A survey of the field of hypercomputation, including discussion of a variety of objections.

Apophaticism -- the view that God is both indescribable and inconceivable -- is one of the great medieval traditions of philosophical thought about God, but it is largely overlooked by analytic philosophers of religion. This paper attempts to rehabilitate apophaticism as a serious philosophical option. We provide a clear formulation of the position, examine what could appropriately be said and thought about God if apophaticism is true, and consider ways to address the charge that apophaticism is self-defeating. In so doing (...) we draw on recent work in the philosophy of language, touching on issues such as the nature of negation, category mistakes, fictionalism, and reductionism. (shrink)

The Turing Test is claimed by many to be a way to test for the presence, in computers, of such ``deep'' phenomena as thought and consciousness. Unfortunately, attempts to build computational systems able to pass TT have devolved into shallow symbol manipulation designed to, by hook or by crook, trick. The human creators of such systems know all too well that they have merely tried to fool those people who interact with their systems into believing that these systems really have (...) minds. And the problem is fundamental: the structure of the TT is such as to cultivate tricksters. A better test is one that insists on a certain restrictive epistemic relation between an artificial agent A, its output o, and the human architect H of A – a relation which, roughly speaking, obtains when H cannot account for how A produced o. We call this test the ``Lovelace Test'' in honor of Lady Lovelace, who believed that only when computers originate things should they be believed to have minds. (shrink)

Accelerating Turing machines are Turing machines of a sort able to perform tasks that are commonly regarded as impossible for Turing machines. For example, they can determine whether or not the decimal representation of contains n consecutive 7s, for any n; solve the Turing-machine halting problem; and decide the predicate calculus. Are accelerating Turing machines, then, logically impossible devices? I argue that they are not. There are implications concerning the nature of effective procedures and the theoretical limits of computability. Contrary (...) to a recent paper by Bringsjord, Bello and Ferrucci, however, the concept of an accelerating Turing machine cannot be used to shove up Searle's Chinese room argument. (shrink)

What are the limits of physical computation? In his ‘Church’s Thesis and Principles for Mechanisms’, Turing’s student Robin Gandy proved that any machine satisfying four idealised physical ‘principles’ is equivalent to some Turing machine. Gandy’s four principles in effect define a class of computing machines (‘Gandy machines’). Our question is: What is the relationship of this class to the class of all (ideal) physical computing machines? Gandy himself suggests that the relationship is identity. We do not share this view. We (...) will point to interesting examples of (ideal) physical machines that fall outside the class of Gandy machines and compute functions that are not Turing-machine computable. (shrink)

This book attempts to explicate and expand upon Frank Ramsey's notion of the realistic spirit. In so doing, it provides a systematic reading of his work, and demonstrates the extent of Ramsey's genius as evinced by both his responses to the Tractatus Logico-Philosophicus , and the impact he had on Wittgenstein's later philosophical insights.

Ebbhinghaus, H., J. Flum, and W. Thomas. 1984. Mathematical Logic. New York, NY: Springer-Verlag. Forster, T. Typescript. The significance of Yablo’s paradox without self-reference. Available from http://www.dpmms.cam.ac.uk. Gold, M. 1965. Limiting recursion. Journal of Symbolic Logic 30: 28–47. Karp, C. 1964. Languages with Expressions of Infinite Length. Amsterdam.

When Ken Malone investigates a case of something causing mental static across the United States, he is teleported to a world that doesn't exist.

Bertrand Russell's 1903 masterpiece "The Principles of Mathematics" places great emphasis on the need to separate propositions from psychological items such as thoughts. In 1919 Russell explicitly retracts this view, however, and defines propositions as "psychological occurrences". These psychological occurrences are held by Russell to be mental images. In this paper, I seek to explain this radical change of heart. I argue that Russell's re-psychologising of the proposition in 1919 can only be understood against the background of his struggle with (...) the problem of the unity of the proposition in earlier work. Once this is recognized, and the solution to the problem offered by the 1919 theory is appreciated, new light is also shed on Russell's naturalism. I go on to compare Russell's psychological "picture theory" with the vehemently anti-psychological picture theory of Wittgenstein's "Tractatus Logico-Philosophicus" and suggest that, once the background of the dispute is brought into clearer focus, Russell's position can be seen to have many advantages over its more celebrated rival. (shrink)

