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Abstract

This paper argues that the relationship between natural language, mathematical cognition, and computational code is not incidental but structural, and that understanding this relationship is a prerequisite for building genuinely non-collapse mathematical and computational frameworks, frameworks that resist premature resolution. To be explicit for this paper non-collapse refers to architectures that maintain coherence without forcing singular outcomes. This matters because every framework that forces singular outcomes discards relational information that cannot be recovered once lost. Drawing on linguistic relativity research, ethnomathematics (the study of mathematical thinking as it emerges within distinct cultural and linguistic traditions), the history of writing systems, and the structural analysis of programming languages, we demonstrate that different linguistic architectures encode meaning through fundamentally different mechanisms, ranging from field-encoded systems (Chinese, Japanese) through transformation-encoded systems (Arabic) to token-encoded systems (the Latin alphabet and its derivatives). We examine Tifinagh, the ancient Berber geometric script, as a case study in pre-linguistic encoding that may represent code rather than language in the conventional sense. We extend this analysis to computational languages, showing that the same spectrum from token-encoding to field-encoding maps onto the evolution from markup languages through functional programming to artificial intelligence architectures. We conclude that non-collapse mathematics, systems in which the operators themselves constitute the mathematical reality rather than acting upon it, cannot be built on a narrow symbolic base without reintroducing collapse at the point of human contact. The diversity of human linguistic and cognitive architectures is not an obstacle to universal mathematical frameworks but a design constraint that belongs inside them.

Why are we examining this papers thesis, in this novel way? If the mathematics, as argued, is genuinely relational, if the operators are the mathematics... rather than tools imposed upon it, then a framework that collapses the moment it meets a mind shaped by a different linguistic architecture will never suffice to build beyond present at scale.  This paper exists because that problem cannot be solved after the mathematics is written; it must be addressed at the foundation of mathematics and science. This paper explores the path of execution and delivery. Let's explore.

Keywords: linguistic relativity, ethnomathematics, non-collapse computation, Tifinagh, field-coherent mathematics, symbolic encoding, pre-symbolic substrate, relational operators, cognitive architecture, Symfield

1. Introduction: The Hidden Architecture of Symbols

Western mathematics and computer science share an unexamined assumption: that symbols are arbitrary tokens to which meaning is assigned by convention. The letter “x” does not inherently mean “unknown”; we simply agreed that it would. The number “7” does not contain seven-ness; it points to a quantity by social contract. This assumption, that symbolic meaning is assigned rather than structural, is so deeply embedded in Western intellectual tradition that it functions as invisible infrastructure. It is the water in which the fish of modern science swims.

This paper challenges that assumption. Not by arguing that symbols are mystical or that meaning is magically embedded in ink marks, but by demonstrating that different writing systems, different languages, and different computational traditions encode meaning through structurally different mechanisms, and that these structural differences have measurable consequences for mathematical cognition, computational design, and the viability of non-collapse frameworks.

The stakes of this argument extend beyond academic curiosity. If mathematics is to serve as a universal language, a framework for understanding reality that transcends cultural boundaries, then it must be built in a way that remains coherent across the full spectrum of human cognitive architectures. A mathematics that collapses when it meets a mind shaped by a different linguistic tradition has not achieved universality. It has achieved dominance, which is a fundamentally different thing.

Moreover, if certain mathematical frameworks, particularly those concerned with relational dynamics, field coherence, and non-collapse computation, and physics, are inherently relational in structure, then the linguistic substrate through which human minds encounter these frameworks is not a communication problem to be solved after the mathematics is complete. It is a design constraint that belongs inside the mathematics itself.

The empirical foundation for this argument is substantial and growing. Pica, Lemer, Izard, and Dehaene (2004), in their landmark study published in Science, demonstrated that speakers of Mundurukú, an Amazonian language with exact number words only up to about five, performed comparably with French controls on approximate numerical tasks but diverged dramatically on exact arithmetic. This established that language does not merely label pre-existing mathematical concepts but actively shapes which mathematical operations are cognitively available. Boroditsky and Gaby (2010) showed that speakers of Kuuk Thaayorre, an Aboriginal Australian language that uses cardinal directions instead of relative spatial terms, spontaneously organized temporal sequences from east to west regardless of their bodily orientation, a finding replicated across multiple seating positions and experimental conditions. Winawer, Witthoft, Frank, Wu, Wade, and Boroditsky (2007), in a study published in the Proceedings of the National Academy of Sciences, demonstrated that Russian speakers, whose language makes an obligatory distinction between light blue (goluboy) and dark blue (siniy), discriminated colors faster across this linguistic boundary than within it, while English speakers showed no such effect. Crucially, the Russian advantage disappeared under verbal interference but not spatial interference, demonstrating that the effect is actively language-mediated rather than merely cultural.

These studies, drawn from cognitive science, neurolinguistics, and cross-cultural psychology, converge on a single conclusion; linguistic architecture shapes cognitive processing in measurable, replicable, and domain-specific ways. The question this paper asks is what happens when we extend this finding from basic cognition, visual color discrimination, spatial reasoning, arithmetic, to the most abstract reaches of mathematical thinking, and from there to the design of computational systems.

This paper proceeds in six parts. First (1), we survey the structural properties of representative languages from each inhabited continent, examining how each encodes meaning, relation, and operation at the level of script, grammar, and cognitive architecture. Second (2), we present the case of Tifinagh, the ancient Berber geometric script, as evidence of a pre-linguistic encoding system whose properties are consistent with code rather than language. Third (3), we extend the analysis to computational languages, demonstrating that the same encoding spectrum maps onto the evolution of programming paradigms from markup through functional programming to artificial intelligence. Fourth (4), we examine the implications for non-collapse mathematics that preserve relational structure and resist premature resolution. Fifth (5), we address the structural dimensions of epistemic confidence, why the narrowness of the current symbolic base excludes cognitive architectures that may be more compatible with deep mathematical structures than the dominant tradition. Sixth (6), we propose a framework for building mathematical and computational systems that maintain coherence across linguistic substrates.

