The Evolution of Mathematics
from First Principles
by Jonathan J. Dickau
©2004 - all rights reserved
is a subject which has seen a new Renaissance recently, owing largely to important
discoveries and theoretical developments over the last thirty years, or so.
Advancing the knowledge and theory of Math might appear to be an entirely
creative act, but this isn’t the only way to look at the evolution of Mathematics.
It must be understood that many mathematicians don’t believe they are inventing
anything, but rather discovering and finding meaningful ways to elucidate
some object, property, behavior, or dynamic that has always been real.
If one takes the time to explore some of these recently discovered mathematical
objects, like the Mandelbrot Set, it‘s easy to understand why they feel that
way. It is far easier to imagine that the Mandelbrot Set is something
which has always been there, waiting to be discovered, than it is to conceive
of the prospect that Professor Mandelbrot actually invented it, brilliant
though he may be. It appears to have endless complexity and levels
upon levels of inexhaustible detail, yet it is finite. It offers many
lessons, yet it is created from a procedure where we simply multiply a complex
number by itself (squaring) then add back the original number, and repeat
the process until some limit is reached.
The utter simplicity of the formula,
giving rise to the Mandelbrot Set, shows us that complex structures can arrive
by simple means. We should re-examine the possibility that all of Math
can be united, by exploring the roots, or first principles, upon which the
rest of Math depends. The idea that all of Mathematics could arise from
a simple progression of ideas or concepts was considered disproved by Gödel,
but is still being debated and must be examined more carefully, in my opinion.
Perhaps it is more accurate to say that some traditional views on what forms
the basis for topics in Mathematics may need to be re-evaluated, or seen
as limited examples from a broader factual landscape, as the true grandeur
of the underlying order giving rise to Math is only now being seen clearly.
Mathematicians are discovering properties of reality that have always been
there, now that they have adequate tools to examine them with, and this is
exciting. Perhaps the most interesting piece of the emerging new basis
for Math is Noncommutative Geometry. Alain Connes, and others, have
made amazing discoveries about the dynamics of spaces where the normal laws
of size and distance do not apply, which I believe shape all of Math.
Most essentially, noncommutative spaces are self-evolving, and distance there
becomes a matrix, a spectrum, or an unfolding process, instead of a number.
Studying this field helps us to know the true nature of nothing.
More specifically, applying the laws
of measure theory to noncommutative spaces reveals those regions to be a very
dynamic and interesting realm to examine, according to the understanding that
has emerged. This dynamic nature comes into play whenever we talk about
spaces approaching zero size. I assert that it’s essential to delineate
and understand what emerges from these regions, if we are to make a legitimate
claim to a basis for determining the evolution of Math by tracing its growth
from the concepts and ideals allowing it to arise. The real question
here is “What are the precursors of mathematical structure?” and recent work
shows there are definitive answers. Evolving our ability to characterize
and describe what’s beyond form as we know it, and exploring what must exist
before form can emerge, can improve our understanding of what gives Mathematics
itself the form it has. It appears that noncommutative geometry offers
unique insights into the form mathematicians find when they explore all other
branches of Math. I don’t believe Kurt Gödel had any idea that
a branch of Mathematics might be discovered depicting self-evolving processes,
which therefore speaks to the question of how all processes arise, when he
proposed his famous theorem. His later work suggests, however, that
he hoped that day would come when we can unify the entire subject of Mathematics,
and it offered insights into how one might do this.
To some extent Mathematics before
this point in history has assumed the nature which the Taoist Philosophers
call Tai Ji, or the Grand Ultimate in the realm of comparisons. It
has become the epitome of detailed distinctions delineating subtle nuances.
Taoists believe there is something more primal, however, something beyond
and before this dual condition which they call Wu-Ji, that is beyond all
opposites - being neither hot nor cold, neither light nor dark, and neither
large nor small. The concept of Wu is very important to the Chinese,
in terms of their world-view, and it is pervasive in the Chinese language.
The word for Physics is Wu-Li, in that tongue, and the land of Wu is a very
real place, in the minds of many Chinese people. To this realm, Taoists
ascribe the quality of pure process, without distinct form, and the amazing
dynamics of this grand realm are being explored by those who study noncommutative
geometry and related topics, in my opinion. The people working in this
area of Math are finding a wealth of unexpected knowledge where it was thought
that nothing was possible to find. They are exploring, in detail, the
dynamics of what lies beyond and before any form whatsoever, and what gives
rise to form as we know it.
One might ask if it’s even reasonable
to consider what is empty and dimensionless, without form and void, but I
assert that it’s essential we do understand. Noncommutative geometry
has many useful applications in Modern Physics, and offers insight here.
It arises naturally in certain formulations of String Theory and M-Theory,
and pops up in all sorts of unexpected places by virtue of its utility.
It describes a mechanism that appears to give process its birth, and may be
the start of evolution itself. Accordingly, we may see a re-ordering
of ideas and hierarchies, from the current outlook in Mathematics to a model
of reality reflecting the emerging view of what Math describes when we start
from the dynamics at the beginning of all process. This change in approach
may take the form of a revolution of sorts, rather than a slow and stately
progression for Mathematics, because mathematicians will need to understand
certain concepts quickly in order to begin working with the new Science and
new technologies that will come to the fore as a result. People in Mathematics
tend to believe that there is no center, or origin, to their subject, and
Math is rather fragmented, as a result. I think that over the next
10 to 25 years we will see a re-shuffling process within the academic world,
to more explicitly accommodate the idea that concepts from some topics are
essential antecedents of others, and provide the underlying basis with which
those topics can be described. One could not, for example, define a
set of objects within a boundary, without defining topological distinctions
first. This makes elementary Topology essential to establishing the
fundamental concepts in Set Theory.
What I believe will emerge from this
re-ordering is a much more flexible view, wherein the familiar concepts will
find a new home, as earlier attempts to discern a pattern or hierarchy will
be seen as first or second-order approximations of what we now believe is
real and evident. Of course, some of the important discoveries and developments
of the recent past have aided our understanding of other fields whose true
form was also emerging. Our ability to describe and utilize what has
been discovered about Noncommutative Geometry, for example, is greatly aided
by discoveries, methods, and ideas explored by those studying the Physics
of Loop Quantum Gravity, and String Theory, and by work in other areas of
Math such as Fractal Geometry and Topology. Cosmological questions raised
by many recent Astronomical observations have forced scientists to employ
much of this new work in Math and Physics, to explain what is being observed.
Nor are these isolated examples, as numerous connections and cross-applications
are being found for ‘new’ discoveries and developments in the subject of
Mathematics. When we examine many subjects we thought were understood,
using our expanded palette of knowledge, our new analytical tools, and our
improved understanding of what is evident, we are being compelled to see
them in new ways, but I believe that this re-framing will eventually lead
to a more accurate view overall.
