How Morphisms Map the Mind: A Radical Phenomenalist Perspective

Setting the Stage: Radical Phenomenalism and Internal Meaning

Imagine that everything we ever know is contained within the theater of our own experience. Radical Phenomenalism takes this internalist stance to the extreme: reference and meaning are purely internal, experiential, and predictive. In this view, our mind operates like a self-contained simulation, concerned only with the way one experience leads to another, and any supposed “external reality” beyond experience is a non-issue. Talk of mind-independent things – Kant’s “things-in-themselves,” a purely objective physical world, insentient matter existing beyond anyone’s perception – is considered empty of meaning. These are pseudo-concepts from the radical phenomenalist perspective, since they refer to nothing accessible in experience and thus add nothing to our understanding. As an analogy, a word in a private language that has no connections to any experience would be semantically void – it points nowhere inside the simulation of mind.

This is a bold claim. It echoes certain threads in philosophy of language and mind: for example, Noam Chomsky’s internalist semantics holds that linguistic expressions do not hook onto external objects at all, but only have content in virtue of mental concepts. On such a view, names don’t really refer to external objects, predicates don’t have mind-independent truth conditions, and sentences are not about an outside world – rather, meaning consists in instructions for building concepts or mental representations. Similarly, radical phenomenalism asserts that what we call “reality” is an internal web of experiences; any correspondence to an external world is either a pragmatic convenience or a predictive tool, not a literal semantic link. If meanings live inside the head, how do we ensure our internal world still makes sense? How can a self-contained mind maintain coherence and intelligibility without anything external to latch onto?

The key, this post will argue, is structure. When external reference drops out, structural relationships between experiences step into the breach. The mind can achieve semantic coherence via the patterns and mappings that connect one experience to another. We will explore this idea in depth using the language of category theory – specifically, various kinds of morphisms (structure-preserving mappings) and related constructs (functors, commutative diagrams). These mathematical notions will serve as a rich metaphor (and perhaps more than a metaphor) for how internal experiences relate, predict, and refer to each other. In essence, morphisms give us an “internal geometry” of mind: a way to chart the landscape of experience by its own internal coordinates, with no need for an external compass.

Before diving into the philosophical analysis, let’s begin with a brief primer on what these morphisms are. We’ll start gently – with clear, accessible definitions of the key types of morphisms for those not steeped in abstract mathematics – and then gradually ramp up to a more expert-level discussion of how these structural mappings might illuminate phenomenology.

Morphisms 101: A Quick Primer on Structure-Preserving Maps

In category theory (a branch of mathematics studying abstract structures and relationships), a morphism is a general term for any structure-preserving mapping between two entities (called objects). If that sounds intimidating, think of a morphism as a kind of translator or connector that takes elements of one structure and reliably sends them to elements of another structure without breaking the rules or relationships that define the structure. For example, if you have a bunch of cities connected by roads in one map, a morphism to another map would send cities to cities and ensure that connected cities stay connected in the new map. Morphisms generalize familiar ideas like functions between sets, but also include mappings in algebra, geometry, and beyond – anywhere we care about preserving patterns or operations.

Let’s break down some special kinds of morphisms and related concepts, in plain language:

• Homomorphism: Literally meaning “same structure,” a homomorphism is any mapping between two structures that preserves their operations or relations. If you perform some operation on elements first and then map the result, you get the same outcome as if you had mapped the elements first and then performed the analogous operation in the target structure. In short, the map doesn’t mess up how things relate: structure in the source is reflected as structure in the target. (For instance, adding two numbers then converting them to a new unit gives the same result as converting each number first then adding – that conversion is a homomorphism preserving the addition structure.)
  
  

• Isomorphism: This is a special homomorphism that not only preserves structure but also pairs things up perfectly between two structures, with an exact one-to-one correspondence. An isomorphism is essentially a bijective (one-to-one and onto) homomorphism. If there is an isomorphism between two structures, they are indistinguishable as far as structure is concerned – you could relabel one as the other and everything works the same. In category theory, an isomorphism means there’s an inverse mapping going back (undoing the mapping), so you can travel forth and back without loss. Intuitively, if two internal experiences or mental models are isomorphic, they have the same shape in the mind; they differ only in superficial labels, not in how their parts connect.
  
  

• Endomorphism: “Endo” means within – so an endomorphism is simply a morphism from a structure to itself. It’s a structure-preserving transformation that starts and ends in the same domain. Think of it like a remix or reconfiguration of a single mental state or system that stays within that system. For example, a rotation of a shape is an endomorphism of the shape’s space: you’re mapping the space to itself, preserving distances. In mental terms, an endomorphism might be a way the mind transforms its own state while keeping the overall type of structure intact (like re-framing a memory without changing its key relationships).
  
