At the time this is written, the mathematics and logic of D. G. Leahy are less well known than his philosophical and theological work. This is a curious artifact of the essential humility of his insight, which manifests in a style of writing that appears impenetrably dense, juxtaposing precise detail required for high-quality work with complex multi-word phrases which, when translated into a more normal-seeming English, can be replaced with simple words like “I” or “perfect” or other simple generalizations that prove the conflation errors embedded in our normal day-to-day layers of abstraction constituting our waking minds. For Leahy has identified the root of the root of the root, and designed, or to be more accurate, discovered, a mathematical temple around it.
What is a “mathematical temple?” A temple exceeding all temples, but also underlying them all, for it is at the center of all temples, everywhere simultaneously. Along the way to describing the temple, D.G. Leahy touches on DNA, logic, the fine structure constant — and other perfect numbers which arise from the Laws of Physics, along with elements of topology, geometry, perfect Riemannian spheres in an absolutely ordered imaginary context, tensors, the math of crystals, Fibonacci spirals, there is even some insight into prime numbers, and always, throughout, the absolute dominion of an immensely ever-creative God of infinite goodness who has given us each the absolute ability to create at God level — which level begins after we realize and overcome the logical flaw which exists at the center of the concept of identity. The logical flaw, once unravelled, is that there is no such thing as identity. Meaning, there is no “I,” and this is why there is no ego in Leahy’s writing. He says “the thought presently being expressed” instead of “I” and everybody gets lost. The irony of people who seek to understand him, yet fail because they are seeking what is expressly not there, and thus finding nothing, is a perfect mirror of the moment of clarity which pierces through when you understand he writes without ego.
Speaking of Riemannian spheres, here is a good introduction to Leahy on spheres. Move quickly through the first part of the quote, where he is only setting up the beautiful center of this passage, and remember, as you read — his math is solid:
Insofar as the truth of THE PLACE identical with creation is the measure of an infinite order of squares (as constructed above), or cubes (in the third dimension), each one of which, in the absolute exteriority of the within now actually existing for the first time, here in the form of the foundational segments of the exploded central square (cube) of any square (cube), measures the diameter of a circle (sphere), there exists for the first time in thought, not ‘a single sphere which expands in size without limit’, but an infinite number of infinitely transparent absolute actualities: the sphere of absolute objectivity now existing: the sphere of spheres infinitely newly beginning, the sphere of infinitely new, infinitely separate spheres: the sphere the surface of which is the beginning of an absolutely transparent depth. This is the inception of the infinite proportionality of the body. This is the limit of the infinite expansion of the ‘single sphere’. This is the beginning of the circumference of the infinite circle. This is the line for the first time. This is the time of beginning. This is creation displacing the abyss itself: the body of the Living God in the form of the beginning, depth absolutely surface, the infinite identical with the finite: the absolute incompatibility of the infinitely numbered points of the circumference of the circle: the absolutely transparent circle. The points of the infinite circle are squares between squares: the point, the square, is the side identical with the diagonal. The logical foundation of the last is most simply and directly evident in the structure of the cornerstone: where the extreme 1 of any side, 10ō or ō01 or 1ō0 or 0ō1, never appears as the middle, but is both the extremes and the middle of the 111 diagonal, which has significance if this 111 diagonal can be identified with the sides, which in fact it can be, since the alternate 01ō diagonal which unites the alternate extremes of the sides, so that 0 and ō are only in this diagonal at once extremes of one middle (the diagonal 1), has the discrete arithmetic value of any side, 10ō or ō01 or 1ō0 or 0ō1, as 1 + 0 + ō = 2, while sharing with the 111 diagonal (arithmetically 1 + 1 + 1 = 3) the identical logical value, 01ō = 1 and 111 = 1. The logical identification of 2 and 3, ‘appearance’ and ‘essence’, is the logical identification of side and diagonal, itself at once the finite identified with the infinite.
(From D.G. Leahy, Foundation: Matter the Body Itself.)