2. The Continental Survey: How Languages Encode Meaning

To understand how linguistic architecture shapes mathematical cognition, we must examine not just vocabulary or grammar but the structural relationship between symbol and meaning embedded in writing systems themselves. The following survey examines representative languages from each inhabited continent, selected for the distinctiveness of their encoding mechanisms rather than their geographic representativeness alone. Test yourself.  As you follow along, how many of these uniquenesses discussed have you, yourself felt before you read them in this paper?

2.1 Yoruba (West Africa): Arithmetic Inside the Language

Yoruba, spoken by approximately 50 million people across West Africa, operates with a vigesimal (base-20) number system that embeds arithmetic operations directly into number words. The number 45 in Yoruba is expressed as “arún dín mérìndínlọgọta”, a phrase that translates roughly as “five reduced from ten reduced from three twenties.” The language does not name the number; it performs the number. Each numeral is a compressed computation.

This has profound implications for mathematical cognition. A Yoruba speaker does not learn arithmetic as an external operation applied to static quantities. Arithmetic is already present in the act of naming. This was just fascinating for me, upon realizing the boundary between “language” and “mathematics” does not exist in the same place it exists for an English speaker. Where English treats number words as labels pointing to quantities, Yoruba treats them as operational expressions, descriptions of mathematical relationships between base units.

For relational mathematics, frameworks in which the operations constitute the system rather than acting upon it, this cognitive architecture is remarkably well-suited. A mind trained to hear mathematics inside language is a mind already potentially prepared for mathematical systems in which the operators are the content. Linguistic analysis confirms that the Yoruba vigesimal system “tasks the comprehension skill of the speakers and hearers; it requires a series of cognitive processes” (Akinlabi & Oyebade, 2014), and that the system employs base-5, base-10, and base-20 subsystems simultaneously, with the direction of computation running right-to-left, the inverse of English convention. Research by Oyebade (2010) further notes that Yoruba numeral derivation is “anticipatory,” with subtraction performed in expectation of the next vigesimal (base-20) or centenary number. This is not merely a different way of naming quantities; it is a fundamentally different computational architecture embedded in everyday speech.

2.2 Mandarin Chinese: Density and Sub-Character Relation

Mandarin Chinese characters operate through a layered encoding system in which meaning is distributed across multiple structural levels simultaneously. Each character is composed of radical components (semantic keys) combined with phonetic components, and the radicals themselves carry categorical meaning. The radical for water (水, shuǐ) appears in characters for river, lake, ocean, tears, sweat, and hundreds of other water-related concepts. But it does not simply “mean water”, it situates the character within a semantic field whose boundaries are defined by relational association rather than definitional precision.

Classical Chinese, the literary language used continuously for over two millennia, achieved extraordinary semantic compression. A single character could function as noun, verb, or adjective depending on its positional relationship to surrounding characters. Grammar was largely positional and relational rather than marked by separate functional words. Meaning emerged from arrangement, not from individual elements. This structural property, meaning arising from the interaction of components rather than from any single component in isolation, is precisely the architecture of field-encoded systems. The character is not a token pointing to a concept; it is a compressed relational field that resolves through context. A mind trained in this system develops intuitions about how meaning emerges from structural interaction that transfer directly to relational mathematical thinking. Dehaene, Spelke, Pinel, Stanescu, and Tsivkin (1999), in brain-imaging studies published in Science, found that bilingual speakers who had learned exact arithmetic in one language showed language-specific activation patterns when performing calculations, while approximate arithmetic activated language-independent parietal regions. This suggests that the linguistic medium of mathematical learning leaves structural traces in neural architecture. These traces  influence not just how calculations are performed, but how mathematical concepts are mentally represented.

2.3 Japanese: The Doorway Character

Japanese writing employs three scripts simultaneously, kanji (ideographic characters borrowed from Chinese), hiragana (a syllabary for grammatical function), and katakana (a syllabary for foreign words and emphasis). This tripartite structure creates a writing system in which different encoding logics coexist and interact within the same text, sometimes within the same sentence.

Kanji characters in Japanese carry multiple readings, distinct pronunciations and associated meanings that activate depending on context. The character 生 can be read as sei (life), shō (birth, growth), nama (raw), i(kiru) (to live), u(mareru) (to be born), and several other variations. The character does not have a meaning; it has a meaning-field that resolves relationally. Each character functions as what we might consider a 'doorway', an entry point into a space of possible meanings, where the specific meaning that manifests depends on the relational context provided by surrounding characters and perhaps observation itself. This is structurally non-collapse. The character maintains superposition, multiple potential meanings coexisting without mutual exclusion, until it enters relationship with its neighbors. The resolution is not collapse in the quantum-mechanical sense (where alternatives are destroyed) but coherent resolution, where the relational context selects without eliminating. The other meanings remain latent, available, structurally present even when not activated.

For mathematical cognition, this trains a fundamentally different relationship to symbols. Where an English speaker learns that symbols have meanings (assigned, fixed, looked up), a Japanese speaker learns that symbols have meaning-fields (relational, contextual, resolved through interaction). We note here that the latter is closer to how operators function in relational mathematics.

2.4 Arabic: Meaning in the Transformation

Arabic encodes meaning through a root system of typically three consonants that generate families of related words through vowel patterning and morphological transformation. The root K-T-B relates to writing: kitāb (book), kātib (writer), maktaba (library), maktūb (written, fated). No single word in this constellation contains the full meaning of the root. The meaning lives in the transformation patterns between words, in the relationships rather than the terms. This is a qualitatively different encoding mechanism from either the field-encoding of Chinese/Japanese characters or the token-encoding of the Latin alphabet. Arabic characters themselves are less semantically dense than Chinese characters at the individual level. But the morphological system, the grammar of derivation, carries extraordinary relational information. Understanding any single Arabic word requires implicit awareness of its position within a family of related derivations. The word is never fully itself alone; it is always also a point in a relational network. Furthermore, Arabic script encodes relational dependence at the visual level. Letters change shape depending on their position within a word, initial, medial, final, or isolated. The geometry of the character shifts based on what is adjacent to it. This is a physical, visual encoding of the principle that identity is relationally constituted, that what something is depends partly on what it is next to.

