Abstract
Although number sentences are ostensibly simple, familiar, and applicable, the justification for our arithmetical beliefs has been considered mysterious by the philosophical tradition. In this paper, I argue that such a mystery is due to a preconception of two realities, one mathematical and one nonmathematical, which are alien to each other. My proposal shows that the theory of numbers as properties entails a homogeneous domain in which arithmetical and nonmathematical truth occur. As a result, the possibility of arithmetical knowledge is simply a consequence of the possibility of ordinary knowledge.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
According to (Field 1989, pp. 232–233), his reformulation of the Benacerraf’s problem does not depend on causal epistemology. However, this formulation seems to presuppose a reliabilist epistemology, which is nothing but the evolution of causal epistemology (with its own objections). In any case, without explaining how to determine such reliability, we can assume that Field is again imposing a causal requirement. [See Clarke–Doane 2017]
Benacerraf calls ‘combinatorial’ those theories of mathematical truth according to which “the truth conditions for arithmetic sentences are given as their [...] derivability from specified sets of axioms". [p. 665]
Undoubtedly, different discourses give rise to variations in truth conditions. An illustrative example is the case of empty names. However, these cases do not fall within the relevant fragment of language that concerns Benacerraf. Although the statement ‘Tom Sawyer was born in Missouri’ has the same form as ‘Aristotle was born in Stagira’, it is fairly clear that our epistemic relationship with the statements is different in the relevant sense.
Yablo (2014) considers different ways of being true for a sentence Note that the discussion here is different. Even if there are different ways of being true, that does not imply that there are different kinds of truth.
It might seem that, in this example, a correspondence theory of truth is presupposed. However, this use of ‘correspond’ is not linked to specific semantic theories but rather to speakers’ natural intuition about the relationship between language and its subject matter. While a correspondence theory of truth typically asserts that a statement’s truth or falsity is solely determined by its accuracy in describing the world, in everyday usage, speakers assume a more general connection between language and facts, which is related to their notions of truth falsehood, lying, misinforming, and so on, without delving too deeply into the specifics.
Burgess, for example, simply rejects Benacerraf’s problem as inapplicable to Quine’s program, on the grounds that our beliefs about abstract objects are justified as a whole as part of our best scientific theories, Burgess (1990).
Hartry Field. Realism, Mathematics, and Modality Blackwell, 1989.
The difficulties grounded in singular instantiation are not solved by distinguishing between relations and properties. The advocate of the principle of singularity assumes that places of arguments in relations are also singular. We may interpret (9) as a relation. Nevertheless, the relation also must be plurally instantiated. For example, (9) might be the relation between Ann, Ben, Carlos, and Danna (as such) and No Exit. On the other hand, it is unclear whether (9) is the relation performing No Exit occurring for some individuals. To begin with, the arity of the relation would be indeterminate, it depends on the individuals in it.
Yi’s technical apparatus allows properties to be described using a language of plurals. In this sense, he distinguishes between pure and impure plural properties—Impure plural properties are expansions of what he calls singular bases:
For example, the property that I call be human[s], indicated by ‘is-a-human\(^{P}\)’,is the plural expansion of be a human. Now, any plural expansion is distributive, that is, it is instantiated by some things as such if and only if it is instantiated by every one of them. [p. 187]
The reader interested in the exhaustive formalization can refer to Yi (1999).
The plurality of zero individuals is, as the definition indicates, that which has no elements (of course there are many concepts that determine this plurality. One of them, for example, is the concept of being different from itself).
Although the notion of property at stake is austere, it has been established that properties are identified by how they can be instantiated (regardless of whether they are actually instantiated). A plurality of minus one individuals cannot instantiate any plural property of multiplicity. Certainly, it may be argued that in the strict sense, there are no pluralities of zero individuals either, thus the property be zero is also not well defined. The assertion here is that the property be zero can be instantiated by any plurality that does not have individuals, and we can specify these pluralities. For example, if Be-zero(\(\alpha \)s), then \(\alpha \)s is the plurality such that if \(H(\beta ,\alpha s)\), then \(\beta \ne \beta \). This identification may be controversial under metaphysical considerations, but it succeeds specifying \(\alpha \)s. In contrast, it is unclear how this could be achieved for the property be \(-1\) (while still keeping the simplicity of the plural properties of multiplicity considered here).
