Could There Be Method Behind Kepler’s Cosmic Music?
Round 1
Reviewer 1 Report
Comments and Suggestions for Authors
I think the primary contribution of this article is its focus on projective geometry, as distinguished from Euclidean/analytical geometry, and its articulation of the ways that projective geometry lies at the core of some of Kepler’s central contributions. This is a valuable and helpful intervention, and it is clearly and convincingly written.
I take some issue, however, with the larger framing of this argument. The author frames his intervention as one which departs from a larger approach to Kepler which partitions his work into a modern physics-based empirical astronomy on the one hand and a discardable neo-Platonic metaphysics on the other. This seems to me to a be straw-man opponent, as recent scholarship on Kepler does not, in fact, make this distinction, and instead argues for the coherence of Kepler’s larger worldview, precisely as this article does. Even some of the older literature does this (for example, Judith Field’s book, which is cited here but not adequately engaged with on this question). So does recent work by Boner, Rothman, Osterhage, etc., none of which are cited here. The author might productively look to Reading the Mind of God, a new series of essays on Kepler’s work published this year. The straw-man is not necessary; this article adds to current scholarship articulating how Kepler’s metaphysics, his geometry, and his astronomy are part of a larger coherent worldview.
This is related to another straw-man argument made in the middle of the paper, on pages p. 19-20: that Copernican astronomy is distinguished from earlier “unreflective” astronomy of everyday experience. The author instead argues that “pre-Copernican astronomers were surely, in one sense, as equally Copernican as Copernicus himself.” If this means that they were reflective and highly theoretical, then yes, this is certainly true. But it is also true that no contemporary scholar of the history of astronomy would make the straw man contrast articulated above. There is, in fact, a great deal of literature highlighting the complexity and deeply reflective nature of pre-Copernican astronomy. I would rethink this contrast, or eliminate it entirely.
There are two small areas that I think could be further clarified:
-- The author makes reference to a Christian-Aristotelian cosmology to which he links Kepler’s own literal conception of the music of the spheres. I think he means something different by this than the usual notion of the Christian-Aristotelian cosmology that is typically described in scholarship, and a clearer explanation of his usage would be very helpful.
--page 17: Kepler’s opposition to an infinite universe is not adequately explained here. It is not simply about what is empirically verifiable, but also about his deep anthropocentrism: a belief that the universe is intended to be observed and appreciated by us, and that God would not create an infinite cosmos with regions that were beyond our possible appreciation.
--page 1: “Galileo’s attempts to unify the explanations of celestial and terrestrial phenomena”: It’s not clear to me what this means. Although Galileo certainly used terrestrial analogies to explain celestial phenomena (e.g. with relative motion), he makes no attempt at unification (this is what Newton does).
Finally, there are a number of errors that need to be corrected:
--page 1: In contrast to what the author claims, Newton’s laws are not actually expressed algebraically by Newton: this happens later.
--page 1: It is “Astronomia” Nova, not “Astronomica.”
--p. 13: “and in 1694: is this supposed to refer to the publication date of the Harmony of the World, which is 1619?
--There are numerous typos or grammatical errors throughout the paper. I did not list them all, but noticed several on page 16: e.g. line 718 isn’t a fully sentence; line 720 has an “of” instead of an “and”; line 763 says “observation” instead of “observational”, etc.
--page 17: “Bruno had been executed….for the Copernican heresy as well as others.” This is false. In fact, Copernicanism was not yet heretical (that happened in 1616) and there is no evidence that this was one of the reasons for his execution.
Author Response
Response to reviewer 1
I am very grateful to this reviewer for pointing out inadequacies in the way the general argument is framed. (Of course, I’m delighted with the assessment of it otherwise as a “valuable and helpful intervention … clearly and convincingly written”).
The reviewer correctly points out that much recent scholarship has contradicted the type of view that, in citing the work of Holton and Love, I misleadingly suggest is the accepted view. In correcting this impression, besides the work of Field I have included that of Rothman, Osterhage and others, as suggested in the review and am most grateful for the advice. Nevertheless, I’ve retained the points made in relation to Holton and Love themselves, as it seems to me that such a view of Kepler is still common in non-specialist treatments.
