MIND ALIVE

MetaNeomathenatics

MetaNeomathematics

by

Joe Staley

Mathematics and Neomathematics

Neomathematics occupies the border land between mathematics and imagination, where truth and possibility reflect each other in the dusky light of dawn and sunset. Although some view mathematics as logic and deduction there also exists a genre of mathematical envisioning, mind casting forth nets of form to gather in some of the phenomena floating past upon the currents of time.

Mathematics requires rigor and neomathematics does not. It is clear that mathematics evolved from neomathematics. Is there some point along the historic continuum at which we could say that neomathematics crystallized into mathematics? It probably has happened several time, one could at least example number theory, geometry and probability. Did mathematics arise with the Greeks or should one include the earlier calculations and measurements made by the Egyptians and Babylonians?

We know that numerical calculation, and even a sort of verbal algebra, were performed correctly at least a thousand years before the classical era; archeological evidence, such as the pyramids at Giza, provide evidence of some effective rational equivalent to mathematics still another thousand years earlier. Yet, from another point of view, mathematics can be said to be consciously maintained as a deductive system of thought and proof only from the classical era itself.

It is not so much a question of when in time mathematics came into being but rather what degree of rigor in thought constitutes mathematics as distinct from neomathematics. We could very well say that this distinction has begun but apparently not ended. We seem to be in a historical period wherein the possible separability of rigor still occurs. Mathematics did not, since the classical era, become entirely free of its neomathematical precursors. The eighteenth century development of modern calculus, by Euler for example, uses neomathematical thought to extend the frontiers of mathematics. Some twentieth century mathematicians, such as Thom and Ramanujan, also were neomathematicians first and mathematicians second.

If we could call mathematics as being neomathematics plus rigor, such a question may help us open a field of question: what is the nature of neomathematics? Not just what distinguishes mathematics from neomathematics, which is an important but limited question (we have named the answer to that question in the first sentence of this paragraph) but the still more creative question as to what distinguishes neomathematics from what is not neomathematics.

If I look at a still life and feel the emotive content of the colors then that is not yet neomathematics. If I notice how the two mangoes on the right side of the flower pot are in contradistinction with the single mangoe to the left of the pot, then that becomes an instance of a neomathematical form under the rubric of triadic dualism. Because I had developed that genus of forms into a fairly coherent and intricate formulation, the instance of observation thus raises many relations of connectivity between the particular visual experience and multiple associations of related forms and instances.

Is neomathematics then a web of connectivity, of somehow associated forms and ideas? This is too broad a description, because it would also apply to, for example, novels, poetry, art in a historic context, popular music and baseball.

Like dawn and dusk, neomathematics occupies the spaces between imagination and logic. Neomathematics coils the spring of new creation.

Husserl questioned the origins of geometry, but his essay might as well have been about the origins of mathematics in general. Neomathematics acts in that origin, but it does so now, with the creation of new mathematics. As to Husserl’s question, one might use neomathematics as a lens to look back in time, partly through structure and partly through analogy, to the origins of geometrical and, indeed, mathematical thought.

Using neomathematics, I have found the beginnings of new mathematical substance, although implicitly but most importantly, the most valuable substance is neomathematics itself. Neomathematically, I have ground lenses to see Triadic Dualism, Arithmogeometry, Vu and the Zero Numbers, as well as beginning several smaller excavations (or projecting a few new constructions (depending upon whether mathematics os made (as I think) or found)).

It is good that I comment upon these things because I occupy the source spot from which, if not original geometry or number, at least new forms of geometry and number originate.

Why Neomathematics?

Neomathematics exists in the borderland between mathematics and imaginative projection of thought. When we watch and listen to the consistencies and variations of life and nature, we may notice unexplained patterns, both regular and irregular. What of the shadows and sunlight that play about the summer leaves in a breeze? What are the starts and stops of time in daily life and human action? Using neomathematics we have the freedom to try to formulate such questions at a symbolic level and then to begin the task of organizing the interrelationships of such symbols.

Neomathematics has the freedom to grasp the beginnings of ideas, to cast a rough symbolic formulation of them, to play them out, see how they run. From a formal abstract point of view, Vu could be called incomprehensible and vague, Neordinals could be called trivial, yet when we delve into their own forms and being we find some curious and creative features which may be worth recording.

