Plato’s Mathematical Psychophysics of Color

Round 1

Reviewer 1 Report (Previous Reviewer 2)

Comments and Suggestions for Authors

acknowledgement that the required changes have been fully completed

Reviewer 2 Report (Previous Reviewer 1)

Comments and Suggestions for Authors

I am satisfied with the author’s revisions and have no objection to the publication of this paper.

All best,

This manuscript is a resubmission of an earlier submission. The following is a list of the peer review reports and author responses from that submission.

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

The paper under review explores the possibility of an interesting intersection between classical philosophy, ancient Greek mathematics and contemporary theories of colour perception. It argues that while Aristotle is often regarded as a foundation figure for developing a naturalistic model for human psychology, greater attention must should be directed to Plato. This shift is justified, according to the author, by Plato’s reliance on Pythagorean principles of mathematical harmony, which equipped him with advanced mathematical tools, such as projective geometry, linear algebra and lattice theory. Within this context, the paper concludes that Plato’s approach offers a foundational basis for understanding colour phaenomena, as later explored by Goethe in the nineteenth century.

In my assessment, the paper demonstrates some notable strengths while also presenting areas that warrant further consideration. I will begin with what I see as its strengths:

a.      The central thesis—advocating for Plato’s mathematical contributions to the psychophysics of colour over Aristotle’s—is both innovative and thought-provoking. Equally compelling is the proposal to re-evaluate Plato’s late dialogues as early precursors to modern psychophysical theories.

 

b.      The author of this paper exhibits extensive interdisciplinary knowledge. The paper successfully integrates diverse fields, including philosophy, history of ancient and modern philosophy, history of ancient and modern mathematics, modern theories of light and vision, etc. Such thematic breadth underscores the author’s capacity to navigate complex, multifaceted subject matter—a task that is both challenging and commendable.

Let me now address areas that I believe warrant further attention:

a.      While the paper attempts to establish a clear lineage of ideas leading to contemporary theories, it risks overextending the relevance of Plato’s contributions by presenting them as direct precursors to modern psychophysical frameworks. In other words, while the attempt to demonstrate continuity between ancient and modern thought is commendable, it may inadvertently conflate historical contexts, potentially misrepresenting the intentions of both ancient (Plato and Aristotle) and contemporary thinkers. For instance, linking Plato’s use of harmonic ratios to modern concepts such as group theory or projective geometry introduces anachronism; these ancient ideas were not formulated with such modern structures in mind. Similarly, the application of terms like "group homomorphism" and "non-Euclidean geometry" imposes contemporary interpretative frameworks onto ancient Greek mathematics. Likewise, Wittgenstein’s exploration of language games operates within an entirely different epistemological (and ontological, in fact) paradigm, emphasizing the limits and structures of language rather than metaphysical and mathematical harmonics. The paper notably lacks a discussion of the historical development of optics from Aristotle to modernity, overlooking significant contributions from not only later Greek thinkers but also from ancient Arabic and Latin sources. These texts, which bridge the gap between classical antiquity and modern thought, are essential for understanding the evolution of ideas about light and colour perception. Incorporating these contributions would offer a more comprehensive historical context, enrich the paper’s analysis and potentially support the author’s thesis.

b.      The use of the term “algebra” in discussion of Plato’s or Aristotle’s work raises concerns, as algebra, in its formalized form as a method for solving arithmetical problems did not exist during their time—there is an enormous recent bibliography on the topic; see for example, the latest edition of Diophantus by Oaks and Christianidis. The invocation of algebraic concepts, such as ‘group theory’ or ‘homomorphism’ conflates contemporary mathematical frameworks with ancient philosophical explorations. Similarly, the application of terms like ‘lattice theory’ or ‘abstract algebra’ to describe Plato’s framework suggests a level of abstraction and generalization that ancient mathematics simply did not achieve (see, l. 65: ‘an applied form of abstract algebra’, l.364-5 ‘algebraic irrationals’, and l.303-307 the correlation between harmonic/arithmetic means to homomorphism).

