Abstract
This paper consists of two parts and has two aims. The first is to elaborate on the historical-philosophical background of Cantor’s notions of infinity in the context of Spinoza’s influence on him. To achieve this aim, in the first part I compare Spinoza’s and Cantor’s conceptions of actual infinity along with their mathematical implications. Explaining the metaphysical, conceptual, and methodological aspects of Cantor’s expansion of the orthodox finitist conception of number of his time, I discuss how he adopts Spinoza’s motifs to overcome the challenges on his way to proposing his transfinite infinites. To achieve the second aim, i.e., to investigate the philosophical foundations of Cantor’s transfinite infinites, I follow a three-step outline analyzing Cantor’s three principles along with their metaphysical and epistemological-linguistic aspects. I analyze the strengths and weak sides of Lakoff and Núñez’s Basic Metaphor of Infinity and Pantsar’s Process → Object Metaphor, then compare their interpretations with mine. I conclude that the metaphors can be improved when their source domains are limited to the target domains they apply to and when supported by a thorough analysis of the historical-philosophical background of their target domains.
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Notes
The notion of AcI in Deo influences Cantor’s thought mostly in the context of self-referential paradoxes. For more on why the AcI in Deo is not suitable type of AcI for Cantor’s project, see Jané, 1995.
The translation of Cantor’s Natura naturante or Spinoza’s Natura Naturata as Nature naturing sounds awkward in English. Yet, when Spinoza’s attitude against the idea of creation by a transcendent God is considered (2002a, 225 [prop. 15, schol.]), Nature as self-generating can be proposed as a better alternative, of course on the condition of that we remember that Natura Naturans and Natura Naturata do not imply any ontological division in Spinoza’s system.
All quotes from Cantor’s Gesammelte (1932) and Briefe (1991) are my translations.
To translate into Spinoza’s jargon, the absence of the AcI in abstracto corresponds to the absence of the mediate infinite mode of the attribute of thought. The absence of such a mode, or the AcI in abstracto in this context, is not directly relevant to the matter at hand and it is an unresolved problem among Spinoza scholars. For some solutions that are open to discussion and that might be helpful for further comparative studies between Cantor’s and Spinoza’s thoughts see Schmaltz, 1997, and Pinheiro, 2015.
Note that in Spinoza’s epistemology, imagination is directly related to the perceptions, and is classified as the first type of knowledge, which is the cause of all falsity (see 2002a, 266 − 68 [prop. 40 & 41]).
Spinoza himself discusses the issue in a body-part analogy (see Let. 32 in 2002b, 848–851).
Despite the distinctness of the concept of infinity being a concern for Lakoff and Núñez, it is left unexplained in their account (see 2000). Let me save the details to the second part.
One referee recommended substituting ‘inaccessibility’ with ‘irreducibility’. However, I chose to retain ‘irreducibility’ as the central theme due to potential confusion associated with the specific meaning of ‘inaccessible’. In mathematical terms, a number, e.g., a cardinal one, is considered inaccessible if it cannot be derived through specific set-theoretic operations on smaller, accessible numbers. As I aim to elucidate its construction can be constructed using alternative tools like higher-order thinking, while ω remains inaccessible through set-theoretical means such as the successor function, its construction can still be explained from a philosophical perspective. The preference for ‘irreducibility’ stems from its implication that, although ω eludes some set-theoretical tools, it remains accessible and accountable through the philosophical framework discussed below. Additionally, Cantor’s conception of ω as a set that cannot be reduced to the totality of all the numbers of the lower class is not only due to Spinoza’s influence on Cantor but also due Plato’s influence on Cantor’s conception sets. To describe his notion of set, Cantor quotes Plato’s essential concepts, such as idea, mikton, apeiron, peras, etc. (see Cantor, 2005, 916, fn.1). For a full-fledged account of Plato’s influence on Cantor’s conception of set, see Hauser, 2005.
In this context, the attributes can be considered as the ways God expresses His essence (see Spinoza, 2002a, 222 [prop. 11]). According to Spinoza, there are infinitely many attributes of God, but we know only two of them, namely the attribute of extension (2002a, 245 [prop. 2]) and the attribute of thought (2002a, 245 [prop. 1]). For example, God is an extended being for Spinoza and it is through the attribute of extension that God expresses this essential feature of Himself and in the form of extended modi, i.e., the physical objects. Likewise, God is thinking being and expresses this essential feature by the attribute of thought and in the form of the thought entities.
Note that Spinoza’s finitist conception of number is distinguished from that of the orthodox finitists because he contends that, as we cannot determine e.g., which five objects should be taken as the transient reality of the number 5, no number has transient reality (see Melamed, 2000).
Cantor may seem to treat transient reality as less important than immanent reality. This is misleading. Consider the following statement about the potential infinity of our understanding and the transfinite: “the potential infinite is only an auxiliary and relational concept and always points to an underlying transfinitum, without which it can neither be nor be thought” (Cantor, 1932a, 391). As shall be seen below, Cantor’s notion of the continuity of the continuum is fundamental to his conception of the potentially infinite processes.
