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MA301Ch1 - CHAPTER 1 Numbers proof and all that jazz There...

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CHAPTER 1 Numbers, proof and ‘all that jazz’. There is a fundamental difference between mathematics and other sciences. In most sciences, one does experiments to determine laws. A “law” will remain a law, only so long as it is not contradicted by experimental evidence. Newtonian physics was accepted as valid until it was contradicted by experiment, resulting in the discovery of the theory of relativity. Mathematics, on the other hand, is based on absolute certainty . A mathematician may feel that some mathematical law is true on the basis of, say, a thousand experiments. He/she will not accept it as true, however, until it is absolutely certain that it can never fail. Achieving this kind of certainty requires constructing a logical argument showing the law’s validity–i.e. constructing a proof. There is, however, a problem with the notion that everything should be proved. If we insist on proving everything then we initially know nothing , and, if we know nothing, how can we prove anything? We have no place to begin. Clearly, we must have some body of information that we know to be true on which to base our proofs. So what can we assume known and what must be proved? Before the time of Euclid, the answer to this question was personal and subjective. You were allowed to assume anything that you could bully your listener into believing. If I could get you to agree that all integers are even (which is false) I could use it to prove all sorts of other wonderful (and equally false) things. This often led to many mistakes, so much so that it was very difficult to know what was true and what was not. Euclid solved this problem for geometry by stating an explicit collection of “self evident” properties (called axioms) which were as- sumed without proof. Furthermore, the axioms are the only properties that were to be assumed without proof. All other properties must be proved using either the axioms or their consequences. 5
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6 1. NUMBERS, PROOF AND ‘ALL THAT JAZZ’. Since the time of Euclid, lists of axioms for many fields of math- ematics, such as set theory, logic, and numbers have been compiled. In these notes, we present one of the standard lists of axioms for the real numbers, which are the numbers used in calculus. Thus, we are stating “up front,” those properties that we are allowed to assume without proof. As will be seen, the list is rather long and will be cov- ered over several sections. We begin with the field axioms , which describe those properties of numbers that do not relate to inequalities. In principle, every number fact we use should be proved using only our axioms. In fact, in these notes, we usually adopt a much looser standard. As the reader will see, proving everything directly from the axioms would take so long that we would never progress beyond this section! It is, however, important that the reader prove a number of basic number facts using only the axioms in order to appreciate their significance.
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