×

Leibniz in Cantor’s paradise: a dialogue on the actual infinite. (English) Zbl 1448.01010

De Risi, Vincenzo (ed.), Leibniz and the structure of sciences. Modern perspectives on the history of logic, mathematics, epistemology. Based on the summer school on Leibniz, Leipzig and Hannover, Germany, July 7–16, 2016, and the conference on “Leibniz and the Sciences”, Leipzig, Germany, November 14–16, 2016. Cham: Springer. Boston Stud. Philos. Hist. Sci. 337, 71-109 (2019).
The main part of this paper consists of an imaginary dialogue between Leibniz and Cantor on the theme of the infinity. The two protagonists compare and comment on their reciprocal opinions in a series of ideas exchanges where the direct quotations from Leibniz’s and Cantor’s works are alternated with reasonings developed by the author. To distinguish the arguments drawn directly from Leibniz’s and Cantor’s writings from his own insertions, the author uses two different typographical characters. After the abstract, the dialogue is preceded by a brief introduction and closed by a conclusion named ‘afterword’ by the author.
The thesis of this paper is that, relying on the scholastic distinction between syncategorematic and categorematic infinite, and specifically on the concept of syncategorematic infinite, Leibniz devised a third kind of infinity which is neither a potential infinity nor Cantor’s transfinite, though being an actual infinite. In the author’s mind, the syncategorematic actual infinite allows Leibniz: 1) to show that his conception of matter, shared by Cantor, according to which matter is actually divided at infinity, is consistent; 2) to prove that, in contrast, Cantor’s transfinite is neither compatible with the actually infinite division of matter, nor (which is even more important) is coherent from a mathematical standpoint.
Content. At the beginning of the dialogue Cantor claims that Leibniz and he himself shares the idea that the actual infinity pertains to matter. Leibniz clarifies that, in his view each part of matter is actually divided at infinity and that, hence, “[…] the least particle ought to be considered as a world full of an infinity of different creatures” (p. 73). Therefore, Cantor has no doubt that Leibniz will agree with his concept of transfinite applied to matter (monads of corporeal matter are denumerable, monads of aethereal matter have the cardinality of continuum). However, Leibniz claims that he believes neither in the existence of transfinite numbers nor in that of and infinite line nor in any infinite quantity considered as a whole. For him the actual infinite exists insofar as multiplicities exist (as those of all things) which surpasses every finite number. In this sense, they are actually infinite. But no number can be assigned to them (p. 74). Cantor objects that Leibniz is now sustaining a pure potential concept of infinity. To explain that this is not the case, Leibniz refers to the scholastic distinction between categorematic and syncategorematic infinite. In particular, referring to the syncategorematic meaning of the term “infinite”, Leibniz highlighted that matter is actually divided at infinite, namely, “there are not so many divisions that there are not more” (p. 75). But no number can express the quantity of such divisions because they exceed any number. Cantor objects that the syncategorematic infinite is nothing but a potential infinite. However, Leibniz claims that the notion of syncategorematic allows us to admit the actual infinite without admitting an infinite while. The only infinite whole (“hypercategorematic”) is the indivisible and undivided Absolute (p. 77). Nonetheless, Leibniz claims, there is a second kind of actual infinity: the maximum, namely the greatest in its kind. Example: an infinite line which cannot be furtherly produced, or the infinite space, which cannot be furtherly extended. However, Leibniz agrees with Cantor that “there is no maximum in things” (p. 78). Finally, there is the infinite at the lowest degree, which indicates a quantity to which no finite number can be assigned as the area included between a hyperbola and its asymptotes. Now the argument arrives at one of his culminating points: Cantor claims the existence of the transfinite numbers (he begins with \(\aleph_0\)). First of all, he shows that the classical arguments against the existence of such numbers are untenable (Aristotle, Tongiorgi, p. 79). Leibniz clarifies with an analogy his idea of the actual division of matter. Given the series \(a_n=(\frac12)^n\), all its terms might correspond to an actual infinite division of matter. This does not mean that an infinitieth term exists (p. 81). Cantor agrees with