REVIEW 3 major objections 4 minor 6 references
Reply to a "Comment on 'Physics without determinism: Alternative interpretations of classical physics' "
T0 review · 3 major / 4 minor · reviewed 2026-08-27 · deepseek-v4-flash
Pith's one-line read The paper defends finite information quantities against a published objection by showing the criticism only hits independent bits, and proposes correlated majority-vote bits as a repair.
desk verdict An honest reply that concedes the comment's main point and sketches a plausible rescue for FIQs, but the rescue is only a sketch: no information measure for correlated bits, no proof of arithmetic closure, and an open parameter k. 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 central object is the finite information quantity (FIQ): a quantity $Q\in[0,1]$ written in binary as $Q=0.Q_1Q_2Q_3\cdots$, where each bit $Q_k$ carries an objective propensity $q_k$ to be 1, and the vector of propensities is required to have finite total information content. The earlier, criticized version added a sufficient condition: beyond some point all bits are equally likely to be 0 or 1, which made the bits mutually independent. The paper's repair replaces that condition with a time-indexed majority-vote construction: at instant $n$, the $n$th bit is the majority of the $k$ most recent random bits $r(n-k+1),\ldots,r(n)$. This keeps the information finite at every instant while introducing correlations between bits, and the paper claims ordinary arithmetic on such quantities yields new FIQs.
What would settle it
Take two FIQs whose bits are majority votes over the last $k$ random bits, add them with binary arithmetic, and compute the information content of the resulting propensity vector; if the total information diverges as $n$ grows for every odd $k$, the proposed repair is false.
Extended reading notes
Core claim
The paper's central claim is that the concept of a finite information quantity remains valid despite the criticism: the objection only refutes the sufficient condition that a FIQ's bits be mutually independent, not the necessary condition that its propensity vector have finite information content. The proposed repair defines the $n$th bit of a FIQ as the majority vote of the last $k$ random bits $r(n-k+1),\ldots,r(n)$, with $k$ odd, so at each time instant the quantity is fixed by finite information while its bits are correlated. The authors assert that two such FIQs can be added and multiplied with the usual arithmetic rules—as in a change of units—and that the result is again a FIQ, extending to all standard arithmetic. They explicitly leave open the characterization of the odd integer $k$.
Load-bearing premise
The repair stands or falls on the unproved assertion that majority-vote FIQs are closed under addition and multiplication while retaining finite information; the paper admits at the end that the characterization of the odd integer $k$ remains open.
Editorial extensions
If this is right
- If the repair holds, classical physics can be described with indeterministic quantities that still obey ordinary arithmetic, so changing units does not force a quantity to acquire infinite information.
- The model becomes explicitly time-dependent: each FIQ bit at time $n$ is determined by a finite block of past random bits, so indeterminism appears as a continuing stream of genuinely new information.
- The proposal strengthens the case that intuitionistic mathematics, where real numbers are unfinished processes rather than completed infinite objects, is a suitable language for physical quantities.
- The necessary condition of finite information content remains the minimal definition of a FIQ, so future models can drop the independence assumption without abandoning the framework.
Reading between the lines
- The majority-vote construction effectively makes every FIQ a discrete-time stochastic process with finite memory; the authors do not spell this out, but it would allow the model to be probed through autocorrelation or fluctuation statistics.
- Closure under addition and multiplication is the crux: if the set of majority-vote FIQs is closed under arithmetic, it constitutes a concrete 'finite-information real' number system; proving this is the obvious open problem the reply leaves.
- The unspecified window size $k$ could be read as a physical correlation length or resolution timescale; comparing the noise statistics of coupled quantities might bound $k$ in a given dynamical system.
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. This short note is a reply by Del Santo and Gisin to a comment by Callegaro et al. on their earlier paper introducing 'finite information quantities' (FIQs). The authors concede that the comment correctly identifies a weakness in the previously proposed sufficient condition for a FIQ, namely the assumption of mutually independent bits, which fails under basic arithmetic operations such as a change of units. They maintain, however, that the necessary condition for a FIQ—finite information content of the vector of propensities—remains a valid and useful concept. To overcome the criticism, they sketch a new construction, inspired by intuitionistic mathematics, in which the bits of a FIQ are generated by a majority vote of the last k random bits produced by nature at discrete times, thereby introducing correlations among bits. They assert without proof that two such FIQs can be added and multiplied using usual arithmetic rules and that the procedure extends to all standard arithmetic, while admitting that the characterization of the integer k remains open.
