The thesis critically examines the question of the philosophical coherence of finitism, the view which seeks to interpret mathematics without postulating an actual infinity of mathematical objects. It is argued that a widely accepted characterization of finitism, most recently expounded by Tait, is inadequate, and a new characterization based on the notion of elementary abstraction is proposed. It is further argued that the notion of elementary abstraction better explains the bearing of Godel's incompleteness theorems on the issue of the (...) coherence of finitism. By abstraction is meant a procedure by which one recognizes or establishes that some infinite process exhibits a distinctive uniformity. If the procedure does not require one to conceive of the integers in any other way than as the result of simple iterations, we call such abstraction elementary. Some six different formal models exploring a variety of different ways in which this notion can be made precise are proposed as possible formal characterizations of finitistic truth. It is proved that all these different models are equivalent with respect to finitistically meaningful sentences. This is taken as strong evidence that the notion of elementary abstraction is determinate enough to provide a basis for a coherent philosophical formulation of finitism. (shrink)

Though Spinoza's definition of God at the beginning of the Ethics unequivocally asserts that God has infinitely many attributes, the reader of the Ethics will find only two of these attributes discussed in any detail in Parts Two through Five of the book. Addressing this intriguing gap between the infinity of attributes asserted in E1d6 and the discussion merely of the two attributes of Extension and Thought in the rest of the book, Jonathan Bennett writes: Spinoza seems to imply (...) that there are other [attributes] – he says indeed that God or Nature has “infinite attributes.” Surprising as it may seem, there are reasons to think that by this Spinoza did not mean anything entailing that there are more than two attributes. In this paper I will argue that Bennett’s claim is fundamentally wrong and deeply misleading. I do think, however, that addressing Bennett’s challenge will help us better understand Spinoza’s notion of infinity. I will begin by summarizing Bennett’s arguments. I will then turn to examine briefly the textual evidence for and against his reading. Then I will respond to each of Bennett’s arguments, and conclude by pointing out theoretical considerations which, I believe, simply refute his reading. (shrink)

Time may be infinite in both directions. If it is, then, if persons could live at most once in all of time, the probability that you would be alive now would be zero. But if persons can live more than once, the probability that you would be alive now would be nonzero. Since you are alive now, with certainty, either the past is finite, or persons can live more than once.

In lieu of an abstract, here is a brief excerpt of the content:Flipping the DeckOn Totality and Infinity’s Transcendental/Empirical PuzzleJack Marsh (bio)How does one perceive a transcendental condition?— Martin Kavka... if it is legitimate to hold Levinas to the standards that he himself imposes on certain other philosophers.— Robert BernasconiI do not believe that there is a transparency possible in method. Nor that philosophy might be possible as transparency.— Emmanuel LevinasThe question of the precise methodological status of the face (...) has yet to be satisfactorily resolved. The best evidence for this claim perhaps lies in texts intended to introduce Levinas to a general philosophical audience. If even first-rate philosophers wrestle still with the puzzle Totality and Infinity bequeaths us, we seem to still await a full and precise diagnosis. For example, Michael Morgan holds the “face-to-face and the responsible self — they are one thing characterized from [End Page 79] two perspectives — are” “the transcendental condition for meaningful human life,” yet elsewhere concedes, “it is unavoidable that in some sense or other the face-to-face does occur as an actually lived experience.” If Morgan’s own nuanced treatment of this question still lands him in equivocation, the problem is certainly not Morgan’s own.1 As Atterton notes, Morgan must finally hold that Levinas’s method is a “transcendental enterprise of a certain sort.”2 But of what sort? Indeed, this question has been asked or analyzed in one way or another from the beginning of Totality and Infinity’s English-language reception.3 Of course, Derrida was the first to broach the question of Totality and Infinity’s general intelligibility, alleging that Levinas ultimately succumbs to a form of empiricism. Of course, de Boer was the first to mount a rigorous Cartesian reading of the text as an (apparently) “ethical transcendental philosophy.” Of course, Bernasconi’s response to de Boer, and his creative repetition of Derrida’s critical point, was the first to give the problem a precise formulation.4 Derrida, de Boer, and Bernasconi hand us the now canonical problem of what I will call the text’s “transcendental/empirico-metaphysical puzzle.”5 The issue boils down to Levinas’s dual insistence that the face “cannot... be stated in terms of experience,” yet is nevertheless “reflected within experience” (TI 25, 23, 67, 260). How should we understand Levinas’s references to various “formal structure[s]” — “of language,” “of interpellation,” “of the idea of infinity” — next to talk of “pure” or “concrete moral experience” (195, 79, 64, 53)? Does the Other occur in “pure experience,” or is she “deduced” as a condition of possibility for experience in general? In this essay, I will propose an internal explanation for Totality and Infinity’s longstanding methodological puzzle. By proposing to “explain” this puzzle, I am making at least three distinct claims: to show (1) that the text suffers from genuine incoherence, thus confirming the general criticism of Derrida, Bernasconi, and others, (2) precisely how this incoherence follows from the details of Levinas’s own explicit method, and (3) precisely why, philosophically, he may have constructed and performed his method as he did.6 By proposing to explain the book’s incoherence, I can [End Page 80] also make specific hermeneutic predictions. If Totality and Infinity’s transcendental/empirico-metaphysical puzzle is genuinely aporetic, it follows that no resolution to it is possible on the text’s own terms.7 As such, any attempt to resolve the puzzle must either (1) augment the text’s own descriptions, (2) utilize other philosophical resources to render its claims coherent, or (3) precisely attempt resolution in either a transcendental or empirico-metaphysical direction. As it turns out, extant scholarship reflects just these hermeneutic strategies.Over the years many philosophers have contributed to this longstanding debate, variously arguing for or against the general intelligibility of Levinas’s account of the face. On the one hand, scholarship that questions the text’s intelligibility does so from a variety of methodological perspectives.8 On the other hand, scholarship that mounts a defense of the text seems to proceed in one of three ways: by (1) apparently repeating Totality and Infinity’s methodological problems, (2) anticipating Levinas’s later work, read... (shrink)

According to infinitism, all justification comes from an infinite series of reasons. Peter Klein defends infinitism as the correct solution to the regress problem by rejecting two alternative solutions: foundationalism and coherentism. I focus on Klein's argument against foundationalism, which relies on the premise that there is no justification without meta-justification. This premise is incompatible with dogmatic foundationalism as defended by Michael Huemer and Time Pryor. It does not, however, conflict with non-dogmatic foundationalism. Whereas dogmatic foundationalism rejects the need for (...) any form of meta-justification, non-dogmatic foundationalism merely rejects Laurence BonJour's claim that meta-justification must come from beliefs. Unlike its dogmatic counterpart, non-dogmatic foundationalism can allow for basic beliefs to receive meta-justification from non-doxastic sources such as experiences and memories. Construed thus, non-dogmatic foundationalism is compatible with Klein's principle that there is no justification without meta-justification. I conclude that Klein's rejection of foundationalism. fails. Nevertheless, I agree with Klein that when in response to a skeptical challenge we engage in the activity of defending our beliefs, the number of reasons we can give is at least in principle infinite. I argue that this type of infinity is benign because, when we continue to give reasons, we will eventually merely repeat previously stated reasons. Consequently, I reject Klein's claim that the more reasons we give the more we increase the justification of our beliefs. (shrink)

