Abstract
Rigid designators designate whatever they do in all possible worlds. Mathematical definite descriptions are usually considered paradigmatic examples of such expressions. The main aim of the present paper is to challenge this view. It is argued that mathematical definite descriptions cannot be rigid in the same sense as ordinary empirical definite descriptions because—assuming that mathematical facts are not determined by goings on in possible worlds—mathematical descriptions designate whatever they do independently of possible worlds. Nevertheless, there is a widespread practice of treating mathematical definite descriptions as rigid. Apart from this, there might be theoretical reasons for admitting that they are rigid in some sense (though not in the same sense as ordinary empirical definite descriptions). The second part of the paper suggests a way out. Borrowing from ideas proposed by Kit Fine, it develops and defends an extended notion of rigidity, which can be applied to mathematical definite descriptions. Importantly, this notion is fully compatible with the claim that mathematical facts are independent of possible worlds.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
Some of these notions are designed to delimit just a subclass of rigid designators; examples are such notions as de jure rigidity, de facto rigidity, obstinate rigidity and persistent rigidity. These particularities will not be discussed here.
According to Kripke’s standard definition, ‘a designator rigidly designates a certain object if it designates that object wherever the object exists’ (Kripke 1980, 48–49). This definition differs slightly from that proposed by Putnam (1975, 231) and Linsky (1977, 51), for whom a rigid designator designates the same object in all possible worlds in which it designates anything at all. The latter definition is more general than the former and yields different results in some cases. Again, such details are put to the side here.
I am merely going to discuss the rigidity of singular expressions, leaving aside general terms and their prospects of being rigid.
In claiming that mathematical entities are necessary existents, I simply follow suit. Many philosophers think that mathematical entities exist (in some sense) in all possible worlds. My considerations, however, neither prove nor disprove this presupposition, so I remain silent on this issue.
What I have in mind here are purely mathematical definite descriptions that are prime examples of rigid designators. Descriptions such as ‘the number that is equal both to the third power of 2 and to the number of planets in the solar system’, ‘the square of the number of planets in the solar system’, ‘the actual third power of 2’ and ‘the smallest two-digit prime number’ (for the last three examples I am indebted to Zsófia Zvolenszky) are not purely mathematical; I shall discuss them later on. I take the notion of a purely mathematical definite description to be intuitively clear. Not all definite descriptions that designate mathematical entities are mathematical descriptions in the sense I have in mind. For example, ‘the number of planets in the solar system’ designates a number, i.e. a mathematical entity, but it does so because the number has a certain non-mathematical property (corresponding to the quantity of certain non-mathematical objects). On the other hand, ‘the third power of 2’ is a mathematical description in the proper sense. Similarly, I take the notion of a mathematical entity (such as a number) and a mathematical property (such as the property of being a prime number) to be intuitively clear.
Designation is often made relative not just to possible worlds but also to times, so a definite description is more appropriately said to designate something in a possible world w and at a time t. I ignore this fact, however, because rigidity pertains only to the modal dimension. The same holds for other relations discussed below, such as exemplification.
Fine distinguishes between worldly and unworldly sentences. Empirical sentences belong to the former category, while mathematical sentences belong to the latter. Fine’s category of unworldly sentences is, however, much broader and covers cases that might appear problematic, such as ‘Socrates is self-identical’ and ‘Socrates is a man’. I will not discuss such examples here because they are irrelevant to my present purposes.
Strictly speaking, this relation is quaternary because empirical objects exemplify properties in possible worlds and at times. For the sake of simplicity, however, I ignore the temporal parameter as I did in the case of designation.