Neomathematics may also be entered by a critical analysis of mathematical concepts. There are some very commonly accepted concepts which deserve closer attention. Although the difference between a line and a curve is well prospected, what does it mean to be any such thing as one of the genus including line and curve? Although it is easy to distinguish the finite from the infinite in most cases, what of the borderland between them? Can we explore that region and make some sense of it? Zero, one and two are commonly accepted numbers but there are vast gulfs of metaphysical, and even practical, questioning which open as soon as any of these or their relationships are looked into.

Neomathematics can attempt to explore regions of thought which are opened up by asking inconvenient questions. What is it like to be the endpoint of a line? What could be the inner structure of a point? In what senses does mathematics have being and existence?

Neomathematics has a valid truth structure. It is what it is and does not pretend what it is not. Neomathematics is open, imaginative, liquid as the mercury of consciousness. Neomathematics written down is the track of the mind through the blank snow of an empty page.

Neomathematics is an art form. It is thought informed by possibility.

Neomathematics is a living habitation. It is a home built home, subject to weather and change, a place to sleep and wake. Mathematics is larger and more solid; it is a great pyramid carefully constructed by living mathematicians as a monument of sober truth, a permanent abode for the dead. But what is now mathematics did not begin in stone; it must also have begun as a flimsy reed hut.

When the first man or woman noticed itself counting, it is very likely that the origins of number and the names of numbers in that language were even then lost among the ancestors, as timeless as the afternoon sun. After all, it is quite basic to notice whether one has three or four children. And geometry must also have mattered very early, in the rounded shapes of gourds for drinking, the straight shafts of spears, circles of stone around a fire pit, the full moon and the iris of the eye. Then sometime men began to count their herds, comparing the number of animals, remembering that number from year to next in order to know the growth or decline. Six or seven thousand years ago people began recording numbers on clay which was then ahrdened and sealed in clay containers. Then, if not already, they began measuring fields and constructing linear buildings. Far more complicated problems were solved by the Babylonians thierty five hundred years ago and then finally the Greeks began to arrange mathematical ideas in sequences which they called proofs, where the most obvious ideas were placed first and the more complex ideas were said to follow of necessity from the original ones. It was then that the first stone of the mathematical building was cut. The Greeks built a temple of Euclidean geometry out of their proofs, strong but finite and all in proportion and ratio, except the distance between the opposite corners. For almost two thousand more years the fields around the temple were left to grow what fruit they might, and the tree of the arabic numerals and the dark and useful zero arose. Then, in the sixteenth century of the common era, new and sharpened symbolic tools were devised, which were then turned on the raw stone to shape it and stack it, and the great construction arose. It grew beside, around and eventually encompassed the old temple of geometry; it had secret passageways of the infinitesimal and the infinite; it was built ever larger and now stands vastly more impressive than the pyramids at Giza. Indeed, for a century it has stood so large that no man can know its passageways and now is so vast that probably no man or woman can even discern its outline. And today, thousands work away, cutting new stone, enclosing ever longer corridors and higher aspects, while millions, almost all of mankind, learn to tread its more accessible passageways. I wonder whither it goeth, whether the pyramid of mathematics will ever have a capstone and whether it will last as long as the pyramids of Egypt. Meanwhile I too have walked some miles of the stoney interior but now I play in the sand outside and sometimes build rude but airy structures as comfort from the sun and desert cold.

Mathematical Thought

What is there about mathematics which distinguished mathematical thought from other thought, from thought in general? Kant proposed that mathematics was based upon intuitive mental constructs of space and time, others have considered mathematics to be equivalent to logic, we may try a more directly phenomenological approach.

Mathematics does often share with logic the sense of compelling necessity through a sequence of thoughts. Space and time intuitions, or imaginations, do commonly appear in at least some branches of mathematics. Neither of these relationships is identical with mathematics. Recalling Aristotle’s principle of identity, we can best say that what mathematics is is mathematics and what mathematical thought is is mathematical thought and, although this may seem at first sight trivial, it does have the consequence of preventing us from making the mistake of identifying some thing which is not itself mathematics, so called, with mathematics.

Since to simply call mathematical thought mathematical is trivial and we are not allowing to define one thing as equivalent to another thing (i.e. mathematics = logic), then what can we say that is nontrivial but true about mathematics or mathematical thought.

