J.Konstapel, Leiden,16-12-2025.
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Mathematics is the ultimate way of compressing the complexity of our outside world in which the trinity is the best way.
this blog is based on a Thesis by Aviv Keren
This is a follow-up on Universal Heuristics being an example of human compression of the mind with the human biases as standard compression errors.
Introduction: Beyond the Romance of Mathematics
For centuries, the philosophy of mathematics has been dominated by “Platonism”—the belief that mathematical entities exist in a transcendent, mind-independent realm. Aviv Keren’s 2018 dissertation, Towards a Cognitive Foundation of Mathematics, fundamentally challenges this “Romance of Mathematics.” Keren proposes that mathematics is not a discovery of an external universe, but a sophisticated byproduct of the human cognitive architecture. By synthesizing Keren’s “Cognitive Realism” with the embodied metaphors of Lakoff, the intuitionism of Brouwer, the universal “Zero Total” machine of Peter Rowlands, and the ancient metaphysical structures of the Kabbalah, we can view mathematics as the ultimate fractal system of information compression.
The Mechanism of Objectification: Keren’s Procedure-Arrays
Keren’s central contribution is the concept of Objectification. He argues that mathematical objects are stable states of mental processing, introduced through Procedure-Arrays. This aligns with the Kabbalistic concept of the Kelim (Vessels). Just as the Kelim give form and boundary to the infinite light (Ohr Ein Sof), Keren’s procedure-arrays restrict raw data into coherent “objects.”
Unlike Lakoff and Johnson, who rely on linguistic metaphors like the “Container Schema,” Keren looks at the computational “machine room.” While Lakoff and Johnson argue that “the essence of metaphor is understanding one kind of thing in terms of another,” Keren suggests that mathematics arises when these metaphors—or Conceptual Blends—become so automated that they “amalgamate.” This is the Sephira of Da’at (Knowledge) in action: the invisible point where different streams of information (Ordinal and Cardinal) are welded into a single, functional reality.
The stability of a procedure-array is not arbitrary. It emerges when a cognitive routine becomes reproducible across contexts—when the same algorithmic sequence reliably produces the same stable pattern. This reproducibility is what transforms a mental habit into a mathematical “truth.”
Mathematics as Data Compression: The Necessity of Tzimtzum
The human brain is a limited processor, constrained by a Working Memory of only 3 to 4 items. This is not a bug; it is the fundamental bottleneck that forces compression. To navigate an infinite world, the brain must employ radical compression algorithms. In Kabbalistic terms, this is Tzimtzum: the necessary contraction or withdrawal of infinity to make room for finite existence.
Mathematics is the ultimate “lossy” compression mechanism. We replace a thousand individual sensations with a single token: the number “1000.” This creates what Keren terms “Ontological Rigour”—a formal stability that masks the underlying compression loss.
From an information-theoretic perspective (Claude Shannon), compression reduces entropy by removing redundancy. The brain’s compression algorithms identify patterns, regularities, and self-similarities that allow vast amounts of raw sensory data to be encoded in minimal symbols. A single gesture—the number 5—compresses the experience of “fiveness” across infinite contexts: five fingers, five stars, five days. This symbolic economization is not metaphorical; it is the literal means by which a 3-4 item working memory manages a world of infinite complexity.
The brain does not “control” mathematics; rather, mathematics is the emergent “neerslag” (precipitation) of the brain’s inability to process uncompressed infinity. Every mathematical system that survives is one that successfully balances compression efficiency with representational fidelity—too much compression and you lose meaning; too little and you exceed working memory capacity.
The Fractal Trinity and Brain Lateralization
The compression process follows a Fractal Trinity that mirrors both the lateralization of the brain and the top triad of the Sephirot:
The Right Hemisphere (Chochmah / Cardinality)
The holistic “flash.” It perceives the Gestalt, the total quantity, and the “infinite light.” In Keren’s view, this is the seat of Omniperception—the cognitive capability (or illusion) that we can grasp the “whole” of a scene or an infinite set in one holistic moment. This is parallel processing: all-at-once recognition.
