Paolo Mancosu provides an original investigation of historical and systematic aspects of the notions of abstraction and infinity and their interaction. A familiar way of introducing concepts in mathematics rests on so-called definitions by abstraction. An example of this is Hume's Principle, which introduces the concept of number by stating that two concepts have the same number if and only if the objects falling under each one of them can be put in one-one correspondence. This principle is at the core (...) of neo-logicism. In the first two chapters of the book, Mancosu provides a historical analysis of the mathematical uses and foundational discussion of definitions by abstraction up to Frege, Peano, and Russell. Chapter one shows that abstraction principles were quite widespread in the mathematical practice that preceded Frege's discussion of them and the second chapter provides the first contextual analysis of Frege's discussion of abstraction principles in section 64 of the Grundlagen. In the second part of the book, Mancosu discusses a novel approach to measuring the size of infinite sets known as the theory of numerosities and shows how this new development leads to deep mathematical, historical, and philosophical problems. The final chapter of the book explore how this theory of numerosities can be exploited to provide surprisingly novel perspectives on neo-logicism. (shrink)

Renaissance philosopher, mathematician, and theologian Nicholas of Cusa (1401-1464) said that there is no proportion between the finite mind and the infinite. He is fond of saying reason cannot fully comprehend the infinite. That our best hope for attaining a vision and understanding of infinite things is by mathematics and by the use of contemplating symbols, which help us grasp "the absolute infinite". By the late 19th century, there is a decisive intervention in mathematics and its philosophy: the philosophical mathematician (...) Georg Cantor (1845-1918) says that between the realm of the finite and the absolute infinite, there is an intermediate realm partaking in properties in a certain sense of both the finite and the infinite: the transfinite realm. Like the finite, the transfinite realm is a realm of mathematical objects, numbers, and knowledge. Like the absolute infinite, the transfinite is a form of infinity insofar as transfinite sets and numbers transcend any finite number. Echoing Cusanus and neo-Platonism, Cantor says that the transfinite sequence of all ordinals is a symbol of the absolutely infinite, that is, God. This paper considers how the doctrine of symbolism and the philosophers' different commitments to the laws of logic, especially the Law of Non-Contradiction (LNC), enables these thinkers to articulate a transcendental apophatic approach to divinity. (shrink)

MancosuPaolo.* *ion and Infinity. Oxford University Press, 2016. ISBN: 978-0-19-872462-9. Pp. viii + 222.

Infinity is an intriguing topic, with connections to religion, philosophy, metaphysics, logic, and physics as well as mathematics. Its history goes back to ancient times, with especially important contributions from Euclid, Aristotle, Eudoxus, and Archimedes. The infinitely large is intimately related to the infinitely small. Cosmologists consider sweeping questions about whether space and time are infinite. Philosophers and mathematicians ranging from Zeno to Russell have posed numerous paradoxes about infinity and infinitesimals. Many vital areas of mathematics rest upon some version (...) of infinity. The most obvious, and the first context in which major new techniques depended on formulating infinite processes, is calculus. But there are many others, for example Fourier analysis and fractals.In this Very Short Introduction, Ian Stewart discusses infinity in mathematics while also drawing in the various other aspects of infinity and explaining some of the major problems and insights arising from this concept. He argues that working with infinity is not just an abstract, intellectual exercise but that it is instead a concept with important practical everyday applications, and considers how mathematicians use infinity and infinitesimals to answer questions or supply techniques that do not appear to involve the infinite.ABOUT THE SERIES: The Very Short Introductions series from Oxford University Press contains hundreds of titles in almost every subject area. These pocket-sized books are the perfect way to get ahead in a new subject quickly. Our expert authors combine facts, analysis, perspective, new ideas, and enthusiasm to make interesting and challenging topics highly readable. (shrink)

The concept of linguistic infinity has had a central role to play in foundational debates within theoretical linguistics since its more formal inception in the mid-twentieth century. The conceptualist tradition, marshalled in by Chomsky and others, holds that infinity is a core explanandum and a link to the formal sciences. Realism/Platonism takes this further to argue that linguistics is in fact a formal science with an abstract ontology. In this paper, I argue that a central misconstrual of formal apparatus (...) of recursive operations such as the set-theoretic operation merge has led to a mathematisation of the object of inquiry, producing a strong analogy with discrete mathematics and especially arithmetic. The main product of this error has been the assumption that natural, like some formal, languages are discretely infinite. I will offer an alternative means of capturing the insights and observations related to this posit in terms of scientific modelling. My chief aim will be to draw from the larger philosophy of science literature in order to offer a position of grammars as models compatible with various foundational interpretations of linguistics while being informed by contemporary ideas on scientific modelling for the natural and social sciences. (shrink)