It is widely recognized that this construal of properties (and relations) faces serious drawbacks. Most importantly, co-extensional mathematical properties are identified with the same set and necessarily co-extensional empirical properties turn out to be identified with the same intension. Nevertheless, despite having considerable limitations, this simple explication is often used, because it can meet various theoretical requirements and can be perfectly sufficient for a number of purposes. I took the liberty of using this explication too, because the question of distinguishing various properties with the same extension is not pressing for the aims pursued in this paper. The crucial point I wish to highlight is that whichever explication of properties is adopted, it should be recognized that if possible worlds are assigned some role or other in explicating empirical properties, they cannot be assigned such a role in the case of explicating mathematical properties.
The difference between mathematical and empirical discourses is clearly captured in the treatment provided by Transparent Intensional Logic (TIL), as developed for example in Tichý (1988) and Duží et al. (2010). Based on a partial typed lambda calculus, the theory evades intensionalization (and temporalization) at the level of logical syntax in the case of mathematical expressions, which means that they do not contain (lambda bound) variables for possible worlds (and times); see, for example, the list of mathematical examples in Duží et al. (2010, 50–51). On the other hand, intensionalization (and temporalization) is mandatory in empirical discourse. (Incidentally, the previous description is not quite accurate because intensionalization (and temporalization) in TIL occurs at the level of meanings—so-called constructions—rather than expressions; variables themselves are treated as a special kind of construction rather than expressions.)
The adjective ‘empirical’ in ‘empirical extension determining procedure’ should be understood as modifying ‘extension’ rather than ‘procedure’. The same applies to ‘mathematical’ in ‘mathematical extension determining procedure’, to be introduced below.
The notions of empirical extension determining procedure and mathematical extension determining procedure are given here without precise definition or explication. When suitably elaborated, they may come very close to the notion of construction employed in TIL; cf. Tichý (1988) and Duží et al. (2010).
Consider ‘The third power of 2 is 8’. If true, this sentence implies that ‘the third power of 2’ designates 8. Since the sentence is a (world-independent) transcendental truth, ‘the third power of 2’ must designate 8 in a world-independent way as well.
When I say that the speaker designates the same object by using a mathematical description, I mean that she designates an object that is the semantic reference of the description in Kripke’s sense rather than the speaker’s reference; cf. Kripke (1977/2011).
A similar approach can be applied to the other examples mentioned in footnote 5. Take ‘the square of the number of planets in the solar system’. It is a hybrid definite description, so it designates 64 with respect to (but neither in nor at) the actual world. It is hybrid because ‘the number of planets in the solar system’ is an empirical definite description, but being a square of is a pure mathematical function. The description’s designation is determined in the following way. ‘The number of planets in the solar system’ designates 8 in—and, derivatively, with respect to—the actual world. Being a square of is a function that assigns numbers to numbers, where this assignment is determined solely on the basis of how things are with numbers in the mathematical realm, independently of possible worlds. In particular, it assigns 64 to 8; consequently, the property of being equal to the square of the number of planets in the solar system has the singleton {64} as its extension with respect to the actual world. Applying the singularizer to this set, its sole member is determined as the referent of ‘the square of the number of planets in the solar system’ with respect to the actual world. Obviously, it may designate some other number in other possible worlds—if there is a world in which the solar system has only 7 planets, it will designate 49 with respect to that world.
Next, ‘the actual third power of 2’ designates 8 with respect to all possible worlds provided ‘actual’ is understood in a slightly modified sense. ‘Actual’ is usually supposed to be a rigidifying device that guarantees that if ‘the F’ designates something in the actual world, ‘the actual F’ designates that thing, if anything, in all possible worlds. What we need is a more general sense according to which ‘actual’ is a rigidifying device that guarantees that if ‘the F’ designates something in/at/with respect to the actual world, ‘the actual F’ designates that thing, if anything, in/at/with respect to all possible worlds. Given this modification, the derivation of the conclusion that ‘the actual third power of 2’ designates 8 with respect to all possible worlds is relatively straightforward.
