David Peat

38 From Certainty to Uncertaintymore, the structure of such a system is fully explicit and transparent sothat it is easy to spot any error of logic. By reducing every argument toa string of logical symbols it should then be possible to analyze proofsabout the foundations of mathematics in a thoroughly rigorous way.But which proofs are to be examined? So far we’ve only dealt withcounting, but mathematics consists of more than just numbers. Whatabout the calculus, geometry, algebra, and so on? How, precisely, isgeometry to be reduced to strings of logical symbols? To see how, let’sgo back to Euclid and his Elements of Geometry. His theorems deal withcongruent triangles, bisecting circles, and so on. But Descartes showedthat every point on the plane can be defined by two numbers, its x andy coordinates. Likewise, a line can be written down as an equation—the straight line y = 3x or the curve y = x 2 .Following Descartes, geometrical figures can be represented by algebraicequations. This means that theorems in geometry can be reducedto solutions and properties of these equations. The whole ofgeometry, along with all its proofs, can be reduced to algebra. In turn,algebra can be reduced to theorems about numbers. And theoremsabout numbers can be expressed using symbolic logic. Proceeding inthis fashion all of mathematics can be reduced to algebra and the rulesof algebra analyzed according to symbolic logic.So far so good. But then the mathematician David Hilbert pointedout that by reducing geometry to algebra, mathematicians had simplyshifted the burden of proof to algebra. David Hilbert argued that itmade more sense to make each and every aspect of mathematics formallyconsistent in its own right. Rather than proving geometry viaalgebra or interpreting points in space as numbers, each branch ofmathematics should be reduced to a formal system of symbols.Hilbert’s ProgramHilbert went further by asking why we need to interpret geometry interms of algebra. In a truly pure mathematics the meaning of thesevarious symbols shouldn’t really matter. Mathematics is simply the