Notes for Ludwig Darmstaedter, pp. 366–7
The neo-logicist program preserves logicism’s spirit of reducing the truths of arithmetic to logical statements that can be known without appealing to experience, thus solving the epistemic problem for arithmetical sentences. Frege’s concerns lead him to draw the axioms of arithmetic from the so-called Basic law V, which was supposed to achieve the ontology required by his metaphysical program:
As Russell showed, this principle presupposes that:
which is false. The neologicist proposal suggests adopting Hume’s Principle as the abstraction principle—since it does not presuppose (ii)—and deriving Peano axioms from it:
The contemporary debate addresses whether (iii) is a logical principle in such a way that it solves the epistemic problem for arithmetic.
Many thanks to the referee who suggested the contrast with neo-Fregeans to illuminate and specify the current proposal. Their comments have been quite insightful. I also thank them for the suggestion of the following abstraction-like rule:
‘The property ‘be n’ is identical to the property ‘be m’ if and only if for any plurality \(\alpha s\) and for any plurality \(\beta s\), the \(\alpha s\) are n and the \(\beta s\) are m if and only if the \(\alpha s\) and \(\beta s\) are equinumerous.
Note that pluralities do not constitute the equivalence class that number would be. The relevant relationship between equinumerous pluralities is that they both are plural instances of the same property of multiplicity. As the reviewer remarks, the statement above is true of properties; the process of going from predicate instantiation to properties is already reasonably widely accepted and at least only as contentious as the existence of properties is.
This question was discussed with Mario Gomez-Torrente during the Mathematics, Modality and Knowledge Symposium, at the CLMPST, 2023.
In Yablo’s framework, the literal content is what the statement would mean, if taken literally; the real content is what we in fact communicate with the statement. The literal content of “She has butterflies in her stomach” places various insects in her digestive tract; the real content only claims that she is nervous. The literal content of “The number of planets is even" says something about a relation between planets and a certain type of abstract object; the real content says only that the planets are evenly-numbered.
References
Balaguer, M. (1995). A Platonist epistemology. Synthese, 103(3), 303–325.
Balaguer, M. (1998). Platonism and anti-platonism in mathematics. Oxford University Press.
Baroody, A.J., Wilkins, J.L.M., Tiilikainen, S. (2003). The development of children’s understanding of additive commutativity: from protoquantitative concept to general concept? A.J. Baroody & A. Dowker (Eds.), The development of arithmetic concepts and skills: constructing adaptive expertise (pp. 127–160). Lawrence Erl–baum Associates.
Benacerraf, P. (1973). Mathematical truth. Journal of Philosophy, 70(19), 661–679.
Burgess, J. (1990). Epistemology & nominalism. A. Irvine (Ed.), Physicalism in mathematics (pp. 1–15). Dordrecht: Springer Netherlands.
Canobi, K. H., & Bethune, N. E. (2008). Number words in young children’s conceptual and procedural knowledge of addition, subtraction and inversion. Cognition, 108(3), 675–686.
Carey, S. (2009). The origin of concepts. Oxford University Press.
Chateubriand, O. (2016). Números como propiedades de segundo orden. J. Ferreirós (Ed.), El árbol de los números. cognición, lógica y práctica matemática (pp. 112–130). Sevilla: Universidad de Sevilla.
Clarke–Doane, J. (2017). What is the Benacerraf problem? F. Pataut (Ed.), New perspectives on the philosophy of paul benacerraf: Truth, objects, infinity. Springer Verlag.
Dehaene, S. (1997). The number sense: How the mind creates mathematics. Oxford University Press.
Feigenson, L., Dehaene, S., & Spelke, E. (2004). Core systems of number. Trends in Cognitive Sciences, 8(7), 307–314.
