On Asymmetric Systems
The Strange Loops of Ideology
“Let me tell you why you’re here. You’re here because you know something. What you know you can’t explain, but you feel it. You’ve felt it your entire life, that there’s something wrong with the world. You don’t know what it is, but it’s there, like a splinter in your mind, driving you mad. It is this feeling that has brought you to me. Do you know what I’m talking about?”
— Morpheus, The Matrix
Conventional wisdom often tells us that perfection is unattainable. Common adages, such as “you can’t have your cake and eat it too”, seem to indicate a fundamental impossibility of a flawless totality — this impossibility, moreover, seems to be a byproduct not of mere low probability, but rather of logical necessity.
Though “conventional wisdom”, of course, often turns out to be dead wrong, I believe we may be onto something in this case. In what follows, I will attempt to show how great thinkers from modern mathematician Kurt Gödel to contemporary philosopher Slavoj Žižek interrogated this topic via the analysis of systems, and particularly what I like to call asymmetric systems.
Asymmetric systems are what I call systems that are their own undoing, whose consistency and completeness cannot both be assured due to their very nature. These are unavoidable systems, systems that are deeply embedded into our lives and give them meaning. We will soon analyze these systems in great depth, but not first without proper context.
If “conventional wisdom” already tells us that there is a fundamental impossibility of this flawless totality, if it is the case that everyone already knows this, why write this article? Because, in the words of philosopher Hegel from The Phenomenology of Mind, “what is ‘familiarly known’ is not properly known, just for the reason that it is ‘familiar’” (Hegel, 17).
(That being said, this is a heavy read. If you want the gist, check out my conclusion.)
My goal, now, is to help supplement what you already know, to see this “familiar knowledge” in a new light.
Systems
The concept of a “system” pervades nearly every field, from “Biological Systems” to “Systems Programming”. While these are wonderful systems, and perhaps in some way relevant to the topic at hand, these are not the sorts of systems I will draw your attention to.
I would like to focus on two types of systems: formal and ideological.
Formal Systems
When I speak of formal systems, I mean systems of logical or mathematical nature that define (1) axioms and (2) rules of inference, by which (3) theorems are produced. Axioms are fundamental, unquestioned truths, and theorems are valid statements derived from the axioms, according to the rules of inference. One such formal system is outlined in the Principia Mathematica, written in the early 20th century by mathematicians Alfred North Whitehead and Bertrand Russell. In the words of Douglas Hofstadter from his seminal work Gödel, Escher, Bach: An Eternal Golden Braid, the Principia Mathematica set out to “derive all of mathematics from logic… without contradictions” (Hofstadter, 31), a noble goal. In an ironic twist of fate, however, the Principia prompted Kurt Gödel to prove that a formal system cannot be both complete and consistent, thereby destroying Whitehead’s and Russell’s dream forever.
Before we discuss what this means and why this matters, I would like to touch upon Turing machines, and how they give us another perspective on the same issue. Turing machines, invented by Alan Turing, are essentially another depiction of a formal system, with axioms, rules of inference, and theorems. In a broad sense, a Turing machine is an abstract vision of a computer, and it can be shown that its behavior simulates that of formal systems¹. The general concept is as follows:
- A formal system has axioms; a Turing machine has input.
- A formal system has rules of inference; a Turing machine has transition rules.
- A formal system has theorems; a Turing machine has output.
But of course, why do we care what the properties of a Turing machine are? In general, we find that Turing machines can give us insight on computation. Of course, Turing machines and real computers are not one and the same, and they do not work the same way. However, we find a direct correspondence in what sorts of problems are computable in real computers and in Turing machines — anything a real computer can compute, a Turing machine can compute as well. In other words, anything a Turing machine cannot compute, a real computer cannot compute. Thus, we find that the nature of computation is closely linked to formal systems, so when we find limitations of formal systems, we should also find these limitations in computation.
Ideological Systems
Now, when I speak of ideological systems, I refer more to what is generally called “ideology”. I call them “systems” simply to highlight what I believe is an isomorphism between ideological systems and formal systems. Informally, when I refer to “ideology” or “ideological systems”, I intend to convey the sense of a system of beliefs and ideals with regard to politics, economics, and policy.