Moreover, I have used this opportunity to try to sharpen my thesis as I have chosen (for this context) to remain agnostic concerning the relations of Kepler’s physics to his broader metaphysics (this is too involved to try to unpick in this context). Rather, I now distinguish the role played by “harmonic” conceptions at the level of Kepler’s observational astronomy from that of the “music of the spheres” thesis. In fact, I think the “new focus” was actually present in most of the detail of the original submission but was not sufficiently differentiated from the “metaphysical” issues. In short, I think this has helped clarify the argument of the paper.
To this end I have distinguished what I describe as two paths of Platonic “harmonic” notions into modernity. One that I call Platonic” or “post-Platonic” as effectively starting with Apollonius’s work of the conic sections, and passing to Arab mathematicians in the 9th, 10th, and 11th centuries, to Kepler himself, and to Desargues, the inventor of projective geometry. The other, more commonly understood “neo-Platonic tradition” stretches more from late neo-Platonists such as Nicomachus of Gerasa and Proclus, in whom mathematics was intricately bound up with metaphysical and mystico-religious ideas. I see Kepler as inheriting both these traditions, but for my thesis, it is the former that is directly relevant, and I attempt to keep it at a distance from the latter considerations to simplify the essay.
I have also corrected the other “straw-man argument” by effectively deleting the offending passages and simplifying the contrast between Kepler and Bruno. In relation to this I have also abandoned that dimension of the earlier version that contrasted “Platonic” and “Aristotelian” metaphysical dimensions in Kepler’s own work. While I’m still convinced of this stand of argument, again, addressing it here simply complicates and confuses the basic thesis—Kepler’s employing “harmonic” concepts by way of inheriting the “amalgam” of practices constituting earlier astronomy which can be broadly understood in isolation from “metaphysical” and “mystical” notions.
The reviewer is also concerned with my earlier use of the Kepler-Bruno distinction, correctly noting that Kepler’s anthropocentric approach is bound up with his conception of the epistemic capacities with which humans were endowed by God. While this is certainly true, in the paper, newly focussed as described above, I skirt issues of this type. A fuller treatment of this, so as to bring in Kepler’s more metaphysical views—which of course would be more true to the intentions of Kepler himself—I have put off to a later date.
Finally, I’m grateful for the reviewer’s corrections to a number of mistakes and have corrected them all accordingly.
Reviewer 2 Report
Comments and Suggestions for Authors
This paper is not well conceived. The idea to understand if the space of Kepler's universe is a projective space is interesting, but there is almost no evidence drawn from Kepler's works in favour of this thesis.
There are many digressions which are useless for the thesis of the author and that, in many cases, are wrong. In this context why to write a succint history of projective geometry, which, furthermore, is full of mistakes? The long digression on the history of Greek astronomy is also useless. All these topics are well known and there is no need to make a summary which is necessarily superficial.
The author should write a section where he explains the possible sources of Kepler as to the theory of projections. The main section should concern a deep research on the relations between Kepler's theory of harmony and his possible conception of projective space. A profound analysis of Kepler's works is necessary to justify the main thesis.
With regard to the thesis the author claims at lines 16-18. I find no evidence in its favour drawn from Kepler's workd.
Specific observations:
Lines 147-148: It is very debatable if all the studies on projections before Desargues' work can be defined as projective geometry. As a matter of fact, I do not think so. Projective geometry is something different from having some principles on how projecting a three dimensional figure on a plane.
Lines 150-151: This is false: projective geometry was edified at the beginning of the 19th century (Poncelet, Brianchon, Gergonne, ecc,). These authors clearly distinguish a geometry dealing with graphic properties and one dealing with metric properties. Klein inserted projective geometry within his theory of the transformation groups.
Line 155: This is absolutely false. Projective geometry has nothing to do with non-Euclidean geometries. Non-Euclidean geometries has a metric geometry and a projective geometry, exacly as Eucliedean geometry.