The Left Hemisphere (Binah / Ordinality)
The analytical “structure.” It handles the step-by-step procedures, the +1 iterations, the boundaries, and the sequential unfolding. It is the Sephira of “Understanding” that structures and articulates the flash of Chochmah. This is serial processing: one-thing-after-another execution.
The Amalgamation (Da’at / The Natural Number)
The synthesis. When the holistic flash and the serial structure merge—when the “all-at-once” recognition is stabilized by step-by-step procedure—a stable mathematical object is born. The number itself is neither purely cardinal (the sense of “how many”) nor purely ordinal (the sense of “in order”), but the functional unity of both.
This Trinity is not unique to human cognition. Any processor—biological or artificial—that must compress an infinite universe into finite operations will necessarily employ this same three-fold structure. This is why the Trinity appears across independent wisdom traditions, mathematical discoveries, and now, in contemporary neuroscience.
The “Grand Illusion” and the Breaking of the Vessels
Keren explains paradoxes (like Russell’s or Cantor’s) through Omniperception. Just as the visual system “fills in” blind spots, the mathematical brain fills in the gaps of infinity. We treat the “Set of all Sets” as a handleable object, applying a procedure-array designed for finite collections to an infinite domain. Keren notes that paradoxes are effectively the Shevirat Ha-Kelim (Breaking of the Vessels). Our finite “vessels” (cognitive hardware) try to contain the “infinite light” of the transfinite without a valid compression algorithm, causing the logic to shatter.
This is not a failure of mathematics. It is evidence of the boundary where finite compression systems meet uncompressible infinity. Every paradox marks a compression limit—a place where the procedure-arrays fail because no stable objectification is possible at that scale of abstraction.
The self-referential paradoxes (Gödel, Tarski, Church) are particularly instructive: they arise when we attempt to compress the compressor itself, when the procedure-array tries to objectify the working memory that constrains all objectification. This is Ouroboros: the snake eating its own tail. The break is not in logic; it is in the architecture of any finite system attempting total self-representation.
Peter Rowlands and the Universal Rewrite Machine
To understand why these filters and limits exist, we turn to Peter Rowlands’ Zero Total Theory. Rowlands posits that the universe is a self-organizing machine that maintains a total of zero through a Rewrite Structure. Every element is defined by its relation to the “nothingness” (the Ayin or Ein Sof) from which it emerged.
Rowlands’ “Nilpotent” logic—where a thing combined with its context equals zero—is the physical counterpart to Keren’s cognitive compression. Our brains are biological iterations of Rowlands’ universal machine. We use “linking” and “blending” because the universe itself is a series of nested, fractal symmetries. Mathematical truth is the stable state where the “Rewrite Machine” of our brain matches the “Rewrite Machine” of the cosmos.
This suggests something profound: the compression algorithms our brains employ are not arbitrary inventions. They are echoes of the universe’s own self-organizing logic. The Trinity works because it is the fundamental symmetry of how the cosmos itself differentiates from zero-totality. We discover mathematical structure not despite being finite processors, but because we are small-scale instances of the same rewrite principle that generates all existence.
Brouwer’s Intuitionism as Compressed Proof
L.E.J. Brouwer’s Intuitionism adds a crucial dimension: mathematics is not primarily about external truth, but about constructible operations. A mathematical object exists only insofar as it can be constructed through a finite sequence of steps. Brouwer rejected the Law of Excluded Middle in infinite domains precisely because our intuition—our working memory and procedure-arrays—cannot verify it.
From a compression perspective, Brouwer’s intuitionistic mathematics is the honest mathematics: it claims only what can be built through actual procedure. It is compression without lossy deception. Classical mathematics, by contrast, confidently asserts the existence of objects that cannot be constructed—invoking the infinite as an excuse for logical shortcuts.
