David W. Henderson
Department of Mathematics
Cornell University
Ithaca, NY 14853-7901
e-mail: [email protected]
Cecilia Hoyles in first plenary told us that whenever we examine someone's conceptions of proof we should learn something about their background -- what have they been taught about mathematics. So, I thought it was appropriate that I start by telling you something of my background, since I am going to talk with you for the next hour about my views of proof as gleaned from my experiences as student, teacher, mathematician and general experiencer of the world.
I have always loved geometry and was thinking about geometric kinds of things since I was very young as evidenced by drawings that I made when I was 6 which my mother saved. But I did not realize that the geometry, which I loved, was mathematics. I was not calling it geometry -- I was calling it drawing or design or not calling it anything and just doing it. I did not like mathematics in school because it seemed very dead to me -- just memorizing techniques for computing things and I was not very good at memorizing. I especially did not like my high school geometry course with its formal two-column proofs. But I kept doing geometry in various forms in art classes, out exploring nature, or by becoming involved in photography. This continued on into the university where I was a joint physics and philosophy major and took only those mathematics courses which were required for physics majors. I became absorbed in geometry-based aspects of physics: mechanics, optics, electricity and magnetism, and relativity. On the other hand, my first mathematics research paper (on the geometry of Venn diagrams for more than 4 classes) evolved from a course on the philosophy of logic. There were no geometry courses except for analytic geometry and linear algebra, which only lightly touched on anything geometric. So, it was not until my fourth and last year at the university that I switched into mathematics because I was finally convinced that the geometry that I loved really was a part of mathematics. This is not an uncommon story among research geometers. Since high school, I have never taken a course in geometry because there were no geometry courses offered at the two universities which I attended. So, in some ways I may have had the advantage of not having taken a geometry course!
But I was educated in a very formal tradition -- in fact, mathematics was I think the most formal that it has been about the time that I was studying at the university in the late 1950's and early 1960's. One of the evidences for this was the number of geometry courses offered at colleges and universities -- there were practically none anywhere at that time except for a few geometry courses for perspective school teachers and still at many institutions such courses are the only geometry courses offered.
I am the same generation as most of the faculty now in mathematics departments in North America (most of us are 50-65 years old) because we were hired to teach the baby boomers in the 1960's. So now my generation is clogging up most of the tenure faculty positions all over North America, and in the USA we will not be required to retire because the Supreme Court has ruled recently that it is unconstitutional to have mandatory retirement ages. Almost all of the mathematicians in my generation had a very formal training in mathematics. This has affected us and affected mathematics and will continue to affect mathematics because my generation now has the positions of authority in mathematics.
I want to mention specifically one mathematician in my generation and that is Ted Koscynski, the suspected Unabomber. He had very much the same kind of university mathematics education that I had. Both of us at the time were socially inept and it was difficult for us to get to know people. In fact, in some ways, this was encouraged in our training all the way through graduate school and certainly was not a hindrance in any way. My thesis advisor talked to me and a few of the other men graduate students and said to us that it would be very important for us to find wives who would take care of all of our social responsibilities, so that we would not have to deal with social things and could put all of our energy into mathematics. Fortunately, one of the things that helped save me was that I did not take his advice -- I got married but not to such a wife. The Unabomber talks about similar things which happened to him.
Both Ted Koscynski and I accepted tenure track professorial positions at major research mathematics departments (Berkeley and Cornell). And we were initially both successful with professional mathematics. Then, in the early 1970's, both Ted Koscynski and I quit mathematics. I got angry with mathematics -- I got very furious about what mathematics had done to me. It is too complicated to go into all my feelings then (even if I could retrace them accurately) -- but if someone came up to me at that time and called me a mathematician I felt strongly like punching them in the face! The evidence indicates that Ted Koscynski had a different more violent reaction but his writings express feelings very similar to the ones I had at that time. He and I both went into the forest and built a cabin and lived there alone and we isolated ourselves. But, there was a huge difference -- I made a constructive positive breakthrough and Ted Koscynski didn't.
It was geometry, the many friends I made, and my family that brought about this breakthrough and in many ways saved my life. I got back into geometry. Before this, I had not been teaching geometry -- I had been teaching geometric topology and such courses but all of my teaching up to then had been very formal. There was one geometry course at Cornell at the time -- the one for perspective secondary school mathematics teachers. It was not considered to be a real mathematics course and I considered myself to be a real mathematician so I did not have any interest in teaching it. But at that time, when I thought I was quitting mathematics, I needed to teach a little in order to have enough money to survive in my cabin, so I took a leave without pay and then occasionally came back and taught for some money. (Fortunately I did not burn any bridges.) So I started teaching this geometry course for perspective teachers. In the first three years that I taught the course, while living in the forest, three mathematics educators familiar to most of you were in the class, David Pimm (Open University), Jere Confrey (Cornell), and Fran Rosamond (National University, San Diego). This geometry course was essentially all that I was teaching for a few years. A lot of what I am going to talk about are my experiences with that course and what happened since then. This course (and my new friends) pulled me out of the fire that Ted Koscynski never got out of.
