REVIEW 3 major objections 6 minor 4 references
On the independence problem of Newton's first law
T0 review · 3 major / 6 minor · reviewed 2026-08-06 · deepseek-v4-flash
Pith's one-line read The paper argues that Newton's first law was not redundant: in the Principia's Euclidean mathematics, zero magnitudes cannot enter a proportion, so the second law simply did not cover the no-force case.
desk verdict A genuinely new formal explanation for why Newton kept the first law separate, built on Euclid's exclusion of zero from proportion; the historical necessity claim is plausible but overstated. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The load-bearing object is Euclid's definition of proportion (Elements, Book V, Definitions 4-6): magnitudes have a ratio only if, when multiplied, they are capable of exceeding one another, and proportional magnitudes are those with the same ratio. Because a zero magnitude cannot exceed or be exceeded by anything when multiplied, it has no ratio; the second law's statement that change of motion is proportional to force is therefore undefined at zero. The paper also relies on the Euclidean convention that zero is the absence of a magnitude, a separate case from any nonzero magnitude, and on evidence that Newton treated ratios as geometric relations rather than numbers.
What would settle it
A single clear passage in the Principia in which Newton applies the second law, or the term "proportion," to a zero force, zero change of motion, or a vanished quantity as a legitimate proportion would refute the formal explanation; the paper itself cites only the Scholium where Newton denies that vanished quantities have an ultimate proportion.
Extended reading notes
Core claim
On the paper's own terms, the central discovery is that the independence problem dissolves once the second law is read through Euclid's Book V definitions: a zero magnitude has no ratio to anything, and therefore cannot be in proportion, so "change in motion is proportional to motive force impressed" applies only when both magnitudes are nonzero. Since the case of no force and no change of motion is excluded from the second law by definition, Newton needed a separate axiom for it, and the first law supplied exactly that. The paper argues from the Principia's text, from Newton's treatment of vanishing ratios in Book I, and from historical evidence that Newton kept the Euclidean conception of proportion as a relation among nonzero geometric magnitudes rather than an algebraic relation among numbers.
Load-bearing premise
The argument stands or falls on reading Newton's word "proportional" strictly by Euclid's definitions, which exclude zero magnitudes from any ratio; if Newton used "proportional" in a looser, proto-algebraic sense, the second law could cover the zero case and the formal explanation collapses.
Editorial extensions
If this is right
- If the formal explanation is right, Newton's first law is not a redundant corollary of the second; the redundancy is an artifact of translating the Euclidean proportion statement into the algebraic equation $F=ma$.
- Newton's silence about the apparent redundancy is explained: to a 17th-century Euclidean reader, no contradiction or duplication arose, because zero simply fell outside the second law.
- The explanation predicts that any use of the second law in the Principia concerns nonzero forces and nonzero changes of motion, so no passage should apply it to the zero case.
- The logical explanation implies that the first law is the more fundamental axiom: alternative force laws such as $F=\sqrt{m}a$ or $F=ma^2$ are compatible with it, and indeed relativity replaced the second law while leaving the first intact.
Reading between the lines
- An implication the authors leave implicit: the same zero-exclusion would apply to any other proportionality law expressed in Euclid's idiom, so the argument could be tested by checking other Principia statements for a parallel gap at zero.
- If the formal explanation is correct, modern textbook presentations that derive the first law from $F=ma$ teach a redundancy that Newton's own formalism never had; the independence problem may be a translation artifact rather than a historical puzzle.
- A testable extension would be a complete search of Newton's manuscripts for any instance where he treats a zero magnitude as admitting a ratio or proportion; one clear instance would weaken the historical claim.
- The same Euclidean convention may bear on debates about whether Newton's other axioms, such as the third law, have hidden scope conditions for zero magnitudes, a question the paper does not address.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper addresses the 'independence problem' of Newton's first law: why Newton stated Law I as a separate axiom when substituting F=0 into the modern algebraic form of Law II (F=ma) seems to make Law I a corollary. The central contribution is a 'formal explanation': Newton's Law II is not an algebraic equation but a verbal statement that change in motion is proportional to impressed force, and because Euclid's definitions of ratio and proportion (Elements V, Defs. 4–6) exclude zero magnitudes, Law II does not apply to the case of zero force or zero change of motion. The first law was therefore needed as a separate axiom to cover that case. The paper also proposes a secondary 'logical explanation' (Law I expresses a more general principle that could survive a changed Law II), and it provides a comprehensive review and plausibility assessment of earlier proposed solutions.
