Mathematical Truth

Mathematical Truth

by : Aria Ratmandanu 

             The first steps towards an understanding of the real influences controlling Nature required a disentangling of the true from the purely suppositional. But the ancients needed to achieve something else first, before they would be in any position to do this reliably for their understanding of Nature. What they had to do first was to discover how to disentangle the true from the suppositional in mathematics. A procedure was required for telling whether a given mathematical assertion is or is not to be trusted as true. Until that preliminary issue could be settled in a reasonable way, there would be little hope of seriously addressing those more difficult problems concerning forces that control the behavior of the world and whatever their relations might be to mathematical truth. This realization that the key to the understanding of Nature lay within an unassailable mathematics was perhaps the first major breakthrough in science. 

              Although mathematical truths of various kinds had been surmised since ancient Egyptian and Babylonian times, it was not until the great Greek philosophers Thales of Miletus (c.625–547 bc) and Pythagoras of Samos (c.572–497 bc) began to introduce the notion of mathematical proof that the first firm foundation stone of mathematical understanding—and therefore of science itself—was laid. Thales may have been the first to introduce this notion of proof, but it seems to have been the Pythagoreans who first made important use of it to establish things that were not otherwise obvious. Pythagoras also appeared to have a strong vision of the importance of number, and of arithmetical concepts, in governing the actions of the physical world. It is said that a big factor in this realization was his noticing that the most beautiful harmonies produced by lyres or flutes corresponded to the simplest fractional ratios between the lengths of vibrating strings or pipes. He is said to have introduced the ‘Pythagorean scale’, the numerical ratios of what we now know to be frequencies determining the principal intervals on which Western music is essentially based.2 The famous Pythagorean theorem, asserting that the square on the hypotenuse of a right-angled triangle is equal to the sum of the squares on the other two sides, perhaps more than anything else, showed that indeed there is a precise relationship between the arithmetic of numbers and the geometry of physical space.

             He had a considerable band of followers—the Pythagoreans—situated in the city of Croton, in what is now southern Italy, but their influence on the outside world was hindered by the fact that the members of the Pythagorean brotherhood were all sworn to secrecy. Accordingly, almost all of their detailed conclusions have been lost. Nonetheless, some of these conclusions were leaked out, with unfortunate consequences for the ‘moles’—on at least one occasion, death by drowning! In the long run, the influence of the Pythagoreans on the progress of human thought has been enormous. For the first time, with mathematical proof, it was possible to make significant assertions of an unassailable nature, so that they would hold just as true even today as at the time that they were made, no matter how our knowledge of the world has progressed since then. The truly timeless nature of mathematics was beginning to be revealed. 

          But what is a mathematical proof ? A proof, in mathematics, is an impeccable argument, using only the methods of pure logical reasoning, which enables one to infer the validity of a given mathematical assertion from the pre-established validity of other mathematical assertions, or from some particular primitive assertions—the axioms—whose validity is taken to be self-evident. Once such a mathematical assertion has been established in this way, it is referred to as a theorem. 

        Many of the theorems that the Pythagoreans were concerned with were geometrical in nature; others were assertions simply about numbers. Those that were concerned merely with numbers have a perfectly unambiguous validity today, just as they did in the time of Pythagoras. What about the geometrical theorems that the Pythagoreans had obtained using their procedures of mathematical proof? They too have a clear validity today, but now there is a complicating issue. It is an issue whose nature is more obvious to us from our modern vantage point than it was at that time of Pythagoras. The ancients knew of only one kind of geometry, namely that which we now refer to as Euclidean geometry, but now we know of many other types. Thus, in considering the geometrical theorems of ancient Greek times, it becomes important to specify that the notion of geometry being referred to is indeed Euclid’s geometry. 

           Euclidean geometry is a specific mathematical structure, with its own specific axioms (including some less assured assertions referred to as postulates), which provided an excellent approximation to a particular aspect of the physical world. That was the aspect of reality, well familiar to the ancient Greeks, which referred to the laws governing the geometry of rigid objects and their relations to other rigid objects, as they are moved around in 3- dimensional space. Certain of these properties were so familiar and self- consistent that they tended to become regarded as ‘self-evident’ mathematical truths and were taken as axioms (or postulates). As we shall be seeing that Einstein’s general relativity—and even the Minkowskian spacetime of special relativity—provides geometries for the physical universe that are diVerent from, and yet more accurate than, the geometry of Euclid, despite the fact that the Euclidean geometry of the ancients was already extraordinarily accurate. Thus, we must be careful, when considering geometrical assertions, whether to trust the ‘axioms’ as being, in any sense, actually true. 

       But what does ‘true’ mean, in this context? The difficulty was well appreciated by the great ancient Greek philosopher Plato, who lived in Athens from c.429 to 347 bc, about a century after Pythagoras. Plato made it clear that the mathematical propositions—the things that could be regarded as unassailably true—referred not to actual physical objects (like the approximate squares, triangles, circles, spheres, and cubes that might be constructed from marks in the sand, or from wood or stone) but to certain idealized entities. He envisaged that these ideal entities inhabited a different world, distinct from the physical world. Today, we might refer to this world as the Platonic world of mathematical forms. Physical structures, such as squares, circles, or triangles cut from papyrus, or marked on a Flat surface, or perhaps cubes, tetrahedra, or spheres carved from marble, might conform to these ideals very closely, but only approximately. The actual mathematical squares, cubes, circles, spheres, triangles, etc., would not be part of the physical world, but would be inhabitants of Plato’s idealized mathematical world of forms.