Chris Angelico <ros...@gmail.com>:
> On Fri, Jul 7, 2017 at 6:43 PM, Marko Rauhamaa <ma...@pacujo.net> wrote:
>> Python's integer object 0 might be equated with the (mathematical)
>> natural number 18974387634. Python code would have no way of
>> introspecting that natural number.
>>
>> The execution model would determine what properties object 18974387634
>> would have.
>
> Then what's the point of that number? If you can't see it from Python
> code, it's not part of the language semantics.
Excellent question!!!
In fact, it is a very frustrating question. You can only define the
semantics of Python (in this case) by providing an *arbitrary* mapping
to an imaginary abstract machine. There's no way to define the objective
abstraction.
Metamathematicians grappled with the same problem a century ago when
they tried to define natural numbers. Their promising start collapsed
because of the Russel paradox. To their great disappointment, they had
to choose an arbitrary set-theoretical model to be the standard:
0 = {}
1 = {0}
2 = {0, 1}
3 = {0, 1, 2}
etc
In fact, today's mathematicians couldn't care less what natural numbers
are. They have captured all relevant characteristics in a number axioms,
and those suffice to generate all interesting mathematics.
Marko
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