2 Deviation Games

HI vs. AI: Thought, Desire, and the Deviation Game

Aware of the complexities of dealing with “thinking” in proposing his famous test “the Turing test,” concerning digital computers’ capacity for intelligent behavior, Alan Turing conceived of this test as an “imitation game,” without assuming that digital computers think in the way humans do (Turing 1950). In considering whether digital AI could in principle think and learn like humans, this memo proposes a new concept, that of “deviation game,” an unprogrammable deviation game. (Hereafter a deviation game will refer to an unprogrammable deviation game. Programmable deviation games are always based on imitation games.) This concept is, I suggest, fundamental to human thinking or intelligence (HI), and, hence, crucial in considering the capacity or incapacity of AI’s algorithmic information processing for reaching human level (HLAI). Human intelligence, I conjecture, is never reducible to algorithms or programs, even in mathematical thinking, which, in its creative aspects, has been defined by deviation games. I also shall relate deviation games to Gilles Deleuze and Félix Guattari’s (DG) concept of thought, which designates a creative process within thinking responsible for art, philosophy, and mathematics and science. Thought, I argue, enables deviation games, and is also essential in considering the question of learning, including mathematics learning, and the role of AI therein.

I consider thought in conjunction with the idea of desire, as “thought-desire,” by taking up and juxtaposing two concepts of desire extending from psychoanalytic theory. The first, that of Sigmund Freud and Jacques Lacan, defines desire statically by the structure of desire that is, in effect, captured in the imitation game; and the second, adopted here, that of DG’s concept of desire, defined, dynamically, through schizophrenic desire and “desiring machines.” Freud and Lacan, as well as DG, saw the unconscious as thinking [Gedanken] and most thinking as unconscious (Lacan 1981). My hypothesis (I claim no more) is that deviation games emerge in human thinking and learning because there is no thinking without desire, and without dreams, as effects of desire. If an AI device, digital or other, cannot dream, it cannot think.

Accordingly, (1) acknowledging remarkable achievements of digital technology, one should not confuse the workings of programming, no matter how sophisticated, with thinking, and (2) if computers or any AI devices are not capable of desire and, thus, cannot dream, they cannot think in the way humans do, who are capable of thought and unprogrammable deviation games enabled by thought. One can extend this problematic to other forms of AI, such as a biologically based one, sometimes represented by “androids” in science fiction. My argument also poses the question as to whether it is possible to create artificial beings that could do all that human beings can without in fact being human, which would include having emotions, desires, or dreams, and thus the unconscious. Consciousness, a commonly debated subject concerning AI, belongs to this list as well. The subject will, however, be put aside, apart from noting a manifest difficulty of assuming that digital AIs are capable of consciousness or self-consciousness in view of the overall argument given here.

A clarification is in order. By “the imitation game,” Turing referred to the overall capacity of digital AIs to imitate all possible effects of HI, without an actual mimesis of the specific manner of HI’s thinking which leads to these effects. Nevertheless, the concepts of imitation and deviation, as considered here, invade Turing’s argument. They do so because of the individuality or even singularity of thinking and thought, the singularity essentially linked to the unconscious and manifested in our capacity to dream. It is this singularity that enables thought’s counter-mimetic deviation game, which leads to unprogrammable deviational effects, such as the invention of new mathematical concepts. The same effect can be produced by a different deviation game, as some new mathematical concepts were, or reproduced through learning and thus partial and partially (but only partially) programmable mimesis of the same process. These facts, however, do not diminish the essential role of a deviation game in the workings of thought. The question is the potential capacity of AIs to create unprogrammable deviational effects, which is, I conjecture, impossible, unless AIs acquire the unconscious and become shaped by its effects, such as dreams. This, more fundamental, question also grounds the question of learning and the role of AI in it, as defined by the relationships and conflicts between imitation and deviation games, insofar as our systems of learning tend to suppress human individuality and especially singularity. This point as such is not new. This memo, however, explores deeper reasons for this situation by using the concepts of thought, desire, and deviation games, which, together, enable a creative resistance learning. This is, I argue, what digital AIs lack. The situation becomes more intriguing when one considers how human education—good or bad—seems to resist deviation games.