We investigate in detail the electromagnetic fields of a uniformly accelerated charge, in order to ascertain whether such a charge does ‘emit’ radiation, especially in view of the Poynting flow computed at large distances and taken as an evidence of radiation emitted by the charge. In this context, certain important aspects of the fields need to be taken into account. First and foremost is the fact that in the case of a uniformly accelerated charge, one cannot ignore the velocity (...) fields. This then leads to other equally vital points. The net field energy turns out to be exactly the same as that of a non-accelerated charge having a uniform velocity equal to the instantaneous velocity of the uniformly accelerated charge. Further, the Poynting vector, seen with respect to the ’present’ location of the uniformly accelerated charge, during the deceleration phase, possesses everywhere a radial component pointing inward toward the charge, becoming nil when the charge becomes momentarily stationary, and during the acceleration phase, points away from the charge position. Last, but not least, when the leading spherical front of the relativistically beamed Poynting flux, advances forward at a large time t to a far-off distance \, the charge too is not lagging far behind. In fact, these relativistically beamed fields, increasingly resemble fields of a charge moving in an inertial frame with a uniform velocity \, with a convective flow of fields in that frame along with the movement of the charge. There is no other Poynting flow in the far-zones that could be termed as radiation emitted by the charge which, in turn, is fully consistent with the absence of radiation reaction and is also fully conversant with the strong principle of equivalence. (shrink)

The regress of reasons threatens an epistemic agent’s right to claim that any beliefs are justified. In response, Peter Klein’s infinitism argues that an infinite series of supporting reasons of the right type not only is not vicious but can make for epistemic justification. In order to resist the sceptic, infinitism needs to provide reason to think that there is at least one justified belief in the world. Under an infinitist conception this involves showing that at least one belief is (...) supported by an infinite series of supporting reasons. This paper argues that showing this makes problems for the infinitist. The finite minds problem that prevents completion of an infinite series is well documented. This paper examines alternative attempts to provide evidence of infinity that the infinitist might take, whether by using a notion of justification without infinite reasons or by altering the notion of evidence. It concludes that both of these fail and consequently infinitism is unable to offer a solution to Agrippa’s trilemma. (shrink)

We will discuss a mathematical proof found in Wittgenstein’s Nachlass, a constructive version of Euler’s proof of the infinity of prime numbers. Although it does not amount to much, this proof allows us to see that Wittgenstein had at least some mathematical skills. At the very last, the proof shows that Wittgenstein was concerned with mathematical practice and it also gives further evidence in support of the claim that, after all, he held a constructivist stance, at least during (...) the transitional period of his thought (1929-33). (shrink)

There are possible worlds in which time is circular and finite in duration, forming a loop of, say, 12,000 years. There are also possible worlds in which time is linear and infinite in both directions, and in which history is repetitive, consisting of infinitely many 12,000 year epochs, each two of which are exactly alike with respect to all intrinsic, purely qualitative properties. Could one ever have empirical evidence that one inhabits a world of the first kind rather than (...) a world of the second kind? We argue for the affirmative answer, contra Quine (1979), Newton-Smith (1980), and Bergström (2013). Our argument for that conclusion differs from an argument for the same conclusion due to Susan Weir, reported by Richard Sorabji (1988). Weir’s argument is probabilistic and explicitly requires having evidence against determinism. Our argument is a direct appeal to the simplicity of laws, and it involves no probabilistic component. It is modeled on Shoemaker’s (1969) argument that one could have evidence of time without change. (shrink)

This volume is based on the talks given at the Workshop on Infinity and Truth held at the Institute for Mathematical Sciences, National University of Singapore, from 25 to 29 July 2011. The chapters are by leading experts in mathematical and philosophical logic that examine various aspects of the foundations of mathematics. The theme of the volume focuses on two basic foundational questions: (i) What is the nature of mathematical truth and how does one resolve questions that are formally (...) unsolvable within the Zermelo Fraenkel Set Theory with the Axiom of Choice, and (ii) Do the discoveries in mathematics provide evidence favoring one philosophical view over others? These issues are discussed from the vantage point of recent progresses in foundational studies. The final chapter features questions proposed by the participants of the Workshop that will drive foundational research. The wide range of topics covered here will be of benefit to students, researchers and mathematicians interested in the foundations of mathematics. (shrink)

A number of philosophers of science and statisticians have attempted to justify conclusions drawn from a finite sequence of evidence by appealing to results about what happens if the length of that sequence tends to infinity. If their justifications are to be successful, they need to rely on the finite sequence being either indefinitely increasing or of a large size. These assumptions are often not met in practice. This paper analyzes a simple model of collecting evidence and (...) finds that the practice of collecting only very small sets of evidence before taking a question to be settled is rationally justified. This shows that the appeal to long run results can be used neither to explain the success of actual scientific practice nor to give a rational reconstruction of that practice. (shrink)

We introduce a novel mechanism to eliminate endgame effects in repeated prisoner’s dilemma experiments. In the main phase of a supergame our mechanism generates more persistent cooperation than finite horizon or random continuation rules. Moreover, we find evidence for cooperation-enhancing “active/reactive” strategies which concentrate in the initial phase of a supergame as subjects gain experience.

Splendour and misery of exaggeration : an introduction -- Thinking as gesture : exaggeration and philosophy -- Trust me : exaggeration and enlightenment -- Odd moves : exaggeration and irony -- The violence of destruction : exaggeration and infinity -- ? and ? end? : exaggeration and politics -- Being guilty : exaggeration and factuality -- Flight simulator : exaggeration and trauma -- The obvious : exaggeration and self-evidence -- Blow job : exaggeration and institution -- Old opera (...) : exaggeration and art -- Life lines : exaggeration and event -- Lexical note. (shrink)

The way Leibniz applied his philosophy to mathematics has been the subject of longstanding debates. A key piece of evidence is his letter to Masson on bodies. We offer an interpretation of this often misunderstood text, dealing with the status of infinite divisibility in nature, rather than in mathematics. In line with this distinction, we offer a reading of the fictionality of infinitesimals. The letter has been claimed to support a reading of infinitesimals according to which they are logical (...) fictions, contradictory in their definition, and thus absolutely impossible. The advocates of such a reading have lumped infinitesimals with infinite wholes, which are rejected by Leibniz as contradicting the part–whole principle. Far from supporting this reading, the letter is arguably consistent with the view that infinitesimals, as inassignable quantities, are mentis fictiones, i.e., (well-founded) fictions usable in mathematics, but possibly contrary to the Leibnizian principle of the harmony of things and not necessarily idealizing anything in rerum natura. Unlike infinite wholes, infinitesimals—as well as imaginary roots and other well-founded fictions—may involve accidental (as opposed to absolute) impossibilities, in accordance with the Leibnizian theories of knowledge and modality. (shrink)