Lines 183-184: Theory of perspective can exist without projective geometry. It is sufficient to think that the ideas of perspective by Alberti, Pier della Francesca, and so on were developed relying upon classical geometrical means as the theory of proportions.
Lines 237-238: This is a strong interpretation given by Rashed and it is not to given for granted. If the author agrees with this interpretation, he has to clarify that it is an interpretation not an incontrovertible truth.
Lines 241-245: Projective geometry is not connected with the negation of the fifth postulate, but with the consideration of graphic properties, in contrast to affine and metrical ones, whereas parallelism is an affine property. It is extraneous to the projective context. Whereas the negation of the fifth postulate is the base of non-Euclidean geometries.
Line 247: the fundamental reference for projective geometry is Poncelet.
Lines 258-259: The double ratio was discovered by Chasles and Steiner, not by Monge and Lazare Carnot.
Lines 294-298: Projective coordinates, which were completely independent from metric references, were discovered before Klein's assertion. It happened in the 1830s (Chasles, Moebius, Pluecker).
Lines 301-302: The value of the cross ratio is -1, as written in the note 19, but the sign "minus" completely changes the geometrical features of cross ratio and it cannot be ignored and replaced by +1.
Lines 442-448: This is true from a mathematical point of view, but what is the connection with Kepler's work? Does the author think that Kepler though of polar and polarity as developed in the 19th century? It seems to me a strong idea which needs a profound justification.
Line 540: It was validated 50 years before Felix Klein (I repeat Chasles, Steiner).
Line 554: Replace 1694 with 1594.
Line 733: Replace Geordano with Giordano
Line 736-738: What means that Kepler was "self-consciously relying on those projective features..."? Certainly he knew stereographic projection, but what do you mean with your assertion?
Author Response
Notes reviewer 2.
I appreciate the detail that this review has gone into, especially in relation to my short history of projective geometry. There are a number of clear mistakes that I have now corrected and I’m grateful for the guidance in this. However, quite a few of the purported mistakes that my short synopsis is apparently “full of” are, I think, more either matters of the reviewer reading things into the text that are simply not there, or disagreements about emphasis. I treat a number of these below, but first want to respond to the criticism that the paper overall is poorly conceived and that it should have been approached in the manner the reviewer sets out in the third paragraph.
In the paper I invoke a style of history of science summed up by Stephen Gaukroger in a passage I cite (I have now expanded on this in a footnote in relation to ways of doing the history of science). This, I think, has just been ignored. Standardly, earlier approaches to the history of science identified the content of science with a body of systematically related propositions asserted by an author, but in relation to the history of astronomy I have concentrated more on certain practices of observation, measurement mediated by specific instruments. My argument is that Kepler inherited these practices as part of an “amalgam” and they are such that have certain assumptions—here, projective conceptions of geometry—built into them. In retrospect and with hindsight we can identify these, but Kepler was not himself in the position to make such conceptual distinctions clear. He believes the geometry he is applying is Euclidean, but oddly thinks of Euclid as a Pythagorean.
The reviewer simply dismisses the type of material central to my case as “digressions which are useless for the thesis” and which consist of topics which are “well known”. Dismissing them in this way it is hardly surprising that he or she finds “no evidence” for the thesis being argued for. Those “digressions” are simply where the evidence is located.
Regarding the originality of these digressions. Individually taken, many issues treated indeed may be well known to specialists. (I have obviously drawn upon, and acknowledged, the work of many specialists.) However, I have not written this essay specifically for specialists who, say, might be expected to know the history of projective geometry or, the history of the astrolabe in premodern astronomy. The essay is written for a broader readership who may find the central thesis interesting, and it is written at a level of generality that attempts to link issues that specialists themselves often do not link. The thesis, as I’ve now reshaped it for clarity, is that Kepler’s embrace of the idea that the cosmos shows “harmonic” features that may perhaps be built into the ensemble of astronomical practices he inherited from Greek astronomy and that serve functions that are quite distinct from those relevant to his explicitly religious cosmology. Presumably some others have related the modern “harmonic” cross-ratio to its musical origins, but after reading a large amount of secondary literature I have yet to see this point clearly made.