Chaos, Thought, and Probability

In What Is Philosophy?, DG propose the concept of “thought” versus merely thinking (in the sense of mental states), as a confrontation between the brain and chaos. They argue that art, science, and philosophy are different, if interrelated, primary creative modes of thought: such creative thought works differently in each, by creating new concepts in philosophy, new compositions in art, and new formal and propositional structures in mathematics and science. Human thinking is capable of thought, and that which is not capable of thought cannot reach the level of human thinking.

DG define chaos “not so much by its disorder but by the infinite speed with which every form taking shape in it vanishes. It is a void that is not a nothingness but a virtual, containing all possible particles and drawing out all possible forms, which spring up only to disappear immediately, without consistency or reference, without consequence” (Deleuze and Guattari 1994, 118). This concept of chaos is unusual. As the invocation of “particles” indicates, it is derived from quantum theory, which also connects it to randomness and probability in their radical form, explained further in this memo. Disorder, implying randomness, is still partly a DG concept, even if (“not so much”) secondary to the virtual. The virtual is a form of the real, which ultimately escapes thought and never becomes actual, an alternative form of the real. The virtual is also the efficacy of desire in DG’s sense, as against that of Freud or Lacan. The latter see this efficacy as defined by the permanence of a structure, such as the Oedipal (triangular) structure, rather than the dynamism of desiring machines. Moving with the infinite mental speed or rather velocity, as a vector defining the direction and speed of thought, the virtual gives rise to thought and enables it to create the actual. Desiring machines are not conventional technological devices. It is possible to read DG’s concept as extending beyond human desiring machines, but for the present purposes, I assume that the concept only applies to humans and possibly some animals.

While chaos is a grand enemy of thought, it is also a great friend of thought, even its greatest friend, and its most important ally in its “more profound” struggle, that against opinion, doxa, an enemy only, which is “like a sort of ‘umbrella’ that protects us from chaos . . . [as] the misfortune of people comes from opinion” (Deleuze and Guattari 1994, 206). Programming us, opinions are akin to algorithms, although both must be originally created by thought. While capable of generating opinions, the brain is ultimately a desiring machine of thought, a multiplicity of such machines, which “plunge us into chaos” in art, philosophy, and mathematics and science. The spectrum of the intelligence-like effects produced by digital systems corresponds to a miniscule portion of those produced by the human brain. Thus, it is not clear how capable computers are or could be of dealing with randomness and of probabilistic reasoning, let alone of acts that are random or contingent, defined by the interplay of randomness and determinability. Probabilistic reasoning, too, only deals with this interplay, because purely random events cannot be estimated even probabilistically. Both are essential capacities of the human brain (and possibly some animal brains). AI, thus far, has not involved contingent, only programmed, acts and has not dealt with contingency, or only insofar as certain apparently contingent events are simulated by determinate processes, and thus are, at bottom, not contingent. The contingency and probability at stake in (unprogrammable) deviation games have, in parallel with quantum physics, to do with events that cannot be entirely determined by the past, even if unknown to us. This lack of knowledge would make this contingency merely a practical, epistemological matter. By contrast, the quantum-like, or even more radical, contingency of thought is fundamental, ontological, and hence unprogrammable. It requires a different form of estimating and, when possible, mathematically calculating probabilities. A computer can be programmed to calculate such probabilities, for example, in quantum theory, but these calculations are not themselves contingent.

If there is no opening toward an unprogrammable deviation game, which entails this kind of contingency and probability, there is no creative thinking, in any domain, but only a proliferation of opinion and forms of programming according to preestablished rules, which are essentially equivalent to, or are generative of, opinions. Although I mean the term “programming” in a broader sense, I do imply that the recent development of and claims for such AIs, as, now ubiquitous, ChatGPT and its avatars, have nothing to do with thinking, let alone thought, but are merely forms of programming, which are products of and an invitation to imitation games, or limited, programmable form of deviation within imitation. They propagate opinion-like knowledge, suppressing deviation games of thought. Even if such programs use entities derived from the deviation-game thinking gathered from internet archives, they convert them to preestablished information or rigid articulations of ideas. Their use in mathematics is a separate issue. An intriguing recent discussion concerning such attempts, and yet difficulties, of approximated mathematical thinking processes by AI is offered in Romero-Paredes et al. (2024). In any event, this use does not appear to seriously challenge my point concerning the difference between programming and creative thinking, enabled by deviation games.

