JB: Sounds like we're talking about an anthropic principle of mathematics here.
HERSH: Maybe so; I never thought of that. I had a serious argument with a friend of mine at the University of New Mexico, a philosopher of science. She said "There are nine planets; there were nine planets before there were any people. That means there was the number nine, before we had any people."
There is a difficulty that has to be clarified. We do see mathematical things, like small numbers, in physical reality. And that seems to contradict the idea that numbers are social entities. The way to straighten this out has been pointed out by others also. We use number words in two different ways: as nouns and adjectives. This is an important observation. We say nine apples, nine is an adjective. If it's an objective fact that there are nine apples on the table, that's just as objective as the fact that the apples are red, or that they're ripe, or anything else about them, that's a fact. And there's really no special difficulty about that. Things become difficult when we switch unconsciously, and carelessly, between this real-world adjective interpretation of math words like nine, and the pure abstraction that we talk about in math class.
That's not really the same nine. Although there's of course a correlation and a connection. But the number nine as an abstract object, as part of a number system, is a human possession, a human creation, it doesn't exist without us. The possible existence of collections of nine objects is a physical thing, which certainly exists without us. The two kinds of nine are different.
Like I can say a plate is round, an objective fact, but the conception of roundness, mathematical roundness, is something else.
Sad to say, philosophy is definitely an optional activity; most people, including mathematicians, don't even know if they have a philosophy, or what their philosophy is. Certainly what they do would not be affected by a philosophical controversy. This is true in many other fields. To be a practitioner is one thing; to be a philosopher is another. To justify philosophical activity one must go to a deeper level, for instance as in Socrates' remark about the unexamined life. It's pathetic to be a mathematician all your life and never worry, or think, or care, what that means. Many people do it. I compare this to a salmon swimming upstream. He knows how to swim upstream, but he doesn't know what he's doing or why.
JB: How does having a philosophy of mathematics affect its teaching?
HERSH: The philosophy of mathematics is very pertinent to the teaching of mathematics. What's wrong with mathematics teaching is not particular to this country. People are very critical about math teaching in the United States nowadays, as if it was just an American problem. But even though some other countries get higher test scores, the fundamental mis-teaching and bad teaching of mathematics is international, it's standard. In some ways we're not as bad as some other countries. But I don't want to get into that right now.
Let me state three possible philosophical attitudes towards mathematics.
Platonism says mathematics is about some abstract entities which are independent of humanity.
Formalism says mathematics is nothing but calculations. There's no meaning to it at all. You just come out with the right answer by following the rules.
Humanism sees mathematics as part of human culture and human history.
It's hard to come to rigorous conclusions about this kind of thing, but I feel it's almost obvious that Platonism and Formalism are anti-educational, and interfere with understanding, and Humanism at least doesn't hurt and could be beneficial.
Formalism is connected with rote, the traditional method which is still common in many parts of the world. Here's an algorithm; practice it for a while; now here's another one. That's certainly what makes a lot of people hate mathematics. (I don't mean that mathematicians who are formalists advocate teaching by rote. But the formalist conception of mathematics fits naturally with the rote method of instruction.)
There are various kinds of Platonists. Some are good teachers, some are bad. But the Platonist idea, that, as my friend Phil Davis puts it, Pi is in the sky, helps to make mathematics intimidating and remote. It can be an excuse for a pupil's failure to learn, or for a teacher's saying, "Some people just don't get it."
The humanistic philosophy brings mathematics down to earth, makes it accessible psychologically, and increases the likelihood that someone can learn it, because it's just one of the things that people do. This is a matter of opinion; there's no data, no tests. But I'm convinced it is the case.