Bill Page

@David Casperson: David,

I would like to recall something that you wrote on Thu, Apr 3, 2008 at 1:33 PM (in part):

I think that a far more fundamental question is "what is the relationship between the object- and the meta-language?" In this
case the meta-language is Aldor in which we express ideas and
algorithms about mathematics of interest.

As category theory is very expressive it is tempting to view all mathematics as applied category theory and say that the objects that we want to talk about: rings, Lie algebras, etc., are categories. That would make category theory the object
language. This doesn't necessarily imply anything about the meta-language, except that it ought to be able to manipulate the things of the object language easily.
...
Translated back into Aldor, the category theoretic constructions
that we want to talk about need not necessarily be the category
theoretic constructs that we want to use to talk about category theory.

------

I think you are absolutely correct to raise this issue. Why indeed should we attempt to design a language like Aldor at all? Afterall, in the final analysis all programming languages are essentially equally expressive - they all are (in principle) Turing-complete. So there is nothing we can do in one that we cannot (in principle) do in another.

I think the answer has to do with the expressiveness of the language. That is, how efficiently can we express the ideas (programs) that we need to express? It seems very likely to me that the most efficient language for this purpose is the metalanguage itself (otherwise it seems likely that if a more expressive language existed, mathematics would quickly adopt it).

So the closer our object language is to our metalanguage, the more "expressive" it is. It seems that the only thing that might prevent us from achieving this optimum is that the meta-language is not in general be entirely computable (or else it is in some essential respect inconsistent).

But category theory itself appears to have some kind of universal applicability. So I wonder if you would agree that in spite of having "translated back into Aldor" as you said above, that we can (and should) remain in category in so much as the formal semantics of Aldor can be specified in categorical terms?

I wonder however exactly what cateogory theory is most appropriate to describe such a programming language? One possible answer I think is topos theory - or by another name: algebraic set theory. Choosing this as the basis for the underlying semantics of the Aldor language would have numerous implications.

Bill.

	Bill Wood

@Bill Page: Sorry not to get back to you sooner; it's been a busy month. My "strategy" for finding out what Aldor is good for is to ask you guys! For example, if you are familiar with the ProjectEuler site, would Aldor be a good language for solving their usual math-intensive programming problems? I've used lisp for most of the problems I've worked, along with the ECLiPSe Prolog+Constraint Logic Programming system and Python for a couple of problems. I gather from recent postings here that some view Aldor as a language to program abstract algebra computations in. What about semi-numeric computation, such as solving congruences or computing with continued fractions? I saw your reference to category theory; has anyone mentioned using Aldor to support some of the reasoning about programs that Haskellers talk about?

Well there's a start. Thanks for giving me the opportunity to ask questions.

-- Bill Wood

	Bill Page

@Oziewicz Zbigniew: Zbigniew, I am very happy now there are at least two people here who claim to know something about category theory! :-) We might wish that Aldor (and computer algebra systems in general) had more direct support for doing categorical computations by at least there is already the beginning of this subject by Saul Youssef:

http://axiom-wiki.newsynthesis.org/SandBoxAldorCategoryTheory

Concerning Frobenius algebra in relativity, I think we must consider first non-commutative algebras. I am not so sure about the Aldor libraries in this regard, but at least Axiom has non-commutative polynomials. There are some simple example calculations here:

http://axiom-wiki.newsynthesis.org/SandBoxNoncommutativePolynomials

Cheers,
Bill Page.

	Bill Page

@Bill Wood: Hello Bill,

Welcome to the Algebraist!

My main motivation for posting this message is the goal of "finding out what Aldor is good for" that you listed as one of your projects. I think that is a very good question. :-) When you have a moment it would be great if you could jot down some notes here (write a blog entry?) about how you intend to find an answer to that question... seriously. It seems to me that developers often do not spend enough time (re-)thinking basic questions like this, so sometimes even to us it seems difficult to answer the question. And it is not so easy to suggest to someone new what path they should follow to find out more about Aldor.

Regards,
Bill Page.

	Bill Page

@William Sit: Hello William,

Thanks for joining in! :-)

One thing that I would really like to discuss here is the relationship between Axiom and Aldor. In particular: What can Axiom do for Aldor? As opposed to the perhaps more obvious question of: What Aldor can offer Axiom users...

What do you think?

[Repeated for the record ... ]