*In this second part of our genealogy, we move not forward in time, but look back to an encounter that took place between two foundational figures in logic and mathematics, in an attempt to identify the conflicting role of contradiction, misunderstanding, failure, and disagreement in the queer history of computation. While again these figures are well known, the encounter between them is often dismissed as a missed connection and a failed opportunity. As such, it is often relegated to an uninteresting footnote in the history of mathematics. By reengaging this encounter I hope to blur the lines between computing, philosophy, and mathematics, and to disrupt the narrative trajectory that would see Turing as the single foundational figure within this history.*

**An Encounter**

In the spring of 1939, Ludwig Wittgenstein taught a course at the University of Cambridge on the foundations of mathematics, a topic that occupied much of his work from 1922 through to the end of the Second World War. That same semester Wittgenstein was finally elected chair of philosophy at the university, acquiring British citizenship soon thereafter. At fifty years old, he was an established figure in analytic philosophy, having published his groundbreaking *Tractatus Logico-Philosophicus* almost twenty years prior, and having written extensively on the work of Gödel, Russell, and Whitehead. While Wittgenstein is considered by many to be the most important philosopher of the 20^{th} century, he published very little in his lifetime, and much of his thought and character can only be derived from what survives of his lectures, notes, and seminars. Still less is known of his sexuality, and until the 1980s it was a subject rarely discussed among colleagues or in the many biographies written about his life and work.[i] Even now that Wittgenstein's homosexuality has been largely acknowledged, most scholars are hesitant to imply a connection between his philosophy and his sexuality – that is, between his work and his inner state, emotions, or personality. If, however, in a contemporary light we understand queerness as a structuring mode of desiring, we might view Wittgenstein's thought not as emerging *from* his sexuality, but as structured by the way in which it shaped his mode of being in the world.

Wittgenstein with Francis Skinner in Cambridge ca. 1933

Wittgenstein is widely regarded to have fallen in love with three men; David Pinset[ii] in 1912, Francis Skinner in 1930, and Ben Richards in the late 1940s.[iii] While it is clear these were relationships of love and affection, the extent to which they were physical is often contested. What seems to make many Wittgenstein scholars uncomfortable in confronting his homosexuality is that it conflicts with the ascetic, almost priestly view of a man so revered by contemporary philosophy. As Bruce Duffy suggests in a 1988 *New York Times* article on the life of Wittgenstein, "In their effort to put forth a plain, unvarnished record of what Wittgenstein did and said, some of these memoirs have almost the feeling of gospels – hushed, reverential, proprietary."[iv] The philosopher – or indeed, the mathematician – as a carnal, sexual being produces a seemingly irresolvable contradiction. Even those accounts that do concede his affection for other men often suggest that those feelings were purely aesthetic or emotional, and were never acted upon. That said, in perhaps the most controversial section of his 1973 biography of Wittgenstein, W. W. Bartley suggests that the philosopher frequently engaged in a kind of anonymous cross-class sexual contact facilitated by public cruising spaces such as parks and high streets.

By walking for ten minutes to the east . . . he could quickly reach the parkland meadows of the Prater, where rough young men were ready to cater to him sexually. Once he had discovered this place, Wittgenstein found to his horror that he could scarcely keep away from it . . . Wittgenstein found he much preferred the sort of rough blunt homosexual youth that he could find strolling in the paths and alleys of the Prater to those ostensibly more refined young men who frequented the Sirk Ecke in the Kärntnerstrasse and the neighboring bars at the edge of the inner city.[v]

These kinds of exceptional spaces as sites for anonymous sexual encounters continue well into the 20^{th} century, and are instrumental in the structure of being and interaction that the author Samuel Delany identifies as *contact*:

[C]ontact is also the intercourse—physical and conversational—that blooms in and as “casual sex” in public rest rooms, sex movies, public parks, singles bars, and sex clubs, on street corners with heavy hustling traffic, and in the adjoining motels or the apartments of one or another participant, from which nonsexual friendships and/or acquaintances lasting for decades or a lifetime may spring . . . a relation that, a decade later, has devolved into a smile or a nod, even when (to quote Swinburne) 'You have forgotten my kisses, / And I have forgotten your name.'[vi]

Bartley's sources have been called into question by many historians, but it is less the detail of his description than the acknowledgement of an embodied sexuality that is significant to this history; it is the difficulty we often have in finding the sexual in the everyday, in the lived work of a person beyond these exceptional moments of contact. While such effects may be invisible or to a degree, unknowable, that does not mean they aren't real and do not have a direct effect on the world.