this view, but claims his classical and well-known position, according to which there is no contradiction in thinking that there are infinite sequences (as that of the natural numbers) which have an infinite number as their limit (p. 81). Cantor explains his concept of transfinite number, until arriving at his idea that the transfinite numbers are mode existing from eternity in God’s intellect (pp. 81–83). Before arguing against the existence of the transfinite numbers, Leibniz clarifies the meaning in which the syncategorematic infinite can be actual. He considers the prime numbers. Euclid proved that their quantity is infinite because, supposing it to be finite a contradiction derives. However, this does not mean that you are legitimate to consider the prime numbers as a whole and to assign them a transfinite number. Here, Cantor clarifies to be perfectly aware that Euclid’s proof in itself does not demonstrate that the whole of primes exists, but that without such assumption, the syncategorematic infinity would have no basis on which it might exist (p. 85). After this reciprocal explanation, Leibniz declares to be ready to show: 1) Cantor’s transfinite numbers are not applicable to Leibniz’s concept of matter, shared by Cantor, 2) they are in themselves contradictory. Leibniz starts from three postulates: “(1) Any part of matter (or body) is actually divided into further parts. (2) Each such body is the aggregate of the parts into which it is divided. (3) Each part of a given body is the result of a division of that body or of a part of that body” (p. 86). By (1) and (2) immediately follows any body to be actually divided in the syncategorematic sense (pp. 86–87). Cantor agrees with this line or reasoning, but also argues that it is only natural to think of all the divisions and to assign them a number; a transfinite number, in fact (p. 87). Now, Leibniz explains this reasoning, granted \(\omega\) to be the first ordinal transfinite number: suppose the \(\omega\)-th division of matter \(P_\omega\) exists, then it derives from the division \(P_{\omega-1}\) and since \(\omega\) is the least transfinite number should be finite, against a well-known Cantor’s idea (p. 88). In other terms, the infinite able to express Leibniz’s theory of matter should admit the ordinary recursion, whereas Cantor’s transfinite ordinal do not (p. 89). Cantor does not agree: the divisions \(\{P_1, P_2, P_3,\dots\}\) are actual infinite in their totality, but each step indicates a finite division. Considered as a process, the infinite in question is only potential. Hence, something like \(\omega-1\) does not exist and Leibniz’s syncategorematic infinite is a potential one. Leibniz, then, relying upon a scholastic distinction argues that the universal quantifier can be used is collective term “for all the elements of a collection” or in a distributive meaning “for each element of a collection”. Therefore, something can be enunciated distributively on all numbers (pp. 89–90). Leibniz continues claiming that the concept of the number of all numbers is contradictory because the axiom that the whole is bigger that a part is not fulfilled (example of the bijection between even numbers and all the numbers). Cantor, then, explains his famous concept of cardinality or power of a set and highlights that the even numbers and all the numbers have the same cardinality, though being the second set richer in entities than the first one (p. 91). Leibniz refuses to consider the 1-1 correspondence between two multiplicities as a criterion of equality unless a previous proof that such multiplicities can be coherently considered as wholes is shown (p. 92). He judges the axiom part-whole as a fundamental criterion to distinguish whether a multiplicity can be considered as a whole. Obviously, Cantor claims that, as to infinite sets, the axiom part-whole is not applicable (p. 92). Therefore, until this moment there are two positions: Leibniz accepts the part-whole axiom and thinks that the actual infinity is admissible, but only in its distributive meaning, namely no infinity can be considered as a whole and to an infinite collection no number can be ascribed. Whereas Cantor considers, under his well-known conditions, that the infinite collections can constitute a set to which a transfinite number is ascribed and that the axiom part-whole is not valid for infinite sets. Cantor claims his thesis that a new concept can be introduced in mathematics if it is self-consistent and does not contradict the other parts of mathematics. Since his notion of transfinite numbers satisfies this requisite, it is legitimate. As a confirmation of the exactness of his view, Cantor shows