The last point is about the extensional and intensional conceptions of infinity and about how transfinite numbers determine what they determine as mathematical concepts. More about these are in due course.
I use the traditional A/B method in citing Kant’s Critique of Pure Reason and the quotes are from Kant, 1998.
See Hamilton, 1837.
This claim has its roots in Spinoza. Discussing and adopting Spinoza’s conception of adequate ideas and his parallelism doctrine, Cantor, quoting Zeller, says that his position about the adequacy of ideas is also Platonic and Parmenidean in a specific sense: “Only conceptual knowledge should (according to Plato) provide true knowledge. However, as much truth is due to our ideas—Plato shares this assumption with others (Parmenides)—just as much reality must be due to their object, and vice versa” (1932a, 206–207). Yet, the combination of Platonic and Parmenidean elements in a Spinozist position has a strong implication, the ideatum of the adequate ideas can be processes because Spinoza’s metaphysics is more comprehensive than Parmenides’ as it includes the notions of motion and rest, and the adequate ideas about them. In Cantor’s frame, this theme transforms into the relation of reality to knowability, where “to the extent that something is, it is also knowable” (1932a, 206–207). It might sound as if the reality of something has the priority over its knowledge, yet for Cantor, “the range of definable quantities is not closed with finite quantities, and the limits of our knowledge [Erkenntnis] can be expanded [by definitions]” (1932a, 176). As shown below, this theme plays an important role in Cantor’s generation of the higher-level cardinalities and the ordinals.
References
Bartha, P. (2022). “Analogy and Analogical Reasoning” in The Stanford Encyclopedia of Philosophy. (ed. Edward N. Zalta). Stanford University. https://plato.stanford.edu/archives/sum2022/entries/reasoning-analogy/ (retrieved 18.08.2022).
Beck, J. (2017). Can bootstrapping explain concept learning? Cognition, 158, 110–121.
Bussotti, P., & Tapp, C. (2009). The influence of Spinoza’s concept of infinity on Cantor’s set theory. Studies in History and Philosophy of Science Part A, 40(1), 25–35. https://doi.org/10.1016/j.shpsa.2008.12.010.
Cantor, G. (1932a). Über unendliche lineare Punktmannigfaltigkeiten. In E. Zermelo (Ed.), Gesammelte Abhandlungen Mathematischen Und Philosophischen Inhalts (pp. 139–247). Julius Springer.
Cantor, G. (1932b). Über die verschiedenen Standpunkte in bezug auf das aktuelle Unendliche. In E. Zermelo (Ed.), Gesammelte Abhandlungen Mathematischen Und Philosophischen Inhalts (pp. 370–378). Julius Springer.
Cantor, G. (1932c). Mitteilungen Zur Lehre Vom Transfiniten. In E. Zermelo (Ed.), Gesammelte Abhandlungen Mathematischen Und Philosophischen Inhalts (pp. 378–440). Julius Springer.
Cantor, G. (1932d). Die kleinste transfinite Kardinalzahl Alef-null. In E. Zermelo (Ed.), Gesammelte Abhandlungen Mathematischen Und Philosophischen Inhalts (pp. 292–296). Julius Springer.
Cantor, G. (1991). Briefe. (eds. Herbert Meschkowski and Winfried Nilson). Berlin: Springer-Verlag.
Cantor, G. (2005). Foundations of a general theory of manifolds: A mathematico-philosophical investigation into the theory of the infinite. In W. Ewald (Ed.), From Kant to Hilbert (II vol., pp. 878–920). Oxford University Press.
Carey, S. (2004). Bootstrapping & the origin of concepts. Daedalus, 133(1), 59–68.
Dubinsky, E. (1991). Reflective abstraction in advanced mathematical thinking. In D. Tall (Ed.), Advanced mathematical thinking (pp. 95–126). Springer.
Dubinsky, E., Weller, K., McDonald, M. A., & Brown, A. (2005a). Some historical issues and paradoxes regarding the concept of infinity: An apos-based analysis: Part 1. Educational Studies in Mathematics. 58, 335–359.
Dubinsky, E., Weller, K., McDonald, M. A., & Brown, A. (2005b). Some historical issues and paradoxes regarding the concept of infinity: An APOS analysis: Part 2. Educational Studies in Mathematics. 60 (2), 253–266.
Fraenkel, A. A. H. (1928). Einleitung in die mengelehre (3rd ed.). Springer.
Gray, E., & Tall, D. (1994). Duality, ambiguity and flexibility: A proceptual view of simple arithmetic. The Journal for Research in Mathematics Education, 26(2), 115–141.
Hamilton, W. R. (1837). “From the Theory of conjugate functions, or algebraic couples: with a preliminary and elementary essay on algebra as the science of pure time” in From Kant to Hilbert: A Source Book in the Foundations of Mathematics (vol. 1), (ed., William Ewald). (1999). Oxford: Clarendon Press, 369–375.
