1.3: The Axiomatic Method

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Eric Lehman, F. Thomson Leighton, & Alberty R. Meyer
Google and Massachusetts Institute of Technology via MIT OpenCourseWare

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The standard procedure for establishing truth in mathematics was invented by Euclid, a mathematician working in Alexandria, Egypt around 300 BC. His idea was to begin with five assumptions about geometry, which seemed undeniable based on direct experience. (For example, “There is a straight line segment between every pair of points”.) Propositions like these that are simply accepted as true are called axioms.

Starting from these axioms, Euclid established the truth of many additional propositions by providing “proofs.” A proof is a sequence of logical deductions from axioms and previously proved statements that concludes with the proposition in question. You probably wrote many proofs in high school geometry class, and you’ll see a lot more in this text.

There are several common terms for a proposition that has been proved. The different terms hint at the role of the proposition within a larger body of work.

Important true propositions are called theorems.
A lemma is a preliminary proposition useful for proving later propositions.
A corollary is a proposition that follows in just a few logical steps from a theorem.

These definitions are not precise. In fact, sometimes a good lemma turns out to be far more important than the theorem it was originally used to prove.

Euclid’s axiom-and-proof approach, now called the axiomatic method, remains the foundation for mathematics today. In fact, just a handful of axioms, called the Zermelo-Fraenkel with Choice axioms(ZFC), together with a few logical deduction rules, appear to be sufficient to derive essentially all of mathematics. We’ll examine these in Chapter 7.