David Peat

On Incompleteness 37What is particularly interesting about the theorems in Euclid’s systemis that, on the one hand, they were proved logically from the axioms,and on the other hand these same theorems could be tested practicallywith facts about the real world and the space in which we live.Euclid’s method was enormously important, both because of its appealto logic and because its theorems agreed exactly with experience.His theorems were true both within the mind and when surveyed inthe field.Mathematics Abstracts ItselfThen, in the nineteenth century, mathematicians began to ask, Whathappens if we change one of Euclid’s axioms—just for fun? Supposewe suggest that parallel lines do meet at a point? Such a new axiom hasno reference to the space in which we live. The key question was, Evenwith a change in one of the axioms, does the entire system still form alogically consistent, but alternative, geometry? Would this geometry betrue in some alternative science fiction universe?In short, mathematicians began to wonder about abstract axiomaticsystems, systems that no longer corresponded with reality. Clearlyin such totally abstract systems the issue of consistency is of paramountimportance. How do we know, for example, that this alternative geometryis not free from internal contradictions?The Power of LogicThe issue of consistency in mathematics has always been resolved byan appeal to logic. The philosopher Leibniz, for example, had arguedthat logic is the ideal language for philosophers. But the traditionallogic of ancient Greece, Rome, and the early Middle Ages relies onpurely verbal arguments: If I assume A then B must follow. Or, A thingcannot be both “A” and “not A” at the same time. Leibniz thereforeproposed that verbal statements should be replaced by strings of symbols.Thus was symbolic logic born. A string of symbols says the samething as verbal statements but in a more economical way, and, what’s