FORMAL DEDUCTIVE SYSTEMS
Global formal deductive systems can be very powerful and are important in certain areas (for example in the study of computer algorithms and in the study of questions in the foundations of mathematics). Local formal deductive systems can be important and powerful in many areas of mathematics (for example group theory.) But many people hold the belief that mathematics is only the study of formal systems. These beliefs are wide-spread especially, I find, among people who are not mathematicians or teachers of mathematics. Let me give some descriptions of formal mathematics. For example, in FOCUS: The Newsletter of the Mathematical Association of America a professor of computer science wrote:
... one of the most remarkable gifts human civilization has inherited from ancient Greece in the notion of mathematical proof [and] the basic scheme of Euclid's Elements ... This scheme was formalized around the turn of the century and, ever since ... mathematicians have rested assured that all their ingenious proofs could, in principle, be transformed into a dull string of symbols which could then be verified mechanically. One of the basic features of this paradigm is that proofs are fragile: a single, minute mistake (e.g. an incorrectly copied sign) invalidates the entire proof. [Babai 1992]
This is the kind of view of mathematics that I learned when I was in school and the university.
Here's a more recent description that just appeared in the past year in the American Mathematical Monthly in an article (by another professor of computer science and member of my mathematical generation) about a new reform teaching technique and text for discrete mathematics which is based on a "computational" formal approach which uses uninterpreted formal manipulations which have been stripped of meaning:
...most students are troubled by the prospect of uninterpreted manipulation. They want to think about the meanings of mathematical statements. Having meanings for objects is a "safety net", which students feel, prevents them from performing nonsensical manipulations. Unfortunately, the use of the "meaning" safety net does not scale well to complicated problems. Skill in performing uninterpreted syntactic manipulation does. [Gries 1995]
He literally means to get rid of the meaning. He takes literally the formalist view of mathematics that the meaning is not important. He goes further to say that the meaning actually gets in the way. I was at one of his talks when he was explaining his new teaching method and he gave a proof of some result in discrete mathematics and I tried to follow the meaning through from the hypothesis to the conclusion, because the hypothesis and conclusion did have meaning. I tried to follow that meaning through the proof in order to see the connections, but I failed to do so. I brought this up at the end and he said something close to: "Yes! That's precisely the idea! We have managed to get the meaning out of the way so that it doesn't confuse the students so they are now better able to do mathematics."
Now let me give another description of mathematics. This was written by Jean Dieudonné in an article which was written in response to an article by René Thom in which Thom was talking about intuition and how it was important to bring in and foster intuition in the schools.
I am convinced that, since 1700, 90 per cent of the new methods and concepts introduced in mathematics were imagined by four or five men in the eighteenth century, about thirty in the nineteenth, and certainly not more than a hundred since the beginning of our century. These creative scientists are distinguished by a vivid imagination coupled with a deep understanding of the material they study. This combination deserves to be called "intuition." ...
In most cases [the transmission of knowledge] will be entrusted to professors who are adequately educated and prepared to understand the proofs. As most of them will not be gifted with the exceptional "intuition" of the creators, the only way they can arrive at a reasonably good understanding of mathematics and pass it on to their students will be through a careful presentation of their material, in which definitions, hypotheses, and arguments are precise enough to avoid any misunderstanding, and possible fallacies and pitfalls are pointed out whenever the need arises. [Dieudonné 1973]
Both of the first two quotes were from computer scientists who down-play the role of meaning and intuition in mathematics. Now, Dieudonné who certainly is a mathematician and a very good one, pointed out that intuition and imagination are very important but that there only a few people (apparently, men) who have that intuition and that for the rest of us it is necessary for mathematics to be put down in a very precise formal way. Dieudonné has two claims to fame that are connected to this. One is he was the founder of the Bourbaki movement which was an attempt (which was never finished) to formalize all of mathematics. The other which is more significant for this gathering is that about the time of this article he was chair of the ICMI (International Commission on Mathematics Instruction) and chair of it at the time that the "New Math" was being spread around the world. He has always been involved in education.
Another example of descriptions of mathematics: Mathematica©, the computer program, when it first came out was advertised as a program that can do all of mathematics -- remember those early ads? If you believe a strictly formal view of mathematics then that claim was believable and many people did believe it.
Confining geometry within formal deductive systems is harmful
Now I want to talk about how I see the view (which I take as starting roughly a hundred years ago) that mathematics is just formal systems has been harmful. I see this view as harmful because:
-- it encourages what I think are incorrect beliefs such as those above that mathematics is only the study of formal systems. Of course, people can have disagreements as to what mathematics is, but I think that most of the people in this room do not believe that mathematics is just formal systems. And let me make it clear that I believe that formal systems do have a place in mathematics and that they are very useful and very powerful in many ways. Formal systems are very important in computer science because that is what a computer does -- deal with formal systems. So it is not surprising that it was professors of computer science that made the statements that I have put here. Formal systems have certainly been very important in various parts of algebra and analysis and topology (which was my area of research) which flourished in this century. But geometry virtually disappeared as evidenced by the fact that there were almost no undergraduate geometry courses in 1970. That trend has now reversed. For example, now at Cornell there are eight undergraduate geometry courses and of those eight there is only one-half of one of them that deals with axiomatic systems. So things are changing.
-- much interesting and useful geometry is either not taught at all or is presented in a way that is inaccessible to most students. For example, spherical geometry was in the high school and university curriculum (or, at least in the textbooks) of 100 years ago. Now, of course, high schools in those days were more elite institutions than they are today, but spherical geometry is almost entirely absent from our courses and textbooks now. Why is it that it disappear? It is not because it is not useful: Spherical geometry is very applicable -- navigation on the surface of the earth, the geometry of visual perception, the geometry of astronomical observations, surveying on a scale of several kilometers. Spherical geometry is a very useful geometry, but we do not teach it anymore -- why? I think the reason is that spherical geometry is very difficult to formalize -- there is no convenient axiom system for spherical geometry. There is an axiom system for spherical geometry (Borzuk did it just before the Second World War) -- it is in a book that is in many mathematics libraries but it rarely has been used -- It is not a useful axiom system. "Non-Euclidean" geometry has been often taught in undergraduate geometry courses, but it has always been "the" non-Euclidean geometry, hyperbolic geometry, which has a relatively simple axiom system and which has only been around for about 160 years. Spherical geometry which is very old (the Babylonians and Greeks studied it) is rarely taught. I can not think of any reasonable explanation for why spherical geometry disappeared except that it does not fit into formalism. This is one of the reasons that I have it in my geometry course. When freed from the confines of formal systems it is possible to present spherical geometry in ways that are based on geometric experiences and intuitions. See [Henderson, 1996a].