  

• Automorphism: This is an endomorphism that is also an isomorphism. In other words, it’s a reversible transformation of a structure that lies within that same structure. Automorphisms are essentially symmetries: you move things around internally and end up with a state that, from the perspective of structure, looks exactly like where you started. (Imagine renaming all the characters in a story – you’ve permuted labels, but the plot structure is unchanged. That’s an automorphism of the story’s structure.) The set of all automorphisms of a given object actually forms a group in the mathematical sense, capturing the idea that you can compose symmetry operations and invert them. Within a mind, automorphisms could represent all the ways you can internally re-describe or reorient your worldview without altering its predictive content or logical structure – a kind of mental self-symmetry.
  
  

• Monomorphism: Often called mono or monic map for short, this is basically the category-theoretic way to say “injective mapping” (one-to-one) in a broad context. A monomorphism is a morphism that doesn’t identify two distinct source elements as the same thing in the target – no collapsing of distinct points. In everyday terms, a mono is an embedding: you can consider the source as a substructure of the target because the map doesn’t lose information about distinctness. If one experience maps monomorphically into another structure in the mind, it means each element of that experience has a unique counterpart in the larger structure – nothing gets accidentally merged. Monomorphisms ensure clarity and non-ambiguity in reference: one internal element points to one internal meaning without conflating it with others.
  
  

• Epimorphism: Dually, an epi (epimorphism) is like a surjective mapping (onto) in category theory. It covers the entire target – every element in the target structure comes from something in the source via the mapping. An epimorphism is a way of saying the target is no “bigger” in relevant structure than what the source can account for. Epimorphisms are like surjective functions, but again generalized. If we think in terms of knowledge or experience, an epimorphic map from structure A onto structure B would mean B doesn’t have any structural feature that didn’t come from A. It’s a kind of coverage or completeness guarantee. In internal terms, perhaps if a certain set of core experiences epimorphically generates a whole conceptual framework, that framework has no gaps unaccounted for – every concept corresponds to or is “covered by” some experience or combination of experiences.
  
  

• Functor (Functorial Mapping): All the above are morphisms within a single level of structure (mapping objects within one category). A functor goes one level higher – it’s a mapping between categories. Concretely, if you have Category X and Category Y (each with their own objects and morphisms), a functor F: X → Y assigns to each object in X an object in Y, and to each morphism in X a morphism in Y, in such a way that structure is preserved at the category level. Functors carry entire networks of relationships from one domain to another. Think of it as a translation not just of words, but of grammar and syntax from one language to another, preserving the form of sentences. In our context, a functor could represent a systematic translation from one “mode of experience” to another – for example, from the domain of perceptions to the domain of concepts, or from thought to language. A functor ensures that if experience A leads to experience B in the first domain, then the translated versions of A and B (e.g. their conceptual counterparts) maintain the same relationship in the target domain. Essentially, functorial mappings carry over the whole structure of connections, ensuring consistency across different realms of internal life.
  
  

• Commutative Diagram: This term might sound esoteric, but it’s a simple idea with a fancy name. A commutative diagram is just a diagram of objects (points/nodes) with arrows (morphisms) between them that has a special property: no matter which path of arrows you follow from one object to another, you end up with the same result. The diagram “commutes” means all the routes are consistent. In a picture, if you have a square of four objects and arrows going around, the condition for commutativity is that going right then down is equivalent to going down then right. In plainer terms, different ways of relating or transforming objects agree with each other – there is no contradiction. Commutative diagrams are the formal way to say “this structure is self-consistent.” In human terms, imagine you have two ways to get from a raw sensation to a decision: path 1 goes through conscious reasoning, path 2 goes through intuitive pattern-matching. If both paths yield the same decision outcome, we have a kind of commutative diagram in cognition – a reassuring consistency. Commutativity is crucial for internal coherence: it means the web of experiences doesn’t give you mixed messages depending on how you traverse it. In our internal simulation, all roads lead to Rome – or at least to the same Rome – so the mind isn’t torn apart by discrepancies.
  
  

With these definitions in mind, we have a toolbox of structural concepts to apply. Next, we’ll delve into how these morphisms (and functors, diagrams, etc.) can serve as the scaffolding for internal reference and meaning. The goal is to see how an inner world of experiences can hang together and make sense purely through structural mappings, without any external hook. We will examine each type of morphism in turn, not as cold abstractions, but as vivid models for phenomenological relations: isomorphisms as mirrors of mind, endomorphisms as inner transformations, functors as bridges between realms of thought, and so on. Along the way, we’ll maintain an analytic tone but occasionally venture into metaphor – because sometimes a poetic image can illuminate a concept in a way a formula cannot.

Isomorphism: When Experiences Mirror Each Other

In a purely internal world, one powerful source of meaning is recognition of sameness. When can we say two experiences mean the same thing? Radical Phenomenalism would answer: only when they play the same role in our predictive experiential framework. This is where isomorphism enters the picture. An isomorphism between two internal structures (say, two concepts or two mental models) implies a perfect structural correspondence – a one-to-one mapping that preserves all relations. If experience A is isomorphic to experience B, then A and B are, in effect, the same pattern wearing different guises.

