Section 2–1 Introduction

(Idealization / Exception / Approximation)

According to Feynman, the scientific method includes observation, reason, and experiment. This is similar to Bacon’s scientific method that proposes testing and refining hypotheses by observing, measuring, and experimenting. During an address titled What is Science?, Feynman slightly disagrees with Bacon’s method and explains that “He [Bacon] spoke of making observations, but omitted the vital factor of judgment about what to observe and what to pay attention to (Feynman, 1999, p. 173).” Feynman points out that we do not merely observe because there is also judgment or reasoning involved. In this section, Feynman suggests three ways in understanding physical laws: idealization (or simplification), exception (or violation), and approximation (or imprecision).

1. Idealization (or simplification):

“First, there may be situations where nature has arranged, or we arrange nature, to be simple and to have so few parts that we can predict exactly what will happen, and thus we can check how our rules work (Feynman et al., 1963, section 2.1 Introduction).”

Feynman imagines the physical world to be like a great chess game being played by the gods, and physicists are observers of the game. Although they do not know the rules of the game, they would like to deduce and understand the rules. To understand the rules or physical laws, he provides the following analogy: In one corner of the chess board that has only a few chess pieces, we can deduce the rules exactly. This first way of understanding physical laws may be described as idealization or simplification when physicists focus on a simple phenomenon or problem having only a few physical variables. Similarly, in Newton’s first law of motion, we idealize a physical world in which there is no resultant external force acting on an object. Physicists imagine the object to be under ideal conditions such that they can solve the problem exactly.

In the previous lecture, Feynman has explained that a picture of water that is magnified a billion times could be idealized in several ways (Feynman et al., 1963, section 1.2 Matter is made of atoms). First, the water molecules in the picture are simplified by having sharp edges. To be more accurate, the edges could be blurred because the particles are almost everywhere and they may be found in locations further away having lower probabilities as predicted by quantum mechanics. Next, for the reason of simplicity, the water molecules are sketched in a two-dimensional arrangement instead of three dimensions. To be more realistic, one may also use an animation that shows the motion of particles in a three-dimensional world.

Historically speaking, Galileo’s method of idealization in experiment marks an important milestone in modern science. For example, Galileo states four claims on the period of simple pendulum: (1) The law of length: the period varies with the square of its length, (2) the law of amplitude independence: the period is independent of its amplitude, (3) the law of weight independence: the period is independent of its weight, and (4) the law of isochrony: all periods are the same if the length is the same. However, we should not expect empirical data to exactly fit these physical laws because they are based on simplifying assumptions such as an ideal world that has no friction. Galilean idealization is the practice of introducing assumptions in physical models with the goal of simplifying theories such that exact calculations are possible.

2. Exception (or violation):

“A second good way to check rules is in terms of less specific rules derived from them. For example, the rule on the move of a bishop on a chessboard is that it moves only on the diagonal… the most interesting phenomena are of course in the new places, the places where the rules do not work — not the places where they do work! (Feynman et al., 1963, section 2.1 Introduction).”

It is possible that a rule that works well for a long time until we discover an exception to this rule. In general, the most interesting phenomena are not those that obey rules, but rather the exceptions or violations that lead physicists to develop new rules that are even more fundamental. In his autobiography Surely you’re Joking, Mr. Feynman, Feynman writes that “the discovery of parity law violation was made, experimentally, by Wu, and this opened up a whole bunch of new possibilities for beta decay theory (Feynman, 1997, p. 248).” Simply put, the violation of parity law implies that there is yet another fundamental law of physics to be discovered. In fact, this violation contributes to an understanding of another new rule: the law of weak interactions.

The exceptions or violations in classical physics have also resulted in revolutions in understanding physics. For example, the wave theory of light fails to account for the blackbody spectrum and the stability of atoms accurately. These failures led to the development of new rules that are now known as quantum mechanics of atoms. In other words, it is important to identify the exceptions or limitations of classical physics and understand to what extent the physical laws are applicable in most physical phenomena. On the other hand, the failure of Newtonian mechanics to account for the null results of the Michelson-Morley experiment has helped Einstein to develop a new rule, the special theory of relativity. Furthermore, the failure of the special theory of relativity to account for the precession of Mercury’s orbit has prompted Einstein to develop another new rule, the general theory of relativity.

Note: If you listen to the CD of this Feynman’s lecture, he did mention the word “violation,” but this word is omitted in this section of The Feynman Lectures. Of course, you can find this word in other chapters of the book.

3. Approximation (or imprecision):“The third way to tell whether our ideas are right is relatively crude but probably the most powerful of them all. That is, by rough approximation (Feynman et al., 1963, section 2.1 Introduction).”

Interestingly, Feynman has cited Alexander Aleksandrovich Alekhine, a world class chess champion because it is difficult to completely understand his reasons for moving a particular chess piece. In a general sense, one may roughly guess his motive is to protect the king during the game. Similarly, it is difficult to have a physical model that describes the motion of a complicated object exactly. As an example, physicists may apply Newton’s second law of motion that is approximately correct to describe the motion of a spacecraft. A prediction of the spacecraft’s motion by using the equation F = ma has a greater magnitude of error if its speed is higher or approaching the speed of light.

Similarly, the mass of a chair can be defined only approximately. Feynman explains that some atoms evaporate from the chair and sometimes a few atoms (such as dirt) fall on it and get mixed with the paint. Strictly speaking, it is difficult to define a chair precisely because we are unable to tell exactly which atoms are chair, which atoms are air, which atoms are dirt, or which atoms are paint that belongs to the chair. Feynman imagines that a student may object because he does not like this imprecision, and he prefers to define everything exactly. Importantly, Feynman explains that “[i]f you insist upon a precise definition of force, you will never get it! First, because Newton’s Second Law is not exact, and second, because in order to understand physical laws you must understand that they are all some kind of approximation (Feynman et al., 1963, section 12–1 What is a force?).”

Questions for discussion:

Physics teachers may discuss the idea of idealization (or simplification), exception, and approximation in understanding physics as found in the following passages in section 1.2 Matter is made of atoms (Feynman et al., 1963).

1. “… This is a picture of water magnified a billion times but idealized in several ways. In the first place, the particles are drawn in a simple manner with sharp edges, which is inaccurate.”

2. “… This minimum amount of motion that atoms can have is not enough to melt a substance, with one exception: helium.”

3. “… Nevertheless, to an excellent approximation, if the density is low enough that there are not many atoms, the pressure is proportional to the density.”

The moral of the lesson: Physical laws are formulated by using idealizations and approximations, but physicists should be cognizant of exceptions or violations in these laws. 

In other words, physicists need to cheat (idealizing assumptions), tweak (approximating answers), and confess their sins whether intentional (idealizations and approximations) or unintentional (exceptions or violations).

Note: For those who are interested in the differences between approximation and idealization, they may want to read this article: Norton, J. D. (2012). Approximation and idealization: Why the difference matters. Philosophy of Science, 79(2), 207-232.

References:

1. Feynman, R. P. (1997). Surely you’re Joking, Mr. Feynman. New York: Norton.

2. Feynman, R. P. (1999). The Pleasure of Finding Things Out: The Best Short Works of Richard P. Feynman. Cambridge, MA: Perseus.

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.