Plato's statement that "God eternally geometrizes" is the theme of Constance Reid's book A Long Way From Euclid. Euclid is unquestionably our great model for logical thought. The powerful words of the Declaration of Independence, "We hold these Truths to be self-evident, that all Men are created equal..." are the axioms for an argument modeled on Euclid's Elements.
Given that this book is written for a general audience, the reader may wonder how much mathematics there is in it. Quite a bit actually. The first half of the book is primarily on Euclid's Elements, with excursions into modern mathematics. The journey begins with the Pythagorean Theorem and the proof that the square root of 2 is irrational. The second chapter has an interesting discussion on what it means to make a good selection of axioms. In the next few chapters we are introduced to primes and square numbers and their "wholly unexpected" relationships. There is an explanation of why the number of primes is infinite and how every composite number can be expressed uniquely as a product of primes (the Fundamental Theorem of Arithmetic). After meeting the Two, Three, and Four Square Theorems we reach Gauss's famous Law of Quadratic Reciprocity. This is followed by a discussion of Eudoxus' Theory of Proportions (book five of the Elements) and how this led to the 19th century description of irrational numbers via Dedekind cuts. We get a lot of information on doubling the cube and trisecting the angle, and a little on squaring the circle. We also meet Descartes and his book La Géométrie, in which he describes how to express a geometric figure in algebraic terms, thus inventing analytic geometry.
Calculus students will benefit from reading Chapter 6. Reid correctly emphasizes that it was Archimedes' method of exhaustion for computing areas and volumes of curves and solids that set the ground work for calculus. After Archimedes many mathematicians found ingenious ways of computing areas and volumes and equations of tangent lines. Newton and Leibniz are credited with inventing calculus because they proved independently that these two seemingly disparate operations are the opposites of each other (the Fundamental Theorem of Calculus).
Reid's writing style is quite entertaining. For example, the struggle by 16th century mathematicians to understand complex numbers is portrayed as follows:
We can compare the negative numbers to things like debts and temperatures below zero and the years before the birth of Christ, but the number i we can compare to nothing in everyday life. It was for this reason that mathematicians, although they went right along using i to solve equations, felt a little guilty about what they were doing. God, they felt, had made the whole numbers. If He had wanted man to have them, He would have made negative numbers and given them square roots!
The book has many nice quotes. Duality in projective geometry is summarized as "an enchanted realm where thought is double and flows in parallel streams," a beautiful and surprisingly accurate metaphor. Euclidean geometry is the "geometry of God's mind" (Reid) and can be described as "nothing, intricately drawn nowhere" (Edna St. Vincent Millay). And as to Euclid's fifth postulate, "whatever else this postulate may be, self-evident it is not" (J. L. Coolidge).
There are unfortunately many errors in the book. For example, on page 21, Reid writes "Proposition 1 of Book VII, for instance, gives us the standard method — still known as Euclid's algorithm — for finding the greatest common measure." This should be Proposition 12. Proposition 1 gives the division algorithm. On the same page "...Book X, the third and final book on numbers," should be Book IX.
In more than one place Reid writes "Their geometry was based on an axiom which stated in essence that parallel lines never meet" (page 60) and "the fifth postulate, which makes a statement very roughly equivalent to our common statement that 'parallel lines never meet'" (page 152). The intention of these statements is to convey to the reader that there is a problem with the fifth postulate, but they are incorrect. Euclid defines parallel lines in Definition 23 of Book 1 as "Parallel straight lines are straight lines which, being in the same plane and being produced indefinitely in both directions, do not meet one another in either direction." The fifth postulate, in all its different forms, does not say "parallel lines never meet." What it says in the original form is "if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles." The best known equivalent version states "Given a line and a point not on the line, it is possible to draw exactly one line through the given point parallel to the given line."
On page 149, we have "...let us take a few of the truths which the Egyptians arrived at from experience and which the Greeks deduced in logical fashion from their axioms and postulates. A straight line is the shortest distance between two points." This is Euclid's first postulate "To draw a straight line from any point to any point." Although it does not have the word "shortest" or "unique" in it, Euclid assumes it does and uses it in other propositions.
In the second half of the book we meet the different geometries that emerged during and after the Renaissance such as projective, elliptic, and hyperbolic geometries. Chapter 11 on four-dimensional geometry and Chapter 12 on topology are explained in ways that even a child can understand. Elementary school teachers will find nice ideas for mathematical experiments. Chapter 13 on the definition of a geometry has a nice explanation on groups suitable for students in an algebra class. Chapter 14 on countable and uncountable sets and Chapter 17 on truth tables are recommended for students taking what is typically called a discrete math or foundations of math course.
There is an error on page 157 in the statement "The true surface of hyperbolic geometry — not just a portion but an entire surface — is what is called the pseudosphere, a world of two unending trumpets." The pseudosphere is one of several models for hyperbolic geometry. It is the surface of revolution obtained by rotating the tractix about the y-axis. This surface has constant negative curvature contrasting it with the sphere which has constant positive curvature. Hence it is called the pseudosphere.