Leibniz the indisputably correct proof concerning the rational numbers’ denumerability (p. 94). Leibniz confirms his view: Cantor’s proof shows that the rational numbers are countable, but not that they form a totality (ivi, pp. 94–95). The discussion continues with various skirmishes, but without any new decisive argument in favour of the two contenders. Cantor shows Leibniz his diagonal proof of the not denumerability of the real numbers included in the interval \((0,1)\) (p. 96). Again, Leibniz thinks that this proof does not imply that the set of real numbers has a cardinality greater than the set of the natural numbers because this conclusion is based on the assumption that such multiplicities exist as wholes, as sets, which is what he judges contradictory (p. 96). Leibniz claims that, once founded the diagonal number, you can add it to the list which Cantor supposes to express all the real numbers and to repeat the reasoning. For, Leibniz states “So your conclusion should not have been that M is nondenumerable, but that it has no complete enumeration, and is therefore not a denumerable totality” (p. 97). Obviously, Cantor answers that his proof supposes ad absurdum that the all real numbers between \(0\) and \(1\) belong to his list. It seems to me that, in this phase, the discussion does not move a step because the problem is always the same: to accept or not to accept actually infinite totalities. There is then a brief discussion on the fact that Cantor refuses Euclid’s axiom according to which the part is less than the whole, whereas Leibniz accepts (pp. 98–99). Leibniz tries now another line of attack: he claims that Cantor admits that some multiplicities cannot be coherently thought of as a whole. Cantor agrees and explains his well-known ideas on the inconsistent multiplicities (pp. 99–100). He presents the paradox of the biggest ordinal, also known as Burali-Forti paradox, but discovered by Cantor himself. This paradox shows that the multiplicity of all the ordinal numbers cannot constitute a new whole, a set. Leibniz returns to his argument that no all of the collections can be considered as wholes. In particular, this is possible only if a real definition of a collection can be offered. Leibniz considers a definition to be real if the defined thing is possible and implies no contradiction. The recursive definitions in mathematics are real, but, according to Leibniz, the transfinite numbers are not passible of recursive definitions (p. 103). After that, Leibniz considers all the monads. Cantor agrees that there are actually infinite monads. Leibniz is going to prove that the totality of all the true units (corporeal monads) as a whole does not exist. He called U the supposed set of all the real unities. According to the genetic formation of the sets invoked by Cantor U cannot be included in U itself. Therefore, U is and is not the aggregate of all the real units. Given this contradiction, U cannot exist (p. 104). Cantor answers that this simply proves that the collections of all the corporeal monads is not a monad. Now, Leibniz moves a critics strictly connected to the previous ones. He wonders why Cantor is sure that the infinite sets and the associated transfinite numbers do not lead to any contradiction as it is the case with the class of all the ordinals numbers (p. 105). Cantor answers that the consistency of the finite sets too is assumed by an axiom. The dialog ends here. The author adds an ‘afterword’ as a conclusion where the main points of the Leibniz-Cantor dialogue are summarised and where he expounds his opinions: he thinks that Leibniz’s syncategorematic actual infinite is a clear alternative to transfinite and potential infinite, which might offer a foundation to mathematics (p. 106). A second point is that “This conception of the infinite as not involving infinite number is appropriate to Leibniz’s conception of the actual infinite division of matter; whereas Cantor’s transfinite is not, since one cannot get to an \(\omega\)-th part by recursively dividing” (p. 106). The idea that, in mathematics, the real definitions are connected to a recursive process implies, in the author’s mind, his philosophy of mathematics to be close to constructivism and intuitionism, though the temporal level is not involved. Cantor’s diagonal arguments is acceptable only if one is available to admit that the natural numbers form a set (pp. 107–108). Finally, the author claims that Leibniz argues that the impossibility to consider a collection of unities as a new unity can be obtained through a reasoning analogous to the one used by Cantor to prove the impossibility that the collections of all the ordinals is a set.
