Evolution of Abstraction in Physics and Mathematics

Posted on 10 June 2008

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One of the key characteristics of the evolution of mathematics and physics has been a progressing increase in the degree of abstraction and generality. First, let’s consider how the description of nature as physics has developed toward increasing generalization. The ancient description of planetary motion set forth concrete geometric models in which the planets moved along circular paths. A major step beyond these concrete models was Kepler’s laws of planetary motion. These general laws set forth a mathematical description of the orbits of the planets. Kepler’s first law states that the orbit of each planet is an ellipse with the sun at one focus. The second law relates states that each planet moves in its orbit such that the line from the sun to the planet sweeps out equal areas in equal times. The third law sets forth a fixed relationship between the size of each planet’s orbit and its period. These laws are not specific to any one planet, but are general laws, with each particular planet being one of the many possible solutions to these laws. After Kepler, the next major development came with Newton who found more general laws of motion. These laws were not limited to planetary motion, but applied to all types of motion, both celestial and terrestrial. Combined with his universal law of gravitation, Newton was able to derive Kepler’s laws as a special case. Einstein generalized further with his general theory of relativity, from which Newton’s laws can be derived as a special case. A similar pattern took place in the development of  quantum physics. First there were concrete phenomenological models of the atomic spectral lines. Then, analogous to Kepler’s laws, there was the general model of the Bohr atom from which these phenomenological models could be derived as special cases. The major step then came with quantum mechanics. Like Newton’s breakthrough, these laws were much more general (not limited to atoms), but the Bohr model could be derived from these general laws as a special case. Physics still awaits a generalization that encompasses both quantum theory and general relativity. The evolutionary pattern, however, is clear: as physics progresses, the laws become more general and comprehensive, with the prior laws and models being consequences of these general laws as applied to limited situations.

With this progressive generalization in the formulation of physical laws, there comes a natural increase in the degree of abstraction from concrete, measurable physical quantities. For example, the early planetary models directly described relationships between concrete observational quantities such as position and time. The laws of Kepler described properties of generic orbits rather than setting forth a model of specific orbits. By fixing the general parameters of Kepler’s laws to specific values, a concrete model could be derived as a special case. Even more abstract, Newton’s laws described relationships between derived quantities such as acceleration, force, and gravity. These physical quantities are not themselves directly observable, but are abstractions derived from directly measurable quantities such as position and time. Moreover, solutions to Newton’s laws are themselves general laws, e.g, the general curve of a parabola describing the general motion of a projectile in a uniform gravitational field, or the general curve of an ellipse describing the general motion of a body in a central gravitational force. Special cases of these general solutions would give the particular parabola for a projectile given a particular initial velocity in a gravitational field of a particular strength, or the particular ellipse for a planet having a particular mass orbiting a star of a particular mass. The observed positions would then be particular points along these curves that describe the whole set of potentially observable positions. As these examples illustrate, the concrete observations are separated from the general laws by successive levels of abstraction, and returning from the general laws to the concrete observational data involves successive restrictions and limitations of the general laws to more specific conditions.

A similar pattern of increasing abstraction takes place in the mathematics used in physics. For example, in the geometrical models of planetary motion, motions of planets are represented by geometrical shapes, e.g., the observed position of a planet is a point on an ellipse (which is essentially a law relating all the observed points to each other). The representation then shifted to numbers, so that the position of a planet is represented by numerical coordinates, and the motion of a planet in the ellipse is represented by an algebraic equation whose solutions are the coordinates. With Newton, the equations of motion are differential equations whose solutions are algebraic equations, whose solutions are coordinates. So, the mathematics increases in abstraction as well.

A similar pattern can be found in the development of pure mathematics. Consider, for example, the notion of number. This began with whole numbers: 1, 2, 3, 4, etc. The notion of number then generalized to include negative numbers and fractions. It then expanded further to include irrational numbers as well. There are even more abstract number systems beyond this, such as the complex numbers, surreal numbers, and transfinite numbers. It is significant that, each successive expansion of the notion of number can be seen to resolve a problem that arises with the more limited notion of number. For example, when performing subtraction of whole numbers, sometime the result is a whole number (e.g., 9-4=5) but sometimes it is not (e.g., 4-9=?). Extending the notion of number to include negative numbers resolves this problem and allows the subtraction operation to always have a solution. Similarly, the operation of taking the square root of a rational number does not always result in a rational number (e.g., the square root of two, or the square root of minus one). Extending the notion of number to include irrational numbers and imaginary numbers allows the operation of taking the square root to always have a solution.

Another pattern in the evolution of mathematics is that a solution to a problem at one level shifts the problem to a deeper level. For example, the problem of calculating the slope of a tangent to a curve or the area under a curve resulted in the development of calculus. But when calculus was first introduced, there were paradoxes of infinitesimals that were not clear. Many of the calculus techniques worked in most cases, but in some cases they resulted in contradictions, which raised doubts about the validity of its foundations. It worked and was very powerful, and was accepted as very important, but one could not be certain that it always would work. There was not clarity about when it would and would not work. Motivated by these problems, mathematicians eventually established a rigorous foundation for calculus using notions of limits. Later, these concepts themselves were also found to have problems related to ambiguity about the nature of the continuum and real numbers. Limits, it turned out, could not be rigorously defined without a rigorous mathematics of the continuum. This motivated the development of the mathematics of the infinite, and set theory. But that too ran into paradoxes at a deeper level, such as Russell’s paradox. This historical development shows how the resolution of a problem at one level opens up deeper questions at a more profound level.

What is the explanation for this evolutionary pattern toward more generality and abstraction? The basic motivation at the root of scientific inquiry is the quest for a coherent understanding of the whole cosmos. Presented with a diverse array of phenomena, the holy grail of all natural philosophy has always been to see this multiplicity as aspects of an underlying unity, to unite the one and the many, variance with invariance, the eternal with the ephemeral. This search for the unity behind diversity, is a search for the relationship and ordered harmony between the diverse parts within the unity. Although originally conceived as a search for the basic metaphysical substance, the key was that the search for unity in diversity is a method for discovering orderly relationships among apparently separate parts, thereby revealing a harmony and unity previously not seen.

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Posted in: Science