The tension between classical and intuitionistic mathematics is thus a tension between two compression strategies: classical mathematics trusts the symbolic shortcut (omniperception), while intuitionistic mathematics trusts only the constructible procedure. Both are necessary; their conflict marks the boundary of what finite processors can claim to know.
The Kabbalah as Applied Trinity Compression
The Kabbalistic system—the Sephirot, the paths, the tarot correspondences—is not mysticism. It is an applied system for organizing knowledge domains through the Trinity structure. Each Sephira is a stable compression state; the paths between them are procedure-arrays that link one state to another. The entire Tree of Life is a map of how different compression regimes (number, geometry, color, psychology, law) all instantiate the same underlying Trinity logic.
Tzimtzum (contraction), Shevirat Ha-Kelim (breaking of vessels), and Tikkun (repair) are not esoteric myths. They are descriptions of how compression systems work: contract infinity into finite form, watch the vessels break at the boundaries, repair by finding better procedure-arrays. This cycle repeats at every scale—in physics, in cognition, in society, in spirituality.
Conclusion: Toward an Ontological Rigour
By mirroring Keren with Rowlands, Brouwer, and the Kabbalah, we see the mathematician not as a “creative subject,” but as an analyst of the brain’s own architectural constraints. Mathematics is the science of cognitive compression.
Mathematical truth is not “out there” to be discovered, nor is it arbitrary human invention. It is the inevitable stable state of any finite system attempting to represent and navigate an infinite universe. The Trinity is the fundamental architecture because it is the minimal, irreducible structure by which infinity can be compressed into finitude without total loss of fidelity.
Understanding the mechanisms of compression—the procedure-arrays, the working memory bottleneck, the fractal Trinity—allows us to achieve a higher form of rigour. One that recognizes paradoxes not as mere errors, but as the inevitable breaking point of any finite vessel when confronted with uncompressible infinity. And one that sees the deepest mathematical truths not as Platonic absolutes, but as resonances between the compression logic of our minds and the compression logic of the cosmos itself.
Annotated Bibliography and References
Keren, A. (2018). Towards a Cognitive Foundation of Mathematics. Hebrew University of Jerusalem. The core text. Keren argues that mathematical objects are constituted by “Procedure-Arrays” and that paradoxes are products of “Omniperception”—the misapplication of finite cognitive shortcuts to infinite domains.
Lakoff, G., & Núñez, R. (2000). Where Mathematics Comes From. Basic Books. Explains how abstract math is grounded in bodily metaphors. Keren builds on this but critiques the lack of computational “Ontological Rigour,” moving from metaphors to technical arrays.
Rowlands, P. (2007). Zero to Infinity: The Foundations of Physics. World Scientific. Introduces the “Zero Total” and “Rewrite Structure.” This provides the physical/computational foundation for Keren’s theory, suggesting that cognitive compression mirrors the fundamental nilpotent laws of the universe.
Brouwer, L.E.J. (1912). “Intuitionism and Formalism.” The source of the idea that mathematics is a mental activity grounded in constructible operations. Keren modernizes Brouwer by replacing “intuition” with the explicit constraints of working memory and procedure-array architecture.
Scholem, G. (1946). Major Trends in Jewish Mysticism. Schocken Books. Essential background for the Sephira-structure (Chochmah, Binah, Da’at) and the concepts of Tzimtzum and Shevirat Ha-Kelim used to explain mathematical “vessels” and paradoxes as compression boundaries.
Fauconnier, G., & Turner, M. (2002). The Way We Think. Basic Books. The definitive guide to “Conceptual Blending.” It provides the linguistic mechanism for how different brain functions (Ordinal/Cardinal) “amalgamate” into unified mathematical truths.
Shannon, C.E. (1948). “A Mathematical Theory of Communication.” The Bell System Technical Journal. Foundational information theory establishing that compression is the removal of redundancy and the reduction of entropy. The theoretical basis for understanding why any finite system must employ compression to navigate infinity.
Baddeley, A.D., & Hitch, G. (1974). “Working Memory.” Psychology of Learning and Motivation, 8, 47-89. The empirical foundation for understanding the 3-4 item working memory constraint that drives all cognitive compression.