In this paper, I explore Wittgenstein’s conception of a hypothesis as articulated in Chapters XII and XXII of ‘Philosophical Remarks’. First, I argue that in Chapter XII, Wittgenstein draws on his account of infinity to begin to challenge the view that all hypotheses can be proven by empirical evidence. I then argue that in Chapter XXII that Wittgenstein sharpens this conception of hypotheses claiming that no hypotheses can be verified. Finally, I suggest that Wittgenstein’s conception of a hypothesis (...) relates to his practical view of how language functions. (shrink)

This article is devoted to a phenomenological analysis and interpretation of the basic concepts involved in phenomenology. The first concept that deserves our attention is that of “apodicticity” along with the related concept of “apodictic evidence”. The next concept is that of “correlation,” which manifests its apodictic character in bringing together all transcendental facts. With the transition from traditional discourses to modern discourse, the concept of apodicticity continues to deepen; at the same time, it is accepted that some metaphysical (...) presuppositions continue to lurk at the foundation of this concept. Apodicticity, as the principle of knowledge, is an idea that lies in the infinity of universal experience, which includes both agreed and mismatched moments. This mismatch, in turn, refers to correlative connections between the transcendental Ego, the world and intersubjectivity. Transcendental subjectivity has a correlative nature, which is due to its self-transcending character, resulting in gaps and distinctions being found within subjectivity. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:“Even the Papuan is a Man and not a Beast”: Husserl on Universalism and the Relativity of CulturesDermot Moran (bio)“[A]nd in this broad sense even the Papuan is a man and not a beast.” ([U]nd in diesem weiten Sinne ist auch der Papua Mensch und nicht Tier, Husserl, Crisis, 290/Hua. VI.337–38)1“Reason is the specific characteristic of man, as a being living in personal activities and habitualities.” (Vernunft ist das (...) Spezifische des Menschen, als in personalen Aktivitäten und Habitualitäten lebenden Wesens, Husserl, Crisis, 338/Hua. VI.272)1. Particular Historicities and Universal ReasonIn this paper2 i shall explore—and suggest a way of resolving—the evident tensions to be found in Husserl’s Crisis of European Sciences (and associated texts) [End Page 463] between his commitment to the universality of reason as the goal or telos of European humanity (founded on the ancient Greek “breakthrough” to philosophy and science) and his recognition of the empirical plurality and “relativity” (Relativität) of individual peoples and nations (e.g. Indian, Chinese, Papuan, Bantu) locked into their own particular “socialities” (Sozialitäten), communal worlds, and historical trajectories (what Husserl broadly calls “historicities,” Geschichtlichkeiten, Historizitäten). 3 I shall examine the complex relations and tensions between Husserl’s conception of universality, whereby the same reason functions in every human as animal rationale, “no matter how primitive he is” (“Origin of Geometry,” Crisis, 378/ Hua. VI.385), and his concept of the self-enclosed particularity of individual peoples with their own cultural forms. Indeed, Husserl often emphasizes that the most prominent feature of cultural plurality is precisely its relativity: “relativity belongs to the normal course of life.”4 I shall evaluate Husserl’s response, which defends the project of realizing the ideal of a critical universal rationality, by situating his discussion in terms of the cultural conflict of the time with the rising National Socialist commitment to racial particularism, and by showing Husserl’s commitment to the inherent universality of the one shared life-world.In his research manuscripts of the 1920s and 1930s Husserl frequently discusses the complex relationships that exist between different cultures and traditions; different cultures have their specific historicities (Crisis, 274/Hua. VI.320), dialects, norms, ways of life, and so on. This is simply a matter of fact. There are, furthermore, some well-known and controversial passages in Husserl’s “Vienna Lecture” of May 1935 and also in his Crisis texts (both in the main 1936 published text, Part One [§6] and in the then unpublished Crisis Part Three A §365) where Husserl speaks of the universality inherent in European philosophical culture of the logos and contrasts it with various other communal forms, which are, in his view, merely “empirical-anthropological” types, enclosed in their particular historicities and relativities. Indeed, in this context, Husserl regularly invokes the idea of the “relativity of everything historical” (die Relativität alles Historischen, Crisis, 373/Hua. VI.382). In addition, there are a number of related texts, collected in the Crisis supplementary volume (Hua. XXIX),6 in the Intersubjectivity volumes (especially Hua. XV),7 as well as in the recent volume on the life-world [End Page 464] (Hua. XXXIX),8 which discuss the empirical differences between peoples and also the layers and strata of social groups, nations, and even the idea of larger international collectivities or “supernations” (Übernationen), such as Europe, or the League of Nations. Furthermore, there are—as Husserl indicates in his 1935 letter to the prominent French anthropologist Lucien Lévy-Bruhl—even cultures that are completely “self-enclosed” (abgeschlossene) unities, cut off, some of which know no history. According to Husserl, into this classical world of closed cultures, the ancient Greeks bring a new form of universality, one that leads them, through the grasp of idealization to infinite tasks, to break through the finite horizons of their environing world (Umwelt), which presents itself to them as a “near-world” (Nahwelt, Crisis, 324/Hua. VI.303; Hua. XXVII.228), and to arrive at the highly refined concept of the “true world” or the “scientific world [which] is a purposeful structure (Zweckgebilde) extending to infinity” (Crisis, 382... (shrink)

In lieu of an abstract, here is a brief excerpt of the content:PLOTINUS AND THE APEIRON OF PLATO'S PARMENIDES JOHN H. HEISER Niagara Unwersity Niagara University, New York WE USE THE TERM "infinite" so freely to designate what supposedly transcends something called ' the finite " that one might imagine the concept to be entirely unproblematic. Greek philosophy's difficulty even entertaining such an idea then appears as a sort of myopia, which we in our superior enlightenment have escaped. I propose (...) to examine the way in which one Greek mind, that of Hotinus, came to entertain rthe idea. There may be no better wiay to bring out the difficulties, as well as the possibilities, inherent in a concept than to observe a first class philosophical mind wrestling with it. I also hope rto show that we have a lot to learn from Plotinus. Plotinus's doctrine of the infinite has of course been examined before, on more than one occasion.1 But those who have done so, judging from what they say or do not say, appear to find nothing puzzling in the concept of " infinite perfection " or" infinite being." "We" have come to understand exactly what it means (or at least we bandy the term about 1 For instance: A. H. Armstrong, "Plotinus' Doctrine of the Infinite and its Significance for Christian Thought," Downside Review 73 (1955) : 47-58; Leo J. Sweeney, S.J., "Infinity in Plotinus," Gregorianum 38 (1957) : 515535 ; 713-732; W. Norris Clarke, S.J., "Infinity in Plotinus: a Reply,'' Gregorianum 40 (1959): 75-98; John Whittaker, "Philological Comments on the Neoplatonic Notion of Infinity,'' in The Significance of Neoplatonism, ed. R. Baine Harris (Norfolk, Va.: International Society for Neoplatonic Studies, 1976), pp. 155-172. John M. Rist devotes the third chapter, "The Plotinian One," of his Plotinus: The Road to Reality (New York: Cambridge Uninrsity Press, 1967) largely to the infinity of the One. 58 54 JOHN H. HEISER as if we did). Greek difficulties are understandable (ailthough it is hal'd not to be patronizing about them), but once we have taken note of these difficulties the only question of interest would seem to be whether Plotinus learned to discount them and to talk as we do. In my view, the only way to appreciate Plotinus's aichievement, and to learn from him, is to take these difficulties with great seriousness and to stmggle through them with Plotinus. Another point most commentators overlook is something I take to be both a frucit and a clue to much of what Plotinus says about the Infinite Source of the universe. This is that, in explaining why the One should be called " infinite," Plotinu!! is presenting his e:xiegesis of a perplexing passage in Plato's Parmenides.2 Rather than following his own 1ine of thought to the conc1usion that the One must be infinite, we see Plotinus casting about for some meaning he can assign to a term he is confronted with. I happen to think that his exegesis of Plato is mistaken. This raises the interesting possibility that Plotinus 's breakthrough, if it is a breakthrough and not an aberration, resulted by accident from a mis.take. Before examining Greek views on the infinite, I should like to play Socrates for a moment and examine the concept itself of " infinite perfection." A bit of dialectic might help dispel some of the complacency that surrounds it. As it stands, "infinitely perfect" is at least verbally self-contradictory, as if one were to say " endlessly finished." This may explain why even Plotinus never put it quite thaJt way. But even if we can get around that, what could it mean for something to be 2 :M:ost commentators simply ignore the point, but Professor Armstrong, who is usually so alert to sources, makes the surprising statement that "there is no clear evidence that Plato ever thought or spoke of the One or Good as in any sense apeiron" ("Doctrine of the Infinite," p. 48). Professor Rist goes further and claims that "the 'beyond Being' of Plato must mean a finite Being in some way beyond other finite beings" (Road to Reality, p. 24). But Rist appears to assume he and his readers share a common understanding of what it... (shrink)

