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The Infinite and the Infinitesimal

Zeno» threw Greek thought into confusion with a series of paradoxes» concerning the infinite and infinitesimal» properties implicit in a continuum» of points in space or time. Eudoxus» was generally regarded as having resolved the paradoxes, by saying that in mathematics we do not have to get exactly to a solution, but merely get as close to it as we like, but the problem of infinity appeared again after the invention of the calculus. It was resolved, principally by Weierstrass», who put Eudoxus’ solution into precise mathematical language and gave us the mathematical theory of limits (analysis»). However, if one studies analysis carefully, one recognises that it does not justify the use of the infinite or infinitesimal, but rather shows how to carefully step around the issue. The statement of a limit is, strictly speaking, a statement about approximation. Nor does non-standard analysis» afford a way out; it is based on empirically untestable axioms and any problematic statement about standard numbers is simply replaced by a problematic statement about non-standard numbers. Non-standard analysis is probably best viewed as a clever way of taking limits.

Infinity was later introduced into mathematics by Cantor», a step which lead to the redefinition of mathematics and, ultimately, to its separation from physics. Before the early twentieth century mathematics was assumed to be abstracted from Nature, but now it is given an axiomatic foundation in set theory. A mathematical structure is said to exist if its axioms lead to no contradiction. However, this means that the “existence” of a mathematical structure does not imply that there is any structure in Nature which obeys the axioms of that structure.

For example, the axiom of infinity allows mathematicians to talk about the set of whole numbers. This set is infinite, because for any whole number you can think of, there is always a larger whole number (e.g. the one you get by adding 1). But that does not imply that there is an infinite number of objects in the universe. If a set can be put into correspondence with the whole numbers, mathematicians say the set is countable, but actually you cannot count to infinity because it would take an infinite lifetime to do so. Strictly speaking, we have no empirical or scientific way to say that infinity exists.

Infinite sets have strange properties. For example, the set of even numbers contains half the set of whole numbers, but we can put it into direct correspondence with the whole numbers, 1↔2, 2↔4, 3↔6, etc. But this means that the set of even numbers has as many elements as the set of whole numbers. Thus, an infinite set is twice as large as itself. Of course, this is silly. That is to say, it is perfectly okay within the context of pure mathematical structures for which the criterion is that they don’t lead to a contradiction, but, in my view at least, it is not okay when talking about physical reality. It was precisely this sort of problem which troubled Zeno, when talking about a continuum of points in space or time.

Even today, strictly speaking, mathematics does not model a physical continuum. The most it can do is talk about an approximation. We cannot write down a typical real number», like π or e, but only the symbols for the number, or we can write down as many decimal places as time or space permits. Empiricism» in science does not strictly allow us to obtain the limit for such quantities. There is a mathematical idea of a limit, through successive approximation, and in many cases we can calculate what the result of successive approximation would be, but we cannot actually carry out an infinite sequence of successive approximations, because it would require infinite time and resource to do so. We know that there is no such thing as a perfect mathematical line in Nature; a mathematical line has no width, but that is not true of any line we can draw. Likewise we know that more precise measurement of the position of a particle does not enable us to identify it's position, but merely leads to the uncertainties of quantum theory. In order to understand those uncertainties, rather than base science mathematical assumptions which cannot be empirically justified, we should re-examine the foundations of mathematics and of physics to see what can be justified.

Atoms and the Void

There is room for argument and doubt about what was actually meant by Democritus» when he advocated the theory of atoms and the void. Few of Democritus’ writings survive intact, and none of those of his teacher, Leucippus», who was responsible for the original theory. The view given here is based not on a specific description of the original theory, but on the belief that the theory was introduced to resolve the problems caused by the paradoxes of Zeno. If this is so, then Leucippus’ idea subsequently became corrupted first by Democritus himself, and later in the hands of the Epicureans» from whom we derive most of our accounts. The problems raised by Zeno and the notion of a physical continuum are as relevant today as they were in Greek times. Relational quantum gravity has its philosophical roots in the same problems. Probably more relevant to most readers, relational quantum gravity has been developed in the context of a knowledge of modern scientific theory, and provides us with explanations for observed phenomena where standard theory has none, an interpretation of quantum theory, and a unification of qed and general relativity.

Leucippus observed that everything we find in Nature is divisible. We may split an object into parts, and then split those parts again. He reasoned that If we do not allow infinite divisibility, then ultimately we must come down to an indivisible object, an atom (literally uncuttable). Of course, Dalton»’s modern atom turned out to be divisible again, but modern science does have indivisible, or ‘elementary’ particles», electrons, photons, quarks, etc., and there are very good, theoretical as well as empirical, reasons for thinking that these particles cannot be further subdivided. In practice, it is not necessary to assume the existence of an elementary particle on metaphysical grounds. If there is such a thing as an elementary particle, then its theoretical properties can be determined, and if something in Nature exhibits precisely those properties, then we will claim that it is an elementary particle.

Leucippus placed his atoms in the void. Exactly what was meant by the void is subject to some debate, but if the theory was proposed as an answer to Zeno then the void was not a space continuum, as it was later understood by the Epicureans. It may have been the void in the sense of Parminedes, meaning a complete absence of properties. Parminedes had criticised this notion of the void, saying that something with no properties cannot be said to have existence. If this is what Leucippus intende, then Democritus corrupted the notion by saying that “size” is a property of an atom. This invokes Parminedes paradox - if the void has no properties then atoms would all be jammed together so that motion would be impossible.