In an attempt to speed up the process,
I will offer this philosopher’s view of what the emerging landscape of Math
might be. Thereby, I can offer some food for thought to those whose
work it will be either to carry out this design, and prove it more technically,
or to disprove the validity of my framework. Please understand that
I am not claiming to have the ultimate knowledge of this subject matter, but
what I have come to know leads me to believe that the emerging picture must
ultimately have many of the elements I am about to describe. It is
comforting for me to know that I am not alone, in this process-oriented view,
however. Stephen Wolfram’s book “A New Kind of Science” suggests a
picture of the universe and the evolution of knowledge about it, which has
many of the same elements as my view. Most importantly, we both feel
that understanding the emerging palette of processes available to evolve form
with is just as important as tracing the evolution of form itself, and is
perhaps the key to understanding that process of evolution. A process
is necessary for the creation of form, and developing a repertoire of procedures
is a necessary step to evolving complex forms, so we need to examine how such
processes and procedures might emerge.
What I propose is that we’ll soon
see a “New Kind of Mathematics”, which will more firmly establish the primacy
of certain topics, and may move down the hierarchal level of other ideas and
disciplines, as there will be a shake-out to establish what processes are
truly essential to all form, and what evolution of process is necessary to
give rise to the diverse and varied universe of thought that constitutes all
of Mathematics. My belief is fostered, in part, by my knowledge of
noncommutative geometry. Noncommutative spaces have amazing attributes
with no real parallel in the character and nature of ordinary spaces, but
analogous properties, nonetheless. As a result, Noncommutative Geometry
(or NCG) offers some of the best insight Math gives us about many types of
spaces, including those which have no definite contents and/or no clearly
defined dimensionality. This is important, when one’s goal is to start
from absolutely nothing, and to trace the building process of Math from its
essential roots. The mathematical concept of a point is relevant to
this discussion, as it is a distinct example of nothing. It has no size,
being infinitely small, and occupies a specific location in space, defining
that position. It may seem insignificant, but it appears that this
is not the case, at all. The idea of an object of zero size is quite
Cosmology suggests that this is exactly
the realm we need to examine, if we wish to fully understand the origin of
the universe. We must remember that, paradoxically, the universe at
the moment of its origin (more precisely, at the Planck instant) was smaller
than any object we can observe today. This tiny size is referred to
as the Planck Length, or the Planck Dimension. Before the first moment
we can speak of things having a measurable size, however, is when things really
get interesting. The picture of a universe with no size at all becomes
something both wonderful and terrifying, as it appears to have infinite energy
and encompass infinite possibilities. Instead of being a boring and
featureless place, an infinitesmal universe with no contents turns out to
be the most interesting place of all, possessing infinite symmetry, numberless
degrees of freedom, and unlimited potential. Physicists find a wealth
of possibilities in a point of zero size, or virtually none. The origin,
or ‘zero-point,’ of the universe may hold the key for understanding all else.
Physicists have also found that a vacuum is far from empty, no matter what
measures are taken to evacuate a vessel in a laboratory. Virtual particles
appear and disappear in supposedly empty space, creating ‘zero-point’ energy.
It suggests that emptiness almost wants to become something.
Similarly, I feel that we need to
go to the source, and properly acknowledge the true zero-point in our study
of Mathematics. To know the necessary pre-requisites for familiar Mathematical
concepts, and the precursors of mathematical reasoning, we need to start from
the very beginning, and consider nothing itself. We therefore need
to examine what emerges from the dynamics of that which appears to be dimensionless
or formless, either having no measurable size, or being indistinguishable
from its surroundings. Understanding of the origins of Mathematics requires
that we fully understand how variability gives rise to variation and thence
to form. Thus, I believe NCG is a key, or central concept from which
all else mathematical emerges. To quote Alain Connes “Noncommutative
measure spaces evolve with time!” This self-evolving dynamic of noncommutative
spaces is a means by which it is possible to evolve the processes necessary
to create topological distinctions, measurability, and other familiar properties
of form and space, in my opinion. If some other mechanism, such as spin
foam networks, is essentially responsible for bringing our universe out of
the infinitesmal realm, into observability, the story is very much the same.
Dynamic processes at work in the world of the infinitesmal are the basis
for variability in the remainder of the universe. Similarly, Mathematics
itself has a basis in ‘zero-space’ and its properties, and it will have to
incorporate the underpinnings of the processes which define it.
What I am asserting is that many hierarchies
within the subject of Mathematics have been built based upon the historical
context, or the chronological order of introduction of new concepts, and need
to be re-examined, ultimately being re-thought to accommodate a more process-oriented
view. Even though the progression of knowledge within a subject is
seen as dependent upon past discoveries and developments, that which is being
revealed has an existence that predates its discovery. I imagine it
will be hard even for some mathematicians, though they already regard some
abstract concepts as concrete objects, to adapt their mindset to a new order
of things where process is primary, and knowing the antecedents is essential.
Maybe it won’t be difficult. Perhaps we’ll observe no severe changes
in paradigm, but the overall impact of the new ways of thinking, and the
new approach to classifying things, will create broad and sweeping changes
within the subject of Mathematics. Should we be teaching Set Theory
to four year olds, in order to better prepare them to learn how to count?
Should we find other innovative ways of making the new approach to Math something
which can truly be understood by our young people? That would be helpful,
but perhaps it is as meaningful to make sure that adults who want or need
to understand have an introduction to various topics in higher Math, so they
can understand the basis for this changing outlook.
In my view, when we start from absolutely
nothing we already have a miraculous place to begin an unfolding process.
Next we need to chart a way from noncommutative spaces into the realm of familiar
concepts, spaces, and forms. Luckily, we find that NCG offers many
insights into familiar spaces, the concepts which define them, and the forms
which inhabit them. Furthermore, it provides a framework which can
serve as a bridge between the categories known as smooth, topological, and
measurable spaces and/or objects. As I stated, there are other ways
to evolve dimensionless spaces into dimensionality, but the elegant tools
of NCG seem designed for this task. Simply put; smoothness refers to
something being continuous, or continuously varying, such that there are no
folds, no knees or kinks, no sharp edges or points (cusps), and no discontinuities
where one part is separated from another. When forms or spaces are topological,
they have surfaces and/or the nature of surfaces and surface-bearing objects,
such as having insides and outsides, flatness or concave/convex curvature,
and so on. Measurability has a fairly strict definition in Mathematics,
relating to Calculus, and pertaining to the idea that a surface can be covered
by an array of rectangles, which can be made arbitrarily small in one or
On some level, it may seem silly to
ask “Which came first, Set Theory or Topology, Geometry, Number Theory, or
Calculus?” Even in the context of the question “Where does Mathematics
ultimately come from?,” we need to be somewhat bold to assert that one area
of study is somehow beyond and before another, but that’s what philosophers
are for and why I’m writing about this. Still, if the true nature of
nothing has been largely unsuspected in the world of Mathematics before NCG,
it seems only logical that we should apply what we now know about nothing
to help us better understand things. How does this help us understand
evolution? It is quite reasonable to consider nothing as an essential
starting place, and NCG appears to be an essential tool for fully understanding
empty spaces. In Physics, there is a strong parallel in the new understanding
we’ve gained about the nature of a vacuum. A vacuum was once defined
by its emptiness, but physicists no longer see a vacuum as empty. It
is now seen as a dynamic place instead. This has forced them to carefully
consider the role vacuum energy plays in Cosmology, Particle Physics, and
several other areas.
