The mathematician who—incidentally—helped mathematicians to stop worrying and love the computer.
By Keith Devlin @KeithDevlin@fediscience.org, @profkeithdevlin.bsky.social
Yes, this month’s title is a reference to the iconic 1964 movie Dr. Strangelove.
Walk into any mathematics professor’s office and you’ll see a computer. Indeed, the chances are high that, even if said mathematician is not actively using it, it’s switched on. Back in 1985—the year AFTER the Macintosh launched—that computer would have been the exception to the norm.
Huh? What? Unless you were active in mathematics back in the 1980s, you surely have no idea what I am talking about.
Fig 1. John von Neumann (1903-57) in the US and Alan Turing (1912-54) in the UK established theoretical mathematical results that helped pave the way for digital computers and were involved in building two of the first such.
After all, computers were essentially created by mathematicians (right?), with the likes of John von Neumann (in the US) and Alan Turing (in the UK) not only developing mathematics that showed theoretically how to design digital computers, but being themselves involved in the design and construction of two of the first such devices all the way back in the 1950s. Surely, being in on the birth, mathematicians were ahead of the curve when it came to using them?
Well, no. That’s not how things transpired—at least for pure mathematics. Academics and teachers in all the sciences, in engineering, in economics, in business and finance, in library science, and even in mathematics education, quickly jumped on the computer bandwagon as soon as universities and colleges started to acquire, first minicomputers (hands up all those for whom the first computer they used was a DEC VAX), and then not long after, a personal computer.
But among mathematicians, for the most part it was only those in applied mathematics who worked with computers.
Fig 2. So much can be said about this great Harris cartoon.
The vast majority of pure mathematicians spent most of their time in day-to-day research activities that had remained largely unchanged for over two millennia. (See Figure 2.) Their tools were paper and pencil and a blackboard (with chalk), and those remain today the primary tools of choice for most pure mathematic research. Their work progressed under a widespread, though unstated, assumption that computers could not possibly play a role in the construction of proofs of theorems.
That assumption had been given a significant jolt in 1976, with the announcement by two mathematicians in the United States, the American Kenneth Appel and the German Wolfgang Haken, that they had made essential use of a computer to solve the famous Four Color Problem. (Figure 3.)
Although their approach involved a logical argument, it required a computer to look for map colorings for 1,476 particular map configurations that their proof generated. It would be a prohibitive task for humans, but their computer completed the task in a few months. (Today’s computers could do it in seconds.)
The Appel-Haken proof showed that mathematics had entered a new era, but it was, everyone assumed, a one-off, or at least highly unusual. For the vast majority of mathematicians, things largely remained the same. There were only two tasks for which they used a computer.
The most common was email, after it began to take off (mostly in universities at first) in the 1980s. (The first email was sent in 1971; the “.com” domain was not introduced until 1985, with most of us who were already on the Internet saying, “Why would anyone want a commercial domain?”)
The other task where mathematicians were increasingly turning to a computer was the preparation of manuscripts. In 1978, Stanford Computer Science Professor Donald Knuth released the first version of his mathematical typesetting system TeX, and a few bold mathematicians started to use it to write their papers, but the learning curve was high. Usage started to grow significantly after LaTeX came out in 1984, providing a much more user-friendly front-end to TeX.
Fig 3. The proof of the Four Color Theorem involved essential use of a computer to check a large number of cases. The standard, brain-driven part of the proof reduced an infinite number of possible maps to a finite number (initially 1,472, but subsequently reduced to a lower number).
Mastering LaTeX was still a significant undertaking, however, but the arrival of what-you-see-is-what-you-get, mathematical word-processing systems that offered drop down menus of alternative alphabets and mathematical symbols, such as MathType (initial release in 1987) made manuscript preparation much easier, albeit without the “perfect” page layout offered by LaTeX.
So, by the late 1980s, mathematicians’ offices frequently contained a personal computer. Though email use was increasing dramatically, manuscript preparation was still restricted to a hardy group of pioneers. And hardly any pure mathematicians were using it in their research or their teaching.
While being (of course) well aware of the nature of mathematical research, some in MAA and AMS leadership circle worried that the community was in danger of being left behind in what was becoming a rapidly developing area, with a variety of new tools coming out: more powerful graphing calculators, computer algebra systems (for instance, Mathematica and Maple were both in the process of being released), and a whole host of others. Many of those research tools could be (and were being) used in university-level education, and other tools were produced specifically for education. Both the MAA and the AMS created Electronic Services Advisory Boards to guide them as they moved into new territory. (I served on both.)
Fig 4. Prof Richard Palais of Brandeis University.
Enter one Professor Richard Palais, a differential geometer (then) at Brandeis University. He earned bachelors (1952), masters (1954), and doctoral (1956) degrees from Harvard, writing his PhD thesis A Global Formulation of the Lie Theory of Transformation Groups under the guidance of Andrew Gleason and George Mackey. Later in his career he became increasingly interested in mathematical visualization.