The way Hofstadter, Gödel, and Turing look at formal systems is very similar to how Slovenian philosopher Slavoj Žižek and philosopher-economist Karl Marx view ideological systems. Both types of systems aim to solve the question of Truth. With formal systems, we seek to prove or disprove certain propositions, and to find new theorems which may gain practical application in the sciences or engineering. With ideological systems, we seek to dictate what is moral, what the correct political structure is, and so on — to influence the public’s perception of the social reality.
But both types of systems are vulnerable in that they are fundamentally flawed, that the application of these systems in a particular manner ensures their undoing. Just as a formal system cannot be both complete and consistent, an ideological system cannot be applied both universally and consistently. Žižek, via psychoanalyst Jacques Lacan, calls these rifts symptoms. Hofstadter calls them strange loops.
Strange Loops
So what, then, is a strange loop? This, even Hofstadter, who coined the term, has difficulty explaining. His primary method of instruction is through the use of examples. In an attempt to give a more direct definition in his 2007 book I Am A Strange Loop, he gives us this:
What I mean by “strange loop” is — here goes a first stab, anyway — not a physical circuit but an abstract loop in which, in the series of stages that constitute the cycling-around, there is a shift from one level of abstraction (or structure) to another, which feels like an upwards movement in a hierarchy, and yet somehow the successive “upward” shifts turn out to give rise to a closed cycle. That is, despite one’s sense of departing ever further from one’s origin, one winds up, to one’s shock, exactly where one had started out. In short, a strange loop is a paradoxical level-crossing feedback loop.
To dissect this definition, let us take an example of a strange loop, and interpret the definition with this example.
A classic form of a strange loop is the Epimenides Paradox, posed by the ancient Cretan philosopher Epimenides: “all Cretans are liars”. This is another form of what is called a liar paradox, which can be expressed as the statement:
This sentence is false.
What is it that makes this sentence strange, and what is it that makes it loopy?
The “loop” part of “strange loop” comes from self-reference: the liar paradox makes a statement about the liar paradox, which makes a statement about the liar paradox, and so on. However, I contend, there are other sentences that have these loops without being “strange” in any way, such as:
This sentence has five words.
There is no paradox here, just a true statement. Paradox is the source of a strange loop’s strangeness, but not just any sort of paradox. The liar paradox involves a series of steps, each perfectly valid in its own right. However, when evaluating the series of steps as a whole, we find the paradox, that “despite one’s sense of departing ever further from one’s origin, one winds up, to one’s shock, exactly where one had started out.” What are these steps? Denoting the liar paradox by the letter L:
- Initially reading L as true, we find “L is false”
- Now that L has been evaluated as false, we find “L is true”
- Now reading L as true, we again find “L is false”
And so on. This is what is meant by Hofstadter’s “paradoxical level-crossing feedback loop.” This is what is meant by the term “strange loop”.
Beyond Epimenides, we also find strange loops in the work of Kurt Gödel, who turned the world of mathematics on its head with his incompleteness theorems. In his 1931 paper, entitled On Formally Undecidable Propositions of Principia Mathematica and Related Systems, he outlined his First Incompleteness Theorem and his Second Incompleteness Theorem. We will focus on the first theorem, which, as summarized by The Stanford Encyclopedia of Philosophy, states:
Any consistent formal system F within which a certain amount of elementary arithmetic can be carried out is incomplete; i.e., there are statements of the language of F which can neither be proved nor disproved in F.
Here is the general idea behind the proof of this theorem: Gödel constructs a “sentence” in F, called G(F), that mathematically encodes the statement: “this sentence is not provable in F.” Sounds awfully like the liar paradox, doesn’t it? Theoretical computer scientist Scott Aaronson explains the implications of the construction of G(F) in a blog post:
If F proves G(F), then F proves both that F proves G(F) and that F doesn’t prove G(F), so F is inconsistent (and hence also unsound). Meanwhile, if F proves Not(G(F)), then it “believes” there’s a proof of G(F). So either that proof exists (in which case it would render F inconsistent, by the previous argument), or else it doesn’t exist (in which case F is unsound). The conclusion is that, assuming F is powerful enough to express sentences like G(F) in the first place, it can’t be both sound and complete (that is, it can’t prove all and only the true arithmetical statements)².
