Introduction

       A definition may be defined as a determination of a concept (Rickert 1888), an explanation of the meaning of a word (Belnap 1993), or “a convention, hence neither true nor false but at most practical or impractical, convenient or inconvenient (Mahner 1998, p. 415).” In my study of The Feynman Lectures on Physics, Feynman has the tendency of mentioning the word define and problematizing definitions of physical concepts (Wong, Chu, & Yap, 2014). In addition, Feynman challenges the conventional, preconceived notions of many physical concepts in his lectures such as Alix G. Mautner Memorial Lectures, John Danz Lectures, and Messenger Lectures. It should be insightful for students to understand how Feynman would answer and analyze “examination questions” from a perspective of definitions. Feynman also seems to be fond of discussing the following four definitional problems: precision, context, circularity, and completeness in knowledge.

1. Problems of Precision

The learning of a concept is related to understanding the attributes of its definition and differentiating the attributes (Klausmeier, 1990). If the definitions are imprecise, they may contribute to alternative conceptions of students (Wong, Chu, and Yap, 2016). Definitions can be considered as imprecise if one or more attributes are not well-defined.Conversely, in The Feynman Lectures on Physics, Feynman opines that “[w]e can’t define anything precisely. If we attempt to, we get into that paralysis of thought that comes to philosophers, who sit opposite each other, one saying to the other: you don’t know what you are talking about! The second one says: what do you mean by ‘talking’? What do you mean by ‘you’? What do you mean by ‘know’?” (Feynman et al., 1963, section 8-1 Description of motion). In other words, it is good to know about the difficulty of defining physical concepts precisely.

Additionally, based on Einstein’s principle of equivalence, an observer in an enclosed room is unable to distinguish whether the force acting on the observer is due to the gravity or acceleration. In Feynman’s Lectures on Gravitation for advanced graduate students and postdoctoral fellows, Feynman (1995) explains that “it will not be possible to define a ‘true’ gravity, since we cannot ever define precisely how much of an observed force is given by gravity and how much is due to an acceleration (p. 92).” That is, we may not be able to determine, to what extent, the force is due to gravity or acceleration in the windowless room. (Please refer to Table 1 for examples of Feynman’s comments on definition which is related to problem of precision.)

Table 1 Feynman’s comments on definitions that are related to precision

S/N	Examples of comments that are related to the problem of precision
1	John Danz Lectures: Words can be meaningless. If they are used in such a way that no sharp conclusions can be drawn, as in my example of ‘oomph,’ then the proposition they state is almost meaningless, because you can explain almost anything by the assertion that things have a tendency to motility. A great deal has been made of this by philosophers, who say that words must be defined extremely precisely. Actually, I disagree somewhat with this; I think that extreme precision of definition is often not worthwhile, and sometimes it is not possible - in fact mostly it is not possible, but I will not get into that argument here (Feynman, 1998, p. 20).
2	Surely, you’re Joking, Mr. Feynman: The definitions weren't accurate. Everything was a little bit ambiguous - they weren’t smart enough to understand what was meant by “rigor.” They were faking it. They were teaching something they didn't understand, and which was, in fact, useless, at that time, for the child (Feynman, 1997, p. 292).
3	The Feynman Lectures on Physics: So the mass of a chair can be defined only approximately. In the same way, to define the mass of a single object is impossible, because there are not any single, left-alone objects in the world - 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, Volume I, 12-3).

2. Problems of Context

To prevent miscommunication, we should avoid the use of a single term for different concepts. However, many words used in physics have alternative definitions. For example, the word force may have multiple definitions in different subject disciplines and in the everyday context.

In the 1985 Alix G. Mautner Memorial Lectures, Feynman mentions that “[a]nother possibility, especially if the lecturer is a physicist, is that he uses ordinary words in a funny way. Physicists often use ordinary words such as ‘work’ or ‘action’ or ‘energy’ or even, as you shall see, ‘light’ for some technical purpose. Thus, when I talk about ‘work’ in physics, I don’t mean the same thing as when I talk about ‘work’ on the street (Feynman, 1985, p. 10)”. However, students may conceptualize the terms used in physics based on their usage in everyday context.

The other problem is that technical terms can be defined differently within the same subject, such as physics, and in various subject disciplines. For instance, “[i]f you are going to be an engineer and work on the design of transformers, magnets, and such, you will have to watch out. You will find many books which use for H the definition of Equation (36.14) rather than our definition of Equation (36.12), and many other books — especially handbooks about magnetic materials — that relate B and H the way we have done (Feynman et al, 1964, section 36-2 The Field H). Moreover, there are disagreements on the correct definition for many scientific concepts when scientists have their own reasoning or argument in different fields of study. (Please refer to Table 2 for examples of Feynman’s comments on definition which is related to problem of context.)

