Processes of Knowledge
George Towner · PhilPapers (PhilPapers Foundation) · 2001
Reality Has the potential of this book as a starting point for exploring reality now been used up? The answer is no; there is at least one more universe with which it is associated. Let us turn to one of the dialogues, called the Timaeus. It starts with a summary of part of the Republic, after which one of the persons of the dialogue, Critias, recounts the legend of Atlantis. Here there is no problem of understanding, even though as far as we presently know Atlantis never actually existed. The physical references—the size of the island, the earthquake and flood that destroyed it, the mud remaining where houses had been—are all comprehensible because they refer to the sorts of things we encounter in physical reality. The references to behavior—the bravery of her warriors, the magnanimity of her leaders—are similarly comprehensible in terms of the behavioral reality we understand. But then Timaeus starts unfolding an elaborate cosmogony, including a scheme for associating the elements of Empedokles (fire, air, water, and earth) with what are now known as the “Platonic solids.” Geometric solids bounded by identical regular polygons were a novelty in Plato’s day; it was relatively recently that Euclid had described some 1.1 Orders of Reality 21 of them in the thirteenth book of his Geometry. Theaetetus, who was contemporary with Plato, was said to have proved that there could be only five such. The theory of regular solids was a largely unused intellectual tool, much like the tensor calculus in Einstein’s day. Intrigued by the solids’ property of decomposing into one another under simple geometric transformations, Plato assigned four of them to what were then the “traditional” physical elements: the tetrahedron to fire, the cube to earth, the octahedron to air, and the icosahedron to water. The dodecahedron was taken to represent the whole cosmos. A geometric calculus could then be formulated in which the decomposition of each solid into sets of the others would parallel the transmutations that were thought to occur among the physical elements. All this is set forth in the dialogue. I mention this theory not for its intrinsic explanatory value, although it enjoyed a lengthy vogue during the Middle Ages. I mention it to illustrate this question: how do we understand the Platonic solids that it discusses? Are they part of physical reality or are they part of behavioral reality? Of course it is easy to manufacture physical objects “in geometric shapes”—a “cube of sugar,” for example. But a “cube of sugar” is not in any sense a geometric cube, because the sugar does not have any of the properties required of the geometer’s object. Its faces are not perfectly flat, its edges do not meet in exact points, and so on. When we prove a theorem about a geometric object we never refer to any physical thing; in fact it is just as easy for us to prove theorems about shapes that cannot be represented physically at all, such as the tesseract. When we create physical things “in geometric shapes” as an aid to visualization, it is always clear that they are not perfect. Since perfect correspondence to description is a necessary property of anything subject to geometric proof, such things cannot be physical objects. This argument has been stated many times before, but it is easily forgotten. A subtler explication for geometric objects is that they are figments of behavior. In this view Platonic solids, for instance, exist just to the extent that we think about them. Certainly all we know about them (and about all other entities of geometry, mathematics, and logic) we have learned through strictly mental operations. The proof that there are only five possible regular convex solids does not require that we examine the shapes of all possible things, or indeed that we use our senses in any way. It follows from the axioms of geom22 Processes of Knowledge etry by logical processes. It is a truth we acquire by sitting quietly in a chair and thinking: the sort of knowledge some classical philosophers called “ a priori. ” Because the whole process begins and ends in behavior, it is natural to suppose that it refers only to more behavior—that Plato’s statements about the tetrahedron, for example, refer only to an idea that was thought up and publicized by Euclid. To be sure, an element of behavioral choice lies at the beginning of any logical discipline. This was nicely illustrated in the nineteenth century when the mathematician Riemann (and later Minkowski) showed it was possible to construct consistent but different geometries by altering the fifth postulate of Euclid’s system, the famous “parallel postulate.” The resulting “non-Euclidean geometries” were actually generalizations of Euclid’s system, introducing certain constants to create a more detailed characterization of space. Euclid’s parallel postulate had amounted to a tacit assumption that these constants were zero. By assigning them various values in what is now called a “curvature tensor,” it became possible to describe different varieties of space, each with different geometric properties. For instance, the sum of the angles of any triangle (which Euclid assumed must always be 180°) varies in non-Euclidean space as a function of the curvature tensor. Thus it seemed that Euclid’s “ a priori knowledge” had been wrong, particularly after Einstein showed in 1915 that actual astronomical space could usefully be described as non-Euclidean: that we could associate nonzero values of the curvature tensor in physical space with the phenomenon of gravity. It seemed that Euclid had unwittingly regarded a behavioral decision—to regard space in one way and not in any other—as a geometric truth. The actual situation, however, is this. No one has ever successfully argued that Euclid’s theorems do not follow from his definitions, axioms, and postulates. What is argued is that some of these beginnings are not as “self-evident” as Euclid thought they were. Once we admit them, the rest follows. The behavioral factor in geometry (and generally in any abstract discipline) is exhausted at the very beginning, when we formulate descriptions of what we are going to think about and how we are going to express our conclusions. After that the conclusions are independent of behavior. But this does not mean that the conclusions are obvious, or that we always think of them. The conclusions of a logical system do not necessarily “lie within” the premises in the sense that it merely takes a little juggling to expose them all. 1.1 Orders of Reality 23 In 1895, for example, Peano published an axiomatic basis for mathematics that can be conveniently summarized on a single page; but the consequences that can be deduced in his system are so voluminous it is unlikely they will ever be fully determined. In terms of the knowledge generated by a discipline such as mathematics, all the development of our understanding occurs after the initial formal decisions have been made. Only a few mathematicians spend their careers thinking about foundations: they are like prospectors who spot a vein of ore and say “dig here!” Following them come armies of other mathematicians who mine the lode, who devote generation after generation to exploring the consequences of the few basic ideas with which they started. Thus it is proper to treat abstract disciplines (such as mathematics) as processes of developing the consequences of initial decisions, rather than of making the decisions themselves, in which case behavior ceases to determine the results. This does not mean that no decisions are made in abstract research. At every point it is necessary to decide where to look next, to judge which consequences of the premises are important and which are trivial. But such decisions do not change the conclusions; they only influence which conclusions are sought. The independence of mathematical truths with respect to our behavior stands out clearly in some of the classic problems in the field. For example, consider the statement that every even number is the sum of two primes. This proposition, known as the “binary Goldbach conjecture,” has puzzled mathematicians for more than 250 years. Computer surveys have shown it to be true for numbers up to 15 digits long, but no general proof has been found. Yet it is either true or not true. Anyone could achieve instant fame by stating a proof or by finding a number that refutes it. No one has, but someone might do either tomorrow. The point is that an immense amount of speculation about this matter has so far failed to resolve it, which could not have been the case had its verification merely involved examining our behavior. There is a “hard reality” here, outside our thoughts about it.