Prelude to the Syntactic-Calculus of Figurative-Language and Categorical-Imperatives

Before we delve deeper into our mathematical and scientific pictograms — those symbolic expressions that reflect the unity of art and reasoning — we wished to offer a few preliminary remarks upon the principles that underlie all our teachings at PAT Tutoring. These principles form the foundation of our learning philosophy and guide the creation of every structure and exercise that follows. These being our Euclidian Elements of ‘figurative-language.’ We debated amongst ourselves whether it be best to commence with our description of the pre-hadronic series of molecular-argumentation first, or to lay out the basic structure of synthetic-reasoning as pertains to the edifice of mathematics. We landed on canonizing the requisite laws and relations of the syntactic-calculus of figurative language in which, while more obscure and esoteric to the uninitiated reader, each of our future systems will be based. 

It is from this point, we conjecture, that the themes and structure of mathematics, as well as all other structures built from the edifice of human reason are liable to be modeled and transformed relative to the inherent structure of metaphor through the extension of abstracted symmetry-groups which we so term ‘categorical-imperatives.’ Through this structure not only are we then able to represent the extensions and expressions of identity and being as functions of differentiation, we may also perform a syntactic-calculus upon each of these ideas which allow us to find the instantaneous-relation of each limit, and to either prove or refine each idea along its naturally emergent ‘analogous-asymptote.’  

‘Analogous,’ in Greek, ‘the logic – again,’ is more than mere anagram, or poetic-device, just as the anagram is not a mere jumbling of characters, nor the pangram an expression of all letters. Nor is the metaphor simply what it appears at first. ‘Metaphor’ succinctly taken as the ‘meta-form,’ the being which utilizes its own inherent identity-function, or the linearization of its being to conjugate into a higher-order dimensional invariant, or symmetry group, within the synthetic-function. 

(By synthetic-function, we mean the operation of reason upon perception, forming higher-order correspondences of meaning — much as a function in mathematics operates upon variables to generate new relations.) 

Not only are we thereby led to accept that all poetic-device is not only a class of objects representing tools in linguistics, but within our own conception, or rationale, as well. We recognize imagery as much as in a visual context, or cortex, rather than simply a mere linguistic operator; we hear onomatopoeia before we translate its use to the paper.  We shall venture between the verbiage of both physics and the sciences, mathematics, as well as linguistics and poetry in general, as we recognize all thought as an ever-evolving field or manifold, in which our conceptions play a large role.  

If all which the mind perceives may be taken into a metaphorical context of representation, we quickly move into the Kantian-realm of ‘figurative-imagination’ or in our case, ‘figurative-language.’  

As Wittgenstein would have noted, our language forms (our language of forms) the complete rational picture which dictates the bounds of our experience, so too do the derivative of our notions determine the form they take within our minds as they are abstracted into these symmetry-categories. Not only this, however, when taken as beings of abstracted-meaning, a sense of synesthesia, or fugue (mixing metaphor of sense), occurs in which we recognize that the operations which our mind plays upon our perceptions allow for all further mental calculations to occur relative to the natural bounds of energy, matter, and perception (discrete-quanta).  

(A true out-of-body experience as the ‘I’ (id) in conception is conjugated relative to the being-in-itself within our perception.) 

In brief, we shall describe these calculations to refine ideas at their limits of either substantial being in intuition, or at the upper bounds (maxima) of reasoning which does not collapse in on itself in an irrational unity. We shall define such ideas as either discrete or continuous, either proven or approximated at their local limits. It shall be shown that each synthetic-function retains its own correlate synthetic-function, with a similar degree-of-freedom, or leading degree or power. We will also endeavor to express the complex-conjugation of each synthetic-function or metaphor and perform complex-analyses of the functions under question, as well as show that any incongruity in reasoning stems from the mixing of metaphor leading to indefinite-functions. 

To recapitulate, or alliterate, we endeavor to show that all perceptions of the mind undergo the same abstractions into relative degrees of symmetry-groups, in which symmetry of the deeper invariant structure is then preserved as functions are metaphorically transformed along the complex-manifold of the mind in order to form conjugate synthetic-functions in which all syntactic-calculus, or operations of the rational-rationale, are carried out in order to form new ideas, or to reason from given principles in the forms of deduction or induction. This elucidates the malleability (in metallic-ratio, of transition elements) of all perception into figurative-language which may be expressed and transformed in the guise of or in relation to another as each categorical-group retains its own canonical symmetry or degree-of-freedom, or participation in the idea. Through this, metaphor becomes not merely a poetic device, but a rational operator within the manifold of thought — a form of syntactic calculus through which new ideas emerge, are derived, and refined from first principles.  