Two Concepts of Desire and Lines of Flight of Thought

Structures of desire, such as that of the triangular Oedipal desire, introduced by Freud, are more rigid. It took Lacan, and the influence of (Bourbaki’s) structuralism in mathematics, to think of desire in terms of structure, while also, against his own (and Freud’s) grain, opening the way to the idea of desiring machines, which transform structures. The Oedipal desire is still a product of desiring machines of a special “neurotic” type, that reproduce and enforce the Oedipal desire, and operate against creative, deviational desiring machines which transform desire from one point of fulfillment to another. DG call these machines “schizophrenic,” although “multi-trajectorial” would be a better term. The Oedipal structure of desire is defined by the infinite distance from a fulfillment, the distance ensured by the Oedipal machine in the familial economy and, in part through it, social and political economies. Once Oedipal machines are in charge, there is no transformation of desire along trajectories of life, only compensatory (substituting) mechanisms for the lack of fulfillment. By contrast, schizophrenic desiring machines are unique to each of us: They define unique, deviational, trajectories of movement according to changing patterns, rather than only by mimetic variations on the same pattern, as do Oedipal machines. As conglomerates of schizophrenic desiring machines, we do not imitate; we deviate, contingently and sometimes just randomly. We create new trajectories of thought-desire, “lines of flight,” and change them unless we surrender to Oedipal or Oedipal-like machines that make us follow the same structures of desires, opinions, or preprogrammed views.

There is always an unfulfillment of a desire, but in two essentially different ways. In the case of Oedipal desire, this unfulfillment is defined by the infinite distance from fulfillment, imposed by the structure of Oedipal desire; in the case of schizophrenic desire by the fact that “a point” of fulfillment is a new trajectory of desire to be pursued by a desiring machine. Only death stops this pursuit. A creative thought is always in flight: It cannot be contained by an algorithm or determinable set of algorithms, although it might use algorithms or programs in its movement.

Deviation Games and the Pedagogy of Thought

Any new mathematics is a creation of thought (thought-desire) by means of one or another, greater or smaller, deviation game. If one adopts this view, mathematical education becomes a pedagogy of thought, in this case what DG refer to as “the pedagogy of concepts” in their sense, in which an introduction of a concept always entails learning of this concept by other humans (Deleuze and Guattari 1994, 112). They reserve this concept of concept primarily for philosophy, but it is, I would argue, extendable, along with the pedagogy of concepts, to mathematics and science (Plotnitsky 2022). Each concept is a unique invention of thought, rather than a generalization from known particulars, a reductive abstraction more often taken to characterize the process of making concepts. This pedagogy does not exclude algorithms or calculations, or imitation games, but is not based on them. DG juxtapose the concept as a creation of thought to the concept as conceived in commercial professional training, shaped, I would argue, by algorithmic learning and the imitation game (Deleuze and Guattari 1994, 12). Accordingly, huge corporate investments into AI and AI-based training are not surprising, where the learning plays imitation games rather than deviation games.

Professional training, which extends far beyond commercial spaces, and merely capitalizes (in either sense), is an important subject in the present context, including in the context of mathematical education, not the least at the high-level where mathematical invention should be cultivated. I would like to address this subject in closing, via remarks by Mikhail Gromov, one of the greatest contemporary mathematicians, on Niels Henrik Abel (1802–1829), most famous for giving the first complete proof of the impossibility of solving the general quintic equations in radicals and other momentous contributions to analysis. Gromov sees Abel as “a major figure, if not the major figure, in changing the course of mathematics from what could be visualized and immediately experienced to the next level, a level of deeper and more fundamental structure” (Raussen and Skau 2010, 401). Abel changed mathematical thought. According to Gromov:

[Abel] changed the perspective on how we ask questions. . . . It is obvious that the work of Abel and his way of thinking about spaces and functions has changed mathematics. . . . [T]he concept of underlying symmetries of structures comes very much from his work. We still follow that development. . . . . This continued with Galois theory and in the development of Lie group theory, . . . and, in modern times, it was done at a higher level, in particular by Grothendieck. This will continue, and we have to go through all that to see where it brings us before we go on to the next stage. It is the basis of all we do now in mathematics. (Raussen and Skau 2010, 401)

Abel’s mathematics emerged in the deviation game, which led him to discovering symmetries across different forms of difference, setting structures into motion. In responding to the question concerning mathematical education and education in general, Gromov gives an unexpected answer, by using Abel as an example:

Raussen and Skau: Education is apparently a key factor. You have earlier expressed your distress about realizing that the minds of gifted youths are not developed effectively enough. Any ideas about how education should change to get better adapted to very different minds?

