21 Thinking Softly with Incomputability

Algorithm

Soft thought is nomenclature for the ways digital algorithms learn about the “immanent ingression” of incomputability within computational culture (Parisi 2013, 20). Put differently, soft thought characterizes the ubiquitous modes of learning in algorithmic programming constituted in mathematical incompleteness, algorithmic randomness, and incomputable probabilities. This memoranda provides a brief history regarding ideas about “incomputability,” and the axiomatic limits of mathematics. Kurt Gödel’s (1931 [1992]) incompleteness theories showed that for any formal axiomatic system (e.g., mathematics, logic) there will always be statements that are true in a system but not provable. His incompleteness theorems challenged attempts to explain everything in the world in mathematical terms—a persistent perspective referred to (loosely) as the theory of everything (Chaitin 2005).

Gödel’s (1931 [1992]) incompleteness theories were designed to challenge David Hilbert’s project, which was, in turn, designed to identify the axiomatic foundations for various parts of mathematics, including the mathematics of probability. Gödel showed this was impossible, and hence, his work illustrated the limits of axiomatic approaches to math, but not necessarily the limits of math to explain the world. Theories of incompleteness insist that mathematics is not a complete axiomatic system, and therefore the world (sociality, learning, education) cannot be reduced to mathematics to establish what is provable and true, and what is not. In fact, according to mathematician and information theorist Gregory Chaitin (2006), Gödel’s incompleteness theorems provide conclusive evidence “that a theory of everything for all of mathematics cannot exist” (79).

Alan Turing utilized Gödel’s incompleteness theories and developed his halting problem that demonstrated the impossibility of determining what sorts of machines—and today, what sorts of algorithms—will halt given a certain input, and what sorts of algorithms will not halt. Turing’s (1936) halting problem indicated that there is no formal axiomatic system that can determine in advance whether a program will stop or continue its calculations within a discrete period of time. In effect, Turing’s halting problem illustrated the incompleteness of algorithms developed on the premises of formal axiomatic systems. Importantly, Turing’s halting problem illustrated the irreducibility of complexity to the finite operations of algorithmic code. While Turing’s halting problem is a tremendous achievement in theoretical computer science (that utilized Cantor’s diagonal argument), it has not meaningfully influenced many mathematicians and computer scientists developing large language models (e.g., ChatGPT). Nevertheless, it remains a significant example of mathematical logic (or meta-mathematics) and has influenced the philosophy of mathematics.

In 2006, Gregory Chaitin produced Omega as digital proof of Turing’s computational halting problem. Omega is the name coined by Chaitin (2006) to identify the infinities of incomputable probabilities. Omega is an infinite and patternless sequence of 0s and 1s produced through an endless algorithm. As such, Omega is a random real number that sketches the axiomatic limits of formal mathematical reasoning. As a result, Omega demonstrated the irreducibility of infinite complexity (or, incompleteness) of mathematics. For Chaitin (2005), the ontology of incomputability “is much more serious than people think . . . perhaps mathematics should be pursued somewhat more in the spirit of experimental science rather than always demanding proofs for everything. Maybe, rather than attempting to prove results . . . mathematicians should accept that they may not be provable and simply accept [incompleteness and incomputability] as an axiom” (para 43). Incomputability challenges ideas that our world can be reduced to mathematics. In other words, incompleteness and incomputability provide ontological counter-evidence to postulates about a “theory of everything.”

Incomputable

The incomputable—as a philosophical term—was developed by Luciana Parisi (2013) in her book Contagious Architecture. Borrowing from Gödel, Turing, and Chaitin, Parisi (2013) argues that traditional understandings of computation are radically reductionist and too often based on insufficient understandings of the axiomatic limits of mathematics and logics. Instead, Parisi (2013) argued that algorithmic computation is “infected” with incompleteness, paradox, and randomness (20). Parisi (2013) notes that the “system of governance defined by the digital world of data can therefore no longer rely upon the smooth programming of tasks, the exact reproduction of rules, and the optimization of conducts, habits, and behaviors. Randomness has become the condition of programming culture” (11; emphasis from the original). Alexander Galloway (2021) put it this way: “Incomputability is defined more by the limits to computation, more by the uncomputable, than by a positive set of capacities. Or, as Beatrice Fazi wryly put it, ‘the founding paradox of computer science is that it is a field that is defined by what it cannot do, rather than by what it can do’” (20).

Soft Thought

Parisi (2013) explains that “soft thought stems from the immanent ingression of incomputable data into digital programming” (20). She continues, “soft thought pertains to the existence of modes of thought, decision making, and mentality that do not exist in direct relation to human thinking. These modes of thought (of which soft thought is only one configuration) maintain a certain degree of autonomy from cognition demonstrated by their social inconsistencies. . . . Soft thought is not there to be understood as a new cognitive function or as a transcendent form of rationality, but to reveal that programming culture is infected by incomputable thought that are yet to be accounted for.” Affirmatively, soft thought is situated as a particular, perhaps unique, mode of thought (there are others). It is directly attributable to “the conceptual prehension of infinite data that defines computational actualities” (Parisi 2013, 400). Further, soft thought is distinguished from both cognition and rationality, and soft modes of thought do not have direct analogues in human thought.