Two fundamental categories of any ontology are the category of objects and the category of universals. We discuss the question whether either of these categories can be infinite or not. In the category of objects, the subcategory of physical objects is examined within the context of different cosmological theories regarding the different kinds of fundamental objects in the universe. Abstract objects are discussed in terms of sets and the intensional objects of conceptual realism. The category of universals is discussed (...) in terms of the three major theories of universals: nominalism, realism, and conceptualism. The finitude of mind pertains only to conceptualism. We consider the question of whether or not this finitude precludes impredicative concept formation. An explication of potential infinity, especially as applied to concepts and expressions, is given. We also briefly discuss a logic of plural objects, or groups of single objects (individuals), which is based on Bertrand Russell’s (1903, The principles of mathematics, 2nd edn. (1938). Norton & Co, NY) notion of a class as many. The universal class as many does not exist in this logic if there are two or more single objects; but the issue is undecided if there is just one individual. We note that adding plural objects (groups) to an ontology with a countable infinity of individuals (single objects) does not generate an uncountable infinity of classes as many. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Approaching Infinity:Dignity in Arthur Koestler's Darkness at NoonRoger BerkowitzIn his allegorical novel Darkness at Noon, Arthur Koestler tells of Rubashov, a founding father of an unnamed Party in an unnamed state.1 Jailed by the current Party leader, "Number One," and pressed to recant his deviationist views, Rubashov resists. At first, he resolves to go to his death to preserve his integrity. Later, Rubashov recognizes that to hold to (...) his own truth when it endangers the goals of political reform is politically irresponsible. He decides to recant.The aristocratic soldier in the neighboring cell is appalled by Rubashov's self-betrayal. "Honour," the aristocrat insists, "is to live and die for one's belief." Rubashov disagrees with the aristocratic idea of honor and responds that honour is "to be useful without vanity," which provokes his neighbor to erupt: "honour is decency—not usefulness." Rubashov answers: "We have replaced decency by reason" (Koestler, pp. 177–78).Today, we rationalist readers of Darkness at Noon are left with the question of the competing demands of usefulness and decency, of reason and dignity. Reason makes its arguments in familiar terms: to prevent a genocide, we will bomb the aggressors; in the name of rationalized health care, some will not get the care they need; for reasons of state, some will be tortured. In politics, as Max Weber teaches, rational ends demand a slow, powerful boring through of hard boards (Weber, p. 93).2 Thus reason—and Koestler unfolds reason as calculating rationality in its most Machiavellian garb—requires the willingness to employ sober and rational means to achieve desired ends. In common parlance, the end justifies the means. Against the nostalgia for a politics of decency [End Page 296] and dignity, reason counsels that to make an omelet, you need to break a few eggs.But what lies behind the opposing claim to decency? What is it that pulls us to resist the claim of reason when reason counsels war, torture, or even bureaucratic neutrality? What, to paraphrase Hannah Arendt, is to be said for the eggs? (Arendt, pp. 270–84).3We may invoke conscience, duty, and decency. We might invoke civil and human rights. We condemn realpolitik and affirm a Judeo-Christian world order. In short, we invoke the dignity of man and affirm one's humanity as an inalienable core that cannot be breached. The limitations of such invocations are by now clear. At a time of recurrent genocides, democratically sanctioned torture, and suicide bombings of civilians, the stirring claims of the dignity of mankind ring hollow. Yet the collapse of the Western tradition of dignity does not necessitate the loss of dignity itself. The failure to secure dignity on religious or rationalist grounds need not send us fleeing, our eyes covered in horror and our heads between our legs. What is needed is a confrontation with our situation, one that seeks to comprehend both the impulse to affirm an inviolable core to humanity and our failure to do so.The ambition of Darkness at Noon lies in its bald effort to affirm the truth of human decency even as it recognizes that man's humanity is a bald-faced fiction. When Rubashov teeters on the precipice between reason and decency, his "eye" tooth aches. He rubs his eyeglasses. He steps only on whole squares in the floor, avoiding the cracks. Tooth, glass, and wholeness refer to the problem of the singular "I," an entity so mysterious that Rubashov follows Nietzsche in naming it the "grammatical fiction." What is the "I," the conscience, the self? How is it that the fictional "I" can and must have such an essential role in modern politics? Where, if anywhere, lies the relevance of the "I" for politics?Darkness at Noon is acclaimed as one of the most important books of the twentieth century—it is number eight on the Modern Library's list of the Hundred Best Novels of the twentieth century. Yet, given its enormous impact, its philosophical ruminations, and its political theme, the novel has garnered surprisingly little academic consideration. It is worth asking, therefore, why Darkness at Noon has not been taken seriously by the academic... (shrink)

