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This chapter discusses how Kurt Gödel found his theorem. He started to study physics at the University of Vienna in 1924, then changed to mathematics in 1926. That same year, he started attending the meetings of the Vienna Circle. These were weekly gatherings on philosophical topics that were headed by the philosopher Moritz Schlick. The philosophy of the circle came to be known as logical empiricism and had an enormous effect on the world of philosophy. Gödel later wanted to emphasize that he by no means shared all of the philosophical ideas of the circle. In the meetings, Gödel came to know the philosopher Rudolf Carnap and the mathematician Karl Menger, in whose mathematical colloquium he later presented many of his results.

Sets out Kurt Gödel's slingshot (collapsing) argument. The original argument—or, at least, the premisses of the argument that Neale attributes to Gödel—can be found in a fleeting footnote to a discussion of the relationship between Bertrand Russell's Theory of Descriptions and Theory of Facts. Usually each theory is viewed as quite independent of the other, but Gödel argues otherwise: that the viability of the latter depends upon the viability of the former (or at least upon the viability of some non‐referential theory of descriptions). Neale summarizes Gödel's standpoint as follows: ‘if a true sentence stands for a fact, then in order to avoid the collapse of all facts into one, the friend of facts must give up either (a) an intuitive and straightforward Fregean Principle of Composition or (b) the idea that definite descriptions are expressions that purport to stand for things.’ The sections of the chapter are: Elimination (linguistic and ontological); Facts and Descriptions; Identities in the Matrix; and The Eleatic One (the collapse of all facts into one).

Introduces the criticisms put forward by philosophers such as Donald Davidson and Richard Rorty to the idea that one thing might represent another: that thoughts, utterances, and inscriptions are said to have content by virtue of their power to represent reality; and that those that do the job accurately are true, they correspond to the facts, or mirror reality—they are representations of reality. The author then outlines the deductive proof that he will present in the book to show that Davidson's and Rorty's criticisms are unfounded. The content of the proof is based on the work of Kurt Gödel and W. V. Quine, and demonstrates conclusively that (i) any supposedly non‐truth‐functional operation must satisfy an exacting logical condition in order to avoid collapsing into a truth‐function, and (ii) any theory of facts, states of affairs, situations, or propositions must satisfy a corresponding condition if such entities are not to collapse into a unity. The three main sections of the chapter: examine the case against representations made by Davidson and Rorty, including Davidson's notorious argument against facts; discuss Rorty's critique of representational philosophy (his anti‐representationalism); and consider collapsing (or slingshot) arguments that discredit the existence of facts (and are used in the critique of the fact–representation distinction), as put forward by Gottlob Frege, and developed by Alonzo Church, Quine, Davidson, and Gödel. The last section of the chapter outlines the structure of the book.

Chs. 9 (and 8) convert the two basic forms of slingshot (collapsing) argument—one used by Alonzo Church, W. V. Quine, and Donald Davidson, the other by Kurt Gödel—into knock‐down deductive proofs that Donald Davidson's and Richard Rorty's cases against facts and the representation of facts are unfounded, and their slingshot arguments for discrediting the existence of facts unsatisfactory. The proofs are agnostic on key semantic issues; in particular, they assume no particular account of reference and do not even assume that sentences have references. Using the same procedure as in Chs. 8 and 9 shows that a slingshot argument engendered by Gödel's suggestions can be converted into a proof that delivers an exacting constraint on non‐extensional discourse—the Descriptive Constraint—and that is more general than anything Gödel appears to have had in mind. The proof of this constraint is constructed without appeal to logical equivalences, without assuming that a semantic theory must treat sentences as having references, without presupposing anything contentious about the semantics of singular terms, and without commitment to any particular semantics for definite descriptions. A constraint on facts, situations, states of affairs, and propositions—indeed on anything that is expressed or represented sententially—drops out as a trivial consequence of a constraint on non‐extensional discourse. The three sections of the chapter are: Principles of Conversion for Descriptions; Gödel's Proof in Quinean Format; and A Stronger Slingshot?