In this paper, elementary but hitherto overlooked connections are established between Wittgenstein's remarks on mathematics, written during his transitional period, and free-variable finitism. After giving a brief description of theTractatus Logico-Philosophicus on quantifiers and generality, I present in the first section Wittgenstein's rejection of quantification theory and his account of general arithmetical propositions, to use modern jargon, as claims (as opposed to statements). As in Skolem's primitive recursive arithmetic and Goodstein's equational calculus, Wittgenstein represented generality by the use of free (...) variables. This has the effect that negation of unbounded universal and existential propositions cannot be expressed. This is claimed in the second section to be the basis for Wittgenstein's criticism of the universal validity of the law of excluded middle. In the last section, there is a brief discussion of Wittgenstein's remarks on real numbers. These show a preference, in line with finitism, for a recursive version of the continuum. (shrink)

One kind of structuralism holds that mathematics is about structures, conceived as a type of abstract entity. Another denies that it is about any distinctively mathematical entities at all—even abstract structures; rather it gives purely general information about what holds of any collection of entities conforming to the axioms of the theory. Of these, pure structuralism is most plausibly taken to enjoy significant advantages over platonism. But in what appears to be its most plausible—modalised—version, even restricted to elementary arithmetic, it (...) requires defence of a very strong possibility claim: that there could be a completed w-sequence of concrete objects. There are very serious epistemological difficulties in the way of providing the requisite defence. (shrink)

This paper surveys a wide range of proposed hypermachines, examining the resources that they require and the capabilities that they possess. 2005 Elsevier Inc. All rights reserved.

Cora Diamond has recently criticised as mere legend the interpretation of a quip of Ramsey's, contained in the epigraph below, which takes him to be objecting to or rejecting Wittgenstein's Tractarian distinction between saying and showing. Whilst I agree with Diamond's discussion of the legend, I argue that her interpretation of the quip has little evidential support, and runs foul of a criticism sometimes made against intuitionism. Rather than seeing Ramsey as making a claim about the nature of propositions, as (...) Diamond does, we should understand him as making a claim about the grammar of the logical connectives. Such a view coheres with the extant evidence of the nature of Wittgenstein's and Ramsey's 1929 philosophical encounters. It is also compatible with attributing to Ramsey a recognition of Wittgenstein's distinction and with denying that criticising it is the lesson of the quip. (shrink)

Does what guides a pastry chef stand on par, from the standpoint of contemporary computer science, with what guides a supercomputer? Did Betty Crocker, when telling us how to bake a cake, provide an effective procedure, in the sense of `effective' used in computer science? According to Cleland, the answer in both cases is ``Yes''. One consequence of Cleland's affirmative answer is supposed to be that hypercomputation is, to use her phrase, ``theoretically viable''. Unfortunately, though we applaud Cleland's ``gadfly philosophizing'' (...) (as, in fact, seminal), we believe that unless such a modus operandi is married to formal philosophy, nothing conclusive will be produced (as evidenced by the problems plaguing Cleland's work that we uncover). Herein, we attempt to pull off not the complete marriage for hypercomputation, but perhaps at least the beginning of a courtship that others can subsequently help along. (shrink)

The focus is on two texts by Wittgenstein where ‘forms of life’ constitute the pivot of an extended argument: ‘Cause and Effect’ and the discussion of colour concepts in ‘Remarks on the Philosophy of Psychology’. The author argues that forms of life are above all Wittgenstein's response to the question what it is to analyse a concept. The remark that forms of life are ‘given’ and must be ‘accepted’ is a natural corollary of Wittgenstein’s antireductionism and his idea of philosophy (...) as a descriptive enterprise. Wittgenstein is, however, not offering forms of life as the ultimate foundation of our statements about causation or colour. He shifts the focus to the questioning activity itself. Our inquiries and descriptions imply conceptions of how to look and what to accept as verification. Forms of life are given, not because they cannot be analysed further, but because the investigation will take them as given. Comparisons are made, on the one hand, with G.H. von Wright’s interventionist account of the concept of causation and, on the other hand, with the two currently dominant interpretations of ‘forms of life’: the ‘linguistic community’ view and the ‘naturalist’ view. (shrink)

