What is the number one?
Answer by Jürgen Lawrenz
I’m going to disappoint you. This question is the sort that everyone should be able to answer – including an untutored child. But as you asked for a philosopher to respond, you must have some inkling about the deep ramifications of identifying just what Number 1 truly identifies. You could look it up in Aristotle’s Metaphysics, Book Delta, ch. 6. He gives a rational answer that should serve you in all but the most extreme of counterintuitive situations. But if you are really determined to get to the bottom of it, you must read Russell & Whitehead, Principles of Mathematics. But let me warn you that it’s not an easy text to get to grips with. For a start, there is very little English in it; and then the authors felt obliged to construct a foolproof system of presuppositions, which takes them over 300 pages before they actually come to a definition of ‘1’. You will excuse me, I hope, if I refuse to give the ‘gist’ of 300 pages in one paragraph. But these are your choices – or else you stick with everyone’s convenience and take the ‘1’ for granted!
Answer by Geoffrey Klempner
There’s an easier book than Russell and Whitehead’s Principia by the man who invented modern logic, Gottlob Frege: Foundations of Arithmetic, translated by the Oxford ‘ordinary language’ philosopher, J.L. Austin — the last person you’d expect to translate a work full of arcane symbolism.
In fact, Frege’s book is a joy to read and a great introduction to the philosophy of mathematics.
Frege recognized that there are two questions to ask about numbers, generally. The first question is about what we are describing when we say, for example, that there is one can of beans in the cupboard. The second, and more metaphysical question, is what we are referring to when we use the names, ‘one’, ‘two’, ‘three’ etc.
IS the can of beans one? But it is also many (a few thousand). It is also trillions (of molecules). Using his new logic, Frege showed how, when numbers are used to count, the number is a property of a concept, e.g. ‘can of beans’, rather than an object. To say that there is one can of beans, and only one, is to say that there is an object x which has the property ‘can of beans in the cupboard’, and if there is any y with that property, then y=x.
The problem is that we have simply used the concept of identity to explain the concept of one. The only response to this is to admit that the two notions are fundamentally the same. The very possibility of there being ‘objects’ presupposes the idea of identity. That’s a logical point of some significance.
The second, metaphysical question, what objects numbers refer to, led Frege to put forward a theory that numbers are classes. The number one is the class of all classes which have exactly one member. Similarly for two, three etc. Frege’s idea here was that the notion of a class is more fundamental than the notion of number, and so counts as a genuine explanation.
Frege discovered, late in his career, that the definition he had proposed led to an insoluble paradox — named ‘Russell’s paradox’ after the famous philosopher who grappled with it.
A significant portion of the history of 20th century mathematical logic has been concerned with the various responses to Russell’s paradox, offering a theory of what numbers ‘are’. However, there are sceptics who would say that Frege’s first answer, what we are doing when we call something ‘one’, is all one needs to say. The metaphysical question what the number one is, in itself, doesn’t have an answer; or, equally, there are any number of equally valid ‘answers’.