On standard accounts of responsibility, one is thought to be responsible for one's own actions or affairs. Levinas' philosophy speaks of a responsibility that goes beyond my actions and their consequences to an infinite, irrecusable, asymmetrical responsibility for the other human. In the dissertation, I present a defense of Levinasian responsibility and argue that distinctive of Levinas' thought as an ethics is the manner in which it maintains the absolute and unexceptionable character of responsibility, while simultaneously putting into question every (...) possible ground or justification for this claim. The central chapters of the dissertation consider the connection between the formal structure of transcendence and the deformalization of this notion as responsibility. After discussing the notions of transcendence employed in the phenomenology of Husserl and Heidegger, I examine Levinas' criticism that transcendence in their sense remains an immanence insofar it continues to tie the conditions of intelligibility to modes of understanding and comprehension. Levinas' reconception of transcendence as a relation to exteriority and absolute alterity is then examined as it is developed first in his early writings , and then in Totality and Infinity through the tropes of the idea of infinity, metaphysical desire, and the face. The aim of these chapters is not only to elucidate in detail the conception of transcendence, but to examine why Levinas thinks that transcendence is already an ethical event. While commentators have sometimes claimed that ethics in Levinas' sense is neither normative nor descriptive, I show that the question of normativity--of how ethical claims are binding on us--is central to the problematic of responsibility, though in a way which ultimately forces us to revise our understanding of both normativity and ethics. Levinas contests the possibility of decisive evidence that would prove the truth of our ethical obligations. What I show is how the call for proof or justification always already distorts and passes over the meaning of responsibility. (shrink)

Most interpreters hold that Kant rejects actually infinite tota synthetica as conceptually impossible. This view is attributed to Kant to relieve him of the charge that the first antinomy’s thesis argument presupposes transcendental idealism. I argue that important textual evidence speaks against this view, and Kant in fact affirms the conceptual possibility of actually infinite tota synthetica. While this means the first antinomy may not be decisive as an indirect argument for idealism, it gives us a better account of (...) how our ideas of the unconditioned generate the antinomies, and it allows us to see important and often overlooked elements in Kant’s account of the infinite. (shrink)

The Death of the Past argues that critical problems in the philosophy of history, such as the the truth of historical texts, how texts relate to the past that they are about, and the nature of historical explanation, can be successfully investigated if we accept the claim that historical writing is historicist--perspectival (from the standpoint of the historian) rather than purporting to be like an eyewitness account (as in the first-person "presentist" views critiqued by Enzo Traverso). This approach admits all (...) relevant evidence as subject to analysis in order to determine the widest point of view, as opposed to the biased "eyewitness" approach, which observes only partially. The link between the past itself and historical knowledge is radically broken, giving the latter the wisdom of hindsight across a broader temporal field from the events examined up to the present. The truth about the past is only accessible from a later point in time; "the past" itself is not a reality for the historian; it no longer exists. Frank Ankersmit offers a surprising foundation for the historicist notion of history: Leibniz' monadological universe, in which each monad or substance has its own point of view. There is no deeper ontology to which they owe their status--just an infinity of monads and a corresponding infinity of points of view on other monads. So with historical representations, statements about the past that can be either true or false but not depictions of the past itself. In striking contrast to Hayden White's view, Ankersmit insists that historical writing is no less rational than the sciences-- mathematical and scientific writing has its counterpart in the writing of history. (shrink)

The author looks to "deep structures" or "submerged" models to give insight into Malebranche's thought. He finds two such models: the traditional model of "substance," according to which things are organized in a hierarchy of genera and species, and the model of "number" which emerged in the seventeenth century, and in accordance with which relations are manipulated to establish further relations. The two archetypal concepts, the one old, the other new, create a tension in many areas of thought. This tension (...) is examined first as it occurs in Descartes. With Malebranche the polarity becomes even more evident. The concept of number is at the base primarily of his epistemology and theory of science, that of substance primarily in his metaphysics and theology. The two spheres are however mutually affecting. Arithmetic and algebra provide Malebranche with "the true logic that serves to discover truth and produce in the mind all the extension of which it is capable." Algebra is "the most fruitful and certain of all sciences... with it the mind is capable of knowing all that can be known with certitude and evidence." Accordingly truths will be of the sort found in the relations typified in algebra as opposed to the subject-predicate relation appropriate to an ontology of substances generically conceived. Malebranche's "ideas," taken ontologically, are viewed by him as substantive entities. Epistemologically, however, they have the nature of relations of magnitude. "The order of truths as relations between ideas was equally," says Hobart, "an order of ideas that were themselves relations--in the same way that the relations between numbers were in effect relations between relations 'compounded to infinity'." The traditional concept of substance to which Hobart refers is primarily Platonic, for he says that in accordance with it, "The world of forms... is the truest and hence the most real, contrasting with the perceptible world in which things are not so perfect. This conception of reality enabled and indeed required the gradations of perfection from the perceptible and imperfect to the real and perfect, those infinite qualitative gradations that comprised the existence and plenitude of the great chain of being." Malebranche's attitude to this is seen by the author to be ambiguous. On the one hand, under the influence of the mathematical and relational way of conceiving truth, a true relation is both real and existent, and a false one neither real nor existent. Hence instead of thinking of reality in terms of gradations of qualities, it is taken simply as the existent, and therefore as merely disjunctive with the non-existent. Nothing can be predicated of the non-existent. At the same time Malebranche wants to retain the gradations of perfection. Thus he says, "But one can distinguish in God two kinds of truths or relations: relations of magnitude and relations of perfection," the latter being assigned to the realm of the practical and moral. The final chapter entitled "The Conflict between 'Substance' and 'Number'" is concerned with how this conflict affected Malebranche's understanding of the relation between science and religion. The conflict is described as occurring within the concepts of infinity, necessity and being in their relation to the divine nature. Finally there is the question of the relation of reason and faith, and here Hobart finds that a dual sense of faith and a dual sense of reason compete with one another in Malebranche's thought. This book is a subtle and novel treatment of the great Oratorian.--Robert McRae, University of Toronto. (shrink)