Regarding the specific criticisms of my short reconstruction of the history of projective geometry:
In regard to lines 147-148, I am interpreted as (and criticised for) claiming or perhaps implying that projective geometry existed before Desargues, but no such claim is made or implied. What I say is that projective geometry “had roots in ancient Greece, Medieval Arabic Astronomy …”. This general claim, on my understanding, is unproblematically true. Pappus’s theorem is standardly taken as the first theorem of projective geometry.
Re lines 150-151. I write that “Projective geometry would only start to be differentiated from Descartes’s geometry in the decades after Kepler’s death in the work of Gerard Desargues and Blaise Pascal, and its final separation from Euclidean geometry would wait until the “Erlangen Program” of Felix Klein in the final decades of the nineteenth century.” This one sentence covers three centuries and, of course, cuts many corners—there is the whole history of projective geometry in between, especially that of the nineteenth century. But why deem it simply “false” to say that it achieved its “final separation” with Klein’s well-known Erlangen Program, given that Klein, certainly on the basis of discoveries made by others, was to differentiate the various geometries in terms of their central invariants? Nothing said in this one sentence or in related passages contradicts the roles the reviewer gives to “Poncelet, Brianchon, Gergonne, etc” in this history, nor to what is said about Chasles, Moebius and Pluecker and others (re lines 294-298).
Re line 247, it is asserted that “the fundamental reference for projective geometry in the nineteenth century is Poncelet”. I concede that Poncelet is standardly described as “the” originator of projective geometry in France and omitting his name was clearly a mistake which I have rectified. Nevertheless, Poncelet had drawn up the work of his former teachers, Monge and Carnot, the latter having introduced the idea of harmonic conjugates in relation to the “complete quadrilatal” in Géométrie de Position of 1803, Theorem VIII, p. 282. Carnot may not have given this double ratio the significance given to it later by Chasles and Steiner, but in the light of Carnot’s book, surely the simple assertion that Chasles and Steiner “discovered” the double ratio (re lines 258-259) is itself extremely questionable.
Any attempt to give a sketch of the history of a topic as complicated as projective geometry will need to construe this history in a simplified way, and some attempt I believe is necessary to make the paper intelligible to the audience to which it is directed. I am grateful for the inadequacies of the original to have been pointed out in this way, enabling me to improve the account. Nevertheless, I believe the assessment of the original as “full of mistakes” is exaggerated.
Reviewer 3 Report
Comments and Suggestions for Authors
The paper gives general information about the history of mathematics at the level of better textbooks on the subject. The historical information is surely known to any serious scholar who is working on Kepler, and the paper does not give any new historical insights about Kepler. No specific argument concerning Kepler's ideas is being made.
Author Response
Response to reviewer 3.
I’m somewhat puzzled by this review which asserts that I offer no new historical insights about Kepler nor do I offer any arguments. (My own concern was that I was being somewhat too ambitious in a work of this length.) I would certainly be very grateful for references to any other interpretative works that argue for my main thesis—that Kepler, besides his well-known advocacy of the thesis of the “music of the sphere”, actually employs harmonic features methodologically within his empirical astronomy, features due to the projective geometry implicit in his optics. I would be very pleased to be able to draw on the work of others who have gone down this path before, but I have not encountered such work.
In particular, I’m certainly aware of, and cite in the essay, the suggestions that Kepler’s optics anticipates later projective geometry, but I am yet to come across a clear statement of the thesis that the invariant of projective geometry itself can be understood as a generalization of the musical double-ratios that Plato had employed in the Timaeus. If this is a view “surely known to any serious scholar”, then why I have not encountered it in the voluminous literature about Kepler that I have consulted, or about the history of astronomy in general, I find a mystery. Even were this criticism to be true, my project of making these connections explicit for a broader audience would, I suspect, still be justified.
That my retelling of this purportedly often told story is at the “level of better textbooks” on the subject I take as a compliment. This level of historical exposition seems to me to be about right to a work of this generality that is directed beyond specialists to a wider audience. (Of course, I don’t accept that I am simply repeating views that are commonplace.)
