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In search of \(\aleph _{0}\): how infinity can be created

  • S.I.: Infinity
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Abstract

In this paper I develop a philosophical account of actual mathematical infinity that does not demand ontologically or epistemologically problematic assumptions. The account is based on a simple metaphor in which we think of indefinitely continuing processes as defining objects. It is shown that such a metaphor is valid in terms of mathematical practice, as well as in line with empirical data on arithmetical cognition.

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Notes

  1. Physics 208a6.

  2. Historically, the most famous empiricist account of mathematics is Mill (1843) System of Logic.

  3. For more extensive information about the state of empirical study of numerical cognition, Nieder (2011), Dehaene (2011) and Dehaene and Brannon (2011) are good places to start. For a more detailed philosophical interpretation of the empirical results, see Pantsar (2014).

  4. It has been suggested that subitizing and the ANS could be part of the same ability, since ANS allows accurate estimation of small quantities. Expirements show, however, that subitizing and ANS have different characteristics, thus supporting the idea that they are separate systems (Revkin et al. 2008).

  5. This account is developed in more detail in Pantsar (2014).

  6. Of course one important question to ask is whether the results concerning monkeys can be applied to the human brain. There is a lot of evidence of this. The same areas in the human brain activate as in monkeys (Piazza et al. 2007) and college students have shown similar patterns as monkeys in number-ordering and quantity estimation tasks (Cantlon and Brannon 2006).

  7. In Nunez (2005) this account is amended and BMI is characterized as a “double-scope conceptual blend”, after Fauconnier and Turner (2002), This means that BMI has two input spaces, one coming from Completed Iterative Processes and the other from Endless Iterative Processes. The idea is that that both inputs are used in BMI to reach endless processes with final resultant states. Instead of Basic Metaphor of Infinity, in this updated account BMI stands for the Basic Mapping of Infinity. While the new account is conceptually more coherent, I do not see the difference as great enough to require independent treatment. Hence, in this paper I focus on the account of BMI given in Lakoff and Nunez (2000).

  8. The Conceptual Metaphor Theory has received a lot of criticism, as well. See Kovecses (2008) for a review of some of the most important problems and a proposal for a solution.

  9. In this regard, the closest relative to the approach here is probably the account of Sfard (1991), who also emphasizes the primary nature of processes in the process-objects duality.

  10. I want to thank the people at the Munich Center for Mathematical Philosophy at the Ludwig-Maximilians-Universität München, in particular Professor Hannes Leitgeb, for providing an inspiring environment for writing this paper. I’d also like to thank the participants of the conference “Foundations of the Formal Sciences VIII: History & Philosophy of Infinity” at the University of Cambridge for all their valuable comments and questions. I am in great debt to the two anonymous referees for suggestions on how to develop the paper. Finally, I want to thank the Academy of Finland for generously supporting this research.

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  1. Department of Philosophy, History, Culture and Art Studies, University of Helsinki, P. O. Box 24, 00014, Helsinki, Finland

    Markus Pantsar

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  1. Markus Pantsar
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Pantsar, M. In search of \(\aleph _{0}\): how infinity can be created. Synthese 192, 2489–2511 (2015). https://doi.org/10.1007/s11229-015-0775-4

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