Is math too tidy to be useful? I tend to agree with the reviewer of Deborah Stone's book, Counting…, Eugenia Cheng's assessment, which is that it's likely that the book may give the philistines/anti-math/anti-science brigade ammunition to criticize mathematics and science, but, in my opinion, that's a risk worth taking. The trouble is that the matter of math's 'tidiness' has, to my knowledge, never been discussed philosophically in mathematics textbooks (none that I've seen anyway and that's been quite a few), and it should be for the very reason that mathematical results can be misinterpreted and or deliberately misused (the lies, damned lies and statistics argument), and also the fact that many people aren't clued up on how to apply mathematics (that being separate from actually doing it).

My position is that in some instances math is too tidy, in others it's perfect for the job, and others still it's either inadequate or requires heavy going—that's lots of mathematical slog required that few people actually know how to perform. For instance, just about every technical and science person has used tables of integrals in calculus wherein the integrals have already been worked out because they're too hard to resolve on an everyday/as needed basis. Moreover, in many problems such as in QED, relativity, turbulence etc. math can get fiendishly hard or overwhelmingly involved even for scientists. We've all heard stories about Einstein having to first understand Curbastro's Ricci curvature and Minkowski space; then there's the devilish Naiver-Stokes equations for turbulence and so on, and so on. In these very complex problems, it seems to me that math hasn't evolved sufficiently to make life easy and it's likely that it never will.

At the other extreme, 'is math too tidy' then I'd suggest for some jobs it is. There's many examples but I'll give just a couple from personal experience. A toolmaker I know uses a computer program to calculate gear ratios to select several pre-made gears from a box of 100 or so. He then installs them into a thread grinder to grind threads (the ratio of gears then sets the thread's parameters). The strange thing is that the programmer never truncated the results/output, so all ratios are displayed to a ridiculous 12-digit accuracy.

Over the years, I've tried to tell him that the program's super high accuracy is nonsense and bears little relevance to practical machining but he won't have a bar of it, he's convinced that 12 digits are necessary otherwise the programmer would not have written the program to display that precision. (Incidentally, the program was specifically written for the purpose of practical machining, not some theoretical application.) Trying to convince him that an accuracy of 6 or 7 digits is at least an order of magnitude more than he will ever need—even when cutting very long threads where helical errors progressively compound as the cutting proceeds—but it's a complete waste of my time. Despite being comparatively bright (his machining is excellent), he has little or no concept of rounding or relevant accuracy.

Another example: years ago, I was involved with writing specs for a bespoke truck scale/weighbridge for an industrial application but I was not involved in designing or building its electronics. Despite my recommendations, the circuit designer who designed and built the electronics—which all worked fine—except that he used an eight-digit readout to indicate the tonnage. I pointed out to him that eight digits implied an accuracy that, very conservatively, was at least three orders of magnitude more accurate than the bridge's load cells were and thus the display was both misleading and ergonomically unsound. An argument of sorts ensued but he wouldn't budge. I even pulled out a Fluke-77 multimeter and showed him how Fluke did precision and rounding on its multimeters but to no avail!

I've seen many similar instances over the years. To me, this seems to indicate that there's a significant systemic failure in the way we teach mathematics. When it comes to rounding and accuracy, it is not enough to assume that people will either use common sense and or mathematics itself, thus we have to labor the point about what it's all about and we need to do so very early on in school.

I've not yet read Deborah Stone's book, Counting… but I gather this is the kind of problem she's driving at.