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6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / m. One

[status and nature of the number one]

7 ideas
For Pythagoreans 'one' is not a number, but the foundation of numbers [Pythagoras, by Watson]
     Full Idea: For Pythagoreans, one, 1, is not a true number but the 'essence' of number, out of which the number system emerges.
     From: report of Pythagoras (reports [c.530 BCE], Ch.8) by Peter Watson - Ideas Ch.8
     A reaction: I think this is right! Counting and numbers only arise once the concept of individuality and identity have arisen. Counting to one is no more than observing the law of identity. 'Two' is the big adventure.
The one in number just is the particular [Aristotle]
     Full Idea: It makes no difference whether we speak of the particular or the one in number. For by the one in number we mean the particular.
     From: Aristotle (Metaphysics [c.324 BCE], 0999b33)
     A reaction: This is the Greek view of 'one', quite different from the Frege or Dedekind view. I prefer the Greek view, because 'one' is the place where numbers plug into the world, and the one indispensable feature of numbers is that they can count particulars.
A unit is that according to which each existing thing is said to be one [Euclid]
     Full Idea: A unit is that according to which each existing thing is said to be one.
     From: Euclid (Elements of Geometry [c.290 BCE], 7 Def 1)
     A reaction: See Frege's 'Grundlagen' §29-44 for a sustained critique of this. Frege is good, but there must be something right about the Euclid idea. If I count stone, paper and scissors as three, each must first qualify to be counted as one. Psychology creeps in.
The idea of 'one' is the simplest, most obvious and most widespread idea [Locke]
     Full Idea: Among all the ideas we have, as there is none suggested to the mind by more ways, so there is none more simple than that of unity, or one; ..every idea in our understanding, every thought of our minds brings this idea along with it.
     From: John Locke (Essay Conc Human Understanding (2nd Ed) [1694], 2.16.01)
     A reaction: What does Locke mean by 'suggested' to the mind? I take it that this phenomenon of psychology (or of reality, if you like) is the foundation of mathematics, making one clearly prior to zero.
We can say 'a and b are F' if F is 'wise', but not if it is 'one' [Frege]
     Full Idea: We combine 'Solon was wise' and 'Thales was wise' into 'Solon and Thales were wise', but we can't say 'Solon and Thales were one', which implies that 'one' is not a property in the same way 'wise' is.
     From: Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884], §29)
     A reaction: Maybe 'one' is still a property, but of a different sort. However, Frege builds up a very persuasive case that just because numbers function as adjectives it does not follow that they are properties. See Idea 8637.
One is the Number which belongs to the concept "identical with 0" [Frege]
     Full Idea: One is the Number which belongs to the concept "identical with 0".
     From: Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884], §77)
     A reaction: This follows from Idea 8653, which defined zero. Zero is the number of a non-existent set, and one is how many sets you have when you have only got zero. Very clever.
Discovering that 1 is a number was difficult [Russell]
     Full Idea: The discovery that 1 is a number must have been difficult.
     From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], I)
     A reaction: Interesting that he calls it a 'discovery'. I am tempted to call it a 'decision'.