This dialogue discusses some possible objections to meaning through isomorphism for abstractions. It follows a prior dialogue that delved deeper into the counter-claim that understanding of meaning requires more than symbols and syntax and algorithms, which is the point of the Chinese Room argument.
Soulum: When we last met, you alleged that the reduction of meaning to isomorphism is sufficient for the abstract parts of human mental life. Some examples of these parts are logic and mathematics, ethics and religion, introspection and philosophy. How can meaning of that level of abstraction arise from trifling "isomorphisms" with human experience?
Isolder: Would you admit that the words, and in some cases formalized symbols, that communicate abstract meanings have definitions? A definition is an excellent specimen of isomorphism. It expresses relations between the defined word and other words. Since the listener then knows which parts of his or her knowledge match up to the words in the definition, the listener can apply the corresponding relations to those parts of knowledge and thereby match up the defined word to another part of his or her knowledge. And at that point the definition has succeeded in its aim of enlarging the language-knowledge isomorphism. As I tried to say before, all that's necessary is that the isomorphisms be cumulative. The level of abstraction is incidental.
Soulum: Aren't you mistaking a symbol's definition in words for the symbol's actual and original meaning? It's undeniable that a symbol's definition is often communicated in terms of other symbols, but you're not addressing the fundamental difference of abstract meaning, which is the abstract symbol's creation. Non-abstract symbol creation is nothing more in theory than slapping a name on something in plain view. Making new combinations of existing abstractions is not much harder; tell me two numbers and I can mechanically produce a third. But from where do these numbers come? What is the "isomorphism" for the number one? The abstractions I speak of are more or less essential to human lifestyles, so the abstractions must be meaningful and the meaning must go beyond mere definitions. Your answer "Anything with a definition therefore has an isomorphism-based meaning" is far too broad for my taste. I could play with words all day long, assigning new definitions right and left, but my newly-defined words could still be self-evidently fictional and useless.
Isolder: Of course. I asked about definitions in order to show that abstract meaning is communicated via isomorphisms. That doesn't imply that every possible definition communicates a valid isomorphism to reality. Nor does it imply that abstractions are created by purely chaotic symbol combinations. Rather, a human forms a new abstraction by isomorphic mental operations: comparisons, extensions, generalizations, analyses. Each of these mental operations might result or not in a new abstraction of significant meaning. Isomorphisms also function as the verification of the new abstraction's significant meaning.
Soulum: It's abundantly clear that you have a one-track mind and its track is isomorphism. Instead of verbally treating every issue as a nail for your isomorphism hammer, why not put it to a specific test by returning to my question about the isomorphisms for the number one?
Isolder: Fine. The number one is isomorphic to a generalization of human experiences, in the category of quantities. In their lives, probably motivated by social interactions such as trades, humans developed a generalized way to specify quantities. It would've been greatly inefficient and bothersome to use individual symbols for "one stone", "two stones", "three stones". And once the quantity symbol had been broken apart from the object symbol, it would've been similarly tedious to use individualized sets of quantity symbols for each object symbol; hence the symbol for "one" stone could be reused as the symbol for "one" feather. The mental isomorphism of quantity between these situations thus became a linguistic isomorphism between the situations. The number one is abstract and mathematicians throughout the centuries have assigned it logically rigorous definitions in terms of other numbers and functions, but its isomorphic connection to real scenarios ensures that its meaning is much more relevant than the free wordplay you mentioned a moment ago.
Soulum: You're describing hypothetical historical events, and you give the impression that the number one is dependent on language usage. I believe that you continue to be purposely obtuse about the essential difference of my point of view. While one is useful in many contexts, its existence is independent and verifiable without reference to fleeting bits of matter. Humans discovered one. Isn't it an astounding coincidence that so many cultures happened to include symbols for one?
Isolder: Is it also an astounding coincidence that so many cultures happened to include symbols for any other concept? You may as well be impressed by the preponderance of circular wheels. We shouldn't be surprised by the independent invention of useful abstractions; think of how many times separate mathematicians have simultaneously reached the same conclusion but stated it using distinct words and symbols that are isomorphic to one another. Moreover, note the history of the number zero, the number "e", the number "i". Cultures got along fine without these luxuries for centuries (although practical "pi" has been around for a while). The pace of mathematical invention sped up when it started to become an end in itself. Yet even "pure" mathematical work is somewhat mythical. Mathematics has always been motivated, whether to prove abstruse conjectures or solve engineering problems. One of the spurs for growth was the need for more sophisticated tools just to describe scientific discoveries. Can a student know physics without knowing calculus, tensors, and vector spaces? You could say that a physics student with a "conceptual" understanding only has approximate mental isomorphisms for the movements of reality.