Gromov: Again I think you have to study it. There are no absolutes. Look at the number of people like Abel who were born two hundred years ago. Now there are no more Abels. On the other hand, the number of educated people has grown tremendously. It means that they have not been educated properly because where are those people like Abel? It means that they have been destroyed. The education destroys these potential geniuses—we do not have them! This means that education does not serve this particular function. The crucial point is that you have to treat everybody in a different way. That is not happening today. We don’t have more great people now than we had one hundred, two hundred, or five hundred years ago, starting from the Renaissance, in spite of a much larger population. This is probably due to education. . . .

Raussen and Skau: You point out that we don’t have anybody of Abel’s stature today, or at least very few of them. Is that because we, in our educational system, are not clever enough to take care of those who are exceptionally gifted because they may have strange ideas, remote from mainstream?

Gromov: The question of education is not obvious. There are some experiments on animals that indicate that the way you teach an animal is not the way you think it happens. The learning mechanism of the brain is very different from how we think it works: like in physics, there are hidden mechanisms. We superimpose our view from everyday experience, which may be completely distorted. Because of that, we can distort the potentially exceptional abilities of some children. There are two opposite goals education is supposed to achieve: firstly, to teach people to conform to the society they live in; on the other hand, to give them freedom to develop in the best possible way. These are opposite purposes, and they are always in collision with each other. This creates the result that some people get suppressed in the process of adapting them to society. You cannot avoid this kind of collision of goals, but we have to find a balance between the two, and that is not easy, on all levels of education. (Raussen and Skau, 2010, 402)

This conflict is at the core of the Enlightenment ideal of education, which we still aspire to. Whether or not he had in mind this ideal, Gromov describes it by referring to giving people “freedom to [individually] develop in the best possible way.” This ideal and, with it, the conflict in question emerged during the Enlightenment, especially with Kant, who introduced the term [Aufklärung], by grounding this ideal in his Critique of Practical Reason, the title of his second Critique (Kant 2015). Two influential examples of advocating this ideal may be cited. The first, before (and influencing) Kant, is Rousseau’s Émile, or de l’éducation (1762), which emphasized the importance of the freedom for an individual to develop in the best possible way, and the difficulty achieving the aim of doing so in a society (Rousseau 1997). The second, following (and yet displacing) Kant, Friedrich Schiller’s Letters on the Aesthetic Education of Man (1795) (Schiller 2004), aimed at reconciling both aims within a framework, which, however, poses major questions of its own (de Man 1996). Within such inheritances, how might a deviation game become part of current education practices?

Abel and others to whom Gromov is referring by an apt collective name “Abels” were not uneducated; quite the contrary. Nor were Abel and Galois merely self-educated in mathematics, even if they were more so than others. But if one considers such figures as Carl Friedrich Gauss and Bernhard Riemann, one finds their revolutionary mathematics emerged from reasonably conventional mathematical educations. Gromov must have been aware of these facts. His point was, I surmise, different, and it may be expressed in terms of deviation games. Our education discourages orientations toward a deviation-game thinking, not the least because while transforming thinking it challenges majority thinking. As I argue, however, without the deviational minority thinking in mathematics, along with the freedom to put it into practice, good mathematicians, let alone the likes of Abel and Galois, are not likely to emerge.

There is also the question whether a very good education removes the necessity of figures like Abels, or perhaps that we in fact have more numerous Abels at work in mathematics than it appears, and that the very juxtaposition to these previous epochs prior to widespread public education is misleading. Even so, I would argue that the present-day Abels or near Abels are only possible in learning environments that encourage deviation games. Mathematics learning within imitation-game practices will always be lacking the line of flight that requires deviational thinking, thought, defined by the exterior, however internal this exterior may be. The question becomes that of the place of AI in the process of learning and doing mathematics through deviation games, if one argues, as I do here, that AIs do not think and, as programmed devices, are only capable of programmable imitation games. It is of course possible for humans to dream of creating AI desiring machines capable of (real) dreams and deviation games. I doubt that such AI entities can be digital. What would these nonhuman desiring machines be, if not human? But then, we are far from entirely certain who we are, as humans, either.

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