A number on numbers

A crank is someone who sees the truth through a fool's eyes. It is a barb I hope the following will not earn, although that is, I dare say, preferable to the mute indifference I expect it to elicit. Without diagrams and notation, the elucidation of my ideas will be difficult, and the dismissal of them easy. Ay any rate, I know this: monumental failure is better than trivial failure.

What I am trying to do is to derive the principles of arithmetic from non-arithmetic premises. It is incumbent on me to mention briefly why this should be desirable. Of course, each mathematician has their agenda, but what they share to a greater or lesser degree is the compunction to rid themselves of irrational numbers, which imperil the elegance and simplicity of mathematics and decree that the spatial world is infinitely divisible. This might seem jejune or even simply uninteresting, but one only need reflect for a moment to see that numbers are not unproblematic things. It is often assumed that 1+1=2 is a necessary truth, but one can only verify this by counting, and this is hostage to experience. As Roger Penrose has said, one cannot rule out of court the possibility that every time we have calculated 1+1+2 we have made a mistake. This vitiates, or at least strengthens, Mill's suggestion that the propositions of mathematics are essentially inductive, although, admittedly, it is induction of a very general and rarefied sort.

Three attempts have been made to reduce arithmetic to logic: by Frege and Von Neumann, by Russell and Whitehead, and by the Italian logician Peano. I think all of them are unsatisfactory. I will trace the three attempts in outline, crossing, as I do so, the unsteady bridge that joins mathematics and philosophy.

The easiest to dispense with is that of Peano. Every number is a successor of a successor – except zero, which is the successor of none, and 1, which is a successor of only zero. So 3 is written sss0. But this, whatever its value as a heuristic, simply displaces the problem, because one is still required to rely on one's counting ability.

Frege was the pioneer behind what is commonly called logicism, the view that mathematics just is logic; courtesy of set theory, he very nearly completed the task. Here a set is a collection of objects, enumerable in any order, and divisible in subsets of sets and subsets of subsets. If there is a one-to-one correspondence between the members of two sets, then they are equivalent. For example the set of the number two is the set of all pairs. But this still relies on counting, so Frege invoked a law which is assured by logic: the set of all things which are not equal to themselves, which of course, contains nothing, and so is designated the null set; and this corresponds to the number zero. The number corresponds to the set of all things with one member, which is the null set. And so on. But this does not work, as Russell pointed out: for how does one treat the set of all sets that are not members of themselves? Since it is not a member of itself, it belongs in the set of all things that are not members of themselves. And this is a contradiction. Needless to say, when Frege found out about this paradox, he was devastated.

Russell tried to avert this problem by recourse to what he called a theory of types. Here, numbers were not simply aggregates of things, but comprised an ascending hierarchy. The laws of arithmetic were really semantic rules for picking out numbers: so 1+1 is semantically equivalent to '2', and so on. But this fails to account for the sense in which numbers are composed out of one another, that arithmetic is prior to number: and this seems a high price to pay.

Any suggestions?