Paradox

PYTHAGORAS’S SEARCH FOR THE COMMON DENOMINATOR 23Fig. 2.1means of pebbles arranged on a flat surface. Square numberswere constructed by surrounding one pebble with gnomons.A gnomon is a set of units that resembles a carpenter’s square(fig. 2.1). This notation probably helped Pythagoras solve thearithmetical problem of finding triangles that have thesquare of one side equal to the sum of the squares of the othertwo. But it also suggests a way of bringing more and more ofreality under the control of numbers. By adding larger andlarger gnomons, one brings larger and larger regions into thespace surrounding the original “one.”The numbers are the whole figure including the space asorganized by the pebbles or dots. If there were no spacebetween the dots, there would just be a single big dot.Pythagoras thought of big numbers as spatially bigger. Thus,all of reality is encompassed by the natural numbers.Pythagoras’s metaphysical mathematics embodied anaesthetic appreciation for beautiful arguments. Some of thePythagoreans’ lovely proofs are immortalized in Euclid’sElements.