Section VI. Encountering Other Philosophies of RealitySection VII. Encountering the Philosophers of Reality, andSection VIII. The Mathematics of Relational Becoming.
II
Part I - A Philosophy of Relational RealityPart II - Relational Reality in Wider Conversation
The division marks more than an expansion of subjects. It marks a change in the direction of inquiry.
Part I will complete that philosophical development by placing relational reality into conversation with alternative philosophies and with philosophers who have understood reality differently.
Part II will then carry the inquiry into wider conversations, beginning with the more exacting language of mathematics.
The purpose is not simply to defend what has already been developed, but to discover what survives comparison, what requires revision, and what may yet become something other than we presently imagine.
III
Next, I may interweave between sections VI-VII and VIII as I develop Part I and II's conversations. I may compose one-or-two math essays and then one-or-two philosophy essays.
Why? First, because I am eager to present both sections as quickly as possible... but I also fear the math section may be a bit complex, if not boring, for some? or many? readers. So by interleaving the essays it will break up the math treatises so that non-math readers don't become overwhelmed.
However, the maths must be done - and they will be done as clearly as one can when discussing higher level mathematics in relationship to a systematic philosophy.
IV
Throughout months of developing relational reality I had not once thought of interpreting it through mathematics. And then, I did think about it. One moment I didn't think about it... and the next moment it consumed me. How odd! And so, I began to wonder whether some of GRR's language might be translatable mathematically.
As backstory, ages earlier I had taught high school mathematics. Before that, I had studied graduate-level mathematics during my junior year at the University of Michigan. I also studied organic chem and physics, did three years of Attic Greek, Psych, Sociology, the Humanities, took Freshman engineering, entry computer programming, and several ancient history classes. My semesters were also filled with weeknight intramural sports, Campus Crusade for Christ ministries, local church ministries, and attending Michigan athletic contests since West Quad was near to the athletic stadiums. Probably the only thing I didn't have time for were girls - though I had my share of friends.
Now how the question of translating a philosophy mathematically came to mind is also quite coincidental. I was simply thinking about polar coordinates in 3D space. That brought to mind vectors, scalars, tensors, and multidimensional geometry. Those queries then led in turn to thinking about transformations in multidimensional spaces and nonlinear dynamical systems. Areas I haven't thought about for many years. And because of that I knew I would need AI to help me.
So I next turned to OpenAI's ChatGPT (I'm using SOL 5.6 at present until Astra becomes available) to help me investigate whether a mathematical language might be developed for relational reality. As we did research together I asked about transformal spaces and we shortly thereafter adopted the working designation RTD - Relational Transformal Dynamics. Consequently, the math for RTD will be worked out experimentally throughout Section VIII's new mathematical section I've recently added. Which is also why I updated the R&M Index and will eventually split it in two.
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| Philosopher Alain Badiou |
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| British Mathematician and Process Philosopher AN Whitehead (right) with Bertrand Russell (left), whom Whitehead had earlier taught at Cambridge. |
VI
What I hope to accomplish with RTD is considerably more modest than developing a new mathematics. My simple hope is to develop a sufficient mathematical language general enough to describe important dimensions of relational reality's dynamic of becoming, while remaining grounded in the mathematics that already exists. If that effort proves useful, perhaps specialists might eventually take its developing framework and test, correct, modify, extend, or adapt it within their own areas of study.
My working intuition tells me that if reality is relational and relationally becoming, then perhaps significant dimensions of that becoming might be stated mathematically. That is the question Section VIII will explore. Whether the attempt will succeed, I do not know. But the question seems important enough that it should be attempted.
VII
In summary,
a philosophy of reality has been developed.
It is now being placed into wider conversation.
First, with other philosophies of reality.
Then, with philosophers who have developed significant understandings of reality.
And finally, in a rather unexpected turn, with mathematics itself.
Perhaps mathematics will confirm some of what has been imagined.
Perhaps it will correct our theory of Generative Relational Realism.
Perhaps it will show that some of our relational philosophical language has been less precise than we thought.
And perhaps, somewhere in the movement between relation and transformation, mathematics may allow us to see something about relational becoming that philosophy alone had not yet made visible.
The examples below are not offered as predecessors of Relational Transformal Dynamics in any strict sense. Their philosophies differ profoundly from one another and from GRR. Rather, they demonstrate something more modest and historically important: philosophy and mathematics have repeatedly entered into serious conversation.
I. Mathematics and Philosophy in the Same Thinker
René Descartes
Descartes provides one of the earliest modern examples of a thinker whose mathematical and philosophical achievements were both substantial. His development of analytic geometry helped unite algebraic representation with geometrical form, while his philosophical writings pursued questions of knowledge, mind, matter, certainty, and reality.
Yet the two enterprises should not simply be collapsed into one another. The Meditations is not a mathematical derivation of Cartesian metaphysics. Descartes is important here because mathematical and metaphysical reasoning inhabited the work of the same thinker while retaining distinguishable methods.
Gottfried Wilhelm Leibniz
Leibniz represents an even deeper conjunction. He independently developed differential and integral calculus, contributed to binary arithmetic and formal logic, and simultaneously pursued a metaphysics of substances, relations, possibility, necessity, and possible worlds.