I found D. G. Leahy several times before I truly discovered him. I expect others will do the same, but as I arrive on the scene in late 2019, realizing by a long and lazy but increasingly intense spiral of an intellectual journey leading right into the core of Leahy’s insights, my best summary is astonishment at how little Leahy is known for his mathematics. When I first encountered him, a little research turned up that he was the author of a paper on skin, yes, epidermis. It was a very technical article, went immediately over my head, but I categorized Leahy as similar to one of my other crackpot-but-may-be-right friends who was expert in some technically very obscure area in addition to his unique philosophical, historical, and spiritual narratives explored in a web of documents online. Little did I know that his research on skin is directly related to his mathematical insights, and he was studying the microstructure of the skin because he understood the unique role our skin plays in separating us from the rest of the 3-dimensional world. Still over my head, but at least I see a glimpse of where he’s headed.
That spiral of an intellectual journey is important because I believe it contains the most complete and yet succinct introductory path to understanding the true depth of Leahy’s profoundly deep achievement: identifying the core mathematical structures which underly all of reality, perception, and that realization which brings us to the very edge of creation, where creation is happening now in all dimensions. To summarize him is not easy; the mind-bending complexities of quantum superposition and entanglement come to mind; his writing is so dense because he is unusually precise but also intermingling concepts which reach deep into numerous fields of study, with philosophy being only the most easily-accessible of his writing so far. Other fields will spend decades slowly becoming aware of his penetrating insights.
From bipolar to tripolar, a path is forged
The most direct way may be an echo of the one upon which I stumbled, or was led. The path which led me to Leahy begins in the late 1990s when I first began studying something that I at first called “trinary logic.” At the time, there was not much on this subject available on the Internet, but there was at least a fascinating dive into the structure of the Aymaran language, which is arguably the only language on earth that carries a ternary logic structure (this particular point is debated.) For me it was a fascinating entry into the peculiar relationship between three and infinity and logic and language. It was a couple years before I discovered that the closest analog to my idiosyncratic ideas on trinary logic (in the world of actual logic) was something called ternary logic. Of course I discovered Donald Knuth’s famous insight into balanced ternary as “the most beautiful number system,” which propelled me further. So I began studying ternary logic, always informally, casually, but also always meditating and thinking about it in spare moments. It was a few more years before I found the logical flaws within ternary logic itself, deconstructed it, realized that its flaws were based on a binary understanding of ternary, and slowly began rebuilding it with an awareness that there were serious problems with how we think of zero, halving, and division itself. The full journey is out of scope for this brief narrative, where it is sufficient to point out that I developed a deep interest in the relationship between three, zero, and what I would eventually realize was infinity. Tesla’s famous insight regarding the mystery hidden within “3, 6, and 9” led me, over a lot of time and more meditation, to understand that, of all the YouTube videos on the subject — and a new one seems to be added a few times a year — none of them capture what Tesla was talking about. No one was even close to deciphering it in a way that correlated with my own studies, which I believed, being founded on an obsession with the number three, were important, and I should at least find traces in the “true” interpretation.
I developed a profound hard-wired emotional intimacy with ternary logic when I realized that doctors who were treating symptoms of a bipolar disorder did so in a manner which was itself bipolar. I gradually developed the realization that “bipolar people are not bipolar, instead, the whole culture is bipolar, and bipolar people somehow reveal this flaw with their approach to life.”
Abandoning the bipolar model incrementally — because sudden transitions strengthened it — I eventually began identifying myself as “tripolar, with an underdeveloped third pole” and slowly began charting a course out of the extremely polarized symptoms, walking on a dry land of psychological hope I was creating between the two edges of the Red Sea of madness. I began to be comfortable with “walking in the margins,” where lost power is regained in tiny increments hidden in the margins of life, not by large moves through the middle. I began writing notes to myself from the manic state to myself in the depressed state, and the other way again, and slowly began building a link across the corpus callosum of my mind, creating a central core of reliable communication between over-divided hemispheres, which arrived peripatetically at first, but slowly, incrementally, gradually, grew to a point of stability within my consciousness, a place I protected, a place where I could go and find peace, a place to meditate. For years, I spent a lot of time in forests, learning by proximity to the spiderweb and curliqued physics of Nature herself, which are quite different from the orthogonal physics of Euclidean space, buried in all languages in Western Civilization and therefore nearly impossible to see objectively. It was many years before I even began to realize that the idea of subject and object revealed an essentially binary structure which was probably not real.
