Devanagari int literals [was Re: Should non-security 2.7 bugs be fixed?]

[Marko Rauhamaa]

Chris Angelico <rosuav at gmail.com>:

> On Tue, Jul 21, 2015 at 7:10 PM, Marko Rauhamaa <marko at pacujo.net> wrote:
>> This is getting deep. It is an embarrassing metamathematical fact
>> that numbers cannot be defined. At least, mathematicians gave up
>> trying a century ago.
>>
>>    In mathematics, the essence of counting a set and finding a result
>>    n, is that it establishes a one to one correspondence (or
>>    bijection) of the set with the set of numbers {1, 2, ..., n}.
>>    <URL: https://en.wikipedia.org/wiki/Counting#Counting_in_mathematics>
>
> AIUI, zero is defined as the cardinality of the empty set, one is
> defined as the cardinality of the set containing the empty set, two is
> defined as the cardinality of the set containing the empty set and the
> set containing the set containing the empty set... which makes
> mathematics the only language *more verbose* than the Shakespeare
> Programming Language in its definition of fundamental constants.

   There are two approaches to cardinality – one which compares sets
   directly using bijections and injections, and another which uses
   cardinal numbers.
   <URL: https://en.wikipedia.org/wiki/Cardinality>

The first approach is comparative, the second approach is quantitative.
You must be referring to the latter meaning (cardinality ~ cardinal
number). However:

   In mathematics, cardinal numbers, or cardinals for short, are a
   generalization of the natural numbers [...]
   <URL: https://en.wikipedia.org/wiki/Cardinal_number>

IOW, cardinal numbers assume natural numbers as a given. Thus, your
definition of natural numbers leads to a circular definition.

Frege et al tried to do the natural thing and defined natural numbers as
equivalence classes:

   0 = the set of sets with no elements
   1 = the set of sets with a single element
   2 = the set of sets with precisely two elements
   etc

Unfortunately, the natural thing leads to a contradiction and must be
abandoned.

Nowadays, mathematicians are content with working with one prototypical
chain of beads:

   0 = ∅
   1 = { 0 }
   2 = { 0, 1 }
   3 = { 0, 1, 2 }
   etc

IOW:

   0 = ∅
   σ(n) = n ∪ { n }

and forget about the true essence of numbers.


Marko