Think of an isomorphism as holding a polished mirror up to the mind: the pattern in the mirror may look flipped or translated, but it’s fundamentally identical to the original. For instance, a complex dream scenario might later be recognized as structurally identical to a real-life social situation you encountered – the characters and settings differ, yet the web of relationships (tensions, alliances, outcomes) line up exactly. That realization (“Ah, the dream was just like that office conflict I had!”) is the discovery of an isomorphism between two experiential structures. According to radical phenomenalist semantics, this structural mirroring is what confers mutual meaning. The dream scenario can “refer” to the office conflict only to the extent that there’s an internal isomorphic mapping between them. Strip away any notion of an objective office or actual people – what made the dream meaningful was that it shared a structure with another experience.

Cognitive science gives credence to the importance of such structure-preserving mappings. In studies of analogical reasoning, researchers have found that people tend to align situations via relational structure more than surface details. There’s even a principle called the “structural constraint of isomorphism,” which encourages mappings that maximize the consistency of relationships between two scenarios. In other words, we naturally seek isomorphic correspondences – we try to map one story or problem onto another in a way that all the key relations match up because that yields the deepest insight. This is structural resonance at work: two different experiences resonate (like two tuning forks humming in harmony) when their pattern of relations can be overlaid perfectly. Such resonance doesn’t depend on any external anchor; it’s an internal echo where one pattern calls out and another answers with the same shape.

Under radical phenomenalism, isomorphisms define internal equivalences. They say, effectively, “these two experiences are the same meaningful structure, even if their raw feel differs.” This could form the basis of an internal notion of truth or reference: a thought is “true” to a perception if the structure of the thought matches the structure of the perception. Likewise, a memory refers to a past event not by reaching into a past external world, but by being (ideally) isomorphic to the original experience of that event. When there’s a perfect structural match, the mind treats the two as interchangeable in inference and prediction. Isomorphism thus underwrites the predictive power of analogies and metaphors in our internal model – a good metaphor isn’t just poetic; it reveals an isomorphism in relational structure that lets us apply knowledge from one domain of experience to another seamlessly.

Of course, in practice, perfect isomorphisms are rare – our internal experiences seldom line up 100%. More often we deal with partial mappings, which brings us to homomorphisms.

Homomorphism: Preserving Patterns Amid Difference

Homomorphisms are the more common, down-to-earth cousins of isomorphisms. They preserve structure too, but not necessarily all of it, and they don’t require a perfect one-to-one correspondence. In a homomorphism, some distinct elements might coalesce and some of the source’s nuances might get lost. Why are homomorphisms important for modeling phenomenology? Because much of what the mind does is take one complex pattern and translate it into another form, retaining as much meaningful structure as possible while perhaps discarding irrelevant detail. This is basically what understanding and abstraction are: finding a simpler or more general pattern that still captures the gist of varied experiences.

Consider how we form a concept from individual perceptions. You see many different chairs, from armchairs to barstools, and your mind abstracts the concept chair. The concept doesn’t preserve every detail (color, material, exact shape) from each instance, but it does preserve the structural relations that matter – perhaps the relation of “supporting a sitting human’s weight” or “having a surface at a certain height for sitting.” We can say the mapping from any particular chair perception to the internal concept of CHAIR is a homomorphism: it forgets some differences but preserves the structural features relevant to sitting functionality. All those different perceptions map into one concept, identifying common structure. In technical terms, the concept is a quotient structure of the experiences (many distinct experiences map to one abstract node – a homomorphism can collapse multiple source elements into one target element while preserving relations). This is how internal reference works in phenomenalism: a concept refers to all its instances by being structurally compatible with each of them via homomorphic mappings.

Homomorphisms thus enable generalization and prediction. If situation X is homomorphic to situation Y – meaning Y has at least all the relational structure of X (possibly more) – then patterns we learned in X can apply to Y. The mind often operates with homomorphic images: a mental model of the world might be a simplified (compressed) version of raw experience that still preserves causal or spatial relations well enough to be useful. These compressions are homomorphic projections from the full rich experience to a leaner model.

From a semantic viewpoint, we can say that an internal concept means what it does by virtue of being the common structure-preserving image of many experiences. In radical phenomenalism, the concept doesn’t point to a transcendent Platonic form or an external category in the world; its meaning is literally the pattern that all those experiences share internally. The homomorphism ensures that what was true in each specific instance remains true in the concept. We might even call this predictive coding of meaning – the concept predicts key aspects of any new instance because the mapping guarantees those aspects correspond. Notably, the brain’s predictive processing framework suggests something similar: the brain continually abstracts and updates an internal model that can generate (predict) sensory inputs. That model doesn’t capture every pixel of experience; it captures the structural regularities and discards noise. In category-theoretic terms, the brain is performing homomorphic compression of experience into a generative model. The better the homomorphism (the more structure preserved), the more successful the prediction and the richer the meaning carried forward.

