Numbers and digits (was Re: bases)

[Edward Cherlin <edward.cherlin.sy.67@aya.yale.edu>]

On Sunday 05 May 2002 11:53, And Rosta wrote:
> > From: John Cowan [mailto:cowan@ccil.org]
> > Sent: 11 November 2001 17:43
> >
> > And Rosta scripsit:
> > > And, similarly, just because Lojban has noncompositional words
> > > for zero through to 15 does not mean Lojban by default uses
> > > base 16.
> >
> > Note that dau does not represent the *number* 10, but the *digit*
> > 10; a fine distinction, but important.
>
> What is the distinction?
>
> --And.

It is the usual problem of use vs. mention. "li dau" uses the digit 
"dau" in mentioning the number 10.

We use digits and other elements, such as a negative sign, as part of 
the representation of numbers that we wish to mention. It is 
particularly easy to confuse the categories when talking about 
single-digit numbers. 

In the following, the form 'A' is a single character, and "A" is a 
string containing one character. (This notation is well-known and 
represents an essential distinction in many programming languages. It 
is usually difficult to get it across to novices, for the same 
reasons that sometimes make it difficult to explain the difference 
between a digit and a number.) I differentiate numbers from 
characters or strings with the prefix "number" or by their appearance 
in a mathematical expression, and write numbers in decimal notation. 

The representation of a number in a particular notation is not the 
number. Thus

~("10" = 10)
~("0A" = 10)
~("dau" = 10)
~("pano" = 10)

The digit '1' is not the number 1, and the digit "dau" is not the 
number 10. The number 10 can be represented by the digit sequence 
"dau" (containing one digit) in an appropriate context, such as 
following "li". The number 160 can be represented by the digit 
sequence "dauno" containing two digits. In this case, as in any 
positional notation, a digit in the rightmost column represents a 
certain value, and the same digit in the next column to the left 
represents that value multiplied by the current base. Thus in 
"dauno"(base 16), we can say that "dau" has as value the number 160, 
but in "dauno"(base 12), its value is the number 120. "dau" is not 
itself the number 10, or the number 160, or the number 120. It is a 
digit representing those values and others in context.

We can say that "dau" represents the number 10 in certain contexts, 
as long as we are aware that this relationship is contextual, and not 
one of identity. It is correct to say that the number represented by 
the string "dau" is 10, but it is nonsense to say that "dau" is the 
number 10.

li dau du li pano //Correct in some contexts
zo dau du li pano //Not just false; a category error

-- 
Edward Cherlin
Generalist
"A knot! Oh, do let me help to undo it!"
--Alice in Wonderland