TOPCOM, An Interview of Prof. David Foulis

Topology Atlas Document # topc-65.htm | Production Editor: Thomas M. Zachariah

An Interview of Prof. David Foulis

Interview from Volume 3, #2, of TopCom

What follows is the text of an interview of Prof. David Foulis, who retired recently from the University of Massechusetts at Amherst. It was forwarded to Topological Commentary by his colleague Melvin Janowitz with the permission of David Foulis, whose research interests are primarily in lattice theory and logic. He has strong interests in teaching as well and is the co-authorof a calculus text. His personal views on the current state of the teaching of mathematics and its effect on the intellectual atmosphere of colleges and universities should be of interest to mathematicians of all kinds. Commentary from our readers is solicited.

Q. What led you to become a mathematician?

A. 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.

Q. How has mathematics changed during your career?

A. There were always mathematical fads which, for no particular reason that I could see, changed from time to time. When I was a graduate student, the fad was algebraic topology, so much so that its partisans arrogantly referred to it simply as topology. Later, nonstandard analysis, then classification of simple groups, then catastrophy theory were all the rage. These days, as a con- sequence of modern technology (read, computers), dynamical systems and chaos theory are in fashion. The idea of a mathematician as a creator and user of abstract patterns--the very idea that once at- tracted me to mathematics--is rapidly being replaced by the notion that the job of mathematicians is to solve problems using the latest gadgetry. Hilbert once remarked that no one can ever drive us from the paradise that Cantor created. I am sometimes afraid that technology may do just that!

Q. How has the teaching of undergraduate mathematics changed?

A. When I was a university student, the instructor of a class was like the captain of a ship. I once saw a professor eject a student from a classroom because he didn't like the color of the poor fellow's shirt (chartreuse as I recall)! The notion that a student might be in a position to "evaluate" a professor would have been unthinkable. In lieu of formal "evaluations," serious students simply made it their business to find out from fellow students which professors were talented teachers, and which were uninspired (or worse). Nowadays students are "empowered" and young instructors who care about merit raises and promotions are obliged to indulge in rampant grade infla- tion lest their careers be put into jeopardy by hostile student evalu- ations.
Just as mathematical fads blossomed and faded, so did various notions of mathematical pedagogy. For a while, it was the so called new math, which wasn't necessarily a bad idea per se, but turned out to be a disaster because so many teachers couldn't handle it. The launch of Sputnik and the subsequent determination of the United States to land a man on the moon generated an enormous surge of interest in technology, science, and mathematics. Toy astronauts, and even plastic scientists in white coats replaced toy soldiers in children's stores. There was an influx of majors in the mathematics departments of our colleges and universities. The very best and brightest students in our calculus classes were math majors, followed by physics majors, followed by en- gineering students. For mathematicians and mathematics teachers, these were the golden years. There was no talk of curriculum reform--and no need for it.
For reasons that I do not fully understand, there has been a steady decline in the number of competent, committed, and ambitious under- graduate math majors. Perhaps breakthroughs in molecular biology, advances in polymer science, the new emphasis on environmental sci- ences, and so on have begun to attract scientifically inclined students to what they perceive as more exciting fields of study. Many students now declaring majors have little talent for mathematics and even less interest in it (nor in any other academic field for that matter). Now the chemists, physicists, and engineers are usually the best students in our undergraduate mathematics classes.
Various groups of mathematicians and mathematics teachers, perceiving that something is amiss, have opined that a twofold solution is at hand: curriculum reform is necessary and technology will save the day. Implementation of these alleged solutions has wrought profound changes in the way our introductory undergraduate mathematics classes are taught. Lectures, in the conventional sense, are anathema. There is heavy reliance on graphing calculators and computer software packages to solve problems. Instructors manipulate calculators connected to projection devices, then assume their roles as "guides by the side" while students work on problems or projects, either individually or in groups. Definitions, theorems, and proofs are barely mentioned, if at all. Were I a young student today, I doubt that I would have responded favorably to these changes.

