The musings of a previously unemployed Jewish Freemason. I write about the job search, about Judaism, and about Freemasonry.

Friday, October 26, 2012

Elementary Versus Simple

Euclid calls his book The Elements (Στοιχεῖα) because it covers what he felt to be the elementary ideas in geometry. Paul uses the term (in the Textus Receptus) in his Epistle to the Hebrews:
"[Jesus was c]alled of God an high priest after the order of Melchisedec. Of whom we have many things to say, and hard to be uttered, seeing ye are dull of hearing. For though by this time you ought to be teachers, you have need again for someone to teach you the elementary principles (στοιχεῖα) of the oracles of God, and you have come to need milk and not solid food. Anyone who lives on milk, being still an infant, is not acquainted with the teaching about righteousness. But solid food is for the mature, who by constant use have trained themselves to distinguish good from evil." [Hebrews 5: 10-14].

So it could be claimed that Euclid regarded his book as the milk, and not solid food. This might not be the most productive attitude to take towards The Elements without a few caveats, so I will make them here.

Mathematics is a difficult subject; indeed it might be the most difficult mental discipline yet discovered. So the "milk" is sufficiently difficult to grasp that it might be counter-productive to regard it as food for infants except insofar as a baby's teeth don't grow in until she has subsided on milk for some time. In this sense, Euclid is trying to help the reader develop teeth, and that is not an easy thing to do. Much of The Elements is very difficult, and requires patience, persistence and deliberation in order to master.

The zenith of Greek mathematics was in spatial geometry, the use of algebraic curves to solve various equations, and in number theory. Much of this work is considerably more difficult than anything found in The Elements (Apollonius of Perga's Conic Sections comes to mind here). But more important than the level of difficulty is the order of presentation. Euclid teaches, in this book, what he feels should come first in a mathematics education. He starts with ten axioms (he would say five common notions and five postulates, but I will cover the difference between these terms in another post). From these, he builds his method of constructions, which he will use to prove various results.

Modern readers of Euclid can be discouraged by the fact that many of the results in The Elements are not easy. They assume that what comes first pedagogically should be easier than that which builds upon that which comes first, and while that is often true, there is no reason why that should always be so.

This was clear to Nobel Prize winning physicist Richard Feynman, who in his two-year course in basic physics given at Cal Tech, included a lecture with an elementary proof that Newton's inverse square law of gravitational attraction implied that bodies in space orbit larger bodies with an elliptical orbit with the larger body at one focus. In writing the lecture, Feynman only used mathematics that Newton and Kepler would have had access to (excluding calculus). The result uses some very subtle results from the theory of conic sections (which were known to Apollonius of Perga, as well as Newton and Kepler), and uses the techniques of Greek geometry, but it remains a very difficult proof. He warned his listeners that elementary does not mean simple. And I do likewise.

So, what did Euclid consider to be elementary? The first book of The Elements is devoted to two important tasks: creating a system of definitions and axioms sufficient to the task of creating all geometry, and proving the Pythagorean Theorem in the famous 47th Proposition. The second book is devoted to algebraic results that use geometrical constructions. The third book is devoted to the geometry of circles. The fourth book covers the construction of various equilateral polygons. The fifth book introduces the concepts of magnitudes and proportions. The sixth book explores similar (or proportionate) geometrical figures. The seventh book defines the number one and from there all whole numbers, including odd and even numbers, prime and composite numbers, addition, subtraction, multiplication, division, squares and cubes of numbers, greatest common divisors and least common multiples, what modern mathematicians call the theory of numbers. The next two books cover more number theory. The tenth book covers the theory of irrational numbers, or what Euclid called incommensurate magnitudes. The eleventh book introduces solid (or three-dimensional) geometry and the theory of solids. The twelfth book covers how to measure volumes. The thirteenth book introduces the five Platonic solids, and shows that there are only five such solids.

When one asks a child what is the most basic fact in mathematics, they are likely to tell you that 1 + 1 = 2. But Euclid doesn't cover that until the seventh book. To Euclid, lines, angles, circles, triangles and other planar figures are more elementary than numbers. To Euclid, the Pythagorean Theorem is the first deep result in mathematics (although the pons asinorum precedes it), and Euclid streamlines his presentation to present the Pythagorean Theorem (and the pons asinorum) as succinctly as possible and yet have the framework created hold up the rest of geometry.

The pons asinorum (Latin for "bridge of asses"), the proof that in an isosceles triangle, the two angles at the base of the triangle are congruent, is the first theorem in The Elements (Book I, Proposition 5) not used as a stepping stone to something bigger (such a proposition is called a lemma in modern parlance). Incidentally, the term pons asinorum has come to mean any test that separates the less intelligent from the more intelligent. The assumption was that if a student hit the wall with the pons asinorum, chances were he was unsuited for higher learning. In that sense, the pons asinorum is elementary, but not simple.

No comments:

Post a Comment