Wittgenstein's conception of infinity can be seen as continuing the tradition of the potential infinite that begins with Aristotle. Transfinite cardinals in set theory might seem to render the potential infinite defunct with the actual infinite now given mathematical legitimacy. But Wittgenstein's remarks on set theory argue that the philosophical notion of the actual infinite remains philosophical and is not given a mathematical status as a result of set theory. The philosophical notion of the actual infinite is not to be (...) found in the mathematics of set theory, only in a certain associated philosophy – what Wittgenstein calls a certain kind of “prose”. (shrink)

A "concept" in the sense favoured by Wittgenstein is a paradigm for a transition between parts of a notational system. A concept-determining sentence such as "There is no reddish green" registers the absence of such a transition. This suggests a plausible account of what is perceived in an experiment that was first designed by Crane and Piantanida, who claim to have induced perceptions of reddish green. I shall propose a redescription of the relevant phenomena, invoking only ordinary colour concepts. This (...) redescription is not ruled out by anything the experimenters say. It accounts for certain peculiarities in both their descriptions and their subjects', and suggests that instead of discovering forbidden colours the experimenters introduced a new use of "-ish". Still, there is a point in speaking of "reddish green" in their context, which can be motivated by invoking what Wittgenstein calls a "physiognomy". (shrink)

The Archimedean Axiom is often held to be an intuitively obvious truth about the geometry of physical space. After a general discussion of the varieties of geometrical intuition that have been proposed, I single out one variety which we can plausibly be held to have and then argue that it does not underwrite the intuitive obviousness of the Archimedean Axiom. Generalizing that result, I conclude that the Axiom is not intuitively obvious.

_ Source: _Volume 6, Issue 2-3, pp 143 - 164 In §513 of _On Certainty_ Wittgenstein asks “What if something _really unheard-of_ happened?” But with this question he is not asking us to make a forecast, a prediction, or some sort of empirico-psychological prophecy about our possible reactions. As I will attempt to show, the question regarding the unheard-of is part of Wittgenstein’s philosophical method—which is to say, it is one of the instruments with which he combats what he sees (...) as the principal source of the confusions of philosophy: mistaking the grammatical for the empirical or, as he also says, the conceptual for the factual. In this sense the question regarding the unheard-of can shed some light on the grammatical status of what he calls “hinges.”. (shrink)

Is it logically possible to perform a "superfeat"? That is, is it logically possible to complete, in a finite time, an infinite sequence of distinct acts? In opposition to the received view, I argue that all physical superfeats have kinematic features that make them logically impossible.

Alan Turing anticipated many areas of current research incomputer and cognitive science. This article outlines his contributionsto Artificial Intelligence, connectionism, hypercomputation, andArtificial Life, and also describes Turing's pioneering role in thedevelopment of electronic stored-program digital computers. It locatesthe origins of Artificial Intelligence in postwar Britain. It examinesthe intellectual connections between the work of Turing and ofWittgenstein in respect of their views on cognition, on machineintelligence, and on the relation between provability and truth. Wecriticise widespread and influential misunderstandings of theChurch–Turing thesis (...) and of the halting theorem. We also explore theidea of hypercomputation, outlining a number of notional machines thatcompute the uncomputable. (shrink)