The Plenum

In relational quantum gravity, Parminedes paradox is avoided because matter is described in terms of elementary point-like, or sizeless, particles, in the sense that a point is that which has no size, and size cannot have meaning in the absence of the properties of space. The notion of the void is useful, not as a physical entity, but for visualisation. Relational quantum gravity describes a structure consisting of elementary particles, their interactions, and nothing else. To understand the properties of that structure we can draw diagrams on paper, but it must be understood that the geometrical properties of the paper have no bearing on the properties of the structure drawn. Mathematically these diagrams are graphs, in which only the nodes and lines have meaning.

Relationism

Descartes» gave the first clear expression of relationism» by observing that we cannot say where something is unless we say where it is with respect to other matter. He concluded that location is determined by contact, motion means change of location, and that empty space is an illusion. From the relationist perspective, three dimensional space is not a fundamental structure into which matter is placed, but is the way in which the mind organises data from sense perceptions. This organisation is possible as a result of real physical properties and relationships in matter arising from interactions between particles.

Newtonian mechanics contradicted this view, by introducing absolute space and absolute time, but Newton» himself was careful to distinguish absolute quantities, hypothesised to exist, from relative space and relative time which we actually measure. Using absolute space and time, Newton» was able to describe mechanics in three laws. In contrast, the mathematical implications of relationism were, and have remained, obscure. Nonetheless, a strictly empirical view would allow only the relative, or measured, quantities, not absolutes which cannot be measured, and in special relativity Einstein» encapsulated a part of the implication of relationism, if only in the sense of relativity of motion.

Relational quantum gravity seeks also to incorporate relativity of position. This is done using quantum logic», introduced by Garrett Birkhoff» and John von Neumann». Quantum logic» is sometimes described as applying counter-intuitive truth values to simple propositions, but in relational quantum gravity it is regarded as a consistent and intuitive extension of two-valued logic and classical probability theory and is a natural formalisation of statements about measurements, to include statements about hypothetical measurements, which are not performed, “If we were to measure the position of a particle, it would be found at x”.

Measurement

Relational quantum gravity distinguishes measurement from a simple count of a number of objects, and defines it to mean a count of units of the measured quantity. Measurement always invokes comparison between the matter under study, and some specially, but arbitrarily, chosen reference matter. For example, mass» is determined by comparison with the International Prototype Kilogram, maintained in Paris. The second» is defined in terms of the period of a particular radiation of caesium-133. The metre» is defined in terms of the second and the speed of light.

All measured quantities can be reduced to functions of time, position and mass. For example, measurement of motion requires a sequence of measurements of position over a period of time; force is defined from the change of motion induced in a mass; electric current is defined in terms of force; charge in terms of time and current; energy is defined in terms of force and distance; temperature is a measure of stored energy, and so on.

In classical mechanics it is assumed that, at each time, each particle has a position in space, independent of whether we measure it or not. Position is given by numbers, and measurement is the process of determining what the numbers are. There are two fundamental problems with this view. First, there is no prior reason to think that, at a fundamental level, the structure of Nature can be placed into a perfect correspondence with the numerical structure, Rⁿ. Second, it does not correspond to what we actually do when we measure position.

Reference Frames

Before we can quantify the behaviour of matter and write down equations to describe physical laws, we must first establish a coordinate system».


Usually the first coordinate denotes the time of the event, and the other three coordinates denote its position.

In practice, before we can determine position, we must choose particular matter, a reference frame». Then we determine position with respect to our reference frame. Reference matter is, to a large degree, arbitrary, and is itself subject to measurement with respect to other matter. Whatever reference matter is used it must include the apparatus used for measurement, some form of clock, axes, and some means of determining distance, such as a ruler or radar. From a relational perspective, it is only meaningful to discuss coordinates with respect to a reference frame.

The General Principle of Relativity

If time and position are not properties of prior space or spacetime, but only of relationships found in matter, then it follows that the fundamental properties of elementary particles have no dependency on time or position. This is expressed in the principle that The fundamental behaviour of matter is always and everywhere the same. Incorporated in this law is the notion that local coordinate systems may always be established by an observer in the same way and we may infer the general principle of relativity»:


The general principle of relativity is extremely powerful. Once put into a mathematical form, it is responsible for much of the mathematical structure of general relativity. It is also the fundamental principle on which relational quantum gravity is based.

Relational Quantum Gravity

The difference between general relativity and relational quantum gravity is that in general relativity coordinate systems are taken for granted; it is assumed that a coordinate system exists at every point in spacetime. In relational quantum gravity coordinate systems only exist because of the actions of an observer. Coordinate systems are necessary for the quantitative description of physical processes. Even for a qualitative description, just of what we see, the eyes are already measuring everything in our vision. Physical processes can take place whether or not they are described, but the description of physical processes depends upon whatever physical processes are required to establish coordinates. An intrinsic uncertainty in measurement follows because measurement must ultimately be based on processes which are not themselves measured.

Before we can talk about coordinates we must study how an observer sets up a coordinate system by means of physical measurement processes. Einstein showed how the way in which spacetime coordinates are defined through measurement leads to special relativity. Relational quantum gravity is based on the same ideas, but deals also with uncertainty in measurement. Because of this uncertainty, there is no such thing as an exact physical coordinate system. Relational quantum gravity shows that the uncertainty in coordinates is the origin also of quantum uncertainty. Quantum theory is essentially a “black box” theory of measurement. We have an initial state, usually determined by the way in which an experiment has been set up, and a final measurement result. We then ask the question “What is the probability that a given initial state will lead to a given measurement result?”.

Everything else is mathematics. It turns out that this is all we need to determine a mathematical structure which deals correctly with probabilities, incorporates the general principle of relativity, and describes how we set up coordinate systems in practice. This is the structure which I call relational quantum gravity. It explains the appearance of diffraction patterns and shows that wave function collapse is no more mysterious than the change in probability when a result is known. To see how it works, we need to start by looking at Einstein’s fundamental ideas and at the mathematical properties which follow from them.

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