First we can make a distinction between mathematics and mathematical thought. We might try a description such as the following: mathematical thought is what occurs in thought when one is thinking mathematically, about mathematics or about mathematical thought.

Mathematical thought may occur in any of several modes. It may be best to attempt to list some of the modes separately and then try to consider what their relationships to each other and to what is to be called mathematical thought may be.

Inner visualization of geometric forms. I can inner visualize a line or curve, a two surface in three space, a regular polygon of several sides, irregular polygons of up to four or five sides, polyhedra imaged in perspective, memories of looking at fractal approximations, similarly for catastrophe forms, small scatters of dots (representing, I suppose, points) and some other forms which I have visualized (e.g, a four sphere projected into various three space slices visualized in perspective).

Inner visualization of symbolic forms. Much or most of the actual substance of what is usually called mathematics? If I see a written ‘1' inside of my awareness, what does that portend? If I see another ‘1', preceded by a ‘+’, then am I entitled to visualize a written/imaged ‘2'?

A feeling of necessary connection. There seems to be a short circuit in the mind that sometimes occurs between two thoughts, perhaps is if their were synaptic energy from one thought making occur synaptic energy from the other thought.

A mental feeling of acceptance, acquiescence, or agreement. Sometimes the mind rests in satisfaction, as if that particular effort were come to completion, as if there were a happy boundary that said “it is time for something new”, as if one held one’s opposite shoulders with one’s hands. As if there were no further current necessity.

A logical disquiet. As when it is said that A follows B but my mind does not acquiesce in that situation. A feeling of impropriety. As if this is not that. A circle is not a square and a pentagon or a decahedron is not necessarily the shape of reality, not even if, or because, you say so.

An ecstasy of new invention. You have to feel this for yourself.

A sense of questioning. Is? Is that? Is that true?

A vagueness on the periphery. O.K., but if you extend that concept a little further, even until it becomes somewhat doubtful, perhaps, you begin to see over a border, but the border is permeable, and perhaps the concept or imagination can or could be let go to further wander into the transborder, the exterior or deep interior....

A sense of wonder. How can such wonderful things be? Have they been thought by humans, as I think(because that makes for a richer and deeper connection to wonder), or were they somehow transmitted from some exterior intelligences?

A sense of connection, wherein two, or more?, thoughts become one.

A sense of exasperation and futility when a sequence of thoughts seems incomplete, wrong or incomprehensible. Is it me or the writer-speaker-lecturer? Or, perhaps, is there some deeper meaning or connection, or disconnection, which the author has missed or left unspoken?

A sense of satisfaction and completion wherein thought, imagination, or a sequence of thoughts and imaginations, is accepted by the mind in the sense of acceptance.

Happiness. There is some aspect of the human mind, a t least of some human minds, which makes either original mathematical thinking, following of known mathematical thinking, and mathematical thought exercises, a wonderfully pleasant sensual opportunity of the mind.

Tradition. It is good to be part of all so much that has been thought and written, by people acting, at least mathematically, with good will and intelligence; also there is a communal consensus to what is known and what is not that leads one to comfortably receive a feeling of acceptance if one habituates oneself to the acceptable and a perhaps more exhilarating feeling of rebellion and independence if one refuses and transgresses against the acceptable.

A sense of integrated symbolism, wherein certain arrangements of the symbols cause other arrangements of the symbols to be accepted by the mind. For example, the symbols ‘1', ‘+’ and ‘2' when associated in the form ‘1+2' can induce the mental occurrence and acquiescence in the symbol association ‘=3'.

There is a school of thought, called formalism, in which the symbols and their formal rules of association are mathematics (rather than part of mathematics). This was a useful position for clarifying such structures; it allowed the mind to look with an intense gaze directly upon the symbols resting on a page, without distraction by their supposed ‘meaning’, except, of course, in a formal sense relating to other symbols. From my point of view, however, formalism, however aesthetically attractive, leaves out those extensions of mathematics, whether of geometric imagination or practical calculation, game or profit, which does occur.

