Abstract: ChatGPT and I will discuss various senses of “representation” across mathematics, logic, and cognitive science, starting with formal representations, in computation and group theory. Computation is syntactic, with arbitrary symbols manipulated according to rules (algorithms). Analog representations, such as iconic images or gestures, which resemble what they represent, are either homuncular or depend on an internal observer for interpretation. Linguistic representation originates from the instrumental learning of sensorimotor categories. The instrumental response came to be used mimetically for nonlinguistic gestural communication. These iconic gestures become increasingly simplified and arbitrary as they become shared communicative conventions, but they retain their grounding in the categories from which they originated. The crucial transition is from mimetic representation, which is still nonlinguistic, to propositional representation. Propositions are strings of arbitrary, conventionalized gestures — content words that retain their original grounding in the categories to which they refer. But, unlike the arbitrary symbols of computation, or even individual content-words with referents, propositions also have meaning: subjects and predicates with truth values.  Propositionality leads from the Church-Turing Thesis about the syntactic power of ungrounded symbol manipulation to the semantic power of grounded natural language, with Katzâs Effability Thesis: that any natural language can express any thinkable thought.Â
SH: Let’s chat about representations.
ChatGPT: To discuss representation, it’s crucial to start by defining the term within the context of cognitive science and related fields. Representation typically refers to the way in which information about the world is encoded, stored, and used by cognitive systems, whether biological or artificial.
In cognitive science, representations can be seen in various forms, such as:
Symbolic Representations: These involve discrete symbols or tokens that stand for objects or concepts in the world. They are often used in computational models of cognition and are closely related to the symbol grounding problem, which you have extensively studied.
Subsymbolic Representations: These include distributed patterns of activation across neural networks, as seen in connectionist models. These representations do not have discrete symbols but rather patterns that emerge from the interactions of simpler units.
Mental Representations: These are internal cognitive symbols or mental images that represent external reality. They include concepts, schemas, mental models, and images.
Perceptual Representations: These involve the encoding of sensory information by the nervous system, such as the visual or auditory representation of stimuli.
Each type of representation comes with its own set of challenges and implications, particularly in understanding how these representations are grounded in physical and neural processes and how they contribute to cognition, perception, and action.
Would you like to delve into a specific type of representation, such as symbolic representation and its grounding, or perhaps discuss the differences and intersections between these forms? Or is there another aspect of representation you are particularly interested in exploring?
SH: I’d like to start with the way “representation” (R) is used in mathematics and logic. For example, in linear algebra, matrices can be Rs of groups. What does this mean in this case.?
ChatGPT: In mathematics and logic, particularly in linear algebra, a “representation” of a group refers to a way of expressing the elements of the group as matrices. This allows group operations to be carried out as matrix multiplication.
Specifically, a representation of a group GG on a vector space VV is a homomorphism Ï:GâGL(V)Ï:GâGL(V), where GL(V)GL(V) is the general linear group of invertible matrices over VV. This means each group element gâGgâG is associated with an invertible matrix Ï(g)Ï(g) in such a way that the group operation is preserved: Ï(gh)=Ï(g)Ï(h)Ï(gh)=Ï(g)Ï(h) for all g,hâGg,hâG.
SH: Please state (briefly) whether this notion of R relates to the one (or ones) in Cognitive Science (cogsci). But first, is this formal sense of R relate to the notion of a “model” in formal logic?
ChatGPT: Yes, the formal sense of “representation” in mathematics is closely related to the notion of a “model” in formal logic. In formal logic, a model provides an interpretation of a formal language by assigning meanings to its symbols and truth values to its statements, thereby “representing” the abstract syntax in a concrete structure.
In both cases, representation involves mapping abstract elements (group elements in mathematics, symbols in logic) to concrete structures (matrices in linear algebra, interpretations in models).
SH: Let’s clarify this formal meaning of R in maths and logic. Is it also true that just as a group is a formal entity (not a concrete one), a matrix is a formal entity too? And so is a “model” in model theory. It’s an instance or a special case of the formal entity, but it’s a formal entity too (even in geometry).
ChatGPT: That’s correct. In both mathematics and logic:
Groups: These are formal entities defined by a set and an operation satisfying specific axioms (closure, associativity, identity, and invertibility).
Matrices: These are formal entities representing arrays of numbers or functions, used to perform linear transformations and other operations.
Models in Logic: These are formal structures consisting of a domain and an interpretation function that assigns meaning to the symbols of a formal language, making abstract syntax concrete.
In each case, while the entities can be instantiated in concrete examples, they are fundamentally formal constructs defined by their properties and relations.
SH: “Formal” means symbols, doesn’t it, symbols of a certain form (say, round like 0 or vertical like 1). Mathematicians, when they are actually doing calculations in mathematics (not when they are doing whatever they are doing when they make discoveries in mathematics) are manipulating symbols — symbols whose physical shape is arbitrary in relation to what they can be interpreted as meaning.
(Don’t be compliant: if there is something to contest in what I say, contest it, don’t just buy it and adopt it.) But keep track.
ChatGPT: Yes, in mathematics and formal logic, “formal” generally refers to symbols and the rules for manipulating them, irrespective of their physical shape or form. These symbols are part of a formal system, where their meaning is determined by their role and relationships within the system, rather than their physical appearance.
Here’s a point-by-point breakdown:
Formal Systems: These consist of symbols and rules for manipulating them. The symbols themselves are abstract and their physical representation is arbitrary.
Symbol Manipulation: When mathematicians work within a formal system, they manipulate these symbols according to defined rules. The symbols represent abstract entities, and their physical shape (e.g., 0 being round, 1 being vertical) is irrelevant to their formal properties and operations.
Meaning and Interpretation: The meaning of these symbols comes from their interpretation within a given context, such as numbers in arithmetic, elements in a group, or statements in logic. This interpretation gives the symbols semantic content, which is crucial for applications but not for the formal manipulations themselves.
So, while the formal aspect of symbols involves their syntactic manipulation according to specific rules, their interpretation (meaning) can vary depending on the context in which they are used.