Q. What about teaching at the graduate level?

A. When I was in graduate school, most of my fellow students were work- ing toward their Ph.D. degrees with the goal of becoming research mathematicians. Nearly every student was supported as a TA, although the NSF was beginning to award more and more graduate fellowships. We were vaguely aware that there was something called the "university administration," but believed that it was populated by people who kept track of our transcripts and by people called deans whose exact func- tion we were unsure of. We were oblivious to the existence of faculty committees and would have regarded a proposal to serve as a student representaive on such a committee as a joke.
Before starting serious work on a Ph.D. thesis, a student had to pass qualifying exams in algebra, real analysis, complex analysis, and topology. It was agreed that proficiency in these four areas would be required for subsequent work in any field of pure or applied mathe- matics. Conversely, displayed competence in these four areas alone was taken as an indicator of future success as a professional mathe- matician. Students were provided with specific lists of material that they were to master prior to the exams, and it was made perfectly clear that it was the obligation of the student to learn this material even if it was not covered in classes. Students who did not perform well on the qualifying exams were given Master's degrees and sent on their way, but the demand for mathematicians was strong and even those with terminal Master's degrees found employment either as teachers in small colleges, in industry, or with the government.
Much of this has changed. The contemporary graduate student is ex- pected to be a proficient user of TeX and to have set up an internet home page with appropriately fetching wallpaper. He or she is ex- pected to serve as student representative on various faculty commit- tees, to be active in the GEO, and to keep informed about the latest machinations of the administration. Many graduate students are work- ing toward terminal Master's degrees and have little or no interest in mathematical research.
Nowadays, a graduate student faces a formidable panoply of exams: class finals, preliminary exams, qualifying exams, oral exams, de- fense of thesis, and so on. For each group of exams, numerous choices are available, tailored to the individual student's preferences. This complex and bureaucratic system of examinations is defended as the only fair and objective way to make decisions affecting the student's academic future. The graduate faculty is expected to spend many hours composing a multitude of exams, grading them, and meeting in committee to decide who passes and who fails. Again there are lists of material to be covered on exams, but now the professors are specifically direct- ed to teach this material in their classes. Thus, in most introductory graduate level courses, the professors are obliged to "teach to the examination." Neither students nor faculty decry these practices. I could not abide by them and preferred for many years to teach at the undergraduate level where there was considerably more freedom to exer- cise my own judgment and initiative. (Even this freedom is being eroded, but that's another story.)
My most satisfying interaction with graduate students has been in directing Ph.D. theses and in seminars where we could follow our interests without being stifled by formal academic criteria.

Q. Any regrets about your retiring?

A. Until recent years I enjoyed teaching and relished the op- portunity to share my love of mathematics with young and devoted students. The changes that I have mentioned above, the bureaucratic regimentation, the increasing numbers of ill prepared, indolent, and often outright hostile students (and other unfortunate develop- ments that I would prefer not to comment on here) have steadily de- creased my enthusiasm for teaching and made retirement seem more and more enticing. Of course, I very much miss teaching as it used to be.
On the other hand, my interest in mathematics and mathematical re- search has not diminished. My primary research interest centers 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.
Whatever remaining regrets I may have for being disconnected from the classroom are balanced by the realization that I now have more time to study, to contemplate, to do mathematics, and to enjoy the freedom that Cantor once perceived as the essence of mathematics.

