(02) NEWCOMB GREENLEAF - STAN AND ERRETT BISHOP

Transcription

1 (02) NEWCOMB GREENLEAF - STAN AND ERRETT BISHOP FORWARD: Errett Bishop, who single-handedly (following Brouwer) developed the revolutionary logical alternative, 'Constructive' mathematics, an algorithmic basis for the whole of mathematics denying the law of the excluded middle, was one of Stan's great friends, a highly cherished intimate since their student days at the U of Chicago. Ever (intellectually) rebuffed by the mathematical community who little understood him, Bishop's work was determinedly kept alive, indeed thrived, owing in whole to Stan's efforts and dedication during the 70's at the New Mexico State University. Newcomb Greenleaf, now a committed 'Constructivist', gives us an absorbing account of this period together with background as to his close relationship with Stan and his path to the 'Constructivist' light. Completing his 1961 PhD at Princeton under Serge Lang, the topic 'Local Zeros of Global Forms', Newc moved on to be a Peirce Instructor at Harvard, arriving in 1964 at the U of Rochester as part of Leonard Gillman's programme to build the math department into a world class outfit. Meeting Stan in 1965, they developed a close relationship, Stan's influences continuing to the present. Three years in, Newc departed for the U of Texas, where he gradually came under the spell of bishop's work, albeit the only mathematician in a department of 100 to do so. Seven years in the wilderness

2 Newc left for Boulder to work with the Tibetan teacher Chogyam Trungpa, subsequent to which, after a short stay at a computer graphics firm, he joined the computer science department at Columbia. Today he enjoys a congenial teaching position at Goddard College, finally finding 'my dream teaching job'. He might also be working on his book, working title 'Bible or Cookbook? An Algorithmic Primer to the Book of Math'. In 1992 Newc published in 'Constructivity in Computer Science' a very interesting 'Bringing Mathematics Education Into the Algorithmic Age'. Signal your interest and I will send it to you. Further he cogently puts forward the constructivist viewpoint in a segment on YOUTUBE: 'Nondual Mathematics: A Tragedy in Three Acts'.

3 Stan Tennenbaum and Errett Bishop Newcomb Greenleaf 1. Stan at Rochester After three intense years as a Peirce Instructor at Harvard, where I felt over my head and out of my league, I joined the mathematics department of the University of Rochester in 1964, a year before Stan arrived. I thrived: confidence returned as research picked up and teaching ripened. But my marriage did not thrive and in 1965 I returned to Rochester for the fall semester as a single father, just as Stan and Carol moved into the neighborhood. I was initially drawn into the Tennenbaum orbit more through Carol, who leapt in and became a source of maternal warmth for my two sons, 9 and 10, making sure that we dined with them at least once a week. The three of us still remember the remarkable dinner-table conversations which Stan facilitated in a way that brought everyone in, from second graders to visiting luminaries. Stan had an immediate effect on the whole math department, enlivening it in many dimensions. He carried with him the intellectual spirit of the University of Chicago in the heyday of Robert Hutchins, along with a beguiling mix of melancholy and joy. He was passionately interested in ideas, and passionately interested in people. He displayed a generosity that made him many friends, and an uncompromising idealism that could get him in deep trouble. For a sizable group of students he was a major lifelong influence, vividly recalled 50 years later. For the next three years Stan was the colleague from whom I learned the most, about mathematics, about life, and particularly about teaching, which he took both very seriously and very playfully. From Stan I learned to come out from behind the authority of The Professor and be real. Now, teaching at Goddard, I would say that Stan taught me to model imperfection, which empowers students to make mistakes and

4 admit ignorance without fear, and thus to explore courageously and joyfully. Stan could embody imperfection perfectly. As I write this I realize how much he prepared me to teach at Goddard. Stan had another important influence on me: he introduced me to the constructive mathematics of Errett Bishop, which ultimately changed my life profoundly. Stan and Errett had been close as graduate students at Chicago, so Stan paid attention in 1966 when Errett became a polarizing figure in the world of mathematics. Stan was fascinated by the revolution in mathematical thought that Bishop proposed, and by the uncomprehending reaction of the mathematics community, and we often spoke of it. But I remained with the uncomprehending majority, and my conversion to Bishop s program came several years later after both Stan and I were gone from Rochester. Below I ll describe the significant role that Stan later played in Bishop s revolution. But first, since Bishop is so little known today, I ll introduce him and his failed revolution, in which Stan played a significant part. 2. Errett In 1966 Errett Bishop, at age 38, was at the peak of his mathematical powers, a brilliant star in the mathematical firmament. But he had long been aware that he naturally thought differently about mathematics, basing it more securely on computation. He had gone along with the established classical mathematics because mathematics is a social activity, and it was the only game in town. He knew that a predecessor, the Dutch mathematician L. E. J. Brouwer ( ) had similar understandings and had failed to create a viable mathematical world with them. Several years earlier Bishop had a vision of how to fix Brouwer, and over a prodigious two-year period he succeeded in bringing a full constructive mathematics into being, succeeding where Brouwer had failed.