Finally, consider ‘the smallest two-digit prime number’ (let us assume that there is a suppressed reference to the decimal system in the description). This is a bit trickier because it partly deals with the linguistic representations of numbers in the language. The description invokes two properties, namely having two digits (or, more precisely, being represented by a two-digit numeral) and being a prime number. The latter property has a set of numbers as its extension at—and, derivatively, with respect to—possible worlds. The former property is metalinguistic; assuming that metalinguistic properties are empirical, an object instantiates it in a possible world provided its linguistic representation (in a given language) is a two-digit numeral in that world. So, this property has a set of entities as its extension in—and, derivatively, with respect to—possible worlds. Now, the intersection of the set that is the latter property’s extension at a possible world w with the set that is the former property’s extension in w can be identified with the extension of the property of being a two-digit prime number with respect to w. ‘The smallest two-digit prime number’ designates the smallest member of the set—i.e. 11—with respect to w because the smallest function, when applied to a set of numbers, selects the smallest member of the set.
In fact, there is perhaps no expression that would designate something non-rigidly at possible worlds. This is because of the way in which the notion of designation at has been introduced: if an expression designates an object independently of possible worlds, it does so at all possible worlds. This specification does not allow for situations in which an expression designates one object at one world and another at another world. On the other hand, the relations of designation in and with respect to permit of non-rigidity.
The term ‘actual’ is used here as a rigidifying device.
References
Duží, M., Jespersen, B., & Materna, P. (2010). Procedural semantics for hyperintensional logic: foundations and applications of Transparent Intensional Logic. Dordrecht: Springer.
Fine, K. (2005). Modality and tense: philosophical papers. Oxford: Clarendon Press.
Kripke, S. (1971/2011). Identity and necessity. Reprinted in S. Kripke, Philosophical troubles (pp. 1–26). Oxford: Oxford University Press.
Kripke, S. (1977/2011). Speaker’s reference and semantic reference. Reprinted in S. Kripke, Philosophical troubles (pp. 99–124). Oxford: Oxford University Press.
Kripke, S. (1980). Naming and necessity. Cambridge (Mass.): Harvard University Press.
LaPorte, J. (2011). Rigid designators. In E. N. Zalta (Ed.), The Stanford Encyclopedia of Philosophy (Summer 2011 Edition). URL = <http://plato.stanford.edu/archives/sum2011/entries/rigid-designators/>.
LaPorte, J. (2013). Rigid designation and theoretical identities. Oxford: Oxford University Press.
Linsky, L. (1977). Names and descriptions. Chicago and London: University of Chicago Press.
McGinn, C. (1982). Rigid designation and semantic value. The Philosophical Quarterly, 32(127), 97–115.
Putnam, H. (1975). The meaning of ‘meaning’. In H. Putnam (Ed.), Mind, language, and reality (pp. 215–271). Cambridge: Cambridge University Press.
Salmon, N. (1981). Reference and essence. Princeton: Princeton University Press.
Soames, S. (2005). Reference and description. Princeton and Oxford: Princeton University Press.
Stanley, J. (1997). Names and rigid designation. In B. Hale & C. Wright (Eds.), A companion to the philosophy of language (pp. 555–585). Oxford: Blackwell.
Textor, M. (1998). Rigidity and de jure rigidity. Teorema, 17(1), 45–59.
Tichý, P. (1988). The foundations of Frege’s logic. Berlin and New York: de Gruyter.
Acknowledgements
A version of this paper was read at Bucharest Colloquium in Analytic Philosophy 2015 – Meaning and Reference (University of Bucharest, 19-21 June 2015). I would like to thank the audience present at the colloquium for a stimulating discussion. I am further indebted to Lukáš Bielik, Pavel Cmorej, Daniela Glavaničová, Miloš Kosterec, Martin Vacek and Zsófia Zvolenszky for their valuable comments on a previous version of this paper. I also wish to express my gratitude to anonymous referees of Philosophia for their criticisms and suggestions.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Zouhar, M. Rigidity in Mathematical Discourse. Philosophia 45, 1381–1394 (2017). https://doi.org/10.1007/s11406-016-9807-7
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11406-016-9807-7