Sunday, May 23, 2010

meaning through isomorphism

Moreover, Gödel's construction revealed in a crystal-clear way that the line between "direct" and "indirect" self-reference (indeed, between direct and indirect reference, and that's even more important!) is completely blurry, because his construction pinpoints the essential role played by isomorphism (another name for coding) in the establishment of reference and meaning. Gödel's work is, to me, the most beautiful possible demonstration of how meaning emerges from and only from isomorphism, and of how any notion of "direct" meaning (i.e., codeless meaning) is incoherent. In brief, it shows that semantics is an emergent quality of complex syntax, which harks back to my earlier remark in the Post Scriptum to Chapter 1, namely: "Content is fancy form."
prelude and the problem

Months ago I finally finished reading Douglas Hofstadter's Metamagical Themas. Since it's a collection of columns about varied topics, I don't plan to comment on it with the same level of enthusiasm that I applied to I am a Strange Loop (if I were, I would've gotten around to it much sooner!). But one of its recurring ideas, also raised in Hofstadter's other books, is so fruitful that I can't resist rambling about it at excessive length: meaning through isomorphism. I'd further claim that its importance rivals and complements that of self-reference, which usually has the spotlight in commentary about Hofstadter's ideas.

The universal philosophical issue or "problem" of meaning is easily explained. It's undeniable that people experience meanings and that a meaning is a relation. One chunk of stuff "means" another chunk of stuff; noun _____ represents, defines, symbolizes, or analogizes noun _______. But how can people reconcile or harmonize this experience of meaning with a universe that, according to the best means of detection and reason, consists of pieces of matter whose interactions are consistently indifferent to relations of meaning? Does/Can meaning really exist? Assuming it does, then what is meaning, where is meaning, and how does meaning originate? I'll work my way back to this later.


The preceding questions have many proposed answers. I'm convinced that Hofstadter's description of meaning through isomorphism is a pretty good one. A mathematical isomorphism has a rigorous logical definition, but in the looser sense intended here, an isomorphism is simply matching one or more parts of one aggregate with parts of another aggregate such that one or more relations between the parts of one aggregate remain valid between the matched parts in the other aggregate. In a word, relations in an isomorphism are "preserved". (In passing, note the circular definitions that an "aggregate" is a collection of "parts" and "parts" are anything in an "aggregate" collection.)

To take an elementary example, if one aggregate is the set of numbers (4, 8, 15) and the other aggregate is the set of numbers (16, 23, 42), then an isomorphism that preserves the relation "greater-than" could match 4 to 16 and 8 to 23 and 15 to 42 because 8 is greater than 4 and 23 is greater than 16, 15 is greater than 8 and 42 is greater than 23, etc. (Naturally, the relation of "subtraction" is NOT preserved since 8 - 4 = 4 and 23 - 16 = 7.)

At first glance, this may seem like dodging the question of meaning. Why should anyone care that, through a greater-than isomorphism, (4,8,15) "means" (16,23,42)? Well, that depends on the situation. Hypothetically, if someone's purpose involved the greater-than relation and he or she could more easily manipulate numbers less than 16, then that person could work on (4,8,15) and use the isomorphism to apply the results to (16,23,42). Imagine the depressing story of a pitiful calculating device that can only store its numbers with 4 bits each but its task is to find the maximum of 16, 23, 42. Still too trivial? Then ponder a comparable isomorphism between number sequences: taking the sequence 0..255, matching 0..127 to itself, and matching 128..255 to the sequence of -128..-1 . Now go read about the computer storage of signed integers to find out why this comparable isomorphism isn't a toy example at all.