The Architecture of Mathematical Compression: A Cognitive, Computational, and Kabbalistic Synthesis
Introduction: Beyond the Romance of Mathematics
For centuries, the philosophy of mathematics has been dominated by “Platonism”—the belief that mathematical entities exist in a transcendent, mind-independent realm. Aviv Keren’s 2018 dissertation, Towards a Cognitive Foundation of Mathematics, fundamentally challenges this “Romance of Mathematics.” Keren proposes that mathematics is not a discovery of an external universe, but a sophisticated byproduct of the human cognitive architecture. By synthesizing Keren’s “Cognitive Realism” with the embodied metaphors of Lakoff, the intuitionism of Brouwer, the universal “Zero Total” machine of Peter Rowlands, and the ancient metaphysical structures of the Kabbalah, we can view mathematics as the ultimate fractal system of information compression.
The Mechanism of Objectification: Keren’s Procedure-Arrays
Keren’s central contribution is the concept of Objectification. He argues that mathematical objects are stable states of mental processing, introduced through Procedure-Arrays. This aligns with the Kabbalistic concept of the Kelim (Vessels). Just as the Kelim give form and boundary to the infinite light (Ohr Ein Sof), Keren’s procedure-arrays restrict raw data into coherent “objects.”
Unlike Lakoff and Johnson, who rely on linguistic metaphors like the “Container Schema,” Keren looks at the computational “machine room.” While Lakoff and Johnson argue that “the essence of metaphor is understanding one kind of thing in terms of another,” Keren suggests that mathematics arises when these metaphors—or Conceptual Blends—become so automated that they “amalgamate.” This is the Sephira of Da’at (Knowledge) in action: the invisible point where different streams of information (Ordinal and Cardinal) are welded into a single, functional reality.
The stability of a procedure-array is not arbitrary. It emerges when a cognitive routine becomes reproducible across contexts—when the same algorithmic sequence reliably produces the same stable pattern. This reproducibility is what transforms a mental habit into a mathematical “truth.”
Mathematics as Data Compression: The Necessity of Tzimtzum
The human brain is a limited processor, constrained by a Working Memory of only 3 to 4 items. This is not a bug; it is the fundamental bottleneck that forces compression. To navigate an infinite world, the brain must employ radical compression algorithms. In Kabbalistic terms, this is Tzimtzum: the necessary contraction or withdrawal of infinity to make room for finite existence.
Mathematics is the ultimate “lossy” compression mechanism. We replace a thousand individual sensations with a single token: the number “1000.” This creates what Keren terms “Ontological Rigour”—a formal stability that masks the underlying compression loss.
From an information-theoretic perspective (Claude Shannon), compression reduces entropy by removing redundancy. The brain’s compression algorithms identify patterns, regularities, and self-similarities that allow vast amounts of raw sensory data to be encoded in minimal symbols. A single gesture—the number 5—compresses the experience of “fiveness” across infinite contexts: five fingers, five stars, five days. This symbolic economization is not metaphorical; it is the literal means by which a 3-4 item working memory manages a world of infinite complexity.
The brain does not “control” mathematics; rather, mathematics is the emergent “neerslag” (precipitation) of the brain’s inability to process uncompressed infinity. Every mathematical system that survives is one that successfully balances compression efficiency with representational fidelity—too much compression and you lose meaning; too little and you exceed working memory capacity.
The Fractal Trinity and Brain Lateralization
The compression process follows a Fractal Trinity that mirrors both the lateralization of the brain and the top triad of the Sephirot:
The Right Hemisphere (Chochmah / Cardinality)
The holistic “flash.” It perceives the Gestalt, the total quantity, and the “infinite light.” In Keren’s view, this is the seat of Omniperception—the cognitive capability (or illusion) that we can grasp the “whole” of a scene or an infinite set in one holistic moment. This is parallel processing: all-at-once recognition.