The introduction to which follows below: 

Prelude to the Syntactic Calculus and Analogous Asymptotes 

Abstract 

We propose that all reasoning, perception, and linguistic form may be modeled as transformations within a unified syntactic calculus. Each transformation expresses an underlying symmetry relation—what we call an analogous asymptote—where metaphor, mathematics, and cognition converge. This paper outlines the axioms of this framework as the philosophical foundation of PAT Tutoring’s learning model. 

I. Foundational Assumption 

Axiom 1 (Self-Evident Principle).
All education and inquiry arise from invariant relations between form and meaning.
These invariants precede content; they define the conditions of learnability. 

Our pedagogical and philosophical work therefore begins with the premise that the structures of mathematics and language share a single geometry of reasoning. Whether these structures grant perception an ontological completeness or merely express an underlying, invariant field remains the central question.
We adopt the latter view: perception is an expression of a deeper symmetry. 

II. Synthetic Functions and Categorical Imperatives 

Definition 1 (Synthetic Function).
A synthetic function is the operation by which reason acts upon perception to produce a new correspondence of meaning, analogous to a mathematical function acting on variables. 

Definition 2 (Categorical Imperative).
In this context, a categorical imperative denotes a symmetry-group of conceptual transformations that preserves the identity of meaning under variation of expression. 

Through these functions, ideas can be differentiated, integrated, and linearly transformed within a common manifold of cognition. The calculus of these transformations constitutes our syntactic calculus. 

III. Metaphor as Transformation 

Proposition 1.
Every genuine metaphor establishes an isomorphism between two conceptual manifolds. 

The Greek analogos—“in ratio again”—reveals that analogy is not a poetic ornament but a logical recurrence of form. Likewise, metaphor (“meta-phorá,” the carrying beyond) signifies a self-referential function of identity. Metaphor therefore acts as a meta-form, conjugating being into higher-order symmetry classes. 

IV. Language, Perception, and the Cognitive Manifold 

Imagery, sound, and symbol form a single perceptual field. We hear onomatopoeia before we write it; vision precedes notation. Within this manifold, each linguistic operation corresponds to a transformation of energy and attention—mental work governed by limits of coherence analogous to physical conservation laws. 

Thus, figurative language, in the Kantian sense of figurative imagination, becomes a topological interface between intuition and concept. As Wittgenstein observed, “the limits of my language mean the limits of my world”; our framework extends this: the differential of language defines the curvature of thought. 

V. The Calculus of Reason 

We identify two principal operations within the syntactic calculus: 

1. Differentiation — Refinement of an idea at its limit of being or perception. 

2. Integration — Unification of ideas under higher symmetries of relation. 

Each function admits a conjugate counterpart, forming complex pairs whose interference patterns yield either stability (proof) or oscillation (approximation). Incongruities in reasoning arise when metaphors of incompatible symmetry mix, producing indefinite or divergent functions. 

VI. Summary and Outlook 

All perceptions of the mind undergo abstraction into degrees of symmetry.
Reasoning, then, is the evolution of these symmetries through metaphorical transformation across the complex manifold of thought. The syntactic calculus provides the formal grammar of this evolution; the analogous asymptote marks its limit and potential. 

Future work will formalize these relations as explicit operators acting on conceptual fields, laying the groundwork for a complete archetypal mechanics of cognition. 

Autumn Reflection — PAT Tutoring 

10-17-25

As we move through the fall season, we are reminded of the ever-evolving nature of education itself. Just as the winds shift and the leaves turn, so too does language transform — carrying with it the dynamism of thought and the Stoic calm that steadies us amid change.

Linguistics, like the seasons, embodies flux. Meaning arises as a relative function — a localized expression within the bounds of our own language, refined at its limits like boundary conditions in the phase-space of learning. Our words are both tools and terrains, mapping the complex plane of understanding through simile, syntax, and shared imagination.

The changing colors remind us that knowledge, too, transforms — that each leaflet from the Tree of Learning has converted as much energy into meaning as it possibly could. Whether our tongue be localized, terrestrialized, or even colonized, the weather of thought shifts in all climates. Our ideas are continually terraformed by perception itself — a testament to the inexhaustible landscape of human understanding.

10-21-25

I spent some time this morning with my fiancé over coffee debating the mythic nature of reality which I have recently found myself sunk deeply into. It was a late-night reading of Borges to my daughter which spurned my own thoughts down a recursive train of reasoning which had not ceased even following the interplay of my own symbolic dreams. I recognize the manner in which the information which I integrate insatiably within the manifold of my own mind must necessarily conform to the Anti-deSitter space contained there-in, and thus, I must perceive a certain homogeneity of expressions, causes and effects, as functions of a natural underlying symmetry, which Schopenhauer abhorred, and Kant attempted to categorize.  