Kant's theory of transcendental ideas can be conceived as a sort of model theory for an empirical first-order object theory. The main features of Kant's theory of transcendental ideas (especially its antinomies and their solutions) can be recognized, in a modified way, in a religious discourse as exemplified in the dialogue of Jesus and the Samaritan woman (John 4). In this way, what is by Kant meant merely as regulative ideas obtains a sort of objective reality and becomes a religiously (...) founded metaphysics. A metaphysical theory of religious dialogue is formalized on the basis of an extended justification logic of evidence and wish agents. (shrink)

Our evidence can be about different subject matters. In fact, necessarily equivalent pieces of evidence can be about different subject matters. Does the hyperintensionality of ‘aboutness’ engender any hyperintensionality at the level of rational credence? In this paper, I present a case which seems to suggest that the answer is ‘yes’. In particular, I argue that our intuitive notions of independent evidence and inadmissible evidence are sensitive to aboutness in a hyperintensional way. We are thus left (...) with a paradox. While there is strong reason to think that rational credence cannot make such hyperintensional distinctions, our intuitive judgements about certain cases seem to demand that it does. (shrink)

A part of the “return to religion” now evident in European philosophy, this book represents the culmination of the career of a leading phenomenological thinker whose earlier works trace a trajectory from Marx through a genealogy of psychoanalysis that interprets Descartes’s “I think, I am” as “I feel myself thinking, I am.” In this book, Henry does not ask whether Christianity is “true” or “false.” Rather, what is in question here is what Christianity considers as truth, what kind of truth (...) it offers to people, what it endeavors to communicate to them, not as a theoretical and indifferent truth, but as the essential truth that by some mysterious affinity is suitable for them, to the point that it alone is capable of ensuring them salvation. In the process, Henry inevitably argues against the concept of truth that dominates modern thought and determines, in its multiple implications, the world in which we live. Henry argues that Christ undoes “the truth of the world,” that He is an access to the infinity of self-love, to a radical subjectivity that admits no outside, to the immanence of affective life found beyond the despair fatally attached to all objectifying thought. The Kingdom of God accomplishes itself in the here and now through the love of Christ in what Henry calls “the auto-affection of Life.” In this condition, he argues, all problems of lack, ambivalence, and false projection are resolved. (shrink)

Thomas Sebeok and Noam Chomsky are the acknowledged founding fathers of two research fields which are known respectively as Biosemiotics and Biolinguistics and which have been developed in parallel during the past 50 years. Both fields claim that language has biological roots and must be studied as a natural phenomenon, thus bringing to an end the old divide between nature and culture. In addition to this common goal, there are many other important similarities between them. Their definitions of language, for (...) example, have much in common, despite the use of different terminologies. They both regard language as a faculty, or a modelling system, that appeared rapidly in the history of life and probably evolved as an exaptation from previous animal systems. Both accept that the fundamental characteristic of language is recursion, the ability to generate an unlimited number of structures from a finite set of elements (the property of ‘discrete infinity’). Both accept that human beings are born with a predisposition to acquire language in a few years and without apparent efforts (the innate component of language). In addition to similarities, however, there are also substantial differences between the two fields, and it is an historical fact that Sebeok and Chomsky made no attempt at resolving them. Biosemiotics and Biolinguistics have become two separate disciplines, and yet in the case of language they are studying the same phenomenon, so it should be possible to bring them together. Here it is shown that this is indeed the case. A convergence of the two fields does require a few basic readjustments in each of them, but leads to a unified framework that keeps the best of both disciplines and is in agreement with the experimental evidence. What is particularly important is that such a framework suggests immediately a new approach to the origin of language. More precisely, it suggests that the brain wiring processes that take place in all phases of human ontogenesis (embryonic, foetal, infant and child development) are based on organic codes, and it is the step-by-step appearance of these brain-wiring codes, in a condition that is referred to as cerebra bifida, that holds the key to the origin of language. (shrink)

Yitzhak Melamed here offers a new and systematic interpretation of the core of Spinoza's metaphysics. In the first part of the book, he proposes a new reading of the metaphysics of substance in Spinoza: he argues that for Spinoza modes both inhere in and are predicated of God. Using extensive textual evidence, he shows that Spinoza considered modes to be God's propria. He goes on to clarify Spinoza's understanding of infinity, mereological relations, infinite modes, and the flow of (...) finite things from God's essence. In the second part of the book, Melamed relies on this interpretation of the substance-mode relation and the nature of infinite modes and puts forward two interrelated theses about the structure of the attribute of Thought and its overarching role in Spinoza's metaphysics. First, he shows that Spinoza had not one, but two independent doctrines of parallelism. Then, in his final main thesis, Melamed argues that, for Spinoza, ideas have a multifaceted structure that allows one and the same idea to represent the infinitely many modes which are parallel to it in the infinitely many attributes. Thought turns out to be coextensive with the whole of nature. Spinoza cannot embrace an idealist reduction of Extension to Thought because of his commitment to the conceptual separation of the attributes. Yet, within Spinoza's metaphysics, Thought clearly has primacy over the other attributes insofar as it is the only attribute which is as elaborate, as complex, and, in some senses, as powerful as God. (shrink)

The axiom of infinity states that infinite sets exist. I will argue that this axiom lacks justification. I start by showing that the axiom is not self-evident, so it needs separate justification. Following Maddy’s :481–511, 1988) distinction, I argue that the axiom of infinity lacks both intrinsic and extrinsic justification. Crucial to my project is Skolem’s From Frege to Gödel: a source book in mathematical logic, 1879–1931, Cambridge, Harvard University Press, pp. 290–301, 1922) distinction between a theory of (...) real sets, and a theory of objects that theory calls “sets”. While Dedekind’s argument fails, his approach was correct: the axiom of infinity needs a justification it currently lacks. This epistemic situation is at variance with everyday mathematical practice. A dilemma ensues: should we relax epistemic standards or insist, in a skeptical vein, that a foundational problem has been ignored? (shrink)

A bold and all-embracing exploration of the nature and progress of knowledge from one of today's great thinkers. Throughout history, mankind has struggled to understand life's mysteries, from the mundane to the seemingly miraculous. In this important new book, David Deutsch, an award-winning pioneer in the field of quantum computation, argues that explanations have a fundamental place in the universe. They have unlimited scope and power to cause change, and the quest to improve them is the basic regulating principle not (...) only of science but of all successful human endeavor. This stream of ever improving explanations has infinite reach, according to Deutsch: we are subject only to the laws of physics, and they impose no upper boundary to what we can eventually understand, control, and achieve. In his previous book, The Fabric of Reality, Deutsch describe the four deepest strands of existing knowledge-the theories of evolution, quantum physics, knowledge, and computation-arguing jointly they reveal a unified fabric of reality. In this new book, he applies that worldview to a wide range of issues and unsolved problems, from creativity and free will to the origin and future of the human species. Filled with startling new conclusions about human choice, optimism, scientific explanation, and the evolution of culture, The Beginning of Infinity is a groundbreaking book that will become a classic of its kind. (shrink)