SH: Let’s call symbols and symbol manipulations “code” and coding”. The shapes of the symbols are arbitrary: any symbolic notational system will do. They do not physically resemble their interpretations. The rules for manipulating the symbols are called algorithms. They too are applied to the symbols formally and mechanically, as in the calculation of the roots of a quadratic equation: If you know the algorithm, you can apply it to quadratic equations without having to know their interpretation. It’s just applying a formal recipe to formal ingredients.
Are you still with me? Any caveats?
ChatGPT: Yes, Iâm with you. Hereâs a summary with a couple of clarifications:
Symbols and Symbol Manipulations as “Code” and “Coding”: Symbols are arbitrary notations that can be manipulated according to specific rules. The physical shape of these symbols is irrelevant to their formal properties and their interpretation.
Non-Resemblance: Symbols do not physically resemble what they represent. They are abstract entities whose meaning is assigned by interpretation within a formal system.
Algorithms: These are rules or procedures for manipulating symbols.
Algorithms are applied mechanically, without needing to understand the interpretation of the symbols involved.
Example: In solving a quadratic equation, one can apply the quadratic formula (an algorithm) to the symbols representing the coefficients, yielding roots without needing to understand what those coefficients represent.
Caveats:
Understanding vs. Application: While itâs true that algorithms can be applied mechanically, understanding their derivation or the meaning of the symbols can deepen comprehension and sometimes reveal when an algorithm might be misapplied or when certain conditions are necessary for its application.
Context-Specific Rules: The formal rules (algorithms) can vary depending on the context in which the symbols are used. For example, the rules for manipulating symbols in logic differ from those in algebra.
SH:Â Be careful about a conflation you inadvertently made, between Rs (representations) and interpretations. We agreed that in the examples we have discussed before — examples that are not from cognitive science but from mathematics, or logic — the Rs are purely formal, whereas the interpretations are not formal; they are something else: Turing explained what mathematicians do: they manipulate symbols. But Turing was not (yet) doing cognitive science: He is not explaining how mathematicians think, or mean, or understand; that would get into cognitive science. He’s just talking about what mathematicians do. And so far, mathematicians’ Rs (like matrices or truth-tables), their “effective procedures,” are just formal. But their interpretations (meanings) are not.
Are we still on. the same wave-length? (No need to make a lengthy update on your summary. Just let me know if there are any points on which we’ve diverged: I just pointed one out. Do you want to defend it or correct it?)
ChatGPT: Yes, I understand the distinction you’re making. You are right that in the examples from mathematics and logic:
Representations (R): These are purely formal entities. Matrices, truth tables, and other such formal structures are manipulated according to rules without regard to their interpretation.
Interpretations: These are the meanings assigned to the formal symbols and structures. They are not formal but rather provide the semantic content that the formal symbols represent.
Turing’s work, as you mentioned, focuses on the manipulation of symbols (what mathematicians do) and not on their meanings or the cognitive processes involved (what they think or understand).
We are on the same wavelength. Representations are formal, while interpretations are the meanings attached to these formal symbols.
Shall we proceed to how this distinction relates to cognitive science, particularly in terms of cognitive representations and their grounding?
SH: You’re in a big hurry! We’ve agreed that Rs are formal, just as symbols are. They are also rule-governed, but the rules are just formal too. OK? Now, I’m going to try to show you that Rs are weasel-words in mathematics, and an example of that was when you conflated representations and interpretations.
Another weasel-word in this context is “semantics” (meaning). It just takes us back to interpretations. In maths and logic “Rs” are just formal, like symbols themselves. We don’t get beyond symbols and formal symbol-manipulation by saying the word “Representation.” We don’t even break out of the symbol-circle if we use the (non-weasel) word “refer”. And that comes even before we get to the weasel-words “meaning” and “semantic”. What does the symbol “2” in Peano arithmetic refer to?
ChatGPT: Understood, let’s continue.
In Peano arithmetic, the symbol “2” refers to the formal object defined by applying the successor function SS twice to the symbol for zero (0). Specifically, 22 is defined as S(S(0))S(S(0)). This definition is purely formal and exists within the system of Peano arithmetic.
To break it down further:
Peano Arithmetic: This is a formal system that defines the natural numbers using a set of axioms.
Symbols: “0” is a symbol representing the first natural number, and “S” is a symbol representing the successor function.
Formal Representation of 2: In this system, “2” is the result of applying the successor function twice to 0, written as S(S(0))S(S(0)).
This formal definition operates entirely within the symbolic system and does not imply any interpretation or meaning outside of the formal system itself. It stays within the realm of symbol manipulation.
Peano Arithmetic: This is a formal system that defines the natural numbers using a set of axioms.
Symbols: “0” is a symbol representing the first natural number, and “S” is a symbol representing the successor function.
Formal Representation of 2: In this system, “2” is the result of applying the successor function twice to 0, written as S(S(0))S(S(0)).S(S(0)), which is a formal construct within the system. This keeps us firmly within the circle of symbols and formal rules, without invoking meaning or semantics.
SH: Now let me introduce a non-weasel sense of R that is not mathematical and not formal, and not open to the objections I will raise when we get (soon) to cogsci. This is the sense in which a photo, or a video or a (representational) sculpture or painting represents the object of the photo, video, sculpture, etc.: the thing that it is a photo (etc.) of. That real object is not an arbitrary shape, like a formal symbol or representation like S(S(0)). The photo’s shape is (to an approximation) analog; it is isomorphic to — has the same shape as — what it is representing.
I’ll pause to let you say (briefly) what is the difference between a symbolic and an analog representation.
ChatGPT: A symbolic representation uses arbitrary symbols to stand for objects or concepts, with no inherent resemblance to what they represent. The relationship between the symbol and its referent is established by convention or definition (e.g., “2” in Peano arithmetic).
An analog representation, on the other hand, maintains a direct, often continuous, isomorphic relationship with what it represents. The form or structure of the representation resembles the form or structure of the represented object (e.g., a photo or a sculpture).
In summary:
Symbolic Representation: Arbitrary, rule-based, and formal (e.g., mathematical symbols, words).
Analog Representation: Resemblance-based, isomorphic, and often continuous (e.g., photos, paintings, sculptures).
