Thinking About Abstractions

I liked Steven Pinker’s New York Times Magazine article on the “moral sense” a great deal, though I to some extent share Will Wilkinson’s concern that Pinker winds up trying to steal a base. I don’t, however, think the objections raised at The American Scene by Peter Suderman, Matt Frost, and Jim Manzi quite hold up.

I think if you want to properly understand what Pinker’s up to, it’s worth thinking about something else: Math. When we do math, we talk a lot about numbers. We don’t talk about numerals, the concrete typographical signifiers of numbers. “V” is a numeral (a Roman numeral) as is “5” but they both stand for the number that you get when you add the number represented by the numeral “4” to the number represented by the numeral “1.” In short, unlike numerals, number are abstract entities. From a certain point of view, this can make the whole enterprise of math come to seem very mysterious. If the numbers are abstract, how can we interact with them causally? And if we can’t interact with them causally, how can know anything about them? One can easily stumble into the view that either all this math talk is just so much BS, or else that there is some heavenly Realm of the Numbers where they live and send us messages through the ether.

An alternative approach would be to step back and decide to consider some different questions. This whole subject of what the numbers are came up because there seems to be so much math happening. Indeed, the math is pretty damn useful. So we could ask ourselves some questions about the role this plays — about what math lets us do. And we can ask questions about the sociology of math, about how the human institution of “doing math” works. If we do this, we’ll notice that the math gets done and it evidently gets done without anyone interacting with an abstract Realm of the Numbers.

Then we can ask ourselves how all this math started getting done. We can look at the history of math. And we can look at ways of teaching math and ways of learning it. We could even devise some experiments to see what happens in a person’s brain when he answers “7 times 6 is 42.” We could see if it looks different for people who’ve memorized their multiplication tables than it does for people who are thinking to themselves “7 plus 7 is 14, plus 7 is 21, plus 7 is 28, plus 7 is 35, plus 7 is 42” (I bet it does). Maybe we can learn a thing or two about the evolution of this business.

The practical gains to our ability to do math that arise from this process of reflection might be enormous. As medieval Europeans we might reflect that our understanding of math originates in our ability to count, and thus that our numerals are based on a basic tally system with “|” for 1, “|||” for 3, and so forth. But math can do so much more than count! But it can do it easier and better if you adopt the zero and a whole different way of working with numerals.

Now, you can imagine someone objecting — “you’re just talking about numerals, but math is about numbers your study of the history of our numerals couldn’t possibly improve our math.” That person would, of course, be incredibly wrong.

That right there brings us pretty far afield from Pinker, but I think it basically sets one up for the proposition that the study of moral psychology — examination of the issues of how reasoning about morality works in practice — could shed some light on moral issues even though, yes, “I wanted to know about what you should do not a bunch of descriptions of what people do in practice.”