Comments on: Diversionary History

By: Alan Liddell

Alan Liddell — Wed, 29 May 2013 01:28:58 +0000

You don’t. Zero is not a counting number. Some definitions of the set N of natural numbers contain zero, some don’t — the crux is the application.

By: Handle

Handle — Wed, 29 May 2013 01:25:30 +0000

But how do you represent 0?

By: Alan Liddell

Alan Liddell — Wed, 29 May 2013 00:52:22 +0000

@admin General formula (more or less) found. It's piecewise, depending on odd or even, and has (several) other quirks, but we can definitely tease base-10 natural numbers into these "superbinaries".

By: admin

admin — Wed, 29 May 2013 00:39:12 +0000

‘Zero Messiah’ makes the point more provocatively.

By: admin

admin — Wed, 29 May 2013 00:38:20 +0000

The Babylonians had inconsistent modules, starting with decimal, and then switching to 60. Without rigidly consistent modules, an efficient positional system is impossible.

By: James A. Donald

James A. Donald — Tue, 28 May 2013 23:24:20 +0000

The Babylonians started off with an oral system where they had a word for a hundred, a word for a sixty, a word for a ten, and a word for a one, or at least that is how they did things when they were writing words in text as text, much as we would say “three hundred and seven” in text, and 307 in a numeric context.

Where we would say one hundred and ninety two, they would say hundred, sixty, ten, ten, ten, one, one.

Perhaps if they had had words for the numbers one to sixty, or words for the numbers one to ten, they would have wound up doing things your way, and today, we would all be doing things that way.

So if we, having words all the way up to a score, had invented positional notation, would have wound up with your system, base twenty.

By: northanger

northanger — Tue, 28 May 2013 23:14:33 +0000

Zero as Messiah.

By: admin

admin — Tue, 28 May 2013 22:27:45 +0000

“… we seem to be operating under the assumption here that if, say, the Babylonians had a counting system that needed a placeholder, they could have just scrapped it for one that doesn’t.” — I don’t think anybody is arguing (or assuming) that.
Rather:
(a) Place value is conceptually independent of zero.
(b) There is no logical reason why a simple positional notation, without zero, could not have arisen.

By: Chevalier de Johnstone

Chevalier de Johnstone — Tue, 28 May 2013 16:30:41 +0000

Perhaps I am missing something, but we seem to be operating under the assumption here that if, say, the Babylonians had a counting system that needed a placeholder, they could have just scrapped it for one that doesn’t. Isn’t it more likely that the notation system, once invented and adopted, displays first-mover technology advantages and is resistant to replacement by some other technology, even if better? Since in the Babylonian base-60 cuneiform system it is not possible to represent some numbers without a placeholder, they invented a placeholder. Yes, they could have thrown out the existing notation system and invented something new, just as we could all be typing on Dvorak keyboards right now.

To say that the positional function of zero is historically superfluous because it’s possible to invent a notation system that doesn’t use it seems to me like saying the function of the accent mark in, say, Spanish or French is superfluous because people could just change their pronunciation to avoid needing an accent mark notation. If we adopt this version of an objective analytical approach then a whole lot of cultural relics are superfluous…but so what?

By: admin

admin — Tue, 28 May 2013 14:51:07 +0000

Goedel clicks in more naturally a little further down the rabbit hole …