Friday, 11 July 2014
The Necessary Nature of Numbers
We can say a world in which there are no animals is also a world in which there is no horses. That appears to be a necessary truth in that at no world at which there are no animals can there be horses. Is it, however, only a conceptual truth and not a metaphysical truth? Is it a truth about our concepts, or concept-kinds, in that contained within the concept [horse] is the concept [animal]? Thus a horse must be an animal, conceptually speaking. Or is it a truth about kinds as they are in themselves – independently of minds and concepts? However, we can only get at horses and animals through our concepts or through our classifications and categories. And they are mind-dependent.
Some have argued that there may well be a world at which 2 + 1 equals 4. Is that possible? Where would the number 3 come at this world? Could it come after, say, 4? In that case, 4 would be 4 + 1. If that number shifted, then so too would all the others. If 4 were 4 + 1, then 5 would be 4 + 2 or 5 + 1. Alternatively, perhaps there is a world in which 3 is simply missing and 4 is the immediate successor of 2. Wouldn’t that pattern need to be repeated? Not necessarily. If it were, then 6 may also be the immediate successor of 4. Would that mean that this is effectively a different arithmetic to our own, or perhaps not arithmetic at all? Can there actually be alternative arithmetics in the way that there are alternative geometries (despite the fact that ‘alternative’ geometries do not necessarily contradict each other).
Despite all that, we can say that what makes a natural number the number it is, is its position in the number series. This seems to unequivocally rule out the possibilities so far discussed. That is, 3 is 3 precisely because it comes after 2 and before 4. Change the series and you change everything. Thus you couldn't even call 3 ‘3’ at this possible world. 3 is its positions in the natural number series. It gains its identity or meaning through its position in the natural number series (which is the basis for arithmetic).
Alternatively, we can simply say that "if per impossible 3 and 4 switched places, 4 would now just be 3" (133). This world may use the inscription ‘4’; though that inscription would still actually be the number 3. Even if it this isn’t a case of inscriptions versus real numbers, their 4, not their ‘4’, would still be 3 (or do we mean ‘our’ 3?).