How many whales?

We are back in Los Angeles, where Rita continues to work on research for her post-doc, so Helena has to deal with lots of new places again.  In the hotel where we are staying, there are whales on the bottom of the bathtub, and when she got ready for her bath last night, she took an unmistakable joy in counting them: "Uma, duas, três, quatro baleias..." and then in English: "One, two three, four whales."  Her counting has gotten quite good of over the last couple of weeks, and she normally makes it to about 14 without a mistake (though seven often gets skipped, for no very clear reason.).

Lots of experts in child development say that children don't really understand numbers until they are much older, so I have spent quite a lot of time with Helena trying to get at how she thinks about counting.  At first, it seemed that she understood numbers not really as numbers, but as a series: just in the same way that she has stacking bowls, where the one marked with a bird must come first, followed by the caterpillar and the bunny, one, two, and three are a series.  That is, after all, what counting is about.

Does she understand, though, that there are four whales, and not just that the word "four" is attached to the last of the whales that she counts in her series?  The classic test with children is to ask them how many things there are, and not to allow them to count one by one: evidently, most kids just say one, two, or "lots."  Helena doesn't seem to accept this test, though (or I don't know how to do it): she always goes back and counts.

Now, Immanuel Kant at one point set off to see if any human knowledge was truly a priori, by which he meant that it was guaranteed to be true without us having to trust our unreliable senses.  He looked to math as an example, and came to claim that math doesn't really require any inputs from the world.  All math, he says, is based on sequence (basically what Helena does as she counts), and sequence is based on time.  Since time is one of the universal, transcendental characteristics of the interaction of all human minds with the world, we can say that mathematical truths are a priori.  (Since space is also one of those transcendental categories, Kant also believes that geometry is a priori, but non-Euclidian thinking might make that a harder argument to accept).

In the end, I think that Helena, even if she is "just" counting a series, is doing math.  But she is also using these series, like any other way of organizing the relationship between her thoughts and the world, as a way to deal with the unknown: a new place, full of new things.  When there are "four" whales instead of "a lot", she can find a big, unknown world just a little easier to understand and deal with.