Some of the mental forms we have mentioned are parts of general thought, whether mathematical or not, especially the more emotional forms. Some of the forms do appear to be more closely associated with mathematics than others, including the associations of symbols and space time imagery. There seems to be, however, no hard and fast separation between the more and the less mathematical thought forms; all of them may occur in mathematical thought and outside of mathematical thought.

Let us imagine a person of ordinary intelligence, A, who has been taught basic arithmetic in the schools. A has been taught that 5 plus 3 is equal to 8 and A has also been taught that 7 times 9 is equal to 63. Being intelligent, A is able to visualize a collection of 5 objects being combined with a collection of 3 objects and that the resulting collection has 8 objects. Being of normal intelligence, A is unable to directly visualize, although A may have in several ways confirmed, that 7 collections each containing 9 objects provides a collection containing 63 objects. A knows the addition problem by a clear intuition but knows the multiplication problem by less direct means. Which is the mathematical thought form? And what if A, being young, has only believed, or perhaps only expressed, that 7 times 9 equals 63 because A has been told to do so.

Why Neomathematics is Valid

Although neomathematics was born long before the full explication of the ideas presented in Mind Alive, neomathematics was informed by the, perhaps implicit, recognition of the central content of that essay. I knew what I was doing more than two decades before I was able to fully explain it.

Although the validity of neomathematics may depend upon the validity of the central ideas presented in Mind Alive, such may not complete the requirements for the validity of neomathematics. Well, this is not exactly true, since I have the freedom to define what I mean by neomathematics; it would be the relation between neomathematics and mathematics which might be a question. Yes, that is a question, but it is not such as to call neomathematics, as itself, into question. That which is, is what it is.

Among the many varieties of nonsense there is a genus which parades a pseudomathematical mask. Nonsense math is not neomathematics; they differ. I will give a short example of each form of thought in order to illustrate the difference.

First some nonsense: ‘Our number system should be based upon twenty instead of ten because twenty is the sum of seven and thirteen. Seven has been shown through the ages to be a powerful number: God created the world in seven days and , at least philosophically, there are seven planets (Pluto is not really a planet but rather a Kuiper object and Earth is not really a planet in the astronomical sense, since planet originally meant wanderer and the Earth does not wander.) Thirteen is the counterpoint to seven: this can be seen since thirteen minus seven leaves six, the perfect number and it is also clear from the long known magic power of thirteen.’

Then the other: ‘In our universe, or at least in our common arithmetic, we can clearly distinguish between sums and products. For example, if we have three objects and add two more objects then we have five objects whereas is we have two copies of a set of three objects then we have six objects and this distinction is not arbitrary, since if we match the sum and product collections one by one we will find that the product collection has one left over. Perhaps, however, one can imagine a universe in which the physical concatenation of three objects with two objects gave the same result as two copies of three objects. In any case, such speculation might lead us to wander if perhaps we could develop a continuum of operations between addition and multiplication and perhaps set up some sort of symbolical or neomatheamatical procedure to investigate such a system.’

For me, there is a distinction between the two paragraphs above and what we are doing in the neomathematics has much relationship with the type of thought such as might be initiated in the second of the paragraphs and not with that which might be associated with the first paragraph.

The Neomathematical Program

I have material to bring forth, from the general concepts of neomathematics through the relatively short explicit expositions of neomathematical concepts on through the longer developments of Vu, Twists, Neordinals, Zero, and Triadic Dualism.

1. Triadic Dualism 108 pages

2. Neomathematicon 96 pages

3. Vu 26 pages

4. Neordinals 41 pages

5. Zero 24 pages

6. Total 315 pages

These manuscripts were written authentically in their time. If I were to rewrite them today I would no doubt change them both in structure and detail; for better and for worse, I am older now than when I wrote them. Now I would be able to improve some of the details, even if only because I now write with a word processing program on a computer and have access to graphical and programming tools, while then I wrote these manuscripts by hand, as I still often do, but then used a typewriter to bring them into print, with ruler, compass and template for the illustrations. I cannot recopy them into more modern form without changing their substance, as if I were not their author; then, I could simply replace the ‘their’s with ‘there’s when appropriate, similarly correcting other typographical errors; but, as I am the author, I would be indubitably compelled to restructure the thought processes also and not necessarily to the advantage of the spirit and unity of the individual writings. Therefore I have decided to present these manuscripts in their original form, via scanner, only adding such extra explanatory text as seems appropriate. So I here apologize for the inconveniences of the primitive form.

