was already aware of the "paradox" and we know that Cantor was "familiar" with the medieval conceptions of the infinite : see Michael Hallett, Cantorian Set Theory and Limitation of Size (1984), see detailed Subject Index, with no occurences of Galileo's name. Thus, I personally support your conjecture about the lack of Galileo influence on Cantor. $\endgroup$ Commented Nov 13, 2014 at 21:01
  • $\begingroup$ Whoa, what an impressingly well-researched question! I'm not sure anyone here will be able to tell you anything you don't know yet, but I'm eager to see if something will come out of this question! :) $\endgroup$
    – Danu
    Commented Nov 13, 2014 at 21:15
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    $\begingroup$ You appear to have more references in your question than on all the answers given in the past 24 hours combined! Really nice question; welcome to the site! $\endgroup$
    – HDE 226868
    Commented Nov 13, 2014 at 21:33
  • $\begingroup$ @Mauro ALLEGRANZA: Various date restricted google-book searches (which I forgot to mention) for words such as Galileo, infinito, Cantor, mengenlehre, etc. don't result in anything either, but maybe someone else will find something that I missed. Hermann Hankel might have known about Galileo's writings on infinity, and I know that Cantor was influenced to some extent by Hankel (Cantor wrote an anonymous review of Hankel's 1870 monograph "Untersuchungen über die unendlich ..."), but even if Hankel wrote about Galileo I doubt Cantor in the early 1870s was sufficiently widely read to see it. $\endgroup$ Commented Nov 13, 2014 at 21:50
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    $\begingroup$ A suggestion: It might be better in terms of thread structure to break this (wonderfully-researched) question into two parts: A question and an answer. $\endgroup$
    – mweiss
    Commented Nov 14, 2014 at 15:37

2 Answers 2


This morning [13 November 2014] I spent several hours going through many Cantor-related papers and books that I have, and I am now nearly convinced that Galileo probably had no influence on Cantor and Galileo had very little if any influence on other mathematicians. For instance, I believe there was no German translation of Galileo's "Two New Sciences" until 1890. (I am not very sure about this, however.)

I have recorded the relevant things I found this morning as excerpts and comments associated with the bibliography that follows. The items in this bibliography are not (by a long way) the only papers and books I looked at. I selected these items for several reasons: for their relevance to the issue with Galileo (and with Bolzano), for their interest to people interested in the posted question What motivated Cantor to invent set theory, and for their interest to people interested in 19th century infinity issues. For the record, the other papers and books I looked through were by Irving Henry Anellis, Roger Lee Cooke, José Ferreirós, Abraham Adolf [Adolph] Halevi Fraenkel, Michael F. Hallett, Thomas William Hawkins, Ernest William Hobson, Arie Hinkis, Phillip Eugene Johnson, Philip Edward Bertrand Jourdain, Akihiro Kanamori, Fyodor Andreyevich Medvedev, and many others.

[1] Carl Benjamin Boyer, The Concepts of the Calculus. A Critical and Historical Discussion of the Derivative and the Integral, Columbia University Press, 1939, vii + 346 pages.

Reprinted by Hafner Publishing Company in 1949 (xii + 346 pages) and by Dover Publications in 1959 (title changed to The History of the Calculus and Its Conceptual Development, xii + 346 pages). (from pp. 270-271) "[Bolzano's] view in this respect resembles that of Galileo, to whom he referred in this connection. [footnote 11, omitted here] Although he denied the existence of infinitely large and infinitely small magnitudes, he maintained, with Galileo, the possibility of an actual infinity with respect to aggregation. He remarked, with respect to such assemblages, the paradox which Galileo had pointed out: that the part could in this case be put into one-to-one correspondence with the whole. [...] [Bolzano's] work remained largely unnoticed until rediscovered by Hermann Hankel more than a half century later."

[2] Georg Ferdinand Ludwig Philip Cantor, Ueber unendliche, lineare Punktmannichfaltigkeiten 5 [On infinite, linear point sets 5], Mathematische Annalen 21 (1883), 545-591.