Table 2 Feynman’s comments on definitions that are related to context

S/N	Examples of comments that are related to the problem of context
1	Feynman lectures on computation: It is important to stress that the meaning of the word ‘information’ here differs from that in ordinary usage – it is not totally distinct from this, but generally “information” in our sense tell us nothing about the usefulness of or interest in a message, and is strictly an academic term. There are lots of words like this in science: the meaning of the words “work” in physics and “function” in mathematics bears little relationship to their colloquial meanings. We will return to the concept of information and define it more rigorously latter (Feynman, Hey, & Allen, 1998, p. 118).
2	Messenger Lectures: It is easy to understand how an object can be symmetrical, but how can a physical law have a symmetry? Of course, it cannot, but physicists delight themselves by using ordinary words for something else (Feynman, 1965, p. 84).
3	Surely, you’re Joking, Mr. Feynman: In it Whitehead kept using the words “essential object” in a particular technical way that presumably he had defined, but that I didn't understand. After some discussion as to what “essential object” meant, the professor leading the seminar said something meant to clarify things and drew something that looked like lightning bolts on the blackboard. “Mr. Feynman,” he said, “would you say an electron is an ‘essential object’?”... One man stood up and said, “A brick as an individual, specific brick. That is what Whitehead means by an essential object.” Another man said, “No, it isn’t the individual brick that is an essential object; it’s the general character that all bricks have in common - their ‘brickiness’ - that is the essential object.” Another guy got up and said, “No, it's not in the bricks themselves. ‘Essential object’ means the idea in the mind that you get when you think of bricks.” (Feynman, 1997, pp. 69-70)

3. Problems of Circularity

In general, a problem of circularity occurs if a concept is defined by means of a word, and this word is defined by the former concept. In The Feynman Lectures on Physics, an example provided is “Webster defines ‘a time’ as ‘a period’, and the latter as ‘a time’, which does not seem to be very useful (Feynman et al., 1963, Vol. I, section 5-2 Time).” Circular definitions pose a problem whether the definitions provide any useful knowledge.

Similarly, if we define weight operationally as “what the weighing scale read”, it may also have a problem of circularity. By checking the dictionary, we may find that the weighing scale is defined as “a balance used for weighing.” That is, defining a physical quantity by the measuring equipment does not definitely help students to conceptualize the nature of this physical quantity. Circularity occurs in the definitions because the equipment is defined by the same physical quantity which this equipment is used to measure.

Furthermore, definitions of two physical concepts are circular if they are both defined by the same equation. There are at least three possibilities for these problems of circularity in defining mass: (1) mass (m) is defined as W/g, and weight (W) is defined as mg; (2) mass is defined as the product of the density of the object (r) and its volume (V), and density is defined as the quotient of mass by its volume (r = m/V); (3) mass is defined as the ratio of force (F) over acceleration (a), and force is defined as the product of mass and acceleration (F = ma). On the other hand, mass is also defined as “quantity of matter,” and the matter is sometimes simply defined as something that “has mass.”  (Please refer to Table 3 for examples of Feynman’s comments on definition which is related to the problem of circularity.)

Table 3 Feynman’s comments on definitions that are related to circularity

S/N	Examples of comments that are related to the problem of circularity
1	The Feynman Lectures on Physics: We could also define force to mean that a moving object with no force acting on it continues to move with constant velocity in a straight line. If we then observe an object not moving in a straight line with a constant velocity, we might say that there is a force on it. 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?).
2	Messenger Lectures: There are many different kinds of atoms, and if you replace one by one of a different kind it makes a difference, but if you replace one by the same kind it makes no difference, which looks like a circular definition. But the real meaning of the thing is that there are atoms of the same kind (Feynman, 1965, p. 95).
3	Lectures on Gravitation: Is it possible to make meaningful physical statements without defining the nature of forces? We may recall the situation in Newtonian mechanics. It is often said that Newton’s law, Fx = mẍ, is simply a definition of forces, so that it has no real physics - that it involves circular reasoning (Feynman, 1995, p. 90).

4. Problems of Completeness in Knowledge

Importantly, physicists are not always completely sure of scientific knowledge. Thus, it is possible to classify knowledge as “known knowns,” “known unknowns,” and “unknown unknowns.” (Logan, 2009).  For example, ‘known unknowns’ refer to knowledge which we know that we still do not know, and ‘unknown unknowns’ refers to knowledge which we are unaware that we do not know. In general, physics teachers may explain that scientific research is usually about experimenting among the “knowns and unknowns.” Physics teachers and students should analyze definitions of physical concepts from the perspective of incompleteness in knowledge.

As an example, Feynman explains that “whether or not one can define absolute velocity is the same as the problem of whether or not one can detect in an experiment, without looking outside, whether he is moving. In other words, whether or not a thing is measurable is not something to be decided a priori by thought alone, but something that can be decided only by experiment (Feynman et al., 1963, section 16-1 Relativity and the philosophers).” Essentially, definitions of physics concepts should be verifiable by experiment, and thus, they are influenced by the experimental results.