As time passed, I was beginning to understand that Aristotlean logic, based on the Law of Excluded Middle as it is, was fundamentally flawed, and came up with a working theory that Aristotle had authored one of the most destructive forces in history. Now, years later, I am beginning to understand the wisdom of his role, a necessary one when you understand how Greek thought prepared the way for Christ to reveal the essential trinity embedded within heavenly logic. He spoke within a culture which was still absorbing the advent of categorical, binary logic, a logic which was designed from its inception to eventually collapse and reveal within a truth which could not be seen for centuries, when the two aspects of zero, the not-nothing, and the unity, were eventually revealed as Aristotlean logic finally began to introspect and thus collapse upon itself. At the time, though, I only understood that binary logic was broken somehow, and out of this brokenness came a broken legal system leading inevitably to war and conflict, an endless zero-sum game between everyone and everyone else, culminating in the very modern threat of the current winner of the zero-sum game gaining the ability to press a single button and destroying all life on earth.
We had to return our culture to its pre-logical ternary logic roots, I could see, but knew I didn’t know enough to begin piecing it all together. I continued seeking.
The scapegoat mechanism reveals how merciless ZERO-SUM logic GETs
About then I discovered, and slowly re-invented, Rene Girard’s ideas on the scapegoat mechanism. The re-invention came because, after the initial discovery, it was several years before I had a chance to actually read Girard’s writings. So for years I operated on a bare skeleton key of his insights, based upon a brief mention in an overheard conversation of how the scapegoat mechanism worked. This small seed though, offered me a way to understand the misery of my condition at the time. As I planted the seed, it grew into a powerful oak tree, eventually carrying me through some of the worst of my chaotic transitions from bipolar madness into tripolar equalibrium which has, thankfully, been the core of my nature for about a dozen years. This was a comfort to me: “It is okay to suffer personally as a consequence of circumstances that are out of my control because I am playing the role of scapegoat and thus others benefit.” I could see that the role changed over time, and there hope was born — I didn’t have to always be this way.
I thus developed my own idiosyncratic understanding of Girard’s scapegoat mechanism. For the sake of my own sanity, I determined that I was playing the role of scapegoat for some very intense, very complex drama within my birth family, along with eight other siblings and post-modern fundamentalist Christian theology. With this lever, I began charting my course out with deeper and deeper insights on the polarized nature of “the hero and the scapegoat,” the greatest and the least of any domain, and their eternal struggle to maintain control over the middle point between them, where the normal person lives.
The center of the Bell Curve is heavily populated with normal people who don’t explore the edges where people like I lived at the time. I learned to stop seeking exceptionalism, to be comfortable with being ordinary, instead of trying so hard to escape it. Once I realized the majesty of the middle, the place of Jesus Christ, who appeared neither as a glowing 30-foot-tall winged giant with who commanded obedience (“the hero”), nor as a lesser creature which adored us and always suffered (“the scapegoat”), but as a man, in the middle, as much like normal as the Creator can be while in mortality, and yet also entirely and absolutely divine in every possible sense of the word. An outer shell of normal hiding an inner world of literally everything that ever was, is, or will be. I developed a keen understanding of the region which to everyone else did not exist. I was studying what exists within the area specifically excluded by the Law of Excluded Middle.
When I finally read Rene Girard, it was years later, after acquiring a job, rent, family, the basic elements of stability. As these came into my life, enough that I could start reading philosophy again, I sought Girard and quickly adjusted my inner ideas on the scapegoat mechanism — and its related mathematical structures — to what he actually wrote, which was far more psychological, less rigidly structured, more mythological than I realized. Not so mathematical. But vast. He introduced me to a whole new world, the largest I knew within contemporary thought, and I became comfortable with seeing hidden in his structure the glimpses of extremely penetrating insight that would take centuries to properly unfurl.
Trying to understand how a response to the scapegoat mechanism would translate into the nature of a people who live in a Zionic utopia, I studied social organization theory for clues on how to implement Girard’s ideas in a culture, and moved onward on my journey, feeling certain that Girard had discovered the biggest insight into the nature of Christ since the time of Christ. Then, I stumbled upon a brief passage in Girard, where he casually mentions how implementation for the entire system of new language, of new thought, begins in a very simple way: Once the scapegoat mechanism is properly described to a group of people, it stops working. It’s a subconcious force, operating in groups, which disappears as soon as the light of ordinary consciousness shines upon it. People automatically stop scapegoating as soon as they realize what they’re doing.