[3] Georg Ferdinand Ludwig Philip Cantor, Grundlagen einer allgemeinen Mannigfaltigkeitslehre: Ein mathematisch-philosophischer Versuch in der Lehre des Unendlichen [Foundations of a General Theory of Manifolds: A Mathematical-Philosophical Essay in the Theory of the Infinite], B. G. Teubner (Leipzig), 1883, 47 pages.

This is a separate printing of Cantor's paper above (pp. 545-591) in which a half-page preface and 4 footnotes have been added (these are in addition to the endnotes of the Mathematische Annalen, which also appear here). These additions are omitted in the 1932 publication of Cantor's "Collected Works".

[4] Louis [Ludovicus] Couturat, De L'Infini Mathématique [On the Mathematical Infinite], Félix Alcan (Paris), 1896, xxiv + 667 + 1 (errata) pages.

"Galileo" does not appear in this book according to a word search I did of the .pdf file. This work is one of two manuscripts that Couturat wrote and submitted for his Ph.D. at the Sorbonne (Paris). The other work is De Platonicis Mythis (1896, v + 119 pages), a literary work written in Latin that deals with the writings of the Greek philosopher Plato (specifically, with trying to distinguish dogmatic passages from irony and allegory passages).

[5] Joseph Warren Dauben, C. S. Peirce's philosophy of infinite sets, Mathematics Magazine 50 #3 (May 1977), 123-135.

[6] Joseph Warren Dauben, Georg Cantor: The personal matrix of his mathematics, Isis 69 #249 (December 1978), 534-550.

[7] William Bragg Ewald, From Kant to Hilbert: A Source Book in the Foundations of Mathematics, two volumes, Clarendon Press, 1996, xviii + 1340 pages (both volumes).

A discussion of Cantor [2] and [3] is given on pp. 878-881 and an English translation (by Ewald) of Cantor [3] is given on pp. 881-920.

(from p. 889) "If we look about in history, we find that similar opinions were often held; they are already to be found in Aristotle." After discussing Aristotle, Cantor goes on to mention (pages refer to Ewald's translation; it is entirely possible that the page lists for each person named is not complete) Locke (p. 890), Descartes (p. 890), Spinoza (pp. 890-892), Leibniz (pp. 890-895), Kant (p. 892). Bolzano is mentioned on p. 895:

(top of p. 895) Bolzano is perhaps the only one for whom the proper-infinite numbers are legitimate (at any rate, he speaks about them a great deal); but I absolutely do not agree with the manner in which he handles them without being able to give a correct definition, and I regard, for example, §§29-33 of that book as unsupported and erroneous. The author lacks two things which are necessary for a genuine grasp of the concept of determinate-infinite numbers: both the general concept of power and the precise concept of Anzahl. To be sure, both appear in germ in isolated passages and as special cases. But he does not work his way through to full clarity and exactness, which explains many inconsistencies and even many errors in this valuable book. Without these two concepts, I am convinced, one can not make further progress in the theory of manifolds. The same is true, I believe, for the fields that are a part of the theory of manifolds or that have the most intimate contact with it--for example, modern function theory on the one hand and logic and epistemology on the other.

[8] Ivor Grattan-Guinness, The correspondence between Georg Cantor and Philip Jourdain, Jahresbericht der Deutschen Mathematiker-Vereinigung 73 #3 (20 September 1971), 111-130.

The following letter from Philip Edward Bertrand Jourdain (dated 3 January 1901) to Cantor appears on pp. 112-112.

[All subsequent additions using square brackets are by Grattan-Guinness, and "connexion" and "emphasised" were in the original] Dear Sir, In some researches on the early history of Manifold-theory, I came across a paper by A. de Morgan "On $\infty$ and on the sign of equality" [[32]] written in 1864 (quite independently of the earlier work of Bolzano), which appears to be of some importance in this connexion. de M. was an upholder of the "eigentlich Unendlich"; and showed that the substitution of "increase without limit" ("uneigentlich Un.") is not always safe. He emphasised the existence of the notion of infinity like space and time forms in the Kantian sense, and the depend[ence] on it of the concept of finite [sic]. The most important is his argument for the conception of an inf[inite] 'multitude' and his clear distinction of 'conception' from 'image'. He often reminds one of Bolzano. He also notices, en passant, the correspondence betw[een] 2 inf[inite] manifolds but not so clearly or with such a full sense of its importance as Bolzano (§80 of Paradoxien [[2]]). His remarks on the history of inf[inity], especially on Aristotle, may, I think, interest you in connexion with your memoir in Math. Ann. Bd. XXI [[8]]. As I think you might like to see this paper, I should have great pleasure in sending you a separate copy of it on hearing from you. Yours sincerely, Philip E. B. Jourdain.