5 Brouwer had, however succeeded in reforming logic so that mathematics could be based on computation, and Bishop used the Intuitionist logic Brouwer created, Below I ll show in detail why numerical meaning requires a modification of classical logic. Errett had been invited to give a major address at the 1966 meeting of the International Congress of Mathematicians. It was expected that he would speak of his deep technical research in function algebras or several complex variables. Instead, he gave a rather elementary talk entitled A Constructivization of Abstract Mathematical Analysis, a passionate but properly muted call for a revolution in the way that we normally do and think about mathematics, classical mathematics. A friend who attended the talk described how it was received. Errett was a clear expositor with a warm personality, and during the talk there was much smiling and nodding. But after the talk the smiles gradually changed to frowns and nodding to head scratching, and the next day it just didn t make much sense, except for a few. It was a pattern that was to haunt Bishop thereafter, a toxic mix of recognition and incomprehension. Bishop started with very high expectations. In 1967 Foundations of Constructive Analysis (FCA) appeared with Chapter 1 called A Constructivist Manifesto. In his 1970 review in the AMS Bulletin, Gabriel Stolzenberg asserted: He [Bishop] is not joking when he suggests that classical mathematics, as presently practiced, will probably cease to exist as an independent discipline once the implications and advantages of the constructivist program are realized. After more than two years of grappling with this mathematics, comparing it with the classical system, and looking back into the historical origins of each, I fully agree with this prediction.

6 Bishop s prediction, seconded by Stolzenberg, could hardly have been farther off the mark While he was invited to speak at all the major universities and many conferences, he rarely felt that he was understood, and very few joined his cause. His Ph. D. students couldn t get good academic jobs and his few disciples had difficulties getting their papers published. Bishop s campaign won one and only one significant victory, which took place in the math department of New Mexico State University, and was entirely the work of Stan Tennenbaum. There was one group that found Bishop s approach to math natural: computer scientists. I experienced this when I spent 8 years teaching CS at Columbia (back in the days when there was a shortage of Ph. Ds in CS). But Errett was fixated on mathematicians and wrote only for them, so that even in computer science he is not well known. Bishop hid the algorithmic inspiration of his vision to make his mathematics look ordinary. The computer science pioneer D. E. Knuth said of, FCA: The interesting thing about this book is that it reads essentially like ordinary mathematics, yet it is entirely algorithmic in nature if you look between the lines. By making his approach look normal to mathematicians who didn t care, Bishop tended to hide its algorithmic nature from from the computer scientists who appreciated it. 3. Stan and Errett Here s a quote from Fred Richman s wonderful essay Confessions of a formalist Platonist intuitionist. When I returned to New Mexico State from a sabbatical leave at Florida Atlantic University, people there were talking about Errett Bishop's book Foundations of constructive analysis. Stanley Tennenbaum had visited the previous semester and conducted a very popular seminar on the subject.

7 This was remarkable at more than one level. To start with, constructive mathematics makes most mathematicians uneasy, for it questions Aristotle s Law of Excluded Middle (LEM), which asserts that every meaningful proposition is either true or false (although we may not know which). This is a sacred cow that generally operates at a subconscious level. If you challenge it, you may encounter intense opposition, as in this famous quote from David Hilbert: Depriving a mathematician of the use of Tertium non datur is tantamount to denying a boxer the use of his fists or an astronomer his telescope. Tertium non datur refers to Aristotle s Law of Excluded Middle. Hilbert, the leading mathematician of the time, was attacking Brouwer. He had tried doing math without LEM, but found it difficult. As we ll discuss below, it is difficult to switch from classical to constructive logic though easy to go the other way. The symbolism of the boxer s manhood represented by his fists and the astronomer erecting his telescope is almost embarrassingly Freudian. Stan not only attracted a lively quorum to his seminar, he took the core of that group so deeply into Bishop s thinking that they turned into a research group in constructive mathematics that remained prolific for over a decade. After Stan left, Fred Richman returned and joined the group, soon becoming its intellectual sparkplug. Fred and Douglas Bridges, a Brit who moved to New Zealand, stood out in the small band of mathematicians around the globe who tried to realize Bishop s vision. And without the support of the NMSU group, even Bridges might have chosen a different direction. This was a totally unique event. Except for NMSU, there was no place where Bishop s vision lived, only lonely constructivists in classical departments. 3. Logic versus Arithmetic At this point I m going to leave the stories of Stan and Errett to interject a simple explanation of the central issue, the incompatibility of two core

