A few weeks ago I had a post on different ways of counting infinities; the main point was that two of the basic principles that hold for counting finite collections cannot be both transferred over to the case of measuring infinite collections. Now, as a matter of fact I am equally (if not more) interested in the question of counting finite collections at the most basic level, both from the point of view of the foundations of mathematics (‘but what are numbers?’) and from the point of view of how numerical cognition emerges in humans. In fact, to me, these two questions are deeply related.
In a lecture I’ve given a couple of times to non-academic, non-philosophical audiences (so-called ‘outreach lectures’) called ‘What are numbers for people who do not count?’, my starting point is the classic Dedekindian question, ‘What are numbers?’ But instead of going metaphysical, I examine people’s actual counting habits (including among cultures that have very few number words). The idea is that Benacerraf’s (1973) challenge of how we can have epistemic access to these elusive entities, numbers, should be addressed in an empirically informed way, including data from developmental psychology and from anthropological studies (among others). There is a sense in which all there is to explain is the socially enforced practice of counting, which then gives rise to basic arithmetic (from there on, to the rest of mathematics). And here again, Wittgenstein was on the right track with the following observation in the Remarks on the Foundations of Mathematics:
This is how our children learn sums; for one makes them put down three beans and then another three beans and then count what is there. If the result at one time were 5, at another 7 (say because, as we should now say, one sometimes got added, and one sometimes vanished of itself), then the first thing we said would be that beans were no good for teaching sums. But if the same thing happened with sticks, fingers, lines and most other things, that would be the end of all sums.
“But shouldn’t we then still have 2 + 2 = 4?” – This sentence would have become unusable. (RFM, § 37)