Homomorphisms also allow partial reference. Suppose we use a metaphor: “Time is a river.” This is not an isomorphism (time and rivers are very different in many respects), but it is a homomorphism mapping some aspects of river onto time – e.g. the one-dimensional flow, irreversibility, and varying currents. The metaphor preserves the flow-structure while ignoring other parts (water, physical banks, etc.). In a radical phenomenalist sense, the metaphor has meaning insofar as it preserves a pattern of experience (our sense of time passing shares structural features with our experience of watching a river). The phrase “time is a river” resonates internally because of that structural overlap, not because the words correspond to objective things “out there.” We interpret it entirely by aligning internal structures (the relationship between past and future aligns with the relationship between upstream and downstream, for example).

In summary, homomorphisms model how the mind achieves meaningful similarity and abstraction. They let us say “this is like that in the following ways” and give us internal referents that are less strict than isomorphism but still carry over important structure. While an isomorphism is like a perfect translation, a homomorphism is more like a faithful summary or analogy – not identical, but true to the spirit.

Endomorphism and Automorphism: Internal Transformations and Symmetries

Next, we turn inward with endomorphisms and automorphisms – mappings that the mind applies to itself. These capture the idea of dynamic self-modeling: the mind can transform its own state or perspective and yet remain the same mind with the same structure at some level. In a self-contained simulation, this ability is crucial. It’s what allows us to imagine alternatives, to shift viewpoints, and to examine our own thoughts, all without stepping outside consciousness.

An endomorphism is any structure-preserving map from an object to itself. In the theater of the mind, an endomorphism could be a mental operation that you perform on an experience or idea that stays within your experiential universe. For example, when you reframe a memory – perhaps considering it from a different emotional angle or placing it in a new context – you’re applying a transformation to that memory while keeping it the same memory at core. As long as the key relationships remain (the sequence of events, the cause and effect), you have an endomorphic mapping: Memory (old framing) → Memory (new framing). The structure (events and their relations) stays, but some aspect (your interpretation or emphasis) changes. Endomorphisms thus represent internal transitions: going from one state of mind to another related state. They allow the simulation to evolve in time while preserving some invariants.

Now, an automorphism is a special kind of endomorphism – one that’s also an isomorphism. This means it’s a transformation you can undo and that leaves the structure entirely intact (it’s a symmetry of the system). Automorphisms in a cognitive context might correspond to things like perspective shifts or renamings that change nothing essential. Consider how a mathematical truth is true in any coordinate system; changing coordinates is an automorphism of the mathematical structure. Analogously, if you have a worldview or a conceptual schema, rotating it metaphorically (viewing it from another standpoint) might be an automorphism if nothing fundamental changes in what predictions it makes. For example, you could swap two synonymous terms in your internal narrative (say, “happiness” with “joy” throughout your thoughts) – if those concepts are truly synonymous in your mind, this substitution is an automorphism; it’s a relabeling that leaves all relational structure and inferences the same. The mind often tests its own consistency by such substitutions: “If I call this by a different name, do I still understand it?” When the answer is yes, we’ve found an automorphism – a reassuring symmetry indicating that the meaning didn’t depend on the particular sensory details or labels but only on the underlying structure.

Automorphisms correspond to internal symmetries or invariances. These are extremely important in radical phenomenalism because they reveal what aspects of experience are merely ornamental and what aspects are structural. If you can permute some elements of your experience (like changing the order of unrelated events in a story you tell yourself) and it makes no difference to the overall significance, that permutation is an automorphism. It tells you that only the structural pattern of causation or logic matters, not the specific sequence in time or the particular examples used.

Moreover, automorphisms relate to the idea of self-consistency and identity. The fact that automorphisms of an object form a group (mathematically)means you can compose multiple symmetries and still have a symmetry. In a mind, this could mean you can layer perspective shifts and re-labelings and still come back to a viewpoint that is effectively equivalent to the original. If the simulation is richly symmetrical, it might be robust: you can shuffle many superficial things and the core predictions stay the same, which suggests a strong internal coherence. On the flip side, if a supposed symmetry breaks something (imagine trying to replace a concept with a synonym and suddenly your understanding falters – maybe they weren’t true synonyms after all), then you’ve discovered a structural difference you didn’t know was there. The search for automorphisms is thus a search for what can change without changing meaning. It’s analogous to a mind testing the limits of its own worldview by hypothetical alterations, seeing what’s invariant. Those invariants define the stable semantic content.

In essence, endomorphisms and automorphisms give us a language for mind’s self-relations. Endomorphisms model the ever-shifting flow of experience that yet retains threads of continuity (I’m the same “I” as a moment ago, in some structured way, even though my thoughts have moved), and automorphisms model the hidden symmetries of thought that make certain transformations trivial (like rephrasing a thought in different words). A radically phenomenal internal world relies on these self-mappings to navigate itself. The only “reality check” it has is internal: can I transform this experience into that one and back (automorphism)? Can I project this state forward in time and still find consistency (an endomorphic evolution)? Category theory assures us that such questions can be rigorously framed in terms of these morphisms.

Monomorphism and Epimorphism: Embedding and Covering Internal Worlds

Moving on, we examine two opposed, but complementary, notions: monomorphisms and epimorphisms. These describe how one structure sits inside another or maps onto another. In phenomenological modeling, they help clarify ideas of parts and wholes, specific instances and general coverage within the internal realm.

