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REVIEW 2 major objections 4 minor 53 references

Physical quantities as a partially additive field

T0 review · 2 major / 4 minor · reviewed 2026-08-09 · deepseek-v4-flash

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Pith's one-line read Physical dimensions and units can be derived from a field whose addition is only partially defined, rather than being postulated as extra structure.

desk verdict A clean new algebraic axiomatization of physical quantities whose central derivation leans on a strong associativity convention that deserves a more explicit physical defense. read the letter →

arxiv 2502.00967 v1 pith:U2YMBMTP submitted 2025-02-03 math-ph math.MPphysics.hist-ph

Georgy Alymov This is my paper · ORCID
classification math-phmath.MPphysics.hist-ph MSC 08A55
keywords physicalquantitiesquantitycalculusdimensionalanalysispartialadditionpartiallyadditivefieldunitsystemdimensionsaxiomatization
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper tries to show that the familiar hierarchy of physical quantities—dimensions, units, numerical values—can be derived from one algebraic structure instead of being put in by hand. The structure is a field whose addition is allowed to be undefined for some pairs, called a partially additive field. The claim is that, from this small modification alone, dimensions appear as the equivalence classes of quantities that can be added, dimensionless quantities form an ordinary field, and every quantity factors uniquely into a dimensionless value and a unit. If the derivation works, quantity calculus becomes a theorem about a single algebraic object rather than a collection of separate postulates.

What carries the argument

The central object is the partially additive field: a set with total multiplication, partial addition, and a separate additive zero $0_a$ for each element $a$, with all axioms read in the strong sense that an equation holds only if both sides are defined and equal or both sides are undefined. The load-bearing mechanism is Lemma 3, which proves that $a+b$ is defined if and only if $0_a = 0_b$; this transitivity of summability turns dimensions into equivalence classes. Theorem 11 then uses multiplicative invertibility of nonzero elements to factor every quantity as a dimensionless value times a unit, and the unit-system axioms in Section 3 convert that factorization into ordinary quantity calculus whenever a coherent unit system exists.

What would settle it

Take the axioms of a partially additive field but replace strong associativity by the weaker convention that $(a+b)+c = a+(b+c)$ is required only when both sides are defined; a model with two elements that share a zero yet are not summable would satisfy the weaker axioms but violate Lemma 3, showing that the derivation of dimensions as summability classes collapses under that reading.

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Extended reading notes

Core claim

On the paper's own terms, the discovery is that the textbook scaffolding of quantity calculus—dimensions, units, numerical values—is not an additional layer on top of field arithmetic: it is forced by making addition partial. Theorem 9 shows that the elements summable with the multiplicative identity $1$ form a field, namely the field of dimensionless quantities. Lemma 3 shows that $a+b$ is defined exactly when the additive zeros of $a$ and $b$ coincide, so mutual summability is an equivalence relation; Theorem 11 then represents every element as $a = v_a \times u_a$ with $v_a$ dimensionless and $u_a$ a unit. Corollary 14 identifies dimensions with the quotient of nonzero elements by nonzero dimensionless elements, making the dimensions a commutative group. The paper takes this as evidence that partial operations give a more economical axiomatization of physical quantities than postulating dimensions and units separately.

Load-bearing premise

The argument depends on reading every equality that involves an undefined sum in the strong sense: an identity holds only if both sides are defined and equal or both sides are undefined; if associativity were required only when both sides are defined, Lemma 3 would no longer force dimensions to be equivalence classes.

Editorial extensions

If this is right

  • Dimension ceases to be primitive: two quantities have the same dimension exactly when their sum is defined, so 'you can only add apples to apples' becomes a theorem rather than a rule.
  • Every quantity is built from a dimensionless numerical value and a unit, and the usual rule that numerical values add only for equal units falls out of Theorem 11.
  • The dimensionless quantities themselves form an ordinary field, so the dimensionless field can be chosen freely—real, complex, computable, or rational functions—without changing the dimensional scaffolding.
  • A coherent unit system exists whenever the nonzero dimensionless elements form a cotorsion group, which covers the standard choices $\mathbb{R}\setminus\{0\}$ and $\mathbb{C}\setminus\{0\}$, so familiar arithmetic with physical quantities is recovered.
  • Making multiplication partial as well (the fieldoid generalization) puts non-multipliable quantities into completely disjoint algebras, which the paper offers as an explanation of why non-multipliable quantities do not appear together in one physical theory.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Beyond the paper: the strong-associativity reading is doing real logical work; if one adopts the weaker partial-algebra convention that associativity is required only when both sides are defined, the derivation of dimensions from summability collapses.
  • Beyond the paper: because the existence of a coherent unit system depends on the cotorsion of the dimensionless multiplicative group, the framework suggests that familiar real and complex scalars are convenient but not uniquely privileged choices.
  • Beyond the paper: a natural next step, not taken here, is to axiomatize exponentiation of quantities so that non-integer and transcendental powers of dimensions become theorems rather than additional postulates, which would connect directly to fractional calculus and critical phenomena.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 4 minor