Our journey arrives in the twentieth century with Hilbert's corrections to Euclid's axioms. Surprising as it may be to some, Euclid made several assumptions that were not explicitly stated in his axioms. Hilbert starts his Grundlagen der Geometrie with three distinct undefined entities: points, lines, and planes. The thrust of his formulation is that one does not have to attribute a physical interpretation for them, but this does not come across in what Reid writes. Hilbert has been quoted as saying "One must be able to say at all times — instead of points, lines, and planes — tables, chairs, and beer mugs."
In a good axiomatic system, the axioms must be independent (no one of them is a logical consequence of the other), the body of theorems obtained from the axioms must be consistent (no two theorems deducible from them can be mutually contradictory), and complete (every theorem of the system is deducible from the axioms) . After Hilbert modified Euclid's axioms, he also proved the independence of the axioms and the consistency of the system. However, his proof of consistency relied on the consistency of arithmetic, whose consistency was an open question.
A good axiom system must also be decidable. This means there exists a method for solving all possible problems in the system. The book ends with a discussion of Tarski's theorem that plane Euclidean geometry is decidable and Gödel's famous theorems that demolished the idea that "absolute consistency of a mathematical system could be established within that system" and that "for certain classes of problems, there can be no general method of solving all the problems in the class." Incidentally, Gödel's first and second incompleteness theorems are two of the most misunderstood results in mathematics, so the reader who wants more information should read, for example, Gödel, Escher, Bach: An Eternal Golden Braid by Douglas Hofstadter.
It is worth noting that the path from ancient Greek geometry to modern geometry went east before returning to the west. While geometry originated in the west, algebra flourished in the east and Descartes' analytic geometry was born from the mingling of the two. Without algebra we wouldn't have zero and the decimal number system nor the concept of a variable. So it is a pity that in Reid's book the origins of algebra merit nothing more than a footnote on page 67. In their book What is Mathematics, Courant and Robbins put it this way: "For almost two thousand years the weight of Greek geometrical tradition retarded the inevitable evolution of the number concept and of algebraic manipulation, which later formed the basis of modern science."
It may also help to keep in mind that the book was written in 1963 and recently republished by Dover. The computer is called the "great electronic computer" and viewed as "a mere drudge of the Queen of Mathematics." It is somewhat better than that now, perhaps more like a knight in the Queen's court. Just as the telescope enables us see further away than our eyes can, and the microscope enables us see further in, computers allow us to visualize objects that previously existed only in the imagination of a few. Take for example, the ubiquitous graphing tools, with which we can see two and three dimensional views of complicated functions. It is as if we have another eye with which to view mathematics.
Some of the problems listed as unsolved have been settled. For example, Hilbert's tenth problem (page 279) on finding a general method of solution for all Diophantine equations was settled by Matiyasevich in 1970. A theorem he proved implies that no general method can be found. His work was based on the work of Julia Robinson, Reid's sister. Fermat's last theorem (page 7) was solved by Andrew Wiles in 1994. In 2002 Agrawal, Kayal, and Saxena gave a polynomial time deterministic algorithm to test whether or not a number is prime (page 275). The largest known prime is now considerably larger than the one listed on page 275 and is available at the Prime Pages listed in the references.
Hilbert's tenth problem (page 279) on finding a general method of solution for all Diophantine equations was settled by Matiyasevich in 1970. A theorem he proved implies that no general method can be found. His work was based on the work of Julia Robinson, Reid's sister.
Finally, Reid states on page 28 that every domain of mathematics is ruled by the axiomatic method. This point of view may have its supporters, but not everyone agrees. Here is a contrasting opinion by Courant and Robbins:
Constructive thinking, guided by intuition, is the true source of mathematical dynamics. Although the axiomatic form is an ideal, it is a dangerous fallacy to believe that axiomatics constitutes the essence of mathematics. The constructive intuition of the mathematician brings to mathematics a non-deductive and irrational element which makes it comparable to music and art.
Overall this book is worth reading and is accessible to college and even high school students. It would have been nice to see a list of references and some of the mathematical statements are not as precise as they should be. It is difficult to explain modern mathematics in lay language without heavy use of symbols. But then again, perhaps that is a limitation we may be imposing on ourselves.
- What is Mathematics? An elementary approach to ideas and methods by Richard Courant and Herbert Robbins, revised by Ian Stewart, Oxford University Press, 1996.
- History of Mathematics: An introduction by David M. Burton, Wm. C. Brown Publishers, 1991.
- A History of Mathematics: An introduction by Victor J. Katz, Addison Wesley, 1998.
- From Five Fingers to Infinity: A journey through the history of mathematics edited by Frank J. Swetz, Open Court.
- Gödel, Escher, Bach: An Eternal Golden Braid, by Douglas Hofstadter, Basic Books.
- The Prime Pages, the Great Internet Mersenne Prime Search and Prime Pursuit: Constructing an efficient prime number detector by Ivars Peterson
- Euclid's Elements
- The Pseudosphere from the Geometry Center at the University of Minnesota
Sandra Kingan (firstname.lastname@example.org
) is assistant professor of mathematics at the Pennsylvania State University, Capital College.