ABSTRACT According to an influential line of argument, the past must be finite because no infinite series can be formed by successive addition. The present paper pinpoints the non sequitur at the heart of this argument, disentangles the ambiguities that disguise it, and dismantles the misleading picture of ‘traversing the infinite’ that gives the argument so much of its allure. Finally, the paper critically explores the related argument that a beginningless series of past events is impossible because there could (...) be no explanation of its having been ‘completed’ at one time rather than another. (shrink)

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)

An analysis of the counter-intuitive properties of infinity as understood differently in mathematics, classical physics and quantum physics allows the consideration of various paradoxes under a new light (e.g. Zeno’s dichotomy, Torricelli’s trumpet, and the weirdness of quantum physics). It provides strong support for the reality of abstractness and mathematical Platonism, and a plausible reason why there is something rather than nothing in the concrete universe. The conclusions are far reaching for science and philosophy.

In lieu of an abstract, here is a brief excerpt of the content:242 HISTORY OF PHILOSOPHY his political and religious predispositions prodded him to demonstrate that the roots of modern science were in the Christian Middle Ages. Sarton's particular foibles are best understood by referring them to his pacifist commitments and the moralistic assumption that the values of science are transferable to other human endeavors. Categories such as inductivism, conventionalism and Popperianism are of little help in gaining historical understanding. (...) For this, they are in fact Platonic ideals which have little heuristic value. I would venture that a truly historical historiography of science would aid us in spotting errors in the past and help us in the future. My guess is that a good analysis would reveal a major source of trouble to have stemmed from preconceived notions which the history of science was meant to illustrate. Surely some of them have been philosophical in character, and even Baconian or conventionalist. But until recent times, most of these preconceptions have stemmed from religious, nationalistic, or personal motivations, or from scientists who wanted support for a particular concept. I agree with Agassi only in the suggestion that a parti pris was at the root of our troubles. Why, then, substitute a Popperian preconception for a conventionalist one? Why not, on the contrary, combat their distorting influence and reduce their impact? This is in fact what has happened in recent times. As the history of science has fallen into the hands of a gradually organized profession, the blatant motivations --of any type--which colored past historical writings have slowly receded. By a constant and critical confrontation of opposing views among specialists, the purpose for examining the history of science has slowly been transformed. The older external motivations have given way to an interest in the subject for its own sake and a gradual transformation has been in the making for the last few decades. As happens in other human endeavors, professionalization has already raised the standards in the discipline and there is every reason to think it will continue to do so in the future. We need not really be as pessimistic as Agassi, nor embrace his particular salvation. ROGEa HAm~ University o] Cali]ornia, Berkeley A MEDIEVAL ANALYSIS OF INFINITY IN THE COURSEof arguing that time must have had a beginning, St. Bonaventure (1221-74) put forward a demonstration which in some ways seems to anticipate Cantor's definition of infinity.1 Bonaventure pointed out that, if the world were infinitely old, then there would have been an infinite number of annual revolutions of the sun around the ecliptic. But during each such period there occur (roughly) twelve revolutions of the moon, i.e. lunar months or lunations. Thus there would be one infinity which was twelve times another--which, says Bonaventure, is impossible. This attempted proof might be formulated as the following For other adumbrationsof this definition,see I. Thomas,"A TwelfthCentury Paradoxof the Infinite,"Journal o] Symbolic Logic, XXIII (1958), 133-134, and W. & M. Kneale, The Development of Logic (Oxford,1962),p. 440. NOTES AND DISCUSSIONS 243 reductio ad absurdum, in which a revolution of the sun is specified as the period between successive vernal equinoxes. If there could be an infinite set of past lunations, then clearly it could be put in a one-to-one correspondence with a proper sub-set of itself--viz., with the set of past lunar months during which vernal equinoxes occurred. But this consequence is preposterous; no set could be so correlated with its own sub-set. Therefore the set of past lunar months cannot be infinite, and the world must have had a beginning. The passage, which is found in Bonaventure's commentary on the Sentences of Peter Lombard, reads as follows :2 Si mundus est aeternus, revolutiones solis in orbe suo sunt infmitae ; rursus, pro una revolutions solis necesse est fuisse duodecim ipsius lunae: ergo plus revoluta est luna quam sol; et sol infinities: ergo infinitorum ex ea parle, qua infinita sunt, est reperire excessum. Hoe autem est impossibile : ergo etc. PA~-r~asoN BROWN Harpur College, State University o] New York Sententiarum L/b. H, Dist. I, P.I, Art. I, Quaest. II, as found in S. Bonaventurme, Opera Omnla... (shrink)

