We would like to promote the development and use of the Aldor programming language by facilitating the interaction between developers and users of the language.

See: http://algebraist.origo.ethz.ch
and http://axiom-wiki.newsynthesis.org

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Bill Page posted Hopf Algebra = Group + Monad
Here is an example of some work that I would really like to do in Axiom and/or Aldor Hopf Algebra = Group + Monad http://sigfpe.blogspot.com/2008/08/hopf-algebra-group-monad.html by sigfpe on the b... [read more]

Thu, Aug 7 at 8:21AM (0 comments)

Bill Page responded to Project Euler

The statement at the Project Euler website includes: "Although mathematics will help you arrive at elegant and efficient methods, the use of a computer and programming skills will be required to solve most problems.". So I think the intention is q...


Tue, Jun 17 at 1:48PM (2 comments)

In his post: "Should 3+4 = 7?" on
http://lambda-the-ultimate.org/node/2...
Stephen Watt explained the rational for why Vector(3+4) is not equivalent to Vector(7) in Aldor (I agree the same applies to Axiom):

"The other stance th...


Tue, Jun 17 at 8:20AM (2 comments)

Bill Page left a note for 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.


Mon, Jun 16 at 7:50PM

William Sit responded to blog on Axiom

Thanks. I was more interested in the tesseract, and I left some comments there. http://amca01.wordpress.com/2008/05/1...

Also the wordpress.com site seems notorious for poor security:
http://en.wikipedia.org/wiki/Wo...


Sun, Jun 15 at 10:44PM (1 comment)

Bill Page responded to Rig

My main motivation for looking at this is a possible re-design of the Axiom library. Many times when I think about the mathematical taxonomy implemented there, I worry that it is too ad hoc. In a computer algebra system it seems to make sense to s...


Sun, Jun 15 at 10:08PM (2 comments)

Bill Wood left a note for 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


Sun, Apr 13 at 4:26PM

Bill Page left a note for 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/Sa...

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/Sa...

Cheers,
Bill Page.


Sat, Mar 29 at 8:46PM

We are now indexed in Technorati Profile.

Fri, Mar 28 at 12:52PM (0 comments)

Bill Page left a note for 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.


Sat, Mar 22 at 3:08PM

---------- Forwarded message ---------- From: Bill Page Date: Fri, Mar 21, 2008 at 5:07 PM Subject: Re: What Axiom can do for Aldor To: William Sit William, With your permission, I woul... [read more]

Sat, Mar 22 at 7:38AM (0 comments)

Bill Page left a note for 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 ... ]


Fri, Mar 21 at 12:58PM

Bill Page posted Please join the network!
I hope you will consider becoming part of an active community to support the development of the Aldor programming language.

Thu, Mar 20 at 9:20PM (0 comments)