# Physics textbooks that distinguish between laws and definitions?

Often when I am learning physics I start to think about whether the laws I'm learning are mere definitions or experimentally determined, and usually the textbook does not make this clear. As Thomas Kuhn writes in The Structure of Scientific Revolutions,

These generalizations look like laws of nature, but their function for group members is not often that alone. Sometimes it is: for ex­ample the Joule-Lenz Law, $H = RI^2$. When that law was discovered, community members already knew what $H$, $R$, and $I$ stood for, and these generalizations simply told them something about the behavior of heat, current, and resistance that they had not known before. But more often, as discussion earlier in the book indicates, symbolic gen­eralizations simultaneously serve a second function, one that is ordi­narily sharply separated in analyses by philosophers of science. Like $f = ma$ or $I = V/R$, they function in part as laws but also in part as definitions of some of the symbols they deploy. Furthermore, the bal­ance between their inseparable legislative and definitional force shifts over time. In another context these points would repay detailed anal­ysis, for the nature of the commitment to a law is very different from that of commitment to a definition. Laws are often corrigible piecemeal, but definitions, being tautologies, are not. For example, part of what the acceptance of Ohm's Law demanded was a redefinition of both 'current' and 'resistance'; if those terms had continued to mean what they had meant before, Ohm's Law could not have been right; that is why it was so strenuously opposed as, say, the Joule-Lenz Law was not.

Another example is conservation of momentum; is momentum conserved because it is defined as a quantity that is conserved or because it has been observed to be conserved?

Are there any physics textbooks (maybe for classical mechanics) that take a more axiomatic approach, clearly distinguishing definitions from laws determined by experiment?

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• Are there any physics textbooks (maybe for classical mechanics) that take a more axiomatic approach, clearly distinguishing definitions from laws determined by experiment? "Physics textbooks" is pretty broad. Are we talking about freshman texts? Graduate-level texts like Jackson? Best-selling freshman texts like Halliday are written for people who don't care about these issues. An axiomatic approach is not synonymous with rigor, and axiomatizations are non-unique. Something that's a definition in one axiomatization can be a law in another axiomatization. For freshman mechanics, try Kleppner. – Ben Crowell Sep 11 '13 at 5:24
• I am asking as an undergraduate, so a textbook at that level would be preferable. I understand that axiomatizations need not be unique; I wonder though, how can you be rigorous without indicating which laws are axioms and which are empirical? – Hypercube Sep 11 '13 at 5:30
• @Hypercube Well in classical mechanics one is best advised to use the Lagrangian or Hamiltonian formulation. This provides a basic mathematical template for analyzing any physical system in the universe. The constraints and structure of the Lagrangian thus constitute the experimentally determined axioms. I don't know about books on the freshman level, but I believe Goldstein's book on Mechanics covers the Lagrangian approach really well. It also delves it classical field theory. – dj_mummy Sep 11 '13 at 7:08
• I am asking as an undergraduate, so a textbook at that level would be preferable. Upper division? Lower division? how can you be rigorous without indicating which laws are axioms and which are empirical? I don't really follow you here. Axioms are typically empirical. For instance, the reason Euclid chose the parallel postulate as an axiom was that it seemed to be valid empirically. With hindsight, this was because he was an intelligent being living in a region of relatively low spacetime curvature. – Ben Crowell Sep 11 '13 at 15:53
• Lower division. In any case, I have started Goldstein and it's excellent so far. Thank you! – Hypercube Sep 13 '13 at 6:28

An axiomatic approach is not as valuable as you think.

This is true even in math...

Some math textbooks will define the algebra of real 2x2 matrices as "four real numbers in a 2x2 grid with the following addition and multiplication rules", and then later they will prove that matrices are equivalent to the linear operators on a 2D real vector space with a specific basis.

Other math textbooks will define the algebra of real 2x2 matrices as "the set of linear operators on a 2D real vector space with a specific basis", and then later they will prove that matrices are equivalent to four real numbers in a 2x2 grid with the following addition and multiplication rules.

Neither presentation is incorrect. It's just that one or the other presentation might work better pedagogically and in context.

After the presentation of this definition and proof, the textbook would hopefully tell you that the other presentation is also possible. These are two alternate ways to define the same thing.

In physics you have the same situation. See my answer to What's the best definition of energy.

I think what you're really looking for is not a "textbook that distinguishes laws from definitions", but a "textbook that is very clearly written and that emphasizes the historical or modern experiments that validate and motivate these concepts."

By the way you won't and can't get a whole consistent and correct picture in a lower-division physics course. The truth will emerge, clearer and clearer, as you take more courses, such as Lagrangian mechanics and quantum mechanics and quantum field theory etc. etc. They will shed new light on all these concepts, letting you see them from different angles.

The situation (definitions vs. axiomatic interpretation of laws, theories, etc.) is even less satisfying as you suggest in your question.

When Newton gave us his second law, F=ma, these quantities are defined in terms of length, time, mass, and energy, but I would point out that it does so without ever defining what a length or a time actually is. Einstein's later formulation of Special Relativity is proof that evidently, length and time are not exactly what Newton thought they were (as in, relative to the observer in different inertial reference frames).

This answer is not meant to discourage you from making either axiomatic structures in math or in physics, nor to try and define something new that you may observe in either discipline. Just be aware, there are limitations of both that are associated with the fact that you are doing so with your manifestly finite mind, and this will require you to use a certain amount of circular reasoning, no matter how airtight your math, definitions, or axioms might be.