-- important notions in mathematics are formally defined in ways that separate them from the students' experiences. For example, the new Chicago Mathematics Curriculum for American secondary schools (which has many good things in it and is now the fastest growing curriculum in the USA) defines a rotation as the product of two reflections. Now that is an interesting fact (theorem) about rotations. But what does a student think when he or she comes to that as the definition of what a rotation is? It is very difficult to relate the product of two reflections with our experiences of rotations such as opening a door or riding a merry-go-round. One of the problems is that our intuition of rotations is a dynamic thing -- it is actually a motion. Whereas to think of rotation as the product of two reflections is a static thing -- it is the result of the rotation motion that is equal to the product of two reflections. If I were a student and saw this definition in the textbook I would say that this geometry is not relating to what I know geometry is and I would feel that the text is telling me that my experiences and intuitions are not important. It appears that the main reason for using this definition is that it is convenient formally in the deductive system in which the geometry in the text is confined. Also, differential geometry (the geometry of curves and surfaces, the geometry of the configuration spaces of mechanical systems, the geometry of our physical space/time) has extremely difficult formalisms which make it inaccessible to most students and even, I suspect, most mathematicians are uncomfortable with the formalisms of differential geometry. Some people have called it the most complicated formalism in all of mathematics. But yet differential geometry is basically about straight lines and parallelism -- very intuitive notions. When we insist on formalizing differential geometry then it becomes inaccessible -- even more so because there is no agreed upon formalism. My second book, [Henderson, 1996b], is an attempt make differential geometry accessible by basing it on geometric experiences and intuitions, as opposed to basing it on standard algebraic and analytic formalisms.
-- many important and useful questions are not asked. This was something that really surprised me when I started teaching this geometry and started listening to the perspective teachers who were taking the course. There are a lot of questions that student have that we never ask in mathematics classes. For instance, the reliance on a formal Euclidean deductive system rarely allows for questions such as "What do we mean when we say that something is straight?". We normally don't ask that in any classes, even though we talk about straight lines all the time. We just write down some axiom or we just say "everyone knows what 'straight' is". In differential geometry the formalism has attempted to get at what the meaning of straight is, but in a way that is not accessible. But one can ask the question about what it means to be straight, you can ask that of students. I've done it with first graders -- they can come up with good discussions. One of the results of this is that when spherical geometry or other geometries are talked about, usually they are just presented with some statement like: "We will define the straight lines to be the great circles on the sphere". But that is ridiculous, the great circles are the straight lines on the sphere, you do not have to define them. If you have a notion of what straightness is, then you can imagine a bug crawling around on the sphere and ask how would the bug go if the bug wanted to go straight -- You can convince yourself that it is the great circles. But we can not even ask those questions in a formal context. Also the connections between linear algebra, geometric transformations, symmetries, and Euclidean geometry are very difficult to talk about in a formal system; in fact (I don't know if I want to say impossible or not) but it is not conveniently done and often not done at in a formal system. Remember the example above of defining a rotation to be the product of two reflections.
{Other questions which we ignore: "Why is Side-Angle-Side true on the plane (but not on the sphere)?", "What is the geometric meaning of tangency?", and "How do we experience the connections between ?"}.
-- mathematicians are being harmed. I have already talked about how mathematicians are being harmed -- I was harmed by the over emphasis on formalism -- so was Ted Koscynski. And I'm sure that you know of examples (at your own university or around your own university) of mathematicians, roughly my generation, who have more or less dropped out of society -- there are a lot of them around, people who have been good mathematician, who had been successful in the system back in 1950's and 1960's. So it has been harmful to mathematicians.
-- students are being harmed. When a student's experiences lead her/him to understand a piece of mathematics in a way that is not contained in the formal system, then the student is likely to lose confidence in her/his own thinking and understanding even when it is backed up by what I will call alive geometric reasoning. Deductive systems do not encourage alive matheamtical reasoning (which in my experiences with students and teachers is a natural human process) and thus they serve to deaden human beings whose thinking and understandings are forced to reside in these systems. We now have machines that can do the computations and formal manipulations of deductive systems: we need more alive human reasoning.
Here are some examples:
One of the things that I clearly remember from the beginning of my teaching of the geometry course is the following: I was teaching the Vertical Angle Theorem and its standard proof:
I can still remember one of the students who was very shy and wouldn't speak up in class, but I was having the students do writing. She wrote on her paper something to the effect of:
All you have to do is do a half-turn. Take this point here (p) and rotate everything about this point half of a full revolution. We have already discussed that straight line have half-turn symmetry and so each line goes onto itself and a goes onto b.