The utter simplicity of the solution, the realization that all war would end instantly and world peace would begin if everyone became aware of a simple thing, came like a bolt of lightning — hmm, that allegory carries the strength but too much suddenness. It was more like finding a pearl of great price hidden in a field, causing me to begin preparing to purchase the field around the discovery, in order to preserve and protect this gem.
How to eliminate the ego and still communicate in a fairly normal manner
Girard was the state-of-the-art for my journey for years, and I was imagining developing the mathematical language to communicate his ideas, along with the crazy mathematical insights on ternary logic, when, not long ago, I found Korzybski. Again, I had encountered his material from time to time before, but a few weeks ago I finally had some time to dig in a little more. Because I was already seeking a structure eliminating ego by making a single, simple change to group dynamics, I was able to rapidly see the amazing power and simplicity of the delta between semantics and Korzybski’s General Semantics: the word “is.”
I soon discovered Korzybski’s insight into the Identity flaw was another way of saying what I had seen years before but had not been able to apply it to language so concisely and powerfully as he did. The word “I” and the verb “to be” were all that needed to be adjusted in language, and the whole thing would be more coherent! As I researched, I found a small group of people who were working on a form of set theory that arises from General Semantics, but as far as I can tell, there are no other mentions, and this work is not yet as workable as Korzybski. It has some important insights which a fuller realization will carry. Meanwhile, on the main GS website, there is a list of all the ways GS interacts with all other fields of study, but there is an entry missing, among the dozens: mathematics. The link between mathematics and General Semantics is non-existent! What???
I immediately began wondering how it would look, knowing already that the triadic structures of C. S. Peirce would play a role, as well as the ternary logic of Łukasiewicz, neither of which I had studied in enough detail yet, but which I knew well enough to know their ideas would be involved. Perhaps the mathematics of tensors, Riemannian spheres, maybe even the ternary ideas being explored by the Iota Tangle, would be where to begin . . .
I did not look long. I soon found D. G. Leahy, only a couple weeks ago, and an incredible fountain of insights began pouring in. Because of Girard’s structure which is at essence, very simple and operates everywhere, and Korzybski’s insight into language and its ego-identity flaw, and my own desire to unravel the true meaning of three, zero, infinity, halving, division, and unity, I was uniquely prepared to see Leahy as close as one can get on his own terms, and realize with wonder that he did all the math required. He found the root of the root of the root, of both logic and mathematics, not to mention his forays into physics.
What was impenetrable to me previously began to make sense: I had an infrastructure, a language, and a lot of internal puzzling over the same paradoxes with multiple forms of zero, the riddles of water, helixes, the “nothing that is not nothing,” that Leahy is solving, enough to recognize what he created is spot on, and so far ahead of me that I am in awe at what I see undiscovered by others within his writing. I was trying to do with 1,2,1/2, and 3, what he has done with implementing two zeros, way ahead of me. Others see glimpses, like Bill Shilito’s site on zero and one, but Leahy has it all.
It is all I can do, at this point, to simply be an evangelist for his mathematical approach, which is an essentially humble elimination of ego and the advent of a new foundation for mathematics and logic. His math dovetails with Korzybski’s semantics, and it all falls into place when you see it. But who am I? I am obscure, way off in some backwater of the Internet which pedantic search engine crawlers and philosophy undergrads who still actually care about obscure things know, but about which nobody else knows or cares. But that will change in time. What happens next is mostly inevitable, all I can do is help it a little. If Cantor discovered “a paradise in mathematics,” Leahy has discovered “the eternal fountain of diamonds out of which paradises are created,” in the center of the infinite now of the thinking which creates everything always. Absolutely everything, always.
The moment of realization, dawning for a few days, crystallized when I saw Leahy discussing his ideas, in an interview with Todd Carter that happened a few years ago, not long before he died. He spoke in a normal, rational manner, and I realized suddenly that it was possible for ordinary people to understand the thought presently occurring, and it simply remained for people like me to help build paths in that direction. A much easier challenge than to do what he has done.
So here it is. This essay is the first draft of a rough “Eureka,” and I haven’t yet got to the part where I talk properly about the role of humility in mathematics, toward which I began aiming as I began writing. But I have at least put to rest my poor mathematical mind’s worry that I may accidentally die without having publically pointed people toward Leahy. This is enough to get started.
These words may be awkward, and there are pieces missing which will be added in, smoothing out the path as time passes, but here it is, a few insights toward helping others on the massive project of unravelling the dense truth buried within Leahy’s writing.