This paper investigates the determinacy of mathematics. We begin by clarifying how we are understanding the notion of determinacy before turning to the questions of whether and how famous independence results bear on issues of determinacy in mathematics. From there, we pose a metasemantic challenge for those who believe that mathematical language is determinate, motivate two important constraints on attempts to meet our challenge, and then use these constraints to develop an argument against determinacy and discuss a particularly popular approach (...) to resolving indeterminacy, before offering some brief closing reflections. We believe our discussion poses a serious challenge for most philosophical theories of mathematics, since it puts considerable pressure on all views that accept a non-trivial amount of determinacy for even basic arithmetic. (shrink)

Artificial Neural Networks have reached “grandmaster” and even “super-human” performance across a variety of games, from those involving perfect information, such as Go, to those involving imperfect information, such as “Starcraft”. Such technological developments from artificial intelligence (AI) labs have ushered concomitant applications across the world of business, where an “AI” brand-tag is quickly becoming ubiquitous. A corollary of such widespread commercial deployment is that when AI gets things wrong—an autonomous vehicle crashes, a chatbot exhibits “racist” behavior, automated credit-scoring processes (...) “discriminate” on gender, etc.—there are often significant financial, legal, and brand consequences, and the incident becomes major news. As Judea Pearl sees it, the underlying reason for such mistakes is that “... all the impressive achievements of deep learning amount to just curve fitting.” The key, as Pearl suggests, is to replace “reasoning by association” with “causal reasoning” —the ability to infer causes from observed phenomena. It is a point that was echoed by Gary Marcus and Ernest Davis in a recent piece for theNew York Times: “we need to stop building computer systems that merely get better and better at detecting statistical patterns in data sets—often using an approach known as ‘Deep Learning’—and start building computer systems that from the moment of their assembly innately grasp three basic concepts: time, space, and causality.” In this paper, foregrounding what in 1949 Gilbert Ryle termed “a category mistake”, I will offer an alternative explanation for AI errors; it is not so much that AI machinery cannot “grasp” causality, but that AI machinery (qua computation) cannot understand anything at all. (shrink)

Though my ultimate concern is with issues in epistemology and metaphysics, let me phrase the central question I will pursue in terms evocative of philosophy of religion: What are the implications of our logic-in particular, of Cantor and G6del-for the possibility of omniscience?

A survey of more philosophical applications of Gödel's incompleteness results.

Traces the development of recursive functions from their origins in the late nineteenth century to the mid-1930s, with particular emphasis on the work and influence of Kurt Gödel.

Sections 3.16 and 3.23 of Roger Penrose's Shadows of the mind (Oxford, Oxford University Press, 1994) contain a subtle and intriguing new argument against mechanism, the thesis that the human mind can be accurately modeled by a Turing machine. The argument, based on the incompleteness theorem, is designed to meet standard objections to the original Lucas-Penrose formulations. The new argument, however, seems to invoke an unrestricted truth predicate (and an unrestricted knowability predicate). If so, its premises are inconsistent. The usual (...) ways of restricting the predicates either invalidate Penrose's reasoning or require presuppositions that the mechanist can reject. (shrink)

We construct a machine that knows its own code, at the price of not knowing its own factivity.

We study the structure of families of theories in the language of arithmetic extended to allow these families to refer to one another and to themselves. If a theory contains schemata expressing its own truth and expressing a specific Turing index for itself, and contains some other mild axioms, then that theory is untrue. We exhibit some families of true self-referential theories that barely avoid this forbidden pattern.