This chapter investigates the possibility of knowing a structure (an isomorphism type) more directly than as the structure of all models of this or that categorical theory. It starts with grasp of small simple finite structures by means of visual templates and then explains how visual grasp of structures may be possible without templates. The case is then made for the possibility of visual cognition of an infinite structure, an ω-sequence, though this is visual cognition of a more abstract kind, relying on the distinction between a visual category specification and a visual image. It is argued that this kind of cognition is not available for many other structures such as that of the closed unit interval of reals under ‘less-than’, but it is available for some ordinal structures beyond ω. The chapter uses this to explain and support some contentious claims of Kurt Gödel.

Chs. 8 and 9 convert the two basic forms of slingshot (collapsing) argument—one used by Alonzo Church, W. V. Quine, and Donald Davidson, the other by Kurt Gödel—into knock‐down deductive proofs that Donald Davidson's and Richard Rorty's cases against facts and the representation of facts are unfounded, and their slingshot arguments for discrediting the existence of facts unsatisfactory. The proofs are agnostic on key semantic issues; in particular, they assume no particular account of reference and do not even assume that sentences have references. Using this procedure, Ch. 8 shows that cleaned‐up versions of the slingshot arguments used by Davidson, Church, and Quine demonstrate theses weaker than those their authors were seeking—the impossibility of facts and non‐truth‐functional (e.g. modal and causal) connectives. This is primarily because the arguments depend upon (a) logical equivalences and (b) theories of definite descriptions (or class abstracts) that must satisfy certain semantic conditions if the aforementioned logical equivalences are to obtain. The four sections of the chapter are: Introductory Remarks; The Quinean Strategy; An Incomplete Connective Proof; and A Complete Connective Proof.

This chapter explores a universal notion of computation, first by describing Charles Babbage's vision of a mechanical device that can perform any calculation as well as David Hilbert's dream of a mechanical procedure capable of proving or refuting any mathematical claim. It then considers the universality of modern computers and the undecidability of certain problems, explores diagonalization and the Halting Problem, and discusses Kurt Gödel's Incompleteness Theorem. It also looks at the three models of computation proposed in the early twentieth century — partial recursive functions, the lambda-calculus, and Turing machines — and show that they are all equivalent to each other and can carry out any conceivable computation. The chapter concludes by considering universal computation and undecidability in tilings of the plane, products of fractions, and the motions of a chaotic system.

Chapter 9 explores the connections between the Vienna Circle of positivism and the esoteric milieu. It shows how the founders of logical positivism, such as Otto Neurath, presented their philosophy as a kind of magical revival. It also demonstrates that other positivists—such as Rudolf Carnap, Hans Hahn, and Kurt Gödel—had a pro-found preoccupation with ghosts and the paranormal. Taken as a whole, the book demonstrates how magic, like metaphysics, also haunts the beginnings of analytic philosophy. It also undoes notions of a dry and apolitical positivism, by describing positivist anti-metaphysics in terms of ideological critique.

The mathematician Kurt Gödel showed in his Incompleteness Theorem in the early 1930s that there are some statements in mathematics that are true but cannot be proven. Whether statements are true is important in the twenty-first century, an age of ‘fake news’ and alternative facts. Patent documents are true and accurate as they are examined and can be challenged for accuracy. This chapter outlines the patenting procedure. It also highlights the role of patents as a source of information alongside other sources. Accurate and true information is important for people with interests in engineering, physical sciences and life sciences. Patent infringement and patent trolls (non-practicing entities) are described. The following technical areas are grouped together to describe how they developed over time and how they may develop in the twenty-first century: communications, computing including quantum computing, life sciences including gene editing (CRISPR), transport and unexpected consequences of technological change.

This chapter argues against the widely held view that the theory of relativity entails that space and time are fundamentally alike. It solves the Problem of Motion by defending a hybrid view that combines a modal account of time with substantivalism about space. This view recovers space-time at a higher level of abstraction, rather than treat it as a fundamental entity. The chapter closes with a brief discussion of the relation between space-time symmetries and the structure of space and time.