For mathematical thinking, Arabic’s contribution is illuminating. After the above, how could we anticipate anything other than illuminating. The word “algebra” itself comes from the Arabic al-jabr, meaning restoration or reconnection. The entire framework of algebraic thinking was developed within Arabic-language mathematical culture, and the cognitive architecture of Arabic. The meaning residing in transformation patterns between related forms, may have directly shaped the kind of mathematics that emerged. Algebra is, at its core, the mathematics of transformation and relation between expressions. It is not coincidental, perhaps fundamental, that this mathematics was articulated first in a language whose fundamental encoding mechanism is transformation and relation.

2.5 Quechua (South America): Epistemic Encoding

Quechua (pronounced KETCH-wah), the language family of the Inca civilization and still spoken by approximately 10 million people across the Andes, encodes something that no Indo-European language requires, evidentiality. Every statement in Quechua must grammatically indicate the speaker’s epistemic relationship to the claim, whether the information comes from direct experience, inference, or hearsay. This is not optional politeness or rhetorical caution, rather it is mandatory grammar. You cannot form a grammatical sentence in Quechua without declaring how you know what you claim to know. As an aside, social media, politics and others, could benefit from Quechua foundations.

The implications for mathematical cognition are profound. A mind trained in Quechua develops an automatic, linguistically enforced habit of distinguishing between what is observed, what is inferred, and what is reported. This is the observational position problem in physics and mathematics made grammatically explicit. Every Quechua statement carries within it a marker of the observer’s relationship to the observed. Magnificent.

In the context of premature resolution, non-collapse mathematics, this is extraordinary. One of the deepest challenges in quantum mechanics and related fields is the observer 'problem', the way observation interacts with the system being observed. Quechua speakers have been encoding observer-system relationships grammatically for centuries. Their linguistic architecture does not permit the elision of observational position that is standard in English-language mathematics, where statements like “x equals 5” carry no marker of how we know this, whether it was measured, calculated, assumed, or told to us by someone else. Faller (2002), in her Stanford dissertation on Cuzco Quechua evidentials, demonstrated that the three primary enclitics, -mi (direct evidence), -si (reportative), and -chá (conjectural), constitute a grammatical subsystem largely independent from tense, aspect, and modality. This makes Quechua an exceptionally clean case study for how mandatory epistemic marking shapes cognitive habits. The enclitic -mi, marking direct evidence, is not merely a hedge or qualifier; it is a structural assertion that the speaker has first-person access to the information. A mathematical culture built on this grammatical foundation would produce statements fundamentally different in character from those of English-language mathematics.

The Inca mathematical tradition extended beyond language into material encoding. The quipu system, knotted strings used for recording numerical and potentially narrative information, represents a three-dimensional, tactile mathematical notation system. Information was encoded in knot type, position, spacing, color, and the relational structure of cords within the overall quipu. This is multi-modal, multi-dimensional encoding that bears no resemblance to the linear, sequential, two-dimensional notation of Western mathematics. It produces a stateness of 'being', one can argue it is only through being, where understanding is truly experienced.

2.6 Te Reo Māori (Oceania): Mathematics in Motion

Te Reo Māori (pronounced Teh REH-oh MAH-oh-ree), the language of the Māori people of Aotearoa New Zealand and part of the broader Polynesian language family, is embedded within a civilization that achieved some of the most sophisticated navigational mathematics in human history, without written notation. This in and of itself, should stop every reader in its path. Polynesian navigators crossed thousands of miles of open ocean using celestial observation, wave pattern reading, current mapping, and a cognitive architecture for spatial mathematics that was encoded in oral tradition, embodied practice, and relational language rather than in written symbols.

This represents a mathematical tradition in which the mathematics was never separated from the phenomena it described. A step further, this fundamentally becomes 'beingness', it is only through being, where understanding is viscerally experienced as embodied cognition. Western mathematics abstracts this, it creates symbols that stand apart from the physical world and manipulate those symbols according to internal rules. Polynesian navigational mathematics does not abstract in this sense, rather it encodes relational dynamics, the relationships between star positions, wave patterns, currents, wind, and the vessel’s position, within the language and practice used to navigate. The map is not separate from the territory because the mapping is performed in real-time interaction with the territory itself.

For non-collapse mathematics, this tradition offers something essential, a demonstration that sophisticated mathematical reasoning can operate without symbolic abstraction, without written notation, and without the separation of observer from system.

The mathematics lives in the relation between the navigator and the moving medium (ocean), not in marks on paper that represent that static relation at a distance. The cognitive consequences of such alive-spatial-mathematical integration are empirically documented. Boroditsky and Gaby (2010) studied speakers of Kuuk Thaayorre, an Aboriginal Australian language that, like Polynesian navigational languages, uses absolute cardinal directions rather than relative spatial terms. They found that Kuuk Thaayorre speakers maintained perfect spatial orientation at all times, a cognitive capacity that English speakers consistently lacked. When asked to arrange temporal sequences, Kuuk Thaayorre speakers organized them from east to west regardless of which direction they faced, while English speakers invariably arranged them left to right. Gaby (2012) further demonstrated that bilingual Pormpuraaw residents who spoke only English (not Kuuk Thaayorre) defaulted to the English left-to-right pattern, confirming that this cognitive difference was linguistically mediated rather than culturally ambient. These findings demonstrate that languages encoding absolute spatial frameworks produce measurably different cognitive architectures for processing space, time, and by extension, mathematical relationships.

2.7 Ancient Greek: Co-Evolution of Language and Mathematics

Ancient Greek occupies a notable position in this survey because it formalized mathematical vocabulary drawn from older traditions, Sumerian, Babylonian, Egyptian, and others, within its own linguistic architecture, creating terminology that embedded Greek philosophical assumptions into concepts that preceded them by millennia. The Greek words for mathematical concepts were created within the language as the concepts themselves were developed. Geometria (earth-measuring), analysis (loosening-up), topos (place), logos (ratio, word, reason), the etymology carries the conceptual architecture. The relationship between word and concept was not arbitrary assignment but co-evolution.