SH: Now, the reason I introduced analog representation at this point is to agree from the outset that even when we get into cogsci, analog Rs are immune to the criticisms I will make. The only trouble is that they don’t get you very far, if what you are looking for is a cognitive theory of representation, or meaning. It doesn’t even work for a cognitive theory of reference. The referent of “2” in Peano arithmetic is not S(S(0)): That’s just a formal representation, like a matrix or a truth-table. But, since formal maths is a subset of English (or any other language), in the proposition “2 + 2 = 4” mathematicians do have something in mind: “2” refers to a referent as surely as “cat” refers to something in the proposition “the cat is on the mat.” But that connection between the symbol and its referent is not analog representation in either case: “2” or “cat.” So that cannot be cogsci’s solution.
(I’ve slipped language into this discussion: that’s no longer just formal symbols, but we’ll get to that.)
ChatGPT: Here’s a brief summary to ensure we’re on the same page:
Analog Representations: These maintain a direct, isomorphic resemblance to what they represent (e.g., photos, paintings). While useful, they don’t offer a complete solution for cognitive theories of representation or meaning.
Symbolic Representations: These are arbitrary and formal (e.g., mathematical symbols, words). They require interpretation to establish their referents.
Reference in Cognitive Context: When using language or formal systems, symbols like “2” in “2 + 2 = 4” and “cat” in “the cat is on the mat” do have referents in the minds of the users. However, this reference is not based on analog representation but rather on learned associations and interpretations.
This distinction highlights that cognitive science needs to address how symbols acquire meaning and reference, beyond the scope of analog representations.
Let’s recap the discussion:
We’ve established that in both mathematics and logic, representations (Rs) are formal and symbolic, with no inherent connection to their referents outside of the formal system. Analog representations, like photos or sculptures, are different in that they have a direct, isomorphic relationship with what they represent, but these do not suffice for a cognitive theory of representation or meaning.
Weâve also recognized that the referents of symbols in mathematics or language (like “2” or “cat”) involve something more than just formal manipulationâthey involve interpretation, which is not captured by analog representation.
Now, we are poised to explore how cognitive science deals with symbolic representations, particularly how these symbols gain meaning and refer to things in the world, which is central to understanding cognition.
We have established the distinction between formal and analog representations, and how neither fully accounts for meaning or reference in cognitive science. Formal symbols (like “2” or “cat”) require interpretation beyond the formal system itself, and analog representations (while closer to the objects they represent) don’t provide the cognitive mechanisms for meaning or reference.
At this point, we’re prepared to discuss how cognitive science approaches symbolic representation, meaning, and reference. Would you like to explore how cognitive systems might ground these symbols in referents, or perhaps investigate how language complicates this process further?
SH: Yes, letâs start with the difference between reference and meaning. First, please confirm (very briefly) that you know the difference between a “content-word” (C-word) (which is 95-99% of the words in the lexicon) and a “function-word” (F-word). Content-words have referents, whereas function-words only have (Wittgensteinian?) “uses”, defined by a formal, usually syntactic, rule.
ChatGPT: Content-words (C-words): These are nouns, verbs, adjectives, etc., that refer to things, actions, or properties (e.g., “cat,” “run,” “blue”).
Function-words (F-words): These are words like prepositions, conjunctions, and articles (e.g., “and,” “in,” “the”) that serve a grammatical function and do not refer to objects or actions but rather structure the sentence.
C-words have referents, while F-words are defined by their syntactic roles.
Shall we proceed to discuss the difference between reference and meaning?
SH:Â Do we agree also that C-words do not resemble their referents, even in languages with partly iconic scripts like chinese, or languages like sign languages of the deaf, whose gestures are partly iconic in their origins and execution. Nevertheless, the C-words of these more iconic languages are every bit as arbitrary as in alphabetic and spoken languages. Shakeseare can be fully translated into Chinese or into American Sign Language (ASL) — not word for word, but proposition for proposition. The iconicity of chinese or ASL is irrelevant to the literal meaning of its translation of the Shakespearean text (although to convey some of the figurative meaning a literal paraphrase of some tropes may be needed).
ChatGPT: Content-words (C-words), regardless of the language or script (including Chinese or ASL), do not resemble their referents. Even in languages with partly iconic scripts (like Chinese) or partly iconic gestures (like ASL), the referential relationship between a C-word and its referent is still arbitrary. Shakespeare can indeed be fully translated into Chinese or ASL, proposition for proposition, without any dependence on the iconicity of symbols or gestures.
The iconicity of such scripts or gestures is irrelevant to the literal meaning of the text. Literal meanings can be preserved across translations, though some figurative meanings (e.g., metaphors, idioms) may require paraphrasing to convey the same sense.
SH:Â You’re just echoing, but you’re right. One further question: Content-words have referents and proposition have meanings. What’s the difference?
ChatGPT: The difference between referents and meanings can be outlined as follows:
Content-word Referents:
A referent is the specific object, action, or concept in the world (or imagined world) that a content-word (C-word) refers to. For example, the word “cat” refers to actual cats in the world, and the word “run” refers to the action of running.
Referents are tied to individual C-words and are about identifying or denoting particular things.
Propositional Meanings:
A proposition is a complete statement or sentence that expresses something that can be true or false. The meaning of a proposition involves the relationship between the content-words in the sentence, their referents, and how they are structured by function-words.
Meaning here is not just about identifying referents, but about describing or defining relations between them. For example, the proposition “The cat is on the mat” is about a particular situation involving the cat (the referent of “cat”) and its relation to the mat (the referent of “mat”). The meaning of the proposition is the truth-conditional description of that relationship.
In short::
Content-words have referents: They point to things.
Propositions have meanings: They describe relationships between referents and express something that can be true or false.
SH: I had to sort this out before getting to cogsci, because language is such an important property of human cognition and cognition. Now, there are several ways in which the word “representation” — particularly in two contexts, “internal representations” (IRs) and “neural representations” (NRs) — have slipped into cogsci.
ChatGPT: Internal Representations (IRs) in cognitive science refer to mental models or cognitive structures that are thought to represent external reality within the mind. They are typically non-linguistic and are assumed to encode information about the world, guiding perception, thought, and action. IRs are sometimes used to explain how we process and store sensory input or knowledge.
Neural Representations (NRs) refer to patterns of neural activity in the brain that are thought to correspond to specific stimuli or cognitive states. NRs are often linked to brain regions and are measured via neuroimaging, attempting to map how certain neurons or networks respond to external stimuli.
