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)

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)

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)

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)

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 identity conception of truth holds that a thinkable is true just in case it is a fact. As such, it sets itself against correspondence theories of truth, while respecting the substantive role played by truth in respect of enquiry. In this article, I motivate and develop that view, and, in so doing, promote a particular conception of sense. This allows me to defend the view from two substantial criticisms. First, that the identity conception of truth is incoherent in respect (...) of its treatment of objects in the realm of reference, and, second, that it is committed to a view of the world in which ordinary objects have no place. (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)

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)

There is, in the Ramsey Archive at the Hillman Library of the University of Pittsburgh, a note, written in 1929, in Ramsey's hand and mostly in German, consisting of twenty paragraphs the contents...

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.

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)

A major challenge in the philosophy of mathematics is to explain how mathematical language can pick out unique structures and acquire determinate content. In recent work, Button and Walsh have introduced a view they call ‘internalism’, according to which mathematical content is explained by internal categoricity results formulated and proven in second-order logic. In this paper, we critically examine the internalist response to the challenge and discuss the philosophical significance of internal categoricity results. Surprisingly, as we argue, while internalism arguably (...) explains how we pick out unique mathematical structures, this does not suffice to account for the determinacy of mathematical discourse. (shrink)

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)

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.