To not borrow too much from analogy, even as all form is figurative representation which expresses itself through figurative-speech, we discussed the lack of mythologizing in the maths and sciences, the language and figures or forms of speech which are conjured by the mind as the ‘I’ of consciousness (the third-eye) embodies these dynamic fields and is beholden to the same natural laws as ontologic-space.  

Of course, I discussed my own philosophical maxims, such that all ontic energy is discrete and quantized and that all interplay of subjective-superlatives must belong to the conjuring topography of the mental-manifold, which, much like a tongue-twister of body-language (shown not said), is prone to tie itself in syntax knots, or ‘branes,’ tied together by one inextricable-cord, lest the entire structure fall to ruins and dissipate into mere representation, or an alleviation of syntactic-volume by route of semiotics.  

(Remarkably, string-theory may be that which holds the ‘brain,’ together, as syntax-knots conjure the idea of self-contained stable categories which resonate through so many mediums. The particle-wave duality of course represents the lack of probability-saturation at low extremes, or within condensates, and higher dimensions are called for by the interpolation of higher-order synthetic-functions with stable degrees-of-freedom.) 

I then discussed further the nature of our knowledge, or our natural-knowledge, as our ideas are continuously refined and either proven or approximated at the analogous-asymptote. A barrier in which our human understanding shall never surpass. I spoke of the unending nature of the maths and sciences, as we forever refine this limit, but never reach the true boundary, as the mere act of participation continues to push the horizon even further.  

I was then pushed down a trail, almost as if by an armed guard, who had restricted my own movements and caused me to conceive of the cyclical nature of all being and signs as discrete entities, not of a simple nor composite substance, but to be analyzed as such. I spoke not of reincarnation, a manner to pick up the thread of your previous li(f/v)e’s work, but I spoke of cosmic recurrence, the nature of entropy and the recursion of discrete packets of quanta. I was led to see past the Upanishads and the Buddhist texts, and all philosophers of every age and millennia, and I was led to the conclusion that none were wrong, they simply took the derivative of the differential operator of reason over knowledge at differing points belonging to the asymptote.  

Knowledge was a problem of gauge-measure, and incongruity to be found in the mixing of metaphor leading to indefinite functions, irrational lines of thought which continued indefinitely, incongruous chimeras and monsters who did not hold up to isomorphism. 

I have slowly led myself to a mystical, rather than spiritual destination. I still hold fast to the bastions of science and mathematics as the arbiters of ‘useful-fiction.’ I do believe I no longer have faith in any story besides those entitled fiction. Hume failed to recognize that his book should be gifted to the fire just as well as any other sophistry, however, I make no similar mistake. I hold that all books will be committed to the fire, not for they hold no value, but because they will return to the larger function of nature, in which all is consumed in reference to itself.  

All being is a true Ouroboros, all meaning cyclical and must satisfy the function which it was measured from. The mythics witnessed this mystery, the academics fail to acknowledge it. They seek to remove the ‘I,’ or the man, from mystery as if he is not at the heart of the novel. There can be no Grand Unified Theory without man at the heart of it, for to merge Quantum Field Theory and General Relativity without man to measure would be a mere mixing of metaphor.  

I do not argue for a participatory universe. There was a cosmos without us, we can intuit the fine-tuning of homogeneous causes which lead to our intuition in the first place. Jung even points out the Archetype of our minds, as a differentiation of the manifold, which Campbell’s symbols always point back to. We often recognize the poetry in nature, the ‘coupling-couplets’ of electrons, the invariant structures of math which we have pulled from its own embodiment of these forms. History itself does not repeat, it rhymes. Our theory of knowledge should be no-less poetic. 

 I argue for the viewpoint of man to be a mere differentiation-by-differenciation. A local-logical (linearized) operation of the reasoning within the manifold of the mind. We act upon the space we intuit by means of the space we inhabit and inhibit. Our connexion to the cosmos is no stranger than that of an asteroid, we simply fail to recognize our own refinement of knowledge as a continuous and discrete process, never to be resolved in perpetuity.  

I argue for the myth to be placed back into our studies, the recognition of human-hubris to be overcome by the knowledge that all is of one function which must protect itself from logical-fallacy yet may continuously be solved and analyzed as the expression of any metaphor. I argue that nature itself be myth, all books be viewed as fictional representations of true events and closed-systems. I argue we return to viewing the natural-world as a deciphering of hieroglyphs which aid us in our quest along the transverse-axis of conjugate-meaning, a barrier to which we will only find at the local semantic-minima of representation in expression, the discrete quanta which must always recur, never to be destroyed nor created. The substance will always be itself individuated. 

Thus, what began as a discussion of myth returns to itself as myth: a self-consuming, self-creating function of mind and matter.