Significance. The formal explanation, if correct, is an original and elegant resolution of a long-standing puzzle, and it is genuinely historical: it locates Newton's reason in his mathematical language rather than in modern concepts. The paper's strengths are its careful distinction between historical and modern questions, its comprehensive and fair literature review, and its reliance on external evidence (Euclid, the Principia, and independent scholarship by Sylla and Grosholz). The derivation from Euclid's definitions is internally valid, and there is no circularity: the argument does not assume the conclusion. However, the central historical claim is not as secure as Section 4.2 states. The evidence connects Newton to the Euclidean conception of ratio in general, but not specifically to his intent in wording Law II; the discussion of the Scholium on ultimate ratios underplays how Newton extended ratio talk to vanishing quantities. The formal explanation is therefore a well-supported reconstruction rather than a firmly established necessity.
major comments (3)
- [Section 3.1] The load-bearing step is the inference from Euclid's Definition 4 to the claim that Newton's Law II is inapplicable when force or change of motion is zero. The paper shows that Euclid excludes zero magnitudes from ratios and that Newton generally used the Euclidean ratio idiom, but it does not directly show that Newton intended Law II's word 'proportional' to carry that technical exclusion. A proto-algebraic or looser reading of 'proportional' remains possible. The Scholium passage quoted in Section 3.1 concerns ultimate ratios of vanishing quantities, not the exact zero-force case; the paper mentions Newton's reply but does not analyze how the method of first and ultimate ratios, which was expressly designed to assign limiting ratios to evanescent quantities, bears on the status of exact zero. A reader can accept all the cited evidence and still hold that Newton could have treated F=0 as a limiting case. This step needs either direct textual support or a revised, weaker formulation of the claim.
- [Section 4.2] The assertion that the formal explanation 'has been firmly established' overstates what the evidence shows. Sylla (1984) and Grosholz (1987) demonstrate that Newton treated ratios as geometrical magnitudes and used proportion idiom, but neither source directly addresses Law II or the zero-force case. The textual evidence is compatible with the formal explanation, but it does not prove that Newton consciously excluded zero from Law II. The abstract's phrase 'necessitate the inclusion' is likewise stronger than the argument supports. I recommend replacing 'firmly established' with a more calibrated claim such as 'the best-supported explanation' or 'highly plausible', and explicitly acknowledging that the proto-algebraic reading of proportion is a live alternative.
- [Abstract and Section 3.1] Even if Euclidean proportion excludes zero, it does not follow that the first law had to be a separate axiom rather than, say, a corollary appended to Law II or a clause added to Law II's wording. The paper asserts in Section 3.1 that 'the separation into two axioms is the simplest and most natural choice', but that is a rhetorical claim, not a demonstration of necessity. If the paper wishes to maintain that the definitions of Euclidean geometry 'necessitate' the inclusion of the first law, it should argue why a separate law, rather than any other formal device, was required; otherwise the conclusion should be weakened to explain why a separate law was a natural consequence of the Euclidean framework.
minor comments (6)
- [Section 4.1] There is a typo in the paragraph on Grosholz: 'for for thePrincipia' should read 'for thePrincipia'.
- [Section 4.2] The text contains several typographical errors: 'categeory' should be 'category', 'prequisite' should be 'prerequisite', and 'the first is law is redundant' should be 'the first law is redundant'.
- [Section 3.2] The sentence 'It can continue to to hold even if...' contains a doubled 'to'; it should read 'It can continue to hold even if...'.
- [Footnote 5] The quote from Aristotle is attributed as '(Aristotle 2000, 46)' but the reference list entry is 'On The Heavens' with no page-level detail; please check the citation format.
- [Declarations] The heading 'F unding' has an unintended space; it should read 'Funding'.
- [Section 2.2.3] The sentence 'Descartes is identified as the source of the first law and the notion that a uniform motion constitutes a state' lacks a specific citation; please add a supporting reference or clarify the source of this identification.
