Section 12–1 What is a force?

(Circular definitions / Imprecise definitions / Idealizations and approximations)

In this section, Feynman discusses circular definitions of force, imprecise definitions of force, as well as idealizations and approximations involved in defining force.

1. Circular definitions:

“Now such things certainly cannot be the content of physics, because they are definitions going in a circle (Feynman et al., 1963, section 12–1 What is a force?).”

It is inappropriate to define constant momentum in terms of “the sum of the external forces is zero” and then define no net external forces in terms of “constant momentum.” Similarly, a definition of constant velocity as a result of “no force” and a definition of force based on “changing velocity” can be described as circular definitions. It is useless to have two definitions that contain two statements in which the premise is equivalent to the conclusion. We can find circular definitions in elementary textbooks and students would have difficulty learning science concepts (Arons, 1990). In some college textbooks, the concept of mass is defined as the resistance of a body to acceleration using the equation m = F/a, whereas force is the ‘cause’ of acceleration based on the same equation F = ma (Wong, Chu, Yap, 2014).

We can find circular definitions in the dictionary because words are always defined by some other words, which are eventually defined by some of the initially defined words within the same dictionary. For example, Feynman found that Webster defines time as a period, and period as time, which is not useful in explaining the concept of time (Feynman et al., 1963, section 5.1). The problem of circular definitions in the dictionary cannot be completely eliminated. Although dictionaries have shortcomings of circularity, this does not necessarily mean that they are definitely useless. In a sense, the problem of circularity in defining physical concepts is not simply a fallacy but a challenge, which cannot be easily resolved.

2. Imprecision definitions:

“If you insist upon a precise definition of force, you will never get it! (Feynman et al., 1963, section 12–1 What is a force?).”

The word force has some independent properties (e.g., magnitude, direction, and point of application), in addition to the law F = ma; but the specific independent properties of force were not completely described by Newton such that F = ma is an incomplete law. Feynman explains that one of the most important characteristics of force is that it has a material origin. More important, we do not have a precise definition of force because Newton’s Second Law of dynamics is not exact and it involves idealizations and approximations. Interestingly, Feynman clarifies that the concept of force is different from “gorce” that is related to the rate of change of position. In physics education research, the idea of gorce is reported as an alternative conception when students believe that “motion implies a force.”

Feynman is imprecise when he uses the word definition which may mean verbal definition, mathematical definition, or operational definition. In a lecture delivered at The University of Washington at Seattle, he mentions that “I think that extreme precision of definition is often not worthwhile, and sometimes it is not possible—in fact mostly it is not possible… (Feynman, 1998, p. 20). Essentially, he disagrees with philosophers who argue that words must be defined extremely precisely. Physics teachers may elaborate that the definition of force is not precise because of its rich culture. One may use Wilczek’s (2004) words: “physicists and mathematicians including notably Jean d’Alembert (constraint and contact forces), Charles Coulomb (friction), and Leonhard Euler (rigid, elastic, and fluid bodies) made fundamental contributions to what we now comprehend in the culture of force (pp. 11-12).”

3. Approximations and idealizations:

“…every object is a mixture of a lot of things, so we can deal with it only as a series of approximations and idealizations (Feynman et al., 1963, section 12–1 What is a force?).”

Newton’s second law of dynamics is formulated through a series of idealizations and approximations. Firstly, physicists may idealize a chair to be a definite thing in an ideal fashion or assume it to be a point object. Secondly, from an experimental perspective, we idealize the universe using Euclidean geometry and conduct land surveying by assuming a flat space-time instead of curved space-time. Next, Feynman clarifies that the mass of a chair can only be defined approximately because it is difficult to determine exactly which atoms belong to the chair, which atoms are air, which atoms are dirt, or which atoms are paints that belong to the chair—every object is a mixture of a lot of things. On the other hand, the forces on a single object also involve approximations of some kind in the real world.

The concepts of idealization and approximation may overlap in understanding various physics problems. In general, idealization involves constructing models or definitions that are relatively simple as compared to the real world, whereas approximation may refer to mathematical methods (e.g., numerical analysis) that are needed to solve physics problems. Importantly, in section 2.1, Feynman imagines the physical world is like a chess game being played by the gods and suggests three ways in understanding physical laws: idealization (or simplification), approximation (or imprecision) and exception (or violation). According to Feynman, an exception of the law F = ma is that the mass of an object is not constant if it is moving at high speeds. He also adds that F = ma is not really a definition because it is not always exactly true.

Questions for discussion:

1. How would you explain that there are problems of circularity in defining the concept of force?

2. What does Feynman mean when he says that we can never have a precise definition of force?

3. How do physicists conceptualize Newton’s second law of dynamics by using idealizations and approximations?

The moral of the lesson: we should be cognizant of problems of defining force that are related to circularity and imprecisions as well as the need of idealizations and approximations in conceptualizing Newton’s second law of dynamics.

References:

1. Arons, A. B. (1990). A Guide to Introductory Physics Teaching. New York: Wiley.

2. Feynman, R. P. (1998). The meaning of it all: Thoughts of a citizen scientist. Reading, MA: Addison-Wesley.

3. Feynman, R. P., Leighton, R. B., & Sands, M. (1963). The Feynman Lectures on Physics, Vol I: Mainly mechanics, radiation, and heat. Reading, MA: Addison-Wesley.

4. Wilczek, F. (2004). Whence the force of F = ma? I: culture shock. Physics Today, 57(10), 11-12.

5. Wong, C. L., Chu, H. E., & Yap, K. C. (2014). Developing a framework for analyzing definitions: a study of the Feynman lectures. International Journal of Science Education, 36(15), 2481-2513.