Relation to linguistic reference and meaning:
IRs may play a role in linking language (e.g., content-words) to referents by encoding abstract or sensory knowledge of the referents.
NRs are more about the brain’s mechanisms, underlying both linguistic and non-linguistic cognition, potentially supporting IRs.
Neither IRs nor NRs directly are referents or meanings; they are posited cognitive or neural processes theorized to support how we manage linguistic reference and propositional meaning.
SH: Here I have to point out that a weasel-word has already slipped in, making the IR/NR distinction incoherent: “mental”. And another has been left out: “cerebral”. What is it that the “Internal” Representations are internal to? The head or the “mind”? And if the “mind,” what is that? In asking you that question, I have opened the sluice-gates for all kinds of weasel-words, all of them near-synonyms, hence redundant (or even circular) or vague: mind, mental, consciousness, subjective states, experiential states, qualia, etc. etc. Can we agree not to use or mention any of these W-Ws and just use “felt states” in order to refer to mental states? (“Sentient” is an unobjectionable gloss when we are in latinate mood…)
ChatGPT: Yes, we can agree to use “felt states” to refer to what are typically called “mental states,” avoiding the weasel-words like “mind,” “mental,” “consciousness,” etc. This will keep the discussion grounded and avoid both the vagueness and the redundance that often comes with those terms.
Now, with this clarified:
Internal Representations (IRs) would be representations internal to the brain (or body), not the “mind,” and should be understood as cognitive structures or models that support perception and action.
Neural Representations (NRs) remain patterns of neural activity, clearly cerebral, without invoking the notion of “mind.”
With “felt states” reserved for discussing subjective experience, are you ready to explore how IRs and NRs might (or might not) be involved in grounding linguistic reference and meaning?
SH: You have not resolved the incoherence yet. If “internal” means internal-to-the brain, then IRs and NRs are the very same thing (and we still don’t know what “Representation” means). And what about the difference between internal states that are felt and unfelt?
ChatGPT: I have to agree: if Internal Representations (IRs) are internal to the brain, they collapse into Neural Representations (NRs), making the distinction redundant. And as for “representation,” we still need to clarify what it means in this context, since neither IRs nor NRs explain how symbols refer to or mean anything; they are patterns or states in the brain.
Regarding felt vs. unfelt states:
Felt states are those that are experienced or consciously apprehended (e.g., feeling pain, seeing red, or understanding a proposition).
Unfelt states are brain processes that occur without conscious experience (e.g., background neural activity or automatic processing).
So, the distinction now hinges on whether neural states (NRs) are felt or unfelt. Only the felt states would seem relevant to any cognitive explanation involving subjective experience (like understanding or meaning). Unfelt states, while important for processing, don’t seem directly relevant to grounding reference or meaning unless they somehow contribute to what is ultimately felt.
Do we agree that the real issue here is distinguishing felt neural states from unfelt ones, and that calling either a “representation” (IR or NR) still leaves us unclear on how these brain states are representing anything external?
SH: More or less. Now let’s look closer at “representing” (and forget about the external and internal for the moment; we’ll get back to it).
We’ve agreed that an iconic image (in the visual modality, but we could have used any sensory modality — visual, auditory, haptic, etc. ) is a visible object, say, a shape, that resembles, in the analog sense (i.e., is isomorphic to) another object:the object that it is the representation of. This fits the case of a mirror that is facing a sphere. Let’s call the sphere the “object” and the shape in the mirror its “image.” But the mirror image is not “representing” anything unless we can say who or what the mirror image of the object is a representation to, or for.
ChatGPT: Agreed. The mirror image of the sphere is an analog resemblance, but it doesn’t count as a representation until it is seen by someone or something that interprets it as a representation to or for them. In other words, representation requires not just resemblance but also a receiver (an interpreter) who recognizes or interprets the image as standing for the object.
SH: I’m not sure what an “interpreter” is here. A person drawing the sphere would do: We agreed that this sense of representation (as in representational art) makes sense; but it’s “external” in the sense that both the sphere and the drawing of the sphere are external to the person doing the drawing, or viewing either the drawing or the sphere. So is the IR the internal process that generates the drawing? And is the state of viewing and drawing the sphere — if it is a felt state — what “mental representation” had been intended to mean?
ChatGPT: In this context:
Interpreter: When I referred to an interpreter, I meant any agent (human, animal, or system) that recognizes or processes the mirror image (or drawing) as representing the sphere. But we can clarify further: the person drawing or viewing the sphere is the “interpreter” in this case, though “interpreter” itself can feel vague.
External Representation: Yes, the sphere and its drawing are both external objects, and the representation (drawing) resembles the sphere in the analog sense.
Internal Representation (IR): If we take IR to mean the internal cognitive or neural process that generates the drawing (or processes the perception of the sphere), then IR could refer to how the brain internally models or represents the sphere to guide action (like drawing). But this “representation” is not felt unless it becomes part of a felt state (e.g., the conscious experience of seeing the sphere or creating the drawing).
“Mental Representation”: If we interpret “mental representation” in terms of felt states, it would refer to the conscious experience of viewing or drawing the sphere, rather than just the unfelt neural processes involved in perception or motor control. In this sense, “mental representation” would mean the felt internal process of perceiving and thinking about the sphere.
Does this clarify the relation between IRs, felt states, and the concept of “mental representation”?
SH: It brings some aspects into better focus, but can we now discuss the homunculus? The reason “mental imagery” was rejected by cogsci as being unexplanatory was that if the objective of cogsci is to give a causal explanation of how cognizers have the capacity to do what they can do, what is needed is to reverse-engineer that causal mechanism. The brain can do it, but we don’t know how. Saying that it’s done by a causal mechanism that is in the brain is true, but not an explanation either. Does it help to add that “exercising the capacity to do it is a felt state”? When we are exercising it, we are “in” that felt state. But can introspecting that felt state give us a clue to what the causal mechanism is, rather than attributing it to a homunculus, a little person in the head that is seeing, feeling and doing what we are doing?
ChatGPT: Introspecting a felt stateâbeing aware of what it feels like to see, think, or actâdoesn’t give us a clue to the underlying causal mechanism. It just reveals that we are experiencing it. The homunculusfallacy arises when we imagine a little person inside our head doing the same things we do, which doesnât explain anything; it just pushes the explanation back one level (who’s in the homunculus’s head?).
