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).

COMMENTS TO THE AUTHOR:

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