I will list the primary themes or topics which have been introduced through neomathematics.

 Zero Numbers

 Triadic Dualism

 Vu and Twists

 Neordinals

 Subpoints

 Seminfinity

 Single Value Logic

 Superintegers and Hypernumbers

 Neoplane

 Gene Theory

 Endpoints and Transversality

 Symbological Logic

 One

Preface to the Neomathematicon

The Neomathematicon, a neologism, a title, the name of an old writing of mine. It is so old that it was written on a typewriter with the nontextual symbols drawn in by hand.

If I attempted to rewrite The Neomathematicon now, I would not write what I did then; the coherence might be improved but the spirit would not be the same. What is The Neomathematicon and what was the spirit in which it was written.

The Neomathematicon attempts to teach by example, explicating short derivations of neomathematical ideas. It was only decades later, although quite recently to this personal present, that I was able to more fully describe the understandings, in Mind Alive, which liberated the creative possibilities emblazoned in The Neomathematicon.

The Neomathematicon has profound limitations; it is not an mathematical essay, although it certainly concerns mathematics. It is not an essay on the foundations of mathematics, although it concerns that, also. It is not a philosophical treatise, although it might be related to such. It is a record of thought, of thought preformed by mathematics and extending outward therefrom, pushing mathematics, as perhaps the ancients did, out of its accustomed beds of formal exactitude.

Where does thought go, impelled by mathematical desires, when it is not restricted to formal exactitude? Well, The Neomathematicon contains several examples.

It has to be mentioned that the thinker of The Neomathematicon was trained in formal logic and mathematics, began graduate level classes at age 19, and then, after some more years of formal courses, pursued independent studies of logic and the foundations of mathematics in the libraries of West Lafayette, Champaign, Berkeley and Boulder.

On Neomathematics

The mind is alive and thought lives in the mind. Mathematics is a form of thought and is alive in the mind. When mathematics is thought that is a living dynamic process.

The written body of mathematics comes to life when that mathematics is thought in a living mind. Written mathematics serves as a record of thought and as a medium of communication of thought, but its most immediate function is to instigate mathematical thought in living minds. All new mathematics is created, until computers start doing it, in the process of living mathematical thought.

Neomathematics accepts the vital reality of mathematics alive in mind and thought, and uses that reality to push mathematical thought consciously as a real and dynamic phenomenon. By doing so, neomathematics is able to guide mathematical creation along free lines of psychic development. The exact and written mathematics can follow after, neomathematics goes on ahead.

To begin neomathematical thought try the two following exercises and watch your own mental processes.

7. Multiply 23x12 in your head.

8.

9. Using the following diagram as a hint, demonstrate the PythagoranTheorem.

Preneomathematicon

If the substance of the Neomathematicon should be found of interest, then it might also be worth while describing the sources from which neomathematics springs. I can only validly attempt to review my own concepts and their origins in logical and personal experience, an equivalent resourcing of the creations of others being beyond my knowledge.

The Time of Creation

1972-3 Twists and Triadic Dualism

Seminfinity

Subpoints

Superintegers

Hypernumbers

Single Value Logic

The Neoplane

Gene Theory

Symbological Logic

1982 The Neomathematicon

Vu

Neordinals

1985 Triadic Dualism

1987 Zero Numbers

Personal Thought History

Finding my Grandfather’s old algebra text at age seven and reading it during that summer on the steps leading up o our two room apartment in Denver.

Then that same winter my father gave me two books for Christmas, Modern library editions of Plato’s dialogues and the short stories of O’Henry. I found the stories most congenial but had even more fun trying to understand the Plato; I recall finding ‘Meno’ rather unconvincing; it sure looked like Socrates put the answers in the boy’s mouth rather than truly elucidating them from the boy’s inner knowing, although now I can be a little more respectful because, if the boy did not intrinsically know the ends of the thoughts, he did accept, or know, the connective tissue of thinking.

It was about the age of ten or twelve that I became immersed in the question of consciousness, which was to dominate the next fifteen years of my life and illuminate the succeeding twenty years. I would be getting dressed for school, putting on a sock, when I would be swept away with the curious problem of how consciousness could arise from material foundations. I kept trying to deal with this problem until, first, at the age of twenty seven, i semi- solved it, structurally, with the theory which is presented in the text,, ‘Topological Psychophysics’. It was a long time before i fully realized that a structural solution was no solution at all, although I long intuited it in doubt, and placed awareness on a footing prior to material foundation.