Although the concept of ‘true infinity’ is undoubtedly central to Hegel's philosophy, the Anglophone rehabilitation of Hegel as a post-Kantian critical philosopher has avoided any sustained interpretive confrontation with the concept. In this paper, I provide a revisionary reconstruction of Hegelian true infinity by engaging with Martin Hägglund's argument in This Life (2019) for the centrality of finitude to Hegel's philosophy. For Hägglund, Hegel's philosophy effects a ‘secular reconciliation’ with finitude by demonstrating that our mortality is not a negative condition (...) to be overcome; rather, mortality is constitutive of rational social agency or ‘spiritual life’. While Hägglund's interpretive emphasis on finitude prima facie occludes a consideration of the concept of true infinity, I show that Hägglund's understanding of spiritual life in terms of the dynamic of self-maintenance implicitly articulates a revisionary understanding of true infinity as instantiated in and through the activity of finite living rational beings. More specifically, I show that Hägglund's understanding of finitude as constitutive of spiritual life is grounded in three closely interrelated Hegelian meta-concepts, all of which Hegel derives in the second chapter of the Logic (‘Existence’): (a) the distinction between abstract and determinate negation; (b) the form of individuality as constituted through determinate negation; and (c) true infinity as the form of individuality understood in explicitly processual terms. I thus ground a deflationary interpretation of true infinity in Hegel's logical account of determinate individuality, and at the same time, contribute to This Life's ongoing critical reception by articulating its Hegelian logical infrastructure. (shrink)

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)

Abstract Descartes's theory of volition as expressed in his Passions of the Soul is analyzed and outlined. The focus is not on Descartes's proposed answers to questions about the nature and processes of volition, but on his way of formulating questions about the nature of volition. It is argued that the assumptions underlying Descartes's questions have become ?intellectual strait?jackets? for all who are interested in volition: neuroscientists, philosophers and psychologists. It is shown that Descartes's basic assumption?that volition causes change (...) in the brain/mind, not in the world around us?has set in train a series of ?themata? that have dominated studies of the will, severely curtailing our understanding. It is then shown that these Cartesian themata are so limiting and confusing that a number of internally contradictory ideas have actually become mainstays of most theories of volition; in particular, the concepts of unconscious sensations and of involuntary volitions. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:preface Sometimes, even those of us who have organized our entire lives around the transformative possibilities of Black feminist thought can sit back in wonder at the expansiveness of this intergenerational transnational practice.Thisspecialissuetakesamomenttoimbibewhere we have been, where we are, and where we have yet to journey. The contributors to this special issue on, or more precisely, of Black feminist thought find Black feminist thinking in a wide range (...) of times, places, and forms. Each of the contributors invites us to bear witness in a distinct way to the pushes and pulls of critical engagement with Black feminist thought. Biomythic origin point “mitochondrial eve” brainstorms the archive and admonishes those of us looking back to “be a forest of your own / dreams.” Gladys Bentley sends unrequited texts to... Gladys Bentley. Everything is possible here, even what should never have happened. In If You Can’t Be Free, Be a Mystery: In Search of Billie Holiday, Farah Jasmine Griffin, one of the most dedicated and prolific stewards of Black feminist thought as a liberatory practice, cites Rita Dove’s poem “Canary” to evoke the protective strategies Black women employ to hold something (ourselves, each other) sacred in an unfree world.1 The lineage we participate in through our Black 1. Farah Jasmine Griffin, If You Can’t Be Free, Be a Mystery: In Search of Billie Holiday (New York: Ballantine Books, 2001), 155–56. “Be a Mystery”: (The Infinity of) Black Feminist Thought 8Preface feminist intellectual labor is sacred to us. For generations, even before the name “Black Feminist,” it has existed as a form of love, communion, protest, and prophecy, often far from the spotlight. Nurtured by the warmth of imagination, practices became praxis. Griffin and others have ensured a tradition wherein, to invoke Griffin’s paraphrase of Toni Morrison in her tribute to Nellie McKay, “the mothers may soar and the daughters may know their names.”2 We are some of the daughters. But we are living in times that neither we nor our mothers would hardly recognize. Abolitionist queer Black feminists are topping the New York Times bestseller lists. Our scholars and experts speak on the nightly news. Some of what were once our most marginalized proposals have become policies, funding keywords, and even memes. The Movement for Black Lives, one of the largest and most visible social movements of our time, or any time, explicitly connects its vision to the foundational Combahee River Collective statement. We do this work now in the light of recognition. Masses of people think they know what Black feminism is, and yet, this is not the world that we dreamed. We, the current generation of practicing Black feminist thinkers, are faced with the consequences of Black feminist practice feeling like less of a mystery. We celebrate the legacy our foremothers have made possible, and we must also think creatively, lovingly, and sometimes even cryptically about what this moment can and must mother—or bring into being. It has to bring into being something we cannot describe from here, something we have not yet known. It is exhilaratingly beautiful to look upon what’s been made possible, but we know there is so much more than recognition. Undaunted by a world still hellbent on not caring for or loving us, even as it consumes our insights, theories, and modes of organizing, we create worlds filled with an abundance of joy, affirmation, and ingenuity. It is from this robust, ancestral practice of world-making that this special issue emerges. The pieces found here provide insight into one of the most dynamic aspects of Black feminist thought: its generative power. The range of topics included in this special issue encapsulates the breadth of this interdisciplinary, transnational, multimodal, and 2. Farah Jasmine Griffin, “That the Mothers May Soar and the Daughters May Know Their Names: A Retrospective of Black Feminist Literary Criticism,” Signs 32, no. 2 (Winter 2007): 483–507. Preface 9 intergenerational practice. Given the seemingly infinite nodes and entry points of Black feminist thought, it is befitting that this issue coheres because of an unboundedness and porousness among the pieces. The art essays, articles, field reviews, and conversations converse, dance, and harmonize with... (shrink)

