Dear Editor,

Thank you for your review of my manuscript, "Double Conformal Mapping: A Finite Mathematics to Model an Infinite World.", which I submitted to Advances in Applied Clifford Algebras.

In your letter you recommend that special attention be paid to the comments of the 2nd reviewer. However I submit that the comments of reviewer #1 are the more relevant for this manuscript.

Reviewer #1 says:

This paper is well written, highly original and interesting. It is also controversial at many levels. But that is to be expected, because the central thesis is so provocative, and supporting evidence is controversial. It would be unproductive to quibble about any particular point. Therefore, it should be published without change and let readers decide.

Reviewer 1 is right that this paper is highly original, interesting, and controversial at many levels. It is not only the content that is highly original, but also the presentation. Instead of making the point with pages of equations, as is customary in a mathematical journal, the author presents his mathematical thesis in pictures. But that is perfectly appropriate for a paper whose principal message is that mathematics evolved from perception, and that mathematical concepts are often better explained in intuitive pictures than symbolic equations.

Reviewer #2 complains:

AACA is a journal whose readers are mathematicians experts in geometric algebra. Hestenes' model can be better explained using a mathematical description. Moreover, everybody knows in the community the power of this model and in particular the way it linearizes the geometry taking into account of infinity.

Reviewer 2 would prefer pages of equations instead of pictures, and thus misses the whole point, which is that there is an alternative to explaining mathematics in equations, it can also be explained in pictures, and to good effect. Hestenes' well-known model is presented anew but this time in pictures, to demonstrate by example how communication through images provides a deeper more intuitive understanding of the concepts, because, as Geometric Algebra reveals, mathematics is geometric at the root of it, and thus the visual / spatial / geometrical aspect of mathematics is neither coincidental nor inconsequential: the visual imagery is the real math, you cannot understand the equations until you can "see the picture" in your mind that the equations represent, whereas a picture depicts the mathematical concept directly, without need for interpretation.

If the opinion of Reviewer 2 is to be given serious consideration then this paper will surely be rejected, even after any minor or major revisions,  because this reviewer does not comprehend the principal message of the paper, and thus no amount of revisions will satisfy them. And in failing to comprehend the message of the paper this reviewer disqualifies themselves as a reviewer of a concept they cannot comprehend.

Reviewer 1 is a person with a larger perspective who recognizes a paradigmatic proposal when they see one.

Reviewer #1:

… because the central thesis is so provocative, and supporting evidence is controversial.
It would be unproductive to quibble about any particular point.

Reviewer #2 would rather quibble. Detailed responses to each comment are provided below.

Response to Specific Comments of Reviewer 2

Reviewer #2:

The aim of this paper is to describe a new mathematical modeling of perception based on Hestenes' conformal embedding and the so-called Bubble World model. I'm convinced that AACA is not the good Journal for publication. I have several remarks to formulate.


The reviewer makes clear that they are not recommending publication because the subject matter is not appropriate for the AACA. Does the AACA only publish well established ideas presented in the most conventional manner? Or is there room in the AACA for papers that are "highly original and interesting" and "controversial at many levels" ? 

This is an editorial choice. On the one hand the reputation of the journal may be compromised by publishing ideas that are later found to be unsound. On the other hand the journal fails in its primary mission if it rejects for publication what turns out in retrospect to have been one of the most interesting mathematical proposals in a long time. "A ship in harbor is safe, but that is not what ships are built for." (John A. Shedd)

Reviewer #2:

-- The section on the ontology of mathematics is useless for the rest of the paper. This is a long standing and exciting debate (see for instance the book "Matière à pensée" by A. connes, Fields Medal, and J.-P. Changeux, neurobiologist). I don't think that this discussion brings relevant information.


This is indeed a long standing debate, but it remains unresolved to this day, which makes it all the more important for it to be resolved at long last, if at all possible. The advance of the present paper is that it challenges the Platonic ideal as a theory that is un-falsifiable in principle, and thus is not a scientific hypothesis of the true nature and origins of mathematics.  The relevance to  the present discussion is that a biological theory of mathematics seeks to model not external reality, but the internal representation of that reality in the mathematical mind, and thus, it explains the significance of the conformal mapping to the representation of an infinite external reality in a finite bounded model. The section on the ontology of mathematics is not at all "useless" to the rest of the paper, it is indeed the prime motivation that led this author to write this paper in the first place.