In any case, I have certainly learnt from this reviewer that my case must be presented in clearer terms than it was in the original submission, and I believe the rewritten version does this. For this, I’m most grateful to this reviewer.
Reviewer 4 Report
Comments and Suggestions for Authors
The paper discusses the connections between Kepler’s metaphysics and mathematics in his astronomical work in the background of Stephen Gaukroger’s notion of science as “complex amalgam.” The paper main claim can be thus summarized: Kepler's 'neo-Platonic metaphysics', often seen as a separate from his scientific work, was actually embedded in his mathematical and astronomical practice. The mathematics underlying Kepler’s study of the cosmos was two-fold: on the one hand, projective geometry, which the author contrasts to Euclidean geometry and its later modification by Descartes, and on the other, the theory of harmony, which is grounded in Pythagorean arithmetic. However, the latter also has a geometric counterpart, namely the theory of platonic solids. On this basis, Kepler's commitment to the 'music of the spheres' was not merely metaphorical, but was deeply connected to the harmonic theory that had been extended to astronomy by the Pythagoreans and Plato.
In a subsequent section, the author contrasts Kepler's approach to Copernicanism with that of Giordano Bruno. The author argues that Kepler's more conservative view was based on what could be established empirically at the time rather than purely speculative ideas about the universe.
The author sees the principle of duality as an overarching fundamental principle that represents a bridge between projective geometry and the theory of polyhedra. This principle underlies Kepler’s astronomy and cosmology.
As I am not an expert in Kepler, I cannot judge whether this thesis is new or original, but it is certainly fascinating. However, I have some perplexities concerning the historical reconstruction of Cartesian and projective geometries. The rational reconstruction of Cartesian geometry contains some anachronisms and should be checked against existing literature. For instance, the author should consult Henk Bos’ or Marco Panza’s works (among others), which provide solid historical reconstructions of Descartes’ geometry. In particular, the concepts of “variable,” “coordinates” that we normally associate to Cartesian geometry are foreign to Descartes’ geometry as it was originally developed.
I also find that the connection between Kepler’s projective ideas and his cosmological view based on nested solids and spheres should be better explained. Does the pole-polar relation play any role in what seems a crucial result for Kepler’s cosmological view, namely, the impossibility of constructing more than five platonic solids? Answering these questions may also clarify the connection to the transition from two to three dimensions, and the constraints that this transition imposes to constructibility, which require, in my opinion, some further explanation. Moreover, how do projective ideas fit within constraints dependent on the dimensionality of the space?
I believe that a revision that will address these issues is required for the paper to be ready for publication.
Author Response
Response to reviewer 4.
This reviewer starts with a summary of the main features of the original submission—it is accurate and certainly captures the general pitch of the essay, importantly mentioning the orientation provided by Stephen Gaukroger’s view of science as a “complex amalgam”. I’m very grateful for the thought that has gone into it—it is very helpful to see this recounted from the perspective of another pair of eyes. (Of course it is very welcome to see the topic reported as “certainly fascinating”.)
As I have mentioned in response to other reviewers, I have actually now refocussed the essay. It no longer aspires to examine the “connections between Kepler’s metaphysics and mathematics in his astronomical work” as to distinguish the two ways “harmonic” ideas bear on these areas, relating the musical roots of the geometrical ideas he employed in his optics to a Platonic tradition centering on the work of Apollonius as operative in his astronomy in ways that are detachable from the more metaphysico-religious ideas about the “music of the spheres”.
I’m grateful for the reviewer pointing out that I have fallen into the trap of simply equating Descartes’s breakthroughs in geometry with the “analytic geometry” that later developed from it, and I’ve tried to correct this mistake.
