How many number names would be a minimum for progress? What is the ideal base? How far could you get without a zero? Could an alternative for zero be developed?
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How many number names would be a minimum for progress? What is the ideal base? How far could you get without a zero? Could an alternative for zero be developed?
Foster: A Number System Without Zero (1947)
Grant Hutchison
The Roman Empire achieved amazing engineering and command tasks with a clumsy number system that makes multiplication very difficult. No equivalent to the decimal point. Still used for some dates recording.
I suppose the minimum number of names required would depend on the number base - same number of names, otherwise you can't distinguish between members of your basic number set. Beyond that, you can name larger numbers by just concatenating the basic names, like reading out a telephone number. Can't see why any real language would restrict itself to just the minimal set, though.
Grant Hutchison
I think the Sumerian number system is actually a good example of why the actual question may be based on a wrong assumption, because I think in general that, because mathematics notation is not something that we do naturally, we adopt different systems in response to changes. The Sumerian system is called base 60 but it wasn't really a base 60 system, in the sense that there were never 59 different symbols or names for the numerals. Rather, they used base 10, and then base 6 on top of that. And it kind of makes sense, because there are generally two bases that people use naturally, obviously ten from our fingers, and then the twelve system that you can do by putting your thumb on the three parts of each finger. And 60 is a multiple of both 10 and 12 and allows you to do ratios very easily, so it's likely that they adopted it on top of base 10 when they needed it. And then zero was adopted by mathematicians who needed it for calculations, and then spread. So rather than being hobbled by a counting system, I think it is likely the mathematics would adapt. The Sumerians themselves apparently used a form of zero, in the shape of a kind of placeholder for missing digits. In Chinese counting traditionally there was no zero, but it was OK up to 100,000, because there are individual digits for 10, 100, 1000, and 10000. So for example, writing 九万７百四十二 means 90,742, because there is no 千 to indicate a number in the thousands. Beyond 100,000 it becomes clumsy, but probably there wasn't that much reason to count numbers that high. And now they've essentially adopted the Hindu-Arabic system.
60 is the smallest number that can be divided by all of the numbers 1 through 6 (and, of course, their counterparts 10,12,15,20 and 30).
So you can divide it evenly in half, thirds, quarters, fifths and sixths (as well as tenths, twelfths, etc).
The next number up - that's divisible by 7 as well - is 2520. Bit of a jump there.
How many number names are not a limiting factor. The Egyptians had one symbol for every number, which really wasn't helpful for doing math. The Romans had a more adaptable system of naming and regular numerals, but again, it wasn't really helpful for math. Both lacked a formal zero, too. But they did have a name for the idea, which was a quirk. I believe some Romans would use an N for zero, but not all the time and not like we use the digit zero, which was probably madding for math minded people.
232,792.560 is divisible by 1 to 20, but some people can't reach their toes so we should stay with 2520.
133,855,722,000 gets you to 25 if my math is correct, and 104,809,030,326,000 to 30.
To add to my prior comment: https://wals.info/feature/131A#1/26/146 and https://wals.info/chapter/131 (Bernard Comrie. 2013. Numeral Bases.
In: Dryer, Matthew S. & Haspelmath, Martin (eds.) The World Atlas of Language Structures Online. Leipzig: Max Planck Institute for Evolutionary Anthropology. (Available online at http://wals.info/chapter/131, Accessed on 2019-12-23.)
Different languages use decimal (probably most common), quinary (5), vigesimal (20), mixed decimal-vigesimal, base 60, and systems that can't be assigned to any of these (body-part system).
I suspect that the real impediment is how easily the numeric system is extended to very large and very small numbers. Some languages, for example, don't have counting numbers that extend pass twenty or so. The number base, per se, is not that important: it really doesn't matter if the base is 10, 12, 16, balanced ternary, or 11.