A monomorphism (mono) can be thought of as an injection or embedding: one structure injects into another without collapsing distinct elements. In everyday speak, structure A is faithfully represented inside structure B. How does this matter for experiences? Consider memory: when you recall a past experience in detail, you are embedding fragments of that past experience into your present consciousness. If the recall is vivid and accurate, it’s like a monomorphism from the past experience structure into your current experience structure – each element of the memory (people, places, the sequence of events) maps to a corresponding element in your current mental tableau, retaining their distinct identities and relationships. A successful embedding means no confusion: the elements of the remembered scene remain separate and identifiable within the remembering mind, just as they were in the original event. A failure of embedding might be forgetting who said what (distinct roles collapse) or blurring two events together – essentially a non-monomorphic mapping where different source elements ended up merged in the target.

Monomorphisms thus speak to accuracy and specificity in internal reference. If an internal concept or experience A monomorphically maps into a larger context B (within the mind), then A’s distinctions are preserved in B. For example, the concept of apple can embed into your broader fruit knowledge without losing what makes it distinct from orange or pear (so long as the mapping from apple-concept to your fruit-conceptual-schema is monomorphic, “apple” won’t accidentally overlap with “orange”). Monomorphism is the guarantee of no unintended identification. In a self-contained semantics, this is critical: we can’t rely on an external world to tell apart our internal symbols (like pointing to two different real objects); the differentiation must come from how they are embedded in larger mental structures. A monomorphism says “this piece goes into the bigger puzzle in a way that maintains its uniqueness.”

Now, an epimorphism (epi) is roughly the dual: it’s like a surjection or covering map – the source structure covers the target structure entirely. Every part of the target comes from something in the source. In internal terms, if structure A epimorphically maps onto structure B, then B has no element or relation that isn’t accounted for by A. Think of a set of core principles or foundational experiences that generate the rest of your worldview. If indeed every belief or concept you have can be traced back to some combination of those core experiences (no new fundamental categories spontaneously appear), then the mapping from those experiences to your full worldview is epimorphic – they cover the whole thing. This resembles certain philosophical ideas: British empiricists like Hume or Locke might say all ideas are built from original sense impressions. That claim can be interpreted as “the set of sense impressions epimorphically maps onto the set of ideas” – every idea is composed of sensory parts, nothing more.

In practice, epimorphism connects to the notion of explanatory adequacy and predictive completeness. If our internal simulation uses a model (source) to predict an experience (target), we want the model’s structure to be rich enough to generate (surject onto) all aspects of the experience. For example, suppose you have a mental model of how social interactions work, and you encounter a new social situation. If your model can account for everything that happens (everyone’s behavior finds a place in your framework), then effectively your model epimorphically covers the new experience. There was no “alien” element in the situation outside your model’s reach – nothing that didn’t map from some element of your prior understanding. But if someone’s action totally baffles you (you have no source element mapping to that target element), then your prior structure was not epi – the new experience had structure that didn’t come from your model. That signals a gap in your internal knowledge.

In a radical phenomenalist world, epimorphisms represent the ideal that one set of experiences (or an internal theory built from them) can generate or predict all other experiences. It’s like having a coherent circle of knowledge with no loose ends sticking out. Of course, in reality we often encounter loose ends – but then we update our internal model to include them, striving again for an epimorphic coverage of our experiential world. We might say we aim for our worldview to be an epimorphism from foundational principles to all observations. This dovetails with coherentist theories of knowledge, where the coherence (here, a kind of covering relation) within our belief system is what justifies beliefs in the absence of an external foundation.

Monomorphism and epimorphism also interact. If we have a mapping that is both mono and epi (called a bimorphism, and if it’s also in a “nice” category it often becomes an isomorphism), that suggests a perfect fit – a substructure and a quotient simultaneously – but let’s not digress into that. The key takeaway is: monos ensure we don’t mix up or lose distinctions (internal precision), and epis ensure we cover everything intended (internal completeness). Together, they help describe how parts relate to wholes inside the mind. A concept should embed in a theory monomorphically (keeping its identity) and the theory should project onto predictions epimorphically (accounting for all phenomena). These structural virtues keep a purely internal reference system tight and coherent, doing justice to each experience (no unjustified identifications) and to the totality of experiences (no unexplained mysteries left).

Functors: Building Bridges Between Internal Domains

So far, we have discussed morphisms within a single level or domain of experience: mapping one idea to another idea, one experience to another experience, etc. But the mind is not monolithic – it has domains or layers. We have sensory experiences, conceptual understandings, linguistic expressions, emotional responses, and so on. How do these different domains of our inner life relate? If reference is all internal, then a word in our language must somehow connect to a concept in our thought, which in turn connects to some perceptual pattern in our experience. These are not direct one-step morphisms because each domain has its own kind of structure. Instead, we can use the concept of a functor to model these relationships between domains.

