What’s the connection between philosophy of mathematics and teaching of mathematics? Each influences the other. The teaching of mathematics should affect the philosophy of mathematics, in the sense that philosophy of mathematics must be compatible with the fact that mathematics can be taught. A philosophy that obscures the teachability of mathematics is unacceptable. Platonists and formalists ignore this question. If mathematical objects were an other-worldly, nonhuman reality (Platonism), or symbols and formulas whose meaning is irrelevant (formalism), it would be a mystery how we can teach it or learn it. Its teachability is the heart of the humanist conception of mathematics.
In the other direction, the philosophy of mathematics held by the teacher can’t help but affect her teaching. The student takes in the teacher’s philosophy through her ears and the textbook’s philosophy through her eyes. The devastating effect of formalism on teaching has been described by others. (See Khinchin or Ernest.) I haven’t seen the effect of Platonism on teaching described in print. But at a teachers’ meeting I heard this:
“Teacher thinks she perceives other-worldly mathematics. Student is convinced teacher really does perceive other-worldly mathematics. No way does student believe he’s about to perceive other-worldly mathematics.” Platonism can justify a student’s certainty that it’s impossible for her/him to understand mathematics. Platonism can justify the belief that some people can’t learn math. Elitism in education and Platonism in philosophy naturally fit together. Humanist philosophy, on the other hand, links mathematics with people, with society, with history. It can’t do damage the way formalism and Platonism can. It could even do good. It could narrow the gap between pupil and subject matter. Such a result would depend on many other factors. But if other factors are compatible, adoption by teachers of a humanist philosophy of mathematics could benefit mathematics education.
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Political conservatism favors an elite over the lower orders. In mathematics teaching, Platonism suggests that the student either can “see” mathematical reality or she/he can’t.
A humanist/social constructivist/social conceptualist/quasi-empiricist/naturalist/maverick philosophy of mathematics pulls mathematics out of the sky and sets it on earth. This fits with left-wing anti-elitism—its historic striving for universal literacy, universal higher education, universal access to knowledge, and culture. If the Platonist view of number is associated with political conservatism, and the humanist view of number with democratic politics, is that a big surprise?