Significance. If the proposed majority-vote construction could be made rigorous, it would indeed rescue the FIQ concept from the comment's arithmetic-closure objection and would be a valuable contribution to the ongoing discussion on indeterministic physics. The reply is honest and clearly identifies the scope of the original weakness, which is commendable. However, the positive proposal is only a sketch: no information measure for correlated bits is defined, no closure proof is supplied, and the central claim that the FIQ concept remains valid is therefore not demonstrated in this manuscript. The reply functions more as a research agenda than as a working solution, and its significance is accordingly conditional on future work.
major comments (3)
- [Proposed refinement (majority-vote FIQs)] The paper asserts that two majority-vote FIQs 'can be added and multiplied ... resulting in new finite information quantities' and that this 'extends to all standard arithmetic', but provides no proof. Since the necessary condition for a FIQ is finite information content, and no information measure for correlated bits is specified (the measure used in Ref. [2] was explicitly for independent bits), the existence of the proposed objects as FIQs is not established. This is the load-bearing point of the reply's defense, so the central claim that the FIQ concept remains valid is unsupported.
- [Arithmetic closure] Standard binary addition and multiplication involve carries from lower-order bits, so the n-th output bit can depend on the entire lower-order history of the operands, potentially on unbounded history. The majority-vote rule supplies only k bits of memory for each input FIQ, but the paper gives no argument that this bounded memory is preserved under arithmetic operations. Without such an argument, the claimed closure under addition and multiplication is at risk, and the proposed rescue may collapse.
- [Open questions on k] The authors explicitly concede that 'the characterization of the odd integer k' remains open. This admission means the construction is incomplete as presented: without a specified or constrained k, the proposal cannot be tested on concrete physical models, and the concluding claim that 'FIQs remain a promising conceptual tool' is a conjecture rather than a demonstrated result. The reply would need to supply at least a partial characterization of k, or a proof that the construction is insensitive to k, to support its central defense.
minor comments (4)
- [Reference handling in text] In the paragraph on propensities, the text says 'contrarily to what the authors of the comment [2] maintain', but the comment being replied to is numbered [1]; this appears to be a reference error.
- [Notation for propensity vectors] The notation for the propensity vector is inconsistent: '[q1,q2,qk]' should presumably be '[q1,q2,q3,...]', and the sufficient-condition notation '[q1,q2,qk,qM,1/2,1/2,]' contains a missing ellipsis and an apparently spurious qk. These typographical issues obscure the mathematical definitions.
- [Formalization of necessary condition] The necessary condition for a FIQ is phrased informally as the information content being finite 'as expressed by some reasonable measure'. Since the reply's defense rests on this necessary condition, specifying the class of acceptable measures (or at least giving a concrete candidate for correlated bits) would make the argument more precise.
- [Brevity of the constructive proposal] The majority-vote construction is described in two sentences and relies on references to work in preparation. Given that this is the paper's main positive proposal, a short technical appendix with a formal definition and a worked example of addition would substantially strengthen the reply.
Circularity Check
The reply's rescue of FIQs rests on an unproved correlated-bit construction, imported from the authors' own forthcoming work, and on a definitional slide from 'finite information per time step' to 'finite information quantity.'
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self definitional
[Second page, unnumbered paragraph beginning 'One possible way to refine the definition of FIQs' (after the discussion of correlated bits)]
"In this way, at each time instant n, the FIQ is determined by a finite amount of information, and yet the bits of the FIQ are correlated. Two such basic FIQs can be added and multiplied ... resulting in new finite information quantities."
The FIQ definition stated earlier in the reply requires finite information content of the whole vector of propensities: 'It is necessary for a vector of propensities that its information content (as expressed by some reasonable measure) is finite.' The majority-vote construction only shows that each time-step bit is computed from the k last random bits, i.e. that each step uses a finite amount of information. That is not the same as the vector having finite total information content, and the reply leaves the required measure for correlated bits unspecified: 'a different measure of information content is required' is not defined.
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self citation load bearing
[Same paragraph, sentence 'as one of us (N.G.) is currently developing (see [4, 5])']
"One possible way to refine the definition of FIQs such that it fulfills the necessary condition but does not incur in the criticism raised in [1], is to take inspiration from intuitionistic mathematics, as one of us (N.G.) is currently developing (see [4, 5])."