I don't know what your reaction is now but my reaction then was "That's not a proof" and I told her so. Fortunately, though she was shy, she was persistent and stubborn and she kept coming back and insisting that that was a proof. She worked on me for about two weeks and I kept listening to her and struggling with the question, Is that a proof?, because it did not seem like proofs that I had been accustomed to and that I would accept. Finally, she convinced me and now I think it is a great proof and much better than the standard proof which is in most of the textbooks. The standard proof has a lot of underlying assumptions that need to be cleared out and many formal treatments do that -- they put in the "Protractor Postulates" which state the appropriate connections between angles and numbers and then you can do the standard proof. But the proof with the half-turn is just connected to a certain symmetry of straight lines. You can use other symmetries of straight lines to proof this result also, but this one is the cleanest, the simplest. And this proof is not possible in a formal system and it is particularly not possible in a formal system if (because you want to insist on putting everything in a formal system) you define a rotation as the product of two reflections. That particularly won't work here because, if you take one of the lines and reflect through the line and then reflect perpendicular to the line that is equivalent to a half-turn, but there is no pair of reflections that will simultaneous do that to both of these lines, but yet a half-turn clearly preserves both lines. I do not see any reasonable way for that to have been included in any kind of formal system. So, if I had been insisting on formal systems, I would have missed out on the half-turn proof and not learned this bit of mathematics. I almost missed out anyhow and it was only because she was very persistent.
After that experience I started listening more to students and expecting that when they would say things that I didn't understand, that maybe they really did have something (and something that I could learn). I took the attitude that we are not working in a formal system, but that we are doing mathematics the same way that mathematicians mostly do mathematics. (In geometry, mathematicians do not stick inside any particular formal system, we use whatever tools might be appropriate: computers, linear algebra, analysis, symmetries. Mathematicians use symmetries a lot!) As I listen to students I have been learning more and more geometry from the students. I used to be surprised at that and thought it was just because I had not been teaching the course for very long. I thought that after I have taught it for a while then I will know it all and I will not see anything new. Well, what happened is that I have been teaching the course for 22 years now and now 30-40% of the students every semester show me some mathematics that I have never seen before! (These students are some mathematics majors, all the perspective secondary school teachers, most of the mathematics education graduate students, a variety of students.) I would miss out on most of this new geometry if things were being done inside a formal system.
Another example: There are many properties of parallel lines in the plane (for example, any line which traverses two parallel lines will intersect those lines at the same angle) whose proofs depend on the parallel postulate. When we get to that point in the course I let the students come up with their own postulate -- whatever it is that they think is most important to assume that will separate the plane from the sphere. There is a difference between the plane and the sphere and there is some difference that has to do with parallel lines. The students come up with all kinds of different postulates, many of which I think would be much more reasonable to assume than the usual parallel postulates. By the way, Euclid's parallel (fifth) postulate is true on the sphere -- Euclid's parallel postulate is not what distinguishes spherical geometry from plane geometry, contrary to what many books say. I take that as evidence that people who have written such mistakes about spherical geometry have never really looked at a sphere -- they have just been looking at the situation formally and thus made the mistake.
Let me give another example to show that I can think about something that is not just geometric. Here is the proof that is usually given to American secondary school students that 
:
now subtract both sides to get
and thus
.
This proof embodies a very useful technique for figuring out, when you have a repeating decimal, what fraction is equal to that repeating decimal. It is a very useful technique in that context. But I claim that it is not a proof in this context. I claim it is something that is masquerading as a formal proof: It looks like a formal proof, it has steps and x's and all that stuff. I started asking my calculus students at Cornell what they thought, and some of the best high school students in North America come to Cornell. They mostly know this proof, because they learned it; but only about half of them believe it, because they do not believe that 
. To show you why I think that this is masquerading as a proof and really isn't a proof: If we try to make this a little more precise as to just what it is we mean by 
(that is part of the problem here). Well, to most students what 
means is, 0.9, then 0.99, then 0.999,... -- a limit of a sequence (at the time they are expressing this, they might not even know what a sequence is) -- you keep putting on one more 9, you go on for ever -- that is the way that they talk about it. It fits in nicely with calculus to do it that way and to think about 
as the limit of a sequence:
.
If you think of it as this limit and then follow the formal rules for subtracting sequences and multiplying sequences and so on, you come out with the amazing conclusion that:
!
This is true -- not very useful, but it is true. And it has to be that way, because there is an assumption being made here -- the Archimedian Axiom. Way back, Archimedes knew that in talking about numbers it was possible to talk about ones which we now call infinitesimal, and then Archimedes had an axiom or principle which rules out these infinitesimals. The Archimedian Axiom (or Principle) gets stated in various different way but is rarely mentioned these days in the North American undergraduate curricula -- most textbooks (if they mention it at all) relegate it to a brief mention in a footnote or exercise. The usual approach these days is to subsume the Archimedian Axiom under Completeness in a hidden way so that you do not even notice that it is there. I think it is important for the students to know that this is an assumption. They can understand why it is convenient to assume that 
and understand that there are a lot of reasons for making that assumption. But we should tell them that it is an assumption -- and it really is.
Another example -- here is a theorem:
For natural numbers, n x m = m x n.
Now, the usual formal proof which I learned for this theorem is a complicated double mathematical induction. I dutifully learned this proof and was dutifully teaching it when I first started teaching. But here is the prop for another proof (I do not want to say 'another proof' but only 'a prop for a proof'):
Here we think of 3 x 4 as three 4's or four 3's (it seems that most mathematicians think of 3 x 4 as three 4's, but many of my students think of 3 x 4 as four 3's). I find a proof based on this schema as more convincing than the one with the double induction. And this proof can be visualized with having arbitrary numbers of dots, because the whole point is that you do not have to count the dots to know that this is true -- there is symmetry. But it is hard to express in words and put down in a linear fashion on a piece of paper and all that kind of stuff.
