On Thu, Jun 12, 2014 at 8:41 AM, Nils Bruin <nbr...@sfu.ca> wrote:
> On Thursday, June 12, 2014 12:48:37 AM UTC-7, Marc Mezzarobba wrote:
>>
>> Sure. But why not have
>>
>> sage: bool(sin(x)^2+cos(x)^2==1)
>> True
>>
>> return False as well, and let the user test that
>>
>> (rhs-lhs).simplify_full() == 0
>
> That would probably be even desirable if you want to make SR more generally
> usable in sage (imagine, say a cached function with SR elements as
> arguments. You would probably end up with a routine that gets *slower* as
> its cache fills)
Also, I think we're at least consistent here. Do you have any examples where
sage: bool((lhs - rhs).simplify_full() == 0) != bool(lhs == rhs)
The fact that equality of symbolic expressions is undecidable in
general, and even more limited in implementation, is something that we
just have to live with.
>> 1. the behavior of == is already full of subtleties (and bugs),
>> so I don't think arguments such as "4/2==2 is surprising for
>> new users" are very strong;
>
>
> If 4/2==2 were to raise an exception, I agree, but we've already seen that
> internal usage of "==" precludes that option. Having 2/2 != 1 would just
> make too much code that is traditionally valid in all kinds of programming
> languages and computer algebra systems have insidious bugs to be socially
> acceptable.
I'll second this. Are there *any* computer algebra systems (or
programming languages for that matter) out there such that 4/2 != 2 !=
2.0? Code would simply be too hard to write.
>> Yes, it is great to have a rich equality predicate that automatically
>> does the right thing in many cases. The only problem is, I know I cannot
>> trust its output, and I am convinced that I never will because the
>> issues discussed in this thread are basically unfixable.
>
>
> I think that's correct. "equality" is a fluid term in mathematics that may
> appear to have a consistent meaning throughout in mathematics, but in fact
> it's a local decision which equivalence relations will be called "equality"
> in mathematics literature. And even within well-defined domains the thing
> that's decided to mean equality can turn out to be undecidable.
+1
Forcing users to use a method, and using == for something else, would
not solve these issues.
>> I think I would lose all confidence in Sage (which may not be such a bad
>> thing, after all).
>
>
> Or more generally, in math software. This problem is not limited to sage
> (although it may be worse because we get to blame python for some of the
> choices)
>
>> >> sage: {t.parent() for t in {R(42), 42}}
>> >> {Integer Ring}
>> >> sage: {t.parent() for t in {42, R(42)}}
>> >> {Univariate Polynomial Ring in y over Rational Field}
>> >
>> > Totally, just like Python's
>> >
>> > {type(t) for t in {0, 0.0}} == {float}
>>
> You can make these examples more fun by ensuring you can control the order
> in which the set gets constructed:
>
> sage: {t.parent() for t in set([42, R(42)])}
> set([Integer Ring])
> sage: {t.parent() for t in set([R(42), 42])}
> set([Univariate Polynomial Ring in y over Rational Field])
> sage: set([0, 0.0])
> set([0])
> sage: set([0.0, 0])
> set([0.000000000000000])
>
>> (After all, 4/2 and 2 *are not* equal. Otherwise 2.is_invertible() would
>> be True.)
>
> That is not such an uncontroversial statement. With ZZ regarded as a subset
> of QQ (quite a valid thing to do) we should have 4/2 == 2. In that view one
> should as whether 2 is invertible *in ZZ* , so it would be the is_invertible
> method that is flawed.
All the is_X methods are with respect to their parent, by design.
sage: R.<x> = ZZ[]
sage: (x^2 - 2).is_irreducible()
True
sage: R.<x> = RR[]
sage: (x^2 - 2).is_irreducible()
False
sage: R.<x> = QQ[sqrt(2)][]
sage: (x^2 - 2).is_irreducible()
False
- Robert
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