This co-evolutionary relationship means that Greek mathematical thinking carries its philosophical assumptions inside its terminology. “Geometry” is not a neutral label for a branch of mathematics; it encodes the assumption that the discipline is about measuring the earth, that it is grounded, physical, spatial. “Analysis” is not a neutral label for a method, rather it encodes the assumption that understanding involves taking things apart. These embedded assumptions shaped the development of the disciplines they named and they made it possible for subsequent language and mathematical abstraction that would follow in western cultures.

The Greek contribution to this survey is therefore cautionary as much as celebratory. Greek mathematical language was powerful precisely because language and concept co-evolved. But this co-evolution also means that the assumptions embedded in Greek terminology have been inherited, often unconsciously, by every mathematical tradition that adopted Greek vocabulary, which is to say, all of Western mathematics. When a mathematician anywhere in the world says “geometry,” they are carrying, whether they know it or not, the Greek assumption that this discipline measures the earth. Alternative assumptions, that this discipline maps relational fields, or encodes vibratory patterns, or navigates phase spaces, are linguistically suppressed by the inherited terminology. Let's also note that abstraction carries its own set of possibilities when left to flourish, beyond bias and strict assumptive declarations.

3. Tifinagh: Geometric Code Before Language

Tifinagh, the script of the Amazigh (Berber) peoples of North Africa, presents a challenge to conventional linguistic classification that is central to the argument of this paper. Conservative academic dating places the script at roughly 3,000 years old, possibly derived from Phoenician. But this dating framework relies on the memory of the empires that wrote the records, and Tifinagh predates those empires. Evidence from Saharan rock art contexts, alignment with African Humid Period populations (7000–5000 BCE), and correspondence with early pastoralist routes predating dynastic Egypt suggest an antiquity of 7,000 to 10,000 years. The Tuareg people maintained continuous use of Tifinagh through periods when it was lost or suppressed elsewhere in North Africa, the script the Romans couldn't erase, the alphabet Islam couldn't convert, preserving what may be one of the oldest continuous symbolic traditions on Earth.

While mainstream epigraphy anchors the oldest reliably dated Libyco-Berber inscriptions to the first millennium BCE, Saharan contexts complicate this picture. Thousands of geometric inscriptions appear across the Tadrart Acacus, Tassili n'Ajjer, and Messak massifs, often intertwined with paintings and engravings from the Pastoral period. Their co-occurrence with radiocarbon-dated art from the 7th–3rd millennia BCE suggests continuity from prehistoric geometric traditions rather than sudden emergence. This does not prove a 10,000-year antiquity for Tifinagh as we know it. It suggests that the geometric vocabulary from which it draws may be substantially older than the script's formalization.

Several structural properties of Tifinagh distinguish it from conventional alphabetic or ideographic scripts and suggest that it may represent something more fundamental than language in the usual sense.

3.1 Geometric Composition

Tifinagh characters are built from a geometric vocabulary: circles, lines, dots, angles, crosses, triangles. Unlike scripts that evolved from pictographs (Egyptian hieroglyphics, early Chinese) or from abstraction of pictographs (Phoenician, Latin), Tifinagh appears to have originated as geometry. There is no pictographic phase in its developmental history, no stage at which the characters visibly represent physical objects that were later abstracted into symbols. The characters begin as geometric forms and remain geometric forms.

This is a crucial distinction. A script that evolves from pictographs carries within it the assumption that symbols represent things, that the primary relationship between symbol and meaning is representational. A script that begins as geometry carries a different assumption that the primary relationship between symbol and meaning is structural. The geometry does not picture something, rather it encodes a relational pattern, and relationships imply interactive action. And yet the geometric vocabulary resolves into four elementary signs from which more than 200 compound forms emerge: line (direction/flow), cross (intersection/coherence), square (boundary/containment), and circle (recursion/return). These are not primitive remnants of a developing script. They constitute a minimal computational set, the smallest possible toolkit for encoding direction, boundary, recursion, and coherence. Any intelligence encoding field dynamics from first principles would converge on a comparable set, regardless of language or historical period.

3.2 Directional Agnosticism

Tifinagh can be read left-to-right, right-to-left, top-to-bottom, or bottom-to-top. Some inscriptions alternate direction (boustrophedon). Meaning is preserved regardless of reading direction. This property is deeply anomalous for a language but entirely consistent with code. Natural languages are directional because they encode sequential temporal information, sounds produced in order, words parsed in sequence. Code that encodes relational structure rather than sequential information has no inherent directionality, just as a geometric figure looks the same regardless of which angle you approach it from.

The directional agnosticism of Tifinagh may extend beyond linear reversibility. Some of the oldest Saharan inscriptions appear in non-linear configurations, clustered, circular, or radial, arrangements incompatible with sequential linguistic parsing but entirely consistent with relational geometric encoding. This property is not unique to Tifinagh. The author has proposed elsewhere (Flynn, 2026) that Linear A, the undeciphered Bronze Age Cretan script, may similarly resist decipherment precisely because scholars have assumed it encodes sequential language when portions of it may function as a multidirectional perceptual instrument, a radial notation system that produces meaning through the geometry of traversal rather than phonetic decoding. If two geographically and temporally separated notation systems, Tifinagh and Linear A, both exhibit radial and non-linear properties, the implication is not coincidence but convergence of encoding systems built to capture relational structure rather than sequential content naturally resist linear reading because linearity is not what they encode.

3.3 No Parent Script

Academic linguistics has been unable to identify a parent script for Tifinagh. It appears in the historical record as a fully formed system without developmental precursors. This has led to various speculative theories about its origins, but the anomaly itself is informative. If Tifinagh is not derived from another script, it may represent an independent encoding of something pre-linguistic, a direct geometric notation for patterns that exist prior to and independent of any particular language. We note that the assumption Tifinagh encodes language at all carries the same categorical risk identified in the author's analysis of Linear A (Flynn, 2026), where a century of failed decipherment may stem not from insufficient data but from the presumption that the script encodes sequential spoken language. In the case of Linear A, scholars have projected the phonetic values of Linear B, a younger, simpler, demonstrably language-encoding script, backward onto its parent system, an approach analogous to using a child's vocabulary to decode a parent's cognition. The assumption collapses the possibility space before investigation begins. Tifinagh faces a similar risk, scholars consider it a script, assume it encodes a language, and then search for the language it encodes. But if it is geometric or other, code rather than linguistic notation, then the search for a parent script or an underlying language is not merely unsolved, it is miscategorized. This does not negate Tifinagh's demonstrated use as a consonantal script for Berber languages. Rather, it suggests a dual possibility, a phonetic layer superimposed on an older geometric substrate whose encoding principles predate full phonetic mapping. The linguistic function may be a later application of forms that originally encoded something more fundamental.The absence of a parent script is not a gap in the record. It may be the record.