Soulum: Again, the specific human history of the abstractions is a digression. No matter the history, no matter the degree of usefulness, no matter the isomorphic resemblance to anything whatsoever, these abstractions are provably true forever in an intellectually appealing fashion. A quintessential blank-slate hermit could conceive of it all; some artists and writers and mathematicians and moralists have in fact created their masterpieces in almost-total isolation. Any time someone communicates one of these abstractions to me, I don't need to "just accept it" in the same way I must when someone communicates the result of an experiment that I don't have the resources to duplicate. I can prove or disprove the abstraction using the capabilities of my mind, guided perhaps by the sort of finely-honed intuition that escapes artificial imitation.
Isolder: The all-knowing hermit you speak of certainly has supernatural intelligence and creativity, not to mention long life! Setting the hermit aside, I readily acknowledge that individuals can figure out the validity of abstractions for themselves. And you may groan when I assert how they do it: isomorphism. The processes of logic and rigorous proof are repeated forms of isomorphism. "All men are mortal" is obviously a relation between the two ideas "all men" and "mortal". "Aristotle is a man" matches "Aristotle" to "all men". If the match is a true isomorphism then the relation is preserved and "Aristotle is mortal". However, I seriously question the amount of emphasis you place on personal testing of abstractions. Wouldn't you concede that for the most part humans really do "just accept it" in the case of abstractions? A teacher can flatly preach that zero multiplied by another number is always zero, or the teacher can list the other multiples of a number to show that according to the pattern the zeroth multiple "should" be zero. What is the likelihood of the teacher using "algebraic ring structure" to prove that, due to distributivity, the product of the additive identity and any other element is the additive identity?
Soulum: I don't maintain that all abstractions are taught with proofs but only that whoever wishes to perform a check on the abstraction will always get the same answer. Call the written justifications "isomorphisms" if you insist, but in any case the abstractions have sublime logical certainty.
Isolder: Indeed, some abstract questions always get the same answer, and some others like the "purpose of existence" seldom do. I'd go so far as to say that the long-term success of an abstraction stems directly from its degrees of determinism, verifiability, and self-consistency. Without these properties, the feeling of sublime logical certainty isn't worth a lot. Whenever someone learns of an abstraction, the abstraction should come with an isomorphism-based checking/consistency procedure to assure that the abstraction is applied correctly. In short, once the abstraction is set loose from isomorphisms with reality, its meaningfulness can only reside in isomorphisms to yet other abstractions. It's no accident that there's one "right" answer for many abstractions; chances are, the abstractions were designed to be that way. The best abstractions contain few gaps between the internal isomorphisms.
Soulum: I disagree. "Best" is subjective and depends on the particular abstraction's goal. Subtlety and difficulty, interpretation and symbolism, are hallmarks of good fiction, for instance. Where are the isomorphisms there?
Isolder: The isomorphisms are whatever the fiction draws out of its audience. Besides ambiguity, good fiction has relatability, which is why outside of its intended audience it can be unpopular or mystifying. An isomorphism doesn't need to be verbalized to be active. Emotional "resonance" is in that category. I'd argue that some of the most behaviorally-powerful isomorphisms are either genetic or unintentionally imprinted from one generation of a culture to the next. Given the level of instinctive reactions seen in animals, ranging all the way from fish to primates, the presence of passionate and involuntary impulses in humans is an evolutionary cliché. Human brain complexity can eclipse the simplicity of the impulses, but humans continue to sense the primitive isomorphisms behind those impulses. For example, facial expressions and body movements are hugely important to the overall effect of a conversational "connection" because the isomorphisms are largely subconscious.
Soulum: To define "isomorphism" that broadly is a few steps too far, in my opinion. It'd be futile to advance other counterexamples, which you would duly treat as fuel for further contrived isomorphisms. It seems that I can't convince you that the soul fathoms meanings too richly intricate for your crude isomorphism-engine.
Isolder: And I can't convince you that, no matter how deep a perceived meaning is, isomorphism is ultimately the manner in which matter exhibits it. No soul is required, but simply a highly-suitable medium for adaptable isomorphisms such as the human brain.