8 mathematical structures: classical logic with LEM, and numbers with their arithmetic. All the numbers we consider will be positive integers. Bishop, and Brouwer before him, saw numbers as objects that can be added, multiplied, divided, subtracted by expressing them in standard decimal notation. In his farewell address of 1973, Bishop put it like this: The Constructivist Thesis. Every integer can be converted in principle to decimal form by a finite, purely routine, process. Bishop didn t want to sound like a computer scientist, so he always said finite, purely routine, process instead of algorithm, I d prefer to put the Thesis in a style that Bishop often used, where one asks, What must be done to construct a number? and answers with: The Constructivist Thesis-Algorithmic Style. To construct a number you must provide data along with an algorithm which will convert at least in principle the data into a decimal representation, a finite sequence of the digits 0 through 9. The simplest case is when the data itself is a decimal representation, which the algorithm just passes along. In principle means that while you can describe the algorithm, it might not be realistic to start it running and wait for the result. In practice the algorithm is often obvious and not mentioned, as in: Let n be the number of prime numbers less than In either formulation the point is that numbers can be put in decimal form, which allows us to do arithmetic with them (provided we remember our tables and algorithms). We re now going to verify what Brouwer first showed: the conflict between classical logic and arithmetic. LEM introduces numbers that you can t do arithmetic with. Theorem. LEM and the Constructivist Thesis are Inconsistent. Brouwer s Proof: Brouwer uses LEM to construct a number q that cannot be converted to decimal notation because it encodes ignorance:

9 Take your favorite unsolved mathematical problem. There are zillions of them. I ll pick the Riemann Hypothesis about the zeros of the zeta function. If the Riemann Hypothesis is true, then let q = 1. If the Riemann Hypothesis is false, then let q = 0. LEM says that the Riemann Hypothesis is either true or false. So either q = 0 or q = 1. Since 0 and 1 are numbers, q is a number in either case. Proof by cases: q is a number, period. But we have no algorithm to compute the decimal representation of q. No algorithm to determine if the Riemann Hypothesis is true. Contradiction, coming from the Thesis and LEM. It may be worth noting that our construction of q does construct something, just not a number. It can be described as a non-empty subset of {0, 1} with at most one element. Knowing that the statement q is empty is false is of no help in constructing an element of q. Close, but no cigar. This completes the proof that the Constructivist Thesis and LEM are contradictory: you can t have both of them. If you accept LEM, you must admit numbers which do not appear in the tables. Thus the split. Brouwer s intuitionists and Bishop s constructivists chose the Thesis, which gives a clear understanding of numbers. A new logic for doing constructive mathematics was needed. Brouwer defined the proper logic, called intuitionist, for Brouwer s philosophy. But the definition lacked a compelling structure comparable to the truth tables that give shape to classical logic. In the 1930s a beautiful foundation was found for intuitionist logic, called Natural Deduction, which describes the logical connective in terms of rules for introduction and elimination.

10 Most mathematicians have chosen to remain with LEM, and accept that it admits numbers with which we cannot compute, and to try to avoid such numbers on an ad hoc basis. A this point I m going to return to Stan in Las Cruces and consider some qualities of Stan that enabled him to succeed, sometimes with corresponding qualities of Errett that led to his failure. 4. Pluralism It was one secret to Stan s success in Las Cruces that, unlike classical and constructive mathematicians, he did not choose between arithmetic and logic. He was equally at home talking math with Errett Bishop or with Kurt Gödel. Stan was a true pluralist, whereas most mathematicians are monists who believe that there is only one true mathematics. Bishop could sometimes sound pluralist but at heart he was a monist who tried to win converts by showing what was wrong with classical mathematics. But within its own context there is nothing wrong with classical math, it is valid. In A Defence of Mathematical Pluralism (2005) the British mathematician E. B. Davies wrote: We approach the philosophy of mathematics via a discussion of the differences between classical mathematics and constructive mathematics, arguing that each is a valid activity within its own context. Stan s pluralism meant that he did not approach the seminar with any antagonism or tension. What a difference that made. And what a contrast to Bishop, who was invited to give the Colloquium Lectures at the summer meetings of the AMS in A memorial volume, Errett Bishop: Reflections on Him and His Research, published by the AMS in 1984, contained the text of his Colloquium Lectures. Here is how it opens:

