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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.

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.]

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.

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.]

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.

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.

[10] 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.

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

[12] Cassius Jackson Keyser, Galileo and the modern concept of infinity, Bulletin of the American Mathematical Society 11 #9 (June 1905), 499-501.

[15] Augustus De Morgan, On infinity; and on the sign of equality, Transactions of the Cambridge Philosophical Society 11 Part 1 (1865), 1-45.

[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.

[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).