Without any form or observation within
the scope of a particular space, that space takes upon itself a remarkable
ambiguously infinite quality. This appears to be true whether we are
talking about space within the physical universe or mathematical spaces and
absolute relations. In the more trivial sense of this statement, we
know only what we have observed, and therefore can only speak with certainty
about that which is both observable and verifiable through independent observation
or derivation. Our expanded knowledge of the more general case allows
us to state that a dimensionless space has dynamic qualities which make it
appear very much like a space of infinite dimensionality. This makes
reality vastly more interesting, and fun! I am inclined to believe it
is also part of why it exists, at all. Let us consider again the mathematical
concept of a point, and re-visit it to highlight the ambiguous connection
with infinity, considering as well the contributions of new concepts to our
picture. The question we must ask, in this case, is “What is required
to wrap around a point?” The complete answer may surprise many people,
but on some level our first answer has to be “It depends.”
Specifically, it depends upon the
dimensionality of the space that our point of reference is embedded in.
We can draw a circle around a point on a plane, but that doesn’t fully encompass
it in a 3-dimensional space. What about a 0-dimensional space, or an
N-dimensional space? How do we deal with those cases? Upon close
inspection, we find that almost all of the conceptual tools we have for answering
such questions come out of conventional geometry, which assumes that space
exhibits the property of commutativity. That is, normal space obeys
the commutative law of arithmetic, under multiplication and addition.
If I took 3 steps of 2 feet along a line, I’d expect that walking 2 steps
of 3 feet in the opposite direction would bring me to the same spot, but
in noncommutative spaces things are not so simple. Distance has a spectrum,
or is a matrix, rather than a numerical value. There is a sense of
both-and, as well as either-or, when in the infinitesmal realm, and this
makes things tricky. M-Theory, which is largely an outgrowth of String
Theory, has as one of its central concepts the idea of a Membrane, or brane
for short. The concept of a brane is actually fairly easy to grasp,
and it directly addresses the question of ‘wrapping around’ something.
You see, it appears the brane is the most ambiguous, or general, way to define
‘wrapping around’ mathematically. This concept may come to our rescue,
when we are looking to incorporate the concepts of non-commutative geometry
into the world of dimensionless objects.
I like using the most general definition
of the word dimensionless, when I speak about an isolated object in a measureless
space. This is also necessary when considering the earliest moments
of cosmological evolution. Though our concepts for approaching this
realm are relative, we must acknowledge that some absolutes prevail therein,
and ambiguity is one. We now know that the spaces we need to explore, in order
to understand this matter, definitely exhibit a non-commutative geometry,
and we can no longer blithely continue to apply our understanding from a commutative
view of space to the realm of the infinitesmal, to events before the Big
Bang, or even to a simple vacuum. In fact, we need to re-vamp our very
concept of zero and/or nothing, and the nature of zero-space. A truly
dimensionless space has no boundaries or limits. Luckily, we have a
window into that realm, which protects us from the fury of infinity.
The zero-brane (non-minimal point particle) bridges or spans the gap between
the infinitely small and the world of our common experience, as it can wrap
around a dimensionless object nicely (completely obscuring it from view),
and possesses all the aspects of noncommutative geometry we hope to preserve.
This will enable us to study some of the properties that distinguish a featureless
space from a distinct manifestation of form.
It is easy for us to see how certain
concepts interrelate in a conventional space, but somewhat more difficult
to fathom what the noncommutative world might be like. We can refer
to the period at the end of a sentence and ask, “How small must it be, or
how far away do we have to be, before it disappears from view?” In a
world where size and distance don’t exist, however, or are not yet possible
by their conventional definitions, this ceases to be a meaningful question.
Fortunately, we will be able to use new conceptual tools like branes to help
us clarify some issues, and explore the realm of the infinitesmal in greater
detail. There are other essential concepts to grapple with, however,
if we are to really understand how things work at that level of scale.
What is weirder than doing without size and distance (as we know them) is
incorporating the idea that the noncommutative landscape is self-evolving,
or more specifically, that noncommutative measure spaces evolve with time.
This may be what churns the cosmic foam (of quantum spin-foam networks), so
to speak. In my opinion, this phenomenon is a by-product of the simple
fact that what is not separated is, by nature, connected. It’s almost
as though the emptiness is a perfect fluid. This suggests that there
is much more to the dimensionless state than infinitesmal size, though.
Let us examine the idea of size or
dimension further, to see what concepts we need to introduce before it becomes
possible to strictly define extent. If we have a featureless object,
and we place it in the middle of an empty space, we have an enigma.
Even if we were to posit that it’s sitting in a normal space, which is conventional
or commutative in its geometry, we are still presented with a puzzling situation
where, given our normal powers of observation, we could never tell how big
it is without having something to compare it to. An object in the middle
of nowhere, with no distinguishing features to speak of, might look exactly
like a black hole. It might appear to be a point-like particle, an amorphous
blob, or a solid sphere. It might appear not to exist at all, or perhaps
it wouldn’t even be clearly distinguishable from its surroundings. A
simple point in empty space could be invisible, or at least completely indistinguishable
from an absolutely empty space, or a zero-brane. Assuming that we are
speaking in the purely geometrical sense, however, featureless may mean merely
that our object is perfectly regular, in terms of being round like a sphere
or circle, which has a perfectly smooth or featureless surface. But
a featureless object in an empty space of unknown dimension is still enigmatic,
and we will explore this paradox further.
If we assume that our object of reference
is either observable as a distinct point, or is itself a distinct viewpoint
of observation, we can address some of the earlier questions. Likewise,
if we have a topological object which requires dimensional space to spread
out in, we can make several statements about how various contained and extended
spaces might be defined therefrom. So, let us return again to the question
of what it takes to completely wrap around a point. On the surface of
a plane, like a sheet of paper, there are only a few variables, so we can
illustrate this concept fairly simply. Let us begin with a dot in an
empty expanse. Of course, any dot on a page actually has a definite
size, but this is not a bad starting place for our explorations into the subject.
