most natural measurement systems are somewhat duodecimal, with 12 inches to a foot, and 12 ounces to a pound, they are also more intuitive than metric units (and I say this as definitely not-a-Yank). When you go to the butcher shop you ask for a quarter of liver, not for 250 grams of liver (in fact I once did this and the butcher remarked it was an unusual way of asking for it)…
maybe not-a-Yank but obviously not-a-Frog either. Here in Frog country, we definitely order 250g of something (one-fourth-of-a-kilogram would be very weird). I guess it is the same way in most of the civilized world. It is also definitely easier to use than the fraction system (if I want 200g/person for 3 persons, I will order 600g; I suspect that even in fraction-land, ordering 3/5th of something is a bit unusual, right? besides, sorting { 625, 600, 667 } is much easier than sorting the corresponding fractions {5/8, 3/5, 2/3}). (source: am a Frog).
For physics applications, using a number system adapted to the measure system is a good thing (for a simple explanation why: imagine computing the volume, in gallons, of a box whose dimensions are given in feet and inches. That is ugly. The same thing in metric is trivial enough that you can do a mental estimate).
However, it is hard to change the metric system (you have to set up a standard, as in Bureau International des Poids et Mesures), but even harder to change the number system since this one is already an international standard (a bit like we tried with decimal time, which failed miserably since there already was a standard). So it is the measure system which must be adapted to the number system.
The problem is that, historically, the measure system was not coherent in base 12 or 20 or 60 or whatever, but a real mess:
https://commons.wikimedia.org/wiki/File:English_length_units_graph.png
So, in order for base-12 to be really useful, you need to first sort out the measures themselves, which basically requires inventing the metric system.
Fore pure math, the choice of the basis is irrelevant. (Source: am a mathematician; computations in base 10, 2, 16, or
p are really the same. We barely even write any numbers, actually). The big thing is the positional number system (including zero); without indo-arabic digits, it is likely that someone would eventually have added a zero to the Greek letters.
About the history of the development of math, it was actually very much invented “on the fly” depending on the needs. A few examples:
- basic arithmetic (positional number system, four operations, negative numbers) was invented for the accountants (XIII-XVIth centuries),
- probability theory for the insurance companies and gamblers (why am I making these two separate classes, I don't know) (XVIIth-XVIIIth centuries),
- calculus for the physicists (this is useless unless you have a good watch) (XVIIth-XVIIIth centuries),
- numerical methods for analysis were invented with the age of the computer (in a wider sense, as “human computer” and “mechanical computer” and “electromechanical computer” and finally “transistor computer”).
So I think that math developed more or less according to its age and time and that it could not have developed significantly faster. On the other hand, the fact that, in any of these instances, it did not come too late either, shows that this was probably the only way that it could develop.