Thus the basic idea is straightforward but its implications are surprisingly wide-ranging as shown by Hofstadter in his more mind-bending sections. His exemplar is the isomorphism in the incompleteness theorems between numbers and the symbols of a formal logic system, although he returns time and again to descriptions of the human capability for analogy, whether in the contexts of translation or recognition or self-image or creativity. A common thread is the logically-strange tendency to transcend by self-reference, which goes by labels like "quote", "break out", "go meta", "use-mention distinction".

applied to computers

However, apart from complicated self-reference, Hofstadter admits in his serious AI speculations that mere meaning through isomorphism remains simultaneously effortless for people yet bewildering to computer science. People can figure out an original isomorphism that works and then not rely on it beyond suitable limits, but a program can't. Whatever are the underlying mechanisms that originally evolved in the human brain for reaching the "correct" answer in a complex environment, no artificial program has quite the same ultra-flexible Frankenstein mix of (what appear from the outside to be) random walks, data mining,  feedback, decentralization, simulation, and judgment. Given its mental acts, we shouldn't be shocked by the sheer quantity of lobes and connections and structure in the brain, and maybe its operation is less a monolithic program than an entire computer packed with programs interrupting one another to get a chance at directing the body's motor functions.

On the other hand, this conspicuous lack of an AI for pragmatic isomorphisms is all too familiar to application programmers like myself. Our job is to fill the gap by the routine imposition of meaning through isomorphism. That is, we try to create a combination of algorithms and data that's an isomorphism for a specific human problem, like running payroll for a company. In a similar fashion, the total computing system is a stack of increasingly-complicated isomorphisms of the application programmer's work. As Hofstadter writes in his columns on Lisp (and properly-educated programmers know), the top-level program is compiled into an isomorphic representation, then the next level down does its own compilation, and so forth, until the original program is in the executable form appropriate for the relevant non-virtual hardware. The towering edifice is an impressive illustration of the exponential power of isomorphisms to capture and translate meaning into, ultimately, an electrical flow.

(Some technically-minded readers may be questioning which "relations" are preserved by these isomorphisms. After all, one or more parts of the "pipeline" could possibly optimize the original code in any number of ways like function in-lining, variable removal, tail calls... In this case, the relations are abstract but are in the category "the effect of the original":  order of statements that are dependent, values of constants including user-visible strings, access details of I/O performed. When relations like these in the original code aren't preserved, the pieces lower in the stack thereby fail to carry out actual isomorphisms that would express the meaning.)

applied to information theory, communications, art

Of course, meaning through isomorphism isn't the only theoretical framework around for understanding computer processing as a whole. The same claim could be made for information theory, which is throughly successful and in use every day. Fortunately, the two are compatible. Say that the communication channel's sender and receiver each have aggregates, and the message symbolically encoded and sent over the channel is an isomorphism between their aggregates. So then the symbols of the channel's message indicate each match from one aggregate's part to the other aggregate's part. Before the first symbol, the receiver is at maximum uncertainty or entropy about the isomorphism. After the first symbol and each symbol thereafter, the receiver can use knowledge of 1) the matches communicated previously, 2) its own aggregate, and 3) its own aggregate's relations between parts to make increasingly likely guesses about the remaining matches (or correct randomly sent errors on a noisy channel). In accordance with information theory, good "entropy coding" for this message would send no more bits to the receiver than are required in order for the receiver's knowledge of aggregates and relations to infer the rest of the isomorphism. The isomorphisms processed in a computer system allow for lower information entropy and therefore greater compression. The most interesting portions of a codec have the responsibility of using relations among parts of the media stream to trash or reconstruct some parts of the aggregate stream.

Given the compatibility between meaning through isomorphism and information theory, it's unsurprising that communication in general is perhaps its most natural manifestation. A language is an aggregate of spoken or written words. The universe is an aggregate of pure experience (I dare you to be more vague!). Hence a worded message is an isomorphism between language and the universe. Rather, the message is an aggregate of words that were purposefully selected in order to communicate, via the language-universe isomorphism, an aggregate of thoughts about pure experience. The relations preserved by this isomorphism are countless and endlessly varied. In the message "The quick brown fox jumps over the lazy dog", "brown" is an adjective to "fox" so the indicated fox must have the color brown in depictions, "jumps" is a verb to the subject "fox" so the indicated fox must be in the midst of a jumping action, "over" is a preposition connecting the phrase "The quick brown fox jumps" to the phrase "the lazy dog" so the indicated fox must be of higher elevation than the indicated dog. Part of the reason why computer parsing of raw human language is stymied is due to a computer's lack of a human's uncannily deep well of experiences to fuel a feedback loop between the comprehension of syntax and semantics. In practice, the nuanced syntax of sentence structure, word forms, and connectives nevertheless results in highly ambiguous statements that require worldly knowledge and/or context to disentangle.

