Excerpts from
Is Nature Accessible to the Mathematical Physicist?
by
William A. Wallace, O.P.… The primary instrument of a mathematical physics is a demonstrative syllogism composed of two premises, one physical or natural, the other mathematical. The middle term will generally be a metrical concept, that is, one that expresses the result of a measuring process that applies a number or figure to a physical entity, and so pertains to both mathematics and physics. Such a measurement, even though approximate, is regarded as true if its result falls within the limits of error of the measuring process. The middle term may also take on meanings that are partly the same and partly different in the two premises, thus making use of analogous predication. …
Aristotle defines nature as "a principle and cause of being moved or of rest in the thing to which it belongs primarily and in virtue of that thing, but not accidentally" (Physics II.1, 192b21-23). He further identifies it with the two essential principles of natural things he earlier uncovered in the Physics, matter or "underlying nature" (191a8), which I call protomatter, and a "natural form" (192b1) that makes the thing be the kind it is. Aristotle uncovers these principles from an analysis of the way in which substances come to be and pass away in the order of nature. His analysis is based not on mathematics but on ordinary experience, through a study of the qualities of objects as presented to the senses. Protomatter is for him a purely potential and indeterminate principle, whereas natural form is an actualizing or determining principle, one that specifies the object to be a particular natural kind. …
The coming-to-be of a natural substance, for Aristotle, could be brought about by the alteration of its sensible qualities. It was in this way that his commentators came to explain the transmutation of the elements, that is, the natural change of one element into another.3 A schema called the
symbolum was commonly used to detail how this came about. …
First, the four elements have never actually been observed in the universe, and in fact are unobservable because they would require a pure admixture of primary qualities, which is never experienced in bodies that come under sense observation. Yet the bridge to the knowledge of these elements is qualitative knowledge. In other words, it is only through the qualities that are known to exist in macroscopic bodies that one is able to reason to the existence of bodies endowed with idealized qualities that, in some way or other, serve to explain the appearances of composed bodies.
Second, both the four elemental bodies and the substrate that is their basic component, protomatter, may be said to be real, although neither is real in the same way as an existent sensible body. Protomatter is real only in the sense of being a potentiality for assuming various natural forms, whereas the four elements are never completely actual in any composite, but are always in some remiss state corresponding to the various degrees of remission of their primary qualities. …
Third, qualities observed in the macroscopic domain are explained by idealized qualities, and idealized qualities are explained by a quality-less substrate. The substrate also lacks quantity, and thus is radically unpicturable and unobservable. Yet it is knowable by experience from a knowledge of substance and from an analysis of what happens in substantial change. Thus both the elements and the substrate serve as real explanatory principles of chemical transformation and of the composition of bodies. They also meet the rather stringent requirements of the logic of explanation. …
"Hot-cold" and "wet-dry" may seem to be strange couplets with which to start discussing elementary particles, but they are really not far different from the "up-down" and "top-bottom" couplets used in recent quark theory. …
The point has been made that Aristotle's analysis in the
Physics was based on ordinary experience, not on mathematics, but rather on the study of sensible qualities, that is, the qualities of bodies as presented to the senses. Is there any way in which such qualities can be made susceptible to mathematical treatment and thus enter into the reasoning processes of a mixed science? Within Thomism it would seem that this question can be answered in the affirmative. The basis for this reply is St. Thomas's teaching that there is a hierarchical ordering among the accidents found in a natural body, with quantity being the most fundamental, and with the remaining accidents coming to substance through quantity as an intermediate (
quantitate mediante). Being received into substance through quantity, sensible qualities have a quantitative aspect, and it is on this basis that they can be measured. Being measurable, they themselves can take on the formality of metrical concepts, and so serve as middle terms in the syllogisms or demonstrations of the mixed sciences, as mentioned in the preamble to this paper.6
To explain this, it should be noted that physical qualities may be divided into two types depending on their proximity to sense experience. Some qualities are directly sensible, such as heat, color, sound, odor, and taste, all of which can be sensed immediately by the external organs of the body. Others are reductively sensible in that they can be known only through sensible effects; examples of this type are electricity, magnetism, and chemical affinities. Pertaining to this latter division are also motive and resistive powers, such as gravity and resistance to motion, which were already known to Aristotle.
