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 for Mathematics Education*

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)