We spoke earlier of being able to draw a circle around it, and this is a
good step. We should acknowledge that our dot actually is a circle,
or rather a disc, and generalize a bit to say that a closed loop of string
of arbitrary shape can encompass it, assuming it too lies on the plane of
the paper. In the newer terminology, a membrane of one dimension (a
one-brane) can completely wrap around a circle (or disc) on a plane, hiding
it from the view of anyone on that plane.
If it were truly a dimensionless point
(in a flat 2-dimensional space) we could make a circle arbitrarily small,
and still encompass it, so long as we remain on-center. We could also
make a circle encompassing our point of reference as large as we wish, and
still assure that it is contained. We must note, though, that forms
can be relatively large or small only if we are in a space where size and
distance are defined. Regardless of how we would like to have things
arise, size and distance have specific precursors and antecedents. Mathematically
speaking, these concepts seem to emerge from tracing the evolution of form
and the process of observation and comparison, as this leads to the possibility
for counting and measurement. As I asserted earlier, there is no absolute
notion of size in an empty space, and no way to measure a singular object
in the absence of all other form. The idea of objects having relative
size and distance only makes sense for two or more distinct objects with
unique centers (preferably in the same field of view), and this gives us
a good idea of what the precursors of size and distance must be. Topological
distinctions giving objects surfaces are a necessary step in the process.
Beyond this, we need to also have a sense of separation between objects;
that is, we need space. Let us examine how these concepts arise, assuming
Consider now an unbroken expanse,
a space of unknown or undefined dimensionality with no contents whatsoever,
to the limits of observability. To observe from within this space requires
a point of view, and on some level that observation is similar or identical
to other point-like objects and phenomena, in defining a unique and distinct
position in space. That is to say that observation, as we normally understand
and experience it, is inherently centric and/or positional. In pure
Mathematics, the precise nature of the observer is not usually considered,
but we find most often that our observer is assumed to be a point-like entity,
distinct but infinitesmal, and often residing in an imaginary extra dimension
from which the form of a curve, or other figure, can be seen in its entirety.
Thus, when we postulate a point as defined mathematically, in the midst of
an unbroken space as defined above, we somehow automatically incorporate the
assumption of a point-like observer who is situated such that the point of
reference is in its field of view. Note that if our observer had a
fuzzy border, being Gaussian or smooth-edged rather than distinct, our point
of reference would also appear blurry or indistinct, and if our observer assumed
both a distinct surface and a measurable size, our reference point would
appear immeasurably small and disappear from view.
For the moment, let us accept the
idea of an observer who is point-like in nature, being of zero size and able
to take in an encompassing view. Let us further posit that a viewpoint
exists, which allows the observer to be apart from the system under observation,
such that the entire collection of objects and phenomena being inspected can
be observed, or any part. Now, if we place a distinct point within an
unbroken expanse, we still have somewhat of an enigmatic situation, regardless
of what type of geometry we attempt to apply. It would seem, however,
that we are forced to assume that the geometry of this space will be non-commutative,
because in this scenario the precursors of commutativity do not yet exist.
What also appears evident to me is that the dimensionality of such a space
is infinite, on some level, until objects and observations delineate and affirm
a particular dimensionality. We find that this idea is already somewhat
commonplace, in the world of Mathematics. When looking to describe
the position or motion of a single particle, mathematicians often use something
called a Hilbert space of infinite dimensions, where every possible position
into which that particle might move is designated by a different dimension
within that space.
In some sense, every object we can
observe (mathematical or physical) is a projection from infinite dimensions
onto the common framework we know as reality. The implication of this
is that objects are actually defined by how they got to here from there.
The specific form taken encodes the nature of the infinite in some meaningful
way. That is to say that each entity of form is given qualities that
are, in effect, borrowed from infinity or represent the limited versions of
infinite qualities as they are projected onto the world of relative interactions
between objects. What we see at work in the everyday world, therefore,
is both a process of building things up from nothing, and one of projections
of the infinite, paring down from infinity, or selection from an almost numberless
variety of possibilities. These two modes of creation are called Additive
and Formant Synthesis, respectively. The world we see around us is the
product of both of these processes operating simultaneously, over the time
since our universe began. We are likewise products, or projections,
of both nothing and infinity. If we see an object’s measurable extent
in space as being the limited metaphorical equivalent of infinity, then the
concept of duration would be the limited equivalent of eternity, and this
provides a third dynamic wherein things might be sustained, once they arise.
Strictly speaking, it may be more
accurate to say that time was actually the first thing to emerge, in the
form of unfolding process. That would make it the first dynamic.
If what Connes suggests in his work on Noncommutative Geometry is accurate,
then the self evolving nature of noncommutative spaces is very significant
in this connection. The most wondrous thing about noncommutative measure
spaces is that they have a built-in time dynamic. It may be that this
relates to the process by which variability arises and measurability becomes
real, and I feel that this is precisely the case. Even if some of Connes’
observations about noncommutative spaces seem a bit fanciful, or important
only in maintaining the abstract sense of things, the picture emerging from
Loop Quantum Gravity (LQG), M-Theory, and other studies suggests that many
of his ideas may be essential for us to fully understand conventional spaces
and mundane reality. Specifically, LQG also suggests that the unfolding
time dynamic plays an essential role in the unfolding of space, and the emergence
of measurable dimensions, or extents. In both cases, there is the appearance
of something which has an evolution of its own, and an inexorable progression
from initial conditions, which would appear to bring forms and phenomena
into being. This suggests that time, or unfolding process, actually
gave rise to space rather than emerging from, or with it.
What Mathematics and Physics have
focused upon, in large measure, could be called the foreground or the actors
in the unfolding play of the universe. There has always been a fascination
with the background, or stage upon which the play is unfolding, but until
recently we haven’t had the tools needed to explore this matter. Now
both Physics and Mathematics have found new ways to examine the nature of
space itself, and to home in on the very fabric of which space is woven.
On both fronts, investigators are finding a wealth of unexpected information
which was unavailable to us, until now. The knowledge that the universe
has an accelerating expansion, and the ability to observe the cosmic background
radiation in detail, are causing a revolution in Cosmology, borne of the necessity
to adapt theory to encompass what we are actually observing. Physicists
are using this information to discern the fabric upon which the universe
is built, the shape of space itself, and the nature of the process which
gave it birth. In a similar way, the discovery of mathematical ‘entities’
like complex numbers and ‘objects’ like the Mandelbrot Set and the zero brane,
gives mathematicians insight into exactly what the background state (or theoretical
space) of mathematical reality must be, which can create the diverse array
of distinct mathematical ideas we observe.