This article was found among the papers left by Prof. Laugwitz (May 5, 1932–April 17, 2000). The following abstract is extracted from a lecture he gave at the Fourth Austrain Symposion on the History of Mathematics (Neuhofen/ybbs, November 10, 1995).About 100 years ago, the Cantor-Veronese controversy found wide interest and lasted for more than 20 years. It is concerned with “actual infinity” in mathematics. Cantor, supported by Peano and others, believed to have shown the non-existence of infinitely small quantities, (...) and therefore he fought against the infinitely large and small numbers in Veronese’s geometry, but also against the non-archimedean systems of Thomae, du Bois-Reymond, and Stolz. As a positive consequence of the controversy the distinction between Cantor’s transfinite arithmetic and the theory of ordered algebraic structures becomes clear. (shrink)

[Use code AUFLY30 for 30% off on the OUP website.] One central aim of science is to provide explanations of natural phenomena. What role(s) does mathematics play in achieving this aim? How does mathematics contribute to the explanatory power of science? Rules to Infinity defends the thesis, common though perhaps inchoate among many members of the Vienna Circle, that mathematics contributes to the explanatory power of science by expressing conceptual rules, rules which allow the transformation of empirical descriptions. Mathematics should (...) not be thought of as describing, in any substantive sense, an abstract realm of eternal mathematical objects, as traditional platonists have thought. In Rules to Infinity, this view, which I call mathematical normativism, is updated with contemporary philosophical tools, and it is argued that normativism is compatible with mainstream semantic theory. This allows the normativist to accept that there are mathematical truths, while resisting the platonistic idea that there exist abstract mathematical objects that explain such truths. Furthermore, Rules to Infinity defends a particular account of the distinction between scientific explanations that are in some sense distinctively mathematical – those that explain natural phenomena in some uniquely mathematical way – and those that are only standardly mathematical, and it lays out desiderata for any account of this distinction. Normativism is compared with other prominent views in the philosophy of mathematics such as neo-Fregeanism, fictionalism, conventionalism, and structuralism. Rules to Infinity serves as an entry point into debates at the forefront of philosophy of science and mathematics, and it defends novel positions in these debates. (shrink)

If spacetime does not exist, it does so in a very unusual way. It curves in response to massive objects. It warps in response to high velocities. The Substance of Spacetime treats spacetime, not merely as a geometric abstraction, but as a real physical substance, opening a window onto reality that would otherwise be impossible to even contemplate.