-- mathematics is being harmed. Historically, most current-day mathematics was based on geometric explorations, geometric reasonings, and geometric understandings. The developers of our current deductive systems in algebra and analysis explicitly attempted to weed out all references and reliances on geometry and the geometric intuitions on which the algebra and analysis was originally based. When we confine mathematics to these formal systems we teach the students to distrust mathematics, not to value it, and not to use their intuitions in understanding mathematics. Many, many students who have a natural interest in mathematics are lost to mathematics by this process -- I almost was.
HOW SHOULD WE DESCRIBE WHAT IS MATHEMATICS?
David Hilbert is considered to be "the father of formalism" so I checked what he had to say. In 1932, late in his career he wrote in the Preface to Geometry and the Imagination:
In mathematics, as in any scientific research, we find two tendencies present. On the one hand, the tendency toward abstraction seeks to crystallize the logical relations inherent in the maze of material that is being studied, and to correlate the material in a systematic and orderly manner. On the other hand, the tendency toward intuitive understanding fosters a more immediate grasp of the objects one studies, a live rapport with them, so to speak, which stresses the concrete meaning of their relations. [Hilbert's emphasis]
As to geometry, in particular, the abstract tendency has here led to the magnificent systematic theories of Algebraic Geometry, of Riemannian Geometry, and of Topology; these theories make extensive use of abstract reasoning and symbolic calculation in the sense of algebra. Notwithstanding this, it is still as true today as it ever was that intuitive understanding plays a major role in geometry. And such concrete intuition is of great value not only for the research worker, but also for anyone who wishes to study and appreciate the results of research in geometry. [Hilbert 1932]
The last sentence in the first paragraph ("On the other hand ...") is a very nice description of what a lot of us are trying to do and he goes on to say how important this is in mathematics. I even went back to his paper "On the Infinite" -- he does not say that mathematics is formal systems or that all of mathematics should be formalized. He, in fact, says very explicitly there that mathematics is based on intuition and that intuition is an appropriate basis for what he calls "ordinary finite arithmetic". He wanted to introduce the formalization in order to take care of various paradoxes that were coming up in dealing with the infinite, because there seemed to be some problems with intuition around infinite things. He never claimed that mathematics was formal -- that was his followers.
Here is a more recent view expressed by William Thurston, who is director of the Mathematical Sciences Research Institute at Berkeley and one of the most prominent American mathematicians. Thurston rejects the popular formal definition-theorem-proof model as an adequate description of mathematics and states that:
If what we are doing is constructing better ways of thinking, then psychological and social dimensions are essential to a good model for mathematical progress. ...
... The measure of our success is whether what we do enables people to understand and think more clearly and effectively about mathematics. [Thurston 1994]
I will now give a description of mathematics that is what I think Hilbert and Thurston are talking about. I call it "alive mathematical reasoning" where I take the word "alive" from Hilbert's quote:
WHAT IS ALIVE MATHEMATICAL REASONING?
Alive mathematical reasoning includes both abstraction and intuitive understanding as Hilbert says in the above quote.
Alive mathematical reasoning is paying attention to meanings behind the formulas and words -- meanings based on intuition, imagination, and experiences of the world around us. It is not memorizing formulas, theorems, and proofs -- this is again something that computers can do. We, as human beings, can do more. As Tenzin Gyatso, the fourteenth Dalai Lama has said:
"Do not just pay attention to the words;
Instead pay attention to meanings behind the words.
But, do not just pay attention to meanings behind the words;
Instead pay attention to your deep experience of those meanings."
Alive mathematical reasoning includes "living proofs", that is, convincing communications that answers -- Why? It is not formal 2-column proofs -- computers can now do formal proofs in geometry. If something does not communicate and convince and answer--why? then I do not want to consider it a proof. What we need are alive human proofs which
-- "communicate": When we prove something to ourselves, we are not finished until we can communicate it to others. The nature of this communication depends on the community to which one is communicating and it is thus, in part, a social phenomena.
-- "convince": A proof works when it convinces others. Proofs must convince not by coercion or trickery. The best proofs give the listener a way to experience the meanings involved. Of course some persons become convinced too easily, so we are more confident in the proof if it convinces someone who was originally a skeptic. Also, a proof that convinces me may not convince my students.
-- "answer -- Why?": The proof should explain, especially it should explain something that the listener wants to have explained. As an example, my shortest research paper [Henderson 1973] has a very concise simple proof that anyone who understands the terms involved can easily follow logically step-by-step. But, I have received more questions from other mathematicians about that paper than about any of my other research papers and most of the questions were of the kind: "Why is it true?" "Where did it come from?" "How did you see it?" "What does it mean?" They accepted the proof logically but were not satisfied -- it was not alive for them.
One of my colleagues at Cornell was hired directly as a full professor based primarily on a series of papers that he had written even though at the time we knew that most of the theorems in the papers were wrong because of an error in the reasoning. We hired him because these papers contained a wealth of ideas and questions that had opened up a thriving area of mathematical research.
Alive mathematical reasoning is knowing that mathematical definitions, assumptions, etc. vary with the context and with the point of view. Alive reasoning does not contain definitions and assumptions that are fixed in a desire for consistency. It is an observable empirical fact that mathematicians and mathematics textbooks are not consistent with definitions and assumptions. We find this true even when the general context is the same. For example, I looked in the plane geometry textbooks in the Cornell library and found 9 different definitions of the term "angle". Also, calculus textbooks do not agree on whether the function y = f(x) = 1/x is continuous or not continuous; and analysis textbooks have many different axioms for the real numbers that have different intuitive connections and necessitate different proofs.
