My other road to humanistic mathematics is through language. I’ve touched on this before. But first let me say that when I use the phrase “humanistic mathematics” I don’t distinguish between “teaching mathematics humanistically” and “teaching humanistic content”. I think they go very well hand in hand, and in my thinking, should do so when teaching.
I was at Wellesley College on Wednesday and I had a conversation with Karen Lange who is a logician. I described my project and perhaps not surprisingly we started to talk about the formalism of mathematics. As a logician, it was clear to her that no symbol or sets of symbols can capture the full content of a mathematical concept. Furthermore, the symbols we actually use are historical accidents – well perhaps not accidents – but the outcome of a historical process where often many different symbols or formalisms have competed. I’ve been thinking along these lines for a few years now, but haven’t really seen it stated explicitly it in print (except once) so it was encouraging to hear it from an expert.
We only have to think about the formalisms of calculus. The formalism of Leibniz survived. Of the formalism of Newton there only remains the overhead dots for time derivatives (and the primes). This is often convenient, but the formalism of Leibniz has much more inherent strength – it almost works by itself for practical computations. But then, Leibniz was explicitly interested in questions about languages of discourse.
My contention is that:
(1) The symbols and formalism of mathematics cannot in principle capture all aspects of a mathematical concept.
(2) Furthermore, and as a consequence of (1) the formalism is inherently vague to some extent.
(3) To make things worse, symbols are often used in slightly different ways in different contexts. Examples are the equality sign = and the zero symbol 0. And there are many +’s in mathematics, all with related meanings, but still …
(4) The formalisms we use are the outcomes of historical and cultural processes.
Question is: What are the consequences for teaching?