Many results in logic and mathematics rely on techniques that allow for concrete, often visual, representations of abstract concepts. A primary example of this phenomenon in logic is the distinction between syntax and semantics, itself an example of the more general duality in mathematics between algebra and geometry. Such representations, however, often rely on the existence of certain maximal objects having particular properties such as points, possible worlds or Tarskian first-order structures. -/- This dissertation explores an alternative to such (...) representations known as possibility semantics. Its core idea is to replace maximal objects with ordered systems of partial approximations. Although it originates in the semantics of modal logic and the representation of abstract ordered structures, I argue that it has far-reaching mathematical, foundational and philosophical significance, especially in the context of semiconstructive mathematics, a foundational framework that does not assume any fragment of the Axiom of Choice beyond the Axiom of Dependent Choices. -/- The dissertation is divided in two main parts. The first part explores various applications of the mathematical framework underlying possibility semantics to lattice theory and non-classical propositional logics. A major theme is the development of constructive dualities for various categories of lattices, which are related to standard non-constructive dualities via Vietoris constructions. -/- The second part of the dissertation explores the alternative foundational setting of semi-constructive mathematics, focusing on three applications of possibility semantics for classical first-order logic to the philosophy of the mathematical infinite. In particular, we introduce generic powers, a semi-constructive analogue of ultrapowers in classical model theory, and we explore the merits of these structures from a foundational, conceptual and historical viewpoint. (shrink)

Why should some essential properties of geometry (i.e., infinity, symmetry, and dimensionality) be both necessary and desirable in the way that they have been constructed albeit with different modifications over time since time immemorial? Contrary to the conventional wisdom in all history hitherto existing, the essential properties of geometry do not have to be both necessary and desirable. This is not to suggest, of course, that one has nothing to learn from geometry. On the contrary, geometry has contributed to the (...) advancement of knowledge in many ways since its inception as a field of knowledge some millennia ago. The point in this book, however, is to show an alternative (better) way to understand the nature of geometry, which goes beyond human conception, intuition, and imagination, together with worldly experience of course, as its foundation, while learning from them all with theoretical implications for time travel, hyperspace, and other important issues. If true, this seminal view will fundamentally change the way that the nature of abstraction in the thinking process is to be understood, with its enormous implications for the future advancement of knowledge, in a small sense, and what I originally called its "post-human" fate, in a large one. (shrink)

ABSTRACT This article explores the conclusion of Calvin Warren’s book Ontological Terror and the nihilistic suggestion for Black life to reject humanism. In the text’s final chapter, Warren unexpectedly references Danish philosopher Søren Kierkegaard’s leap of faith and reflects on Black life’s enduring spirit through an anti-Black world. This article’s analysis faithfully traces Warren’s nihilistic arguments against humanism and scaffolds them through his reference to Kierkegaard. Utilizing the methods of critical philosophy of race and Black existential philosophy, the first (...) section contextualizes Ontological Terror’s main arguments, places Warren’s reference to Kierkegaard inside a longer Black existential lineage, and grafts Kierkegaard’s concept of the tragic hero onto Warren’s antihumanism. The second section offers a reconstruction of Warren’s ideas through Kierkegaard’s knight of infinite resignation and the leap of faith. This section stresses the importance of both endurance and spirit for Warren’s divestment from humanism. The third section puts Warren’s nihilism in conversation with the work of Black feminist philosophers to offer an alternative way to interpret divestment from humanism. The article concludes by engaging David Marriott’s and Frantz Fanon’s understanding of invention to help rethink the links between nihilism, future-oriented thinking, and Black endurance against anti-Blackness. (shrink)

Although rahner originally maintained the validity of conceptual abstractions and the nonintelligibility of matter, Other works arguing from the ultimate unity of spirit in matter both in god, Their common origin and end, And in the essence of the soul, Led to the affirmation of a certain intelligibility of matter. Rahner's proof for god's existence, Based on the intellectual dynamism that transcends all finite realities, Concepts included, As it seeks fulfillment in the infinite, Is ambiguous. Whether the infinite is god, (...) Matter, Or perpetual transcendence cannot be demonstrated, And the appeal to the supernatural beatific vision is insufficient. "grundkurs" finally argues that the matter entering into human "esse" and knowing demands the affirmation of god's infinite mystery if the world has a sense. Yet rahner simultaneously strives to preserve conceptual knowledge and catholicism's "sacramental vision." in this he is similar to maritain and st. Thomas. (shrink)

