Spring 2011 -- In The Logical Leap, David Harriman draws a fundamental contrast between two historically crucial scientific theories: Newton’s theory of gravity and 20th-century quantum mechanics. Harriman expresses a common attitude: that Newton’s inverse square law of gravitational attraction is “the very archetype of a causal law” (142). Meanwhile quantum mechanics “as a fundamental theory of physics . . . is strangely empty.” “It gives a mathematical recipe . . . but fails to provide causal models of subatomic processes” (248).

Harriman heaps praise on Newton for his logical methods. And he scorns those who’ve interpreted quantum theory in terms of non-objective philosophy. All warranted. But what matters, in the context of studying inductive logic and science, are the logical structures of the theories, which are extremely similar in how they treat fundamental causes.
At the core of quantum theory is Schrodinger’s Equation. It, and the associated theory for interpreting its solutions, is sophisticated. The theory predicts the behavior of matter at the atomic level. Some of the elements that go into the equation—for example, potential energies—are familiar characters from classical physics. Some of what comes out—for example, characteristic frequencies—can be measured and used in science and technology. (When I was a Kodak scientist, we used Schro­dinger’s Equation to design colored dyes.) But there is no theory of the underlying structure of atoms and molecules that explains why Schrodinger’s is the right equation; quantum theory tells us how the quantitative elements of atomic behavior fit together, but it doesn’t explain atomic behavior. It doesn’t tell us why atoms and molecules behave as they do.
 
At the core of Newton’s theory of gravitation is his inverse square law of gravitational attraction—that any material body exerts a force on any other material body that is proportional to the masses of the bodies and inversely proportional to the square of the distance between them. The inverse square law combines with his famous second law of motion (F=ma) to yield equations that describe the trajectories of objects, with each object moving under the gravitational influence of the others. All of the elements—such as mass and acceleration—that go into these equations are familiar from basic physics, and what comes out of the equations—the trajectories—are equally familiar. But there is no theory of the underlying structure of matter that explains why the inverse square law produces the right equations. Newton’s theory tells us how the quantitative elements of classical gravitational theory fit together, but it doesn’t explain gravitational attraction; it doesn’t tell us why material objects exert mutual gravitational attraction.
 
One tantalizing, frequently overlooked, and unexplained element of Newton’s theory is the role of mass. The modern understanding of mass—inertial mass—arrived along with Newton’s three laws of motion. That theory, in fact, provides the full, rigorous definition of the concept as we understand it: inertial mass is the resistance of a body to acceleration. The quantity that appears in the inverse square law is the gravitational mass, a measure of a body’s tendency to attract other bodies gravitationally. Gravitational mass is the analog of electrical charge. It really ought to be called “gravitational charge” but it’s not, because it is quantitatively identical with inertial mass. Physics still has no causal account for this amazing fact. (This fact, that gravitational and inertial mass are the same, is why bodies of different masses fall to Earth at the same rate.) 
  
Nor is there an account for the inverse square feature of the gravitational law. There have been attempts, through the years, at causal accounts that would entail the inverse square aspect. No such account has succeeded. (See Richard Feynman’s The Character of Physical Law for a nice discussion of attempted explanations.) 
Both Newton’s theory of gravitation and quantum theory are, equally, superbly effective mathematical accounts of physical behavior that provide no underlying causal explanations of the phenomena they describe. Human knowledge is finite; at any time, it has a frontier. We work to move that frontier. “I have not been able to discover the cause of these properties of gravity from phenomena, and I frame no hypotheses,” wrote Newton in a classic passage Harriman quotes on page 142. “And to us it is enough that gravity really does exist and act according to the laws which we have explained. . . . ” Until one has identified causes of the phenomena at the frontier, one must accept it as “enough” that one has identified the thus-far fundamental facts that constitute that frontier. If one finds this state of affairs “strangely empty,” there’s only one rational thing to do: more scientific research.

Photo: Richard Feynman

David S. Ross has worked as an industrial mathematician with Eastman Kodak Company, solving problems in imaging science and technology, and with Kaiser Permanente, developing mathematical models of human physiology for medical applications. These days, as an academic applied mathematician, he is involved in research on rumor propagation, on prostate cancer, on wound-healing, and on thermodynamics of mixtures. His mathematical focus has always been applications of differential equations.

Dr. Ross also has an interest in philosophy and the epistemological foundations of math, a topic on which he has lectured and written. He is the author, with Avner Friedman, of the book Mathematical Models in Photographic Science, published by Springer-Verlag. He is working on a textbook on mathematical modeling.

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