Although Kurt Gödel is remembered
for his proof of our inability to ever delineate the subject of Mathematics
in a complete way, he was known to have favored the idea that Mathematics
could, and should, be unified. In one of his last published works, Gödel
suggested that one way we can unify the subject of Mathematics is by studying
the things it has discovered as self-existing entities. The progression
of natural numbers does seem to arise naturally, in this world of objects,
but does this concept have an existence independent of our perception?
If we assume that this is the case, then we are in a position to state some
of the laws of arithmetic, and to ask about other kinds of numbers like the
remaining reals and the imaginary numbers. If we assume that they too
have an existence outside our imagination, we are in a position to create
complex objects, such as the Mandelbrot Set, from simple formulae.
If we accept the idea that these objects are self-existing forms, we are
left feeling that the entire subject of Mathematics (in some form) pre-exists
any concept we might have of it. This is precisely how Gödel believed
we have to look at the subject of Math, if we are to find a means of unifying
its various areas of study, and this is where I am proposing the answers
will be found, as well.
This is to say that while a finite
set of rules may not be found, by which every statement of mathematical fact
might be verified, there is nonetheless a complete formulation of Mathematics
somewhere ‘out there’ in theoretical space. Plato held a similar view,
that the world of ideas has an independent existence, which predates physical
reality. This outlook is helpful to me, by putting certain theoretical
concepts on a par with facts of the observable universe. In truth, I
feel that the basis for our universe is precisely such theoretical realities.
The fact that these same unchanging entities also appear to be what constitutes
Mathematics is my primary reason for writing this paper. I have long
felt that mathematical objects, such as the Mandelbrot Set, have a life of
their own and an influence on the appearance of form here, in the physical
realm. What has become apparent to me more recently is that this gives
us the means to unite the subject of Math, or show how the orderly progression
from simple ideas and concepts can generate all sorts of complex behaviors
and entities, requiring complex theoretical descriptions to elucidate.
At this point, I have generated all
of the conceptual landscape I need to put the entire description of how I
believe Math evolved, in the theoretical realm prior to its discovery, into
a simple list. This list enumerates some of the necessary phases of
evolution for the subject, as a specific example of how evolution proceeds,
or my estimation thereof. By so doing, I also hope to give my best predictions
of how Mathematics will come to evolve in the future, beyond our current
understanding of what Math is, and what it describes.
So here goes....
Certain essential ideas -
0. – Zero is not necessarily nothing.
Nor does emptiness preclude evolution. In the absence of a One, or higher-order
numerical quantities, there is also nothing to separate zero from infinity.
Zero is a representation of empty space (as a null quantity or place keeper),
but our concept of this idea is founded on conventional definitions of number,
extent, and measure. The dynamic nature of nothingness will force us
to consider a broader view.
1. – A unified, or connected, state exists
prior to any distinction, wherein noncommutative geometry prevails, in a primitive
form, because conditions for commutativity do not exist. What is not
separated is united or co-equal, therefore, pervasive within its domain.
All is one, or unified, but evolution proceeds even in the absence of forms
with boundaries. Process, or evolution, creates possibilities.
In effect, time (or a process unfolding in time) gives rise to space, or
rather moves us from non-commutative spaces to more familiar territory.
2. – Topological distinctions precede, or accompany, the concepts of set
and number, defining the limits of countability and computability by creating
boundaries for sets and/or groups. The idea of a topological boundary
as a distinct division containing items or serving as the surface for an object
is essential to the evolution of many other mathematical concepts, including
almost all of the Mathematical knowledge that people are most familiar with.
A circle defines an open space, as well as having an edge or boundary, and
both are useful.
3. – The viewpoint of an observer, even a theoretical one, must be considered
along with the system that is being studied thereby (by adopting a particular
viewpoint), and the creative role thereof must be acknowledged to be part
of that system. The view from above a circle is different from the view
at its center or perimeter, for example. Separation is thus observable,
but also a kind of duality which exists in all bounded forms, and in the distinction
we observe between object and observer, object and background, or background
4. – The existence of multiple distinct objects and/or viewpoints is a necessary
precedent to defining the concepts of numbers and counting, as there would
be nothing to count otherwise. The shape, size, and motions of those
objects correspond to utilized degrees of freedom, in some extended space.
An orderly progression to concepts such as distance and proportion proceeds
somewhat automatically therefrom, leading to the conditions for measurability.
This is also the stage where relativity first becomes possible to define,
since there are independent objects which can move relative to each other.
The basis for familiar arithmetic has its origin in this stage, or this arena,
5. – With the appearance of mathematical order in sufficient measure, the
presence of chaos must also emerge. Distinction, comparison, counting,
and measurement require us to develop techniques like addition, multiplication,
and so on, in order to keep track of things, but this opens the door to so
much more. When we try to find non-integral square roots, things get
interesting, because we need to introduce the imaginary square root of minus
one, called i, and the part-imaginary, or complex numbers. Combining
the tools of simple arithmetic with the concept of complex numbers provides
the basis for fractals such as the Mandelbrot Set, and for many other beautiful,
and interesting forms, some almost unbelievably complicated.
6. – The emergence of complementary descriptions within the subject, for
the same concepts and entities from different viewpoints or abstractions,
becomes a means for unifying the understanding of our objects of study.
This has been seen in how the various ‘flavors’ of String Theory (once thought
to be competing descriptions) are now being weaved together, in what is called
M-Theory. Fundamental dualities actually provide a roadmap, of sorts,
to chart out the process of unification. It can be clearly observed
in Grigori Perelman’s recent work, which may have proved the Poincaré
conjecture (thus unifying the possible shapes of space). As puzzle pieces
get put into place, the shape of adjoining pieces is also known, and those
remaining pieces invite further exploration and discovery, to aid in completing
7. – The process of looking back over what was created provides new insights.
We have arrived back where we were before, but with a new understanding of
what it means to be ‘here,’ in a 3-dimensional space with a commutative geometry,
where our ‘normal’ concepts of number, size, and distance make sense.
We can understand the laws of relation more fully, and realize the special
position each holds in the cosmic hierarchy. We can come to know the
face of process itself, and how that relates Mathematics to Information Theory,
Computing, and the Science of Consciousness and its evolution, as well as
to Physics. Once we have a sense of the overall pattern, we can see
how things truly relate to each other, and know from where they have come.
We may come to know how it is that some pieces of the puzzle have always been
there. We could also learn and accept that we are a part of the picture.
this juncture I will explain some of the basics further, and then move on.