ABSTRACT:Relational leadership invokes an ethics involving a leader’s affective engagement and genuine concern with the interests of others. This ethics faces practical difficulties given it implies a seemingly limitless responsibility to a set of incommensurable ethical demands. This article contributes to addressing the impasse this creates in three ways. First, it clarifies the nature of the tensions involved by theorising relational leadership as caught in an irreconcilable bind between an infinitely demanding ethics and the finite possibilities of a response (...) to those demands. Second, it examines this ethical challenge in acknowledgement of the hierarchical discourses and power dynamics in which leadership relationships are constrained and enacted. Third, it proposes “ethical irony” as a way leaders can respond to the demand for ethics without resulting in either an escape from ethics, or being crushed by its burden. Three dimensions of ethical irony are examined: ironic perspective, ironic performance, and ironic predilection. (shrink)

My aim in this chapter is to extend the Realist account of the foundations of linguistics offered by Postal, Katz and others. I first argue against the idea that naive Platonism can capture the necessary requirements on what I call a ‘mixed realist’ view of linguistics, which takes aspects of Platonism, Nominalism and Mentalism into consideration. I then advocate three desiderata for an appropriate ‘mixed realist’ account of linguistic ontology and foundations, namely (1) linguistic creativity and infinity, (2) linguistics as (...) a theory of types (and not tokens) and (3) independence but structural respect between language and the linguistic competence thereof. My own brand of mixed realism, what I call ante rem realism, is defended along the lines of an ante rem or non-eliminative structuralism, the likes of which has been offered for mathematics by Resnik (1997) and Shapiro (1997). In other words, grammars describe a mind-independent (but not necessarily unconnected) linguistic reality in terms of linguistic patterns or structures also known as natural languages. I further amend this picture to allow for the possibility of a naturalistic account of language acquisition and evolution by arguing against a particular view of the type-token distinction. (shrink)

(No abstract is available for this citation).

There is a wide range of realist but non-Platonist philosophies of mathematics—naturalist or Aristotelian realisms. Held by Aristotle and Mill, they played little part in twentieth century philosophy of mathematics but have been revived recently. They assimilate mathematics to the rest of science. They hold that mathematics is the science of X, where X is some observable feature of the (physical or other non-abstract) world. Choices for X include quantity, structure, pattern, complexity, relations. The article lays out and compares (...) these options, including their accounts of what X is, the examples supporting each theory, and the reasons for identifying the science of X with (most or all of) mathematics. Some comparison of the options is undertaken, but the main aim is to display the spectrum of viable alternatives to Platonism and nominalism. It is explained how these views answer Frege’s widely accepted argument that arithmetic cannot be about real features of the physical world, and arguments that such mathematical objects as large infinities and perfect geometrical figures cannot be physically realized. (shrink)

In this paper, the author derives the Dedekind-Peano axioms for number theory from a consistent and general metaphysical theory of abstract objects. The derivation makes no appeal to primitive mathematical notions, implicit definitions, or a principle of infinity. The theorems proved constitute an important subset of the numbered propositions found in Frege's *Grundgesetze*. The proofs of the theorems reconstruct Frege's derivations, with the exception of the claim that every number has a successor, which is derived from a modal axiom (...) that (philosophical) logicians implicitly accept. In the final section of the paper, there is a brief philosophical discussion of how the present theory relates to the work of other philosophers attempting to reconstruct Frege's conception of numbers and logical objects. (shrink)

A neologicist set theory based on an abstraction principle (NewerV) codifying the iterative conception of set is investigated, and its strength is compared to Boolos's NewV. The new principle, unlike NewV, fails to imply the axiom of replacement, but does secure powerset. Like NewV, however, it also fails to entail the axiom of infinity. A set theory based on the conjunction of these two principles is then examined. It turns out that this set theory, supplemented by a principle stating that (...) there are infinitely many nonsets, captures all (or enough) of standard second-order ZFC. Issues pertaining to the axiom of foundation are also investigated, and I conclude by arguing that this treatment provides the neologicist with the most viable reconstruction of set theory he is likely to obtain. (shrink)

