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?