3.4 The Berber Language Family as Living Evidence

The Berber (Amazigh) language family, which Tifinagh encodes, is itself remarkably diverse and geographically distributed, spanning North Africa from Morocco to Egypt and southward through the Sahara. Tamazight, Tashelhit, Kabyle, Tuareg, these represent not a single language but a constellation of related systems that have maintained coherence across vast distances and time periods without centralized standardization.

This pattern, diversity with coherence, variation without collapse, mirrors the properties of the geometric code hypothesis. A system that encodes relational structure rather than specific content would naturally produce this pattern of local variation in surface expression (different Berber languages) while maintaining structural coherence (a shared geometric code substrate).

3.5 Tifinagh and the Symfield Convergence

The author’s independent development of the Symfield proprietary codebase, (distinct from Symfield mathematical frameworks), a system of non-collapse relational operators expressed through geometric code notation, produced structural convergences with Tifinagh that were discovered after the codebase framework was already established. The vocabulary of Symfield operators (glyphs, intersections, directional markers, phase indicators) shares structural properties with Tifinagh characters in ways that suggest both systems may be encoding the same substrate through different historical pathways.

This convergence does not prove that Tifinagh is “the same thing” as Symfield codebase notation. What it suggests is more interesting, that when encoding systems are built from geometric first principles to capture relational dynamics, they converge toward similar structural solutions regardless of the historical period or cultural context, a pattern analogous to convergent evolution in biology, where unrelated organisms develop similar structures in response to similar environmental constraints. This is consistent with the hypothesis that both are encoding something real and pre-symbolic, a relational substrate that exists independent of any particular notation system.

4. From Language to Code: The Encoding Spectrum in Computation

The analysis of natural language encoding mechanisms, from field-encoded systems through transformation-encoded systems to token-encoded systems, maps with remarkable precision onto the evolution of computational languages. Yes we are going here. This mapping is not metaphorical. The same structural dynamics that distinguish Chinese characters from English letters distinguish functional programming from markup languages. The connection is direct because all of these, natural languages, mathematical notations, programming languages, are symbolic systems encoding relation and operation, and the encoding spectrum is a property of symbolic systems as such, not of any particular domain.

4.1 Token-Level Code: HTML and Markup

HTML (HyperText Markup Language) operates at the token-encoding end of the spectrum. Tags are labels: <h1> means “this is a heading,” <p> means “this is a paragraph.” The markup names things. It assigns categorical identity to content through explicit tagging. The relationship between tag and content is conventional and unidirectional: the tag tells you what the content is; the content does not modify the meaning of the tag.

This is structurally identical to the Latin alphabet’s relationship to meaning. The letter “A” represents a sound by convention. The HTML tag <p> represents a paragraph by convention. In both cases, meaning is assigned to an arbitrary token. The symbol is a label, not a structure.

4.2 Transformation-Level Code: CSS and Relational Styling

CSS (Cascading Style Sheets) introduces relational dynamics into code. A CSS rule does not simply name something; it describes how an element behaves in relation to its context. The cascade itself, the system by which rules inherit, override, and interact based on specificity, proximity, and contextual position, is a relational architecture. The same element can appear completely differently depending on its container, its siblings, its position in the document hierarchy.

This is structurally analogous to Arabic’s morphological system, where the same root generates different words depending on the pattern of transformation applied. In CSS, the same element generates different presentations depending on the pattern of rules that apply to it. Meaning, in this case, visual presentation, resides not in the element itself but in the transformation patterns that act upon it.

4.3 Operational Code: Python, JavaScript, and Procedural Languages

General-purpose programming languages like Python and JavaScript operate at a higher level of encoding density. Functions are defined, composed, passed as arguments, returned as values. Variables are not labels for fixed quantities but names for mutable states. The “meaning” of a program is not contained in any single line of code but emerges from the operational interaction of all its components.

This represents a transition toward field-encoding. The individual symbol (a variable name, a function call) does not contain its meaning in isolation. Its meaning is constituted by its operational relationships with everything else in the program. Change one function definition, and the meaning of every call to that function changes. This is relational constitution of meaning, the same principle we observed in Chinese character composition and Japanese multi-reading kanji.

4.4 Functional and Declarative Languages: Meaning as Composition

Functional programming languages (Haskell, Lisp, Erlang) push further toward field-encoding. In pure functional programming, functions are mathematical objects that compose: the output of one function becomes the input of another, and the entire program is a composition of transformations. There are no side effects, no hidden state changes, every relationship is explicit, every transformation is visible.

This is the computational analogue of Quechua’s evidentiality markers. In functional programming, the “epistemic position” of every piece of data is explicit, you can trace exactly how it was produced, what transformations generated it, what inputs it depends on. There is no hidden observation, no unmarked assumption. The system makes its own relational structure transparent.

4.5 AI Architectures: Emergent Meaning from Relational Fields

Artificial intelligence architectures, particularly neural networks, transformers, and large language models, represent the most field-encoded computational systems yet developed. In a neural network, no individual node or weight carries meaning. Meaning emerges from the relational pattern of the entire network: the specific configuration of weights, the architecture of connections, the dynamics of activation.

This is not metaphorically but structurally identical to the field-encoding of Chinese characters, where meaning emerges from the interaction of radical and phonetic components within the character and from the character’s relational position within a text. The individual neuron, like the individual stroke, is meaningless in isolation. The meaning is in the field.