First, I believe we must understand that zero, as we commonly know it, is
not a quantity which actually exists anywhere in the universe, but rather
it is a conceptual quantity based upon the difference between none and one,
or some, and the relative difference between one, or few, and many.
The real nature of nothingness is almost pregnant, or hungry to become something,
giving rise to processes which create distinctions. Ambiguity is a quality
which pervades the world of nothing, and may be what gave rise to the possibility
for variation, thus to all form as we know it. The matrices present
in noncommutative equations, for the equivalent of position, length (distance),
and area, represent the uncertainty, ambiguity, and graininess (or ‘spectral’
nature) of space itself at the smallest scales. At our ordinary scale
of size these matrices are diagonal, having a nice row of zeros from corner
to corner, and such a matrix will commute just fine, so it can be replaced
by a scalar quantity - that is, by an ordinary number. Both/and, but
not quite either, gives way to either/or-ness, as things become more and
more one way or another, when we approach common levels of scale. A
dimensionless empty space is not bounded in any way, but it exists without
the means to express its theoretical infinitude, and must develop into something
else, in order to express size or dimensionality at all.
The presence of an observer, or any
kind of object (even an infinitesmal one), defines a sense of proximity and
distance, interiority and exteriority, center and periphery, object and background.
This inherent duality carries forward into each and every ensuing phase of
the creative process, whether for a universe or for Mathematics, where making
distinctions is required for the development of other qualities. Before
any distinctions whatsoever, it can be said that all is unified, or unity.
Topological distinctions are the most familiar aspect of the objective world,
however. All objects have surfaces, and the boundary on a circle is
something we can easily see in the same way. It’s also easy enough to
generalize about boundaries being containers for sets of things, and having
an existence of their own, because in the world of everyday life, that’s how
things are. Nonetheless, we also want to understand how they got that
way. Part of the process is finding a theoretical basis, but part of
it is more empirical. Thus, we need to accept that the universe (as
we know it) does exist, and also to assume that the process which gave rise
to the universe, and that reality which gave rise to Mathematics, are one
and the same (or tightly inter-related). The study of Math is necessarily
caught up in the Science of observables, and therefore the question of observation
and observers is indeed germane to the topic of Mathematics.
Likewise, where there is any type
of computation involved, we invoke the sense of a mind in action, or a thinker
of sorts, and we need to acknowledge this aspect of the analysis, in order
to understand the nature of our own reasoning. In some sense, this also
suggests that much of the evolution of Mathematics does involve the work
of mathematicians to develop, or describe it. But it also seems to
suggest that why mathematicians have something to study, in the first place,
is the result of some kind of intelligence at work, or at least some kind
of device or entity computing the evolution of the cosmos. I shall leave
that question alone, however, as it raises too many other questions.
For now, let it suffice for me to suggest that a Quantum Mechanical description
of observation may apply here, or rather observability, where an observer
is not necessary, per se, as even the possibility of observation will shape
the outcome, or result. When what we are talking about is the shape
and extent of Mathematics itself, however, we must acknowledge that our role
is partly creative, and partly one of discovery, which gives us two ruling
forces. One is the nature and shape of the creative process itself,
and the other is the nature of observation and discovery. The fact that
both processes involve unfolding procedures, with a similar evolution, is
what led me to believe it was necessary to write this paper, and other works
which describe the steps in the creative, and observational, processes.
What is unified (or one) in the beginning
becomes dual, then multiple, as the result of any process of evolution resulting
from observation or creative acts. In fact, any process that involves
bounded form, in any manner, is already expressing duality, and multiplicity
is not far behind. With form, however, also comes dimensionality.
It is interesting to note that all familiar objects, and many mathematical
forms, contain space. We could say that they contain a certain amount
of nothing, and that it is actually the extent of this nothingness which gives
bounded forms size, at all. Here again, we are speaking of the emptiness
as though it were actually something, and perhaps it is so in this context,
as well. The idea of something having an extent in space is also kindred
to the concept of having an extended space available, a space to extend into,
or a degree of freedom to occupy and utilize. Through alternating cycles
of observation and exploration, it is possible to evolve the tools for comparison,
by learning to count, then to measure, the basis for normal arithmetic emerges.
It would seem that we need to first distinguish one from none, then we must
discern duality or discover two-ness, and move on to the rest of the counting
numbers. From there, it seems we have the basis for addition, and multiplication
is next. When we add division, to form fractions, then a whole new
class of partial numerical quantities emerges, but mathematicians quickly
realized that there are irrational numbers, as well. Beyond this, there
are many other interesting numerical quantities, but perhaps most important
are complex numbers.
In all fairness, we should start by
saying that complex numbers are part imaginary, or rather that they have both
a real and an imaginary component. In higher Math, the use of imaginary
numbers is quite commonplace, so I’ll define them here. If you take
the number two, and multiply it by itself, you get four. Squaring the
number three yields nine. Accordingly, two is said to be the square
root of four, three the square root of nine, and so on. One can calculate
square roots for other positive numbers, but the square root of a negative
number is said to be imaginary, because squaring any number (negative or positive)
will yield a positive value, at least for any real number. Nonetheless,
we need to introduce the concept of imaginary numbers to solve for any square
root that is not an integer. Imaginary numbers are almost indispensable
in some areas of Math and Physics, because they embody a peculiar ambiguity
which makes them perfect for modeling cyclical phenomena, oscillating objects,
wavelike properties of both matter and energy, and much more. It should
be noted here that matrices used in characterizing non-commutative spaces
usually contain imaginary elements, in addition to real numbers, making size
and distance complex quantities in noncommutative spaces. In my opinion,
it’s the ambiguity in complex numbers that makes all motion possible, in
our universe or anywhere else. They contain a measured amount of both/and,
to go along with the either/or nature of real numbers, allowing for variability
Once we open up the subject of Math
to this extent, we have unlocked a toolbox sufficient to create all kinds
of wonderful and sophisticated forms. Although it has been an orderly
progression to this point, however, one could end up feeling like our toolbox
had been previously owned by Pandora, from Greek mythology, because that same
order contains all of the seeds necessary to create chaos. In a similar
way, many Mathematicians were taken by surprise when the subject of Fractal
Geometry first made its appearance, because the underlying simplicity of
the order behind chaos, regarding the formulas required to create chaotic
forms, was completely unexpected by many, including most of the early researchers
in Fractal Geometry and Chaos Theory themselves. The blossoming of this
field, however, has allowed us to model a wide range of naturally-occurring
forms in a manner that is far more detailed and realistic than was possible
with geometry before fractals. Mountains, clouds, shorelines, and trees
can now be simulated so well that it’s hard to tell a real landscape from
a computer-generated 3-d image, and this is due largely to Fractal Geometry.