After the above letter, Grattan-Guinness says: "Cantor replied at once, and his answer showed that he had not previously read de Morgan's paper". Cantor's reply is in German, not translated. I don't have time to type it now, but I will do so on another day if anyone is interested.

Part of a letter written by Cantor (29 March 1905) to Jourdain (2nd paragraph on p. 124): "With Mr. Weierstrass I had good relations and I possess a most interesting correspondence with him, which I will show to you. Of the conception of enumerability of which he heared from me at Berlin on Christmas holydays 1873 he became at first quite amazed, but one or two days passed over, it became his own and helped him to an unexpected development of his wonderful theory of functions." [Note: I believe Cantor discovered the uncountability of the reals on 7 December 1873. In footnote 17 on p. 124, Grattan-Guinness says it is not clear which of the results of Weierstrass that Cantor was talking about.]

[9] Ivor Grattan-Guinness, The rediscovery of the Cantor-Dedekind correspondence, Jahresbericht der Deutschen Mathematiker-Vereinigung 76 #2-3 (30 December 1974), 104-139.

Grattan-Guinness makes the following remark about a letter by Cantor to Dedekind on p. 125: "The only point worth quoting is Cantor's discovery of a work by Bolzano (presumably his book [2] on the infinite), in a note of 7 October 1882." Bolzano's book was published in 1851, but I believe it was mostly unknown to mathematicians until the 1870s, and even then it was not very widely known. In footnote 18 on p. 125, Grattan-Guinness writes: "A year later Cantor mentioned Bolzano for the first time in his papers […]"

[10] Edward Kasner, Galileo and the modern concept of infinity, Bulletin of the American Mathematical Society 11 #9 (June 1905), 499-501.

[11] Cassius Jackson Keyser, Theorems concerning positive definitions of finite assemblage and infinite assemblage, Bulletin of the American Mathematical Society 7 #5 (February 1901), 218-226.

[12] Cassius Jackson Keyser, Concerning the axiom of infinity and mathematical induction, Bulletin of the American Mathematical Society 9 #8 (May 1903), 424-434.

[13] Cassius Jackson Keyser, The rôle of the concept of infinity in the work of Lucretius, Bulletin of the American Mathematical Society 24 #7 (April 1918), 321-327.

[14] Gregory Harvey Moore, Zermelo's Axiom of Choice: Its Origins, Development and Influence, Studies in the History of Mathematics and Physical Sciences #8, Springer-Verlag, 1982, xiv + 410 pages.

Reprinted by Dover Publications in 2013. Galileo is discussed on p. 23, along with Bolzano (discussed on pp. 23-24). There is no indication of what Cantor might have known or thought of Galileo, but Moore does say the following at the bottom of p. 23 to the top of p. 24: "It is uncertain to what extent Bolzano's views on the infinite influenced Cantor, who discussed the Paradoxes of the Infinite only in 1883. On that occasion he praised Bolzano's book for asserting that the actual infinite exists but criticized it for failing to provide either a concept of infinite number or the concept of "power" based on equipollence. [footnote 5, omitted here] Although he may have adopted the terms Menge (set) and Vielheit (multitude or multiplicity) from Bolzano, Cantor's interest in the infinite had originated much earlier in his career."

[15] Augustus De Morgan, On infinity; and on the sign of equality, Transactions of the Cambridge Philosophical Society 11 Part 1 (1871), 145-189.

Separately published as a booklet by Cambridge University Press in 1865 (same title; i + 45 pages).

[16] Jules Tannery, De l'infini mathématique [On mathematical infinity], Revue Générale des Sciences Pures et Appliquées 8 (1897), 129-140.

[17] Abel Étienne Louis Transon, Sur l'emploi de l'infini en mathématiques [On the use of the infinite in mathematics], Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences (Paris) 73 #6 (1871), 367-369.

