Part 9, Chapter 1, Algebra 1

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"Try as we might, we were again unable to come up with a suitable definition. "Even if I can't define addition, I know what it is when I see it," Recordis muttered. The gremlin laughed again. "You have hit a dead end even earlier than I had anticipated you would. I shall soon be back to claim the kingdom." He spread his cape and a gust of wind blew him out the window. Recordis began to moan, "I wonder if people who know algebra get paid more than people like us who don't," he sobbed. "We will learn algebra, or else we will give our lives in the attempt," the king said. "We will start by giving a precise definition of every term that we use." We were trying to decide which term to define first when the professor suddenly realized what was wrong. "We can't define 'everything'!" she said. "We can only define a word by using other words. Let's say that the word 'number' is the first word we want to define. We can't use any words in its definition, because we would not as yet have defined any words." "The dictionary is able to print a definition for every word," Recordis objected." "But the dictionary must inevitably have circular definitions!" the professor said. "Suppose we define 'big' to mean 'large.' and "large" to mean "big' That's what I call a circular definition, because it leads you around in circles." Sure enough, the dictionary definition of the word "number" also turned out to be circular. The dictionary defined "number" to mean "a symbol representing how many." When we looked up the definition of the word "many, " we found "a large indefinite number." "What this means, " the king said, "is that we have to start with some terms that we will assume we know the meaning of. We call these the undefined terms (in italics.) Then we will make definitions of the other terms by using these undefined terms." We decided that we would let number and addition be undefined terms, since we all had an intuitive idea of what they mean. Then the professor realized that we also needed some behavior rules for these terms"(called axioms or postulates). They would pin down the undefined terms to make sure they behaved the way we thought they did, not proving them but assuming they were true. "We spent the next week working on a list of properties that seemed to describe the behavior of numbers. I won't detail all of the arguing that occurred, but we finally came up with the following list:" (to be started and given to you in the next lessons for tomorrow.)