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The Faulty Logic of the ‘Math Wars’
This mechanical image of calculation is the target of a number of philosophical critiques. Arguably the most famous of these is in Wittgenstein’s later writings. Wittgenstein suggests that analogies between mathematical computations and mechanical processes only seem appealing if we overlook the fact that real machines have parts that bend and melt and are invariably subject to breakdown. If, however, we keep this fact in mind, it seems clear that a psychological mechanism cannot ground my confidence that I am getting things right. That is, it seems clear that, in order to be justified in believing that I have mastered an algorithm, I require a type of mental activity that isn’t simply causally generated. That, according to Wittgenstein, is why it is wrong to understand algorithm-based calculations as expressions of nothing more than “mental mechanisms.” Far from being genuinely mechanical, such calculations involve a distinctive kind of thought.
That the use of
standard algorithms isn’t merely mechanical is not by itself a reason to
teach them. It is important to teach them because, as we already noted,
they are also the most elegant and powerful methods for specific
operations. This means that they are our best representations of
connections among mathematical concepts. Math instruction that does not
teach both that these algorithms work and why they do is denying
students insight into the very discipline it is supposed to be about.