A critique of algorithmic rationalisation offers at best some initial reasons and preliminary ideas. Critique is understood as a reflection on validity. It is limited to an “epistemological investigation” of the limits of the calculable or of what appears “knowable” in the mode of the algorithmic. The argumentation aims at the mathematical foundations of computer science and goes back to the so-called “foundational crisis of mathematics” at the beginning of the 20th century with the attempt to formalise concepts such as (...) calculability, decidability and provability. The Gödel theorems and Turing’s halting problem prove to be essential for any critical approach to “algorithmic rationalisation”. Both, however, do not provide unambiguous results, at best they run towards what later became known as “Gödel’s disjunction”. The chosen path here, however, suggests the opposite way, insofar as, on the one hand, the topos of creativity appear constitutive for what can be regarded as cognitive “algorithmic rationalisation” and which encounters systematic difficulties in the evaluation of non-trivial results. On the other hand, the investigations lead to a comparison between the “mediality” of formally generated structures, which have to distinguish between object-and metalanguages, and the “volatile” differentiality of human thought, which calls for syntactically non-simulatable sense structures. (shrink)

This chapter presents a new semantics for inductive empirical knowledge. The epistemic agent is represented concretely as a learner who processes new inputs through time and who forms new beliefs from those inputs by means of a concrete, computable learning program. The agent’s belief state is represented hyper-intensionally as a set of time-indexed sentences. Knowledge is interpreted as avoidance of error in the limit and as having converged to true belief from the present time onward. Familiar topics are re-examined within (...) the semantics, such as inductive skepticism, the logic of discovery, Duhem’s problem, the articulation of theories by auxiliary hypotheses, the role of serendipity in scientific knowledge, Fitch’s paradox, deductive closure of knowability, whether one can know inductively that one knows inductively, whether one can know inductively that one does not know inductively, and whether expert instruction can spread common inductive knowledge—as opposed to mere, true belief—through a community of gullible pupils. (shrink)

This paper reassesses the criticism of the Lucas-Penrose anti-mechanist argument, based on Gödel’s incompleteness theorems, as formulated by Krajewski : this argument only works with the additional extra-formal assumption that “the human mind is consistent”. Krajewski argues that this assumption cannot be formalized, and therefore that the anti-mechanist argument – which requires the formalization of the whole reasoning process – fails to establish that the human mind is not mechanistic. A similar situation occurs with a corollary to the argument, that (...) the human mind allegedly outperforms machines, because although there is no exhaustive formal definition of natural numbers, mathematicians can successfully work with natural numbers. Again, the corollary requires an extra-formal assumption: “PA is complete” or “the set of all natural numbers exists”. I agree that extra-formal assumptions are necessary in order to validate the anti-mechanist argument and its corollary, and that those assumptions are problematic. However, I argue that formalization is possible and the problem is instead the circularity of reasoning that they cause. The human mind does not prove its own consistency, and outperforms the machine, simply by making the assumption “I am consistent”. Starting from the analysis of circularity, I propose a way of thinking about the interplay between informal and formal in mathematics. (shrink)

The alleged proof of the non-mechanical, or non-computational, character of the human mind based on Gödel’s incompleteness theorem is revisited. Its history is reviewed. The proof, also known as the Lucas argument and the Penrose argument, is refuted. It is claimed, following Gödel himself and other leading logicians, that antimechanism is not implied by Gödel’s theorems alone. The present paper sets out this refutation in its strongest form, demonstrating general theorems implying the inconsistency of Lucas’s arithmetic and the semantic inadequacy (...) of Penrose’s arithmetic. On the other hand, the limitations to our capacity for mechanizing or programming the mind are also indicated, together with two other corollaries of Gödel’s theorems: that we cannot prove that we are consistent, and that we cannot fully describe our notion of a natural number. (shrink)

Reinhardt’s conjecture, a formalization of the statement that a truthful knowing machine can know its own truthfulness and mechanicalness, was proved by Carlson using sophisticated structural results about the ordinals and transfinite induction just beyond the first epsilon number. We prove a weaker version of the conjecture, by elementary methods and transfinite induction up to a smaller ordinal.