How exactly people can decode their own words is a fiendish and glorious enigma that's convinced many speculators to tie language to the essence of humanity. Those who presume a soul frequently equate it to the explicitly verbal segments of intelligence (e.g. "confess with your lips"). It's definitely a truism that, of all earthly creatures, people have the most developed and subtle languages. They can organize their mental and social lives to levels of complexity that contrast with the simplicity of their materials. All their creations in language, art, and other domains can have symbolic depth.

applied to the distinction between form and content

These feats of abstractive composition and interpretation lead to a commonsense division between a work's "surface form" and its "meaningful content". For instance, a surface form of red is said to represent the inner content of the artist's aggression, and a milestone of any artistic genre is the point at which critics begin to tirelessly accuse its artists of betraying the genre's pioneers by producing "mindless copies" that mimic style (form) but without substance (content).

Finally I return to one of the points in the opening quote. The practical classification of an expression's qualities into form and content just described is contradicted by Hofstadter's pithy motto "content is fancy form". In his words, "[...]'content' is just a shorthand way of saying 'form as perceived by a very fancy apparatus capable of making complex and subtle distinctions and abstractions and connections to prior concepts' ". Elsewhere he writes that form and content are on a "continuum", and the gap between is "muddy". Form and content are simply the same expression evaluated on different strata or in different domains. Syntax and semantics are different in degree, not in kind. If someone isn't taken aback by this proposition then he or she might not grasp its import.

I've found it instructive to employ this perspective on John Searle's thought-provoking Chinese room argument. There's a person, completely ignorant of written Chinese, in a closed room armed with only the materials necessary to 1) receive a message in Chinese, 2) manually execute the algorithm of a program that can pass a Turing Test in Chinese, 3) send the reply generated back out of the room. Assume, as in any Turing Test, that the judge outside the room who sends the original message and receives the reply can only rely on the messages to determine whether the unknown correspondent knows Chinese. Since the algorithm in the Chinese room can pass the Turing Test by producing highly-convincing replies, isn't it true that 1) based on the replies the judge will conclude that the person in the Chinese room understands the messages and 2) the judge's conclusion is in direct contradiction to the person's actual level of knowledge of Chinese? If you grant these two points, then the Turing Test criterion for "understanding" fails to find the right answer for the person in the Chinese room. Now change the role of the Chinese room inhabitant from a person+algorithm into a computer program executing the same algorithm. Remember that the person in the Chinese room passed the Turing Test by "shuffling symbols" that had no meaning to him or her. Is it any more reasonable to think that a program that passes a Turing Test is doing anything more than "shuffling Chinese symbols" like the person in the Chinese room? The upshot of the argument is that no matter how well a message's form of linguistic symbols is processed, its content or meaning could still be unknown to the processor; understanding of form does not imply understanding of content so content cannot be form.

As I see it, the meaning-through-isomorphism interpretation leads to a disturbing viewpoint on the Chinese room argument (reminiscent of how the EPR "paradox" led to disturbing but theoretically-consistent results for quantum mechanics). I'm restricted to a single clue to deduce who in the Chinese room argument really knows the meaning/content/semantics of the message: the location of the aggregates to which the message's symbols are isomorphic. The argument postulates upfront that the person doesn't know the Chinese language. In other words, he or she doesn't have any information about the aggregate or relations of the Chinese language, but the message's symbols are parts of that unknown aggregate. Clearly no isomorphism happens in the person and so none of the intended meaning is there. I can agree that the person's lack of the necessary aggregate makes him or her clueless about the messages. But that leaves one possibility: the algorithm is the thing that understands the meaning. In the Chinese room, or indeed in any scenario alike to a Turing Test, in order for the algorithm to form convincing replies at all it must be able to decode the meaning, and to decode the meaning it must have the necessary aggregates to complete the isomorphism. Based on the usual opinion that understanding demonstrates intelligence, for the purposes of the Chinese room, the algorithm is more intelligent than the person. From the standpoint of the message's meaning, the person's participation in the communication is akin to functioning as the algorithm's tool, channel, or messenger (according the customary literary allusion, the algorithm is the person's "Cyrano de Bergerac"). When meaning occurs through isomorphism, there's no logical contradiction. The judge's guess that the person within knows Chinese is nothing more than an honest mistake. Don't blame the messenger for a passing Turing Test.

