Known Unknowns 199

Known Unknowns 199 Numbers It was not until the late 18" century that numbers were properly codified. The mathematician Giuseppe Peano gave us the rules, so they are called Peano axioms. Here are his ‘axioms’ in natural language. Peano Axioms 1. The first number is named zero. 2. Every number hasa next number (called its successor). Example: the next number after one is two. 3. Numbers are singular. Every number with the same name is the same thing. 4. If something is true of a number, it should be true of the next number (the successor number). From this we can prove some very simple things. 1+1=2. Because the next number after 1 is 2 and ‘+1’ means take the successor. (You can see J cheated here a little and did not take 100 pages for the proof.) Back to my poor mother: “Why is the lowest number zero, Mummy?” “Because I say so!” Or, at least “...because Mr. Peano said so.” That’s what an axiom is. “OK, but why is 3 greater than 2”” “Because I said that each number has a thing that comes after it. “But, why can’t 3 come after zero!” “Tt can!” “But then, if 3 is the thing after zero, I could count 0, 3, 2, 4...” “Yes, if you want to...” “Tm sort of lost. Now, you are saying that 3 doesn't really ‘mea’ anything. It just comes after 0” “Yes. You can make up any symbols you like. You just have to remember what you said and be consistent.” The dialogue shows the importance of definition in mathematics. I could define my counting numbers as 0, 1, 2, 3, 4 or as 0, 2, p, 6, ¢, OF = th, {k, 2 or to be really annoying and confusing 0, 3, 1, 2, 4; they are only arbitrary symbols. It helps us to learn the numbers because 1 is a single line, 2 is two lines joined, three is basically three lines looped together, and four is four lines, but we could have used any symbols we cared for. It is the rules for manipulating these symbols that are the important part and give mathematics its meaning. HOUSE_OVERSIGHT_015889