General Principles

General Principles

It is my contention that mathematics is a collection of general principles.

Mathematics has to do with sets and I will start with the set of positive counting numbers. My development is typical of the development of sets used in an algebraic way in general. (See Arithmetic, March 2006)
I start with positive counting numbers because almost everybody counts and generally they start by counting on their fingers.
When a child is given a roll of Life Savers she will count them and see how many she has and thus become aware of positive counting numbers. When all the Life Savers are gone a child will add zero to his set of numbers. When a friend gives him some Life Savers, he adds. When he gives some of his Life Savers away he subtracts. Subtraction is "take away".
He sees that if Ed gives him 3 Life Savers and John gives him 5, he ends up with 8 Life Savers regardless of who gives him Life Savers first. Thus he becomes aware of operations on counting numbers and that they obey certain rules.
In a later mathematics class, the positive integers are introduced more formally and the rules stated more explicitly but the student has everyday experience and her fingers to fall back on.
Most children start adding counting numbers on their fingers and the rules are introduced to make the process more efficient. Since the Arabic symbols 2, 3, 4, 5, 6, 7, 8, 9 give no clue as to how many fingers they represent, they have to be memorized.
In an arithmetic class the numbers are kept small until the student believes in the rules. But when the numbers become large, the numbers lose intuition. A student can't use their fingers to compute 2348 + 4729. Intuitively there isn't any difference between 4728 and 4729, the difference is mathematical. So, the rules must be extended into the mathematical world. Adding 2348 + 4729 must be done formally.

Multiplication is introduced as fast addition, 4 x 5 = 5+5+5+5+5 and a little more memorization is a good idea. Division is introduced as multiple "take aways". 14÷3= how many times 3 can be taken away from 14, 4, and the remainder, 2.
The point is that I have a set, of counting numbers, and have put an arithmetic on them, that is, operations on them.


When he loses a pack of Life Savers he doesn’t have in a game of horse, he becomes aware of negative numbers to some degree.
The idea of something that is less than nothing is an intellectual and philosophical leap. I can recall in algebra class that negative roots of polynomials were dealt with gingerly.
I think that the teacher has to make some sort of intuition for negative numbers. I say that a negative number represents the opposite of whatever a positive number represents. If a positive number represents distance to the right, a negative number represents distance to the left. If a positive number represents time after the clock starts, a negative number represents time before the clock started.
This implies that the meaning of a negative number depends on context and the meaning of a positive number. Since opposites cancel each other it makes sense that 5 + (-5) = 0.
(It also makes sense that all these ideas were eventually abstracted by mathematicians and made independent of context.)

This is where we start, a set, in this case the positive counting numbers, with operations, in this case addition, subtraction (take away), multiplication (fast addition) and division (multiple take away), defined on it. Some rules hold, like 4 x 5 = 5 x 4, 4+5 = 5+4.

But there is a problem. 5 x (-4) = adding up 4 losses five times = -20 makes sense but (-4) x 5 = adding up 5 a minus 4 times doesn’t.
Here I do something that will be done many times when a set of numbers is enlarged and the rules don’t make sense when applied to the new numbers. (-4) x 5 is not defined and if something is undefined it is ok to define it. I define it so that the rules work.

(-4) x 5 = -(4 x 5) = -20 so that (-4) x 5 = 5 x (-4) = -20


This is where most mathematics that leads to computation starts. The sets may be other than the counting numbers and the operations may be other than the standard arithmetic operations but mathematics starts with a set and some operations on it where the operations satisfy some rules.

The next step after integers is fractions. Making intuitive sense of the ratios of integers is not easy. Elsewhere I have gone into the problems of fractions in some detail. (Fractions and Rational Numbers Revisited, January 2006) Here I will just say a few words about the rules.
We have addition, subtraction, multiplication and division of integers and we now have to define those operations on fractions.
The problem is complicated by the fact that every fraction has many, many, many representations. 1/3 = 2/6 = 3/9 = … for example.
Another problem is multiplication. 5 x 1/3 = add up 1/3 (of a pie perhaps) 5 times. But what does 1/3 x 5 mean? How can I add 5 to itself 1/3 of a time?
1/3 x 5 is not defined and we define it so that 1/3 x 5 = 5 x 1/3.
When we introduce new numbers we have to define the operations so that the rules applied to the new numbers is consistent with the rules applied to the old numbers.
It is my opinion that when arithmetic is taught, this process should be pointed out to the students. It is my opinion that the student should be made aware of what is going on and that this would dissipate a lot of mathematics anxiety. The way it is now, as of this writing, arithmetic in particular and mathematics in general is taught in the dark and the students are afraid of the dark.