When talk turns to philosophy of science, it’s common for people to bring up cutting edge topics like quantum mechanics, the Higgs boson, and superstring theory. But as it happens, some fascinating issues arise when we consider much a less esoteric example: Newton’s laws of motion.
Newton’s theory of mechanics, famously expounded in the Principia, was a landmark achievement and a key development in the scientific revolution. But over the years a number of thinkers have noted that lurking at the core of Newton’s laws are some remarkable logical and philosophical inconsistencies. In a 1985 article in the American Journal of Physics, Robert Brehme [1] wrote:
A physical theory should be precise, economical, and logical. Ideally it is expressed as a mathematical law involving entities whose definitions and measures lie outside the law. Most physical theories conform to these criteria. One serious exception, however, is the circular logic that seems to occur in the connection made between Newton’s first two laws of motion, the inertial frame, and force. Simply put, the circularity is that Newton’s laws are said to hold only in an inertial frame, while an inertial frame is defined as any frame in which Newton’s laws hold.
Consider Newton’s second law, which can be expressed using the famous equation F=ma, where F represents force, m represents mass, and a represents acceleration. If this is indeed a scientific law then it can be empirically verified. But here a problem arises, as noted by Leonard Eisenbud in 1958 [2]:
Implicit in his statement of the [second] law is the assumption of the prior existence of a quantitative definition of “force”; unfortunately Newton nowhere gives such a definition.
The same issue applies to mass. As Brehme writes:
… if Newton’s second law is to be a true law and not merely used to define force or mass, a means must exist for determining force and mass independent of one another and independent of the second law.
Eisenbud notes that during the second half of the nineteenth century, Newton’s laws were criticized by physicists, philosophers, and mathematicians. He cites Hertz (1894) [3]:
It is exceedingly difficult to expound to thoughtful hearers the very introduction to mechanics without being occasionally embarrassed, without feeling tempted now and again to apologize, without wishing to get as quickly as possible over the rudiments and on to examples which speak for themselves. I fancy that Newton himself must have felt this embarrassment.
As the theory of relativity was being developed at the beginning of the twentieth century classical mechanics was subjected to further scrutiny. Part of the problem was a lack of clarity. Writing in 1902, Henri Poincaré [4] pointed out that “treatises on mechanics do not clearly distinguish between what is experiment, what is mathematical reasoning, what is convention, and what is hypothesis.”
It might be asked why such considerations should be of concern to us. Could this just be philosophical nitpicking? Indeed Eisenbud has noted that “For almost all practical purposes … [Newton’s] considerations are entirely adequate.” But the critical evaluation of Newton’s laws is important for at least three reasons.
First, note that just three years after Poincaré’s book was published, Einstein published his theory of special relativity. This huge step forward was enabled by the critical examination of Newton’s laws that took place over the preceding decades. In particular, a careful consideration of inertial frames of references was a key element in Einstein’s reasoning.
Second, from the perspective of science teaching, I believe that we do a disservice to inquiring students when we gloss over the logical inconsistencies in Newton’s laws. Science is not a collection of facts to be memorized (unless we’re preparing for a trivia game). Instead, science is a complex interplay of observation, experiment, intuition, deduction, hypothesis, and modeling. Students need to be empowered to question the received wisdom–it may not turn out to be so wise.
Finally, from the perspective of the history and philosophy of science, a critical examination of Newton’s laws raises some profound questions. What is a scientific law compared to a definition? How do sets of definitions interrelate? When and why is a definition useful? How do such definitions arise, and how did Newton arrive at such a profoundly useful and important set of definitions? To note that F=ma is a definition is not to say that it is arbitrary. To what extent is the choice of definition constrained, and how? That these and other such questions arise from consideration of a theory of mechanics originating over 300 years ago and still widely used may give us pause. While the latest theories of physics can raise intriguing issues in philosophy of science, even wellworn theories can provide rich food for thought.
References:

Brehme RW, 1985. On force and the inertial frame. American Journal of Physics 53 , 952.
 Eisenbud L, 1958. On the classical laws of motion. American Journal of Physics 26 , 144.
 Hertz H, 1889. Principles of Mechanics, translated by D.E. Johnes and J.T. Walley. The Macmillan Company, New York.
 Poincaré H, 1902. Science and Hypothesis. Translated by WJ Greenstreet, 1905. The Walter Scott Publishing Company. New York. [Full text pdf available through Project Gutenberg.]