Commentary. The true protagonist of this dialogue is Leibniz. With regard to Cantor, there is not much to say because the author presents in a dialogic form the classical positions expressed by Cantor in the course of his mathematical career. Whereas, the author speaks through Leibniz and has the intention to prove the unsustainability of Cantor’s conception concerning the existence of the transfinite numbers. The arguments expounded against Cantor are not convincing. Let us start from Leibniz’s idea that matter is actually divided at infinity. There is no doubt that this is one of the milestones of Leibniz’s ontology. It is enough to refer to the letter to Malebranche on 22 June 1679, to the Avant-propos of the Nouveaux essais sur l’entendement humain or to Monadology, §57, only to give some examples. It is true that Cantor adopted this view when he claimed that the monads of corporeal matter have the cardinality of the natural numbers, whereas those of the ether have the cardinality of the continuum. However, while in Leibniz this is an important claim of his ontology, in Cantor it is an absolutely marginal part of his conception which has properly nothing to do with his mathematical theory. Rather, it is an odd and naïve application of such a theory to nature. Well, I think that it makes no sense to claim that “matter is divided at infinity”. It is a string of words which seems to have a meaning, but it does not have. Leibniz analogy with the series \(a_n=(\frac12)^n\) is, in fact, a mere analogy, which is of no help to guess what means that matter is actually divided at infinity. Either matter is discrete or it is divisible at infinity in the potential sense of this term. I think that no other reasonable meaning of division of matter can exist. Therefore, it is not possible to sustain a consistent conception of actual infinity relying upon this basis. Under this respect, the distinction, which is undoubtedly refined, between the two meaning of the syncategorematic infinite is of no help and the interpretation of syncategorematic infinity as an actual one is subject to similar critics to which the infinite division of matter is subject. Obviously, several positions are possible with regard to the actual infinite: it is possible to claim that actually infinite totalities exist, but no actual number do, à la Spinoza; or, as Leibniz does in this dialogue, it is possible to assume a series of axioms which make the existence of actual infinite set impossible and, thence, also that of transfinite numbers. Of course, this is a legitimate axiomatic choice, from which the interesting interpretations à la Wittgenstein of the diagonal argument follows. However, it is also possible, as Cantor did, to assume a series of axioms (which Cantor made implicitly) and definitions which allow the existence of both infinite sets and cardinal numbers. The author-Leibniz is absolutely right in claiming that Cantor’s distinction between consistent and inconsistent multiplicities is not clear and that this might spread an odd light on Cantor’s theory of transfinite. However, the axiomatic set theories – born a few years after Cantor’s work – have also the purpose to avoid, with a series of well known axioms, the possibility of inconsistent multiplicities and to permit that of sets. As a matter of fact, the transcription of Cantor set theory into the Zermelo-Fraenkel theory seems to me faithful to Cantor’s original thought. Ergo, several axiomatic choices are possible which are consistent with several different positions in respect to the actual infinite. However, two concerns of this paper seems to me not correct: 1) the idea that through the concept of the actually infinite division of matter and through the highlighted meaning of syncategorematic infinite a consistent conception of the actual infinite may be obtained; 2) Cantor’s conception and theory of transfinite is inconsistent. If, rather than as two theses claimed with absolute certainty the whole discussion of this dialogue is interpreted as a series of proposals on which to discuss, then, under such a perspective, this paper is conceptually rich and stimulating to rethink from a conceptual and historical standpoint several notions expressed in the philosophy of the 17th century as well as their possible links with the scholastic tradition.
For the entire collection see [Zbl 1432.01007].

MSC:

01A50 History of mathematics in the 18th century
01A60 History of mathematics in the 20th century
00A30 Philosophy of mathematics
03-03 History of mathematical logic and foundations
03E10 Ordinal and cardinal numbers

Biographic References:

Leibniz, Gottfried Wilhelm; Cantor, Georg
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Arthur, R.T.W. 2015. Chapter 7: Leibniz’s Actual Infinite in Relation to his Analysis of Matter. In Interrelations Between Mathematics and Philosophy, ed. G. W. Leibniz, 137-156. Springer: Archimedes Series, Norma Goethe, Philip Beeley and David Rabouin. · Zbl 1332.01012
[2] Bassler, O. Bradley. 1998. Leibniz on the Indefinite as Infinite. The Review of Metaphysics 51 (June): 849-874.
[3] Beeley, Philip. 1996. Kontinuität und Mechanismus. Stuttgart: Franz Steiner. · Zbl 0856.01010
[4] Benardete, José A. 1964. Infinity: An Essay in Metaphysics. Oxford: Clarendon Press.
[5] Bolzano, Bernard. 1950. Paradoxes of the Infinite. London: Routledge/Kegan Paul. · Zbl 0039.00506
[6] Cantor, G. 1879-83. Über unendliche lineare Punktmannigfaltigkeiten, 1-4, CGA 139-164.
[7] ———. 1883. Grundlagen einer allgemeinenen Mannigfaltigkeitslehre, CGA 165-209.
[8] ———. 1884. Die Grundlagen der Arithmetik, CGA 440-451.
[9] ———. 1885a. Über verschiedene Theoreme aus der Theorie der Punktmengen in einem n-fach ausgedehnten stetigen Raume G_n, CGA 261-277.
[10] ———. 1885b. Über die verschiedenen Standpunkte in bezug auf das actualle Unendliche, CGA 370-77.
