The Evolution of Mathematics from First Principles

        Mathematics 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 astounding, actually.

        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 both dimensions.

        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 they must.

        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....

        At 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 in form.

        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 itself.

        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.