Circularity Check
No circularity: the formal explanation is anchored in external sources (Euclid, Principia text, Sylla, Grosholz), and no fitted quantity or self-citation chain is load-bearing.
full rationale
Walking the paper's derivation chain: the formal explanation begins with Euclid's Book V definitions of ratio and proportion, argues that a zero magnitude cannot stand in proportion, observes that Newton's second law is stated as a proportion, and concludes that the second law therefore does not cover the zero-force case, leaving that case for the first law. Each link in this chain is supported by external textual evidence: Euclid's definitions are quoted directly, Newton's wording of Law II is quoted, the Scholium on ultimate ratios is quoted, and the historical claim that Newton adhered to the Euclidean 'first tradition' of proportion is supported by citations to Sylla (1984) and Grosholz (1987), neither of whom is an author of this paper. None of these sources already contains the paper's target conclusion that Law I was included because of a technical requirement of Euclidean geometry; they supply premises about what proportion meant, not the conclusion about Newton's reason for separating the laws. The paper also explicitly distinguishes its account from the similar earlier account of Koslow, arguing that Koslow does not identify the root cause in the mathematical language, which is a substantive differentiation rather than a renaming. The 'logical explanation' is presented as a speculative supporting idea and is not used to derive the formal explanation. There are no fitted parameters, no equations that reduce to their own inputs, and no load-bearing self-citations; the only author-identity overlap concerns the authors' own contributions, which are not cited as evidence. The main epistemic weakness is that Section 4.2's claim that the formal explanation 'has been firmly established' overstates the strength of the indirect historical evidence, especially given that Newton's own method of ultimate ratios shows he was comfortable extending ratio concepts to vanishing quantities. This is a concern about interpretive overconfidence and historical correctness, not about circularity.
Assumptions & free parameters
assumptions (3)
- standard math Euclidean proportion requires both magnitudes in a ratio to be non-zero (Euclid, Elements, Book V, Def. 4).
- domain assumption Newton's use of 'proportion' in the second law follows Euclid's Book V definitions rather than a proto-algebraic reading.
- domain assumption Newton maintained a strict separation between zero and non-zero magnitudes in his mathematical treatment of physical quantities.
Cite this review
Pith. "Pith review of On the independence problem of Newton's first law." pith.science (2026). https://pith.science/paper/VZCOBS5D
@misc{pith2026250704282,
author = {Pith},
title = {Pith review of: On the independence problem of Newton's first law},
year = {2026},
howpublished = {\url{https://pith.science/paper/VZCOBS5D}},
note = {Machine review of arXiv:2507.04282}
}
read the original abstract
Newton's laws of motion pose an apparent problem, sometimes referred to as "the independence problem": the first law seems to be a simple consequence of the second law, raising the question of why it was included as a separate law. Numerous answers to this question have been proposed in the literature. The main contribution of this paper is a novel answer which we call "the formal explanation." Unlike previous accounts it relies on mathematical formalism and argues that the definitions of Euclidean geometry necessitate the inclusion of the first law. We provide evidence in support of this claim. A second contribution is a comprehensive review of previously suggested explanations, which so far have often been treated in a fragmented manner, and a discussion of the plausibility of the various answers.
Reference graph
Works this paper leans on
-
[1]
Infomotions, Inc., South Bend, UNITED STATES (2000)
Aristotle: On The Heavens. Infomotions, Inc., South Bend, UNITED STATES (2000). http://ebookcentral.proquest.com/lib/tau/detail.action?docID=3314398 Arons, A.B.: A Guide to Introductory Physics Teaching. Wiley, New York (1990) Bergmann, P.G.: Introduction to the Theory of Relativity. Dover classics of science and mathematics. Dover Publications, New York ...
work page 2000
-
[566]
Cambridge University Press, Cambridge (2014) Koslow, A.: The Law of Inertia: Some Remarks on Its Structure and Significance (1969) Koyr´ e, A.: Newtonian Studies, 1st phoenix ed. edn. Phoenix books. University of Chicago Press, Chicago (1968) Kuhn, T.S.: The Road Since Structure: Philosophical Essays, 1970-1993, with an Autobiographical Interview. Univers...
work page 2014
-
[589]
Metaphysics Research Lab, Stanford University (2020)
Green Lion Press, Santa Fe, N.M (2003) DiSalle, R.: Space and Time: Inertial Frames. Metaphysics Research Lab, Stanford University (2020). https://plato.stanford.edu/archives/win2020/entries/ spacetime-iframes/ Earman, J., Friedman, M.: The meaning and status of Newton’s law of inertia and the nature of gravitational forces. Philosophy of science40(3), 32...
work page 2003
-
[786]
University Science Books, Sausalito, Calif (2005) Zinkernagel, H.: Causal fundamentalism in physics. EPSA philosophical issues in the sciences: Launch of the European philosophy of science association, 311–322 (2010) 17
work page 2005
Reviewed August 6, 2026 · model on record in the stance chip above.
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