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.
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.
David Hilbert[SE: Hilbert, p. iii]
These words were written in 1934 by the "father of Formalism" David Hilbert in the Preface to Geometry and the Imagination by Hilbert and S. Cohn-Vossen. Hilbert has emphasized the point I wish to make in this book:
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 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 understanding. Without understanding we will never be satisfied with understanding we want to expand that understanding and to communicate it to others.
Many formal aspects of mathematics have now been mechanized and this mechanization is widely available on personal computers or even handheld calculators, but the experience of meaning in mathematics is still a human enterprise that is necessary for creative work.
In this book I invite the reader to explore the basic ideas of geometry from a more mature standpoint. I will suggest some of the deeper meanings, larger contexts, and interrelations of the ideas. I am interested in conveying a different approach to mathematics, stimulating the reader 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, or 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 mathematics in the classroom.
Proof as Convincing Argument
That Answers Why?
Much of our view of the nature of mathematics is intertwined with our notion of what is a proof. This is often particularly true with geometry, which has traditionally been taught in high school in the context of "two- column" proofs. The course materials in this book are based on a view of proof as a convincing argument that answers a why-question.
Why is 3 x 2 = 2 x 3 ? To say, "It follows from the Commutative Law" does not answer the why-question. But most people will be convinced by, "I can count three 2's and then two 3's and see that they are both equal to the same six." OK, now why is 2,657,873 x 92,564 = 92,564 x 2,657,873? We cannot count this it is too large. But is there a way to see 3 x 2 = 2 x 3 without counting? Yes.
Figure 0.1 Why is 3 x 2 = 2 x 3 ?
Most people will not have trouble extending this proof to include 2,657,873 x 92,564 = 92,564 x 2,657,873 or the more general n x m = m x n. Note that for the above to make sense I must have a meaning for 3 x 2 and a meaning for 2 x 3 and these meanings must be different. So naturally I have the question: "Why (or in what sense) are these meanings related?" A proof should help me experience relationships between the meanings. In my experience, to perform the formal mathematical induction proof starting from Peano's Axioms does not answer anyone's why-question unless it is such a question as: "Why does the Commutative Law follow from Peano's Axioms?" Most people (other than logicians) have little interest in that question.
As further evidence toward this conclusion, you have probably had the experience of reading a proof and following each step logically but still not being satisfied because the proof did not lead you to experience the answer to your why-question. In fact most proofs in the literature are not written out in such a way that it is possible to follow each step in a logical formal way. Even if they were so written, most proofs would be too long and complicated for a person to check each step. Furthermore, even among mathematics researchers, a formal logical proof that they can follow step-by-step is not always satisfying. For example, my shortest research paper ["A simplicial complex whose product with any ANR is a simplicial complex," General Topology 3 (1973), pp. 8183] 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 sort: "Why is it true?" "Where did it come from?" "How did you see it?" They accepted the proof logically but were not satisfied.
Let us look at another example the Vertical Angle Theorem: If l and l' are straight lines, then the angle a is congruent to the angle b.
Figure 0.2 Vertical Angle Theorem
As we will see in Problems 3.1 and 3.2 of this book, the proof of this theorem that is most convincing to someone depends on the meanings that person has of "angle" and "congruence." Some years ago, after I had been teaching this geometry course for over ten years, several proofs that were convincing to me were presented by students in the class. But one student found the proofs not so convincing and offered a very straightforward simple proof of her own. My first reaction was that her argument could not possibly be a proof it was too simple and did not involve everything in the standard proof. But she persisted patiently for several days and my meanings for angle congruence deepened. Now her proof is much more convincing to me than the standard one. I hope you will have similar experiences while working through this book.
You may ask, "But, at least in plane geometry, isn't an angle an angle? Don't we all agree on what an angle is?" Well, yes and no. Consider this acute angle:
Figure 0.3 Where is the angle?
The angle is somehow at the corner, yet it is difficult to express this formally. As evidence, I looked in all the plane geometry books in the university library and found their definitions for "angle." I found nine different definitions! Each expressed a different meaning or aspect of "angle" and thus, each would potentially lead to a different proof of the Vertical Angle Theorem. We will see this more when we discuss Problems 3.1 and 3.2.
Sometimes we have legitimate why-questions even with respect to statements traditionally accepted as axioms. The Commutative Law above is one possible example. Another one is Side-Angle-Side (or SAS): If two triangles have two sides and the included angle of one congruent to two sides and the included angle of the other, then the triangles are congruent. You can find SAS listed in some geometry textbooks as an axiom to be assumed; in others it is listed as a theorem to be proved and in still others as a definition of the congruency of two triangles. But clearly one can ask, "Why is SAS true in the plane?" This is especially true because SAS is false for (geodesic) triangles on the sphere. So one can naturally ask, "Why is SAS true on the plane but not on the sphere?"
I have been teaching a geometry course based on the material in this book for a long time now (since 1974). One might expect that I have seen everything. But every year, about one-third of the students will show me a meaning or way of looking at the geometry that I have never thought of before and thus my own meaning and experience of geometry deepen. Looking back, I notice that these students who have shown me something new are mostly persons whose cultural backgrounds or race or gender are different from mine; and this is true even though most of the students in the class and I are white males. (For details of this data and further discussion see my article "I Learn Mathematics From My Students Multiculturalism in Action", For the Learning of Mathematics 16 (1996), pp. 3440, or on my webpage.)
You should check this out in your own experience. We should listen carefully to meanings and proofs expressed by all persons. We should also be more critical of many of the standard histories of mathematics and mathematicians, which have a decidedly Eurocentric emphasis.
As we personally experience Conclusions 1, 2, 3 above, we are led to the following conclusion.
Conclusion 4: If I experience 1, 2, and 3, then other persons (for example, my students) are also likely to have similar experiences.