More on the nature of mathematics
The debate is heating up (in a good way) over at FQXi where I posed a question about the nature of mathematics. While presently remaining agnostic on the issue, I want to emphasize why this is even a question at all since a number of people seem fairly wedded to the fact that mathematics is invented. That may still be true, but there are certain aspects of mathematics that simply cannot be invented. For instance, it is well-known that many animal species have the ability to do basic arithmetic (adding and substracting) and one of Pavlov’s experiments with dogs demonstrated that they have a fairly sophisticated ability to distinguish between certain geometrical objects. In fact (and this doesn’t surprise me in the least), monkeys perform about as well at mental addition as college students. Interestingly enough, what sets humans apart appears to be language. Even though chimpanzees, for instance, can outperform humans at tracking numbers and remembering sequencing, the level of specificity possessed by humans is not possessed by any other species. We are the only ones who assign symbols to numbers. I find this utterly fascinating, particularly in light of my recent FQXi essay.
Thus, the questions that come to mind are two-fold. First, perhaps some mathematics is inherent (discovered) but that some is invented. Is the difference discrete or continuous? Is there a gradual change from inherent to invented? Second, is this really a question of language and, if so, how much of mathematics is a language? It clearly can’t all be since monkeys, for example, don’t have language. And what is it about language that can suddenly transform something from real and inherent to something that is invented and beyond the realm of possibility? Is this just further evidence that mathematics is really a language and just a language? If so, what do we call the logical rules of addition and subtraction that seem to be understood by many creatures?