Dear Dr Lehar,

We have received the reports from our advisors on your manuscript, "Double Conformal Mapping: A Finite Mathematics to Model an Infinite World.", which you submitted to Advances in Applied Clifford Algebras.

Based on the advice received, I have decided that your manuscript could be reconsidered for publication should you be prepared to incorporate major revisions. When preparing your revised manuscript, you are asked to carefully consider the reviewer comments (especially that of the 2nd reviewer) which can be found below, and submit a list of responses to each comment. You are kindly requested to also check the website for possible reviewer attachment(s).


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.

Reviewer #2: The aim of this paper is to describe a new mathematical modelling 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 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.

-- 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 recommand 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.

-- 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 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 ?

-- 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.

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

Author's Response