Section 22–1 Addition and multiplication

(Mathematical proof / Mathematical philosophy / Mathematical definitions)

In this section, we can understand addition and multiplication from the perspectives of mathematical proof, mathematical philosophy, and mathematical definitions.

1. Mathematical proof:

“we may find the Pythagorean theorem quite interesting … an interesting fact, a curiously simple thing, which may be appreciated without discussing the question of how to prove it… (Feynman et al., 1963, section 22–1 Addition and multiplication).”

Feynman explains that this chapter will not be developed from a mathematician’s point of view because mathematicians are mainly interested in how various mathematical facts are demonstrated. In addition, they are interested in how many assumptions are absolutely required and what is not required, but they are not interested in the result of what they prove. However, in the words of a mathematician, “[p]roofs are for the mathematician what experimental procedures are for the experimental scientist: in studying them one learns of new ideas, new concepts, new strategies... (Rav, 1999, p. 20).” Interestingly, Paul Erdös knew 37 different beautiful proofs of the Pythagorean theorem when he was seventeen. In The Pythagorean Proposition, Loomis (1986) compiled 367 proofs of the Pythagorean theorem.

Feynman says that we may find the Pythagorean theorem is an interesting fact, which may be appreciated without discussing the question of how to prove it. In his autobiography, Einstein (1949) says that “I remember that an uncle told me the Pythagorean theorem before the holy geometry booklet had come into my hands. After much effort I succeeded in ‘proving’ this theorem on the basis of the similarity of triangles; in doing so it seemed to me ‘evident’ that the relations of the sides of the right-angled triangles would have to be completely determined by one of the acute angles (p. 9).” In A Beautiful Question, Wilczek (2015) reconstructs Einstein’s proof by showing how two similar triangles can be added to form another similar triangle. Every physics student ought to have the pleasure of understanding this beautiful proof.

2. Mathematical philosophy:

“But we are not going in that direction, the direction of mathematical philosophy …   from the assumption that we know what integers are and we know how to count (Feynman et al., 1963, section 22–1 Addition and multiplication).”

Feynman prefers not to include mathematical philosophy from the assumption that we know what integers are and how to count. It is worthwhile to discuss mathematical philosophy that is related to the definition of number. For example, Frege argues that numbers are objects and defines numbers as extensions of concepts. In Gouvêa’s (2008) words, “it is no longer that easy to decide what counts as a ‘number.’ The objects from the original sequence of ‘integer, rational, real, and complex’ are certainly numbers, but so are the p-adics. The quaternions are rarely referred to as ‘numbers,’ on the other hand, though they can be used to coordinatize certain mathematical notions (p. 82).” Mathematicians may not agree with the definition of number, but we can simply define a number as a means for counting instead of a set of things that is more abstract.

It appears that we know what integers are, what zero is, and what it means to increase a number by one unit. As a suggestion, one may mention that a number divided by zero threatened the foundation of mathematics and it remains undefined. Furthermore, Seife (2000) writes that “[z]ero clashed with one of the central tenets of Western philosophy… (p. 25)” because the Greek universe rejects the concept of zero. On the other hand, one may discuss the origin of zero in early Hindu and Buddhist philosophical discourses about the concept of “emptiness” or “void.” Historically, the doctrine of “sunyata” (or void) is one of the profound contributions of philosophy from India. However, physicists may not agree with the definition of vacuum or void.

3. Mathematical definitions:

“Now as a consequence of these definitions it can be easily shown that all of the following relationships are true… (Feynman et al., 1963, section 22–1 Addition and multiplication).”

According to Feynman, after we have defined addition, then we can start with nothing and add a to it, b times in succession, and call the result multiplication of integers (i.e., b ´ a). In addition, we can have a succession of multiplications: if we start with 1 and multiply by a, b times in succession (i.e., a^b). Mathematicians may not like Feynman’s approach and prefer to define the concept of group, ring, and field. They can be informally defined as follows: A group is a set in which we can perform addition or multiplication operation having some properties (i.e., closure, identity, inverse, & associativity). A ring is a group under addition and satisfies many properties of a group for multiplication. A field is a group under both addition and multiplication. In short, we can have different rules for different mathematical objects.

Feynman says that it is very hard to define properties of numbers such as continuity and ordering. Essentially, it is about the concept of the Dedekind cut and ordered continuum that provides a rigorous distinction between rational and irrational numbers. In 1872, Richard Dedekind proposes an arithmetic formulation of the idea of continuity. That is, the real numbers on a line (or number line) form an ordered continuum, such that any two numbers x and y must satisfy one and only one of the conditions: x < y, x = y, or x > y. He also postulates the concept of a cut that separates the continuum into two subsets, say X and Y, such that if x is any member of X and y is any member of Y, then x < y. This cut may correspond to a rational number or an irrational number.

Questions for discussion:

1. How would you prove the Pythagorean theorem using similar triangles?

2. How would you define the concept of numbers?

3. How would you explain the properties of numbers such as continuity and ordering?

The moral of the lesson: numbers have some nice properties such as a + 0 = a, and a.1 = a, but physicists can use them like a tool without the need of proving it.

References:

1. Einstein, A. (1949). Autographical notes (Translated by Schilpp). La Salle, Illinois: Open court.

2. Feynman, R. P., Leighton, R. B., & Sands, M. (1963). The Feynman Lectures on Physics, Vol I: Mainly mechanics, radiation, and heat. Reading, MA: Addison-Wesley.

3. Gouvêa, F. Q. (2008). From Numbers to Number Systems. In T. Gowers, J. Barrow-Green, I. Leader (Eds). The Princeton Companion to Mathematics. Princeton: Princeton University Press.

4. Loomis, E. S. (1968). The Pythagorean Proposition. Reston, VA: National Council of Teachers of Mathematics.

5. Rav, Y. (1999). Why do we prove theorems?. Philosophia Mathematica, 7(1), 5-41. 6. Seife, C. (2000). Zero: The Biography of a Dangerous Idea. New York: Viking.

7. Wilczek, F. (2015). A Beautiful Question: Finding Nature’s Deep Design. New York: Penguin Press.