Alive mathematical reasoning is using a variety of mathematical contexts: 2- and 3-dimensional Euclidean geometry, geometry of surfaces (such as the sphere), transformation geometry, symmetries, graphs, analytic geometry, vector geometry, and so forth. It is not Euclidean geometry as a single formal system. When a mathematician is constructing a proof that needs a mathematical argument she/he is free to use whatever tools work best in the particular situation. Mathematicians do not limit themselves in this way. Also, those who use geometry in applications, do not feel restricted to a single formal system.
Alive mathematical reasoning is combining together all parts of mathematics: geometry, algebra, analysis, number systems, probability, calculus, and so forth.
Alive mathematical reasoning is applying mathematics to the world of experiences.
Alive mathematical reasoning is using physical models, drawings, images in the imagination.
Alive mathematical reasoning is making conjectures, searching for counterexamples, and developing connections.
Alive mathematical reasoning is always asking -- WHY?
BUT WHAT ABOUT CONSISTENCY AND CERTAINTY?
-- formal deductive systems do not gain consistency. For example, is the function f(x) = 1/x continuous? Look in several calculus books. They give different answers! Differential geometry is another example where there is no consensus as to which formalism to use, but yet everyone thinks they are talking about the same ideas. Why?
-- formal deductive systems usually do not gain for us the certainty that we strive for. Formal deductive systems are useful and powerful in some circumstances, for example, in deciding which propositions can be logically deduced from other propositions and whether certain processes or algorithms will always produce the expected result. But, these deductive systems only give us certainty that certain steps (that can in principle be mechanized) can be carried out. They usually do not gain us certainty for the human questions of "Why?" or the human desire for experiencing meanings.
ALIVE MATHEMATICAL REASONING BRINGS BENEFITS TO MATHEMATICS
In my experiences students with alive geometric reasoning are the most creative with mathematics. These are also the students who can step back from their individual courses and see the underlying ideas and strands that run between the different parts of mathematics. They are the ones who become the best mathematicians, teachers, and users of mathematics.
There is research evidence that successful learning takes place for many women and underrepresented students when instruction builds upon personal experiences and provides for a diversity of ideas and perspectives. See, for example [Belenky et al, 1986], [Cheek, 1984], and [Valverde, 1984]. Thus alive mathematical reasoning in school classes may contribute to increasing the numbers of mathematicians who are women and persons from racial and cultural groups that are now underrepresented.
In my own teaching I encourage students to use alive mathematical reasoning and observe how their thinking and creativity is freed and their participation is opened up. See [Lo et al, 1996] As I listen to the alive mathematical reasonings of my students I find that 30-40% of the students show me mathematics that I have not seen before and that (percentage-wise more of these students are women and persons of color than white men. [Henderson, 1996].
CLOSING EXAMPLE
I will conclude with a proof that I learned from a student in a freshman course which is taught in the same style and using some of the same problems as the geometry course. The course was for "students who did not yet feel comfortable with mathematics" and who were social science and humanities majors. There a student, Mariah Magargee, who was an English major; she had been told all the way through high school that she was no good at mathematics and she believed it. I want to share with you her proof that the sum of the angles of a triangle on the sphere is more than 180 degrees. We had previously, in class, been talking about the standard proof that on the plane the sum of the angles of a triangle is always 180 degrees:
In class I stressed that the students should remember that latitudes circles (except for the equator) are not geodesics (straight on the sphere) and I urged them not try to apply the notions of parallel to latitude circles. Mariah ignored my urgings and noted that two latitude circles which are symmetric about the equator of the sphere are parallel in two senses -- first of all they are equidistant from each other and:
Nice proof! I like it. That is Mariah's proof. This is a student who believed that she was no good at mathematics and was told she was no good at mathematics, but she taught me a really nice proof.
REFERENCES:
[Atiyah 1994] M. Atiyah et al, Responses to "Theoretical Mathematics: Toward a cultural synthesis of mathematics and theoretical physics", by A. Jaffe and F. Quinn, Bull. Amer. Math. Soc. 30, 178-207.
[Babai 1992] L. Babai, Transparent Proofs, Focus: The Newsletter of the Mathematical Association of America, 12, n3, p1.
[Belenky et al 1986] Belenky, M. F., Clinchy, B. M., Goldberger, N. R. Tarule, J. M. Women's ways of knowing: The development of self, voice, and mind. New York: Basic Books.
[Cheek 1984] Cheek et al, Increasing Participation of Native Americans in Mathematics, Journal for Research in Mathematics Education, 15, pp107-113.
[Dieudonné 1973] Jean Dieudonné, "Should We Teach Modern Mathematics", American Scientist 77, 16-19.
[Fellows et al 1994] M. Fellows, A.H. Koblitz, N. Koblitz, Cultural Aspects of Mathematics Education Reform, Notices of the A.M.S., 41, 5-9.
[Gries 1995] David Gries, American Mathematical Monthly, .
[Henderson 1973] D. Henderson, A simplicial complex whose product with any ANR is a simplicial complex, General Topology, 3, 81-83.
[Henderson 1981] D. Henderson, Three Papers: Mathematics and Liberation, For the Learning of Mathematics, 1, 12-13.
[Henderson 1996a] D. Henderson, Experiencing Geometry on Plane and Sphere, Prentice-Hall.
[Henderson 1996b] D. Henderson, Differential Geometry: A Geometric Introduction, preliminary version, Prentice-Hall.