11 During the past ten years I have given a number of lectures on the subject of constructive mathematics. My general impression is that I have failed to communicate a real feeling for the philosophical issues involved. Since I am here today, I still have hopes of being able to do so. Part of the difficulty is the fear of seeming to be too negativistic and generating too much hostility. Constructivism is a reaction to certain alleged abuses of classical mathematics. Unpalatable as it may be to have those abuses examined, there is no other way to understand the motivations of the constructivists. The volume also contains Remembrances by Nerode, Metakides, and Constable which document the difficulties that Errett had with the mathematical community, and the contrasting receptivity of computer scientists. 5. Logic Mathematicians are always trained to use classical logic with unrestricted use of the LEM (and equivalents like double negation elimination). One secret to being a good mathematician (and a quick one) is to put the use of logic on a subconscious level. It can be very hard indeed to change subconscious patterns, as any meditator can affirm. As Fred Richman put it in Interview with a Constructive Mathematician: Catching when the law of excluded middle is used is much more difficult. It's been my experience that most mathematicians cannot do it. That's because the law of excluded middle is an ingrained habit at a very low level. It took me along time to reform my logical thinking so that it was naturally constructive and then to make it again quick and subconscious.

12 For Bishop intuitionist logic seemed to come naturally. Perhaps he never made excluded middle such an ingrained habit. In any case he was not strong on giving those trying to learn to think constructively a place to land when they let go of classical logic. He really didn t want to talk about logic, perhaps because constructive mathematicians were often told that they were no longer doing math, but were doing logic! While the truth is that classical and constructive use logic exactly the same way, to prove theorems. They just use different logics. Classical logic is made legitimate by truth tables. Intuitionist logic is legitimized by Natural Deduction with its introduction and elimination rules. But Bishop never presented them as a complete system. He seemingly did not want to give anyone a chance to say that he was doing logic. He provided no safe landing for those who let go of classical logic. With his rich background as a logician Stan managed to provide it in his NMSU seminar, but I don t know if he used Natural Deduction. More importantly, Stan had an uncanny sense for the trouble spots of his students. He would have been very aware that the transition from classical logic to intuitionist logic is a difficult one, a kind of psychoanalysis in which subconscious patterns are exposed, modified, and made habitual again. 6. Meaning One way to describe how the world of mathematics changes when one moves from a classical to an algorithmic framework, is to focus on the subtle shifts in the meaning of such central concepts as: truth, falsity, number, set, element, equality, identity, infinite set, higher infinity, function, proof, existence. The problem is that the meanings of these terms are highly interlocked, in a way that tends to keep meaning stable. So if you learn a new meaning for equality but other terms stay the same, the new meaning for equality will be outvoted by the old meanings, and will no longer

13 make sense. You need to learn enough constructive meanings so that they can form a coherent area of understanding from which you can extend. We ll look at how the meaning changes for some of these terms, for set and element, equality and identity. Classical mathematics starts with a domain of primitive elements with distinct identities. These basic elements are grouped into sets, which in turn can be elements of other sets. The clear fact is that the elements exist first and are collected into sets. In Bishop s approach to sets, it is the set which creates the possibility of elements. To construct a set you must describe what must be done to construct an arbitrary element. The set precedes its elements. Of course the number 3 had been around for a long time before anyone considered a set N of all numbers, but only existed as an element of N after the latter was constructed. Let s consider equality and identity. Classical math tends to conflate them, and to use the equal sign when things are identical. And identity is a global predicate: any two mathematical objects are either the same or not. Constructively, equality is always a convention, as Bishop proudly proclaimed. And there is no universal identity relation. Part of the construction of a set is providing a definition of equality, and the construction must include proofs that the proposed definition is reflexive, symmetric, and transitive. If there is no universal identity relation, what it means if the same symbol is used more than once, as in reflexivity x = x. Bishop suggested that he relied on the repeatability of mathematical constructions, or intentional identity. Mathematical constructions, like scientific experiments, must be repeatable, and in the reflexive equation the second x is a repeat, a copy, of the first. To prove the proposed equality relation is reflexive you must show that if you carry out any