I’m grateful too for the comments on my treatment of the nested spheres and platonic solids and have tried to make this clearer by simplifying the account and attempting to relate more perspicuously the two themes of pole and polar/inversion and the harmonic cross-ratio respectively. I probably haven’t taken this as far as the critique suggests they could be taken—in a sense I think this would take the presentation to a much more complex level and require much deeper levels of analysis. Recently analyses of Plato’s mathematical work have appeared which stress the role of Theaetetus and his work on “anthyphairesis” (I’m thinking of the work of Stelios Negropontis and his colleagues) which could provide clues here about how to proceed, but space has precluded me from following this path further. What I offer is more a snapshot of an area that relates different ideas in a suggestive way but that could be explored further. Nevertheless, I believe that by bringing out the crucial role of the harmonic cross-ratio within the structure of nested dual polyhedra, I have brought together more clearly than is often done the solutions that Kepler offers in Mysterium Cosmographicum on the one hand and Harmonice Mundi on the other.
Round 2
Reviewer 2 Report
Comments and Suggestions for Authors
As a matter of fact, you took into account no one of my suggestions. You add some parts, but they do no go in the direction I indicated. You insisted on your historical and conceptual view of projective geometry, which is wrong. I think that, unless, you improve the parts I indicated and that I indicate once again, your paper cannot be published.
Comments for author File: 
Comments.pdf
Author Response
I take it that the recommendations that I am said to have ignored are contained in the sentences, “The author should write a section where he explains the possible sources of Kepler as to the theory of projections. The main section should concern a deep research on the relations between Kepler's theory of harmony and his possible conception of projective space. A profound analysis of Kepler's works is necessary to justify the main thesis.”
I have, if fact, given an account the possible sources of Kepler’s theory of projections. They are Apollonius’s work on conic sections, which Kepler is described as having read in 1603 (footnote 14), and the approach to optics of Hasan Ibn al-Haytham, the views of whom Kepler engaged with in his Optics: Palipomena to Witelo and optical part of astronomy. The role of these sources have been made more explicit in the revised version by distinguishing two different pathways via which work on harmonic proportions were transmitted from Plato to modern culture. One was via the neo-Platonistists of late antiquity, the other, was via the work on conics by Apollonius of Perga and transmitted more via Arabic mathematics and astronomy in the medieval period.
I assume that reviewer 2 believes all this to be insufficient, insisting that, “deep research” into and “a profound analysis” of Kepler’s works is needed. I am in no way disagreeing with the idea that further inquiry along these lines would be a good thing, but as a criticism of the existing paper this simply ignores the point that I had made in my earlier response to reviewer 2.
There I had written that “my argument is that Kepler inherited these practices as part of an “amalgam” and they are such that have certain assumptions—here, projective conceptions of geometry—built into them” and that “the reviewer simply dismisses the type of material central to my case as ‘digressions which are useless for the thesis’”. As I point out there, by dismissing them in this way, “it is hardly surprising that he or she finds ‘no evidence’ for the thesis being argued for. Those “digressions” are simply where the evidence is located.” I note that this reviewer nowhere challenges my fundamental Gaukrogerian premise that science as an “amalgam” of theories and practices. Reviewer 2’s implicit assumption is that this idea has no place in the history of science, but absolutely no reason is given as to why this is the case. Nevertheless, I have benefited from these criticisms in that I have been forced to make the methodological structure of my paper clearer than it had been. For this I’m grateful to the reviewer. There is a place for profound disagreement in work of this kind in that it can lead to the sharpening of the claims being made.
Round 3
Reviewer 2 Report
Comments and Suggestions for Authors
As I have already told in my previous reviews, this paper suffers of many defects. The author made some improvements, but the structure is the same. I suggest further improvements.
Comments for author File: Comments.pdf
Author Response
It is certainly the case that my paper (like pretty much all academic papers) could be improved. I have earlier addressed some of reviewer 2’s specific criticisms, and my responses have surely improved the paper. I am grateful for these opportunities. However, there are limits to this type of response, since, as I argued in an earlier reply, some of his/her criticisms have been unwarranted in either attributing to me claims I have not made, or themselves presupposing claims which I have argued are mistaken. The reviewer has challenged none of these responses but continues to assert that the paper suffers from “many defects” (not named) that “must be” improved.
There is nothing further in this reviewer’s contributions on which I can further act.