His importance to our inquiry lies partly in the extraordinary breadth of his attempt to understand reasoning formally. Mathematics, logic, metaphysics, and the structure of possibility belonged to a larger intellectual project, even though they cannot simply be identified with one another.
Alfred North Whitehead
Whitehead is especially significant for the present study. Before developing the philosophy of organism associated with Process and Reality, he was already an accomplished mathematician. His early work included A Treatise on Universal Algebra, and with Bertrand Russell he later produced the three volumes of Principia Mathematica.
Whitehead's later process philosophy should not be treated as though it were mathematically derived from Principia Mathematica. Nevertheless, his intellectual biography demonstrates that rigorous mathematical thought and a metaphysics centered upon process, event, relation, and becoming can inhabit the same philosophical life.
For RTD, that precedent is particularly difficult to overlook.
II. Logic, Mathematics, and the Foundations of Thought
Frege transformed modern logic through the development of a formal system capable of representing quantified reasoning with unprecedented precision. His work on the foundations of arithmetic also became inseparable from philosophical questions concerning number, meaning, reference, truth, and logical structure.
Frege therefore represents a different intersection of philosophy and mathematics: not primarily a metaphysics mathematically expressed, but an inquiry in which foundational mathematical questions generated profound philosophical consequences.
Bertrand Russell
Russell's mathematical logic and philosophy were likewise deeply intertwined. His discovery of the paradox bearing his name exposed fundamental difficulties in naïve set theory, while his collaboration with Whitehead on Principia Mathematica attempted a systematic logical reconstruction of mathematics.
Russell subsequently carried habits of logical analysis into epistemology, metaphysics, language, and philosophy of science. Mathematics did not merely decorate his philosophy. It helped shape his conception of philosophical rigor.
III. Mathematical Form as Philosophical Method
Spinoza offers a strikingly different case. His Ethics is famously presented more geometrico - in geometrical order - through definitions, axioms, propositions, demonstrations, corollaries, and scholia.
Spinoza was not thereby producing a mathematical theory of metaphysics. Rather, he borrowed the demonstrative architecture associated with Euclidean geometry as a model of philosophical necessity and systematic reasoning.
His example reminds us of an important distinction for RTD: a philosophy can look mathematical without itself constituting mathematics.
IV. Mathematics as Ontological Resource
Alain Badiou
Badiou represents perhaps the most immediately relevant contemporary example. In Being and Event, he makes the radical philosophical proposal that mathematics - particularly axiomatic set theory - is ontology. Concepts drawn from set theory consequently enter directly into his treatment of multiplicity, situation, event, and being.
Later work extends his mathematical engagement further, including substantial use of category-theoretic ideas.
Whether one accepts Badiou's ontology is not the point here. His importance lies in the seriousness with which he permits established mathematics to enter philosophical reasoning about reality.
For the present project, there is also a personal connection. Having once watched and listened to Badiou move between philosophy and mathematics at the whiteboard while developing these ideas, I had already encountered firsthand a philosopher willing to allow mathematical structures to participate directly in metaphysical thought.
Only much later did that memory return as we began asking whether relational becoming might likewise enter into serious conversation with mathematics.
V. Mathematics and the Limits of Formalization
Ludwig Wittgenstein
Wittgenstein complicates the story. His early Tractatus Logico-Philosophicus employed an austere logical architecture and emerged from sustained engagement with Frege and Russell. His later philosophy became increasingly suspicious of attempts to force language, thought, and human practice into comprehensive formal systems.
His writings on the foundations of mathematics consequently provide a useful counterweight to philosophical enthusiasm for mathematization.
RTD should hear that warning.
The fact that something can be represented mathematically does not establish that its mathematical representation exhausts what the thing is.
VI. Different Ways Mathematics Can Enter Philosophy
These thinkers reveal several importantly different relationships between mathematics and philosophy:
Mathematician and philosopherDescartes, Leibniz, Whitehead, RussellLogic and foundations as philosophical inquiryFrege, RussellMathematical form as philosophical methodSpinozaMathematics explicitly proposed as ontologyBadiouPhilosophical examination of mathematical formalization itselfWittgenstein
These categories overlap, and none should be regarded as a simple ranking of mathematical sophistication. The more important point is that there has never been only one way for mathematics and philosophy to meet.
VII. Where RTD Enters the Conversation
Relational Transformal Dynamics proposes yet another relationship.
RTD does not presently claim that mathematics is ontology, as Badiou does. It does not attempt to derive philosophy from mathematical logic in the manner associated with the logicist project of Russell and Whitehead. It does not merely borrow the appearance of mathematical demonstration, as Spinoza borrowed geometrical form. Nor does it assume that because Whitehead was both mathematician and process philosopher his process metaphysics therefore possesses an implicit mathematical formulation.
Its question is narrower:
Can important dimensions of relational becoming be described mathematically?
That question places RTD within a long intellectual history while leaving its outcome undecided.
We may discover that established mathematics already describes much of what GRR has expressed philosophically. We may discover that some relational concepts resist meaningful formalization. We may find that apparently novel ideas are already familiar within dynamical systems, network theory, topology, probability, information theory, or other mathematical disciplines. And perhaps we may discover places where bringing those established mathematical descriptions together exposes a problem not adequately captured by any one of them.
The precedent provided by these philosophers therefore gives RTD neither proof nor permission to invent mathematics carelessly.
It gives us something better:
reason to ask the mathematical question seriously.
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