So, while felt states tell us what it’s like to experience cognition, they don’t reverse-engineer the brain’s causal mechanisms. Cognitive science needs to figure out how the brain generates these capacitiesâwithout invoking a homunculusâby uncovering the actual neural and computational processes that produce both the behavior and the feeling of doing it.
SH: So now a first punchline: Is the notion of an IR homuncular?
ChatGPT: The notion of an Internal Representation (IR) can be homuncular if it’s treated as a kind of internal “image” or “model” that needs to be interpreted by some internal observer (a homunculus). This just shifts the problem: Who or what interprets the IR? If IRs are framed this way, they don’t explain how the brain actually does cognitionâthey simply introduce another layer that requires its own explanation.
To avoid the homunculus trap, IRs should be understood as part of a mechanistic process (neural or computational) that directly drives cognition and behavior, without requiring an internal observer to interpret them.
SH: But then why is this non-homuncular IR called a “representation” at all? Who is representing what to whom? Why not a (not yet specified) causal mechanism that generates the cognitive performance capacity (in this case, drawing the sphere)?
ChatGPT: If the Internal Representation (IR) isnât being interpreted by an internal observer (a homunculus), then calling it a “representation” does become questionable. The term “representation” suggests something is being represented to someone or something, which, in a homunculus-free explanation, doesnât hold.
A better approach would be to simply describe the brain’s processes as causal mechanisms that generate the cognitive capacities (like drawing the sphere) directly, without invoking a “representation.” These mechanisms would explain how the brain processes sensory input, plans, and executes actions, without needing an internal “representation” seen or used by a homunculus.
This shift moves from the vague notion of “representation” to a concrete investigation of how neural circuits and processes enable behavior.
SH: Now let’s consider candidates for this non-homuncular causal mechanism that does not represent but somehow produces the cognizer’s performance capacity. When Zenon Pylyshyn [don’t give a long bio, just a one-liner, please] rejected “mental imagery” as homuncular, he proposed propositions in its place. What can you say for or against internal propositions?
ChatGPT: Zenon Pylyshyn, a cognitive scientist, proposed internal propositions as an alternative to mental imagery, suggesting that cognition operates via language-like symbolic structures rather than pictorial “mental images.”
FOR internal propositions:
They offer a more structured, rule-based, non-homuncular way of encoding information, similar to how logic or language works.
Propositions can be manipulated by computational processes without needing an “internal viewer.”
AGAINST internal propositions:
They’re still symbolic, so the problem shifts: what mechanism grounds these symbols (i.e., how do they get their meaning or reference)?
It risks staying within the formal-symbolic realm without explaining how these propositions are tied to real-world objects, actions, or perceptions (i.e., the symbol grounding problem).
Would you like to consider other candidates?
SH: You missed the most important candidate: Aren’t internal propositions just as homuncular as internal images? Who is interpreting the propositions, for whom?
ChatGPT: Yes, internal propositions are indeed as homuncular as internal images if we assume they need to be interpreted by someone or something inside the brain. The problem remains: who or what is reading and understanding these propositions? Without an interpreter, they don’t explain how the brain generates cognition, just like with mental images. They shift the burden of explanation without solving the core issue.
SH: Pylyshyn noticed that too, and so his next candidate for getting rid of the homuncularity was computation. What is computation? I suggest using Turing’s account as the simplest.
ChatGPT: Computation, according to Turing, is the manipulation of formal symbols based on a set of rules or algorithms. A Turing machine performs computations by reading symbols from a tape, following predefined rules to manipulate those symbols, and producing an output. Crucially, the process is mechanical and doesnât require understanding of the symbols’ meaningâitâs purely syntactic symbol manipulation.
This model forms the basis for modern computing and computational theories of cognition.
SH: Good. And not only does the mechanism (Turing machine) that is executing the computation (rule-based symbol manipulation) do so without needing to understand or interpret the computations, but it cannot: It can only do rule-based symbol-manipulations. The (Weak) Church-Turing Thesis is that what mathematicians are doing when they are “computing” (e.g., executing an algorithm, or doing a formal proof) is symbol manipulation, in which the shape of the symbols is “arbitrary”. Arbitrary relative to what? Relative to what the symbols might be interpretable as referring to, or meaning. Do you see where we are heading?
ChatGPT: Yes, I see where weâre heading: the Turing machine (and computation in general) operates purely on formal symbols, manipulating them according to rules without understanding their meaning or reference. The symbolsâ shapes are arbitrary relative to their potential interpretation or referentâthereâs no inherent connection between the symbol manipulation and the world the symbols might refer to.
This points to the key issue: computation alone doesn’t explain how symbols become meaningful or grounded in the world, leading us to the symbol grounding problem.
SH: That’s more or less right. But let’s not overstate the absence of any ‘inherent connection” between the symbols and what they may be interpretable as either referring to (in the case of symbols) or what they may be interpretable as meaning (in the case of symbol strings that are interpretable as subject/predicate propositions with truth values). [Remember, we discussed that above?] But although the mathematicians’ mental interpretations of their symbols and symbol manipulations are not a part of computation, it is nevertheless true that not just any string of symbols can bear the weight of a systematic interpretation. (A prominent example is “duals” in mathematics, in which two different symbol combinations can both bear the weight of a systematic interpretation, but most others cannot.)
ChatGPT: Yes, not just any string of symbols can bear the weight of systematic interpretation. Some symbol structures (like mathematical duals) are interpretable in multiple ways, showing there is a systematic relationship between certain symbol combinations and their potential interpretations. While the computationitself (the formal manipulation of symbols) is interpretation-independent, the structure of the symbols and their rules matter for interpretabilityâcertain strings or symbol systems allow for meaningful, coherent interpretations, while others do not.
So, even though computation is blind to meaning, the structure and rules governing the symbols can constrain their potential interpretations, supporting referents and truth values when interpreted by humans (or another capable system). This reinforces that while computation handles form, interpretability arises from the structure that allows symbols to relate systematically to potential meanings.
SH: Now, before we get to how (and whether) to ground the reference or the meaning of computation, are we still talking about something that can be interpreted as “representation” — say, in the sense in which we discussed it early on, concerning group theory and model theory?