Qualities in all these categories can be said to be quantified simply because they are present in quantified bodies. Their quantity can be measured in two ways, giving rise to two measures usually associated with physical quality, namely, extensive and intensive measurements. …
The four-element theory of Aristotle was, of course, the longest lasting theory in the history of science, enduring from the fourth century BC all the way to the early nineteenth century. The "hot-cold" and "wet-dry" couplets of that theory prompted much work with furnaces and solutions in alchemy, and also with Galenic medicine, up to the Renaissance. By the nineteenth century metrical concepts were even available for dealing with heat and fluidity…. But ultimately they proved inadequate for attacking the element problem and had to give way to "gravity-levity" as the preferred couplet for its solution. …
Bohr speculated that when an electron moves farther from the nucleus in this way it absorbs an amount of electromagnetism determined by the different energy levels of the two orbits; when it drops from an outer orbit to an inner one, it emits a similar amount. By formulating a series of rules stating which electron transitions are allowed and which are not, Bohr found that he could explain the emission and absorption spectra of many chemical elements. In effect, he could correlate the wavelength and intensity of the radiation characteristic of a particular element with the jumping of electrons in the atomic model of that element from one orbit or energy state to another.
Further refinements of Bohr's model began with Arnold Sommerfeld's replacement of circular orbits by elliptical orbits. Then, along with that, was the possibility of the electron orbits having various orientations in three-dimensional space, providing an azimuthal quantum number, and its having different angular momenta, giving two more energy states. Another was the introduction of electron spin, that is, the rotation of an electron on its own axis, to make a fourth. In all, therefore, each electron in an atom now had the possibility of existing in four energy states, each denoted by a different quantum number. Yet another was the introduction of a principle by Wolfgang Pauli specifying that no two electrons in an atom could occupy the same energy state at any one time, which was equivalent to stating that no two electrons in an atom can have the same four quantum numbers. …
The four quantum numbers were based on calculations of theoretical physicists using classical mechanics and electromagnetic theory. By what seems a remarkable coincidence, counterparts of their results could be found in four different types of spectral lines of the elements identified by spectroscopists working in different portions of the electromagnetic spectrum. These lines were described by them as strong, principal, diffuse, and fundamental, and designated by four letters (s, p, d, and f), which could be correlated with the four quantum numbers of Bohr's theory. …
With this I return to Aristotle's concept of nature and take up the second principle he identifies with nature, namely, natural or substantial form. … Of itself, protomatter is unintelligible, but when actualized by form it becomes intelligible in the substances we know through sense experience. The case is different with natural form, for the human mind grasps it directly and instinctively. Natural form provides the window through which the world of nature is seen and through which many of the natures inhabiting it can be readily understood.10 …
In
The Modeling of Nature I made the case that the information provided by the periodic table of the elements provides us with a knowledge of their natures far superior to any comparable knowledge we have of living organisms.13 … And all of this information is reducible to sense knowledge, through the use of the metrical concepts we have explained above. What is more, we need not base our knowledge on theoretical entities such as elliptical orbits and spinning electrons. We tend to replace "element" with "atom" and "compound" with "molecule," but it is the elements and the compounds that fall directly under sense experience, not the atoms and the molecules. And when we measure the spectral lines that reveal energy levels, we do so in terms of wavelengths in the electromagnetic spectra of elements we handle, spectra that are themselves visible, either directly or reductively. The middle terms in our reasoning processes are thus both physical and mathematical. …
Do we also know the natures of planets and stars? I discuss this briefly in
Modeling and propose that the answer is "No," and this for the simple reason that they do not have natures in the strict sense.15 Of the heavenly bodies, some, such as planets and asteroids, are mainly solids, whereas stars like our sun are principally hot gases. Earth itself is a mixture or aggregate of many different elements and compounds, held together by the force of gravity. Similarly the sun is a mixture of hydrogen and helium in the gaseous state. The unity of a star would seem to be analogous to the unity of the earth: largely a mass of different substances held together by natural forces of one type or another. And if current models of planets and stars are correct, they can go through a process of evolution and can have a history like many plants and animals. Yet, unlike plants and animals, planets and stars have no natural form, there is no unifying or specifying form within them guiding that history toward some perfective state, as it does in the case of organisms. …
Replacing orbiting electrons with wave functions or other theoretical entities in no way calls into question what has already been said. The observational basis for quantum mechanics, and the information it provides of natures in the non-living, remains exactly the same as previously: energy levels, revealed by spectral lines, and the frequency of transitions from level to the other, revealed by the intensity of spectral lines. The changes over the past fifty years are in the way scientists theorize about what goes on in the interiors of atoms and their nuclei, not about the experimental findings that ground their theorizing. …
At the top [of Figure 6] is shown the famous wave equation formulated by Erwin Schrödinger early in 1926. The variable is the Greek letter psi, and so the equation is referred to as the psi-function. What psi stands for, unfortunately, is the subject of much dispute. Schrödinger originally thought it stood for electric charge distribution within the atom, thus giving it a physical meaning, but he later ruled this out as impossible. In 1929 Max Born gave psi a statistical interpretation, saying that it represented the probability of finding an electron at a particular place within the atom. This view was vigorously rejected by Schrödinger. And finally in the early 1950's David Bohm gave psi a realist or deterministic interpretation, holding that its results are determined by potentials within the atom. So we have three views of the wave equation: an actual physical function, a probability function, or a potentiality function. No physicist currently holds the first view, but the second and third are still the subject of vigorous dispute among physicists and philosophers of science.