The proposed repair is not derived in the reply; it is attributed to a line of work by one of the authors, with Ref. [5] listed as 'Manuscript in preparation.' The reply's conclusion that FIQs remain valid depends on this construction satisfying the necessary condition and being closed under arithmetic, but no proof or independent check is provided. The assertion that 'Two such basic FIQs can be added and multiplied ... resulting in new finite information quantities' is simply stated. A self-citation to an unpublished same-author manuscript cannot serve as independent support, so the central claim is load-bearing on the authors' own unverified framework.
full rationale
The reply correctly identifies that the external Comment attacks the sufficient condition of independent bits, and the necessary condition of finite information content is not itself shown to be false by the Comment. That gives the discussion some independent content. However, the proposed rescue is not a demonstrated derivation: the majority-vote construction is sketched, its information content for correlated bits is left undefined, and its closure under addition and multiplication is asserted rather than proved. In addition, the paper leans on the same authors' unpublished and forthcoming work as the source of the repair. These are real circularity-related weaknesses, but they do not make the entire reply vacuously circular, because the reply does concede the external criticism and the necessary condition remains a meaningful definition independent of the new construction. A score of 4 reflects partial self-citation and a definitional gap in the central repair, without claiming that the main claim is forced purely by definition.
Assumptions & free parameters
free parameters (1)
- k (majority-vote window)
assumptions (4)
- domain assumption Physical quantities are expressed in binary as Q=0.Q1Q2..., with each bit Qk governed by a propensity qk.
- domain assumption A finite information quantity is one whose vector of propensities has finite information content under some reasonable measure.
- domain assumption Propensities take values only in rational numbers and need not satisfy all Kolmogorov axioms.
- ad hoc to paper Nature has the power to continually produce genuinely new information as random bits r(n) at discrete times.
Cite this review
Pith. "Pith review of Reply to a "Comment on 'Physics without determinism: Alternative interpretations of classical physics' "." pith.science (2026). https://pith.science/paper/C6QDFNEQ
@misc{pith2026200912284,
author = {Pith},
title = {Pith review of: Reply to a "Comment on 'Physics without determinism: Alternative interpretations of classical physics' "},
year = {2026},
howpublished = {\url{https://pith.science/paper/C6QDFNEQ}},
note = {Machine review of arXiv:2009.12284}
}
read the original abstract
In this short note we reply to a comment by Callegaro et al. [1] (arXiv:2009.11709) that points out some weakness of the model of indeterministic physics that we proposed in Ref. [2] (Physical Review A, 100(6), p.062107), based on what we named "finite information quantities" (FIQs). While we acknowledge the merit of their criticism, we maintain that it applies only to a concrete example that we discussed in [2], whereas the main concept of FIQ remains valid and suitable for describing indeterministic physical models. We hint at a more sophisticated way to define FIQs which, taking inspiration from intuitionistic mathematics, would allow to overcome the criticisms in [1].
Reference graph
Works this paper leans on
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[2]
Physics without determinism: Alternative interpretat ions of classical physics
Callegaro, L., Pennecchi, F. and Bich, W. (2020). Comment on “Physics without determinism: Alternative interpretat ions of classical physics”. Phys. Rev. A, forthcoming
work page 2020
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[1]
by Callegaro et al. that points out some limits of the model of indeterministic physics that we proposed in Ref. [2]. In particular, their criticism demonstrates a weakness in the def- inition of finite information quantities (FIQs), which were in- troduced in that paper to address the fundamental problem that real numbers –assumed to be the values taken b...
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[3]
Del Santo, F. and Gisin, N. (2019). Physics without determin - ism: Alternative interpretations of classical physics. Phys. Rev. A, 100(6), 062107
work page 2019
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[4]
Gisin, N. (1991). Propensities in a non-deterministic phys ics. Synthese, 89(2), 287-297
work page 1991
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[5]
Gisin, N. (2020). Mathematical languages shape our underst and- ing of time in physics. Nature Physics, 16, 114116
work page 2020
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[6]
Gisin, N. (2020). Indeterminism in Physics and Intuitionis tic Mathematics. Manuscript in preparation
work page 2020
Reviewed August 27, 2026 · model on record in the stance chip above.
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