Postmodern critical assessments of Freud’s theory of mourning disavow the idea of grief’s conclusiveness, insisting that mourning is an interminable process or even a transcendental structure of experience. However, such assessments presuppose an ontological orientation toward finitude that avoids the profound speculative implications of the non-finite status of death in the unconscious. In consequence, mourning comes to assume an indefinite, generic status as a condition of experience instead of a resolutely speculative confrontation with the impossible real of infinitude. Freud’s writings (...) evidence his difficulties with this unconscious disjunction of mourning from finitude, but can be elucidated by turning to Hegel’s critical discussion of “infinite grief,” the melancholy apprehension of finitude as a spurious infinity. Hegel’s analysis in turn anticipates Lacan’s elaboration of the death drive as a concept of unconscious transmission. Here, what begins to emerge is a theory of mourning that concerns the speculative inheritance of an immortal object. (shrink)

A part of the “return to religion” now evident in European philosophy, this book represents the culmination of the career of a leading phenomenological thinker whose earlier works trace a trajectory from Marx through a genealogy of psychoanalysis that interprets Descartes’s “I think, I am” as “I feel myself thinking, I am.” In this book, Henry does not ask whether Christianity is “true” or “false.” Rather, what is in question here is what Christianity considers as truth, what kind of truth (...) it offers to people, what it endeavors to communicate to them, not as a theoretical and indifferent truth, but as the essential truth that by some mysterious affinity is suitable for them, to the point that it alone is capable of ensuring them salvation. In the process, Henry inevitably argues against the concept of truth that dominates modern thought and determines, in its multiple implications, the world in which we live. Henry argues that Christ undoes “the truth of the world,” that He is an access to the infinity of self-love, to a radical subjectivity that admits no outside, to the immanence of affective life found beyond the despair fatally attached to all objectifying thought. The Kingdom of God accomplishes itself in the here and now through the love of Christ in what Henry calls “the auto-affection of Life.” In this condition, he argues, all problems of lack, ambivalence, and false projection are resolved. (shrink)

Ockham realized the value of a careful study and application of the mathematical theory of continuity, even though he possessed no conclusive empirical evidence that it adequately explained the continuity experienced in the world of sense. He also clearly perceived a relation of continuity to infinity, since continuity, as a property of an order which is possible only to a series of terms, must have an infinite number of terms. “A continuous series must have an infinite number of (...) terms.”. (shrink)

This article attempts to broaden the phenomenologically motivated perspective of H. Weyl's Das Kontinuum in the hope of elucidating the differences between the intuitive and mathematical continuum and further providing a deeper phenomenological interpretation. It is known that Weyl sought to develop an arithmetically based theory of continuum with the reasoning that one should be based on the naturally accessible domain of natural numbers and on the classical first-order predicate calculus to found a theory of mathematical continuum free of impredicative (...) circularities only to stumble, to cite a key question, in the evident lack of intuitive support for the notion of points of the continuum. In this motivation, I set out to deal from a Husserlian viewpoint with the general notion of points as appearances reducible to individuals of pre-predicative experience in contrast with the notion of an interval of real numbers taken as an abstraction based on the intuition of time-flowing experience. I argue that the notions of points and of real intervals in the above sense are not by essence related to objective temporality and thus their incompatibility in mathematical terms is ultimately due to deeper constituting reasons independently of any causal and spatio-temporal constraints. (shrink)

Heracleides Ponticus, a pupil of the schools of Plato and Aristotle, who lived from about 390 to 310 B.C., shared the wide interests of many of his pre-Platonic predecessors. Diogenes Laertius gives a long list of his works, many of them now known only by their titles, which he divided into writings on ethics, physics, grammar, music, rhetoric, and history. Like most of his predecessors he gave some attention to the heavens and speculated about the nature of the moon , (...) comets , the infinity of the cosmos ; he was best known in antiquity, in this field, for his suggestion that the phenomena could be saved if the heavens were at rest and the earth revolved about the central axis .2 One of two pieces of evidence for his involvement in anything more than this general, inexact speculation in the field of astronomy is contained in the commentary on Plato's Timaeus written in Latin by Chalcidius somewhere at the beginning of the fourth century A.D.; it is this passage which this paper will discuss. (shrink)

Phenomenology, understood as a philosophy of immanence, has had an ambiguous, uneasy relationship with transcendence, with the wholly other, with the numinous. If phenomenology restricts its evidence to givenness and to what has phenomenality, what becomes of that which is withheld or cannot in principle come to givenness? In this paper I examine attempts to acknowledge the transcendent in the writings of two phenomenologists, Edmund Husserl and Edith Stein (who attempted to fuse phenomenology with Neo-Thomism), and also consider the (...) influence of the existentialist Karl Jaspers, who made transcendence an explicit theme of his writing. I argue that Husserl does recognize the essential experience of transcendence within immanence; even the idea of a physical thinghas “dimensions of infinity” included within it. Similarly, he asserts profoundly that every “outside” is what it is only as understood from the inside. Jaspers toomakes the experience of transcendence central to human existence; it is the very measure of my own depth. For Edith Stein, everything temporal points towardthe timeless structural ground which makes it what it is. Transcendence is an intrinsic part of being itself. Furthermore, the very lack of self-sufficiency of my own self shows that the self requires a ground outside itself, in the transcendent. There is strong convergence between the three thinkers studied on the concept of transcendence, which is indeed a central, if largely unacknowledged, concept in phenomenology both in Husserl and his followers (Stein), but also, throughJaspers, in Heidegger. (shrink)

Draft for presentation at the 14th International Kant-Congress, September 2024. -/- Abstract: Kant claims we intuit infinite space. There’s a problem: Kant thinks full awareness of infinite space requires synthesis—the act of putting representations together and comprehending them as one. But our ability to synthesize is finite. Tobias Rosefeldt has argued in a recent paper that Kant’s notion of decomposing synthesis offers a solution. This talk criticizes Rosefeldt’s approach. First, Rosefeldt is committed to nonconceptual yet determinate awareness of (potentially) infinite (...) space, but I argue that such awareness is (1) prima facie implausible, as well as contravened by textual evidence from Kant’s (2a) definitions of ‘infinity’ and (2b) account of geometrical construction. Second, scrutiny of how Kant thinks awareness of potential infinity is afforded by geometrical construction (e.g., extending a line segment ad indefinitum) indicates no essential role for decomposing synthesis. Familiar synthesis from parts to wholes is sufficient. (shrink)

De Neys argues against assigning exclusive capacities to automatic versus controlled processes. The dual implicit process model provides a theoretical rationale for the exclusivity of automatic threat processing, and corresponding data provide empirical evidence of such exclusivity. De Neys's dismissal of exclusivity is premature and based on a limited sampling of psychological research.

