Aggregation

Is number a property of an aggregate of things? But what is an aggregate? Can the very same things have the same number once disaggregated? Frege (The Foundations of Arithmetic § 23, translation J.L. Austin) writes:

To the question: What is it that the number belongs to as a property? Mill replies as follows: the name of a number connotes, ‘of course, some property belonging to the agglomeration of things which we call by the name; and that property is the characteristic manner in which the agglomeration is made up of, and may be separated into, parts.’

Here the definite article in the phrase "the characteristic manner" is a mistake right away; for there are very various manners in which an agglomeration can be separated into parts, and we cannot say that one alone would be characteristic. For example, a bundle of straw can be separated into parts by cutting all the straws in half, or by splitting it up into single straws, or by dividing it into two bundles. Further, is a heap of a hundred grains of sand made up of parts in exactly the same way as a bundle of 100 straws? And yet we have the same number. The number word ‘one’, again, in the expression ‘one straw’ signally fails to do justice to the way in which the straw is made up of cells or molecules. Still more difficulty is presented by the number 0. Besides, need the straws form any sort of bundle at all in order to be numbered? Must we literally hold a rally of all the blind in Germany before we can attach any sense to the expression ‘the number of blind in Germany’? Are a thousand grains of wheat, when once they have been scattered by the sower, a thousand grains of wheat no longer? Do such things really exist as agglomerations of proofs of a theorem, or agglomerations of events? And yet these too can be numbered. Nor does it make any difference whether the events occur together or thousands of years apart.

Number, concepts and existence

The Maverick Philosopher is agonising about number and existence in this post. It would be simpler if we returned to the original text of Frege which started all this (Die Grundlagen der Arithmetik 1884. Page numbers are to the original edition).

Frege claims with a concept the question is always whether anything, and if so what, falls under it. With a proper name such questions make no sense. (§51, p. 64). He also claims that when you add the definite article to a concept word, it ceases to function as a concept word, although it still so functions with the indefinite article, or in the plural (ibid).

This is part of a section of the Grundlagen where he develops the thesis that number is a property of concepts, not of things. Thus if I say (§46, p. 59) that ‘the King’s carriage is drawn by four horses’, I am ascribing the number 4 to the concept horse that draws the King’s carriage. The number is not a property of the horses, either individually or collectively, but of a concept.

From these two claims, namely that number is a property of concept words, and that proper names are not concept words, it seems to follow that ‘Socrates exists’ makes no sense. If ‘Socrates’ is not a concept word, then it seems no concept corresponds to it, but existence means that some concept is instantiated, so ‘Socrates exists’cannot express existence. This is the difficulty that Bill is grappling with.

But why, from the fact that ‘Socrates’ is not a concept word, does it follow that there is no corresponding concept? Frege has already told us that a concept word ceases to be such when we attach the definite article to it. So while ‘teacher of Plato’ signifies a concept, ‘the teacher of Plato does not. Why can’t the definite noun phrase ‘Socrates’ be the same, except that the definiteness is built into the proper name, rather than a syntactical compound of definite article and concept word. Why can’t ‘Socrates’ be semantically compound? So that it embeds a concept like person identical with Socrates, which with the definite article appended gives us ‘Socrates’?

As I argued in one of the comments, the following three concepts all have a number

C1: {any man at all}
C2: {any man besides Socrates}
C3: {satisfies C1 but not C2}

If the number corresponding to C1 is n, then the number of C2 is n-1. And the number of C3 is of course 1, and if C3 is satisfied, then Socrates exists. Simple.