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 2^{nd} 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.

Author:

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.

Author:

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.

Author:

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.

--
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.

Author:

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 ?

Author:

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.

--
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.

Author:

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.

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

Author:

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.