Although the author acknowledges this issue (see, for example, l.270), the treatment of this problem appears insufficient. Thus, the paper would benefit from a cleaner distinction between the mathematical terminology and frameworks that existed in Plato’s time and the modern concepts being applied.

c.      In relation to the previous point: The paper relies on outdated or non-historical references for topics related to the history of ancient Greek mathematics. There is an extensive recent body of literature on incommensurability, Pythagorean numeracy, and the Delian problem that the author neglects. This oversight includes both primary sources and more recent historical studies, which results in significant inaccuracies. For instance, the paper misrepresents the so-called "practical character" of the solutions to the Delian problem (l. 443) as discussed by pseudo-Eratosthenes (see, the distinction between mechanical solutions (theoretical constructs) and practical solutions in Knorr’s The Ancient Tradition of Geometric Problems). Moreover, some ideas presented in the paper are framed as facts, though they are far from being widely accepted or supported by consensus; for example, the claim that Pythagorean harmonics derive from Mesopotamian thought. Additionally, the paper overlooks an important aspect of Greek mathematics: the distinction they made between irrational magnitudes and numbers. For the Greeks, irrational magnitudes were not considered numbers in the modern sense. This distinction is crucial to understanding why what we now call irrational numbers could not be described were only representable geometrically; see, for example, Meno 388-392.

d.      Certain mathematical sections (e.g., the discussion of group homomorphism and harmonic cross-ratios) are dense and may benefit from simplified explanations or diagrams to make them more accessible to a wider academic audience. Additionally, the paper’s dismissal of Aristotle’s influence on modern thinkers feels somewhat abrupt. The paper could benefit from a lengthier presentation of why other thinkers believe that it was Aristotle who played a more significant role and not Plato.

e.      Some particular points that the author might need to address include: Which passage from Timaeus supports the explicit idea that colours followed the creation of the world’s soul in mathematical sense? Clarifying this point would strengthen the argument. Additionally, the claim that Plato dismisses the practical benefits of mathematics requires further justification as there are several passages in book 7 of the Republic against this idea.

 

In conclusion, while this paper presents an original and thought-provoking idea, it is, in my view, not yet ready to be published. Significant further research is required to refine the historical contexts involved, particularly in relation to the influence of Plato’s work on modern thought. Additionally, the references to modern concepts and frameworks should be reconsidered and more carefully framed to avoid anachronisms. Clarity in terminology is essential, as the author’s use of certain terms—such as "algebra," "group homomorphism," and others—appears to impose modern interpretations onto ancient texts. The author would benefit from rethinking these terms and considering whether they are necessary or helpful in supporting the argument. Overall, a more nuanced and historically informed approach, alongside a careful revision of terminology, would improve the paper significantly.

 

Reviewer 2 Report

Comments and Suggestions for Authors

As I  I cannot share the author's argumentation as a whole (in the perspective  of one who has studied thoroughly the Timaeus  in the context both of Plato's philosophy  and the ancient cosmology and science), I do not have any specific suggestions for it to be improved.  (however, I might also offer a few specific remarks if the author believe it should be helpful) 

My main point is that  I deem it wrong to project onto the cosmology as well as onto the psychology of the Timaeus mathematical notions and operations that did not belong at all to the mathematicians of the 5th and 4th centuries, nor of Greek antiquity in general. The best historians and philosophers of ancient mathematics  believe (at least from Jacob KLein) that the algebric  Revolution (as well as the mathematical analysis later on) arises from that crucial change in the concept of number that was made possible in Early Modern Age by the reception of Greek mathematics  through Arabic Science. Given that, any attempt to find  anticipations of modern mathematical constructions  in the ancient scientific literature should be considered anachronistic: even more so if this attempt  is based on such elaborate and long-winded demonstrations as those found in this paper.

Moreover, one need not look too far to find a scientific background  that sufficiently explains Plato's approach . One only has to look to Pythagorean mathematics, which is known to be the illustrious and undoubted precedent (the author knows this well) for the theory of musical accords which  Plato elaborates on.

Moreover, the author cites no literature on ancient mathematics except for a few titles on the Pythagoreans. Therefore he/she also misses (but this should not be significant?) the essential function of Plato's resorting to mathematical concepts and constructions in his dialogue: a function which was clarified, to take an illustrious example, by Geoffrey Lloyd in an essay of many decades ago on Mathematics and nature in the Timaeus, in which it is definitely shown that in this dialogue mathematics serves, for Plato, as the foundation of a representation of the regularity of the cosmic order. Nothing less, but also nothing more.