14 construction of an element, and then repeat the construction, the proposed relation will declare that the original and the copy are equal. This formally eliminates the possibility that you could construct a set in which equality is determined by flipping a coin. Stan would have gone through the changes in meaning with his seminar, helping them to put enough constructive meaning together so that a coherent world came into focus. It would have been a joint exploration through tricky terrain rather than being called from on high. I first experienced these changes around 1971 when, after several years of wrestling with Bishop s work, I learned to see mathematics constructively, or as I would say now, algorithmically. My conversion would never have occurred save for the patient mentoring of a friend, Gabriel Stolzenberg, with a deep understanding of Bishop. At the end, I found that the appearance of my mathematical world had changed. It was still unquestionably mathematics that I observed, but it had a different texture, more alive, less remote. 7. Brouwer I m going to close with an account of my own difficulty with Brouwer, which was an early barrier to engaging with Bishop s program. when I was in graduate school I had been socialized to believe that Brouwer had been a great young mathematician, who proved the Brouwer Fixed Point Theorem, one of the first deep results in the new field of topology. Then, in later life, he began to worry too deeply about what it all meant, and basically went crazy. Brouwer made a speaking tour of the USA in 1960, and how we graduate students laughed at poor Brouwer who was presenting a counter-example to his own greatest theorem. It was a sad warning to stay away from dangerous ideas that could threaten your sanity. That warning was still fresh when Stan and I began discussing Bishop s constructivism, it was an obstacle that slowed me down. And when Stan went to Las Cruces, that

15 understanding of Brouwer, which was still very widespread would have been something to confront. It s easy to imagine how skillfully Stan would have taken that on. Brouwer s life and work is much better documented now. The story that we had accepted or fabricated about Brouwer s life was completely false. He had the dangerous ideas when he was young, and they appeared in his Ph. D. thesis. Then he realized that nobody would pay attention to them unless he was a recognized mathematician, so he did standard mathematics until he was a famous topologist, and then he came out. I m going to close with Brouwer s counter-example to his own Fixed Point Theorem. That theorem is classically valid in all dimensions, but we ll only consider dimension one, where the theorem is essentially the Intermediate Value Theorem of elementary calculus. Brouwer constructed a continuous function f such that f(0) is negative and f(1) is positive, and you cannot locate a point where f takes the intermediate value 0, without solving some unsolved problem. Here s a graph of Brouwer s function: You may have guessed that between 0.3 and 0.7 the function f is constant, taking a value we ll call zeroey very close to 0, which might

16 be positive or negative or exactly zero. If zeroey is positive then there is a unique value, slightly less than 0.3, negative and the value is slightly more than 0.7. And if zeroey is zero, then f has the value 0 from 0.3 to 0.7. Again our ignorance is encoded in zeroey. Consider our ignorance about the decimal expansion of " works very nicely to construct zeroey. We know virtually nothing about it, but we do have very good algorithms for computing it. For our unsolved problem, let s ask if ever contains 100 consecutive 7s. Start computing the decimal expansion of ", and watching our for 100 consecutive 7s, and outputting a sequence of rational numbers (fractions) to define zeroey, and also keeping track of how many digits of " you ve looked at. As long as you haven t found 100 consecutive 7s, your output is 0. But if you find 100 consecutive 7s, ending in the decimal place n, then change to output to ± 1 10 & where the sign depends on whether n is odd or even, and leave the output there forever forever. To sum up, when you start computing the sequence that defines zeroey, you keep getting 0 but the sequence always retains the possibility of switching to an infinitesimal positive or negative value. But all is not lost for the Intermediate Value Theorem. Brouwer s example captures the essence of functions for which the intermediate value theorem doesn t hold. If a function is never constant, then that function takes all intermediate values. Most functions are never constant, unless they are constructed to be constant over some interval. For instance polynomials, trigonometric functions, etc. The Intermediate Value Theorem becomes much more interesting constructively. I try to imagine how Stan would have presented this example in Las Cruces. I wish I d been there. But I lost touch with Stan after our time in Rochester. There were occasions when I could have contacted him, and regret that I didn t. I particularly wish we d spoken after I finally came to understand Bishop. There would have been so much to talk about.