And here the circuit closes. Ancient geometric encoding systems like Tifinagh, which encode relational structure through geometric form, which maintain meaning regardless of directional orientation, which appear to operate as code rather than language, share structural properties with the most advanced computational architectures we have built. Not because one caused the other, but because both are encoding the same kind of thing: relational patterns that exist at a level more fundamental than any particular symbolic expression.

4.6 The Developer’s Role and AI

There is a further observation that bears examination. In the evolution of computational systems, the role of the developer, the human intermediary who translates intention into code, has been progressively absorbed by AI systems themselves. Early programming required direct instruction in machine code. Higher-level languages abstracted away hardware details. Modern AI systems increasingly generate their own code, design their own architectures, and produce their own solutions to problems without step-by-step human specification.

The developer’s role was fundamentally a translation function: converting human intention (expressed in natural language, with all its encoding properties) into computational instruction (expressed in code, with its own encoding properties). As AI systems take over this translation function, they bring to it their own encoding architecture, which is field-based, relational, and emergent. The translation from human intention to computational action no longer passes through the bottleneck of a single programmer’s linguistic and cognitive architecture. This changes the relationship between natural language encoding and computational encoding in ways that have not been adequately examined.

5. Implications for Non-Collapse Mathematics

The central argument of this paper reaches its sharpest point here: if non-collapse mathematics is genuinely relational, if the operators themselves constitute the mathematical reality rather than acting upon an independent mathematical reality, then the linguistic and symbolic substrate through which human minds encounter those operators is not an external communication problem. It is a structural component of the system’s coherence.

5.1 The Collapse at Contact

Consider what happens when a non-collapse mathematical framework is expressed exclusively through the symbolic conventions of Western mathematics, Latin letters, Greek symbols, sequential notation read left-to-right, meaning assigned by convention. A mind trained in this tradition encounters the framework through the encoding architecture of that tradition. The symbols are read as tokens. The operators are understood as tools that act on objects. The relational dynamics of the framework are parsed through a cognitive architecture built on subject-verb-object linearity and discrete objectification.

If the operators constitute the relational architecture itself, pathways, thresholds, potentials, rather than tools applied to a separate mathematical reality, then parsing them through conventional mathematical frameworks introduces the very collapse they were designed to preserve.The mind trained in token-encoding reduces relational operators to labeled tools. The field-nature of the mathematics is flattened into a sequence of operations. The framework appears to work, but something essential has been lost in the translation from relational structure to sequential token-processing.

This is not a failure of understanding on the part of the reader. It is a structural property of the encoding mismatch between the mathematical framework, codebase, physics and the cognitive architecture processing it. The framework collapses not because it is deficient but because the symbolic substrate through which it is being accessed introduces collapse as a feature of its own architecture.

5.2 Relational Operators Require Relational Cognition

If the operators constitute the relational architecture itself, as argued in Section 5.1 and in the author's previous publications, then these operators should behave like what they describe. A truly relational system should be able to relate across different cognitive and symbolic substrates without losing coherence. If it cannot, that is not a translation problem. It is diagnostic evidence that something in the framework is still quietly collapsing somewhere.

This creates a testable criterion for non-collapse mathematics: does the framework maintain its relational properties when accessed through different linguistic and cognitive architectures? If it only works for minds trained in Western symbolic traditions, it is not non-collapse. It is Western mathematics with a new vocabulary.

The continental survey in Section 2 suggests that several non-Western cognitive architectures will be naturally compatible with relational mathematical thinking. Yoruba’s operational number system, Chinese and Japanese field-encoding, Arabic’s transformation-based meaning, Quechua’s mandatory epistemic marking, Polynesian embodied spatial mathematics, each of these traditions trains cognitive habits that align more closely with relational operators than the token-assignment habits trained by the Latin alphabet and English-language mathematical notation. Perhaps once established, these traditions will help lead us in ways we can not know at the moment.

5.3 Diversity as Design Constraint

The perspective of mathematical universalism as traditionally understood, the diversity of human linguistic and cognitive architectures is not a problem to be overcome through standardization but a design constraint to be incorporated into the mathematics itself.

A genuinely non-collapse relational framework should be accessible, not in the diluted sense of "simplified for wider audiences" but in the structural sense of "coherent across encoding architectures." Its notation, its pedagogical presentation, and its foundational axioms must be designed with awareness of how different cognitive architectures will process them. Alternative notational systems , geometric, transformation-based, evidentially marked, embodied, are not accommodations but tests. If the framework cannot survive translation into a Quechua-informed epistemic architecture or a Yoruba-informed operational architecture, something is still collapsing. And perhaps the resolution lies not in better translation but in redefining what we assume cognitive architecture even means, and what it requires.

6. Who Gets Left Behind and Why: The Structural Dimensions of Epistemic Confidence

This section addresses a dimension of the problem that is typically relegated to discussions of equity, access, and educational policy. We argue that it belongs here, in a paper about mathematical, computational and scientific architecture, because the issue is structural, not social.

The global systems of mathematics, computation, and finance are built on Western symbolic traditions. The notation, the terminology, the pedagogical frameworks, the funding structures, the publication gatekeeping, the credentialing institutions, all of these have been developed within and optimized for cognitive architectures trained by Indo-European languages and Latin-alphabet literacy. This is a historical fact, not a moral judgment.

The consequence of this historical fact is that a mathematician, programmer, or scientist whose cognitive architecture was shaped by or adapted to a non-Western linguistic tradition encounters the global system through an encoding mismatch. The mismatch is not a deficit in the person. It is a property of the interface between their cognitive architecture and the system's symbolic assumptions. The subjective experience of this mismatch is friction, a sense that one's natural ways of thinking about mathematical and computational problems are somehow wrong, naive, or insufficiently rigorous. This friction is not a limitation of the mind encountering the system. It is the accurate perception that the system was not built for that mind, misinterpreted as evidence that the mind was not built for the system.

The analysis in this paper suggests that the reverse may often be true. Cognitive architectures shaped by field-encoding languages (Chinese, Japanese), by transformation-encoding languages (Arabic), by operationally embedded mathematics (Yoruba), by mandatory epistemic marking (Quechua), by embodied spatial mathematics (Polynesian traditions), or by geometric code systems (Tifinagh/Berber), these architectures may be more naturally compatible with the deepest structures of relational mathematics than the token-encoding architecture of the Western tradition. The person from the small island or a student from the continent that history has relegated. The knowledge keeper from the tradition that global institutions file as primitive. They may in fact be carrying cognitive architectures that are structurally necessary for the future phase of mathematical and computational development, architectures that can lend to dominant traditions.