Epistemologists have profited from studies of the ways in which ‘know’ is ambiguous. We can also profit by studying the ways in which ‘know’ is vague. After classifying sources of vagueness for ‘know,’ I spend the second section examining theories of vagueness. With the exception of the theory that vague predicates are incoherent, which I try to refute, we need not take a stand on a particular theory to show that the vagueness of knowledge has substantive epistemological implications. The third (...) section supports the thesis through a survey of ways in which appeals to vagueness can be applied to the field's issues. First, I show how higher order vagueness is evidence against the KK thesis, the incorrigibility of sense data, and the completability of epistemology. An interesting resemblance between infinity and vagueness provides the point of departure for the next type of application. For this resemblance suggests a new way of handling apparently infinite belief structures. The approach is illustrated with an analysis of common knowledge. The third type of application concerns the ways in which disguised sorites creep into our thinking about knowledge. In addition to brief illustrations concerning naive holism, question begging, and an objection to incorrigibilism, I provide more detailed illustrations involving Jonathan Adler's sceptical appeal to epistemic universalizability, the prediction paradox, and William Lycan's objection to Gilbert Harman's social knowledge cases. I conclude with some general remarks on the lessons to be learned from the vagueness of knowledge. (shrink)

This article presents evidence that an anonymous publication of 1573, aLetter sent by a gentleman of England [concerning …] the myraculous starre nowe shyning, was written by Thomas Digges, England's first Copernican. It tells the story of how it arose out of research commissioned by Elizabeth I's privy counsellors in response to the conventional argument of Jean Gosselin, librarian to Henri III of France, that the star was a comet which presaged wars. The text is significant because it seems (...) to contain the observations and opinions that Digges held before he completed his other astronomical treatise, the groundbreakingAlae seu scalae mathematicae. It also casts some light on the development of Digges's radical and puritan views about the star, Copernican astronomy, the infinity of the universe and a belief that the ‘latter days’ of the world had arrived. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Figurative Language and the “Face” in Levinas’s PhilosophyDiane PerpichThe value of images for philosophy lies in their position between two times and their ambiguity.—Levinas, "Reality and Its Shadow"Imagery... occupies the place of theory's impossible.—Le Doeuff, The Philosophical ImaginaryFor many readers, and perhaps above all for Levinas himself, there is something deeply dissatisfying about the account of the "face of the other" in Totality and Infinity and yet the (...) importance of this figure for the major ethical claims of the book can hardly be overstated. The fundamental thesis broached through the notion of the face is the difference between the way in which things are given to consciousness (the order of ontology) and the way in which human beings are encountered (the order of ethics). Whereas things are given to consciousness in sensible experience through the mediation of forms or concepts, the face is present, according to Levinas, in its "refusal to be contained" in a form (Levinas 1969, 194). A passage early in Totality and Infinity defines the face as follows: "The way in which the other presents himself, exceeding the idea of the other in me, we here name face. This mode does not consist in figuring as a theme under my gaze, in spreading itself forth as a set of qualities forming an image. The face of the Other at each moment destroys and overflows the plastic image it leaves me" (50–51, italics in original). A face is thus a very peculiar sort of "phenomenon." In effect, it is non-phenomenal; it does not appear as such and remains exterior to concepts.1 Rhetorically, the face is an image that represents the inadequacy of every image for representing alterity. That is, it represents the impossibility of its own representation and so the problems begin. [End Page 103]In "Violence and Metaphysics," Derrida suggests that Levinas's claim that the absolute alterity of the other refuses conceptualization is undermined from the outset since in order to have meaning as an other, to be recognized and respected as an other, the other must first of all appear: "it is impossible to encounter the alter ego..., impossible to respect it in experience and in language, if this other, in its alterity, does not appear for an ego (in general). One could neither speak, nor have any sense of the totally other, if there was not a phenomenon of the totally other, or evidence of the totally other as such" (Derrida 1978, 123). In what sense can we think or represent absolute alterity if, strictly speaking, it is unthinkable and unrepresentable? And, further, what are we to make of the performative contradiction insofar as Levinas's discourse doesprecisely what it says it is impossible to do? As Derrida notes, even if one cannot thematize the other, "this impossibility and this imperative themselves can be thematized" (123). The difficulty here is that if there "is" an alterity of the type that interests Levinas, then there is no way to represent or speak about it literally—in which case the question arises, in what sense is it? And if one can speak about it, then it is not unrepresentable and thus is not absolute in the sense Levinas says it is.A further contradiction in the notion of the absolute alterity of the face arises owing to the fact that insofar as Levinas's notion of an absolute other forbids us from assigning to the other any determinate predicate, it seems as if all unique, singular faces are the same. As Jean-Luc Marion formulates this problem: "how can one assign an identity to the origin of the appeal such that one can specify which face is involved each time, but without thereby reducing it to a visible phenomenon in the mode of a spectacle?" (Marion 2000, 226). Marion makes a virtue of this feature of Levinas's account, suggesting that "it belongs to the very sense of the appeal that it remain essentially anonymous, and not by default but by excess—the excess of alterity over what it alters" (240). Anonymity, on this view, belongs to the formal structure of interpellation and Marion therein... (shrink)

Many books explain what is known about the universe. This book investigates what cannot be known. Rather than exploring the amazing facts that science, mathematics, and reason have revealed to us, this work studies what science, mathematics, and reason tell us cannot be revealed. In The Outer Limits of Reason, Noson Yanofsky considers what cannot be predicted, described, or known, and what will never be understood. He discusses the limitations of computers, physics, logic, and our own thought processes. Yanofsky describes (...) simple tasks that would take computers trillions of centuries to complete and other problems that computers can never solve; perfectly formed English sentences that make no sense; different levels of infinity; the bizarre world of the quantum; the relevance of relativity theory; the causes of chaos theory; math problems that cannot be solved by normal means; and statements that are true but cannot be proven. He explains the limitations of our intuitions about the world -- our ideas about space, time, and motion, and the complex relationship between the knower and the known. Moving from the concrete to the abstract, from problems of everyday language to straightforward philosophical questions to the formalities of physics and mathematics, Yanofsky demonstrates a myriad of unsolvable problems and paradoxes. Exploring the various limitations of our knowledge, he shows that many of these limitations have a similar pattern and that by investigating these patterns, we can better understand the structure and limitations of reason itself. Yanofsky even attempts to look beyond the borders of reason to see what, if anything, is out there. (shrink)

ABSTRACTTranslated into English in 2004, Vladimir Jankelevitch’s book Music and the Ineffable has made a significant impact in anglophone musicology. I argue that the figure of the feminine is central to his understanding of music and musical ineffability, and use feminist philosophers’ interpretations and critiques of the figure of the feminine in his close friend and colleague Emmanuel Levinas’s work to unpack the gender politics of Jankelevitch’s book and the secondary literature on it. I focus on the figure of the (...) feminine in Time and the Other and Totality and Infinity, which are works we have evidence Jankelevitch read. There, Levinas describes the feminine as a “form of being” that is “inexhaustible, infinite” and thus “refuses thematization positively”. This is nearly identical to Jankelevitch’s definition of musical ineffability as “fertile inexplicability”. Not only is Jankelevitch’s concept of musical ineffability identical to Levinas’s concept of the feminine, it is also couched in gendered terms throughout Music and the Ineffable. “Musical ineffability” in Jankelevitch’s sense is thus a parallel to Levinas’s concept of the feminine. I then read Jankelevitch and his interpreters in musicology through feminist scholarship on Levinas to show how the former two succumb to the same ultimately patriarchal flaws as the latter. For example, just as Levinas prioritizes paternity and the reproduction of patriarchy, muiscologist Carolyn Abbate uses Jankelevitch as the white masculine other in which musicology recognizes itself as both different and the same. (shrink)