So here we have the strange loop, encoded in the statement G(F), that shows us that any formal system strong enough to express G(F) must be “imperfect” — that it cannot prove all and only all truths, as we (and Whitehead and Russell) had wanted from them! Assuming a formal system that can prove any statement to be true or false (i.e. a complete system), G(F) is the kernel buried within the system that dissolves its consistency, both proven true and proven false. Alternatively, if we have a formal system that is consistent, G(F) is “the truth which transcends theoremhood” (Hofstadter, 94), the truth that we will never be able to prove nor disprove. It is clear, then, that we find strange loops even at the heart of mathematics itself.
Recall that I have mentioned that Turing machines and formal systems express the same exact idea, and that what is not computable by a Turing machine will also not be computable by a real computer of any form. So might there be an analog of Gödel’s Incompleteness Theorems for computation? Yes — it is the unsolvability of the halting problem. So what is the halting problem, and why is it unsolvable?
The problem is as follows:
Write a program HALT(P, x) that returns true if program P halts when given input x (in other words if it is possible to compute P(x) in finite time), and false otherwise, for any P and x.
The alternative to P halting on x, of course, is for it to run forever.
Turing actually manages to prove that writing HALT(P, x) is logically impossible: for what if we wrote a program called TURING³ which first called HALT(P, x) with itself as input, and then halted if HALT(P, x) returned false, and ran forever if HALT(P, x) returned true?
We have found the instance where the program HALT, turned upon itself, ceaselessly contradicts itself. This is again a strange loop, eerily similar to the liar paradox!
This is another indication, then, that formal systems are very closely related to computation; this link may be understood at its core as the types of problems that plague both systems: strange loops.
Now, on to ideological systems. Slovenian philosopher Slavoj Žižek has focused much of his work on the critical analysis of ideology, particularly in his book The Sublime Object of Ideology. Here, he proceeds first by explaining the notion, first introduced by the psychoanalyst Jacques Lacan, that “Marx invented the symptom”. This conclusion is similar to that reached by Gödel and Turing, but now with regard to ideological systems rather than formal ones. Žižek defines “the symptom” in psychoanalytic terms:
“A formation whose very consistency implies a certain non-knowledge on the part of the subject” (Žižek, 16).
As we have seen with Hofstadter’s definition of the strange loop, this definition will best be explained with a canonical example. To establish such an example, Žižek focuses on the Marxian analysis of the commodity and its exchange, via philosopher-economist Alfred Sohn-Rethel:
“During the act of exchange, individuals proceed as if the commodity is not submitted to physical, material exchanges; as if it is excluded from the natural cycle of generation and corruption; although on the level of their ‘consciousness’ they ‘know very well’ that this is not the case.” (Žižek, 12)
There is then, some misrecognition at play in the act of exchanging commodities, in believing that the “value”, or “monetary worth” of a commodity is a property of the commodity itself, rather than the product of a social network of relations. Of course, we all still know that “value” is not an inherent property of a commodity, but we act, at an unconscious level, as if we were not aware of this.
Even physical currency itself, Žižek notes, exhibits this behavior. He again cites Sohn-Rethel:
“A coin has it stamped upon its body that it is to serve as a means of exchange and not as an object of use. Its weight and metallic purity are guaranteed by the issuing authority so that, if by the wear and tear of circulation it has lost in weight, full replacement is provided. Its physical matter has visibly become a mere carrier of its social function.” (Žižek, 13)
This is all very well, you may remark, but why is the act of commodity-exchange so important? What does this have to do with “symptoms”, with “ideology”, or with “strange loops”? As it turns out, the commodity provides us an insight into what Žižek calls “the fundamental question of the theory of knowledge”:
“Objective knowledge with universal validity — how is this possible?” (Žižek, 10)
It does so by providing us a definition of the unconscious: that which drives this misrecognition of the social dimension of the act of exchange. And it does so by providing us with a case-study of ideology at its finest.
We have noted that, in the case of commodity-exchange, “the proprietor partaking in the act of exchange… overlooks the universal, socio-synthetic dimension of [the] act, reducing it to a casual encounter of atomized individuals in the market”(Žižek, 15). However, there is something more sinister in this misrecognition, something which makes the commodity-exchange into something more general, into an “ideology”. Sohn-Rethel, and by extension Žižek, argues that this “overlooking” is not only a property of the act of exchange, but is also necessary for the act of exchange to even exist.