Recall that a functor maps objects to objects and morphisms to morphisms from one category to another, preserving the structural relations. Imagine one category that represents perceptual phenomena (call it Percepts), and another category that represents conceptual structures (call it Concepts). A functor F: Percepts → Concepts would assign each perceptual object (say, a particular perceived object or event) a corresponding concept, and each transformation or relation among percepts a corresponding relation among concepts. For example, the perceptual experience of seeing a particular apple might map to the concept “apple”. The act of peeling that apple (a relation between perceptions: the apple and its changing state) might map to a conceptual relation (“an apple can be peeled” or more generally an action concept of peeling applied to apple). If F is a good functor, then the structure of relationships in the perceptual realm is reflected in the conceptual realm. This ensures that our concepts honor the patterns in our perceptions – a fundamental requirement for making sense of our experiences.

Likewise, consider a functor from Concepts to Language (the category of linguistic expressions). This functor G might map the concept “apple” to the word “apple”, the concept “eat” to the word “eat”, and ensure that if concept A is related to concept B in a certain way (e.g. subject-verb-object relation “you – eat – apple”), then G(A) is related to G(B) in the corresponding grammatical way in the sentence (“you eat apple”). Functorial mapping guarantees that language preserves conceptual structure – at least ideally (natural language can be ambiguous or approximate, but our internal semantic interpretation tries to enforce such a mapping).

By chaining functors, we link perception to thought to language: Percepts → Concepts → Language. The composition of functors is a functor, and ideally the whole chain commutes with the actual process of perceiving and then describing. For instance, whether I directly describe what I see (“I see a red apple”) or first recognize it conceptually (perception to concept) and then put it into words (concept to language), the end result is the same described situation – this is a form of commutative diagram spanning multiple domains: perception → language vs. perception → concept → language yield the same correspondence if all mappings align properly. This illustrates how functors and commutative diagrams work together to enforce coherence across levels of abstraction.

In a radical phenomenalist model, functors are the formal way to capture internal reference across different “languages” of the mind. There’s the “language” of sense-data, the “language” of abstract thought, the “language” of imagination, etc. None of these refer to an external world; they refer to each other through systematic mappings. My visual image of an apple invokes my concept APPLE via a functor; that concept invokes auditory imagery of the word “apple” via another functor (if I think of the name), and maybe it even invokes a motor plan (reaching out to grab it) via yet another mapping in the sensorimotor domain. The meaningful coordination of all these internal subsystems is achieved not by an external apple forcing itself into my mind, but by these functorial relations that let each part of the mind talk to the others consistently. If any of these mappings fail – say my concept mapping from perception is off (I mistake the apple for a tomato at first glance) – then there’s a misalignment that usually gets corrected by additional experience (surprise at the taste, etc.). The mind strives to make the diagram commute: all pathways of internal reference should agree, forging one coherent experience out of sight, sound, thought, and action.

An interesting feature of functors is that they can highlight differences in representation while maintaining a core similarity. For example, the concept of time in a story narrative versus the concept of time in a musical score might be related by a functor that maps temporal positions in the story (beginning, middle, end) to temporal positions in the music (start, development, conclusion). The structures are analogous but not identical in content. By mapping them, one can compare the structure of a novel and a symphony (a metaphor often drawn in literary analysis). This is like treating one as a category and the other as another category, then mapping via a functor. If the diagram commutes, patterns in the novel correspond to patterns in the symphony. This shows up in creativity: we say, “This novel has a crescendo and climax like a piece of music” – effectively using a functorial analogy between art forms.

In sum, functors in the internal context are bridges of resonance: they ensure that the orchestra of the mind’s parts plays in harmony. Each section (perception, conception, language, emotion) can have its own instruments and notes, but functorial mappings keep them in tune with one another, preserving the score (structure) across different modalities. The absence of an external conductor (objective reality) is compensated by the consistency of these internal bridges.

Commutative Diagrams: Internal Coherence and Structural Truth

We’ve hinted at commutative diagrams already as we discussed functors, but let’s zoom out and consider why commutativity matters so much for a self-contained mind. A commutative diagram is essentially a consistency check. It says: if you start from some experience or concept X and there are multiple routes to arrive at Y (via different intermediate mental operations or representations), all those routes should agree on Y. If they do, Y is well-defined and meaning is coherent; if they don’t, you have a problem – an internal contradiction or ambiguity.

Imagine a simple example: you have a concept of “what happens if I drop a glass” that you can arrive at in two ways. Path 1: recall a memory of dropping a glass → imagine the sound of it shattering. Path 2: recall the abstract concept “fragile object falls” → logically infer “it breaks” → visualize broken glass and imagine the sound. In a commutative diagram scenario, both path 1 and path 2 should result in the same expected sound and image of a shattered glass. If path 1 gave a different outcome than path 2 (say one mental route predicted it would bounce unharmed while the other predicted it would break), your internal knowledge is incoherent. Something doesn’t commute; the internal references disagree. You’d experience confusion or surprise when one of the routes is actualized.