W ciągu 30 lat dokonała się głęboka przemiana w relacji między Internetem a aktywnością naukową. Nastąpiło przesunięcie funkcji Internetu, który przestał być jedynie instrumentem współpracy akademickiej, a stał się również narzędziem używanym przez media społecznościowe, zezwalające na maksymalną dyfuzję irracjonalizmu i starego faktualizmu. Aby zrozumieć tę przemianę, należy ponownie zbadać mechanizmy zbieżności opinii. Wynika ona z istnienia wspólnego świata, który ukazuje nam te same fakty i determinuje nieustanną rewizję przekonań każdej osoby. W tym procesie asymptotycznej zbieżności opinii nie wymaga się (...) żadnej komunikacji między jednostkami. Oczywiście, jeśli fakty znane przez jednych są komunikowane innym, konwergencja znacznie przyspiesza, i to tak bardzo, że wiedzę musimy ująć jako rezultat zbiorowej aktywności, w której wymiana informacji jest jednym z jej istotnych źródeł. Z jednej strony takie poznawcze współdziałanie wydaje się naturalne i łatwe do implementacji. Biorąc pod uwagę właściwości informacji, które przechowujemy, nawet jeśli je udostępniamy, to dzielenie się informacją przestaje być przedmiotem zwykłych trudności związanych ze współpracą. Z drugiej strony takie współdziałanie jest produktywne z samej natury: współpraca między kimś, kto zna φ, i kimś, kto wie, że φ implikuje ψ u obydwu jako mających wiedzę ψ, której żaden z nich nie posiadał przed wymianą informacji. Wczesny Internet pozwolił na ekstremalną intensywność tej informacyjnej współpracy. Obecnie mamy do czynienia z odmienną sytuacją. Poszczególne wyznaczniki rosnącej porowatości między naukowcami a opinią publiczną wzmocniły rolę wymiany. Nie jest to już wymiana informacji, lecz wymiana opinii. Stronniczość tkwiąca w naturze ludzkiej, a szczególnie potwierdzenie tej stronniczości, prowadzi do odwrócenia relacji między faktami a opiniami. Szukamy faktów potwierdzających podzielane przez nas opinie, wątpiąc w te, które ich nie określają, i tworzymy fakty do potwierdzenia tego, o czym jesteśmy przekonani. Dlatego przechodzimy od współdziałania do przynależności, dzieląc Internet na homogeniczne grupy wyznawców. (shrink)

Résumé. En trente ans, le rapport d’Internet à l’entreprise scientifique a changé. Nous sommes passés d’un instrument de collaboration scientifique à un dispositif de réseaux sociaux qui assure la plus large diffusion à l’irrationalisme et à l’alt-factualisme. Pour comprendre ce changement, il convient de réexaminer les mécanismes de la convergence des opinions. La première source de cette convergence est l’existence d’un monde commun, qui nous expose aux mêmes faits et qui détermine par révision successive des croyances de chacun, sans que (...) la moindre communication soit requise, une convergence asymptotique des opinions, et ce quelles que soient les croyances de départ. Naturellement, si les faits connus par les uns sont communiqués aux autres, cette convergence s’accélère considérablement, si bien que nous devons considérer la connaissance comme le résultat d’une activité collective, et l’échange d’informations comme l’une de ses sources nécessaires. D’une part, cette coopération épistémique est naturelle et facilement implémentable : en vertu des propriétés caractéristiques de l’information, qui est un bien que nous conservons lorsque nous le donnons, le partage informationnel est exempt des difficultés qui affectent généralement la coopération. D’autre part, cette coopération est intrinsèquement productive : la collaboration entre celui qui sait que φ et celui qui sait que φ implique ψ met chacun des deux agents en possession de la connaissance de ψ, qui n’était pourtant détenue par personne avant l’échange. L’internet des débuts donne à cette coopération informationnelle une ampleur inégalée. La situation actuelle est différente. Un certain nombre de facteurs, dont la porosité croissante entre les scientifiques et le large public, ont mis au premier plan l’échange, non des informations factuelles, mais celui des opinions. Les biais notoires de la psychologie humaine, au premier rang desquels le biais de confirmation, tendent à inverser le rapport entre les faits et les opinions. On cherche les faits confirmant les opinions entretenues, on met en doute les faits récalcitrants et l’on crée des faits corroborants. En bref, on passe d’une problématique de la coopération à une logique de l’affiliation, segmentant l’internet en groupes doxastiques homogènes. (shrink)

Las máquinas aceleradas de Turing (ATMs) son dispositivos capaces de ejecutar súper-tareas. Sin embargo, el simple ejercicio de definirlas ha generado varias paradojas. En el presente artículo se definirán las nociones de súper-tarea y ATM de manera exhaustiva y se aclarará qué debe entenderse en un contexto lógico-formal cuando se pregunta por la existencia de un objeto. A partir de la distinción entre posibilidades lógicas y físicas se disolverán las paradojas y se concluirá que las ATMs son posibles y existen (...) como objetos abstractos. (shrink)

