.surely the number of bikes one need cannot be
summed up in such a simple equation as n+1?
For those readers who are not au fait with this equation, it can be defined thusly:
Let the number of bicycles one owns be denoted as n and the number of one bicycles one feels one needs to own as N. For any given value of n, N can be calculated using the equation N = n + 1.
This is an elegant method of describing the phenomenon that the acquisition of bicycles leads almost inevitably to the desire to acquire more bicycles. It is, unfortunately, inaccurate and therefore only suitable for t-shirts and the kind of conversation loaded with references to threads from the days of uk.rec.cycling and urbancyclist-uk (i.e. for those recidivists who remember the ailing mollusc being adopted as a unit of currency sometime in the mid 1990s). It is inadequate for accurately predicting the number of bicycles a person actually wants or needs.
This mathematical conundrum is one that has been tickling the back of my mind for some time, despite Stigler's Law of Eponymy ensuring that any solution I derived would not be named Raven's Theorem (one is not, after all, driven by anything so crass as the desire for mathematical fame). I had hoped that it would be something elegant involving imaginary numbers, however counting machines using the cumulated radians in their wheels does not permit any sensible use of Euler's formula — obviously, given that there will always be a whole number of wheels, the value of x is irrelevant and therefore indeterminable.
Recently I have begun wondering if set theory and transfinite numbers may hold the answer. If B is the set of bicycles that one currently owns and V is the set of bicycles that one would like to own, then #V > #B. The difficulty lies in defining by how many elements V is bigger than B, especially as V need not be finite. There may be only so many bicycles in the world, but our imaginations can include such things as rocket-propelled hovercycle.