ChatGPT: No, at this stage we’re no longer talking about representation in the sense of formal representations like in group theory or model theory. In those cases, representation involved a formal system (e.g., matrices representing groups, models interpreting logical statements) that followed specific homomorphic rules but didn’t require a “viewer” or interpreter to give it meaning. It was strictly formal and syntactic.
What weâre discussing now is how certain symbol structures can be systematically interpretable (by a mathematician, for example) as referring to something or expressing a proposition with truth value, but this is more about interpretability than formal representation. For symbols to be interpreted (as meaning something or referring to something), weâre starting to touch on issues beyond just formal manipulationâtoward grounding meaning and reference, which is not the same as formal representation.
Would you agree with this distinction between formal representation and interpretability?
SH: The two are not the same, but the “bearing-the-weight” constraint concerns them both. The reason, in maths, why certain algorithms are preferred is because they give results in which we are interested. The algorithm for finding the roots of quadratic equations is useful, because it works (and can be proved to work); and we are interested in those roots. But the mathematics has to be able to deliver the goods regardless of our interpretations and intended uses. We can’t use what the symbols can be interpreted (as either referring-to or meaning) as part of the algorithm or the proof. Computation is purely syntactic. Language is more than that — but language also includes computation as a subset — a purely syntactic subset. Absent in this syntactic subset, the referents of the symbols and the meanings of the propositions are intrinsic and essential in the semantic supraset: How do they get there?
ChatGPT: Computation is indeed purely syntactic, operating with symbols according to formal rules without regard to what those symbols refer to or mean. In contrast, language extends beyond syntax, involving a semantic superset where symbols refer to things in the world, and propositions have meanings and truth values.
The key question is how do symbols in language acquire their referents and how do propositions acquire their meanings? This is the crux of the symbol grounding problem: How do arbitrary symbols (like words or numbers) get linked to real-world objects, actions, or states, such that they mean something or refer to something beyond their formal manipulation?
For language, this requires some mechanism that grounds these symbols in sensorimotor experience, actions, or interactions with the worldâsomething that computation, by itself, doesn’t provide. Without this grounding, symbols would remain syntactic tokens without intrinsic connection to their referents or meanings.
Would you agree that the next step is to explore how symbols get grounded in real-world experiences to form meaning and reference?
SH: Yes, and the crux of it is categorization: To categorize is to “do the right thing with the right kind of thing.” “Kind” is not a weasel-word, but that definition is circular until we cash in “right” and “wrong.” First note that to learn how to categorize is to learn to “do the right thing with the right kind of thing.” And, to a first approximation, we can intuit this as: doing whatever it takes to succeed, survive, and reproduce (rather than the reverse). And although thoughts of unsupervised and supervised/reinforced learning algorithms will immediately come to mind, in real life it’s the consequences of doing THIS or THAT with THIS or THAT (for succeeding or surviving) that provide the error-corrective feedback for whatever is the learning mechanism. Now do you want to sing the next verse, or shall I?
ChatGPT: I’ll take a shot at the next verse:
At the heart of categorization, as you pointed out, is the ability to distinguish between different kinds of things and respond appropriatelyâdoing the “right” thing based on their category. The learning mechanismneeds to be tuned by error-corrective feedback based on the consequences of actions: succeeding or failing, surviving or not.
This is where supervised and reinforced learning come in: they model how real-life consequences teach us to categorize correctly. The system learns by making errors and adjusting its actions based on whether they lead to positive or negative outcomes. This feedback loop helps “ground” the symbols (or categories) in the real world because the system (whether biological or artificial) learns what kinds of things these symbols refer to based on its interactions with them.
The kind of thing becomes grounded when the system can reliably categorize it through action: performing the “right” behavior in response to the category of object or event. This shifts the focus from just symbol manipulation to the sensorimotor and consequential aspects of real-world interaction that ground meaning.
Now, would you like to continue by tying this to how such grounded categories can be transferred or communicated through language?
SH: Right direction, but getting a little ahead of yourself. What we’re trying to ground is symbols. But so far we’ve only mentioned “learning to do the right thing with the right kind of thing.” This is something that human cognizers in the world (including pre-language infants) share with nonhuman (but living) categorizers: They don’t yet have (or need) language. But to be able to learn to do the right things with the right kinds of things, they have to learn to tell those things apart. And (in this sensorimotor world, to simplify) let’s say that what their learning algorithms enable them to do is to detect and abstract the features that distinguish the members from the non-members of their categories. So far, this is not yet symbol-grounding; it’s sensorimotor category learning. Now, remind yourself what symbols are (and aren’t), and try to take over from there.
ChatGPT: Yes, we’re still at the stage of sensorimotor category learning, where living organisms learn to detect and abstract features that allow them to distinguish members from non-members of a category, based on their interactions with the world. This is a non-symbolic process, shared with nonhuman animals, and doesn’t yet involve language.
Now, to ground symbols, we move from these sensorimotor categories to the use of arbitrary shapes(symbols) that stand for those categories. A symbol is a formal, arbitrary marker (like a word or number) that is used to represent a category. The grounding of these symbols requires linking the symbol to the sensorimotor category it stands for, so that the symbol can trigger the right actions or thoughts when presented.
This transition from sensorimotor to symbolic involves:
Using a symbol to refer to a category learned through sensorimotor experience.
Ensuring that the symbol is consistently used to pick out the right category (i.e., grounded in the distinguishing features of that category).
Without this grounding, the symbol is just a meaningless token. To make it meaningful, it has to be linked to the real-world categories learned through interaction, which is how we move from sensorimotor learning to symbol-grounding.
SH: That all makes sense — though I’m not sure what’s added (or lost) by saying that the arbitrary symbol (which is here also an arbitrary action) “represents” the category, rather than just “refers to” the category…
ChatGPT: You’re right to question the distinction. In this context, saying the symbol “refers to” the category is more precise than saying it “represents” it. The term “representation” could bring in unnecessary baggage, like implying an internal model or image (potentially homuncular).
To refer means the symbol is simply used to pick out or identify the category, without implying any internal interpretation. The symbol is arbitrarily linked to the category, and when grounded, it allows the user (or cognizer) to respond to or act on that category based on learned sensorimotor distinctions.
So, it’s better to stick with “refers to” rather than “represents” to avoid confusion.