Normally, “the consciousness of [the exchange agent] is taken up with their business and with the empirical appearance of things which pertain to their use” (Žižek, 15). What were to happen, though, if it were the case that an agent realized the full social dimension of their exchange? Sohn-Rethel argues that their action would no longer be ‘exchange’; it would be something else, and the social dimension that springs out of the act of exchange would no longer arise. In short, then, “this non-knowledge of the reality is part of its very essence”: “if we come to ‘know too much’, to pierce the true functioning of social reality, this reality would dissolve itself” (Žižek, 15).
This, then, is what we can call ideological: it is “the social effectivity, the very reproduction of which implies that the individuals ‘do not know what they are doing,’” i.e. the social “being itself in so far as it is supported by ‘false consciousness’” (Žižek, 16). It is now clear that there is a link between what Marx calls “ideology” and Žižek’s “symptom”, which we had defined earlier as that “whose very consistency implies a certain non-knowledge on the part of the subject.”
We now have found how Marx “invented the symptom” of Lacan: both men identified “a certain fissure, an asymmetry, a certain ‘pathological’ imbalance which belies… universalism” (Žižek, 16). Moreover, this asymmetry is not something that can be rectified, “an insufficiency to be abolished by further development”: it is this ‘non-knowledge’ on the part of the participants which ensures the ontological consistency of the social reality, which gives the ideology its existence. In general, what we have uncovered is a “certain logic of exception: every ideological Universal — for example freedom, equality — is ‘false’ in so far as it necessarily includes a specific case which breaks its unity, lays open its falsity” (Žižek, 16).
This principle, now that we have conceived it in its abstract form, can now be clarified with some concrete examples. Take Žižek’s example of freedom. Freedom is, as he explains:
“A universal notion comprising a number of species… but also, by means of a structural necessity, a specific freedom (that of the worker to sell freely his own labor on the market) which subverts this universal notion. That is to say, this freedom is the very opposite of effective freedom: by selling his labour ‘freely’, the worker loses his freedom — the real content of this free act of sale is the worker’s enslavement to capital.” (Žižek, 16–17)
When the “ideological Universal” of freedom is enacted, we see that one particular component of this Universal, the freedom of the market, forces a sort of non-freedom, because now individuals are forced to sell their labor for wages in order to survive. It is in this way that a particular freedom, a symptom, subverts the Universal notion of freedom.
There are more examples of this phenomenon beyond the Marxian fascination with commodities and labor: take any principle that can be extended universally to create an ideological system. Take, for example, the concept of “tolerance”, which I take to be one of the ruling cornerstones of today’s Left, especially in America⁴.
The proposed “goal” of tolerance is to treat all as equal, regardless of background, appearance, or behavior. What then, when there are individuals who jeopardize tolerance itself, via sexism, homophobia, racism, and so on? The Universal of tolerance is stuck in a dilemma: in order to maintain its universality, the solution is to be “intolerant” of this tolerance, to not tolerate this sort of intolerant behavior. However, in maintaining its universality, tolerance undoes its own consistency, in that it no longer is accepting of all. This then, is the symptom of tolerance, that which lays bare its inconsistency: what of tolerating intolerance?
This dilemma is often referred to as the paradox of tolerance, engaging thinkers such as Karl Popper and John Rawls. In reasoning about the paradox, we find ourselves doing a certain traversal of different levels, only to find ourselves back to where we started, in a perpetual state of contradiction, similar to what we found with the liar paradox, with the Gödel sentence, with Turing’s analysis of the Halting problem. This, then, is the strange loop of the ideological Universal, “the paradox of a being which can reproduce itself only in so far as it is misrecognized and overlooked” (Žižek, 25).
Thus is the link between ideological systems and formal systems: the instance that lays bare the paradox of the system, the impossibility of the system’s totality, the strange loop, the symptom. Moreover, this connection is not simply one between auxiliary properties of the two types of systems: both the strange loop and the symptom are essential for their corresponding systems to exist, they are both built-in to the system itself, they are both unavoidable, by the very nature of the systems they inhabit.
