[om] Re: [om-a] critique of OpenMath

[Richard Fateman]

David Carlisle wrote:
> 
> > Since it is done without reference to any pre-existing CDs, I assume it
> > would not be acceptable to OMS.
> 
> Not at all. OM is not a purely formal system in which it is planned to
> build the whole of Mathematics from the ground up. It is possible to
> model the logical frameworks that are needed for such a project in
> OpenMath, and some are doing that.

This a very ambitious project. You (or whoever is trying to do this)
is following in the footsteps of Leibniz, Frege, Bolzano, Cauchy, Weierstrass...
Dedekind, Cantor, Peano, and of course Russell & Whitehead's
Principia Mathematica.
see, for example,
http://plato.stanford.edu/entries/principia-mathematica/

The logicist's goal according to Russell is
"to show that all pure mathematics follows from purely logical premises and uses only concepts
definable
in logical terms."

Should we believe the logicist viewpoint?
pro: computer scientists would like to believe they hold the
key to knowledge. arguably the human brain is some kind of
system that can be modeled from the neuron up by some formal
(perhaps statistical) system, and so all of human endeavor
is logical. (Artificial Intelligence fans have other arguments
as well.)
con:
If you ask a mathematician who is not a logician, if what he/she
does (namely mathematics) can be done by a purely formal system
you will likely be rebuffed.  "Doing mathematics" involves
creativity, esthetics, (and for applied math) knowledge of
the real world. Computers can't do this.

As a practical matter, successfully re-doing set theory from yet another
axiomatic computer-based formal notation is in itself not very
interesting. Contemporary projects (some of which have been
in progress for substantial periods of time, mostly prior to
OpenMath) includ NuPRL. Edinburgh LCF, deBruijn's Automath

See for example,

http://www.cs.cornell.edu/Info/Projects/NuPrl/Intro/FormalMath/formalmath.html

To be up front, and in case it is not apparent, let me say that I am personally skeptical that
formalizing all of mathematics is going to happen any time soon.

Nevertheless, even if it were to be possible, it seems rather remote to me that the key to success
is the use of OpenMath rather than one of the other formalisms used to date.  In fact
OpenMath, so far as I can tell, is far weaker than any of these systems, each of
which is equivalent in formal computational power to a universal Turing machine.
OpenMath is not, so far as I can tell, being essentially a clumsy
context-free grammar.


... <snip> 
> Because at the current stage of the game it's mainly tool writers or
> people with an interest in influencing such people, who are going to be
> reading this stuff.  Currently people using OpenMath and MathML (and to
> a lesser degree, XML in general) are currently people with an interest
> in the issues surrounding such languages. But I don't think that that is
> a reflection of the fact that we are using XML syntax.  I wouldn't be
> any happier thrusting lisp in people's face.

I would be happier because it might be about 1/10 the size.  I tried
looking for the "binary" representation, in the hope that it
would be more compact, but found only a dead link to INRIA.

> 
> > So far as I can tell the reason to use XML is to prevent anyone from
> > doing interesting things.
> 
> It's a point of view, and I have a lot of sympathy with your championing
> of lisp as a language. But if you take a look at the current explosion
> of internet use, would you really want to say that XML prevents
> interesting things being done, and all the interesting stuff is
> happening in lisp?

Undoubtedly there are interesting things being done with XML.  
I had some OTHER interesting things in mind that are being prevented
by adherence to XML.  Many other interesting things are possible
because people are able to use javascript, java, plugins, etc.
Hardly any of this stuff is in lisp, as it happens, though some
of it on the server side is in lisp.

> 
> > How to attack the hard problem?  I don't know for sure, but I'm looking
> > at MINSE and glyphd.  The author of this stuff (Ka-Ping Yee) is now
> > a grad student at UC Berkeley.  He did this quite impressive stuff while
> > an undergrad at Waterloo, and it seems to have been totally ignored.
> > I ignored it myself, but then I was already somewhat disillusioned!
> > Not as disillusioned as Ping, though.
> 
> Minse shares some of the problems with TeX, in that it it leads
> Mathematics into a ghetto (even if it it seems like a nice cosy
> environment). 

Actually, I don't agree with this. MINSE technology is not
restricted to Mathematics at all. 

>I want the Mathematics markup to be part of the mainstream
> document markup not some arcane black box. Document markup today (at
> least) means XML. the fact that there are 1001 valid complaints one may
> make against XML don't particularly concern me. There are 100000001
> complaints one could make against English as a language, but it still
> serves its purpose if the people I'm communicating with speak that
> language.

I suspect that MINSE can accomodate all of XML and more.  It can
do this with browsers that are unaware of XML.  Instead of continually
updating each person's browser to accomodate changes in XML, the
changes can be incorporated in polymediators. These could be
central, regional, or local (on each machine).  

> 
> I grew up with TeX. Everyone I know: my friends, my wife, everyone; they
> all use TeX. However it has dawned on me that there are in fact people
> out there who have mathematical documents that I might want to share and
> that they will never use TeX (or any batch formatter with an explicit
> markup language). A plugin to use TeX, or minse or any other non
> standard language wouldn't help bridge this divide. But in using XML,
> MathML and OpenMath do at least offer a hope. The MathML WG includes
> old TeX hackers like myself, but also representatives from Microsoft
> (and design science who make the equation editor used by Word) and
> assorted other companies and institutions somewhere in the middle of
> that spectrum. XML may be verbose, and apparently lacking in features
> but if it is something on which everyone can agree then that agreement
> is worthwhile I think. You can write mathematics in Microsoft Word
> (MathType) save it as MathML, display it (natively, without any extra
> plugin) in Mozilla or Amaya browsers, or typeset it (directly, without
> using a conversion program) in TeX. I believe this is progress, and I
> don't believe that much progress would have been made if a syntax other
> than XML had been used. Given a browser with embedded XSL (Internet Explorer
> and some builds of Mozilla) the benefits of MathML rendering apply
> equally to OpenMath.

Actually I believe that translations of TeX to the Design Science
standard MathType(?) is far more useful than TeX to OpenMath.  
It allows me to make Powerpoint
slides with math formulas ... I type in TeX and I get an expression
that can, if necessary, be further edited by Word's equation editor.
Translating TeX to OpenMath is fairly implausible unless you happen
to have an appropriate CD.  Translating to MathML is (I guess) often
possible, but if there is some sacrifice in expressibility in doing so
there will still be TeX diehards. Frankly what I've been doing is
TeXing to PDF, in which form math, text, hyperlinks, etc can
coexist fairly well, and most people can see what you've done.
It is naturally scalable.  The disadvantage is that I cannot
produce dynamic html this way (yet). Though MINSE and glyphd
could in principle do most of the job of typesetting math
dynamically so it can be displayed anywhere.


Regards.
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