applied to philosophy and the brain

I called the idea of an intelligent algorithm "disturbing", but the delegation of various "intelligent" tasks isn't novel. Mathematical calculations were one of the first to be handed over to algorithms and devices. Then there's the long list of recommendations for sundry occurrences (do this when there's a fire, do that when someone needs resuscitation). And how much of the typical job is reducible to either rote actions or following new orders whose rationale is unknown?

The disquieting aspect of the imaginary Turing Test algorithm is the unprecedented usurping of the noblest of intellectual pursuits, understanding meaning. A traditional philosopher might declare that to be human is to understand and, furthermore, understanding is an accomplishment that can't be performed by anything else. Rocks don't understand. Plants don't understand. Animals don't understand but many are trainable. In contrast, humans experience a detailed "meaning mirror" of the universe that's symbolized in their languages. The "meaning mirror" has the name "mind" or "soul". Humans can understand the meaning of an expression by its effect on the "mind". In summary, meanings are ethereal inhabitants of minds, and only humans have minds.

Such a traditional explanation is appealing (to some people especially so) but it's complicated because it grants "first-class co-existence" to a purely mental/non-physical world. By making access to the cognitive world a special privilege of humans, it's dispiriting to the prospect of AI ever arising. It's also messy because people tend to embellish the details of the non-physical world in a multitude of opposing directions. It conflicts with the normal and productive method of the sciences, which is to find physical causes for phenomena. It outlines the existence of meanings but at numerous costs.

Dropping the traditional philosopher's explanation leaves the philosophical question of meaning unanswered...unless isomorphism is introduced in its place. Isomorphism furnishes a plausible intellectual underpinning for meaning in a solely materialistic universe. An isomorphism requires only materialistic ingredients for its elements: parts, aggregates, relations, matches.

What materials? The choices are everywhere, as boundless as creativity. Earlier, one set of materials was the software and hardware components of a computer system. In his writings Hofstadter has used DNA as a sterling example whose meaning is the proteins transcribed from it. And in keeping with the section on communication, any usable information channel is a candidate for meaning through isomorphism.

For philosophical concerns, the more relevant set of materials for isomorphism is the human brain. If we're to give up believing in our non-physical realities, we should reasonably expect a competitive brain-based theory of our mental capacities and qualia. I expect the brain's networks to be effective materials for isomorphisms. The combined excitatory and inhibitory network connections seem like prime building blocks for exquisite parts, aggregates, relations, and matches. Connections in general are implicitly essential to the definition of an isomorphism. An aggregate is parts that are connected, a relation is a connection between parts, a match is a connection between parts in separate aggregates.

One can then concede the, for lack of a better name, mind/matter isomorphism: the physical layout of the brain's network is directly responsible for all "non-physical" thoughts we feel. I don't suggest that there's a "grandmother neuron" but that the numerous neurons and even more numerous neuronal junctions, in response to the onset of the relevant stimuli, effect a mental experience of grandmother, whatever that may be. I neither suggest that one brain's network layout of isomorphisms resembles a second brain's except on a gross regional level; the variance among individuals in immediate word-association responses certainly makes a closer resemblance doubtful. I do suggest that, with sufficient prior knowledge about the isomorphism between the specific brain's network and its environment, a "scan" of the merely anatomical changes associated with the formation of a new long-term memory would enable the scanner to know with some certainty what the memory was "about". (I'm skeptical that anyone could figure out a workable procedure for it. Brain interfaces are getting better all the time, but the goal is to clumsily train the brain and the interface to work together, not to accurately read the brain's ephemeral signals.)

applied to objectivity

The proposed isomorphism between a person's brain and his or her encounters with reality puts not only the philosophical categories of "mind" and "matter" in a new light but also the categories of "objective" and "subjective" meaning. Objectively, whenever a scientist examines eroded canyons with sedimentary rock walls, maybe unearthing fossilized water-dwellers, he or she can assert the ancient history of the river that flowed there long ago. The river left traces so the river can be factually inferred. Also objectively, given the full network of a brain (and many secondary clues?), the brain's memories could be deduced in theory. The thoughts of the brain left traces so the thoughts can be factually inferred. At the time people would've called the brain's thoughts subjective, but with enough work the thoughts might as well be called objective.