The cosmic singularity provides negligible evidence for creation in the finite past, and hence theism. A physical theory might have no metric or multiple metrics, so a ‘beginning’ must involve a first moment, not just finite age. Whether one dismisses singularities or takes them seriously, physics licenses no first moment. The analogy between the Big Bang and stellar gravitational collapse indicates that a Creator is required in the first case only if a Destroyer is needed in the second. The (...) need for and progress in quantum gravity and the underdetermination of theories by data make it difficult to take singularities seriously. The singularity exemplifies the sort of gap that is likely to be closed by scientific progress, obviating special divine action. The apparent irrelevance of cardinality to practices of counting infinite sets in classical field theory and Fourier analysis is noted. Introduction The Doctrine of Creation and Its Warrant Cardinality and Sizes of Infinity Modern Cosmology and Creation Tolerance or Intolerance toward Singularities? Leibniz against Incompetent Watchmaker? Induction from Earlier Theories' Breakdown? Stellar Collapse Implies Theistic Destroyer Stacking the Deck for GTR Quantum Gravity Tends to Resolve Singularities Vicious God-of-the-Gaps Character Fluctuating or Inaccessible Warrant Big Bang Cosmology Not Especially Congenial to Faith CiteULike Connotea Del.icio.us What's this? (shrink)

IntroductionJ. Howard Sobel devotes seventy pages of his wide-ranging analysis of theistic arguments to a critique of the cosmological argument. The focus of that critique falls on the argument a contingentia mundi; but he also offers in passing some criticisms of the argument ab initio mundi, or the kalam cosmological argument.Sobel provides the following Statement of the argument:Everything that begins to exist has a cause of its existence.The universe began to exist.Therefore, the universe has a cause of its existence [that (...) did not begin to exist].Sobel will accept the causal premiss only if ‘begins to exist’ means ‘has a first instant of its existence,’ and he disputes the arguments and evidence for.Traditional proponents of the kalam argument sought to justify by means of philosophical arguments against the infinity of the past, while contemporary interest in the argument arises from the empirical evidence of physical cosmology for the truth of. (shrink)

In the late 1940s and early 1950s Lorenzen developed his operative logic and mathematics, a form of constructive mathematics. Nowadays this is mostly seen as the precursor to the more well-known dialogical logic and one could assumed that the same philosophical motivations were present in both works. However we want to show that this is not always the case. In particular, we claim, that Lorenzen’s well-known rejection of the actual infinite as stated in Lorenzen (1957) was not a major motivation (...) for operative logic and mathematics. In this article, we claim that this is in fact not the case. Rather, we argue for a shift that happened in Lorenzen’s treatment of the infinite from the early to the late 1950s. His early motivation for the development of operativism is concerned with a critique of the Cantorian notion of set and related questions about the notion of countability and uncountability; only later, his motivation switches to focusing on the concept of infinity and the debate about actual and potential infinity. (shrink)

Did Leibniz exploit infinitesimals and infinities à la rigueur or only as shorthand for quantified propositions that refer to ordinary Archimedean magnitudes? Hidé Ishiguro defends the latter position, which she reformulates in terms of Russellian logical fictions. Ishiguro does not explain how to reconcile this interpretation with Leibniz’s repeated assertions that infinitesimals violate the Archimedean property (i.e., Euclid’s Elements, V.4). We present textual evidence from Leibniz, as well as historical evidence from the early decades of the calculus, to (...) undermine Ishiguro’s interpretation. Leibniz frequently writes that his infinitesimals are useful fictions, and we agree, but we show that it is best not to understand them as logical fictions; instead, they are better understood as pure fictions. (shrink)

In everyday life and in science we acquire evidence of evidence and based on this new evidence we often change our epistemic states. An assumption underlying such practice is that the following EEE Slogan is correct: 'evidence of evidence is evidence' (Feldman 2007, p. 208). We suggest that evidence of evidence is best understood as higher-order evidence about the epistemic state of agents. In order to model evidence of evidence (...) we introduce a new powerful framework for modelling epistemic states, Dyadic Bayesianism. Based on this framework, we then discuss characterizations of evidence of evidence and argue for one of them. Finally, we show that whether the EEE Slogan holds, depends on the specific kind of evidence of evidence. (shrink)

Fine-tuning in physics and cosmology is often used as evidence that a theory is incomplete. For example, the parameters of the standard model of particle physics are “unnaturally” small, which has driven much of the search for physics beyond the standard model. Of particular interest is the fine-tuning of the universe for life, which suggests that our universe’s ability to create physical life forms is improbable and in need of explanation, perhaps by a multiverse. This claim has been challenged (...) on the grounds that the relevant probability measure cannot be justified because it cannot be normalized, and so small probabilities cannot be inferred. We show how fine-tuning can be formulated within the context of Bayesian theory testing in the physical sciences. The normalizability problem is seen to be a general problem for testing any theory with free parameters, and not a unique problem for fine-tuning. Physical theories in fact avoid such problems in one of two ways. Dimensional parameters are bounded by the Planck scale, avoiding troublesome infinities, and we are not compelled to assume that dimensionless parameters are distributed uniformly, which avoids non-normalizability. (shrink)

How are we to understand the use of probability in Popper’s corroboration function? Popper says logically, but this raises a problem that becomes apparent when his views on logical probability are compared with those of Keynes. Specifically, Popper does not make it clear how we could have access to, or even calculate, probability values in a logical sense. For first, he would likely want to deny the Keynesian distinction between primary and secondary propositions, and the underlying notion of knowledge-by-acquaintance with (...) Neo-Platonic entities. Second, he would presumably reject not only the notion of non-numerical probabilities, but also the claim that the principle of indifference is aught other than a heuristic, given paradoxes such as Bertrand’s. An attempt might be made to solve this problem by appeal to semantic or possible world analyses over the relative truth-values of statements, but this seems to fail due to concerns relating to infinity: if the set of possible worlds in which p is true is infinite, and the set of possible worlds in which q is true is infinite, then no numerical value for P can be reasonably determined by a comparison between those worlds. As such, an all-too-familiar sort of criticism of Popper’s view of science seems to loom large: if the corroboration function only makes sense when the probabilities employed therein are subjective, then what counts as impressive evidence for a theory might seem to be a matter of convention, or even whim. However, we now have a new interpretation of probability at our disposal: the intersubjective interpretation, recently championed by Gillies, according to which probabilities can lie on a spectrum between the aleatory and the epistemic. Might we employ this to preserve the corroboration function, and deliver ourselves from Bayesianism, without succumbing to radical subjectivism? (shrink)