GEOMETRIC PROOF AND KNOWLEDGE WITHOUT AXIOMS At All Levels
David W. Henderson
Cornell University, Ithaca, NY, USA
1. What is geometric intuition? How does it affect and effect our understanding?
2. What role does our imagination have in increasing our knowledge and understanding?
3. How can we be rigorous without axioms?
4. Are precise definitions always desirable in geometry? Are they always possible?
5. Why is spherical geometry (the geometry of our planet) missing from almost all textbooks?
6. What use are geometric axioms? What power do they give us?
7. For what are physical models needed? What about pictures, diagrams, and mental images?
8. How can we have proofs in geometry without axioms? Why do we want to?
9. Can knowledge gained without axioms be superior to knowledge from formal axiom systems?
10. When are we satisfied with our understanding? When are we certain?
11. Can we imagine completed infinity? Does it matter?
12. How can we use geometry to understand the real numbers?
These are questions that I think we should explore and should be a part of any thinking about the teaching of geometry. I have explored them but do not know any full answers. I am a mathematician with research interests in Geometry and Geometric Topology. For the past thirty years I have been teaching geometry mostly to pre-service and in-service secondary school teachers and to other university students, and I have also taught geometry at all levels from primary school students to doctoral students. In the process of this teaching I have learned to encourage my students to write and talk about their understandings of geometry, to listen to their ways of understanding, which in turn has helped me to reflect on my own understanding of geometry. The ideas that follow are the result of this listening and reflecting and my other experiences with geometry.
The inner core of my teaching is contained in the following quote from Tenzin Gyatso, The Fourteenth Dalai Lama (London, 1984):
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.
To me geometric understanding (at all levels from first grade to the university) is based on one's own deep experience of the meanings involved. Only after this experience do formal presentations and study make sense. I believe that axiomatization of geometry is a powerful tool in some situations but that in most situations it is an obstacle to deeper understandings and creativity.
I believe that mathematics is a natural and deep part of human experience and that experiences of meaning in mathematics are accessible to everyone. Much of mathematics is not accessible through formal approaches except to those with specialized learning. However, through the use of non-formal experience and geometric imagery, many levels of meaning in mathematics can be opened up in a way that most human beings can experience and find intellectually challenging and stimulating.
Formalism contains the power of the meaning but not the meaning. It is necessary to bring the power back to the meaning.
A formal proof as we normally conceive of it is not the goal of mathematics it is a tool a means to an end. The goal is experiential understanding. Without understanding we will never be satisfied with understanding we want to expand that understanding and to communicate it to others. This communication naturally involves convincing arguments that answer Why? and which help bring the hearers to their own deep experiences of the meanings. It is these communications that I believe we should call "proofs" often the most convincing communications are not formal. Recent research involving interviews with students in classrooms has shown that students at all levels are capable of deep experiences and thoughtful communication of mathematical ideas.
Many formal aspects of mathematics have now been mechanized and this mechanization is widely available on personal computers or even hand-held calculators, but the experience of meaning in mathematics is still a human enterprise that is necessary for creative work.
I am interested in conveying a different approach to mathematics, stimulating people to take a broader and deeper view of mathematics, and to experience for her- or himself a sense of mathematizing. Through an active participation with these ideas, including exploring and writing about them, people can gain a broader context and experience. This active participation is vital for anyone who wishes to understand mathematics at a deeper level as well as vital for anyone wishing to understand something in their experience through the vehicle of mathematics.
This is particularly true for teachers or prospective teachers who are approaching related topics in the school curriculum. All too often we convey to students that mathematics is a closed system, with a single answer or approach to every problem, and often without a larger context. I believe that even where there are strict curricular constraints, there is room to change the meaning and the experience of the mathematics in the classroom.
The geometric content can be presented through a sequence of inviting and challenging problems which lead the students to deep experiences of the ideas involved. Then as an intimate part of the learning experience the students should be encouraged to communicate, orally and in writing, what they have experienced in a way that is convincing to others.
I follow David Hilbert who said 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. ... 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."
I have just completed a new textbook (to be published August 1995) based on my 20 years of teaching geometry. This text is problems-based and involves the students (and instructors) in active exploration of ideas in plane and spherical geometry. The preliminary version of the text has been in universities in the USA, Canada, Palestine, South Africa, and Portugal, primarily in courses for future secondary school teachers. The text is non-axiomatic but stresses proofs constructed by the students: "Proof" is defined as "a convincing argument that answers Why?"
I am now completing the first draft of a second textbook (based on the same beliefs stated above) that is a "Geometric Foundation for Differential Geometry". This second text is intended for mathematics majors and beginning graduate students who wish an introduction to differential geometry that starts with experiences of basic geometric meanings and then ends with analytic formalizations. I do not believe that it is possible or desirable to do this starting from axioms.
The formalisms of differential geometry are considered by many to be one of the most complicated and inaccessible of all the formal systems in mathematics. It is probably fair to say that most mathematicians do not feel comfortable with their understanding of differential geometry. In addition, there is little agreement about which formalisms to use and about how to describe these formalisms with the result that the starting definitions, notations and analytic descriptions vary widely from textbook to textbook. What all of these different approaches have in common are underlying geometric intuitions of the basic notions such as straightness (geodesic), smooth, tangent, curvature, and parallel transport.
David W. Henderson
Department of Mathematics
Ithaca, NY 14853-7901, USA