The Prater park in Vienna

In the *Tractatus,* Wittgenstein defines truth as a tautology, that is, a result achieved through the mere repetition of the same meaning. While he insists that there exist religious or ethical truths, he argues that they cannot be put into words, that they are unknowable through language, and that claims to express ethical truths through philosophy must fail. Wittgenstein summarizes the *Tractatus* with the maxim: “What can be said at all can be said clearly; and what we cannot talk about we must pass over in silence.”[vii] What does it mean that for Wittgenstein truth is something that can be known but not discussed, that is indescribable? And how does he apply this critique to truths we understand to be beyond language – the truth of the body, or the truth of mathematics?

**Missed Connections**

Back at Cambridge in 1939, another young scholar and philosopher was also beginning his research at the university. After two years working under Alonso Church at the Institute for Advanced Study in Princeton, New Jersey, Alan Turing took up a position as an untenured research fellow at Cambridge, having failed to acquire a full lectureship. Turing and Wittgenstein had been introduced the summer of 1937, but it was not until two years later in 1939 that they would have any meaningful interaction. That spring Turing was also teaching a course on the foundations of mathematics that shared the same name as Wittgenstein's lecture.[viii] Perhaps intrigued, Turing enrolled. Over the course of the semester, Turing engaged in a lengthy dialogue with Wittgenstein, challenging and outright refusing much of Wittgenstein's thoughts on logic and mathematics.[ix] Despite their disagreement, this seems a pivotal moment in the history of computing, in which two queer figures engage with the limits of knowledge and computability, questioning that which exists *outside of* or *beyond*.

As a young man, Wittgenstein had thought logic could provide a solid foundation for a philosophy of mathematics. Now in his fifties, he denied outright there were any mathematical facts to be discovered. For Wittgenstein, a proof in mathematics does not establish the truth of a conclusion, but rather fixes the meaning of certain signs. That is, the "inexorability" of mathematics does not consist of certain knowledge of mathematical truths, but in the fact that mathematical propositions are *grammatical*, a kind of language game through which meaning becomes fixed. One the first day of class, Wittgenstein begins by stating, "I shall try and try again to show that what is called a mathematical *discovery* had much better be called a mathematical *invention*."[x]

The *Erkenntnis* from the Königsberg Congress of 1930

Throughout the semester, Wittgenstein attempts to demonstrate that, if we may identify a single contradiction within a system such as mathematics, it ceases to function and loses all meaning. In one particularly memorable exchange, Wittgenstein puts forth one of his favorite contradictions – known as Epimenides' paradox, or the liar's paradox – in which I make the claim "I am lying," thereby creating a paradox in which if I *am* lying I am telling the truth, and if I am *not* lying I am telling a lie. Such an example may seem like nothing more than a silly logic puzzle, but it is significant that they produce a paradox that cannot be made meaningful to mathematics, and that these contradictions exist outside of any functional or productive applications. This, of course, is an affront to the very practice of mathematical logic. As Andrew Hodges notes, "Getting statements free from contradictions is the very essence of mathematics. Turing perhaps thought Wittgenstein did not take seriously enough the unobvious and difficult questions that had arisen in the attempt to formalize mathematics; Wittgenstein thought Turing did not take seriously the question of why one should want to formalize mathematics at all."[xi]

Wittgenstein uses Turing as a straw man of sorts, tasked with defending the philosophical validity of mathematics as a whole. Over the course of the two-term seminar, one can't help but get the sense that the two men are speaking past one another; that their concerns and interests diverge on a fundamental level. On the whole, Turing argues for a rather conservative approach to mathematics and its use in material applications. Surely, Turing argues, mathematics must be more than language games, as it enables us to build bridges that do not fall down, and to calculate with great precision measurable truths in the world. Yet despite his philosophical refusal, Turing's own work and research during the three years prior to the lectures touches on many of the same themes Wittgenstein was pursuing in his lectures, and addresses those invisible or unknowable truths that escape mathematical calculation through computation. While the two are clearly are at odds over their importance, both are nonetheless explicitly preoccupied with these externalities, these meaningless contradictions.