Within radical phenomenalism, truth and reference become matters of internal coherence: a statement or belief is “true” if it fits into a commutative diagram with your entire experiential network. All the pieces reinforce each other without mismatch. For instance, the “diagram” of perceptions, concepts, and linguistic assertions about “the apple” is commutative if no matter how you translate between seeing the apple, thinking “apple”, and saying “apple”, you get a stable cluster of confirmations (every route – seeing then naming vs. conceptualizing then expecting a certain sight – aligns). If any part fails (like you say “apple” while picturing a banana internally due to some mix-up), the diagram doesn’t commute and you have a semantic error entirely within your own head. We fix such errors by adjusting mappings until consistency is achieved – effectively, making the diagram commute again.

Philosophically, commutative diagrams capture what the American Pragmatists or some coherence theorists of truth intuited: that truth is a property of a whole system of statements/ideas working together without friction. Here we frame it as structural coherence. Structural resonance (our recurring theme) is maximized when every loop closes properly – when going around any cycle of internal relations brings you back to where you started. This is reminiscent of predictive coding again: the brain sends top-down predictions and receives bottom-up sensory data, and the two meet in the middle. Ideally, the predicted signals and the actual signals cancel out (error minimization) – that’s essentially a commutative diagram between the model and the sensation. If there’s a discrepancy (prediction doesn’t match input), the “diagram” of brain signals doesn’t commute and an error is fed back for correction. The brain then updates to restore commutativity. In a fully phenomenalist setting, this correction process isn’t about “getting closer to an external truth” but about making the internal simulation self-consistent and predictively successful. The only standard of truth is that the simulation doesn’t trip over itself and that it continues to anticipate experience correctly (in its own terms).

We can imagine the entirety of one’s mind as a giant diagram with countless nodes (concepts, experiences, memories) and arrows (relations, inferences, mappings). The integrity of the mind requires that for any given start and end point in this network, all ways of connecting them yield a consistent result. When that holds, we have what we might call structural sanity. When it fails, we see phenomena like cognitive dissonance (two paths in the diagram give clashing outcomes), or illusion (perception path vs. higher cognition path disagree), or paradox in thought. The resolution of these issues is always: add or adjust a connection so that the paths can reconcile – essentially, complete the diagram or refine it until it commutes.

A powerful image here is resonance again: when the structure is fully consistent, any given experience will reverberate correctly through all interpretative pathways and yield a single harmonious note – the mind “understands” the experience in a unified way. When something is off, it’s like dissonance – two parts of the mind produce clashing tones. Commutativity is the condition for harmony.

It’s worth noting that in mathematics, important properties are often stated as commutative diagrams – they show that doing things in different orders yields the same outcome, which usually signals an underlying natural relationship or conserved quantity. In an internalist semantics, one might say the only conserved quantity is meaning itself, preserved across transformations. If meaning is to survive the journey from perception to thought to action to memory, all those transformations better align. The commutativity is the guarantor that what we mean stays the same no matter how we express or process it internally.

Implications for Cognition, Knowledge, and Semantic Future

By now, it’s clear that category theory’s morphisms and related constructs are more than abstract math – they provide a rich descriptive framework for an internalist, structural theory of mind. What are the broader implications of modeling phenomenology in this way?

1. Cognition as Category: Thinking of mental states as objects and mental operations as morphisms encourages us to see cognition as structured transformation rather than static representation. The focus shifts from what experiences are in some absolute sense to how experiences relate to one another. This resonates with the dynamic, network-centric view of the brain and mind. It suggests that to understand cognition, we should map out the transformational structure of experience: How does one thought lead to another? Which patterns map onto which? The morphism lens highlights learning as adding new morphisms or objects in the category of mind – essentially expanding the internal category so that more external scenarios can be simulated (in phenomenalist terms, integrating what were “pseudo-concepts” by giving them internal analogues). It’s a very process-oriented view of knowledge: to know something is to have constructed a structural mapping for it within your existing mental category.

2. Epistemology without External Reference: Traditional epistemology worries about how our beliefs correspond to an external world. Radical Phenomenalism flips the script: knowledge is about the consistency and richness of our internal model. Using the morphism framework, we can redefine justification as structural coherence. A belief is justified if it slots into the network (category) without breaking commutativity – i.e., it is an image of other experiences (homomorphic outcome), it fits the functorial translations (makes sense in perception, thought, and language consistently), and it doesn’t identify things that should remain distinct (respects monomorphisms) nor postulate ungrounded new entities (respects epimorphisms from known bases). This is a rigorous way to implement a coherentist epistemology. It also dovetails with predictive processing: a well-justified belief is one that helps minimize prediction errors across the board – essentially ensuring diagrams commute when that belief is included.

