SOME HISTORICAL NOTES From: Stephen Wolfram, A New Kind of Science Notes for Chapter 9: Fundamental Physics Section: The Nature of Space Page 1028 Space and its contents. A number of somewhat different ideas about space were discussed in antiquity. Around 375 BC Plato vaguely suggested that the universe might consist of large numbers of abstract polyhedra. A little later Aristotle proposed that space is set up so as to provide a definite place for everything - and in effect to force it there. But in geometry as developed by Euclid there was at least a mathematical notion of space as a kind of uniform background. And by sometime after 300 BC the Epicureans developed the idea of atoms of matter existing in a mostly featureless void of space. In the Middle Ages there was discussion about how the non-material character of God might fit in with ideas about space. In the early 1600s the concept of inertia developed by Galileo implied that space must have a certain fundamental uniformity. And with the formulation of mechanics by Isaac Newton in 1687 space became increasingly viewed as something purely abstract, quite different in character from material objects which exist in it. Philosophers had meanwhile discussed matter - as opposed to mind - being something characterized by having spatial extent. And for example in 1643 Thomas Hobbes suggested that the whole universe might be made of the same continuous stuff, with different densities of it corresponding to different materials, and geometry being just an abstract idealization of its properties. But in the late 1600s Gottfried Leibniz suggested instead that everything might consist of discrete monads, with space emerging from the pattern of relative distances between them. Yet with the success of Newtonian mechanics such ideas had by the late 1700s been largely forgotten - leading space almost always to be viewed just in simple abstract geometrical terms. The development of non-Euclidean geometry in the mid-1800s nevertheless suggested that even at the level of geometry space could in principle have a complicated structure. But in physics it was still assumed that space itself must have a standard fixed Euclidean form - and that everything in the universe must just exist in this space. By the late 1800s, however, it was widely believed that in addition to ordinary material objects, there must throughout space be a fluid-like ether with certain mechanical and electromagnetic properties. And in the 1860s it was even suggested that perhaps atoms might just correspond to knots in this ether (see page 1048). But this idea soon fell out of favor, and when relativity theory was introduced in 1905 it emphasized relations between material objects and in effect always treated space as just some kind of abstract background, with no real structure of its own. But in 1915 general relativity introduced the idea that space could actually have a varying non-Euclidean geometry - and that this could represent gravity. Yet it was still assumed that matter was something different - that for example had to be represented separately by explicit terms in the Einstein equations. There were nevertheless immediate thoughts that perhaps at least electromagnetism could be like gravity and just arise from features of space. And in 1918 Hermann Weyl suggested that this could happen through local variations of scale or "gauge" in space, while in the 1920s Theodor Kaluza and Oskar Klein suggested that it could be associated with a fifth spacetime dimension of invisibly small extent. And from the 1920s to the 1950s Albert Einstein increasingly considered the possibility that there might be a unified field theory in which all matter would somehow be associated with the geometry of space. His main specific idea was to allow the metric of spacetime to be non-symmetric (see page 1056) and perhaps complex - with its additional components yielding electromagnetism. And he then tried to construct nonlinear field equations that would show no singularities, but would have solutions (perhaps analogous to the geons discussed on page 1058) that would exhibit various discrete features corresponding to particles - and perhaps quantum effects. But with the development of quantum field theory in the 1920s and 1930s most of physics again treated space as fixed and featureless - though now filled with various types of fields, whose excitations were set up to correspond to observed types of particles. Gravity has never fit very well into this framework. But it has always still been expected that in an ultimate quantum theory of gravity space will have to have a structure that is somehow like a quantum field. But when quantum gravity began to be investigated in earnest in the 1980s (see page 1058) most efforts concentrated on the already difficult problem of pure gravity - and did not consider how matter might enter. In the development of ordinary quantum field theories, supergravity theories studied in the 1980s did nominally support particles identified with gravitons, but were still formulated on a fixed background spacetime. And when string theory became popular in the 1980s the idea was again to have strings propagating in a background spacetime - though it turned out that for consistency this spacetime had to satisfy the Einstein equations. Consistency also typically required the basic spacetime to be 10-dimensional - with the reduction to observed 4D spacetime normally assumed to occur through restriction of the other dimensions to some kind of so-called Calabi-Yau manifold of small extent, associated excitations with various particles through an analog of the Kaluza-Klein mechanism. It has always been hoped that this kind of seemingly arbitrary setup would somehow automatically emerge from the underlying theory. And in the late 1990s there seemed to be some signs of this when dualities were discovered in various generalized string theories - notably for example between quantum particle excitations and gravitational black hole configurations. So while it remains impossible to work out all the consequences of string theories, it is conceivable that among the representations of such theories there might be ones in which matter can be viewed as just being associated with features of space. Stephen Wolfram, A New Kind of Science (Wolfram Media, 2002), page 1028. © 2002, Stephen Wolfram, LLC