[Henderson 1996] D. Henderson, I Learn Mathematics From My Students -- Multiculturalism in Action, For the Learning of Mathematics, 16, 34-40.
[Hilbert 1932] D. Hilbert and S. Cohn-Vossen, Geometry and the Imagination, New York: Chelsea Publishing Co., translation copyright 1983.
[Lo et al 1996] Jane-Jane Lo, Kelly Gaddis and David Henderson, Learning Mathematics Through Personal Experiences: A Geometry Course in Action, For the Learning of Mathematics, 16, n.2.
[Thurston 1994] W. P. Thurston, On Proof and Progress in Mathematics, Bull. Amer. Math. Soc. 30, 161-177.
[Valverde 1984] Underachievement and Underrepresentation of Hispanics in Mathematics and Mathematics-Related Careers. Journal for Research in Mathematics Education, 15, 123-133.
QUESTIONS AND ANSWERS:
Question: How might we encourage our students to experience the passion that they have for mathematics -- that passion can include joy, fear, excitement, however you want to interpret the word.
Answer: I only know ways that I have tried and I do not think there is only one way. It seems to me that the most important thing is to expect it to happen and I do that and try to convey that I expect to see that from the students. The other thing that I started noticing when I started to have the students write is that a lot of the students' ideas and their passions are very fragile. There are a lot of students who do not dare to speak it in class, but they will write something. Maybe what they write isn't even directly what they really want to say but they will hint at it. So I have them write and then I respond to their writing and then they respond to my comments and so there is a dialogue that goes on in the writing. That I found to be the most powerful because then if someone just starts having an idea about something that is very tentative and very fragile, I can encourage it. I can encourage it easier in the written dialogue than it can happen in a class situation. That together with just expecting it and encouraging it whenever it happens and validating it is what I find that works for me.
Question: What is formalism good for?
Answer: One of the areas for which formalism is clearly good for is in computer science, in studying the algorithms and proofs in computer science. A huge area in computer science now is how to prove that a program does what you want it to do -- its a formal proof because that is what computers do, they are formal systems. The other place in which formalism is very powerful is in any situation like with groups. Studying groups is a good place to have axioms and build it up formally because there are a lot of different models for what a group is. So you can prove certain results that work for anything that satisfies these particular axioms and there are some examples and you can apply it across all the examples. There are a lot of areas like that in mathematics where that can happen. I think that Euclidean geometry is a particular bad place to apply formalism, because there is essentially only one model of Euclidean geometry and it is not a question of building these things up and then you can apply it somewhere else. Those areas where there actually are different models for a particular axiom system are areas where formal systems are powerful tools. I would also say that sometimes it is useful to use formal systems in areas where you actually have several different axiom systems -- which we do not usually allow in courses, we usually stick with only one. This happens in differential geometry, there are very complicated formalisms for differential geometry but there are a lot of different ones and to play the different ones off each other can be powerful. If you try to stick within one then you lose the geometric meaning, but if you go across them then the only thing which ties them together is the geometric meaning; and that is one way to get at what the geometric meaning is.
Question: How is the way that you teach geometry dependent on whether you have preservice teachers or have mathematics majors?
Answer: The main course that my book is based on has mathematics majors and preservice teachers (who are also mathematics majors) and mathematics education graduate students and then there are miscellaneous people (teachers, artists, or other members of the community who are interested in geometry) -- I do the same thing with all of them. Because most of the feedback is based on the writing that they do, they can respond in different ways, so I can have a different dialogue going on for different students depending on where they are coming from. I have not been able to do this as well in other subjects, but geometry is particularly suited to this because most people do not have much background in geometry, and that evens them out. Also geometry is more accessible concretely and through the intuition. I should tell you about one workshop that I did that was very powerful, one of the most powerful workshop that I have lead. It was in South Africa and they had gotten together a group of about 50 people which included elementary school teachers (many of whom had not finished secondary school, so had very weak mathematics backgrounds and virtually nothing in geometry), secondary school teachers, mathematics education people, and research mathematicians (including the chair of the mathematics department) -- the whole span. I had them work on the same problems in small homogeneous groups, the elementary school teachers worked with each other and the research mathematicians were working with each other. Of course, what they were doing in their small groups was very different, but I then had them report back to the whole group what they had found. Then the research mathematicians had to express it in a way that made sense for the elementary school teachers and the elementary school teachers were able to express what they had found and see that they had found things that the research mathematicians hadn't seen. It was very powerful -- I try to have as much diversity as possible in my class, but I have never had that kind of diversity before or since.
Question: Can or should geometry be taught separately from algebra?
Answer: When I teach geometry, I encourage students to use whatever they find useful to use and, if that is algebra, then -- great. Is that the kind of thing you were meaning by the question?
Question: Should geometry be taught before algebra?
Answer: My own feeling is that geometry comes before algebra -- much of algebra developed out of geometry, historically, and that should not be lost. Mostly I would like to turn the question around: Should algebra be taught separate from geometry? And there my answer would definitely be NO. Whenever I teach linear algebra, geometry is there a lot. And that is true historically -- much of linear algebra was developed to help with the description and study of the geometry of higher dimensional spaces.
Question: Could you elaborate on your comments that formalist mathematics is harmful and destructive? Is this perhaps a function of generation, because Leslie Lee seemed to be really able to identify with your comments because she too wanted to build a cabin?