Interestingly, in Feynman’s letter to Armando Garcia J., dated 11 December 1985, Feynman writes that “I am sure of nothing, and find myself having to say ‘I don’t know’ very often . . . It is fun to find things you thought you knew, and then to discover you didn’t really understand it after all (Feynman, 2005, p. 396).” This may be related to his belief that “there is always the possibility of proving any definite theory wrong (Feynman, 1965, p. 157).”

Table 4 Feynman’s comments on definitions that are related to incomplete knowledge

S/N	Examples of comments that are related to the problem of incomplete knowledge
1	Messenger Lectures: You might call that a condition of geography, analogous to the situation that when I translate in space I must translate everything. In the same sense, you might say that the laws for time are the same and we must move the expansion of the universe with everything else. We could have made another analysis in which we started the universe later; but we do not start the universe, and we have no control over the situation and no way to define that idea experimentally (Feynman, 1965, p. 87).
2	The Present Status of Quantum Electrodynamics: The simplest suggestion for defining Amperes-hypothesis would be to say that in the “fundamental Lagrangian” (the exact form of which, at present, unknown) all gradient operators ∂μ on charged fields are to be replaced by ∂μ - Aμ and no other coupling to Aμ is to be assumed. There are two objections to this formulation. First, we do not know the form of the future theory; no Lagrangian may exist. Is there not some formulation closer to the observed properties of the particles? The second objection is possibly academic; if the Lagrangian were before us, we would probably know exactly what to do (Feynman, 1961, p. 158).
3	The Feynman Lectures on Physics: Earlier in the history of physics, when it was supposed to be very important to define every quantity by direct experiment, people were delighted to discover that they could define what they meant by E and D in a dielectric without having to crawl around between the atoms. The average field E is numerically equal to the field E0 that would be measured in a slot cut parallel to the field. And the field D could be measured by finding E0 in a slot cut normal to the field. But nobody ever measures them that way anyway, so it was just one of those philosophical things (Feynman et al., 1964, section 11-4 Electric fields in cavities of a dielectric).

       Feynman’s analysis of definitions of physics concepts may help students to have a deeper understanding of the physics concepts. It can be beneficial to discuss definitional problems during classroom teaching and assessment. That is, we should not passively teach students definitions of physical concepts by merely providing statements of definitions from textbooks. However, physics teachers may provide incorrect answers for examination questions. This blog provides my reflections on selected questions, and Feynman’s insights based on my readings of his books. Please feel free to provide any comments or suggestions.

References

1. Belnap, N. (1993). On rigorous definitions. Philosophical Studies, 72(2/3), 115–146.

2. Feynman, R. P. (1961). The Present Status of Quantum Electrodynamics. In Brown, L. M. (ed.), Selected papers of Richard Feynman (pp. 134-164). Singapore: World Scientific. 

3. Feynman, R. P. (1965). The character of physical law. Cambridge: MIT Press.

4. Feynman, R. P. (1985). QED: The strange theory of light and matter. Princeton: Princeton University Press.

5. Feynman, R. P. (1995). Lectures on gravitation (B. Hatfield, ed.). Reading, MA: Addison-Wesley.

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

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

8. Feynman, R. P. (2005). Perfectly reasonable deviations from the Beaten track: The letters of Richard P. Feynman (M. Feynman, ed.). New York: Basic Books.

9. Feynman, R. P., Hey, J. G., & Allen, R. W. (1998). Feynman lectures on computation. Reading, Massachusetts: Addison-Wesley.

10. 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.

11. Feynman, R. P., Leighton, R. B., & Sands, M. (1964). The Feynman Lectures on Physics, Vol II: Mainly electromagnetism and matter. Reading, MA: Addison-Wesley.

12. Klausmeier, H. J. (1990). Conceptualizing. In B. F. Jones & L. Idol (Eds.), Dimensions of thinking and cognitive instruction (pp. 93–138). Hillsdale, NJ: Lawrence Erlbaum Associates.

13. Logan, D. C. (2009). Known knowns, known unknowns, unknown unknowns and the propagation of scientific enquiry. Journal of Experimental Botany, 60(3), 712–714.

14. Mahner, M. (1998). Operationalist fallacies in Biology. Science & Education, 7(4), 403–421.

15. Rickert, H. (1888/2000). The Theory of Definitions. In J. C. Sager (ed.) Essays on definition (pp. 191–249). Amsterdam: John Benjamins B. V.

16. 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.

17. Wong, C. L., Chu, H. E., & Yap. K. C. (2016). Are Alternative Conceptions dependent on Researcher’s Methodology and Definition?: A Review of Empirical Studies related to Concepts of Heat. International Journal of Science and Mathematics Education, 14(3), 499-526.