**... mathematical ideas originate in empirics, although the genealogy is sometimes long and obscure. But, once they are so conceived, the subject begins to live a peculiar life of its own and is better compared to a creative one, governed by almost entirely aesthetical motivations, than ... to an empirical science.**(von Neumann, in The Mathematician).*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.**As a mathematical discipline travels far from its empirical source, or still more, if it is a second and third generation only indirectly inspired by ideas coming from "reality" ..... there is a grave danger that the subject will develop along the line of least resistance, that the stream, so far from its source, will separate into a multitude of insignificant branches, and that the discipline will become a disorganized mass of details and complexities.**(von Neumann, continuation of previous quote)*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!-
**It is a truth universally acknowledged that almost all mathematicians are Platonists, at least when they are actually doing mathematics rather than philosophizing about it.**(Gerald B. Folland, in this 2010 article.)*Commentary.*Exactly. Surely we all view the Pythagorean theorem as an exact fact in a certain idealized world. -
**... most nonmathematicians who use mathematics .... are formalists. For them mathematics is the discipline of manipulating symbols according to certain sophisticated rules, and the external reality to which those symbols refer lies not in an abstract universe of sets but in the real-world phenomena that they are studying.**(Gerald B. Folland, continuation of previous quote)*Commentary.*As does the previous quote, this sounds right, though it would be interesting to seek evidence beyond a few quotations. -
**There are three types of mathematics;***pure*and*applied*are not amongst them.*Commentary.*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. **Identifying "mathematics" as theorem-proof mathematics is like identifying "graphic art" with oil painting.***Commentary.*A corollary to the "three types of mathematics" comment.**The body of mathematical knowledge resides in two quite different places; in libraries and in mathematicians' heads.***Commentary.*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.**The metaphor of theorem-proof mathematics as a building is illuminating but not perfect.***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.-
**Seeing an astronomer using a telescope to observe a galaxy, no-one will confuse the telescope with the galaxy. Mathematics differs from science in that there is no clear distinction between the tools and the objects of study.***Commentary.*The thought is commonplace, but the telescope/galaxy analogy is crisper than other formulations I have seen.