The Left Hemisphere (Binah / Ordinality)
The analytical “structure.” It handles the step-by-step procedures, the +1 iterations, the boundaries, and the sequential unfolding. It is the Sephira of “Understanding” that structures and articulates the flash of Chochmah. This is serial processing: one-thing-after-another execution.
The Amalgamation (Da’at / The Natural Number)
The synthesis. When the holistic flash and the serial structure merge—when the “all-at-once” recognition is stabilized by step-by-step procedure—a stable mathematical object is born. The number itself is neither purely cardinal (the sense of “how many”) nor purely ordinal (the sense of “in order”), but the functional unity of both.
This Trinity is not unique to human cognition. Any processor—biological or artificial—that must compress an infinite universe into finite operations will necessarily employ this same three-fold structure. This is why the Trinity appears across independent wisdom traditions, mathematical discoveries, and now, in contemporary neuroscience.
The “Grand Illusion” and the Breaking of the Vessels
Keren explains paradoxes (like Russell’s or Cantor’s) through Omniperception. Just as the visual system “fills in” blind spots, the mathematical brain fills in the gaps of infinity. We treat the “Set of all Sets” as a handleable object, applying a procedure-array designed for finite collections to an infinite domain. Keren notes that paradoxes are effectively the Shevirat Ha-Kelim (Breaking of the Vessels). Our finite “vessels” (cognitive hardware) try to contain the “infinite light” of the transfinite without a valid compression algorithm, causing the logic to shatter.
This is not a failure of mathematics. It is evidence of the boundary where finite compression systems meet uncompressible infinity. Every paradox marks a compression limit—a place where the procedure-arrays fail because no stable objectification is possible at that scale of abstraction.
The self-referential paradoxes (Gödel, Tarski, Church) are particularly instructive: they arise when we attempt to compress the compressor itself, when the procedure-array tries to objectify the working memory that constrains all objectification. This is Ouroboros: the snake eating its own tail. The break is not in logic; it is in the architecture of any finite system attempting total self-representation.
Peter Rowlands and the Universal Rewrite Machine
To understand why these filters and limits exist, we turn to Peter Rowlands’ Zero Total Theory. Rowlands posits that the universe is a self-organizing machine that maintains a total of zero through a Rewrite Structure. Every element is defined by its relation to the “nothingness” (the Ayin or Ein Sof) from which it emerged.
Rowlands’ “Nilpotent” logic—where a thing combined with its context equals zero—is the physical counterpart to Keren’s cognitive compression. Our brains are biological iterations of Rowlands’ universal machine. We use “linking” and “blending” because the universe itself is a series of nested, fractal symmetries. Mathematical truth is the stable state where the “Rewrite Machine” of our brain matches the “Rewrite Machine” of the cosmos.
This suggests something profound: the compression algorithms our brains employ are not arbitrary inventions. They are echoes of the universe’s own self-organizing logic. The Trinity works because it is the fundamental symmetry of how the cosmos itself differentiates from zero-totality. We discover mathematical structure not despite being finite processors, but because we are small-scale instances of the same rewrite principle that generates all existence.
Brouwer’s Intuitionism as Compressed Proof
L.E.J. Brouwer’s Intuitionism adds a crucial dimension: mathematics is not primarily about external truth, but about constructible operations. A mathematical object exists only insofar as it can be constructed through a finite sequence of steps. Brouwer rejected the Law of Excluded Middle in infinite domains precisely because our intuition—our working memory and procedure-arrays—cannot verify it.
From a compression perspective, Brouwer’s intuitionistic mathematics is the honest mathematics: it claims only what can be built through actual procedure. It is compression without lossy deception. Classical mathematics, by contrast, confidently asserts the existence of objects that cannot be constructed—invoking the infinite as an excuse for logical shortcuts.
The tension between classical and intuitionistic mathematics is thus a tension between two compression strategies: classical mathematics trusts the symbolic shortcut (omniperception), while intuitionistic mathematics trusts only the constructible procedure. Both are necessary; their conflict marks the boundary of what finite processors can claim to know.
