Commentary. One of the few well-known quotes that seems to me exactly right and needs no further comment. Except that the whole essay is worth reading.
Commentary. Again, exactly right and needs only one comment: that it is much easier to recognize this effect in other people's subjects than in one's own!
Commentary. Exactly. Surely we all view the Pythagorean theorem as an exact fact in a certain idealized world.
Commentary. As does the previous quote, this sounds right, though it would be interesting to seek evidence beyond a few quotations.
This is a hobby-horse of mine which I shall develop at length elsewhere.
(i) Theorem-proof mathematics seems a much better name than pure mathematics for two reasons: (a) ``pure" refers to motivation, and it's perverse to name an academic discipine after the motivation of its practioners rather than what they actually do; (b) Areas such as computational complexity and much of mathematical economics are abstract theorem-proof mathematics just as are group theory and differential geometry, so why make an artificial distinction?
(ii) The opposite end of the spectrum is what I'll call engaging quantitative aspects of the real world (QARW). Bottom-level quantitative descriptions of aspects of natural science, engineering, human society etc are the domains of their own academic disciplines. But then the analysis of mathematical models (to derive predictions to test experimentally), and the modern statistics paradigm "here's a lot of data -- what does it tell us?", are instances of the intellectually serious pursuit I am calling QARW mathematics, its distinguishing feature being the explicit connection with empirics. A glance at a typical applied mathematics journal shows that most of what's traditionally called "applied mathematics" does not fit this criterion.
(iii) Each of these two types of mathematics is "anchored", the former to the whole preexisting body of theorem-proof mathematics, and the latter to empirical verification in the real world. Inbetween is a third type of mathematics I call unanchored; for instance, someone invents a probability model, does some simulations to investigate its behavior, but doesn't get any feedback from real data.
Commentary. A corollary to the "three types of mathematics" comment.
To make an analogy, when you talk about "baseball" you might be thinking about
(i) what's in the record books -- Joe DiMaggo and all that
or (ii) tonight's MLB game, as perceived by players or spectators
or (iii) baseball outside of MLB.
Similarly, when people assert generalities about mathematics they might be thinking about
(i) known mathematics in books
or (ii) the process of doing mathematical research
or (iii) using known mathematics
and these are sufficiently different that I'm doubtful whether any non-trivial generality applies to all of them.
Commentary. It's a good metaphor in that new mathematics adds on to what's known, and that it's important for new work to be placed appropriately in relation to previous work. In this sense mathematics is an opposite of poetry (say), where each poem stands alone. On the other hand a building has an overall design and purpose; mathematics is a distributed evolving entity.
Commentary. The thought is commonplace, but the telescope/galaxy analogy is crisper than other formulations I have seen.