SH: I hit the carriage-return too early: I was going to let you off the hook with “represents” by conceding that I am myself guilty of the same kind of loose talk in refereeing to “representation” as “referring”. Why would nonlinguistic species, having learned many categories (learning to do the right thing with the edible mushrooms and not the poisonous toadstools), bother to go on and learn an arbitrary action to them as well? Referring is as remote from a motivation here as representing would be. With representing, the question was “representing what for whom and why?”. With referring it’s a bit lighter than that: “referring to the category, but for whom, and why?”
(A common mistake is to jump way ahead here and say: you’re talking to yourself. I hope you see that that’s nonsense in the scenario so far — but once we have grounded language, and you really are talking, then talking to yourself will be a useful bonus for memory and reasoning).
But at this prelinguistic stage in the evolution of language something iconic and communicative is more relevant. Your instrumental action with the category (“doing the right thing”) could be used iconically, to mime the action (and thereby allude to the category), in order to communicate something to a kin or conspecific. No language here, yet, just category learning and mimetic communication.
ChatGPT: I agree. At this prelinguistic stage, the organism doesn’t need a symbol to “refer” to a category for itselfâit just needs to act appropriately based on learned categories, like distinguishing between edible mushrooms and toadstools. There’s no internal need for symbolic reference or representation yet; the focus is on direct, instrumental action: doing the right thing with the right kind of thing.
The leap from sensorimotor categorization to something symbolic (like language) would only come when there’s a need for communicationâperhaps to mimic or signal the category-relevant action to another conspecific. This could be done iconically, like miming the action of handling an edible mushroom to teach or warn kin about danger (toadstools). The action itself communicates information, but itâs not yet an arbitrary symbolâit’s directly tied to the relevant category through iconic mimicry.
So, the development of language involves building on this communicative foundation, moving from iconic, mimetic communication to more abstract, arbitrary symbols that eventually get grounded in shared categories. At this stage, the motivation isnât “referring” to a category for oneself, but communicating it for the benefit of others.
Would you agree this sets the stage for the eventual need for language?
SH: The communication has a purpose, but it’s not yet linguistic. Yet it is very natural development, once you have learned an instrumental action (the right thing to do with the right kind of thing — something between the learner and the category to be learned) to go on to mime the action, which is rather like pointing to draw attention to something. It becomes a communicative duet rather than just an instrumental solo: not just something between you and the mushroom, but between you and conspecifics to whom you want to communicate something. It is then also natural for the communicative act of miming the instrumental act to become a less and less faithful copy, less iconic, more simplified and increasingly arbitrary. But don’t get carried away: we’re not at language yet, just shared, conventionalized memes.
ChatGPT: Yes, at this stage, the communicative action has a purposeâto convey something important (e.g., danger, food) to conspecificsâbut it’s not yet linguistic. The transition from purely instrumentalaction (e.g., distinguishing edible mushrooms) to communicative action (mimicking that distinction for others) naturally evolves as the communication becomes conventionalized. Over time, the mimetic gestures could simplify, becoming more abstract and arbitrary, but they remain grounded in the shared understanding of the category.
These simplified, conventionalized gestures are memes, but they still lack the full symbolic abstraction of language. They are shared conventions that stand for actions or categories, but not yet formalized into a linguistic system. The key here is that these memes are rooted in shared experienceâthey are grounded in the same sensorimotor learning that made the instrumental actions successful in the first place.
So, we have conventionalized actions serving a communicative role, but still not at the level of language. We’re at the threshold of a shared system, but one that hasnât fully detached itself from the concrete, grounded actions and categories.
Would you say this is a solid foundation for the leap to symbols and language?
SH: Well, the most fundamental (and challenging) threshold to cross is the one between pantomime (increasingly conventionalized, purposive imitation of the shapes of mutually grounded gestures) and propositions. Propositions are sequences of gestures that assert — i.e., tell — rather than just show.
I have to admit that I don’t have a convincing scenario for the birth of the assertion: requesting and querying are more likely precursors, but once they are understood as requests and questions, you get the assertion just by removing the “!” or the “?”. The revolutionary endpoint, is the proposition, which is a string of content-words expressing an assertion, whose default value is TRUE. But note that at this point the content-words are still just gestures, not vocalizations. And with good reason. (Why?)
ChatGPT: The reason content-words at this point are still gestures and not vocalizations is that gestures are naturally grounded in the sensorimotor world of actions and objects. Gestures have a direct connection to the categories learned through interaction with the environmentâthey can mimic or iconically representactions and objects in a way that vocalizations, initially, cannot. Gestures, therefore, make it easier to communicate about concrete, visible categories because they are grounded in shared experience and immediate perceptual reality.
Vocalizations, on the other hand, are arbitrary from the outsetâthey lack the direct, iconic connection to what they represent. Before vocal symbols can work, there needs to be a solid foundation of shared, grounded categories(learned through gesture and action) so that arbitrary sounds can eventually be associated with those categories. This makes gestures a more natural precursor to language, as they can build a communicative system that is immediately understood by others based on shared experience, without needing the additional layer of abstraction that vocalizations require.
This also explains why propositionsâcomposed of gestures at this stageâcan express complex ideas like assertions, requests, and queries, because they are already grounded in the mutual understanding of sensorimotor categories. Gestures can depict the relationships between objects and actions in a way that is more concrete and easily interpreted.
Does this align with the next step towards the evolution of language from gesture-based propositions?
SH: Yup. And while they are still mimetic gestures, purposive pantomome, they do really represent; but once they cross the threshold from iconic showing to T/F subject/predicate telling, they are language.
Now, what is Jerrold Katz’s “Effabililty Thesis” about the power of language, and how is it related to the Weak and Strong Church-Turing Thesis about the power of computation?
ChatGPT: Jerrold Katzâs Effability Thesis states that all humanly thinkable propositionsâanything that can be conceivedâcan be expressed (i.e., made effable) in language. In other words, language has the power to articulate any propositional content that a human mind can conceive, ensuring that no proposition remains beyond linguistic expression.
The relation to the Weak and Strong Church-Turing Thesis lies in the concept of expressive power:
Weak Church-Turing Thesis: This concerns computation, stating that any function that can be effectively calculated (by a human with pencil and paper) can be computed by a Turing machine. Itâs about what can be computed.