Reviewer #2:

-- I don't agree with the fact that there exists a biological theory of mathematics. When speaking of "computational mechanism in the brain", one has to explain the neuronal implementation of this mechanism. I recommend for an example of such description in the vision context the paper by J. Petitot : "The neurogeometry of pinwheels as a sub-Riemannian contact structure" in Journal of Physiology, 97 (2003), 265-309. In particular, the "phenomenal perspective" is an inappropriate term in this paper.


Surely the reviewer does not contest the existence of a biological theory of mathematics; that theory was presented in Lakoff et al. (2000) as cited in the paper, and that theory is also supported by the arguments of the present paper. What the reviewer surely means is that he or she contests the truth or validity of the biological theory of mathematics, although he or she does not declare the alternative that they would accept in its stead. A primary message of the present paper is that it is no longer acceptable to support the Platonic theory of math, even by default, without explaining why a dogmatic belief in an un-falsifiable hypothesis of mathematics existing independent of any kind of computational mechanism in the brain should be considered more credible than a scientific  hypothesis of mind as a physical process in the physical mechanism of the brain, consistent with the theory of evolution. Surely the time has come to de-mystify the origins of mathematics, and to do so on a firm scientific footing instead of on dogmatic faith.

The reviewer suggests that it is invalid to suggest that the brain is a computational mechanism without discussion of the neural implementation of that mechanism. I submit that the alternative, that the brain is not a computational mechanism, is itself a hypothesis that requires an explanation of the neural non-implementation of this supposed non-mechanism, or why there is a brain in the first place if the mathematical mind can operate somehow independent of it. If mathematics is not an artifact of the computational mechanism of the brain, then where does mathematics come from and where are its laws written and enforced? The Platonic theory has long outlived its usefulness.

The truth is that nobody understands how the brain really works, no neuronal theory has yet been proposed to plausibly account for our three-dimensional spatial experience nor our mathematical understanding. The question of whether the brain computes our experience is centrally valid to the argument of the present paper, that the operational principles of the  computational processes of the brain can be explored by examining the properties of mathematics.

"Phenomenal perspective" is highly appropriate as a term in a paper that proposes that the properties of phenomenal perspective, i.e. the way that things in the distance appear smaller, while also appearing undiminished in size, are direct evidence for 1: the indirectness of perception, i.e. that the world we see in experience is not one and the same as the external world it presents in effigy, and 2: the utility of a conformal projection for representing an infinite external space within a finite but explicit spatial representation.

While the subject of phenomenology is rarely broached in a mathematical journal, it is centrally relevant in an article that proposes that phenomenal perspective is a direct manifestation of a conformal geometrical representation of surrounding space.

Reviewer #2

-- The description of Hestenes' conformal model is too long and confusing. AACA is a journal whose readers are mathematicians experts in geometric algebra. Hestenes' model can be better explained using a mathematical description. Moreover, everybody knows in the community the power of this model and in particular the way it linearizes the geometry taking into account of infinity.


The presentation of Hestenes' conformal model in the paper was not for the purpose of educating the readership on the "well known" power of this model, but rather to demonstrate how a rather esoteric mathematical concept known only to a select few can be presented with visual images and appeals to intuition, as an alternative to the more conventional presentation in the form of equations. Indeed, the significance of Clifford Algebra is that it reveals that all of algebra is really a branch of geometry, and that our spatial intuitions of mathematical concepts are neither coincidental nor inconsequential, but in fact the spatial understanding of mathematical concepts represents a more basic or primal understanding of the principal concepts of math than the symbology by which we have come to represent them in more conventional  presentations.  I contend that the mathematical argument by visual imagery makes the paper more interesting and even "fun" to read, a rare quality for a mathematical paper, while being no less rigorously "mathematical" than a conventional presentation.