In my mid and late teens I became immersed in mathematics from a creative perspective. Learning it was sometimes a chore but my joy was to make it up; naively, at age 16, more knowingly, after twenty, in full awareness of what I was doing, by age 29, with serious regard after forty and, whatever, after fifty.

I took my first formal course in mathematical logic at Purdue when I was eighteen, the second semester of my sophomore year. That course included a simplified but nonobjectionable, proof of Godel’s incompleteness theorems; which I, personally, immediately applied to theoretical physics, as if what could not be originally proven provided a free space of existential action. But, of course, if mathematics was incomplete, then theoretical physics must consequentially be incomplete, and thus there is left a room for independent consciousness, although I did not fully develop that thesis for decades.

After Godel and the Intuitionists, as well as the state requirement to take uniformed military training, which I did for one semester and then refused, I became doubtful of my classroom assignments, which in America at that time were to be interpreted as absolute demands upon slaves or automatons, rather than as directions for intellectual development.

In my case adolescent preoccupation with questions of being and knowing gradually combined with studies in mathematics to direct my attention almost exclusively to questions of logic and the foundations of mathematics, concerning which I haunted the libraries and classrooms of Purdue, U of I Urbana, and Berkeley fairly consistently between the ages of 19 and 25, sometimes as a registered student and sometimes in a floating capacity. After years of logical and mathematical study both formal (Depauw and Purdue Universities 1961-5, GRE Advanced Mathematics, 880/900, 1967; B.S. Mathematics, University of the State of New York, 1978 [some incongruity]) and independent(Berkeley, Champaign-Urbana,) and development of a geometrical synapse-action theory of consciousness, I developed neomathematics in the 1970's and applied it to the construction of the zero numbers in the 1980's. I read and reread classics of philosophy and logic: Heraclitus, Parmenides, Plato, Aristotle, Descartes, Leibniz, Locke, Berkeley, Hume, Kant, Husserl, Wittgenstein, Heidegger, Russell, Frege, Carnap, Bolzano, Heyting, Brouwer, Quine and, most explosively, Godel. I recall that Godel bothered me with respect to physics more than regarding mathematics. Toward the later of those years I also acted with the international avante-garde and my own predeliction was to redefine life with diverging occupations of time, in one case, for example, imaging action as a six dimensional point which diffracted into various multiplicities of lower dimensional events and then living this experimentally in both personal and artistic being. This might have been acceptable to the whatever but I attempted to found a mathematical theory of consciousness based upon the graph theory of neural interconnectivity, a manuscript eventually completed in 1972 as Topological Psychophysics, still extant. Late night walks and meditations by the fountains of the financial district were exquisite. The points of this are that I had the opportunity to speculate upon my postGodelian results in ‘mathematics’, to write down notes coherent enough to be later incorporated into a 100 page text, THE NEOMATHEMATICON, which is also still extant. Then I had a convergence with Carol, my future wife, and returned to life as we know it.

Several of the neomathematical concepts have particular sources. Triadic dualism (c 1972) arose from symmetry of neural synaptic causal connections in a set of three mutually connected neurons and was introduced in an early essay on the graph theory of thought, Topological Psychophysics. Subpoint theory © 1972) occurred to me while sitting in a coffee shop in San Francisco thinking about the identity relations people had with themselves and interrelations with others. Seminfinities © 1972-3) came about while walking among the fountains of the night along the second story outer deck around the Transamerica Pyramid. The original contradiction leading to zero numbers (© 1983) appeared while sitting in a wooden rocker in an old trailer among the Ponderosa and granite of Cherokee Park, Colorado. While twist algebra also appeared in old San Francisco days © 1972-3), its application as (Vu © 1983) began in Cherokee Park.