Obviously the isomorphism in a human brain is incredibly dense and interwoven, which causes the complexity of the undertaking to have the magnitude of a perfectly accurate measurement and forecast of all the weather in North America. It's too hasty to proclaim it impossible, though. People have managed to translate hieroglyphics and break the Enigma code. The elusiveness of the "right" isomorphism doesn't disqualify it from discovery (well, not counting some exceptions like 1) the perfect elusiveness of "isomorphisms", like one-time pads, that by design have "matches" but preserve precisely zero relations, and 2) degradation/corruption of the aggregate's material medium).

Having seemingly decided that the derivation of meaning through isomorphism places it into the "objective" category, one could be forgiven for attempting to additionally place it into the "unambiguous and undebatable" category. In people's regular conversations, the two often go hand-in-hand; objective facts are brought in to silence the clamor of unfounded opinions. Not so for isomorphisms. In fact, an isomorphism's objective existence is why it can't have any inherent authority or precedence over other isomorphisms. Surely people can agree on any number and flavor of criteria for the selection of an isomorphism, but there's no physical coercion. (I could elaborate on how the personal selection of isomorphisms is supportive of pragmatic philosophy but I won't...)   

x + y = 815. I'm writing a message to send in a bottle, but in what language? I see a person holding two fingers against his forehead and I take for granted that he has a headache. You mention a common acquaintance by the name "Alex" and I conclude you're speaking of "Alexander" not "Alexandra". "Ambiguity" is more or less shorthand for too many isomorphism candidates to pick.

In a much more formal context, computer scientists have established ever-growing sets of problems that are proven to be solvable by the same kind of algorithm. When a fast algorithm solves any problem in the set, it could attack the rest, too. On an algorithmic basis, the problems are isomorphic. A computer scientist searching for a generalized solution to the set of problems doesn't need to "prefer" one to the rest. (He or she need not be too depressed. In most applications a "good enough" or "partial" solution is adequate.)

The "relativity" (non-preference) of objective isomorphisms is mind-blowing to me. It turns the world "inside-out". The Earth is not the center of the physical universe. Neither am I the center of the universe of meaning. After a thunderclap, the impact sets air in motion. The sound wave is one set of molecules jiggling, then the next, then the next. By moving similarly, i.e. isomorphically, aren't the air molecules transmitting the "meaning" of the thunderclap? Eventually, the air in my ear canals moves after being pushed in turn. The movement of the drum corresponds to an isomorphic shift in electrical impulses (yeah, I know I'm simplifying it). The nerve cells isomorphically react and in so doing continue to pass along the "meaning" of the thunderclap into the brain that I like to refer to as "mine". The connections in that brain isomorphically mirror my stored memories, spilling over into my language centers. "I hear thunder." In the relative terms of all these isomorphisms, who's to argue that I'm the origin of my spoken thunder message? But I may not be the terminus, either. What if my statement motivates the people around me to leap into action? Aren't their actions isomorphic to my statement? My message has made their actions meaningful. It started with a thunderclap.

1 comment:

  1. What ho. I challenge you to a blog-off. I have been meaning to write a post about isomorphism since I started writing about six months ago.

    The gist (Hofstadter word!) of what I am going to say is this:

    Through isomorphism, we can link abstract mathematics to concrete things in the world. And that's why math is relevant.

    I don't know exactly what I mean by a blog-off, but since you're clearly interested in the same stuff, maybe you can read and respond to the post when I write it.

    Oh, and unsolicited advice: have you thought about splitting this post up?