Am 24.03.2014 01:58, schrieb Richard Fateman:
Now we must address what is meant by integer. In common lisp, integer
meansarbitrary precision integer.
> Consequently, every rational number IS an integer/integer.
[...]
> It seems to me the meanings of words in sympy should correspond to
> their meanings in mathematics not some hack mockery of the word that
> appears in one or even several programming languages.
Your position seems inconsistent - you derive your idea about the nature
of rational numbers from Common Lisp, yet you denounce exactly that kind
of reasoning as "some hack mockery of the word that appears in [...]
programming languages".
It seems to me the meanings of words in sympy should correspond to their
meanings in mathematics not some hack mockery of the word that appears in
one or even several programming languages. The data structure for IEEE
double-float is used to represent a subset of the rational numbers.
To paraphrase, IEEE numbers are some hack mockery of the word that
appear in hardware implementations.
You could also have a data structure for "rational as the ratio of two
arbitrary precision integers,."
> That's been found fairly useful for symbolic programming.
Isn't that what SymPy does?
----
Just to demonstrate how far away "meaning in mathematics" is from any
computerized representation, including that of Common List, here's a
short (and probably slightly wrong) description of what rational numbers
"are":
Going down to the very foundations, rational numbers "are" the minimum
model that satisfies the field axioms (see model theory for the
definition of "minimum" and why "the" minimum exists for these axioms).
Most importantly, rationals "are" not a pair of integers, because for
integer pairs, (1,2) != (2,4), but for rationals, 1/2 = 2/4.
Integer pairs are merely a possible *representation* of rationals.
And now, for fun, the real numbers:
These "are" the minimum model that satisfies the total and the dense
ordering axioms.
No mention of field properties. The real numbers just happen to have the
rational numbers embedded (where "embedded" is a term with a strict
definition in model theory).
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