And Now, Perhaps, Some Reflection
Through an analysis of the work of Kurt Gödel, Alan Turing, and Slavoj Žižek, along with many others on the way, I have attempted to show the limitations of our systems of knowledge, particularly in formal systems and ideological systems. There is of course, and always will be, more evidence to support this. But rather than discuss further evidence, I think it is time to talk more about ramifications.
There seems to be a sort of “wall” that we hit when constructing systems of knowledge: they cannot be all-encompassing while still maintaining logical consistency. This restriction is exposed by a certain juxtaposition of symmetry of form and asymmetry of content: the ever-self-referential strange loop that contains a paradox hidden between its levels of abstraction, the symptom at the heart of an ideological Universal (it is no coincidence that Žižek described the symptom as “an asymmetry”). This, then, is why I like to group ideological and formal systems under the term asymmetric systems. Hofstadter agrees that this is a key result of his analysis of Gödel’s Incompleteness Theorem and Turing’s Halting Theorem:
“All have the flavor of some ancient fairy tale which warns you that ‘To seek self-knowledge is to embark on a journey which … will always be incomplete, cannot be charted on any map, will never halt, cannot be described.’” (Hofstadter, 692)
We see a similar structure with Escher’s Drawing Hands (the lithograph at the beginning of this article): the visual symmetry — of both hands simultaneously drawing each other, bringing each other into existence — belies the logical asymmetry, the clear impossibility of the situation. This is often the case with Escher’s artwork.
I believe this is precisely what makes his artwork so compelling, that the juxtaposition of symmetry of form and asymmetry of content so closely mirrors our systems of knowledge, representing the incessant human desire for symmetry confronted with the realization of its impossibility. It is clear, then, that in life, we are attracted to symmetry, not only on a visual level, but on a cognitive one as well.
It is this juxtaposition, this confrontation, that follows us throughout our lives without us knowing it, as it is with Neo in The Matrix. The film fulfills our fantasy by finding the source of his “something-is-not-quite-right”, by explaining away these asymmetries (such as déjà vu or the physical limitations of human ability) not as inherent properties of the world itself but rather as properties of a sort of “false perception” — Morpheus provides us a method to “escape” this, to “jump outside” the system, to reveal that the “real world” within the Matrix is void of this asymmetry, and through this to envision a Universal without its symptom, which we know is impossible but for which we always still strive.
This, as it happens, is Žižek’s definition of utopia:
“In short, ‘utopian’ conveys a belief in the possibility of a universality without its symptom, without the point of exception functioning as its internal negation.” (Žižek, 18)
This is where he also criticizes Marx, arguing that even the “universalization of the production of commodities… brings about a symptom,” that “the Marxian perspective [of] utopian socialism” is impossible (Žižek, 18). Beyond Marx, many great thinkers have fallen victim to this desire for an impossible symmetry, such as Russell and Whitehead with the Principia Mathematica.
But the consequences for blindly chasing this utopia could be much worse, as Žižek explains: “the greatest mass murders and holocausts have always been perpetrated in the name of man as harmonious being, of a New Man without antagonistic tension” (Žižek, xxviii). There seems to be harm, then, in obsessively fixating upon on the resolution of our world’s inherent asymmetry, for it can drive one mad.
One other man who had this sort of vision of a utopian resolution for humanity was Hegel, who outlined a procedure of dialectics, a system of iterative progress towards the final stage of humanity, characterized by a rational state of free and equal citizens and culminating in a full and total ‘absolute knowledge’. Žižek provides a radical interpretation of Hegel, which in his own words “runs counter to the accepted notion of ‘absolute knowledge’ as a monster of conceptual totality devouring every contingency”:
“Dialectics is for Hegel a systematic notation of the failure of such attempts [to resolve the fundamental asymmetry] — ‘absolute knowledge’ denotes a subjective position which finally accepts ‘contradiction’ as an internal condition of every identity.” (Žižek, xxix)
Contradiction as an internal condition of every identity.
‘Absolute knowledge’ as an embrace of this condition.
Before concluding, I would like to return one more time to Hofstadter’s Gödel, Escher, Bach. I have only thus far presented to you a piece of it; considered a seminal book in the field of artificial intelligence, much of the book attempts to explain, to some degree, where consciousness comes from, or “how it is that animate being can come out of inanimate matter.” Hofstadter’s answer? Strange loops.