Of course, this raises the question: if everything is internal, what distinguishes fiction from reality or illusion from veridical perception? The answer would lie entirely in structural virtues: a hallucination might be a perfectly vivid object in perception, but it fails commutativity with other senses or past experience (the diagram from vision to touch to memory doesn’t close – you see it but can’t touch it, etc.). A delusion might be a strongly held concept, but it fails some internal functor mapping (perhaps linguistic assertions vs. actual perceptual predictions don’t line up). Thus, error is detectable as internal inconsistency, not reference to an external fact. Over time, a phenomenalist system would “debug” itself by eliminating or compartmentalizing such inconsistencies. In category terms, the internal category evolves, adding or removing morphisms, to become more internally consistent (which, from the outside, looks like “becoming more aligned with reality,” but that’s a redundant description from within the system).

3. The Future of Philosophical Semantics: If we embrace the idea that semantics (meaning) is all about internal structure, category theory offers a new formal language for semantics. Instead of model-theoretic semantics where words correspond to set-theoretic objects in some external model, we might envision a category-theoretic semantics where words and sentences correspond to objects and morphisms in the category of mental concepts. The truth of a sentence would mean that the corresponding diagram in the conceptual category commutes or that a certain morphism exists there. For example, the sentence “All A are B” might be true internally if the object representing concept A maps (via a monomorphism, say) into the object representing concept B – i.e., A is structurally a sub-object of B in the conceptual category. This internal semantic approach could handle intensional contexts, metaphor, and other aspects more naturally by focusing on structural consistency rather than reference to external sets.

Moreover, applying category theory could lead to compositionality via functors: the meaning of complex expressions could be built by functorially mapping the syntactic structure (a category of linguistic expressions) into the semantic structure (category of concepts). In fact, category theory is already used in computational semantics and cognitive science to deal with compositional meaning and analogy. Radical Phenomenalism would use these tools but interpret the resulting structures as literally all there is to meaning – no hidden external referents at all.

4. Structural Resonance as Understanding: We often talk about “grasping the structure” of a problem or “resonating” with a piece of music or art. This theory formalizes that intuition: to understand something is to find a morphism (usually an isomorphism or homomorphism) between that new thing and something already in our mental repertoire. When the structures align, we exclaim “I get it!” – essentially, a commutative diagram has been completed between the new experience and our existing knowledge. This has practical implications for education and communication: teaching becomes the art of creating morphisms from the student’s current knowledge to the target knowledge (perhaps building a chain of functors from an intuitive domain to a more abstract domain). When done right, the student finds an isomorphism where before there was confusion.

5. Limitations and Extensions: A skeptic might point out that category theory describes static structures, whereas the mind is dynamic and sometimes messy. But category theory has its own way of handling dynamics (via functor categories, natural transformations, etc.), and messiness can be approached via open systems (composing categories). Also, one might ask: does this internal structural view really capture the richness of qualia and raw feeling? Radical Phenomenalism would answer that qualia themselves are just nodes in this structural web – their significance lies in the predictive and relational roles they play. The ineffability of “raw feel” doesn’t mean it’s outside structure; it means it’s an atomic object in the category that we haven’t broken down further. But we relate it to other things (this feel morphs into that feel under certain conditions, etc.), so it’s still known by its place in the structure.

Looking forward, one intriguing possibility is using these ideas in artificial intelligence or cognitive modeling. If reference is purely internal, an AI doesn’t need sensors pointed at a “real” world to develop meaning; it could, in principle, simulate its own world and develop an internal category of experiences. Its “understanding” would be measured by the structural complexity and coherence of that internal category. For humans, this offers a lens to understand phenomena like dreams or virtual reality: even cut off from external input, the mind can sustain meaning through internal consistency and structural richness (a dream can feel meaningful if it has narrative coherence – an internal logic – even if it’s not “real”).

Finally, embracing structural mappings as fundamental might help bridge different philosophical views: realists focus on structure in the external world, idealists on experiences – here we say structure itself is the key, and it doesn’t matter whether it’s “out there” or “in here” as long as it’s consistent. Radical Phenomenalism just chooses “in here” as the arena, but it shares with structural realism the idea that relations, not isolated objects, carry the day.

In conclusion, by applying the framework of morphisms, functors, and commutative diagrams to the theater of consciousness, we articulate a vision of meaning that is structural, internal, and self-sufficient. The mind becomes its own universe of forms, with morphic maps ensuring that this universe hangs together. Reference is re-imagined as the resonance between one part of the mind and another, and truth as the integrity of the whole. While this doesn’t deny the practical existence of an external world (we all act as if there is one, and it’s a useful hypothesis), it relegates that concept itself to an internal role – just another model we use, subject to the same structural criteria of coherence. The payoff is a deeply unified view of semantics and phenomenology: understanding the mind as a formal system of transformations might one day allow us to speak about subjective experience with the clarity and precision of mathematics, without losing the poetry of metaphor. After all, mathematics itself, as we’ve seen, can be poetic – what is a commutative diagram if not a silent poem about harmony? And what is an isomorphism if not a perfect rhyme between two structures? In the end, the rational and the metaphorical unite: the internal geometry of mind is at once rigorous and resonant, mapping the many into one and the one into many, again and again, in a self-reflective dance of meaning.

References: (Structural mappings and definitions drawn from category theory; internalist semantics and radical phenomenalism informed by linguistic and cognitive theories.)