In actual practice the wave equation is most used by chemists, and for their purposes it suffices to interpret the equation statistically as a probability function. …
Wave mechanics thus is a potent instrument for studying the functions of electrons within atomic and molecular structures. It can also be used to speculate about electrons and photons when they are conceived as individual wave pulses outside the atom. Here the mathematics becomes complex, partly because of the infinities involved. The problem may perhaps be understood by a simpler example that has its roots in Aristotle. In the Posterior Analytics I,13 Aristotle observes that "it belongs to the physician to know that circular wounds heal more slowly [than other kinds], but it belongs to the geometer to know the reasoned fact" (79a15-16). The reason is that healing occurs along the perimeter of a wound, and the circle has the smallest perimeter for any given area; thus it will heal more slowly. Using calculus, one can also calculate the rate of healing. What is involved is a function which starts with the wound's area, A, and decreases exponentially with the passage of time. The fit is remarkable for all points along the curve except at the end. The mathematical function approaches the x-axis asymptotically, which means that it never reaches the x-axis, or, as some say, it meets the x-axis at infinity. But here nature doesn't obey the mathematics. At some point in time nature closes the wound and A goes to zero.
In mathematical physics, as noted at the beginning of this paper, two premises are ultimately involved in any proof, one a physical premise, the other a mathematical premise. Which premise should take priority in case of conflict? In my view the physical premise must be regulative over the mathematical. This goes contrary to much contemporary discussion of quantum mechanics, where, it seems to me, mathematics is driving the arguments. If physics is not, then the philosophy of nature has little or nothing to contribute to the debate, which perhaps explains why it is consistently ignored. …
The basic mathematics behind matrix mechanics is the same as that behind wave mechanics, but it uses a different formulation, one proposed by Werner Heisenberg in 1925. Heisenberg became concerned about Schrödinger's wave equation because psi was not an observable, and he thought physics should stick to observables. Accordingly, he saw the goal of quantum mechanics to be the computation of two matrices, one a diagonal matrix which would list the observed energy levels of atoms and molecules, the other a related matrix that would list the transition probabilities between the various levels. In both cases one would be concerned with observables or measurements: the wavelengths of spectral lines and their intensities, both of which are available to the physicist. Heisenberg also saw these matrices as grounded in the
potentia of Aristotle, his term for protomatter. As I see it, his view of the psi-function was ultimately a potentiality function, although he is often listed as following the Copenhagen interpretation, which sees it as merely a probability function. …
…we are still left with only three basic particles, the proton, the neutron, and the electron. And none of these can be said to have a nature in the strict sense, although they enter into the composition of the elements and compounds whose natures we do know, as already explained. High-energy physicists provide evidence for the existence of mesons and baryons by the hundreds, of bosons such as the photon and gluons of different types, of leptons such as the electron, the positron, and various kinds of neutrinos. All of these are only transient entities,
entia vialia, to use the Latin expression.20 We may think of them as transient natures, but this is not the sense in which Aristotle intended the term "nature." On the other hand, he did intend nature to mean protomatter. And here we have the clue to the significance of particle physics. Protomatter is the ultimate substrate, the basic potency that underlies the operations of nature. The different groupings of strange properties fail to yield a particle ultimate, but Aristotle's ultimate is still there, the potential correlate of the natures we have come to know in the world of the inorganic.