Field, H. (1980). Science without numbers. Princeton University Press.
Field, H. (1989). Realism, mathematics, and modality. Blackwell.
Frege, G. (1884). Foundations of arithmetic (J.L. Austin, Trans.). L. Nebert.
Gilmore, C. K., & Spelke, E. S. (1992). Children’s understanding of the relation between addition and subtraction. Cognition, 44(1), 43–74.
Gómez-Torrente, M. (2019). Roads to reference. Oxford University Press.
Hale, B. (2000). Reals by abstraction. The Proceedings of the Twentieth World Congress of Philosophy, 6, 197–207.
Heyting, A. (1959). Constructivity in mathematics. Amsterdam: North- Holland Pub. Co.
Hodes, H. T. (1984). Logicism and the ontological commitments of arithmetic. Journal of Philosophy, 81(3), 123–149.
Hofweber, T. (2005). Number determiners, numbers, and arithmetic. Philosophical Review, 114(2), 179–225.
Kripke, S.A. (1992). Logicism, wittgenstein, and de re beliefs about numbers. Unpublished transcript of the Whitehead Lectures delivered at Harvard University in May of 1992 .
Levine, S. C., Jordan, N. C., & Huttenlocher, J. (1992). Development of calculation abilities in young children. Journal of Experimental Child Psychology, 53(1), 72–103.
MacBride, F. (2003). Speaking with shadows: A study of neo-logicism. The British Journal for the Philosophy of Science, 54(1), 103–163.
Maddy, P. (1980). Perception and mathematical intuition. Philosophical Review, 89(2), 163–196.
Maddy, P. (1981). Sets and numbers. Noûs, 15(4), 495–511.
Moltmann, F. (2013). Reference to numbers in natural language. Philosophical Studies, 162(3), 499–536.
Oliver, A., & Smiley, T. (2013). Plural logic. Oxford University Press.
Rayo, A. (2002). Frege’s unofficial arithmetic. The Journal of Symbolic Logic, 67(4), 1623–1638.
Rayo, A. (2006). Beyond plurals. A. Rayo & G. Uzquiano (Eds.), Absolute generality (pp. 220–254). Oxford University Press.
Shapiro, S. (1983). Mathematics and reality. Philosophy of Science, 50(4), 523–548.
Topey, B. (2020). Realism, reliability, and epistemic possibility: On modally interpreting the benacerraf-field challenge. Synthese, 199(1–2), 4415–4436.
Vivanco, M. (2020). Referential uses of Arabic numerals. Manuscrito-International Journal of Philosophy, 43(4), 142–164.
Wiese, H. (2003). Numbers, language, and the human mind. Cambridge University Press.
Wiese, H. (2020). The co-evolution of number concepts and counting words. Lingua, 117(5), 758–772.
Yablo, S. (2014). Aboutness. Princeton University Press.
Yi, B. (1999). Is two a property? Journal of Philosophy, 96(4), 163–190.
Yi, B. (2005). The logic and meaning of plurals. part i. Journal of Philosophical Logic, 34, 459–506.
Acknowledgements
Versions of this material were presented at MIT, UNAM, University of Miami, Vienna (at the “What in the World(s)?!” conference), Prague (at the 16th CLMPST), and elsewhere. I owe thanks to Mario Gomez-Torrente, Carmen Curcó, Otávio Bueno, Max Fernandez, Simon Evnine, Stephen Yablo, Agustín Rayo, Milo Phillips-Brown, and Curtis Miller for their insightful comments and the fruitful discussions held throughout the process of this work.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
I declare that Dr. Otávio Bueno, one of Synthese’s editors, was my doctoral advisor at the University of Miami. However, I clarify that the research developed in this work was carried out entirely independently of the doctoral research developed under Dr. Bueno’s supervision, and the submission followed the rules to ensure a double-blinded review.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Vivanco, M. Numbers as properties. Synthese 202, 114 (2023). https://doi.org/10.1007/s11229-023-04330-z
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s11229-023-04330-z