We argue that the set of humanly known mathematical truths (at any given moment in human history) is finite and so recursive. But if so, then given various fundamental results in mathematical logic and the theory of computation (such as Craig’s in J Symb Log 18(1): 30–32(1953) theorem), the set of humanly known mathematical truths is axiomatizable. Furthermore, given Godel’s (Monash Math Phys 38: 173–198, 1931) First Incompleteness Theorem, then (at any given moment in human history) humanly known mathematics must (...) be either inconsistent or incomplete. Moreover, since humanly known mathematics is axiomatizable, it can be the output of a Turing machine. We then argue that any given mathematical claim that we could possibly know could be the output of a Turing machine, at least in principle. So the Lucas-Penrose (Lucas in Philosophy 36:112–127, 1961; Penrose, in The Emperor’s new mind. Oxford University Press, Oxford (1994)) argument cannot be sound. (shrink)

We first discuss Michael Dummett’s philosophy of mathematics and Robert Brandom’s philosophy of language to demonstrate that inferentialism entails the falsity of Church’s Thesis and, as a consequence, the Computational Theory of Mind. This amounts to an entirely novel critique of mechanism in the philosophy of mind, one we show to have tremendous advantages over the traditional Lucas-Penrose argument.

G. Priest's anti-consistency argument (Priest 1979, 1984, 1987) and J. R. Lucas's anti-mechanist argument (Lucas 1961, 1968, 1970, 1984) both appeal to Gödel incompleteness. By way of refuting them, this paper defends the thesis of quartet compatibility, viz., that the logic of the mind can simultaneously be Gödel incomplete, consistent, mechanical, and recursion complete (capable of all means of recursion). A representational approach is pursued, which owes its origin to works by, among others, J. Myhill (1964), P. Benacerraf (1967), J. (...) Webb (1980, 1983) and M. Arbib (1987). It is shown that the fallacy shared by the two arguments under discussion lies in misidentifying two systems, the one for which the Gödel sentence is constructable and to be proved, and the other in which the Gödel sentence in question is indeed provable. It follows that the logic of the mind can surpass its own Gödelian limitation not by being inconsistent or non-mechanistic, but by being capable of representing stronger systems in itself; and so can a proper machine. The concepts of representational provability, representational maximality, formal system capacity, etc., are discussed. (shrink)

Queen Anne is dead, and it is a fallacy to substitute a definite description for another designator of the same object in stating the content of someone’s propositional attitudes. The fallacy can take subtle forms, as when Godel’s incompleteness theorems are used to argue against mechanistic views of mind. Some instances of the fallacy exemplify a more general logical phenomenon: the set of principles satisfied by one sentential operator can differ from, and even contradict, the set of principles satisfied by (...) another sentential operator coextensive with the first. These notions will be formally defined. The phenomenon will be explored through a series of example, with particular attention to the misapplication of Godel’s results. (shrink)

Over a period of 22 years (1934-1955), the mathematician Kurt Gödel wrote down philosophical remarks, the so-called Maximen Philosophie (Max Phil). They are preserved in 15 notebooks in Gabelsberger shorthand. The first booklet contains general philosophical considerations, booklets two and three consist of Gödel's individual ethics. The following books show that Gödel developed a philosophy of science in which he places his discussions on physics, psychology, biology, mathematics, language, theology and history in the context of a metaphysics. A complete, historical-critical (...) edition of Gödel's Philosophical Notebooks is now being prepared for the first time at the Kurt Gödel Research Centre of the Berlin-Brandenburg Academy of Sciences and Humanities. One volume will be published each year as part of this edition. Volume 5 is particularly interesting because Gödel reveals his satisfaction with it at the beginning of 'Maximen V'. He states that he has arrived at independent thought here, which has not primarily come about through discussion with other authors. Furthermore, in addition to numerous remarkable philosophical considerations, there is a previously unknown interpretation of his (second) incompleteness theorem that goes beyond the mathematical one. (shrink)

In the paper some applications of Gödel's incompleteness theorems to discussions of problems of computer science are presented. In particular the problem of relations between the mind and machine (arguments by J.J.C. Smart and J.R. Lucas) is discussed. Next Gödel's opinion on this issue is studied. Finally some interpretations of Gödel's incompleteness theorems from the point of view of the information theory are presented.