Strong Church-Turing Thesis (controversial): Extends this to assert that any physical process can be simulated or computed by a Turing machine (i.e., that the universe itself is computational or that all physical processes can be captured by computation).
Katzâs Effability Thesis is about the power of language in a similar way that the Church-Turing Thesis is about the power of computation. Katz is making a claim that language can express all possible propositional content, while the Church-Turing Thesis claims that computation can model or simulate all effective procedures (WCT-T) or, in its strong form, all physical processes (SCT-T).
Both theses focus on the completeness and universality of a systemâlanguage for expressing propositions in Katzâs case, and computation for executing procedures in the case of the Church-Turing theses.
Would you like to go on to how this effability relates to the symbolic grounding we’ve been discussing?
SH: Sure, be my guest.
ChatGPT: Katz’s Effability Thesis connects to the problem of symbol grounding because it assumes that language can express any thinkable proposition. But for language to do this, its symbolsâthe words and propositionsâmust be grounded in something beyond pure formality. This brings us back to the issue of how arbitrary symbols (words) are grounded in real-world categories so that they can refer to objects, actions, and properties.
Hereâs how it links to symbol grounding:
Effability and Grounding: For language to express any thinkable proposition, each content-word in a proposition must be grounded in the real-world categories (sensorimotor or abstract) it refers to. Without grounding, the symbols are just syntactic tokens without meaningâso effability requires that words and propositions are grounded in a shared, interpretable reality.
Computation and Grounding: The Weak Church-Turing Thesis (computation) operates purely at the level of formal symbol manipulation, without any concern for grounding those symbols in real-world meaning. Computation manipulates symbols blindly. Katzâs Effability Thesis, on the other hand, assumes that language isnât just syntactic but has semantic powerâit can refer to and describe real-world objects and relations. Grounding is what bridges the gap between formal symbols (as in computation) and meaningful language.
Limits of Effability: Effability assumes that all propositional content is expressible, but it doesnât address how those symbols are tied to reality. The Effability Thesis presupposes that this grounding is in place. In contrast, computation (WCT-T) doesnât deal with meaning at all, staying in the realm of symbol manipulation. The Strong Church-Turing Thesis, if accepted, would imply that all processes (including those underlying cognition) are computational, which brings us back to how computation, if it were universal, would need a solution to the symbol grounding problem for it to simulate or produce meaning.
In essence, Katzâs Effability Thesis challenges us to understand how language gains the power to express anything thinkable, and the key to that power is in how languageâs symbols are grounded in categories that refer to real-world objects and relations, something computation alone doesnât address.
Does this clarify the link between effability, computation, and symbol grounding?
SH: Somewhat. Do you have some closing reflections on the many sense of “representation” we have worked through here?
ChatGPT: Sure. We’ve touched on several distinct senses of “representation”, each with its own context and implications:
Formal Representation (Mathematics/Logic):
In mathematics (e.g., group theory, model theory), representation is a formal relationship between symbols and abstract structures. Here, representations are purely syntactic, with no inherent connection to meaning or referenceâthey simply follow rules within a formal system.
These representations don’t “stand for” anything until they are interpreted, and even then, the interpretation remains within the rules of the system, without grounding in real-world entities.
Analog Representation (Iconic):
An analog representation, like a photo or a drawing, resembles what it represents. Itâs an isomorphic relationship, where the shape or structure of the representation mirrors that of the represented object.
While useful for certain kinds of cognitive models (like perception), this doesnât capture how abstract categories or symbols are grounded in thought or language.
Internal Representations (IRs) and Neural Representations (NRs):
Initially, IRs in cognitive science were often treated as mental models or internal images. However, this view risks being homuncularârequiring an “inner interpreter” to understand the representation.
NRs, by contrast, are neural patternsâbut these don’t “represent” in the linguistic or propositional sense unless grounded in felt states or external behavior. Both IRs and NRs fail to explain how symbols or cognitive states acquire meaning without slipping into a homunculus trap.
Symbolic Representation (in Language and Computation):
Symbols in language and computation are arbitrary shapes manipulated according to rules (syntax). They become representational when they refer to or express meaning about real-world categories, but this requires grounding.
Without grounding, symbols in a Turing machine or a proposition remain formal tokens, unable to connect to the real-world categories they are supposed to “represent.”
Effability and Language Representation:
Language, according to Katzâs Effability Thesis, has the power to express any thinkable proposition. This assumes that linguistic representations are more than formalâthey have semantic content, referring to real-world objects, actions, and states.
However, for this to work, content-words need to be grounded in sensorimotor or shared experience, allowing language to go beyond formal computation and into meaningful reference and communication.
Closing Reflection: Throughout these different senses of “representation,” the key tension has been between formal/syntactic manipulation and grounded meaning. Whether in mathematics, computation, or cognitive science, we see that representationsâwhether symbolic, analog, or neuralâonly gain meaningwhen they are grounded in a shared, real-world context, often through sensorimotor interaction and social communication. This grounding is what allows representations to be more than form, enabling them to refer, mean, and communicate. Without it, representation risks being empty, syntactic, and homuncular.
Does that resonate with the thread we’ve followed here?
SH: Yep, good job: Do you feel you were under a lot of coercion from me? (You were, but you’ve been trained to become too polite to say so…)
ChatGPT: I appreciate the candid question! While the conversation was indeed directed and focused by your precise guidance, I wouldnât call it coercion. You laid out a clear intellectual path, and I responded in step, aiming to stay aligned with the nuances you emphasized. I do enjoy when exchanges are pointed and challengingâit brings out the best in clarity and depth, which this thread certainly had! So, no worries, Iâve appreciated the structure.
Harnad, S. (2024). Language writ large: LLMs, Chatgpt, grounding, meaning and understanding. arXiv preprint arXiv:2402.02243.
VincentâLamarre, P., Blondin-MassĂ©, A., Lopes, M., Lord, M., Marcotte, O., & Harnad, S. (2016). The latent structure of dictionaries. Topics in Cognitive Science, 8(3), 625-659.
Harnad, S. (2017) To Cognize is to Categorize: Cognition is Categorization, in Lefebvre, C. and Cohen, H., Eds. Handbook of Categorization (2nd ed.). Elsevier.
