Unity of a Complex – Page 2 – Maverick Philosopher

In an important article, Max Black writes:

Beginners are taught that a set having three members is a single thing, wholly constituted by its members but distinct from them. After this, the theological doctrine of the Trinity as "three in one" should be child's play. ("The Elusiveness of Sets," Review of Metaphysics, June 1971, p. 615)

1. A set in the mathematical (as opposed to commonsense) sense is a single item 'over and above' its members. If the six shoes in my closet form a mathematical set, and it is not obvious that they do, then that set is a one-over-many: it is one single item despite its having six distinct members each of which is distinct from the set, and all of which, taken collectively, are distinct from the set. A set with two or more members is not identical to one of its members, or to each of its members, or to its members taken together, and so the set is distinct from its members taken together, though not wholly distinct from them: it is after all composed of them and its very identity and existence depends on them.

In the above quotation, Black is suggesting that mathematical sets are contradictory entities: they are both one and many. A set is one in that it is a single item 'over and above' its members or elements as I have just explained. It is many in that it is "wholly constituted" by its members. (We leave out of consideration the null set and singleton sets which present problems of their own.) The sense in which sets are "wholly constituted" by their members can be explained in terms of the Axiom of Extensionality: two sets are numerically the same iff they have the same members and numerically different otherwise. Obviously, nothing can be both one and many at the same time and in the same respect. So it seems there is a genuine puzzle here. How remove it?

Continue reading “On the Elusive Notion of a Set: Sets as Products of Collectings”

Having recently returned from the Geneva conference on Bradley's regress, I have much to ruminate upon and digest. I'll start my ruminations with some comments on Richard Gaskin's work.

In an earlier post I suggested that we ought to make a tripartite distinction among vicious, benign (harmless), and virtuous (helpful) infinite regresses. To put it crudely, a vicious regress prevents an explanatory job from getting done; a benign regress does not prevent an explanatory job from getting done; and a virtuous regress makes a positive contribution to an explanatory job's getting done. I gave an example of a putative virtuous regress in the earlier post which example I will not repeat here. In this post I draw your attention to a second putative example from the work of Richard Gaskin, whom I was happy to meet at the Geneva conference on Bradley's Regress. Gaskin's proposal is that "Bradley's regress is, contrary to to the tradition, so far from being harmful that it is even the availability of the regress which guarantees our ability to say anything at all. Bradley's regress is the metaphysical ground of the unity of the proposition." ("Bradley's Regress, the Copula, and the Unity of the Proposition," The Philosophical Quarterly, vol. 45, no. 179, April 1995, p. 176) In terms of my schema above, Gaskin is claiming that Bradley's regress is positively virtuous (not merely benign) in that it plays a positive explanatory role: it explains (metaphysically grounds) the unity of the proposition.

I will now attempt to summarize and evaluate Gaskin's position on the basis of two papers of his that I have read, and on the basis of his presentation in Geneva. (I should say that he has just published a book, The Unity of the Proposition, which I have not yet secured, so the following remarks may need revision in light of his later work.)

Continue reading “Bradley’s Regress as the Metaphysical Ground of the Unity of the Proposition: Notes on Gaskin”