On Thu, Oct 1, 2015 at 11:00 AM, Bill Page <bill.p...@newsynthesis.org> wrote:
> On 1 October 2015 at 13:07, William Stein <wst...@gmail.com> wrote:
>> On Thu, Oct 1, 2015 at 9:38 AM, Bill Page <bill.p...@newsynthesis.org> wrote:
>>> [Changed thread subject from: Sources of funding - perhaps computer
>>> manufacturers? ]
>>>
>>> What I find hard to swallow is the peculiar mix of "parent",
>>> "category", and Python data types (class system and inheritance). In
>>> spite of the available documentation in the category system in Sage, I
>>
>> Just curious -- Bill Page, did you ever take a graduate mathematics
>> course in "category theory", or at least a course that had a
>> significant portion devoted to it, or at least carefully read (and did
>> exercises!) from a textbook on category theory?
>
> Yes, yes, and yes.
>
>> Because despite
>> being called "abstract nonsense" by some, category is a serious,
>> highly technical, and nontrivial area of mathematics, with interesting
>> theorems, a long development, etc., just like other areas of math.
>>
>
> Yes, certainly. I consider category theory an appropriate foundation
> for a very large part of mathematics.
>
>> I would expect that somebody who doesn't know category theory (the
>> mathematical area) would find the category framework in Sage very
>> confusing, just like somebody who doesn't know calculus might find the
>> symbolic functionality confusing.
>
> No, I find the Sage implementation of these ideas confusing, or at the
> very least overly complicated. I am sure that if I had a good reason
> and enough energy to want to overcome this impression
> (pre-conception?) then I would probably find categories in Sage very
> useful. It certainly seems to be useful to at least a small core of
> developers in Sage.
Cool; thanks for explaining your background.
One other impression you have is that categories were just bolted on
by combinatorics people at the end. However, David Kohel and I
actually implemented the first round of category-related stuff in Sage
right at the very, very beginning -- it was one of the first things we
did, motivated by what David had wished Magma had. And it is has
just been iteratively refined over the users, with Nicolas Thiéry
doing by far the most work on it during his sabbatical a few years
ago. There was also a lot of input coming from the multiple rounds
of rewriting of the coercion model, by me, David Roe, Robert Bradshaw
(especially), and others.
By the way, look at how coercion "works" in Magma:
$ magma
Magma V2.18-5 Thu Oct 1 2015 16:59:12 on compute3-us [Seed = 629019987]
Type ? for help. Type <Ctrl>-D to quit.
> R<x> := PolynomialRing(IntegerRing());
> x + 1/2;
>> x + 1/2;
^
Runtime error in '+': Bad argument types
Argument types given: RngUPolElt[RngInt], FldRatElt
>
Total time: 0.550 seconds, Total memory usage: 9.53MB
-- William
>
> It is important also to keep in mind what are called "categories" in
> Axiom have a best only an indirect connection to mathematical category
> theory. To my knowledge there has been only a bit of work done in
> trying to actually implement some form of category theory in
> Axiom/Aldor.
>
> http://axiom-wiki.newsynthesis.org/CategoryTheoryAndAxiom
>
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--
William (http://wstein.org)
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