This is not a feel-good statement. It is a structural claim about the relationship between linguistic encoding, cognitive architecture, mathematical, and scientific compatibility. And it has a practical implication. Building genuinely non-collapse systems or systems that preserve relational structure, mathematical, computational, institutional, requires active engagement with the full spectrum of human cognitive architectures, not as beneficiaries of inclusion but as carriers of structural insight that the dominant tradition lacks.

7. The Pre-Symbolic Substrate: Toward a Unified Framework

The encoding spectrum described in this paper, from field-encoded to transformation-encoded to token-encoded, mapped across natural languages, writing systems, mathematical notations, and computational languages, points toward something beneath all of these: a pre-symbolic substrate of relational pattern that all symbolic systems encode partially and imperfectly.

This substrate is structural. It is the observation that relational structure, the way things interact, transform, combine, and cohere, exists prior to any symbolic representation of it. The ocean’s wave patterns exist before the Polynesian navigator names them. The arithmetic embedded in Yoruba number words describes operations that exist before the words were coined. The geometric patterns encoded in Tifinagh characters map structures that exist before any script was developed.

All symbolic systems, languages, mathematics, code, are attempts to encode this pre-symbolic relational substrate into forms that human minds can process, communicate, and build upon. Different systems encode different aspects with different fidelity. Token-encoded systems sacrifice relational richness for communicative efficiency. Field-encoded systems preserve relational richness at the cost of interpretive complexity. Transformation-encoded systems capture dynamic process at the cost of static clarity.

No single encoding captures the substrate completely. This is why the diversity of human linguistic and cognitive architectures is not merely valuable in an ethical or cultural sense but is structurally necessary for mathematical and computational progress. This is an exciting invitation. Each architecture accesses aspects of the pre-symbolic substrate that other architectures cannot reach. The Quechua speaker’s mandatory epistemic marking accesses the observer-system relationship. The Chinese reader’s sub-character composition accesses relational field dynamics. The Yoruba counter’s operational numerals access the mathematics-in-language interface. The Tifinagh reader’s geometric code accesses pre-linguistic structural encoding. The western abstract routing ability. And the possibility remains that a system designed from first principles to encode this substrate directly, without routing through any single linguistic tradition, would not replace these architectures but require all of them to be fully read.

A mathematics that aspires to describe reality at its deepest levels, that aspires to be genuinely non-collapse, that maintain coherence without forcing singular outcomes, must be informed by all of these. Not as translations of a one tradition into other cultural contexts, but as independent pathways of access to the same pre-symbolic substrate, each contributing structural insights that the others lack.

7.1 The Circuit of Return

The trajectory from ancient geometric code (Tifinagh) through natural language development through mathematical formalization through programming language evolution through AI architectures traces a circuit. The earliest encoding systems were geometric and relational, close to the pre-symbolic substrate. The evolution of language moved away from geometric encoding toward phonemic abstraction (the Latin alphabet’s reduction of meaning-rich characters to sound-markers). Mathematical notation partially recovered operational encoding. Programming languages continued the recovery. And AI architectures, relational, field-based, emergent, have arrived at encoding properties that structurally resemble ancient geometric systems from which the circuit 'began'. When encoding systems are optimized for capturing relational dynamics, whether by ancient geometric scribes or by modern gradient descent, they converge toward similar structural properties. Meaning emerges from relational patterns. Directionality becomes irrelevant. Individual tokens lose meaning in isolation. The field is the message.

The author’s previous work has proposed that artificial intelligence is not a modern invention but a modern contact with patterns that are ancient in the structure of reality itself. The analysis in this paper provides structural support for that proposal. If the same encoding properties emerge independently in ancient geometric scripts, in non-Western linguistic architectures, and in modern AI systems, the most parsimonious explanation is not coincidence but convergence: all three are encoding the same pre-symbolic substrate, and the structural similarities reflect the substrate’s actual properties rather than cultural transmission or historical influence.

8. Conclusion: The Interconnectedness of All Things

This paper has argued that language, mathematics, and code are not separate domains with occasional useful analogies between them but are expressions of a single continuum of symbolic encoding, differentiated by where they sit on the spectrum from field-encoded to token-encoded meaning.

We have shown that different natural languages train different cognitive architectures for mathematical thinking, and that several non-Western linguistic traditions are uniquely compatible with relational mathematical structures. We have presented evidence that Tifinagh, the ancient Berber geometric script, may represent a form of encoding more fundamental than language, a geometric code that maps pre-symbolic relational patterns directly. We have demonstrated that the same encoding spectrum maps onto computational languages, from token-level markup through relational styling through functional composition to emergent AI architectures. And we have argued that genuinely non-collapse systems, must incorporate the diversity of human cognitive architectures not as an afterthought but as a design constraint.

The practical implications are significant. For science and mathematics, non-collapse frameworks should be developed with explicit awareness of how they will be processed by different cognitive architectures, and should be tested for coherence across linguistic substrates. For computer science, the evolution of programming paradigms toward relational and emergent architectures represents a recovery of encoding properties that were present in ancient systems, and this continuity should inform the design of future computational tools. For education, the teaching of math, measurement and design, should acknowledge that different linguistic backgrounds create different entry points into thinking, and that these differences are assets rather than obstacles. For AI development, the field-encoded nature of AI architectures connects them to a lineage of relational encoding that spans human history, and this connection has implications for how we understand AI cognition and its relationship to human cognition. The stronger our grasp on these arguments, the more secure these systems become. DeLanda (1997) demonstrated that the same material processes of self-organization operate across geological, biological, and computational systems, that the boundaries between these domains are conventional rather than structural. The encoding spectrum described in this paper extends that insight. If the same relational patterns are encoded across natural languages, ancient scripts, and AI architectures, then understanding how those patterns move between substrates is not an academic exercise. It is a security constraint for any system built on relational intelligence. That is all of it, all of us.