The common view of mathematics by the general public as an esoteric subject accessible only to professional mathematicians is an unfortunate consequence of the way that mathematics is taught in school, and how it is presented in mathematical journals. The real promise of Geometric Algebra is that it can serve to open mathematics up to the wider public by presenting mathematical concepts in intuitive spatial terms that are accessible to anyone. If the AACA insists that all papers be translated into the symbolic gobbledegook common to mathematical discourse then it risks missing out on the greatest promise of Geometric Algebra, which is to reveal the beauty and harmony of mathematical concepts to the lay readership as clearly as the appreciation of  art and music. 

Reviewer #2:

-- The advantage of mixing both Hestenes' and Bubble World models is not clear for me. I have a question in this direction: Hestenes' model linearizes Mobius transforms, what are the expressions of these transforms in the new model ?


With this comment the reviewer reveals clearly that he or she has entirely missed the central thesis of the paper. The advantage of mixing Hestenes' and the Bubble Model is 1: to provide a mathematically closed model of the conformal projection, using a novel but backward-compatible interpretation of the concept of closure as a computational projection that can be expressed in a finite projection mechanism, i.e. one that does not require a projection of anything to infinity. And 2:  To relate Hestenes' conformal model to the familiar conformal projection observed in phenomenal perspective whereby objects in the distance appear smaller by perspective, while at the same time appearing undiminished in size.  The fact that Hestenes' conformal projection turns out to be a conformal reflection of the Bubble World model that is familiar to anyone who has ever noticed the warp in phenomenal perspective is a very significant confirmation of the relevance of Hestenes' conformal projection to our own perceptual experience. 

The fact that the Bubble World model is indeed an exact conformal reflection of Hestenes' conformal projection itself confirms that the properties of Hestenes' conformal model, including the linearization of the Möbius transform, are also reflected in the Bubble World model, as demonstrated in Figure 17 of the paper, whose significance Reviewer #2 also surely missed. Figure 17 shows the Pythagorean theorem being demonstrated in both the positively and negatively curved spaces of the Bubble World and Hestenes' conformal mapping, in the same way that it is demonstrated in Euclidean geometry. Without a single equation, the point is made implicitly in the figure, that if there is a coherent point-for-point mapping between the Euclidean world, and the positively- and negatively-curved conformal projections, then anything that can be "proven" or demonstrated in one of those spaces can be equivalently demonstrated in the other two spaces by conformal reflection, and that includes the Möbius transform.

Reviewer #2:

-- The section on non-Euclidean geometries doesn't bring relevant information. Most of professional mathematicians know the story of the fifth postulate. Lots of paper of AACA are devoted to applications of Clifford algebra to physics, e.g. relativity, involving curved spaces.


The novel contribution of the recounting of the story of the fifth postulate in the present paper is that the argument is made exclusively in the form of an intuitive picture (Figure 17). That figure makes the case that if the properties of Euclidean geometry survive a conformal mapping, as the story of the Fifth Postulate confirms, then any theorems that can be demonstrated in Euclidean space have perfectly isomorphically equivalent demonstrations in those other spaces too. It is like observing the proof through a curved mirror: If the curvature is mathematically lawful, then the proof remains valid despite the distortion. The relevance of the fifth postulate to the present paper is that the Bubble World perspective, i.e. the way that we perceive the world around us, is manifestly a conformally warped world with a positive curvature, thus confirming Gauss' fear that we cannot tell whether our world is Euclidean or non-Euclidean. The discovery of the relation between the positively-curved Bubble World, the negatively-curved conformal model, and Euclidean space, can be seen as the culmination of the story of the fifth postulate to its final conclusion, that our perception of the world is indeed a non-Euclidean one, although the world itself that is represented by our perception may still be Euclidean. This is a significant milestone in history of mathematics.

Reviewer #2:

Although containing quite interesting reflections, this paper brings no new significant contribution.


Quite a contrast to the assessment of Reviewer #1:

This paper is well written, highly original and interesting. It is also controversial at many levels. But that is to be expected, because the central thesis is so provocative, and supporting evidence is controversial. It would be unproductive to quibble about any particular point. Therefore, it should be published without change and let readers decide.

Review Round 2