Q.	What led you to become a mathematician?
A.	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.
Q.	How has mathematics changed during your career?
A.	There were always mathematical fads which, for no particular reason that I could see, changed from time to time. When I was a graduate student, the fad was algebraic topology, so much so that its partisans arrogantly referred to it simply as topology. Later, nonstandard analysis, then classification of simple groups, then catastrophy theory were all the rage. These days, as a con- sequence of modern technology (read, computers), dynamical systems and chaos theory are in fashion. The idea of a mathematician as a creator and user of abstract patterns--the very idea that once at- tracted me to mathematics--is rapidly being replaced by the notion that the job of mathematicians is to solve problems using the latest gadgetry. Hilbert once remarked that no one can ever drive us from the paradise that Cantor created. I am sometimes afraid that technology may do just that!
Q.	How has the teaching of undergraduate mathematics changed?
A.	When I was a university student, the instructor of a class was like the captain of a ship. I once saw a professor eject a student from a classroom because he didn't like the color of the poor fellow's shirt (chartreuse as I recall)! The notion that a student might be in a position to "evaluate" a professor would have been unthinkable. In lieu of formal "evaluations," serious students simply made it their business to find out from fellow students which professors were talented teachers, and which were uninspired (or worse). Nowadays students are "empowered" and young instructors who care about merit raises and promotions are obliged to indulge in rampant grade infla- tion lest their careers be put into jeopardy by hostile student evalu- ations. Just as mathematical fads blossomed and faded, so did various notions of mathematical pedagogy. For a while, it was the so called new math, which wasn't necessarily a bad idea per se, but turned out to be a disaster because so many teachers couldn't handle it. The launch of Sputnik and the subsequent determination of the United States to land a man on the moon generated an enormous surge of interest in technology, science, and mathematics. Toy astronauts, and even plastic scientists in white coats replaced toy soldiers in children's stores. There was an influx of majors in the mathematics departments of our colleges and universities. The very best and brightest students in our calculus classes were math majors, followed by physics majors, followed by en- gineering students. For mathematicians and mathematics teachers, these were the golden years. There was no talk of curriculum reform--and no need for it. For reasons that I do not fully understand, there has been a steady decline in the number of competent, committed, and ambitious under- graduate math majors. Perhaps breakthroughs in molecular biology, advances in polymer science, the new emphasis on environmental sci- ences, and so on have begun to attract scientifically inclined students to what they perceive as more exciting fields of study. Many students now declaring majors have little talent for mathematics and even less interest in it (nor in any other academic field for that matter). Now the chemists, physicists, and engineers are usually the best students in our undergraduate mathematics classes. Various groups of mathematicians and mathematics teachers, perceiving that something is amiss, have opined that a twofold solution is at hand: curriculum reform is necessary and technology will save the day. Implementation of these alleged solutions has wrought profound changes in the way our introductory undergraduate mathematics classes are taught. Lectures, in the conventional sense, are anathema. There is heavy reliance on graphing calculators and computer software packages to solve problems. Instructors manipulate calculators connected to projection devices, then assume their roles as "guides by the side" while students work on problems or projects, either individually or in groups. Definitions, theorems, and proofs are barely mentioned, if at all. Were I a young student today, I doubt that I would have responded favorably to these changes.
Q.	What about teaching at the graduate level?
A.	When I was in graduate school, most of my fellow students were work- ing toward their Ph.D. degrees with the goal of becoming research mathematicians. Nearly every student was supported as a TA, although the NSF was beginning to award more and more graduate fellowships. We were vaguely aware that there was something called the "university administration," but believed that it was populated by people who kept track of our transcripts and by people called deans whose exact func- tion we were unsure of. We were oblivious to the existence of faculty committees and would have regarded a proposal to serve as a student representaive on such a committee as a joke. Before starting serious work on a Ph.D. thesis, a student had to pass qualifying exams in algebra, real analysis, complex analysis, and topology. It was agreed that proficiency in these four areas would be required for subsequent work in any field of pure or applied mathe- matics. Conversely, displayed competence in these four areas alone was taken as an indicator of future success as a professional mathe- matician. Students were provided with specific lists of material that they were to master prior to the exams, and it was made perfectly clear that it was the obligation of the student to learn this material even if it was not covered in classes. Students who did not perform well on the qualifying exams were given Master's degrees and sent on their way, but the demand for mathematicians was strong and even those with terminal Master's degrees found employment either as teachers in small colleges, in industry, or with the government. Much of this has changed. The contemporary graduate student is ex- pected to be a proficient user of TeX and to have set up an internet home page with appropriately fetching wallpaper. He or she is ex- pected to serve as student representative on various faculty commit- tees, to be active in the GEO, and to keep informed about the latest machinations of the administration. Many graduate students are work- ing toward terminal Master's degrees and have little or no interest in mathematical research. Nowadays, a graduate student faces a formidable panoply of exams: class finals, preliminary exams, qualifying exams, oral exams, de- fense of thesis, and so on. For each group of exams, numerous choices are available, tailored to the individual student's preferences. This complex and bureaucratic system of examinations is defended as the only fair and objective way to make decisions affecting the student's academic future. The graduate faculty is expected to spend many hours composing a multitude of exams, grading them, and meeting in committee to decide who passes and who fails. Again there are lists of material to be covered on exams, but now the professors are specifically direct- ed to teach this material in their classes. Thus, in most introductory graduate level courses, the professors are obliged to "teach to the examination." Neither students nor faculty decry these practices. I could not abide by them and preferred for many years to teach at the undergraduate level where there was considerably more freedom to exer- cise my own judgment and initiative. (Even this freedom is being eroded, but that's another story.) My most satisfying interaction with graduate students has been in directing Ph.D. theses and in seminars where we could follow our interests without being stifled by formal academic criteria.
Q.	Any regrets about your retiring?
A.	Until recent years I enjoyed teaching and relished the op- portunity to share my love of mathematics with young and devoted students. The changes that I have mentioned above, the bureaucratic regimentation, the increasing numbers of ill prepared, indolent, and often outright hostile students (and other unfortunate develop- ments that I would prefer not to comment on here) have steadily de- creased my enthusiasm for teaching and made retirement seem more and more enticing. Of course, I very much miss teaching as it used to be. On the other hand, my interest in mathematics and mathematical re- search has not diminished. My primary research interest centers 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. Whatever remaining regrets I may have for being disconnected from the classroom are balanced by the realization that I now have more time to study, to contemplate, to do mathematics, and to enjoy the freedom that Cantor once perceived as the essence of mathematics.