Newcomb Greenleaf


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 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’.

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 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.


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 (1881-1966) 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.

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.

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.

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

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

Logic vs. 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 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 1010 .

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:

  • 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.

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.


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 1973. 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:

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.


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.
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.


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 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 of all numbers,
but only existed as an element of
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 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.


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 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 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
π 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 3.14159… ever contains 100 consecutive 7s. Start
computing the decimal expansion of
π, 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
π 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 ± / 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