Mechanism is the thesis that men can be considered as machines, that there is no essential difference between minds and machines.John Lucas has argued that it is a consequence of Gödel's theorem that mechanism is false. Men cannot be considered as machines, because the intellectual capacities of men are superior to that of any machine. Lucas claims that we can do something that no machine can do-namely to produce as true the Gödel-formula of any given machine. But no machine can (...) prove its own Gödel-formula. (shrink)

Disertační práce se zabývá problematikou připisování myšlení jiným entitám, a to pomocí imitační hry navržené v roce 1950 britským filosofem Alanem Turingem. Jeho kritérium, známé v dějinách filosofie jako Turingův test, je podrobeno detailní analýze. Práce popisuje nejen původní námitky samotného Turinga, ale především pozdější diskuse v druhé polovině 20. století. Největší pozornost je věnována těmto kritikám: Lucasova matematická námitka využívající Gödelovu větu o neúplnosti, Searlův argument čínského pokoje konstatující nedostatečnost syntaxe pro sémantiku, Blockův návrh na použití brutální síly pro (...) řešení imitační hry, Frenchova teorie subkognitivních informací a Michieho skepticismus ohledně možnosti umělého vědomí. Závěr práce zachycuje současný stav recepce Turingova testu a představuje pokusy o jeho praktickou realizaci, například v každoroční soutěži o Loebnerovu cenu. Autor práce zastává názor, že ani po více než šedesáti letech od uveřejnění Turingova paradigmatického eseje stále neexistují žádné vážné důvody pro zamítnutí jeho tvrzení. Tradiční komputační funkcionalismus možná není ideální teorií vysvětlující činnost myslí a jako slibnější se může jevit vývoj v neurálních vědách, ale Turingův test je přesto užitečným a snad i jediným nástrojem pro detekci inteligence u lidmi vytvořených strojů. (shrink)

This is a paper for a special issue of Semiotic Studies devoted to Stanislaw Krajewski’s paper. This paper gives some supplementary notes to Krajewski’s on the Anti-Mechanist Arguments based on Gödel’s incompleteness theorem. In Section 3, we give some additional explanations to Section 4–6 in Krajewski’s and classify some misunderstandings of Gödel’s incompleteness theorem related to AntiMechanist Arguments. In Section 4 and 5, we give a more detailed discussion of Gödel’s Disjunctive Thesis, Gödel’s Undemonstrability of Consistency Thesis and the definability (...) of natural numbers as in Section 7–8 in Krajewski’s, describing how recent advances bear on these issues. (shrink)

In this paper I will attempt to explain why the controversy surrounding the alleged refutation of Mechanism by Gödel’s theorem is continuing even after its unanimous refutation by logicians. I will argue that the philosophical point its proponents want to establish is a necessary gap between the intended meaning and its formulation. Such a gap is the main tenet of philosophical hermeneutics. While Gödel’s theorem does not disprove Mechanism, it is nevertheless an important illustration of the hermeneutic principle. The ongoing (...) misunderstanding is therefore based in a distinction between a metalogical illustration of a crucial feature of human understanding, and a logically precise, but wrong claim. The main reason for the confusion is the fact that in order to make the claim logically precise, it must be transformed in a way which destroys its informal value. Part of this transformation is a clear distinction between the Turing Machine as a mathematical object and a machine as a physical device. (shrink)

Lucas-Penrose type arguments have been the focus of many papers in the literature. In the present paper we attempt to evaluate the consequences of Gödel’s incompleteness theorems for the philosophy of the mind. We argue that the best answer to this question was given by Gödel already in 1951 when he realized that either our intellectual capability is not representable by a Turing Machine, or we can never know with mathematical certainty what such a machine is. But his considerations became (...) known only in recent times when many scholars were already aware of Benacerraf’s and Chihara’s analyses on the consequences of Gödel’s incompleteness theorems for the philosophy of the mind. Benacerraf and Chihara, in fact, discussing Lucas’ paper, arrived at the same conclusions as Gödel in the sixties, but in a more formal way. After Penrose’s provocative arguments, Shapiro again shed light on the question. In our paper, after a broad and simple presentation of the contributions to the debate made by different authors, we show how to present Gödel’s argument in a rigorous way, highlighting the necessary philosophical premises of Gödel’s argument and more in general of Gödelian arguments. (shrink)