Turing's most famous work on this subject is *On Computable Numbers*, published in 1936, in which he establishes the definition of computable numbers as "the real numbers whose expressions as a decimal are calculable by finite means," stating that "a number is computable if its decimal can be written down by a machine." Turing expands his thesis, proving that his formalism was sufficiently general to encompass anything that a human being could do when carrying out a definite method. Importantly, Turing also established in this work the limits of computation, identifying the existence of uncomputable problems that cannot be solved through a definite method.[xii] The most famous such problem is the halting problem, in which an algorithm is built to calculate whether a given program will halt and produce a solution, or run forever. If such a program were to exist, we might in turn apply it back onto itself, asking it to find if it will ever halt, and in doing so creating a paradox not dissimilar to that of the liar.

More interesting to this project, however, is a supplementary paper published in 1939, titled *Systems of Logic Based on Ordinals*, in which Turing asks if it is possible to formalize those actions of the mind that do not follow a definite method — mental actions we might call creative or original in nature. There exist certain sets of uncomputable problems which are functionally solvable by human means, but for which there is no definite method for calculating an answer. Here Turing suggests the impossibility of accounting for this intuitive action through computation, stating:

Mathematical reasoning may be regarded rather schematically as the combination of two faculties, which we may call intuition and ingenuity. The activity of the intuition consists in making spontaneous judgments which are not the result of conscious trains of reasoning.[xiii]

It is unclear how such intuition functions, or how to understand and successfully implement it, but Turing's biographer Andrew Hodges suggests that "the evidence is that at this time [Turing] was open to the idea that in moments of 'intuition' the mind appears to do something outside the scope of the Turing machine." That is, outside of computation as Turing has defined it.[xiv]

**Outside**

How then to bring together these two moments of the founding and formalization of computing? In one we have the refusal of the truth mathematics would hope to claim and an investigation of those contradictions that exist within, but are beyond the scope of logical inquiry. In the other there is an investigation of those exceptional sites and the suggestion that there is a process that exists beyond computation that nonetheless allows us to make truthful claims about the world. Two views on the same problem, and a seemingly impassible philosophical divide.

For most historians of mathematics and technology, this encounter is viewed as a failure of recognition, and of the inability of Turing and Wittgenstein to reach across and make contact with one another on these fundamental questions. Much as it is unclear what one may have known about the other's sexuality, or if such similarities were even legible as a form of community or even commonality, there seems here to be a misrecognition, a failure to connect. And yet I would like to suggest that this is precisely the point, that this is precisely what makes this a queer encounter. It is the impossibility of narrativizing this encounter in legible terms, and the way in which this impossibility mirrors the indescribable, external truths that so preoccupied the minds of both men, that unites them. It is in these exceptional spaces outside of formally describable systems – binary code, language, mathematics – that we may identify a queerness at work.

In choosing this, perhaps the earliest moment at which such an inquiry is made possible, it seems meaningful that such questions are being posed by two queer men who met only briefly and, perhaps appropriately, were unable to come to an agreement, or to even understand the questions the other sought to answer. And yet each man's work seeks to investigate the limits of a particular system of knowledge that functions by delimiting the analog world through the construction of a hermetic system; one that rejects those externalities that might otherwise cause it to fail. If we consider queerness simply in terms of sexual preference or as an alternative formation within an established set of desiring modes, then describing any form of computing as "queer" may seem absurd. If instead we understand queerness as a process of self-shattering rather than self-fashioning, then we begin to align it with these exceptional objects and practices that exist beyond the limits of a system such as computation. While it is no doubt true that queerness is not the only means by which we might ask these questions of technology, or through which we might seek an alternative to the universalizing structures of computing technology, it is my suggestion that is an ideal lens through which to examine that which exists outside or beyond, and one that begins here in these earliest moments in the history of computation.

[i] W. W. Bartley III's *Wittgenstein *(1973) devotes 4-5 pages to the philosopher's sexuality, based on interviews conducted in the 1960s and translations of Wittgenstein's own encrypted journals – many of which were destroyed at his own insistence in 1950, a year before his death. Based on these passages the book was attacked vehemently and repeatedly by Wittgenstein's family and colleagues, in the pages of the *New York Times Literary Supplement*, and at the annual Wittgenstein Congress at the Wittgenstein Documentation Center in Kirchberg am Wechsel, Austria. The book was called sensationalist and false despite the availability of multiple documents corroborating Wittgenstein conflicted feeling towards his sexuality.

