# David James Foulis

### Biography

**David Foulis**'s father, James R Foulis (1903-1969), was an American professional golfer who won the Illinois PGA championship several times. In fact the family consisted of many professional golfers and golf course architects. His grandfather, David Foulis (1868-1950), was one of five brothers (David, Jim, Robert, John and Simpson) born in Scotland, three of whom were professional golfers, one, Simpson, was an amateur golfer and one, John, was a bookkeeper and golf ball maker. All five brothers emigrated to the United States. David and Jim took out a patent on the mashie niblick, a wooden shafted club roughly equivalent to a modern seven iron. Jim (1870-1928), won the second US Open Golf Championship in 1896 which was held at Shinnecock Hills Golf Club on Long Island. Since this archive is based in St Andrews, Scotland, we must note that the five Foulis brothers were all born in St Andrews.

David Foulis, the subject of this biography, moved with his family to South Miami, Florida, when he was fifteen years old. There he attended the Ponce de Leon High School and, after graduating, entered the University of Miami in September 1948. At first his interest was in physics and he majored in physics for his Bachelor's degree which he was awarded on 9 June 1952, Magma Cum Laude. However, he had already come into contact with many members of the Mathematics Department at the university and they had stimulated his interest in mathematics. He said [2]:-

When I was an undergraduate physics major, I first became aware of the profound relationship between mathematics and the scientific enterprise. A mathematician was a maker of abstract patterns or models, often suggested by problems arising in the experimental or the descriptive sciences. Somehow, the mathematical structures thus created were endowed with an almost magical power to relate, explain, and predict natural phenomena. However, before I finished my undergraduate degree, I began to understand that the patterns studied by mathematicians originated not only from specific scientific problems, but from philosophical or logical questions, or even purely from intellectual curiosity. Georg Cantor, the creator of set theory, once said that the essence of mathematics is its freedom, a notion that I found to be immensely appealing. The job of a physicist was to study the physical world as it is, but a mathematician could study all possible worlds, constrained only by the requirement of logical self-consistency! Realizing this, I determined to pursue my graduate studies in mathematics.He was particularly influenced by Herman Meyer who was the Chairman of the Mathematics Department. After the award of his Bachelor's degree, Foulis studied for a Master's degree in Mathematics beginning his studies in September 1952. He was awarded the degree by the University of Miami in June 1953. He was appointed as a graduate assistant in the Mathematics department of Tulane University in New Orleans in September 1953, spending one academic year in this position until June 1954. He then went to the University of Chicago as a National Science Foundation fellow, spending the two years September 1954-June 1956 there [1]:-

While at the University of Chicago, he was strongly influenced by Irving Kaplansky and Paul Halmos, both expositors par excellence. That influence persists in his oral presentations as well as his written publications, and manifests itself in a much admired polished clarity in which the presentation of mathematical ideas becomes a kind of poetry.In 1956 he married the mathematician Linda Falcao.

In September 1956 Foulis returned to Tulane University where he began undertaking research for his Ph.D. with Fred Boyer Wright Jr as his thesis advisor. There were others on the staff at Tulane University who had a major role in his mathematical development, particularly Al Clifford who had been appointed to Tulane in 1955. Al Clifford invited Gordon Preston to spend the two years 1956-58 working with him at Tulane University and Preston was also a major influence on Foulis. He was awarded a Ph.D. for his thesis

*Involution Semigroups*which he submitted to Graduate School of Tulane University on 30 July 1958. Foulis gave the following acknowledgement in his thesis:-

The author would like to express his gratitude to Professors A H Clifford, G B Preston and especially F B Wright for their careful reading of the manuscript, their helpful suggestions and criticisms and their time which the author consumed during frequent discussions on the subject matter of this paper.The thesis was approved by Fred Wright, Al Clifford and Gordon Preston. In the Introduction, Foulis explains how he came to study involution semigroups:-

It is well known that many proofs in ring theory depend more heavily upon the multiplicative structure of the ring than upon the additive structure, and this has been the fountainhead for many of the interesting results in the theory of abstract semigroups. recently there has been much interest among algebraists in rings with an antiautomorphism of period two, a ↦ a*, and in particular in Banach *-algebras of one description or another. here, one observes that not only the multiplicative structure, but also the "adjoint" map a ↦ a* play the decisive roles in the proofs, the additive structure and the topological structure (if any) being relegated to a position of secondary importance. These fact suggest to the author that a systematic study of an abstract semigroup equipped with an antiautomorphic involution a ↦ a* might not be amiss.We should mention that the structure that Foulis studied in his thesis ia a generalisation of a group, since in a group the map $a \mapsto a^{-1}$ provides the antiautomorphic involution.