When the history gets written (yes, I am doing that right now), he will be (more widely) known for having played a key role in persuading (pure) mathematicians to embrace the computer in their work. Though his role in that important transformation was an incidental one. Here, briefly, is what happened.
When Knuth released TeX, Palais was one of its first users (and proponents). In 1979 he co-founded the TeX Users Group and become its first president. Among his doctoral students was Leslie Lamport, who subsequently created LaTeX.
In 1987, the AMS asked Palais to write a series of articles for the Notices, surveying the various mathematical word processing systems that were available at the time and talk about TeX/LaTeX. The interest in those articles was sufficiently strong that the AMS decided to launch a regular section “Computers and Mathematics” in the Notices, which was sent to all members ten times a year, with the stated goals being
(1) to reflect, both practically and philosophically, on cases where computers were affecting mathematicians and how they might do so in the future;
(2) to act as an information exchange into what software products were available; and
(3) to publish mathematicians’ reviews of new software. The hope was that this would both provide inspiration for mathematicians to make greater use of computers, and act as an information exchange for the various possibilities computers offered in their work in addition to manuscript preparation.
[ASIDE: The general feeling at the time, as I recall it, was that MAA members were already coming on board, particularly with regard to tools for calculus teaching, but many pure mathematics researchers in the AMS were still not sufficiently motivated. Though as I and many university mathematicians belong to both organizations, this nuance may not have a great deal of substance. Certainly, putting the Palais articles and the new “Computers and Mathematics” section in the AMS Notices did send the message that “This is important to pure mathematics researchers”. I was not party to decisions made by either organization other than serving on the advisory boards.]
In any event, between May 1988 and December 1994, the Notices’ Computers and Mathematics section published 59 feature articles, 19 editorial essays, and 115 reviews of mathematical software packages (31 features, 11 editorials, and 41 reviews with Stanford mathematician Jon Barwise as the first editor and 28 features, 8 editorials, and 74 reviews under me as the second and last). The section closed down at the end of 1994 since we all decided it had done what it set out to do.
The CAM section was not the only initiative the mathematics community set up to help mathematicians get up to speed. Around the same time, a number of mathematicians were developing a new subfield of mathematics called “Experimental Mathematics”. In this field, one of the primary goals in using computers was to formulate conjectures that could subsequently be proved by conventional means—which cast the computer as an additional weapon in the pure mathematician’s armory rather than a completely separate technological endeavor.
In 1992, a new journal with that name as its title was established by the American mathematicians David Epstein, Silvio Levy, and the German-American mathematics publisher Klaus Peters. And in the fall of that year, the Canadian mathematicians Jonathan and Peter Borwein sent me their article “Some Observations on Computer Aided Analysis”, written to introduce their new field to the mathematical community at large, which I published in the October issue of “Computers and Mathematics”.
When he introduced the last CAM section he edited, in February 1991, Barwise had written:
“Whether we like it or not, computers are changing the face of mathematics in radical ways, from research, to teaching, to writing, personal communication, and publication. Over the past couple of years we have seen numerous articles about these developments.
Computers are even forcing us to expand our idea about what constitutes doing mathematics, by making us take much more seriously the role of experimentation in mathematics. (I draw attention to a new journal devoted to experimental mathematics below.)
One view of the future is that mathematics will come to have (or already has) two distinct sides: experimentation, which can exploit the speed and graphics abilities of programs like Maple and Mathematica, to allow us to spot regularities and make conjectures, and proof, very much in the style of today’s mathematics. …
Whether we applaud or abhor all these changes in mathematics, there is no denying them by turning back the clock, any more than there is in the rest of life. Computers are here to stay, just as writing is, and they are changing our subject.”
It is surely obvious from those final remarks Barwise made that in the early days, the computer was seen as something of a threat by some mathematicians, and the “Computers and Mathematics” section was not without its detractors.
Taking over a month later, I began by saying that:
“This column is surely just a passing fad that will die away before long. Not because mathematics will cease to have much connection with computers, but rather, quite the reverse: the use of computers by mathematicians will become so commonplace that no one thinks to mention it any more.”
When I wrapped up the section four years later, I wrote:
“With its midwifery role clearly coming to an end, the time was surely drawing near when “Computers and Mathematics” should come to an end. The change in format of the Notices, which will take place at the end of this year, offered an obvious juncture to wind up the column. …
The disappearance of this column does not mean that the Notices will stop publishing articles on the use of computers in mathematics. Rather, recognizing that the use of computer technology is now just one more aspect of mathematics, the new Notices will no longer single out computer use for special attention. I’ll drink to that.
The child has come of age.”
Indeed it had! (You would not be reading these digitally-transmitted words if it hadn’t.)
NOTE: For readers attending the Joint Mathematics Meeting in Seattle next month, I am chairing a 90-minute panel discussion looking back at the Computers and Mathematics initiative, in which a number of mathematicians who contributed to the section will reflect on that period.
REFERENCE: For further details on the history described above, see the paper I wrote for the 2017 Jon Borwein Commemorative Conference.