[ii] David Hume Pinset was a descendent of the philosopher David Hume, and was a friend and colleague to Wittgenstein, collaborating on research and traveling on holidays with him to Iceland and Norway. In 1918 Pinset was killed in a military flying accident, and Wittgenstein would later dedicate his *Tractatus Logico-Philosophicus* (1922) to his memory.

[iii] Monk, Ray. *Ludwig Wittgenstein: The Duty of Genius*. Free Press, 1990, pp. 583–586.

[iv] Duffy, Bruce. "The Do-it-Yourself Life of Ludwig Wittgenstein" *The New York Times* November 13, 1988. <http://www.nytimes.com/1988/11/13/books/the-do-it-yourself-life-of-ludwig-wittgenstein.html?pagewanted=all&src=pm>.

[v] Bartley: *Wittgenstein*, p. 47.

[vi] Delany, Samuel. *Times Square Red Times Square Blue*. New York: NYU Press, 2001 p. 123-124.

[vii] Wittgenstein, Ludwig. *Tractatus Logico-Philosophicus*. New York: Routledge, 1921-2001, p. 3.

[viii] Diamond, Cora (ed.) *Wittgenstein's Lectures on the Foundations of Mathematics: Cambridge, 1939* Ithaca, NY: Cornell University Press, 1976.

[ix] These encounters have been collected and recorded based on the notes of four students who attended the lecture, and were subsequently edited and published. As such they form an imperfect, but essential archive.

[x] Diamond, *Ibid.* 416.

[xi] Hodges, Andrew. "Alan Turing: One of the Great Philosophers" Web. <http://www.turing.org.uk/philosophy/ex4.html>.

[xii] Turing's work on uncomputability does not emerge from nowhere. It is informed by several decades of debate in the early history of mathematics – what is often referred to as the *foundational crisis of mathematics*, or* Grundlagenkrise der Mathematik –* over the question of whether mathematics had any foundation that could be stated within mathematics itself without suffering from irresolvable paradoxes. This led to competing schools of thought, the most important of which was Hilbert's program, named after the German mathematician David Hilbert. The program proposed to ground all existing theories to a finite, complete set of axioms, and provide a proof that these axioms were consistent. However, in 1931 Kurt Gödel's incompleteness theorems showed that any consistent system with a computable set of axioms which is capable of expressing arithmetic can never be complete, that it is possible to prove a statement to be true that cannot be derived from the formal rules of the system. Turing would take Gödel's work further, applying this theorem to the concept of computability, defined as that which can be stated within a formal system and may therefore be executed by a machine with a procedural grasp of computational logic.

[xiii] This train of thought belongs to a field in the philosophy of mathematics known as "intuitionism."

[xiv] To be clear, these externalities and paradoxes are not simply language games, but can be applied to real world problems as well. One famous example is that of Zeno's paradoxes, formulated by the Greek philosopher Zeno of Elea (ca. 490-430 BCE). In Zeno's dichotomy paradox, he states that "locomotion must arrive at the half-way stage before it arrives at the goal" (Aristotle, *Physics* VI:9, 239b10). In other words, if any possible finite distance may be divided in half, then in order to reach a given goal, a moving object must first get halfway there. Before it can get halfway there, it must get a quarter of the way there, before traveling a quarter it must travel one-eighth, and so on. The resulting solution requires the object to complete an infinite number of tasks, which Zeno maintains is an impossibility – yet clearly in the observable world objects move from location to location and arrive at their destination despite this contradiction. The same limits exist for computation, and have led to a hypothetical computational models that allow for a countably infinite number of algorithmic steps to be completed in finite time. The resulting hypothetical computer is often referred to as a Zeno machine, and is an example of a super-Turing machine – that is, a computer that functions beyond universal Turing computation. It is interesting to note that Zeno, like many Greek men, participated in homosexual *erastes-eromenos* mentor relationships, and was loved and mentored by Parmenides of Elea, the founder of the Eleatic school of philosophy.