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In his book, "Science and Hypothesis" on page no. 98, Henri Poincaré write on subject of defining force:

[.....When we say force is the cause of motion, we are talking metaphysics; and this definition, if we had to be content with it, would be absolutely fruitless, would lead to absolutely nothing. For a definition to be of any use it must tell us how to measure force; and that is quite sufficient, for it is by no means necessary to tell what force is in itself, nor whether it is the cause or the effect of motion. We must therefore first define what is meant by the equality of two forces. When are two forces equal?.......]

But I don't get why is it not necessary to to tell what force is? Force is one of the most important concepts of Newtonian mechanics so why not define it? What did Newton the great, meant by using the word "force"?

  • See R.Westfall, Force in Newton's physics as well as F.De Gandt, Force and Geometry in Newton's Principia. – Mauro ALLEGRANZA Feb 10 at 13:41
  • Force, in N's mechanics is the cause of change of motion; see Def.iii, iv, v – Mauro ALLEGRANZA Feb 10 at 13:42
  • Since the pubblication of N's Principia, the debated raged concerning the "methaphysics of force". See e.g. M.Hesse, Forces and Fields. – Mauro ALLEGRANZA Feb 10 at 13:46
  • See also Poincaeré's Philosophy of Physics as well as Poincaré’s Relationism about P's views concerning "metaphysical entities" in physics: "These equations express relations, and if the equations remain true it is because these relations preserve their reality. They teach us, before and after, that there is such and such a relation between some thing and some other thing; only this something we used to call motion, we now call it electric current. 1/2 – Mauro ALLEGRANZA Feb 10 at 13:57
  • But these names were only images substituted for the real objects which nature will eternally hide from us. The true relations between these real objects are the only reality we can attain to, and the only condition is that the same relations exist between these objects as between the images by which we are forced to replace them. If these relations are known, what does it matter if we deem it convenient to replace one image by another? (1900: “Sur les Rapports de la Physique Expérimentale et de la Physique Mathématique”)" 2/2 – Mauro ALLEGRANZA Feb 10 at 13:59
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Poincaré is advocating for what is now called structural realism. Structural realism says that the mathematical properties of mind-independent objects are real, and they are all that is truly knowable about those objects. Physics deals with (a subset of) mathematical structures. A mathematical structure is a set of objects (or multiple sets of different types of objects), together with mathematical properties, relations, and functions defined on these objects. The world has structure, our theories have structure, and when the structure of our theories match up with the structure of the world, our theories are true. Questions about what force "really is" either reduces to structural questions about the mathematical properties, relations, and functions that force is involved with, or else cannot be known. (Personally, given some kind of loose distinction between primary and secondary qualities, I have a hard time even imagining what an answer to the question could even look like. Are we asking what it's like to be a force from the inside, as though forces somehow have qualia?)

I recall Richard Feynman expressing a similar view about energy. Feynman said that energy is a property of a system, such that if the system is closed, and we fast forward or rewind the system, every time we compute the total energy of the system the value we get will be the same. We could say more about energy, such as how it relates to heat, work, force, etc., but collectively, that just is what energy really is.

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  • Structural realism or conventionalism? – Geremia Feb 14 at 20:37
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This goes along with Feynman's criticism of vocabulary as a part of scientific knowledge, especially at a basic level. His story here illustrates why he thinks words out of context are a waste of time. What really happens, when you make a description, is that you discover what you can predict and understand based on a concept or measure, and that power sells the definition, even if it does not really fit any more ordinary definition you might propose.

We are misled, when a physics class starts with definitions, into thinking those are important basic concepts. Really, they are convenient things to slap labels on, that happen to have explanatory power. Consider Newton's definition of 'work', for instance (the movement of a mass over a distance). It is an important measure. It sort of needs a name, and for a very backward sort of perspective, that is sort of an appropriate name. But is that work? The difficulty of most of the things we consider work (and thus how much work is done by accomplishing them) is not directly related to how much mass moves how far...

So while it is a necessary motivation to look at a goal and start guessing measures that might capture the concepts implicit in the goal, it is a bad idea to become attached to the motivation, because the thing you define might grow and change and end up not really fitting the name or its more basic conceptual frame.

Modern physics is even more disconnected from common-sense definitions. We end up, for instance defining mass as a form of energy. So if you cling to the notion that mass is what makes things fall, you are going to be very confused. If you think mass is what you measure on a scale, you are wrong, we can go to space. If you think of mass as the source of gravity, you are going to have trouble with general relativity, where gravity and acceleration cannot be distinguished.

Advancing theory redefines your terms, as we have continued to modify our perspective on mass. So Poincare's approach is the one that prevails. Science has to be ready to give up common-sense definitions or metaphysical motivations, and go with the flow of theoretical modifications wherever that takes us.

He is mildly overstating his case. The metaphysical concept worked as motivation. Without the notion of force from common-sense experience, we would not have looked where we found what we needed. But to think of that as the proper definition of force for all time would eventually get in our way.

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