Fractals are also used to study a wide range of physical phenomena, especially
those that manifest on the complex boundary between order and chaos.
It has largely been the advance of
computers that allowed us to see some of these forms, which would otherwise
remain hidden behind a lengthy series of computations, and would be too complicated
to plot by hand. However, upon seeing a fractal Fern, one wonders if
maybe the pattern for ferns exists in a purely mathematical sense, prior to
the actual appearance of form, and if a living fern is, perhaps, a product
of that mathematical pattern. The utter simplicity of the formula to
generate a fern shape, and therefore its ease of generation, may make it a
good mold for the forces which shape reality to pour energy into. In
a similar fashion, I have suggested that perhaps the Mandelbrot Set is actually
a mold which helps to create the pattern for evolution in the universe, and
thereby gives rise to a working Cosmology. This is especially pertinent
in the context of a formulation of Physics where physical processes represent
a form of computation. It would seem that all forms and patterns are
utilized, by succeeding generations of creative processes, both in the subject
of Math, and in our universe at large. There is reason to believe that
Mathematics as a whole, and not merely one piece, is what shapes our reality.
This leaves us with the question, however, of what creates the shape of Math
So far, I have given some attention
to points zero through five, and points six and seven have seen the emergence
of their reason to be. Until now, I have stuck to generalities and relatively
safe areas of inquiry, so as not to lose readers who would find some value
in this kind of overview. This paper addresses a shift in mathematical
philosophy rather than focusing on the Math itself, although I do hope to
give those in that field something to think about, and talk about.
In my opinion we need to have still more bridges between disciplines within
Mathematics, and stronger bridges between other subjects like Physics with
Math. Beyond this point, I will dive deeper and reserve the right to
speculate wildly, but with some clear references to current developments in
Mathematical Physics. I acknowledge that I will also be delving into
areas such as epistemology, and other subjects bordering on Mysticism.
You may feel free to assume that the remaining ideas are merely my flights
of fancy, if you like, and deal with that subset of ideas or learning processes
which you feel comfortable with. Rest assured that I will be treating
the subject of how Math came to be, and how it evolves, not merely speculating
wildly about what may be possible.
The next phase, or stage, of the evolutionary
process I am describing involves the resolution of differences in the picture
of reality arising from the use of various interpretational schemas, to describe
the same territory. While both Quantum Mechanics and Relativity Theory
are good at explaining some aspects of our universe, for example, each is
weak in arenas where the other is strong, and there are ways in which the
pictures they present seem to differ, or even disagree. Likewise, each
of the various disciplines in Math make certain key assumptions, but they
have different areas of application, and often differing usage of quite similar
terminology, which is confusing. Terms within a given topic came into
common usage because it was convenient for the arena in which those terms
are generally used. The task of reconciling Mathematics, so that it
all works together well, is partially a matter of dealing with differences
of opinion, or differing areas of application, partially a matter of reconciling
different definitions of terms, and partially a matter of seeing how things
are related – that is, seeing the big picture, in the first place. Grigori
Perelman’s recent work on completing the Thurston geometrization process,
and proving the Poincaré conjecture, appear to have knitted together
a number of areas of inquiry within Math, by showing how broad classes of
objects and spaces are related, or can be classified. It also demonstrates
that all the possible 3-spaces are variations of one thing, the 3-sphere,
which is unique.
Likewise, we see other areas of Math
and Physics being unified, as clever individuals see the ways in which complementary
pieces from differing views can be knitted together, to form a more congruent
picture of the whole. What were once thought of as conflicting formulations
of String Theory are now seen as various ‘modes’ of behavior for one greater
entity, which has come to be called M-Theory, where the M is usually said
to stand for membranes. This terminology arose as an acknowledgement
that there are modes of behavior, in what was being studied, that are less
‘stringy’ and more like surfaces, or ‘branes’. The idea of a brane is
often thought of as the extension of familiar concepts from the study of
surfaces into the realm of higher dimensions, and I will speak about this
connection in a bit. Let us start, however, by returning to some concepts
that are ‘basic’ to understanding this, and to the progression of stages which
I outline above. Once again, let’s re-visit the idea of wrapping around
a point, considering our new insights, and see where that takes us now.
A point is defined in Mathematics
as having no size and defining a specific location in space. If we
regard this conventional idea of a point as valid, and ask what the implications
are of an object of zero size, we are forced to consider a variety of issues.
One particularly interesting facet of this question is that an object can
only have an extent within an extended dimension (or dimensions), and the
full length or width can be observed only from a view orthogonal to the object’s
own extent. What this means is that one can only tell exactly how large
a circle is from a viewpoint that is 90 degrees from a radius and/or situated
exactly ‘above’ its center (with the plane of the paper lying flat before
you). The view from the edge of a circle would be of a line, and from
the center you would see an expanse. If you were able to ‘stand’ upon
the edge of the circle, a bit apart from it, you might see that it is curved.
To get a true idea of both its size and shape, you have to be in precisely
the right place. Again, let us stipulate that Mathematics allows us
to ‘project’ into whatever dimension we need to inhabit, as an observer, to
have a clear view of what we are trying to study. We find that this
object is a ‘line’ when seen from the edge, an ‘expanse’ when seen from the
center, and a ‘circle’ only when we can see its entirety from a distance,
and this is important. This dimensional ambiguity seems to be a natural
attribute of the figure itself, however.
The circle belongs to a family of
figures which mathematicians call spheres. The familiar figure bearing
that name is more precisely called a sphere of two dimensions (or more simply,
a 2-sphere). If we define a sphere as that set of points which is equally
distant from a particular center, within the extended dimensions - the conventional
version of a sphere follows naturally, given a 3-dimensional space to stretch
out in. If we follow the same formula on a 2-d sheet, the figure we
get is a circle. We can also discuss what happens in higher and lower-dimensional
spaces, as this is where things get interesting. The higher-dimensional
analogy is more difficult for most of us to visualize, but mathematicians
and physicists find the generalization of spheres into higher dimensions very
useful indeed. These figures are sometimes called hyperspheres.
Going the other way, we can speak of a 0-sphere (sphere of zero dimensions)
being the set of two points equally distant from a common center on a line.
Note that, in every case, the dimension of the figure’s surface is one less
than the dimension of the enclosed space, or of the extended space it must
reside in. Remember, just as a point has no size, an ideal circle has
a perimeter with no thickness, and an ideal sphere (a 2-sphere, at least)
has a surface with no depth.