However, due to a multitude of personal thoughts and experiences I underwent an involuntary change of mental stance in 1972, of which the only verbal expression I have been able to find is that ‘my mind turned inside out; some good friends came to my aid in that difficult hour; it was scary while the transformation occurred, a transformation never reversed, but has been of deep value in the subsequent years. I now suspect that that transformation was but a return to normal sanity out of the barriers which a challenging early life erects about the self, but both at that time and subsequently I suspected that it led me unto a breach into an exceptionally direct and personally aware experience of mental life, of consciousness, of awareness; in any case, it entirely changed my relationship to mathematics. I began to experience mathematics from the inside out and also of course from the outside in, as always. I was still able to do the normal things such as integrate around a pole and find the residue, et cetera and I found it of extreme value when, many years later and not very many years ago, when pure existence and fatherhood led me into teaching mathematics at a community college, I was able to resurrect so many lost mathematics students by an intuitive perception of their vision. But in my own research and personal development the transformation led to fundamental change, toward holding mathematical ideas, reasoning and calculation changeably floating in front of my primary consciousness but part of and connected with that awareness, not as something learned outside of my self but as something known, evaluated, recreated and held into being by means of my own personal consciousness and thus and therefore changeable, flexible and replaceable by the more direct experiences of my personal consciousness. This is very much like the place that Husserl attempts to describe in his Origins of Geometry.

There is a mental space between mathematics and the imagination. As a result of various personal and intellectual experiences I have occupied that space for many years; this has resulted in a number of more or less original initiatives that were collected under the name Neomathematics. Some of the concepts produced may well be absurd and some of little import but others appear to me to have potency. One of these I worked up into a 100+ page essay, Triadic Dualism; triadic dualism is an addition to the available symmetry forms; it is valid, it may be useful but it is of limited importance. Another conceptual area which I have made some attempt to develop concerns zero numbers; attempting to transform the neomathematical concept of zero numbers into a consistent mathematical theory has led to many frustrations and difficulties; the somewhat limited results which have been achieved are presented in the essay on Zero Numbers which appears on my web page. With so many easier avenues of mind and writing to follow, I may never do the work alone which zero numbers deserve and that is a shame because potentially the zero numbers, if they can be tamed, might well provide an extension of the numbers systems comparable in significance, both as regards mathematics and its applications, to the rational or complex numbers.

Metanotes

Let us try to find the elements which determine the genus of neomathematics:

● pattern

● symbolization

● symbolic relationship

● relationship to external pattern

● algebraization

● generalization

● questions

● extensions

● formalization

● consistency, contradiction, or?

Subpoints

What is a point?

Imagine an isolated point, a point with no external references, thus no externally specified location.

The point is its own location. It is where it is, because it is what it is, and, so far as it knows, it is the only place that is.

The point is identical with itself. The self identity of the point can be called a relation and can be symbolically named. Attached to the symbolic name can be a figure illustrating the relation of the point to itself.

Now we have two entities, the point and the relation of self identity. These are not the same thing. At the very least they differ as the thing and its name, but actually they appear to differ more strongly since the first thing is called a point and the second a relationship.

Among these two differing entities we can find a further series of entities. The point has a relation of otherness to the original relation of self identity, while the original relation of self identity has a relation of self identity with itself. Furthermore, we can construct a still different relation between the point and the relation of otherness and another one between the point and the second level of the relation of self identity. Now that the original relation of self identity is entrapped within a growing series of more complex relations, it may be meet to name a new copy of the relation of self identity of the point with itself; since this relation will also naturally indice a series of relations like the above and a new series between the two series, we again have an opening to define a third primordial self identity relation of the point to itself and so on.

You can see that we quickly grow a rich structure of relational entities, nameable, symbolisable, diagrammatical, and potentially calculable.

Having found this subpoint structure of a single point we come to the possibility of applying it in multiple point situations. The simplest multipoint case being, perhaps, the case of two points, where we immediately perceive the possibility of relations between and among the subpoint structures of the two points, as well as other relations of otherness and categories of similarity.

Eventually we can go further, by applying the subpoint structure to classical situations, such as a point in a line or curve, the endpoint of a line, a point in a higher dimensional space, a point in a circle, the corner point of a triangle, a point in a continuous nowhere differentiable curve, a point in a fractal, etc.

Furthermore, we could consider the possibility of applying some of our subpoint concepts to other situations, such as the unit or Parmenides’ concept of being or the self or Aristotles’ concept of identity (which may be, however, where we started).

Joe Staley

back to Mind Alive