*For the preceding segment of this four-part genealogy, see *Part 1

*For the preceding segment of this four-part genealogy, see *Part 1

*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**

*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

*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.

^{th} century, and are instrumental in the structure of being and interaction that the author Samuel Delany identifies as *contact*:

The Prater park in Vienna

*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**

*outside of* or *beyond*.

*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

*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]

*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.

*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:

[xiii]

**Outside**

*Next segment:* Part 3

[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.

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