Wholes, Parts, and Numbers
Salmon (Nathan)
Source: Salmon (Nathan) - Metaphysics, Mathematics, and Meaning, 2005
Paper - Abstract

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Oxford Scholarship Online

  1. It would appear to be provable that there cannot be exactly two and a half oranges on the table. For the orange-half on the table is itself not an orange. An orange is a whole orange (or nearly enough so), whereas an orange-half, whatever else it is, is not a whole orange (nor even nearly so). Thus, there are only two oranges on the table, together with a third thing that (despite its color, taste, etc.) is no orange.
  2. This paradoxical conclusion is rejected. Instead a non-classical understanding is adopted on which the numerical quantifier ‘there are exactly n’, surprisingly, creates a non-extensional context.

Philosophers Index Abstract
  1. What does "There are exactly n oranges on the table" mean when n denotes a mixed whole/fractional number, like two and one-half?
  2. A puzzle is developed by "proving" that the exact number of oranges on the table cannot be mixed and must instead be whole. Solutions to the puzzle are canvassed and found unsatisfactory.
  3. An alternative solution is proposed on which numbers are properties of "pluralities" – the many rather than the one –relative to a property appropriate to the one's (not the many). This proposal has the consequence that "There are exactly n ..." is a non-extensional operator.

Author’s Introduction
  1. I present here a puzzle that arises in the area of overlap among the philosophy of logic, the philosophy of mathematics, and the philosophy of language. The puzzle also concerns a host of issues in metaphysics, insofar as it crucially involves wholes, their parts, and the relation of part to whole.
  2. Almost entirely nontechnical, the puzzle is disarmingly simple to state. What little technicality I introduce below is mostly of a purely logical nature, and mostly inessential to the puzzle's central thrust.
  3. I discovered the puzzle nearly twenty years ago. (See note 4 below.) It had been my intention since that time to publish the puzzle together with its solution, but finding a solution that I was strongly inclined to accept proved difficult. I have presented the puzzle orally and informally to a number of philosophers, including several of the world's greatest thinkers in the philosophy of logic and the philosophy of mathematics. None offered a solution that strikes me as definitively striking to the heart of the matter. Indeed, I was in no position to make others appreciate the full philosophical significance of the problem.
  4. I present here a couple of my own proposals for its solution, an acknowledgement of some shortcomings of those proposals, and a final nod in the direction of the solution I currently think is best.


Philosophical Perspectives, Vol. 11, Mind, Causation1, and World, 1997, pp. 1-15

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