Answer: Several people here came up to me afterwards and said they had had similar experiences, not just Leslie, and they were all about my age. Our generation was the generation before the baby boomers -- I think the baby boom generation (the ones that went into mathematics and the ones that didn't) was just different and maybe that is part of the reason why my generation is affected. Also I think that mathematics was most formal when we were in school. There seems to be something about our generation; and we are now in charge so we are more visible and that puts more pressure on us. Also, this formalism is only a product of this century, so for a long while the leaders in mathematics who were doing the formalism also knew that it was not all of mathematics -- like the quote I have form Hilbert and that was in the 1930's. But then somehow after the war in the late 1940's and 1950's, most of the mathematicians who had had direct personal contact with what was going before formalism came in had died off and so that may have affected our generation.
Question: How has the harmfullness or destructive nature of formalist mathematics has manifested itself beyond the urges to live in the woods?
Answer: Well, I do not think that the urge to live in the woods is not harmful! I think that it was mainly that when I was gong through school I never grew up socially and I was not effectively encouraged to grow up socially. I was a 'brain' (or 'science whiz') and that particularly substituted for growing up socially. Being immersed in formalism sort of fit that and encouraged that. That's part of it and I think that with the baby boom generation there were lots of external forces that started drawing people out, that was just not around in my days. And people that were more alive and more socially active then I was tended not to go into mathematics, or if they were in mathematics then they dropped out.
Question: Could you picture a world where mathematics majors and graduate students could be extroverted and emotional and deal with people and still do formalism? Do think there is something inherent in formalism?
Answer: I do not know for sure. I look at the graduate students now at Cornell and there are some of them who are like I was but there are also significant numbers of them now who are not I was and who are alive in lots of ways. They are surviving at Cornell and Cornell is basically still a very formal place, but they are also interested in teaching and we have mathematics graduate students who are taking the initiative to do some educational reform. So, yes, I think it makes sense that it fits in with formalism. I should say more about formalism, there is the part of mathematics, the foundations of mathematics, which is specifically studying formal systems and now in lots of places, in particular at Cornell, there is almost no distinction between it and a part of theoretical computer science. That is an active area of research where there is a lot of exciting things going on, I am not talking about that, that is a part of mathematics. I don't know ... Let me tell you one observation that I had that is less true now but it used to be true 10 years ago or so. Almost all of the women graduate students did not go into geometry or geometry related areas, but instead went into very formal areas. I talked with some of these women trying to catch why this was true, because in most of these cases these women were very active outside of mathematics -- they were involved in various social movements, political movements, feminist activities, and other such things going on 10-20 years ago. They expressed to me that they had to separate their lives -- when they were doing mathematics they had to separate it off from the rest of their life and it was easier to do that when they were doing formal mathematics. That seems to fit in with what I am saying. I see that happening less now, for both the men and the women.
Question: We are interested in the resurgence of geometry at Cornell. Can you give us a sense of the constellation of the geometry courses that are available at Cornell and the population that they are for?
Answer: As far as I can tell, Cornell has more geometry courses than anywhere else in the world. The undergraduate geometry courses are:
Answer: First of all, only one of those eight geometry courses deals extensively with axioms and formalism and that is the hyperbolic and projective geometry course and it does not deal with axiom systems totally. But I agree with you that the important thing is, How? I think that as long as you can start with something that is a concrete contextual situation and use that to start building the area of mathematics, then it can be done with any subject; and I think all parts of mathematics have such grounding. Geometry is easier to get into because there is not a tradition of having a long string of prerequisites, this linear sequence of courses and so on, in geometry, so it is easier to jump in different places. But I have done it in an abstract algebra course where, because it was me doing it, I started with symmetries of polyhedra and ended up with Galois theory. The main thing that I try to do is to have a concrete contextual situation where the students are able to experience the meaning of what is going on and so, in that way, it can be more constructive. The other part of it (what I mentioned in my answer to the first question) is eliciting from the students their ideas and their thinking, so that if there is some way that their imagination and intuition can latch on, then I just start pulling out of them what the ideas are and guiding them in the right ways and giving suggestions and writing challenging problems.
Question: Are the other geometry courses at Cornell also not so prerequisite bound?
Answer: None of the geometry courses have any of the other geometry courses as prerequisites.
Question: It is perhaps true that the phenomena is not so much a growing interest in geometry but rather a growing interest in something mathematical that can engage the students in less formal ways?
Answer: I think you may well be right and that geometry is just a particularly convenient or easier area to that in. There is a debate going on in our department as to whether or not students who want to take my courses are wanting to take it because of me or is something about he course. It is hard to gauge that but it seems to be what you say, that they are really looking for a different way in which to engage with mathematics -- something that is less formal, that is not just lecture and exams.
Question: Have you had any resistance to you approach from colleagues, other mathematicians , or from students?
Answer: Yes, all of the above. The geometry courses, fortunately, are not required for mathematics majors except for perspective teachers, so student who do want to do the course just do not take the course and I have had not complaints form the perspective teachers. But, I and some graduate students are trying to put some of these ideas into the calculus now; we have had a few cases of students getting up and stomping out of the room when we introduce small group work and other activities to engage the students. It seems that this happens because they are not there to learn calculus, they are there because they are required to take calculus and they want to do it with an minimum amount of effort to get their passing grade so they can go on to do whatever it is that it is required for. The students who want to learn calculus, they seem to love the new approaches. Sometimes I am teaching one section and there are other sections of the calculus course taught is traditional ways. I am required by the department for my students to have the same assignments and exams, so I am giving them stuff in addition to that. I tell them this right up front and I tell them why -- I think they will learn it better and with more understanding, but they will have every week more assignments than the other sections. Typically, what happens when I announce that in the beginning, a few students drop out and other students hear about it and come in. At the end of the semester the students report that the extra stuff is the best part of the course.