The Kabbalah as Applied Trinity Compression
The Kabbalistic system—the Sephirot, the paths, the tarot correspondences—is not mysticism. It is an applied system for organizing knowledge domains through the Trinity structure. Each Sephira is a stable compression state; the paths between them are procedure-arrays that link one state to another. The entire Tree of Life is a map of how different compression regimes (number, geometry, color, psychology, law) all instantiate the same underlying Trinity logic.
Tzimtzum (contraction), Shevirat Ha-Kelim (breaking of vessels), and Tikkun (repair) are not esoteric myths. They are descriptions of how compression systems work: contract infinity into finite form, watch the vessels break at the boundaries, repair by finding better procedure-arrays. This cycle repeats at every scale—in physics, in cognition, in society, in spirituality.
Conclusion: Toward an Ontological Rigour
By mirroring Keren with Rowlands, Brouwer, and the Kabbalah, we see the mathematician not as a “creative subject,” but as an analyst of the brain’s own architectural constraints. Mathematics is the science of cognitive compression.
Mathematical truth is not “out there” to be discovered, nor is it arbitrary human invention. It is the inevitable stable state of any finite system attempting to represent and navigate an infinite universe. The Trinity is the fundamental architecture because it is the minimal, irreducible structure by which infinity can be compressed into finitude without total loss of fidelity.
Understanding the mechanisms of compression—the procedure-arrays, the working memory bottleneck, the fractal Trinity—allows us to achieve a higher form of rigour. One that recognizes paradoxes not as mere errors, but as the inevitable breaking point of any finite vessel when confronted with uncompressible infinity. And one that sees the deepest mathematical truths not as Platonic absolutes, but as resonances between the compression logic of our minds and the compression logic of the cosmos itself.
Annotated Bibliography and References
Keren, A. (2018). Towards a Cognitive Foundation of Mathematics. Hebrew University of Jerusalem. The core text. Keren argues that mathematical objects are constituted by “Procedure-Arrays” and that paradoxes are products of “Omniperception”—the misapplication of finite cognitive shortcuts to infinite domains.
Lakoff, G., & Núñez, R. (2000). Where Mathematics Comes From. Basic Books. Explains how abstract math is grounded in bodily metaphors. Keren builds on this but critiques the lack of computational “Ontological Rigour,” moving from metaphors to technical arrays.
Rowlands, P. (2007). Zero to Infinity: The Foundations of Physics. World Scientific. Introduces the “Zero Total” and “Rewrite Structure.” This provides the physical/computational foundation for Keren’s theory, suggesting that cognitive compression mirrors the fundamental nilpotent laws of the universe.
Brouwer, L.E.J. (1912). “Intuitionism and Formalism.” The source of the idea that mathematics is a mental activity grounded in constructible operations. Keren modernizes Brouwer by replacing “intuition” with the explicit constraints of working memory and procedure-array architecture.
Scholem, G. (1946). Major Trends in Jewish Mysticism. Schocken Books. Essential background for the Sephira-structure (Chochmah, Binah, Da’at) and the concepts of Tzimtzum and Shevirat Ha-Kelim used to explain mathematical “vessels” and paradoxes as compression boundaries.
Fauconnier, G., & Turner, M. (2002). The Way We Think. Basic Books. The definitive guide to “Conceptual Blending.” It provides the linguistic mechanism for how different brain functions (Ordinal/Cardinal) “amalgamate” into unified mathematical truths.
Shannon, C.E. (1948). “A Mathematical Theory of Communication.” The Bell System Technical Journal. Foundational information theory establishing that compression is the removal of redundancy and the reduction of entropy. The theoretical basis for understanding why any finite system must employ compression to navigate infinity.
Baddeley, A.D., & Hitch, G. (1974). “Working Memory.” Psychology of Learning and Motivation, 8, 47-89. The empirical foundation for understanding the 3-4 item working memory constraint that drives all cognitive compression.