Paper associated with the session dated 7 August 1871.

[18] Giulio Vivanti, Notice historique sur la théorie des ensembles [Historical report on the theory of sets], Bibliotheca Mathematica (2) 6 #1 (1892), 9-25.

The bibliography has 56 items.

[19] Giulio Vivanti, Lista bibliografica della teoria degli aggregati 1893-1899 [Bibliographic list of the theory of sets 1893-1899], Bibliotheca Mathematica (3) 1 (1900), 160-165.

The bibliography has 67 items, which are listed in chronological order.

[20] William Charles Waterhouse, Gauss on infinity, Historia Mathematica 6 #4 (1979), 430-436.

(from p. 435) References to Gauss in connection with set theory apparently all go back to a comment published by Cantor himself. In an article of 1885 (published in 1886) he wrote: It is just two years ago that Herr Rudolf Lipschitz in Bonn drew my attention to a certain place in the Gauss-Schumacher correspondence where the former speaks out against any introduction of actual infinity into mathematics (Letter of 12 July 1831). I answered thoroughly, and on this point did not accept the authority of Gauss, which I respect so highly in all other areas ... [Cantor 1932, 371]. This in isolation makes it seem that Cantor considered Gauss an opponent of his set theory. But as one follows his discussion it becomes clear that this disagreement is only with the words, not with Gauss' actual ideas. The crucial sentence is this: [omitted] Thus Cantor objected not to Gauss' statement in context but to the meaning attributed to it by his own contemporaries.

[21] William Henry Young, The introduction of the mathematical idea of infinity, Mathematical Gazette 4 #67 (December 1907), 147-159.

[22] William Henry Young and Grace Chisholm Young, The Theory of Sets of Points, Cambridge University Press, 1906, xii + 316 pages.

The 2nd edition was published by Chelsea Publishing Company in 1972 (xvi + 326 pages). The 2nd edition was prepared by Rosalind Cecilia Hildegard Tanner and Ivor Grattan-Guinness, and it corrects printing mistakes and simple errors in the original and it includes an appendix of supplementary notes prepared by Grace Chisholm Young. By my count the bibliography has 308 items. "Galileo" does not appear in the Index of Proper Names on p. 320 of the 2nd edition.

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    $\begingroup$ Dave, that was a very thorough job. $\endgroup$ Commented Jul 3, 2017 at 13:55

Cantor was definitively not influenced by Galileo since he believed that Galileo was an opponent of actual infinite numbers. Cantor quoted Galileo as such (but only 15 years after inventing set theory) in Mitteilungen zur Lehre vom Transfiniten 1888 (footnote on p. 417 of the collected works) and in a letter to Hilbert of January 27, 1900.

The footnote refers to a book of Moigno:

Moigno: Imposs. d. nombre act. inf. Paris 1884. Hier werden Galilei, Gerdil, Torricelli, Guldin, Cavalieri, Newton, Leibniz als solche angeführt, welche sogenannte Beweise gegen die Möglichkeit aktual unendlicher Zahlen geführt hätten.

The letter to Hilbert contains the paragraph:

Mein Gegensatz zu Gauss besteht hingegen darin, daß Gauss alle Vielheiten, mit Ausnahme der endlichen, für inconsistent hält (ich meine unbewusst, d. h. ohne den Begriff zu haben) und daher kategorisch und principiell dasjenige Actualunendliche, welches ich Transfinitum nenne, verwirft, mithin auch die transfiniten Zahlen, deren Existenz ich begründet habe, für unmöglich erklärt (V. Brief von Gauss an Schumacher, v. 12. Juli 1831). Dies ist übrigens auch der Standpunct der Aristoteliker, auch von Galilei, Leibniz, Newton, d'Alembert, Cauchy, etc. bis auf Kronecker.

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    $\begingroup$ Hi! Your answer could use a citation (the said footnote?) from your reference! $\endgroup$
    – VicAche
    Commented May 9, 2017 at 10:56
  • $\begingroup$ @Francois Ziegler: Galileo was the first name. $\endgroup$
    – Franz Kurz
    Commented May 18, 2017 at 11:30
  • $\begingroup$ @Claus That's right. $\endgroup$ Commented May 18, 2017 at 12:31

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