Much like strange loops are unavoidable in formal systems, so is the concept of “self”:
“What is so weird in this is that the formal systems where these skeletal ‘selves’ come to exist are built out of nothing but meaningless symbols… Meaning comes in despite one’s best efforts to keep symbols meaningless!” (Hofstadter, P-3)
As a firm believer in the “AI thesis”, Hofstadter argues that “as the intelligence of machines evolves, its underlying mechanisms will gradually converge to the mechanisms underlying human intelligence” (Hofstadter, 574). In other words, the brain can be modeled as a formal system, and “every aspect of thinking can be viewed as a high-level description of a system which, on a low level, is governed by simple, even formal, rules” (Hofstadter, 553).
This notion is radical in that it sees no issue in calling formal systems “conscious”. Of course, the validity of this is heavily debated, but let us take it as true for the sake of speculation. Might it be, then, that since ideological systems — afflicted by their own form of strange loops, symptoms— are homologous to formal systems, as we have learned here, that they too are conscious? That systems of belief, products of human consciousness, are the first form of man-made “artificial” intelligence?
I think yes, and I think that this concept can help us understand the source of modern-day “viral” trends, and even the rise of Donald Trump. But that discussion will have to be for another time.
Footnotes
1. ^ The definition of a Turing machine is as follows:
Consider an infinite memory tape, made up of discrete “cells”, each of which can hold a symbol such as a digit or a letter. The “head” of the machine can read the symbol at one given cell at a time, and according to the symbol that is read, can write a new symbol in this current cell if it so chooses, and then move either one cell to the left or to the right. This process repeats itself until the machine determines it is time to halt. The machine also maintains a “state” of the operation, which may also change between repetitions. The Turing machine is given a starting position for the head and a starting state, along with a finite table of instructions, that given a state and a symbol read at the head, will determine:
- What new symbol will be written,
- Whether to move the head left or right,
- What the new state will be, and
- Whether the program should halt
Now that we’ve defined a Turing machine, what’s the point? First, let’s confirm that the machine is isomorphic to a formal system as I asserted earlier — its axioms are the initial configuration (state, head position, and symbols on the tape) of the machine; its rules of inference are determined by its table of instructions; and its theorems are the reachable configurations of the machine. Thus, when we speak of properties of formal systems, we should at least have the intuition that Turing machines have these properties as well.
2. ^ While Aaronson talks about the soundness and completeness of formal systems, you may have noticed that earlier in this article I discussed Gödel in terms of “completeness and consistency”. The distinction between what Aaronson calls “soundness” what I call “consistency” is a subtle and technical one, though somewhat serious. If you are concerned or intrigued by this distinction, feel free to read Aaronson’s blog post, and rest assured a method called Rosser’s trick resolves this discrepancy.
3. ^ Take the case that the program HALT(P, x) exists. Let us write another program called TURING(P), that is as follows:
Run HALT(P, P), in other words, determine whether the program P halts with itself as its input. If HALT returns true, TURING runs forever (i.e. through some infinite loop). If HALT returns false, TURING terminates immediately.
Now consider the case where we call TURING(TURING). We start by calling HALT(TURING, TURING). There are two cases:
If TURING(TURING) halts, HALT(TURING, TURING) will return true. In turn TURING(TURING) now runs forever, since the result of HALT(P, P) where P = TURING was true. On the other hand, if TURING(TURING) does not halt, HALT(TURING, TURING) returns false, and now TURING(TURING) halts.
Now, the contradiction is apparent. If TURING(TURING) halts, then TURING(TURING) runs forever. If TURING(TURING) runs forever, then TURING(TURING) halts. Therefore, it cannot be the case that HALT(P, x) exists!
4. ^ I’d like to note that the actual issue of tolerance in today’s Left is a much more nuanced issue: I have simplified it here to illustrate my point. However, Žižek’s principle is that the you cannot rid an ideology of the symptom, even through use of exceptions or nuance. Therefore, a full critique of today’s Left could be made (though it would be heavily complicated by the fact that there is no clear consensus on what the explicit tenets of “the Left” are).
