What "linguistic and philosophical problems" might be inherent in trigonometry?

Question

In "A Mathematician’s Lament", Paul Lockhart derides the "status quo" of math education, claiming that "mathematics is an art form done by human beings for pleasure" but instead is taught "devoid of creative expression of any kind". His writing is provocative, and I'm sure his accusations and suggestions could spark lively debate wherever they might circulate. I'm also sure such debate would be against the rules of this site…

My question is not about the essay's arguments per se, but rather what Lockhart might be referring to near the end of his essay, as he criticizes a typical trigonometry class wherein:

The measurement of triangles will be discussed without mention of the transcendental nature of the trigonometric functions, or the consequent linguistic and philosophical problems inherent in making such measurements.

This topic seems like it'd be interesting to learn more about — but what is he alluding to here? What "linguistic and philosophical problems" would he claim come about in trigonometry? Are they inherent in "triangle measurement" itself, or do they come about more specifically with the use of transcendental functions?

Answer 1

Here's my own best guess, even though the grammar Lockhart uses seems to most simply reduce to:

The measurement of triangles will be discussed without … [mentioning] the consequent linguistic and philosophical problems inherent in making such measurements.

My guess is that he really means something more like:

[Trigonometry] will be discussed without mention of the transcendental nature of the trigonometric functions, or the consequent linguistic and philosophical problems inherent in [that class of functions].

Lockhart places a significant emphasis on the history of mathematics in his essay. For just one example (the stem "histor" appears eighteen times in his paper!), he asserts:

Technique in mathematics, as in any art, should be learned in context. The great problems, their history, the creative process— that is the proper setting

So I looked a bit into the history of transcendental functions. They were first defined by Euler, who also had a major influence over our conception of functions in general. I found one page on The function concept to be particularly relevant. I'll isolate one particular thread, which runs through several paragraphs in the original:

[I]n 1748 the concept of a function leapt to prominence in mathematics … due to Euler[, who] divides his functions into different types such as algebraic and transcendental. … However there was a difficulty in Euler's work which was to lead to confusion, for he failed to distinguish between a function and its representation.

There's whiffs of philosophy and linguistics already in this summary ("concept", "types", "representation"), which might come through more strongly in the original sources and books cited. Perhaps the "problems inherent" are semantic, e.g. what does it mean for a function to "transcend", and Lockhart has just fissioned that into its constituent linguistic and philosophical parts.

This answer seems the "easy" one, and certainly not as interesting as awakening dormant existential and semiotic dilemmas, merely holding a ruler up to a three-sided shape! (Like Schrödinger's cat, but now an ethical dilemma concerning the metaphysical welfare of a polygon?) So I hope this is the wrong answer, or at least that someone more competent in math history can give a more definitive one.

Answer 2

What Lockhart seems to be referring to is the fact that in order to express the sine of a rational angle (for example an integer number of degrees) you will in general need transcendental numbers. In other words, working with trigonometric functions forces you to broaden the range of allowable numbers. The history of such "broadenings" is very interesting and Lockhart seems to lament the lost potential for classroom excitement when this history is not explored.