Analytical operations in Mathematics do not discipline the reason; they do not familiarize the student with a chain of syllogisms connected by a manifest necessity at every link: they do not show that many kinds of subjects may be held by such chains: and at the same time, that the possibility of so reasoning on any subject must depend upon our conceiving the subject so distinctly as to be able to lay down axiomatic principles as the basis of our reasoning.
52 With reference to analytical mathematics, the argument in favour of the use of Mathematics as a permanent educational study, loses all its force. If we can only have analytical mathematics in our system of education, we have little reason to wish to have in it any mathematics at all. Our education will be very imperfect without Mathematics, or some substitute for that element; but mere analytical mathematics does not remedy the imperfection. If we can only have analytical mathematics, it is well worth considering whether we may not find a much better educational study to supply its place in Logic, or Jurisprudence. The general belief, for undoubtedly it is a general belief, that Mathematics is a valuable element in education, has arisen through the use of Geometrical Mathematics. If Mathematics had only been presented to men in an analytical form, such a belief could not have arisen. If, in any place of education, Mathematics is studied only in an analytical form, such a belief must soon fade away.
William Whewell, Of a Liberal Education in General, pp. 49-50.
This can be easily misread if one is not familiar with the whole of Whewell's argument. He is not claiming that analytical mathematics -- algebra and calculus -- are inferior mathematics, or that they are ill-suited to discovery of mathematical truths; in fact, he thinks the opposite is true. He is also not saying that they should not be part of education at all. His argument is rather that they are not good as foundations -- education, insofar as it is mathematical, should build up to them, not take them as basic. It's the perfection of analytical mathematics, its capacity for extremely abstract representation, that makes it poorly suited for getting people used to thinking mathematically and rationally. And this is because in analytical mathematics, as such, you don't actually think about problems -- you think about formulas and abstract relations that can be interpreted in many different ways for many different kinds of problems. If people get started too early on this, it is easy for them to start using these x's and y's and formulas concerning them as nothing but a crutch. In geometry, however, which Whewell argues should be a major and foundational part of education, thinking about this particular problem, and what this particular problem requires, and going through it step by step, is an immense part of what you do.
I think it's interesting that our educational systems have generally done what Whewell says they should not do -- focus on analytic mathematics rather than geometry, in both mathematics and the sciences -- and that the result he predicts has in fact come about -- people in general often don't see the point of their mathematics education. Whether the one is caused directly by the other is a trickier question, but it is worth thinking about.
