#### Foreword

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’.

Foreword | Stan at Rochester | Errett | Stan and Errett | Logic vs. Arithmetic | Pluralism | Logic | Meaning | Brouwer

#### 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.

Foreword | Stan at Rochester | Errett | Stan and Errett | Logic vs. Arithmetic | Pluralism | Logic | Meaning | Brouwer

#### 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 (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.

Foreword | Stan at Rochester | Errett | Stan and Errett | Logic vs. Arithmetic | Pluralism | Logic | Meaning | Brouwer

#### 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.

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.

#### 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 10^{10} .

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.

#### 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 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.

#### 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.

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.

#### 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 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 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.

#### 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 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

about.