Parisi argues that programming culture is “infected” with soft thought (2014, 20). Alternatively, she uses the idea of “contagion” to describe the ways soft thought has inserted moments of incompleteness and randomness into our daily lives. These are the “alien rules” of soft thought that prehend incompleteness, randomness, and infinite complexity germane to the axiomatic limits of computation. She explains, the “function of algorithms thus involves not the reduction of data to binary digits, but the ingression of random quantities into computation: a new level of determination that has come to characterize automated modes of organization and control. Far from making the rational system of governance more efficient, this new level of determination forces governance to rely on indeterminate probabilities, and thus to become confronted with data that produce alien rules. These rules are at once discrete and infinite, united and fractalized” (2013, 12). Soft thought should not be understood as a kind of “naive ontology,” and particularly not as (a) an admission of inherent algorithmic fallibility, and definitely not as (b) an ontological ground that insists that, for example, sociality, education, and learning are immune to computation and governance. Rather, the saturated practices of soft thought only increase the amount of incompleteness, paradox, contingency, and randomness in and around our lives.

(Alien) Learning

A broad characterization of some of the conflicting approaches to learning rest on presuppositions of, on one hand, rule-based machine learning (RBML) often evidenced through “if-then” expressions that comprise the mathematics of the prediction model under development. RBML are forms of learning that follow forms of conditional arguments or hypothetical syllogisms often used in deductive logic, and rooted in philosophies of learning often discussed as cognitivism and computationalism. On another hand, in direct opposition to theories of cognitivism and computationalism, forms of algorithmic learning exhibit ideas of interaction and adaptation, described by Parisi (2013) as “open learning” (99). The more interactive and adaptive forms of learning are based on philosophies of learning discussed as emergentism and enactivism. Forms of “open learning” (i.e., emergentism and enactivism) are forms of learning discussed broadly as embodied cognition—the idea that the body or the body’s interactions with the environment constitute (or contribute to) learning. As such, learning is not derived or deduced from a set of (internal) calculations, but emerges from the complex experiences and affective interactions of a body with a system or environment. For Parisi and Portanova (2011), open learning is in direct contrast to forms of cognitivism and computationalism (e.g., RBML) because learning “cannot be programmed since it remains a question of affective consciousness and experience” (para 33).

As a result, the contagion of incomputability interjects inconsistency into learning processes—hence, the idea of alien learning as nonhuman machine learning. However, rather than understand contagion only as recourse to potential expressions of inherent algorithmic fallibility, incomputability signals a postdigital envelope that has “incorporated [randomness and contingency] into the body of computation, not excluded from it” (Galloway 2021, 21). As a result, soft thought accelerates “the indeterminacies of information as a potential source of the unexpected. In other words, the relentless recalculations of data guarantee the possibility of discovering something new” (Majaca and Parisi 2016, para. 10). Chance, risk, and contingency now become integral mediators to learning, and significant means to learn differently.

Galloway (2021) affirms the idea that the incomputable was central to postdigital computation. He argues that the indiscernible and the indeterminate are at the heart of computation: “Indeed part of the history of computation is the history of the uncomputable being colonized by the computable. . . . In a sense, randomness and contingency have become fully industrial. Today the computable is closely intertwined with the uncomputable” (21).

The incomputability of soft thought has become speculative, post-probabilistic, and now generates novelty (Parisi 2013, 61). Algorithms and digital computing have their own capacity to be adaptable and creative in ways that challenge the assumption of “artificiality” and challenge presumptions that computational thought is limiting. Rather, mathematical incomputability has infected, or functions as an inherent “contagion” to, the practices and processes of cognitivism and computationalism. Soft thought, then, becomes the post-probabilistic artifice, environment, or ground for (alien) learning.

Rather than treat the alien learning of incomputability as paranoiac problems to solve, soft thought provides the ground for novel forms of learning, and in ways that “engage [in] a fuller appreciation of [incomputability] in forming interventions that affectively address changes in sociality and subjectivity” (Clough 2018, para. 6). In other words, alien forms of learning predicated on soft thought would assist the development of the next generation of incomputable and post-probabilistic subjects. Antonia Majaca (2018) hints at our postdigital and incomputable curricula when she notes that the incomputable subject would experiment with ideas like “machinic unconscious, psycho-cybernetics, sensitive automata, influencing machines, memory, paranoia (as a condition belonging to two systems of thought simultaneously—reason and unreason), human black box, and the entanglements of these phenomena within the aesthetico-political environments engulfing contemporary techno-epistemic, composite, and imminently incomputable Subjects” (para. 3). Majaca’s incomputable subject both recognizes the demands of soft thought, but more importantly, soft thought provides opportunities for the next iteration of alien learning between cybernetics and subjectivity—iterations infused with randomness, incompleteness, and soft thought.

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