In the case of a circle, or 1-sphere,
it appears completely one dimensional from the periphery. If you are at the
level of the rim, you see a straight line, but other views make the circle
appear 2-d. The ambiguity present in the dimensionality of a simple
figure like this is obvious, but things get even more interesting with higher
dimensions and more complex figures. It would seem that there is some
ambiguity present even in very simple figures and problems. For some
higher-dimensional figures, the sense of interiority and exteriority is almost
entirely relative. So how do we deal with the apparently simple matter
of wrapping around a point? When we wish to completely encircle or encompass
another figure, you will remember, we need to know a bit about both its geometry
and that of the space it resides in. Our point of reference could be
viewed as a closed loop or disc of zero size, on a 2-d surface (or in a 2-d
space). It could also be a normal sphere (a 2-sphere) of zero size,
in a 3-d space. Similarly, we can extend this to say that our point
is an n-sphere, with a radius of zero, in an n+1-dimensional space.
That is to say we can view it as an n dimensional object of zero size.
On the other hand, one might assert that our point is actually a 0-dimensional
object with a size that is definite or ambiguous. It is this 0-dimensional
case which a zero-brane is expressly meant to handle completely, in my view,
by being able to wrap around a point even though there is virtually nothing
extended there to encircle or encompass.
This idea that a single figure would
have different ‘modes of address,’ or a different ‘aspect’ depending upon
how it is approached, is rather curious. Philosophically speaking, even
a point exhibits a similar duality. In an unbroken expanse, it’s the
very model of total or absolute symmetry, but compared to the expanse itself,
it is what breaks the symmetry by establishing a reference frame with a distinct
sense of interiority/exteriority. At least this is so, if the geometry
within that unbroken space truly corresponds to our normal (commutative)
idea of ‘empty’ expanses. Noncommutative geometry allows us to include
additional degrees of freedom in our numbering scheme, by assigning matrices
instead of numbers to positions and distances, where it is necessary.
This approach has proven essential in characterizing spaces in the realm of
the infinitesmal, but yields the ‘ordinary’ result at conventional scales.
Mathematicians were once too enamored of the idea that “nothing is simply
nothing” just as Physicists once believed in “empty space,” but there is a
point at which this concept apparently breaks down. As we approach zero
size, the whole idea of objects with surfaces becomes replaced by a picture
where everything is merged to a degree, and is participating in a cooperative
interaction of evolution, or unfoldment. At this scale, space itself
becomes significantly grainy, foamy, or quantum-mechanical in nature, too.
The uncertainty principle, which places limits on observation in Physics,
might well originate from this condition.
Most humans are quite accustomed to
the idea of being ‘surface dwellers,’ and we have incorporated this idea so
fully into our world-view that it is hard to grasp a view which requires us
to let some of the endemic concepts go. Nonetheless, there is now a
great deal more to work with, if we are hoping to see Mathematics go deeper,
and evolve into something which brings us closer to answering the really big
questions. Noncommutative geometry alone has provided several key insights
into what lies below the surface, and what gives rise to surfaces, that are
both unexpected and unprecedented. It speaks to issues one might not
expect to even arise in a discussion about Math. One might argue that
some of the things I would hope to include in this discussion are not in
the area of Mathematics as well, but I would respond that this one subject
teaches us a wealth about everything. In fact, recent work reveals
deep mathematical connections between Cosmology and Computing, which may
disclose the secrets of Consciousness, as well. Math is universal!
However, what Mathematics teaches us is by nature mathematical, and not grammatical.
Math shows us figures and relations, but it does not show us their significance,
nor does it purport to explain why they are there. Instead, Math gives
us insight into what is possible to create. It also gives us insight
into the creative process itself, or into the nature of processes in general.
Seeing Math as a window into our own
consciousness, or into the process of how intelligence emerges on both the
cosmic and personal level, is just now being explored in earnest, but shows
great promise. Using Mathematics as part of one’s personal theology
would not often lead one astray, in my opinion, but it is a relatively uncommon
belief that Math has anything to do with our belief in a higher power.
On some level, however, Math is the higher power itself, as it is the means
by which all manifested form is created. New insights from recent developments
in Mathematics and Physics seem to prove that intelligence and the creative
process are linked, inextricably, such that form could not arise without some
quantum-level computational process to evolve it. That is, it appears
that the universe came to be because it first began to compute, or process
information. Physicist John Wheeler coined the phrase “It from bit”
a number of years ago, but more recently another physicist, Paola Zizzi, updated
this wording to “It from qubit,” which indicates that they are quantum bits,
rather than being simply ones or zeros. Of course, on could argue that
computational processes, consciousness, and intelligence, are completely separate
and unrelated subjects, but if my perception is accurate, that the so-called
“Observer Effect” underlies Mathematics just as much as Physics, this is
a moot point.
Some might feel that the question
of how the various objects of Mathematics came to arise (in theoretical space,
possibly before our universe began) isn’t about Math, at all, and is best
left to philosophers. The future of the subject, however, lies with
people who are trying to answer exactly this kind of question. I am
decidedly a Platonist, in having a view that mathematical objects do have
a ‘life’ of their own, an independent existence apart from any observer or
their sense-perceptions (possibly apart from the entire universe of phenomena).
Not everyone who knows about higher Math shares my view, and perhaps it would
be a little premature for some of them to do so, but all mathematicians will
soon need to embrace some of these ideas, if I am correct. I am hopeful
that the ideas I’ve shared in this work have provided food for thought, without
introducing too many layers of conjecture to be of value. I don’t know
all the answers, but I hope I’ve raised some interesting questions, and made
the reader think a bit. It is my wish to propel others to seek their
own answers, or to offer possible answers to open questions, but I do not
claim to be an expert in all of these matters. Rather, I am a philosopher
who wants to shed some light on the subject of Mathematics, and I feel that
I can see some changes coming, which will greatly enliven that field of study.
Let us hope that the new developments provide many new perspectives from which
to examine what we have already come to know, and to discover it anew.
Copyright © ’04 – Jonathan J. Dickau – all rights
Single copies of this document for reference
or personal use are allowed, but reproduction
for commercial purposes is not permitted.
Recommended Reading and Sources of Inspiration –
“Mind Tools” and “Infinity and the Mind” by Rudy Rucker
“The Emperor’s New Mind” by Roger Penrose
“The Elegant Universe” by Brian Greene
“A New Kind of Science” by Stephen Wolfram
“The Fractal Geometry of Nature” by Benoit Mandelbrot, and
“Fractals Everywhere” by Michael Barnsley
Additional Reference Material –
Noncommutative Geometry Year 2000 – Alain Connes
Emergent Consciousness from the Early Universe to Our Mind – Paola Zizzi
A Minimal Model for Quantum Gravity – Paola Zizzi
Return to Jonathan's
Mysteries of Science
Return to Jonathan's
this page was first posted
July 14, 2006