Friday, July 20, 2007
Terrence Deaconlinks to this post (0) comments
[Mathematics has] for centuries prompted philosophical debates concerning the origins of abstract form. Consider the operations of addition, subtraction, multiplication, and division in elementary arithmetic. These are, in one sense, cultural creations. They are conventional operations using culturally created tokens that could be embodied in a seeming infinite variety of different patterns of tokens and manipulations. But it is not quite right to say that these operations were ‘invented’. Specific individuals over the history of mathematics did indeed invent the various notation systems we now use to represent mathematical relationships, but often these inventions came as a result of discoveries they made about the representation of quantitative relationships. Mathematical ‘facts’ have a curious kind of existence that has fascinated philosophers since the ancient Greeks. Being represented seems to be an essential constitutive feature of a mathematical entity, but this doesn‘t mean that anything goes. Mathematical representations are precisely limited in form. Generalizations about the representation and manipulation of quantity, once expressed in a precise symbolic formalization, limit and determine how other mathematical generalizations can be represented in the same formal system. For my purpose it is irrelevant whether these kinds mathematical entity are ‘latent in the world’ in some Platonic sense or whether they emerge anew with human efforts to formalize the way we represent quantitative relationships. What matters is that symbolic representations of numerical relationships are at the same time arbitrary conventions and yet subject to non-arbitrary combinatorial consequences. Because of this deep non-arbitrariness, we feel confident that mathematics done anywhere in the universe will have mostly the same form, even if the medium notation were to differ radically.