For over a decade, the hypercomputation movement has produced computational models that in theory solve the algorithmically unsolvable, but they are not physically realizable according to currently accepted physical theories. While opponents to the hypercomputation movement provide arguments against the physical realizability of specific models in order to demonstrate this, these arguments lack the generality to be a satisfactory justification against the construction of any information-processing machine that computes beyond the universal Turing machine. To this end, I present a more (...) mathematically concrete challenge to hypercomputability, and will show that one is immediately led into physical impossibilities, thereby demonstrating the infeasibility of hypercomputers more generally. This gives impetus to propose and justify a more plausible starting point for an extension to the classical paradigm that is physically possible, at least in principle. Instead of attempting to rely on infinities such as idealized limits of infinite time or numerical precision, or some other physically unattainable source, one should focus on extending the classical paradigm to better encapsulate modern computational problems that are not well-expressed/modeled by the closed-system paradigm of the Turing machine. I present the first steps toward this goal by considering contemporary computational problems dealing with intractability and issues surrounding cyber-physical systems, and argue that a reasonable extension to the classical paradigm should focus on these issues in order to be practically viable. (shrink)

Skinner has been criticized for advancing essentialist interpretations of meaning in which meaning is treated as the property of a word or a grammatical form. Such a practice is consistent with a "words and things" view that sought to advance an ideal language as well as with S-R views that presented meaning as the property of a word form. These views imply an essentialist theory of meaning that would be consistent with Skinner's early S-R behaviorism. However, Skinner's more developed account (...) of meaning is based on his later selectionist behaviorism, and this account of meaning is aligned in many respects with the views of Darwin, Peirce, F. C. S. Schiller, Dewey, and Wittgenstein. After adopting a selectionist theory of meaning, it was inconsistent for Skinner to maintain essentialist practices although Skinner did so in response to influences that had little to do with his selectionist theory. Skinner's particular contribution to this pragmatic-selectionist tradition of meaning is the integration of meaning within his three-term contingency for operant behavior. In this account, meaning lies in the probabilistic functional relations of verbal behavior rather than in its form. (shrink)

This thesis is a critical and comparative study of four commentators on the later Wittgenstein’s rule following considerations. As such its primary aim is exegetical, and ultimately the thesis seeks to arrive at an enriched understanding of Wittgenstein’s work through the distillation of the four commentators into what, it is hoped, can be said to approach a definitive interpretation, freed of their individual frailties. -/- The thesis commences by explicating the position of Kripke’s Wittgenstein. He draws our attention to the (...) ‘sceptical problem’ of how we are to resolve the apparently paradoxical situation that whilst we seem to use language meaningfully, there is no fact about us that constitutes our meaning one thing as opposed to something else, and consequently the possibility of our actually meaning anything seems to evaporate. Kripke interprets Wittgenstein as accepting the validity of the sceptical problem, but seeking to establish that the force of the problem is radically diminished because the justification which it has shown to be unobtainable is actually unnecessary for rule following to take place. -/- McDowell tries to show that Kripke is mistaken when he views Wittgenstein as endorsing scepticism in this way, because he sees Kripke as failing to appreciate a section of Philosophical Investigations which suggests that one ought to reject the sceptical paradox by correcting the misunderstanding which gives rise to it. McDowell reads Wittgenstein’s claim as being that we mistakenly think we are caught in a dilemma which requires us either to endorse the sceptical paradox or to subscribe to a mythological picture of rule following; whereas, so the thought goes, we must reject the entire dilemma. -/- Although McDowell’s criticism of Kripke is essentially correct, he is motivated to that criticism by an incorrect reading of Wittgenstein. Central to this misinterpretation is his failure to note Wittgenstein’s belief that universal scepticism is nonsensical. Winch does much to flesh out the nature of Wittgenstein’s claim here, although he makes the mistake of attributing to Kripke that position which the latter finds in Philosophical Investigations. Despite inheriting this error from Winch, Diamond nonetheless improves on his attempt to characterize the shortcomings of Kripke’s reading as an interpretation of Wittgenstein, enabling the thesis to reach a conclusion about Wittgenstein’s understanding of rule following. (shrink)