Harris, E. E. (1995). The substance of Spinoza. Humanities Press.
Hauser, K. (2005). Cantor’s Concept of Set in the light of Plato’s Philebus. The Review of Metaphysics, 63(4), 783–805.
Jané, I. (1995). The role of the absolute infinite in Cantor’s conception of set. Erkenntnis, 42(3), 375–402.
Kant, I. (1998). Critique of Pure Reason, (ed. & trans. Paul Guyer, Allen W. Wood). New York: Cambridge University Press.
Lakoff, G., & Nunez, R. (2000). Where mathematics comes from. Basic Books.
Lucash, F. (1996). The co-extensiveness of the attributes in Spinoza. Southwest Philosophy Review, 12(2), 51–61.
Melamed, Y. Y. (2000). On the exact science of Nonbeings: Spinoza’s view of Mathematics. Iyyun: The Jerusalem Philosophical Quarterly, 49(1), 3–22.
Newstead, A. (2009). Cantor on infinity in Nature, Number, and the Divine mind. American Catholic Philosophical Quarterly, 83(4), 533–553.
Nunez, R. (2005). Creating mathematical infinities: The beauty of transfinite cardinals. Journal of Pragmatics, 37(10), 1717–1741.
Pantsar, M. (2015). In search of ℵ0: How infinity can be created. Synthese (SI: Infinity), 192, 2489–2511. https://doi.org/10.1007/s11229-015-0775-4.
Pantsar, M. (2018). Early numerical cognition and mathemtical processes. Theoria: An International Journal for Theory History and Foundations of Science, 33(2), 285–304.
Pantsar, M. (2021). Bootstrapping of integer concepts: The stronger deviant-nterpretation challenge (and how to solve it). Synthese. 199, 5791–5814.
Pinheiro, U. (2015). Looking for Spinoza’s missing infinite Mode of Thought. The Philosophical Forum, 46(4), 363–376. https://doi.org/10.1111/phil.12083.
Ryle, G. (1949). The conception of mind. Hutchinson.
Schmaltz, T. (1997). Spinoza’s mediate infinite Mode. Journal of the History of Philosophy, 35(2), 199–235.
Sfard, A. (1991). On the dual nature of mathematical conceptions: Reflections on processes and objects as different sides of the same coin. Educational Studies in Mathematics, 22(1), 1–36.
Spinoza, B. (2002a). “Ethics” in Spinoza Complete Works, (ed. Michael L. Morgan), (trans. Samuel Shirley). Indianapolis: Hackett.
Spinoza, B. (2002b). “Letters” in Spinoza Complete Works, (ed. Michael L. Morgan), (trans. Samuel Shirley). Indianapolis: Hackett.
Spinoza, B. (2002c). “Treatise on the Emendation of the Intellect” in Spinoza Complete Works, (ed. Michael L. Morgan), (trans. Samuel Shirley). Indianapolis: Hackett.
Stauffer, L. (1993). Spinoza, Cantor, and infinity. Southern Philosophical Studies, 15, 74–81.
Tapp, C. (2005). Kardinalität Und Kardinäle: Wissenschaftshistorische Aufarbeitung Der Korrespondenz Zwischen Georg Cantor Und Katholischen Theologen Seiner Zeit (p. 53). Steiner. Boethius.
Von Neumann, J. (1923). “On the introduction of transfinite numbers” in From Frege to Gödel: A Source Book in Mathematical Logic, 1879–1931. (ed. Jean van Heijenoort, trans. Beverly Woodward). (1967). Cambridge & Massachusetts: Harvard University Press, 346–355.
Wagner, R. (2012). Infinity Metaphors, idealism, and the Applicability of Mathematics. Iyyun: The Jerusalem Philosophical Quarterly, 61(2), 129–148.
Winter, B., & Yoshimi, J. (2020). Metaphor and the philosophical implications of Embodied Mathematics. Frontiers in Psychology, 11, 569–487. https://doi.org/10.3389/fpsyg.2020.569487.
Zermelo, E. (1908). “Investigations in the foundations of Set Theory I” in From Frege to Gödel: A Source Book in Mathematical Logic, 1879–1931. (Ed. Jean van Heijenoort). (1967). Cambridge and Massachusetts: Harvard University Press, 199–216.
Acknowledgements
This article represents an enhanced section of my unpublished PhD thesis titled “On the Constructions of Pure Numbers: An Epistemic-Constructivist Analysis”, which was conducted under the guidance of Prof. Dr. David GRÜNBERG. I am deeply grateful to the RP reading group members, Maşuk ŞİMŞEK, Koray AKÇAGÜNER, Dilek YARGAN, Hasan ÇAĞATAY, and Can ÇÖTELİ, for their valuable feedback on earlier versions of this paper.
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Birgül, O. Unveiling the philosophical foundations: On Cantor’s transfinite infinites and the metaphorical accounts of infinity. Synthese 202, 164 (2023). https://doi.org/10.1007/s11229-023-04379-w
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DOI: https://doi.org/10.1007/s11229-023-04379-w