The endless relay between numeric and spatial representations (and Riemann’s amazing ability to foreshadow possibilities)
The extent to which an idea in mathematics creates an idea in science is largely underappreciated. It is common to think of mathematics as the tool that one needs to describe the reality explored by physics, as if the mathematics is secondary, or a purely linguistic consideration. But it should be clear that this is only part of the story. The bigger story is highlighted by yet another article I read today about a Foundational Institute award recipient whose work is focused on uniting quantum mechanics and general relativity (as is Bob Coecke’s work, who is also supported by the Institute, and was the subject of my last post). This awardee is David Rideout, a physicist at the University of California, San Diego and his ideas have their seeds in the transformational insights of Riemann.
The fabric of space-time, as it is now understood, breaks down inside a black hole, where huge masses are confined to very small regions creating points of inconceivably infinite density. Another kind of infinity plagues quantum theory. Rafael Sorkin, a physicist at Syracuse University, New York adds that, “The bare electric charge of the electron is very badly infinite” in the quantum equations of electromagnetic force, a fact not reflected in reality. Rideout and others have considered that a way around these difficulties is to imagine space-time as a network of discrete chunks rather than as a continuous fabric. The chunks are very small, a mere 10 to the minus 35 meters across. But in this world, space-time could never shrink to the infinitely small volumes that the mathematics of black holes has to deal with, because there would no longer be the continuity that produces them. To keep this world from being just a “pile of dust,” in contrast to our dynamic universe, this discreteness would have to be combined with causal structure. It turns out that this combination of attributes can be found in an area of mathematics called causal set theory. The hope is to build space-time out of related but discrete pieces.
The project goes like this: Start with a list of every point in space and time, all the way back to the Big Bang. Then add information about the relationship between every pair of points in the set. Luckily, there is only one thing you need to know about the relationship between those two points: Can A affect B? Thanks to Einstein’s theory of special relativity, we know that nothing can travel faster than the speed of light. This means that some pairs of points in space-time will be forever disconnected. So, if the two points are suitably distant in time and space, the answer will be no, one could not have affected the other. If they are close enough, the answer is yes. A causal set is a roster of those “no” and “yes” answers for every pair of points in space and time.
Rideout belief is this,
that causal sets offer a view of time as a process that is occurring—as in quantum theory—with the future being formed as we live our lives and make decisions: “The growth dynamics of causal sets expresses the progression of time as a continual process of becoming, like the unfolding of a flower or the growth of a tree.”
Einstein Online does a really nice job of describing how this ordering brings about a geometry, tracing the idea of discrete space back to Zeno as well as Riemann’s famous 1854 lecture. In this lecture, Riemann addressed the foundations of geometry with a careful scrutiny of the notions: number, magnitude, and manifold. Riemann is quoted in the Einstein Online discussion saying:
“The question of the validity of the presuppositions of geometry in the infinitely small hangs together with the question of the inner ground of the metric relationships of space. In connection with the latter question… the above remark applies, that for a discrete manifold, the principle of its metric relationships is already contained in the concept of the manifold itself, whereas for a continuous manifold, it must come from somewhere else. Therefore, either the reality which underlies physical space must form a discrete manifold or else the basis of its metric relationships should be sought for outside it[...].”
Here Riemann is asking what it is about the structure of space that makes it possible to talk about measurable things like distance, area, volume and angles (the “metric relationships”), and he is contrasting the case in which the deep structure of space is continuous with the case in which it is discrete. In a continuous space (like that of Euclidean geometry) there are between any two points always an infinity of others, and every volume can be divided into smaller and smaller volumes without limit. In a discrete space, in contrast, any bounded region is composed of a finite number of elements or “building blocks”, and the process of subdivision must always come to an end at some stage. Riemann’s point, then, is that a discrete space has metric information built in from the start. It is easy to see the truth of this: For instance, simply counting the elements composing a region of space provides a natural measure of that region’s volume. For a continuous space, in contrast, this possibility to count elements is lacking (you’d just get infinity for the answer) and the origin of the metric relationships has to be explained in some other manner.”
Einstein borrowed Riemann’s notion of a continuous curved space for his theory of general relativity but also foresaw Riemann’s discrete space possibility, renaming Riemann’s discrete manifold a continuum-discontinuum. This quote from Einstein was also provided by Einstein Online (from a published 1954 letter):
“In any case, it seems to me that the alternative continuum-discontinuum is a genuine alternative; i.e. there is no compromise here. In [a discontinuum] theory there cannot be space and time, only numbers[...]. It will be especially difficult to elicit something like a spatio-temporal quasi-order from such a schema. I can not picture to myself how the axiomatic framework of such a physics could look[...]. But I hold it as altogether possible that developments will lead there[...].”
The apparent simplicity of causal set relationships is impressive. In the space-time continuum there is an elaborate mathematical web of point-events that carry information about smoothness, nearness, times, and distances. But the ordering of the elements of a causal set is much simpler, relying largely on a kind of before or after relationship among the elements. Einstein Online writes the slogan Order + Number = Geometry.
Mathematics is the endless penetration of what order, number, and spatial representations can bring to mind. It’s as if it unleashes the inexhaustible powers of cognition, or the baffling reach of our complex nervous system. When I imagine the mind’s movement – from counting, to its reluctance to accept the square root of 2 as a number, to its conflict over infinities and continuums, to its arithmetization of the limit, then finding a generalized notion of space, that later brings it back to counting – it looks like one of the major waterways moving through our water driven planet.