Consider a point p and a line l in some plane, with p not on l:

How many lines are there in the plane that pass through point p and that are parallel to line l?
It seems clear, by what we mean by “point”, “line”, and “plane”, that there is just one such line. This assertion is logically equivalent to Euclid's 5th, or parallel, postulate (in the context of his other postulates).
     In fact, this was seen as so obvious by everyone, mathematicians included, that for two thousand years mathematicians attempted to prove it. After all, since it “is true”, it should be provable. One approach, taken independently by Carl F. Gauss, Janos Bolyai, and Nickolai Lobachevsky, was to assume the negation of the “obvious truth” and attempt to arrive at a logical contradiction. But they did not arrive at a contradiction. Instead, the logical consequences went on and on. They included “theorems”, all provable from the assumptions. In fact, they discovered what Bolyai called a “strange new universe”, and what we today call non-Euclidean geometry.
     The strange thing is that the real, physical universe turns out to be non-Euclidean. Einstein's special theory of relativity uses a geometry developed by H. Minkowski, and his general theory, a geometry of Gauss and G. F. B. Riemann.[1]
     Mathematics is about an imaginative universe—a world of ideas, but the imagination is constrained by logic. The basic idea behind proof in mathematics is that everything is exactly what its definition says it is. A proof that something has a property is a demonstration that the property follows logically from the definition alone. On the intuitive level, definitions serve to lead our imagination. In a formal proof, however, we are not allowed to use attributes of our imaginative ideas that don't follow logically solely from definitions and axioms relating undefined terms. This view of proof, articulated by David Hilbert, is accepted today by the mathematical community, and is the basis for research mathematics and graduate and upper-level undergraduate mathematics courses.
     There is also a very satisfying aspect of “proof” that comes from our ability to picture situations—and to draw inferences from the pictures. In calculus, many, but not all, theorems have satisfactory picture proofs. Picture proofs are satisfying, because they enable us to see [2] the truth of the theorem. Rigorous proof in the sense of Hilbert has an advantage, not shared by picture proofs, that proof outlines are suggested by the very language in which theorems are expressed. Thus both picture proofs and rigorous proofs have advantages. Calculus is best seen using both types of proof.
     A picture is an example of a situation covered by a theorem. The theorem, of course, is not true about the actual picture—the molecules of ink stain on the molecules of paper. It is true about the idealization that we intuit from the picture. When we see that the proof of a theorem follows from a picture, we see that the picture is in some sense completely general—that we can't draw another picture for which the theorem is false. To those mathematicians that are satisfied only by rigorous mathematics, such a situation would merely represent proof by lack of imagination: we can't imagine any situation essentially different from the situation represented by the picture, and we conclude that because we can't imagine it that it doesn't exist. Non-Euclidean geometry, on which Einstein's theory of relativity is based, is an example of a situation where possibilities not imagined are, nevertheless, not logically excluded. Gauss [3] kept his investigations in this area secret for years, because he wanted to avoid controversy, and because he thought that “the Boeotians” [4] would not  understand. It is surely a profound thing that the universe, while it may not be picturable to us, is nevertheless logical, and that following the logical but unpicturable has unlocked deep truths about the universe. In mathematics, unpicturable but logical results are sometimes called counterintuitive.
    Rigorous and picture proofs are both necessary to a good course in calculus and both within grasp [5]. To insist on a rigorous proof, where a picture has made everything transparent, is deadening. Mathematicians with refined intuition know that they could, if pressed, supply such a proof—and it therefore becomes unnecessary to actually do it. We therefore focus the method of outlining proofs on those theorems for which there is no satisfactory picture proof.
     Our purpose is not to cover all theorems of calculus, but to do enough to enable a student to “catch on” to the method [6]. A detailed formal exposition of the method can be found in Introduction to Proof in Abstract Mathematics and Deductive Mathematics—An Introduction to Proof and Discovery. Second Edition.
     In calculus texts, “examples” are given that illustrate computational techniques, the use of certain ideas, or the solution of certain problems. In this text supplement, we give “examples” that illustrate the basic features of the method of discovering a proof outline.

FOOTNOTES

1. See Marvin Greenberg, Euclidean and Non-Euclidean Geometries, W. H. Freeman, San Fransisco, 1980 (return to text)
2. The word “theorem” literally means “object of a vision”. (return to text)
3. One logical consequence of there being more than one line through p parallel to l is that the sum of the number of degrees in the angles of a triangle is less than 180. Gauss made measurements of the angles formed by three prominent points, but his measurements were inconclusive. We know today that the difference between the Euclidean 180 and that predicted for his triangle by relativity would be too small to be picked up by his instruments. (return to text)
4. A term of derision. Today's “Boetians” have derisive terms of their own—claiming courses that depend on rigorous proof have “rigor-mortis”. (return to text)
5. The fundamentals of discovering proof outlines can be picked up in a relatively very short time—compared to the years of study of descriptive mathematics prerequisite to calculus. (return to text)
6. In the American Math. Monthly 102, May 1995, page 401, Charles Wells states: “A colleague of mine in computer science who majored in mathematics as an undergraduate has described how as a student he suddenly caught on that he could do at least B work in most math courses by merely rewriting the definitions of the terms involved in the questions and making a few obvious deductions.” The reason this worked for Prof. Wells's colleague is that definitions are basic for deductive mathematics—and upper-level math courses are taught from a deductive perspective. Our method for discovering proof steps is merely a systematic way of making the deductions—a way that can be taught. (return to text)