Abstract
In some sense, both ontological and epistemological problems related to individuation have been the focal issues in the philosophy of mathematics ever since Frege. However, such an interest becomes manifest in the rise of structuralism as one of the most promising positions in recent philosophy of mathematics. The most recent controversy between Keränen and Shapiro seems to be the culmination of this phenomenon. Rather than taking sides, in this paper, I propose to critically examine some common assumptions shared by both parties. In particular, I shall focus on their assumptions on (1) haecceity as an individual essence, (2) haecceity as a property, (3) the classification of properties, and thereby (4) the search for the principle of individuation in terms of properties. I shall argue that all these assumptions are mistaken and ungrounded from Scotus’ point of view. Further, I will fathom what consequences would follow, if we reject each of these assumptions.
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Notes
Even though our discussion starts with the problems (possibly) peculiar to ante rem structuralism in philosophy of mathematics, it must be of interest to philosophers in other areas such as philosophy of quantum mechanics. In fact, it becomes more and more urgent to share results in all areas of philosophy, where individuation matters. Also, as identity of indiscernibles or haecceitas demonstrates, history of ontology could be instrumental for deeper understanding of our problems. In the same vein, the ante rem aspect of Shapiro’s ante rem structuralism can be discussed in a broader historical context.
See, for example, MacBride (2005, 219), (2006a, 64) and Ladyman (2005, 219) for different formulations of the alleged dilemma. It is not clear which way of formulating the dilemma is the best, but it seems that Keränen and Shapiro are more careful than others at least in that they are more sensitive to the possible roles and functions of identity of indiscernibles in this alleged dilemma. Here I am indebted to an anonymous reviewer for more focused discussion.
Allan B. Wolter cites the following items as examples of such a conflation: Bergmann (1964, pp. 160, 165, 287, 1967, pp. 167, 191, 199, 204, 222), Plantinga (1978, 132), Chisholm (1986, 160), Losonsky (1987, 253), Wolter (1992, xix, n. 26), Park (1990, 377, n. 8)]. Further example would be Rosenkrantz (1993). One anonymous reviewer counts my observation that ‘haecceity has been widely conflated with individual essence’ as correct. Nevertheless, the reviewer thinks it irrelevant, for “Shapiro and Keränen are not concerned with individual essence so there is nothing for them to conflate with”. My response should be obvious: insofar as they assume haecceity as a property, it is their burden to indicate what kind of property that property is. Please see note 11 below for further discussion of connected points.
In Latin, it reads “diversaaliguid-idem entia”. Cf. Metaphysics V, ch. 9, 1018112-13.
Scotus compared an individual and a species in terms of their relationship to what is below each, to what is above which, and to what is on a par with each (Dist. 3, q. 6, nn. 170-2; Park 1990).
This sketch is drawn from Park (2000). A fuller version is presented in Park (1990). Park (1990), entitled as “Haecceitas and the Bare Particular” paid much more attention to refuting philosophers such as Jorge J. E. Gracia, who tend to assimilate haecceitas to bare particulars, than those who assimilate haecceitas to an individual essence.
Please note that by doing this we are thereby fathoming at the same time the possibility of understanding individuation not in terms of properties.
At this stage, one anonymous reviewer complained that “none of the author’s quotations either from Shapiro’s texts of Keränen’s texts…show that Shapiro and Keränen use the expression ‘individual essence’—either explicitly or by implicit reference!” (Emphasis is the reviewer’s) Based on this observation, the reviewer presented a very subtle defense of Shapiro and Keränen: “The denial of individual essence does make sense within the context of Scotus since he is concerned with the contraction of the specific nature. But the assertion of individual essence (projected onto Shapiro and Keränen by author) does not make sense from the point of view of Shapiro and Keränen”. (Emphasis is the reviewer’s) Instinctively, I would like to respond by reminding them that it is their burden to indicate what kind of property haecceity is. If it is not an individual essence, what kind of property do they have in mind in assuming haecceity as a property?
One anonymous reviewer criticizes incisively my failure to provide a link between the first and the second part of this paper. The only consolation is that the same reviewer finds from my discussion in this paragraph “a hint at the direction for getting a new impetus from revisiting the ‘history of ontology’”. The reviewer even raises a series of insightful questions to answer in that direction: “can distinct but property-indiscernible mathematical objects be accounted for in terms of some haecceitistic property? If yes, how could it be taken as non-trivial? Can Scotus’s alternative reading of haecceitas be of any help here?” In the next section, I will try to give at least some partial responses to these questions.
Virtually all anonymous reviewers require such a discussion, even though it is simply beyond my ability to show the future directions of mathematical structuralism. The discussion below would have been impossible without one reviewer, who specifically suggested to touch upon theories of essences.
See also the following quote from Avicenna: “And the quiddities of things may be in individual things, and they may be in the mind; so they have three respects: the respect of quiddity inasmuch as it is that quiddity is not added to one of the two modes of existence, nor to what is attached to the quiddity, insofar as it is in this respect. And quiddity has a respect insofar as it is in individuals. And there accidents which make particular its existence in that are attached to it. And it has a respect insofar as it is in the mind. So there accidents that make particular its existence in that are attached to it; e.g., being a subject and being a predicate, and universality and particularity in predication…”. [Avicenna (1952); quoted from Bäck (1996, pp. 133–134).
Needless to say, what I write in this paragraph is at best highly speculative. Interestingly, one anonymous reviewer anticipated all this: “Scotus’s notion of ‘nature’ is hardly applicable to abstract mathematical entities. But the difficulty is not only that ‘nature’ in Scotus’s sense is tailored to something else than to abstract mathematical entities. As is known, the specific nature for Scotus and for the scholastic-Aristotelian tradition is qualitative, or better, quidditative, but such nature cannot be resolved into a bunch of properties! So it would be difficult to assess Scotus’s position as relevant to the contemporary issue on haecceitas as a property”. I would concede fully the difficulties involved in demonstrating the relevance of Scotus’ theory of common nature to contemporary issues in mathematical structuralism. But wasn’t the reviewer too pessimistic because of the implicitly assumed mathematical Platonism? As is well-known, Scotus and all other Aristotelians are ready to employ the theory of abstraction.
Quine wrote: “The three main mediaeval points of view regarding universals are designated by historians as realism, conceptualism, and nominalism. Essentially these same three doctrines reappear in twentieth-century surveys of the philosophy of mathematics under the new names logicism, intuitionism, and formalism” (Quine 1948).
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Park, W. What if Haecceity is not a Property?. Found Sci 21, 511–526 (2016). https://doi.org/10.1007/s10699-015-9429-8
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DOI: https://doi.org/10.1007/s10699-015-9429-8