Summary. The manuscript introduces 'partially additive fields': sets with total multiplication, partial addition, a per-element zero 0_a, and an equality convention for all axiom equations under which both sides are defined and equal or both are undefined. It proves that summability coincides with equality of zeros (Lemma 3), that dimensionless elements form a field (Theorem 9), that every element factors uniquely as a dimensionless value times a selected unit (Theorem 11), and that dimensions, defined as summability classes, form a commutative group (Corollary 14). It also analyzes coherent unit systems, giving sufficient conditions (Theorem 31), and generalizes to 'fieldoids' where multiplication is also partial (Appendix A). The paper's thesis is that dimensions and units need not be postulated separately but emerge from a partially additive field structure.

Significance. If accepted, the paper offers a clean algebraic unification of quantities, numerical values, units, and dimensions in one structure, with a genuinely derived representation theorem and no fitted parameters. The proofs are generally coherent, and the appendices extend the framework nontrivially. The main caveat is that the claimed 'emergence' of dimensions is conditional on axiomatic choices: the per-element zero axioms and the strong associativity convention. The significance of the contribution therefore lies in the unified structure and in the derived theorems, rather than in an unconditional derivation of dimensional analysis from ordinary field axioms.

major comments (2)
  1. [Definition 1, Lemma 3] The strong associativity convention in Definition 1 is load-bearing: the proof of Lemma 3 uses it to infer, from a = a + (b + (-b)), that the parenthesization (a + b) + (-b) is defined, and conversely to rearrange a + b = a + 0_a + b into (a + b) + 0_b. Under the weaker partial-algebra convention in which associativity applies only when both sides are already defined, summability is not forced to be transitive, and the equivalence classes of Definition 13 need not exist. Since Lemma 3 underpins Theorem 11 and Corollary 14, the strong reading is a substantive axiom rather than a bookkeeping device. The paper should either justify the strong reading as the physically appropriate one (definedness of physical addition is an equivalence relation, so the strong form is exactly the algebraic translation) or explicitly present it as an additional modeling assumption and soften the claim that dimensions 'arise naturally' from field axioms alone.
  2. [Definition 1, Lemma 6, Section 4] The axioms already encode the dimensional partition: each element a is assigned a unique zero 0_a, and Lemmas 3 and 6 identify the summability classes with these zeros; Section 4 explicitly notes the one-to-one correspondence between dimensions and zero quantities. Thus the claim in the abstract and conclusion that dimensions 'do not have to be explicitly postulated' overstates what is derived. The partition of Q into dimensions is present in the zero structure, while what is genuinely derived is the field structure of dimensionless elements, the unit-value representation, and the group structure of dimensions (Corollary 14). I recommend qualifying the 'emergence' language accordingly.
minor comments (4)
  1. [Definition 1] The equality convention 'both sides are defined and equal, or both sides are undefined' should be formalized: since u is an element of Q_u, it should be stated explicitly that 'undefined' means 'equal to u' and that equations with u on both sides are permitted.
  2. [Lemma 26] The displayed identity '0_a × 0_a = 0_a × a + 0_a × (-a) = a × a + (-a) × a + 0_a × (-a)' appears to contain a typo or a missing intermediate step; please check and clarify.
  3. [Section 3] The statement that 'divisible groups, finite groups, and their direct products are cotorsion' is terse; a one-sentence explanation of why cotorsion gives the desired splitting would help readers.
  4. [Theorem 31] Condition 2 of Theorem 31 is a strong global homogeneity assumption; a brief comment on its physical status and on how restrictive it is compared with the alternatives discussed in Section 3 would be useful.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the paper's theorems are explicit consequences of the stated axioms, not disguised restatements of the conclusions.

