[Axiom-mail] Re: Axiom and OpenMath/MathML

[Bill Page]

On Saturday, December 07, 2002 3:54 PM Tim Daly
address@hidden wrote:

>
>> C Y wrote:
>> 
>...(snip)...
>> You might want to read 
>> http://www.cs.berkeley.edu/~fateman/papers/openmathcrit.pdf
>> to see some of his objections to OpenMath.
>
> Well, I've read Fateman's objections and I tend to agree with
> him.

Tim, I am surprized! My reaction to Fateman's paper was
immediately negative and even after re-reading I still get
the impression that it is only a rather poorly executed
hatchet attempt. I suppose that to debate this here in the
Axiom mail list might be considered kind of coping out since
I know that there are supporters of OpenMath who read this,
but I have know idea whether Richard Fateman does. Perhaps
it has been argued in detail elsewhere, but what harm can
it do (except distract Tim from getting Axiom into a
publishable open source state... <grin>).

On

  http://www.cs.berkeley.edu/~fateman/algebra.html

Richard Fateman writes about the above cited paper:

> Here is a somewhat informal paper criticism of the
> OpenMath project. OpenMath's goals are to provide a
> technique for encoding all of mathematics; a kind of
> superset of MathML, which has recently hit the WWW. The
> topic and approach may have been insufficiently
> analytical or technical, but I suspect it was just too
> irritating to the sensibilities of Openmath boosters.
> ISSAC committee for that year declined the paper.

I am sure it was irritating but to suggest that it is merely
insufficiently analytical or technical strikes me as an
exaggeration. In most respects I think Fateman is simply
wrong and his didactic method also seems dubious. He makes
several gratuitous comparisons to, for example, software
reuse, Godel's incompleteness theorem and even Robert Jones'
utopianism. He states that "[OpenMath] does not have any
mandate outside the rather simple application of denoting
what could be trivially done in any programming language
capable of representing attributed trees". Under the
heading "Good and Bad Science" it seems to me that Fateman
illustrates exactly the kind of error of which he
implicitly accuses the OpenMath designers: "palpably false
[assumptions] that means that any possible results of the
investigation are totally disconnected from any possible
relationship with other mathematics or applications."

I suppose there are almost as many philosophies of
mathematics as there are mathematicians, but Fateman is
a computer scientist. Having been in the position in my
distant past of (unsuccessfully) defending a PhD thesis
proposal in front of an audience consisting of members of
the mathematics and computer science department who
habitually sat on opposite sides of the room in spite of
sharing the same floor of the same campus office tower,
I am acutely aware of the extremely polarized views of the
respective roles of those who feel they must include the
word science in their professional titles and those who do
not. So I feel a little "cheated" in this article by
Fateman - perhaps somewhat like how a feminist might feel
being put down by another woman who views her role
in society in more mundane and conventional terms.

It seems as if Fateman is promoting a philosophy that claims
that mathematics is inherently only a human endeavor that is
forever out of the grasp of mere mechanical systems. But
I have always favored a philosophy that is almost
diametrically opposed to this view. I find the
constructivist/intuitionist view of mathematics much more
compatible with the kind of engineering goals that are the
foundations of computer science. To that end, category
theory, and in particular the algebraic approach to set
theory, seem like a much more suitable starting point then
conventional logic and formal systems. 

Tim Daly wrote:

> I don't see how a system like Axiom can send a complex
> type, say List(Polynomial(Fraction(Integer))) to another
> system and get the same type back. This is an important
> issue for Axiom as most of the semantics of an expression
> depend on the type. Most of the other computer algebra
> systems can't handle Axiom's complex type towers so I
> don't expect OpenMath or MathML to do it any time soon.

In my view it is precisely Axiom's success at presenting
an apparently complete and constructive type system
demonstrates the possibility of achieving at least some of
the goals of OpenMath. What OpenMath looks for is some kind
of universality. At a deep level computation universality
guarantees that any program in one language can be re-
written (simulated) in another but of course OpenMath wants
this universality to be expressed at a higher level than
that of the Turing Machine or recursive functions, or for
that matter LISP. So I don't think is an accident that
Axiom was one of the first mathematical software systems
which incorporated some of the OpenMath ideas.

Tim, I think you over state your case when you claim
that "other computer algebra systems can't handle Axiom's
complex type towers". I would not deny that a strongly
typed language such as that implemented by Axiom is more
expressive and more concise than an untyped language like
Maple. But there is nothing inherent in the design of
Maple's programming language that makes in impossible to
implement that same algorithms and data structures as
Axiom - it's just more work in Maple.

In any case, I would consider Axiom in a somewhat different
role. I think that you might agree that if someone where
to re-code an algorithm expressed in Maple into Axiom,
then the problem of re-coding this algorithm a second time
into for example Mathematica, is much simpler. In a sense,
the rigidity imposed by Axiom's type system formalizes a
larger part of the algorithm than is otherwise the case.
Or stated another way: there are few right ways to express
the algorithm in Axiom than in Maple or Mathematica.

So I see OpenMath as attempting to fill this role as a kind
of interlinga for mathematical software. To what extent it
can do that in it's current incarnation is an open question
as far as I am concerned. But I think the attempt is both
worthwhile and likely to be very productive in the long run.

--------

I am rather new to both sides of the argument, so I hope
that I have not badly mis-represent either position and if
I have I would be glad to be corrected.

Regards,
Bill Page.