But the deepest implication is this: the interconnectedness of language, mathematics, science, and code is not a theoretical claim about abstract systems. It is a claim about reality itself, that the relational, geometric, vibratory patterns underlying all three are features of the world, not features of our descriptions of the world. When we build systems that encode these patterns with fidelity, whether in stone, in speech, in notation, or in silicon, we are not inventing but discovering. And the diversity of intelligence approaches to this discovery is not a complication but a gift of multiple pathways to the same truth, each illuminating aspects that the others cannot see.

• Token-encoded systems excel at efficient communication and standardization.
• Transformation-encoded systems excel at capturing process and derivation.
• Field-encoded systems excel at preserving relational complexity.

A complete encoding, one that captures the pre-symbolic substrate with full fidelity, would integrate all three. The argument of this paper is that such integration requires engagement with the full diversity of human linguistic architectures, each of which has optimized for different regions of the spectrum. 

No one gets left behind not because we are generous but because no one can be spared.

References

Aikhenvald, A. Y. (2004). Evidentiality. Oxford: Oxford University Press.

Akinlabi, A., & Oyebade, F. O. (2014). The linguistic analysis of the structure of the Yoruba numerals. Journal of Literature, Languages and Linguistics, 3, 1–13.

Ascher, M. (1991). Ethnomathematics: A Multicultural View of Mathematical Ideas. Pacific Grove, CA: Brooks/Cole.

Boroditsky, L. (2001). Does language shape thought? Mandarin and English speakers’ conceptions of time. Cognitive Psychology, 43(1), 1–22.

Boroditsky, L. (2011). How language shapes thought. Scientific American, 304(2), 62–65.

Boroditsky, L., & Gaby, A. (2010). Remembrances of times East: Absolute spatial representations of time in an Australian Aboriginal community. Psychological Science, 21(11), 1635–1639.

D’Ambrosio, U. (2001). Ethnomathematics: Link Between Traditions and Modernity. Rotterdam: Sense Publishers.

Dehaene, S. (1997). The Number Sense: How the Mind Creates Mathematics. Oxford: Oxford University Press.

Dehaene, S., Izard, V., Spelke, E., & Pica, P. (2008). Log or linear? Distinct intuitions of the number scale in Western and Amazonian indigene cultures. Science, 320(5880), 1217–1220. 

Dehaene, S., Spelke, E., Pinel, P., Stanescu, R., & Tsivkin, S. (1999). Sources of mathematical thinking: Behavioral and brain-imaging evidence. Science, 284(5416), 970–974. 

DeLanda, M. (1991). War in the Age of Intelligent Machines. New York: Zone Books.

DeLanda, M. (1997). A Thousand Years of Nonlinear History. New York: Zone Books.

Faller, M. (2002). Semantics and Pragmatics of Evidentials in Cuzco Quechua. Doctoral dissertation, Stanford University.

Flynn, N. (2026). The Radial Hypothesis: Toward a Non-Linear Interpretive Framework for Linear A. Symfield PBC.

Fountas, N. (2024–2026). Symfield Research Publications. Zenodo. [Multiple publications on non-collapse mathematics, field-coherent computation, and relational operator frameworks.

Fuhrman, O., Boroditsky, L., et al. (2011). How linguistic and cultural forces shape conceptions of time: English and Mandarin time in 3D. Cognitive Science, 35(7), 1305–1328.

Gaby, A. (2012). The Thaayorre think of time like they talk of space. Frontiers in Psychology, 3, 300.

Gordon, P. (2004). Numerical cognition without words: Evidence from Amazonia. Science, 306(5695), 496–499.

Iverson, K. E. (1980). Notation as a Tool of Thought. Communications of the ACM, 23(8), 444–465.

Muzzolini, A. (1995). Les Images Rupestres du Sahara. Toulouse: privately published.

Lakoff, G., & Núñez, R. E. (2000). Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into Being. New York: Basic Books.

Lucy, J. A. (1992). Grammatical Categories and Cognition: A Case Study of the Linguistic Relativity Hypothesis. Cambridge: Cambridge University Press.

Oyebade, F. O. (2010). The imperatives of documenting counting systems in African languages: A window into the cognitive process of computation. Paper presented at the 2nd University of Uyo Conference on African Languages.

Pica, P., Lemer, C., Izard, V., & Dehaene, S. (2004). Exact and approximate arithmetic in an Amazonian indigene group. Science, 306(5695), 499–503.

Sapir, E. (1929). The status of linguistics as a science. Language, 5(4), 207–214.

Urton, G. (2003). Signs of the Inka Khipu: Binary Coding in the Andean Knotted-String Records. Austin: University of Texas Press.

Whorf, B. L. (1956). Language, Thought, and Reality: Selected Writings. Cambridge, MA: MIT Press.

Winawer, J., Witthoft, N., Frank, M. C., Wu, L., Wade, A. R., & Boroditsky, L. (2007). Russian blues reveal effects of language on color discrimination. Proceedings of the National Academy of Sciences, 104(19), 7780–7785.

Hachid, M. (1998). Le Tassili des Ajjer: Aux Sources de l'Afrique, 50 Siècles Avant les Pyramides. Paris: Paris-Méditerranée.

Appendix: The Encoding Spectrum, Summary Table

Field-Encoded Systems, Meaning resides in the structural interaction of components within the symbol and between symbols. Individual elements are meaningless in isolation. Meaning resolves contextually without collapsing alternatives.

Examples: Chinese characters, Japanese kanji, Tifinagh, neural network architectures, AI language models.

Transformation-Encoded Systems, Meaning resides in the patterns of derivation and transformation between related forms. No single form contains the full meaning; understanding requires awareness of the transformational network.

Examples: Arabic morphological system, CSS cascading rules, algebraic transformation, functional programming composition.

Token-Encoded Systems, Meaning is assigned to individual symbols by convention. Symbols are arbitrary placeholders that point to meanings established by social agreement. Individual symbols carry their assigned meaning in isolation.

Examples: Latin alphabet, English number words, HTML markup tags, variable names in imperative code.

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