full rationale

The paper derives dimensions and units from Definition 1's axioms, which contain no dimension, unit, or value primitives. Lemma 3 proves that summability coincides with equality of local zeros, and transitivity follows; only then does Definition 13 introduce dimensions as equivalence classes. Theorems 9 and 11 are proved by explicit algebraic manipulations from those axioms and lemmas. No parameter is fitted to data, no previous result by the same author is cited as load-bearing, and no uniqueness theorem is imported to force a choice. The strong associativity convention in Definition 1 is a genuine axiom rather than a hidden restatement of the target result: it constrains the partial operation, and the paper shows the dimensional structure as a consequence. One may debate whether this axiom is physically motivated, but that is a modeling assumption, not circular reasoning. The paper itself flags speculative parts (footnote 5), which further supports that the derivation is not presented as forced by prior conclusions.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The paper is pure mathematics; there are no fitted numbers and no invented physical entities. The central claim rests on the axioms of Definition 1, especially strong associativity and per-zero nontriviality, plus additional structural axioms introduced for the coherent-unit-system existence proof in Section 3 and Appendix B. All of these are assumptions rather than results, and the coherence axioms in particular are chosen ad hoc to make the existence proof go through.

assumptions (7)
  • domain assumption Physical quantities form a partially additive field (Axioms of Definition 1: partial addition, total multiplication, local unique zeros, global 1, inverses, and nontriviality per zero).
    The paper postulates this structure for physical quantities; it is not derived from physics. The central derivation of dimensions and units depends entirely on this axiomatic modeling choice.
  • ad hoc to paper Strong associativity of partial addition: (a+b)+c = a+(b+c) with the reading 'both sides are defined and equal, or both sides are undefined' (Definition 1).
    This strong-form associativity is load-bearing for Lemma 3, which makes summability equivalent to sharing a zero and defines dimensions as summable classes. Weaker partial-algebra associativity would break the derivation, and the paper gives no independent physical justification for choosing the strong reading.
  • domain assumption Nontriviality: every zero is the zero of some non-zero element, i.e. ∀a∈Z ∃b∈Q\Z with a = 0_b.
    Ensures every dimension class contains non-zero quantities, matching physical intuition but not derived from the other axioms.
  • ad hoc to paper For coherent unit systems, the multiplicative group of non-zero dimensionless elements is cotorsion (Section 3).
    Introduced as a sufficient condition for the existence of a coherent unit system. It covers R\{0} and C\{0} but is an external constraint on the field of dimensionless quantities, not a consequence of the partially additive field axioms.
  • ad hoc to paper Theorem 31 condition 1: no dimensionful roots of dimensionless elements, i.e. for all n and all dimensionful a, a^n is dimensionful.
    Auxiliary axiom used to prove existence of a coherent unit system. It excludes situations such as (4 m^2)^(1/2) = 2 m, which occur in real physics, and is therefore restrictive.
  • ad hoc to paper Theorem 31 condition 2: any two non-zero dimensionful elements of the same dimension agree on the existence of nth roots.
    Auxiliary non-discrimination axiom introduced solely to make the divisibility argument in Theorem 31 work. It is not motivated by physical necessity and functions as a technical condition.
  • standard math Divisible subgroups of an abelian group are direct summands (used in Theorem 31, cited to Fuchs [35]).
    A standard group theory result invoked to decompose the multiplicative group of non-zero elements into dimensionless factors times a unit group.

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Cite this review

Pith. "Pith review of Physical quantities as a partially additive field." pith.science (2026). https://pith.science/paper/U2YMBMTP

@misc{pith2026250200967,
  author       = {Pith},
  title        = {Pith review of: Physical quantities as a partially additive field},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/U2YMBMTP}},
  note         = {Machine review of arXiv:2502.00967}
}
read the original abstract

We generalize the concept of a field by allowing addition to be a partial operation. We show that elements of such a "partially additive field" share many similarities with physical quantities. In particular, they form subsets of mutually summable elements (similar to physical dimensions), dimensionless elements (those summable with 1) form a field, and every element can be uniquely represented as a product of a dimensionless element and any non-zero element of the same dimension (a unit). We also discuss the conditions for the existence of a coherent unit system. In contrast to previous works, our axiomatization encompasses quantities, values, units, and dimensions in a single algebraic structure, illustrating that partial operations may provide a more elegant description of the physical world.

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