Tullis was appointed as an Assistant Professor of Mathematics at Lehigh University for the year 1958-1959. His next appointment was as an Assistant Professor of Mathematics at Wayne State University where he worked for the four years 1959-1963. Following this he was again appointed as an Associate Professor of Mathematics, this time at the University of Florida for the two years 1963-1965. During these years he published

*Baer *-semigroups*(1960),

*Conditions for the modularity of an orthomodular lattice*(1961),

*A note on orthomodular lattices*(1962),

*Relative inverses in Baer *-semigroups*(1963), and

*Semigroups co-ordinatizing orthomodular geometries*(1965). Foulis then moved to the University of Massachusetts, Amherst. In [1] Richard Greechie explains how this came about:-

One of his professors at Miami, Wayman Strother, later became the Head of the Department of Mathematics and Statistics at the University of Massachusetts, Amherst. Wayman invited Dave to join the Mathematics Faculty at the University of Massachusetts, Amherst "to revise the undergraduate curriculum". Within 5 years, the undergraduate curriculum was substantially revised and the world's largest group of specialists in orthomodular lattice theory was centred in Amherst. Wayman once confided to me that he had made an observation when Dave was one of his Advanced Calculus students in Miami; he said, "Dave is someone who does not know how to make a bad proof".Dedication to teaching meant that there was less time for Foulis to undertake research, so his next paper did not appear until 1968 when he published

*Multiplicative elements in Baer *-semigroups*. After this he published a number of papers with Charles Hamilton Randall (1928-1987). Randall, after working in the nuclear industry, obtained a position at the University of Massachusetts and had a close and fruitful collaboration with Foulis for twenty years until his death in 1987. Their collaboration led to the publication of 22 joint articles (a few with an additional third author). A collaboration between someone with interests in lattices and semigroups and someone who had worked in the nuclear industry sounds unlikely but in fact their interests coincided quite closely. This was because Randall had become interested in foundations and Foulis in quantum theory [1]:-

Dave always had a compelling interest in "understanding" quantum physics. These demands of understanding led him from being an undergraduate physics major to focusing on mathematics in graduate school. He developed models (many unpublished) for the foundations of quantum mechanics. When Dave learned of the innovative and independent ideas of Charlie Randall, a close collaboration was born. This led to an important thrust called "The Amherst School" in which he and Charlie, along with other colleagues and students, made profound progress in our understanding of mathematical foundations of empirical studies, in particular of quantum mechanics.Richard Greechie, who was one of Foulis's graduate students gaining a Ph.D. with his thesis

*Orthomodular Lattices*in 1966, recalls how devoted Foulis was to teaching his students. He recounted his favourite story [1]:-

... when I was an aspiring (first year) graduate student who had shown an interest in projections on a Hilbert Space, Dave offered to tell me how (an abstraction of) these structures played a role in the foundations of quantum mechanics. He proceeded to give me a one-on-one lecture that lasted from 7 pm on a Friday evening till 1 am. At 1, Dave said that he couldn't finish that evening and would I like to meet again "tomorrow". The lecture resumed exactly 6 hours later!Foulis retired in 1997 and was made professor emeritus at the University of Massachusetts. Rather than marking an end to his research career, this in many ways marked a new beginning and he has published almost 50 papers in the 15 years since then. These papers are on a broad range of topics such as Ordered Structures, Orthostructures, Foundations of Quantum Mechanics, Foundations of Statistics, and Operator Theory. For example in 2010 he published

*Synaptic algebras*which studied a special class of partially ordered algebraic structures defined by a set of axioms which make them into a spectral order unit normed space and a special Jordan algebra. In 2007 he published papers such as

*Effects, observables, states, and symmetries in physics*and (with Richard J Greechie)

*Quantum logic and partially ordered abelian groups*. Here are Foulis's own comments made after he retired [2]:-

[Now I am retired] my interest in mathematics and mathematical research has not diminished. My primary research interest centres around mathematical models for nonstandard logics, particularly the logics that arise in connection with quantum mechanical systems and the nonmonotonic logics that pertain to inference in expert systems. The study of measures on these logical models is a burgeoning new field called noncommutative measure theory. In September I will be giving an invited lecture on this topic in Italy. I am active in the International Quantum Structures Association, and continue to participate in the annual IQSA conferences. I have recently been appointed Visiting Professor at Florida Atlantic University, where I am participating in a seminar on quantum logic and where I am a member of the Ph.D. committees of two graduate students.Finally, we mention that Foulis published seven undergraduate textbooks. These are

*Fundamental Concepts of Mathematics*(1962), (with Mustafa A Munem)

*Calculus*(1978), (with Mustafa A Munem)

*Calculus: With Analytic Geometry*(1984), (with Mustafa A Munem)

*After Calculus: Algebra*(1988), (with Mustafa A Munem)

*After Calculus: Analysis*(1989), (with Mustafa A Munem)

*Algebra and Trigonometry with Applications*(1991), and (with Mustafa A Munem)

*College Algebra with Applications*(1991).

Foulis has three children David, Dean and Scott. He is now married to the mathematician Hyla Gold who, Greechie writes [1]:-

... enjoys listening to his lectures and, during an International Quantum Structures Association Meeting, can usually be seen in the back of the lecture hall during Dave's presentation. There is no doubt in my mind that she plays a major role as facilitator, which has been an important factor in Dave's outstanding productivity. Hyla wrote the solutions manual to Dave's first Calculus book, providing solutions to approximately 5000 problems, an impressive feat before the availability of graphing calculators. I suspect that Hyla also plays an editorial role before Dave's papers are submitted, and have often been amused by her comments after a lecture, over dinner, such as, "You changed that part about the connection with operator theory".

### References (show)

- R Greechie, David James Foulis,
*Math. Slovaca***62**(6) (2012), 1007-1018. - M Janowitz, An Interview of Prof David Foulis,
*Topological Commentary***3**(2) (15 August 1998).

### Additional Resources (show)

Other websites about David Foulis:

Written by J J O'Connor and E F Robertson

Last Update October 2013

Last Update October 2013