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Hilbert reconstructed Euclid's axioms. Is there an equivalent restructuring of Newton's axioms, or are they considered consistent?

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  • $\begingroup$ Newton's "axioms" are not axioms in the sense of mathematics. They are laws of nature, and their origin is experimental. Which experimental facts to choose as "axioms" of a physical theory is a question of convenience and of our understanding of nature. Modern exposition of classical mechanics is based on the mathematical formulation of Lagrange and Hamilton, rather than Newton's axioms. $\endgroup$ Sep 11, 2023 at 21:20
  • $\begingroup$ See e.g. Patrick Suppes etc, Axiomatic Foundations of Classical Particle Mechanics (1953) and H.A. Simon, The Axiomatization of Classical Mechanics (1954). $\endgroup$ Sep 12, 2023 at 7:15
  • $\begingroup$ And Leo Corry, David Hilbert and the axiomatization of physics (1894–1905) (1997) $\endgroup$ Sep 12, 2023 at 7:16
  • $\begingroup$ I was thinking of a situation where you abstained from comparing with reality in a study of Newtonian mechanics, but I realize from Allegranza's reference Corry that this may be difficult, in that we would still be dependent of some connection to reality supplied by the concepts of space and motion etc. $\endgroup$ Sep 14, 2023 at 13:58

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Would you consider clarifying the question itself to enable a more informative answer than this? Particularly, can you be more specific about what you mean by 'restructuring', in a sense that could equally apply to alterations in Euclid's axioms and/or in Newton's laws of motion?

In all three editions of Newton's 'Principia' the axioms that occur near the beginning are not just 'axioms', they are 'axioms or laws of motion'.

They have an essentially physical aspect that means they can't meaningfully be treated as purely mathematical constructs that might be differently chosen, they were selected for their compatibility with natural phenomena of motion and related physical behavior: and any alteration of them that is contemplated must clearly maintain that compatibility.

Having said that, the 'axioms or laws of motion' have often been re-expressed. They are usually expressed nowadays, especially in modern physics textbooks, in a way that looks very different indeed than their original expression by Newton. Yet it is widely considered that the differences are or ought to be of modernised form and expression only, and not of substance or core meaning. Yet arguments can arise over such questions as whether the modern formulations really do match in substance what Newton meant, and sometimes even what did he mean?

I will try to go further if you would clarify the question.

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  • $\begingroup$ I am not sure what the difference is. Euclid's axioms are not purely mathematical constructs either, they were selected for their compatibility with the properties of physical space. Both are mathematical idealizations of the respective properties/laws. And Newton's axioms can be differently chosen, and supplemented by making hidden assumptions explicit, while preserving the theorems, just as Euclid's were by Hilbert. Isn't that what authors in analytical mechanics did with Newton's? $\endgroup$
    – Conifold
    Sep 12, 2023 at 20:28
  • $\begingroup$ @Conifold I saw the question of difference arising especially in the OP's reference to 'consistency', as if inconsistency had been a motive for reconstructing Euclid's axioms. As far as I'm aware consistency has not been an issue in relation to the laws/axioms of motion. $\endgroup$
    – terry-s
    Sep 17, 2023 at 21:49

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