- To: mathgroup at yoda.ncsa.uiuc.edu
- Subject: Wolfram's Talk
- From: fateman at peoplesparc.Berkeley.EDU (Richard Fateman)
- Date: Fri, 8 Mar 91 13:11:25 PST
Stephen Wolfram: "Mathematica 2.0 and the new paradigm for technical computing" Review by Richard Fateman Dr. Stephen Wolfram, principal designer of the Mathematica program spoke at a Math-department sponsored colloquium at the Univ. of California, Berkeley, on March 7. 1991. Wolfram came prepared with a Mac-II hooked up to a color video projector, and talked while typing into various Mathematica notebooks. Wolfram discussed the history of the subject (that is, how Wolfram used to write large and small programs and decided that he should write a system to make the writing of programs easier). He then demonstrated various aspects of the Mathematica 2.0 system. Highlights were, to me, (a) the inclusion in notebooks of nicely digitized title pages and some contents of various books (Abramowitz & Stegun, Gradsteyn and Ryzhek); (b) the playing (audio) of the Riemann zeta function along the critical line (demonstrating by its rising pitch that the crossings generally get closer together); (c) the animation of a square wheel rolling on a particular curved surface, illustrating how it maintained a level axle. Wolfram did not disappoint those who came to hear hype. Hype: the statement that large tables of integrals would need CD ROMs and therefore algorithms that could do 75% of the largest tables were either equivalent or better. Truth: the largest tables of integrals are less than 1 megabyte of ascii characters. This hardly needs CD ROM storage. But the accuracy of Mma for integration -- even representing the contents of the tables, much less replacing them, seems open to doubt. Representation of special conditions on parameters is a particular weak point in Mma. Many erroneous integrals have been found to result from deep-seated incorrect assumptions in Mma. Hype: the Mma number system is conservative and helps keep track of the loss of accuracy. No need for the user to worry about numerical analysis. Truth: sometimes the accuracy claimed has no relation whatsoever to the accuracy of the computation. Hype: Mathematica has broken new ground and far surpasses its predecessors, namely Fortran, Cobol, and perhaps Pascal and Lisp. Truth: Unmentioned systems among languages include APL, Prolog... Unmentioned symbolic manipulation predecessors include FORMAC, ALTRAN, Mathlab '68, Reduce, Macsyma, ... and up to the present time, Maple, Derive, Theorist, FORM, MAO, PARI... Unmentioned interactive systems for math include MATLAB, MATHCAD, etc. The hype was quite expected, although I think that SW was a bit more defensive of the Accuracy stuff than usual, and he seemed to be alerted to the passing titter in the audience when he mentioned "lexical scope" in version 2.0. A cute touch was SW's modesty ... he said that undoubtedly some people in the audience could provide even more interesting demonstrations of Mma applications. He claimed 100,000 users (plausible, if you include casual ones, though Paul Abbott of WRI in December, 1990 claimed 15,000 licensed systems). SW claimed 10 million lines of code have been written. (How many different lines, though?) and that Mathematica's language was the new language for the future (displacing Pascal, C ..). After about 70 minutes of Mma demo, Wolfram spent about 2 minutes on the second part of his topic -- this "new paradigm"... which consisted of platitudes that approaches to science will, or should, change to a more computational mode. That Mathematica, and only Mathematica (?) has made this possible, was clearly the conclusion the audience was expected to draw. Q&A followed a brief break. The first spate of questions suggested that the audience had responded to the sales pitch as one would expect and asked When/where/how much... Q: Would algorithms be documented? A: Maybe, but they have been concentrating on adding stuff, not documenting. In fact, lack of documentation helped because then WRI was not locked into specifics, and was free to change the relative complexity of approaches. Prof. W. Kahan spoke up (note: Kahan earlier distributed a page of problems Mma 1.2 botches) He asked SW to type 12 characters into Mma. After some prompting from the audience, SW overcame his reluctance. (1/3)^x 3^x Kahan challenged him to simplify it. Mma 2.0 couldn't, but SW inserted some values for x, and each time got "1" or in the case of a complex number, nearly 1. Kahan pointed out that for any real or complex value for x, this expression is 1, but Mma doesn't know it. SW said you could write a pattern and asked Kahan.."What's your point?" Kahan then asked SW to type the same expression but with r instead of 3. With such a substitution, PowerExpand in Mma 2.0 reduces the expression for arbitrary r and x, to 1. Kahan pointed out that for arbitrary r, this expression is NOT 1. SW repeated "What's your point?" Kahan said that he thought Mma was deficient in its capabilities with respect to manipulation because it was devoid of semantics -- ranges for variables, for example, and that pattern matching was an inadequate technique for doing mathematics. That it was "built on a foundation of sand". SW asked, "Does anyone else want to make a speech?" ..... Since my ride home was leaving, I left at this point.. I understand the rest of the questions were of the sort "have you fixed this bug" etc.