[11] ———. 1886. Über die verschiedenen Ansichten in Bezug auf die actualunendlichen Zahlen. Bihand Till Koniglen Svenska Vetenskaps Akademiens Handigar 11 (19): 1-10.
[12] ———. 1887-8. Mitteilungen zur Lehre von Transfinitum, CGA 378-439.
[13] ———. 1890-91. Über eine elementare Frage der Mannigfaltigskeitslehre, CGA 278-281.
[14] ———. 1895-7. Beiträge zur Begründung der transfiniten Mengenlehre, CGA 282-356.
[15] ———. 1962. Gesammelte Abhandlungen, ed. Ernst Zermelo. Berlin, 1932; repr. Hildesheim: Georg Olms. Cited as CGA.
[16] Dauben, Joselph Warren. 1979. Georg Cantor: His Mathematics and Philosophy of the Infinite. Boston: Harvard University Press. · Zbl 0463.01009
[17] Dedekind, R. 1963. (German original, 1872). Essays on the Theory of Numbers. New York: Dover Publications. · Zbl 0112.28101
[18] Dummett, Michael. 1977. Elements of Intuitionism. Oxford: Clarendon Press. · Zbl 0358.02032
[19] Geach, Peter. 1967. Comments on an essay of Abraham Robinson’s. In Problems in the Philosophy of Mathematics, ed. Imre Lakatos, 41-42. Amsterdam: North-Holland Publishing Co.
[20] Grattan-Guinness, Ivor. 1970. An Unpublished Paper by Georg Cantor: Principien einer Theorie der Ordnungstypen; Erste Mittheilung. Acta Mathematica 124: 65-107. · Zbl 0186.29603 · doi:10.1007/BF02394569
[21] ———. 1974. The Rediscovery of the Cantor-Dedekind Correspondence. Jahres-bericht der deutschen Mathematike-Vereinigung 76: 104-139. · Zbl 0292.01017
[22] Hallett, Michael. 1984. Cantorian Set Theory and the Limitation of Size. Oxford: Clarendon Press. · Zbl 0656.03030
[23] Lavine, Shaughan. 1994. Understanding the Infinite. Cambridge, MA: Harvard University Press. · Zbl 0961.03533
[24] Leibniz, G.W. 1981. New Essays on Human Understanding. Trans. Peter Remnant and Jonathan Bennett. Cambridge: Cambridge University Press. Cited as NE.
[25] ———. 1849-63. Leibnizens Mathematische Schriften. C. I. Gerhardt. Berlin/Halle: Asher and Schmidt, reprint ed. Hildesheim: Georg Olms, 1971. 7 vols. Cited as GM.
[26] ———. 1875-90. Die Philosophische Schriften von Gottfried Wilhelm Leibniz, ed. C.I. Gerhardt. Berlin: Weidmann; reprint ed. Hildesheim/New York: Georg Olms, 1978. 7 vols. Cited as GP.
[27] ———. 1923. Sämtliche Schriften und Briefe, ed. Akademie der Wissenschaften der DDR. Darmstadt/Berlin: Akademie-Verlag; cited by series, volume and page, as A VI 2, 123, etc.
[28] ———. 1993. De Quadratura Arithmetica, [1676]. Ed. Eberhard Knobloch. Göttingen: Vandenhoek & Ruprecht.
[29] Levey, Samuel. 1998. Leibniz on Mathematics and the Actually Infinite Division of Matter. The Philosophical Review 107: 49-96. · doi:10.2307/2998315
[30] ———. 1999. Leibniz’s Constructivism and Infinitely Folded Matter 134 162, New Essays on the Rationalists, Rocco Gennaro Charles Huenemann New York, Oxford University Press
[31] Rescher, Nicholas. 1967. The Philosophy of Leibniz. Englewood Cliffs: Prentice Hall. · Zbl 0162.30806
[32] Rucker, R. 1983. Infinity and the Mind: The Science and Philosophy of the Infinite. (originally published by Birkhäuser, 1982). New York: Bantam. · Zbl 0485.03001
[33] Russell, Bertrand. 1903. Principles of Mathematics. New York: W. W. Norton.
[34] Uckelman, Sara L. 2015. The Logic of Categorematic and Syncategorematic Infinity. Synthese 192: 2361-2377. · Zbl 1369.03083 · doi:10.1007/s11229-015-0670-z
[35] Wittgenstein, Ludwig.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.