Axioms

Isaac Newton, in his Mathematical Principles of Natural Philosophy of 1687, used the heading “Axioms (or laws)” for what we know as the three laws of motion. The term “axiom” puzzles me, because an axiom should be a given, requiring no further proof, and used as a premise when reaching a conclusion. But in this case the axiom (or law) seems to be something that Isaac Newton is setting out to prove applies universally, to celestial bodies as well as objects on Earth.

Eric Schliesser is a world authority on the history and philosophy of early modern science. He has an interesting discussion of this topic here: Why does Newton use “Axioms or Laws of Motion”.

What I find interesting is the way that Newton is so careful in providing the logic of his conclusions. A well-known example is where, in describing the effect of gravity, he adds* “The reason for these properties of gravity, however, I have not yet been able to deduce from the phenomena, and I do not contrive [or feign, or frame, in different translations] hypotheses.”

The work is full of careful distinctions. In the General Scholium, at the end of the work, he says that gravity explains the orbit, but not the initial velocity, of planets: “but though these bodies may, indeed, persevere in their orbits by the mere laws of gravity, yet they could by no means have at first derived the regular position of the orbits from those laws.” The work also covers philosophical questions, for example:

  • what is space
  • what causes an object to have “gravity”?

As far as I can tell, these remain open questions today. Brian Cox, in Universal** (box 13), describes a field (like the electromagnetic, or Higgs, or inflaton field) as a mathematical representation of behaviour, but without prescribing the mechanism that creates it.

Isaac Newton, in Book III of Principia, gives four rules of reasoning, or ”philosophy”:

  1. That there ought not to be admitted any more causes of natural things than those which are both true and sufficient to explain their phenomena
  2. Accordingly, to natural effects of the same kind the same causes should be assigned, as far as possible
  3. The qualities of bodies that do not suffer intensification and remission, and that pertain to all bodies upon which experiments can be carried out, are to be taken as qualities of bodies universally
  4. In experimental philosophy, propositions gathered from the phenomena by induction are to be taken as true, whether exactly or approximately, contrary hypotheses notwithstanding, until other phenomena appear through which they are either rendered more accurate or liable to exceptions.

Without these rules, Newton’s calculations would show that the motion of celestial bodies is consistent with terrestrial gravity, but not necessarily as having the same cause.

I think we have been misled over the recent past into thinking that “Philosophy” is esoteric word play without fundamental significance. I prefer to think of philosophy as the method of reasoning we employ to reach useful conclusions. In a similar way, I think we have been misled into thinking that “History” is the study of arcane documents, like Gladstone’s diaries. I prefer to think of history as the narrative of events explaining where we are now.

It is interesting that scientific papers regularly start with statements of both philosophy and history. Here is an example: Hydrodynamic Quantum Analogs, by Professor John W.M. Bush, Professor of Applied Mathematics, MIT: “A decade ago, Yves Couder and Emmanuel Fort discovered that a millimeter-sized droplet may propel itself along the surface of a vibrating fluid bath by virtue of a resonant interaction with its own wave field, and that these walking droplets exhibit several features reminiscent of quantum systems. We here describe the walking-droplet system and, where possible, provide rationale for its quantum-like features. Further, we discuss the physical analogy between this hydrodynamic system and its closest relations in quantum theory, Louis de Broglie’s pilot-wave theory and its modern extensions”.

Indeed, you could almost say it is so obvious as to be hardly worth mentioning that a new theory needs to be grounded in a thorough understanding of past evidence, together with a valid way of reaching the conclusion from the data.

* Newton’s Principia, The Central Argument: Translation, Notes, Expanded Proofs (3rd Edition); Dana Densmore, trans. William H. Donahue; Published 2003 by Green Lion Press; ISBN 9781888009248.

** Universal: A Journey Through the Cosmos: Brian Cox, Jeff Forshaw; Published 2016 by Penguin; ISBN 9781846144363.

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