_Abstract_: In this article, I propose an original re-interpretation of the encounter between deconstruction and psychoanalysis as it is described by Jacques Derrida in his early essay “_Freud and the scene of writing_” (1966). My working hypothesis is that Derrida first reads psychoanalysis as a _partially_ _deconstructive_ human science. To test this hypothesis, I begin by demonstrating that Derrida’s reading draws on the description of deconstructive sciences offered since his early version of_ Grammatology _(1965-66). Second, I explain that it traces (...) across Freud’s work the increasing adequation of the psychoanalytic account of psychism to the model of a somehow spontaneous archiving machine. Finally, I show that, for Derrida, as a consequence of this adequation, psychism – understood as the origin of life, temporalization and the relation to the other – also marks the beginning of the history of technics. As my analyses develop, it should become increasingly evident that any critical assessments of Derrida’s encounter with psychoanalysis must reckon with the overall project of deconstruction in which this encounter is inscribed. _Keywords_: Deconstruction; Life; Psychoanalysis; Technics; World _L’essere-nel-mondo della psiche: le prime interpretazioni freudiane di Derrida_ _Riassunto_: In questo lavoro intendo proporre una re-interpretazione originale dell’incontro tra decostruzione e psicoanalisi come lo descrive Jacques Derrida nel suo saggio “_Freud e la scena della scrittura_” (1966). La mia ipotesi di lavoro è che Derrida abbia letto in prima istanza la psicoanalisi come una scienza umana _in parte_ _decostruttiva_. Per verificare questa ipotesi inizierò mostrando che la lettura di Derrida attinge alla descrizione delle scienze decostruttive offerta sin dalle prime versioni di _Della grammatologia _(1965-66). In secondo luogo proverò a illustrare come si ritrovi nell’opera di Freud un progressivo allinearsi della descrizione psicoanalitica dello psichismo al modello di un apparato di archiviazione più o meno spontaneo. Mostrerò infine che, per Derrida, come conseguenza di questo allineamento, lo psichismo – inteso come origine della vita, temporalizzazione e relazione all’altro – segna anche l’inizio della storia della tecnica. Con lo sviluppo delle mie analisi diverrà sempre più chiaro che una valutazione critica dell’incontro di Derrida con la psicoanalisi debba fare i conti con il progetto complessivo della decostruzione in cui questo stesso incontro si iscrive. _Parole chiave_: Decostruzione; Vita; Psicoanalisi; Tecnica; Mondo. (shrink)

Abstract: Nietzsche and the Early Romantic Concept of Critique. Nietzsche is usually considered a fierce critic of romanticism. It is often overlooked, however, that his criticism mainly concerns the late European romanticism, while his attitude towards early German romanticism is characterized by appreciative ambivalence. In light of the early romantic concept of criticism, this paper exa-mines the affinities between Nietzsche and two important representatives of early philosophical Romanticism: Friedrich Schlegel and Novalis. Specifically, it exhi-bits the systematic continuity a) between (...) the concept of romanticising (romanti-sieren) and Nietzsche’s perspectivism, and b) between the early romantic idea of ‘infinite reflection’ and Nietzsche’s talk of a ‘new infinity’. (shrink)

ABSTRACT Prima facie, the philosophy of Emmanuel Levinas would seem to be inherently averse to literature as an ethical mode. Indeed, in his early work, up to and including Totality and Infinity (1961), literary art is often censured with what amounts to Platonic zeal. However, as I will demonstrate, this criticism stands alongside what is seemingly an incongruous use of literary art as a means of ethical exemplification. By exploring this tension, I will show how the contra-epistemic aesthetic of (...) S. Y. Agnon (1888–1970) can be read within ethical terms consistent with Levinas’s philosophical position. (shrink)

ABSTRACT Because the modern concept of number emerged within a quadrivium that included music alongside arithmetic, geometry, and astronomy, musical considerations affected mathematical developments. Michael Stifel embedded the then‐paradoxical term “irrational numbers” (numerici irrationales) in a musical context (1544), though his philosophical aversion to the “cloud of infinity” surrounding such numbers finally outweighed his musical arguments in their favor. Girolamo Cardano gave the same status to irrational and rational quantities in his algebra (1545), for which his contemporaneous work on (...) music suggested parallels and empirical examples. Nicola Vicentino's attempt to revive ancient “enharmonic” music (1555) required and hence defended the use of “irrational proportions” (proportiones inrationales) as if they were numbers. These developments emerged in richly interactive social and cultural milieus whose participants interwove musical and mathematical interests so closely that their intense controversies about ancient Greek music had repercussions for mathematics as well. The musical interests of Stifel, Cardano, and Vicentino influenced their respective treatments of “irrational numbers.” Practical as well as theoretical music both invited and opened the way for the recognition of a radically new concept of number, even in the teeth of paradox. (shrink)

Abstract During 1745-1755 Bošković explicitly used the concept of scientific theory in three cases: the theory of forces existing in nature, the theory of transformations of geometric loci, and the theory of infinitesimals. The theory first mentioned became the famous theory of natural philosophy in 1758, the second was published in the third volume of his mathematical textbook Elementorum Universae Matheseos (1754), and the third theory was never completed, though Bošković repeatedly announced it from 1741 on. The treatment of (...) continuity and infinity in natural philosophy, geometry and infinitesimal analysis brought about inter?theory relations in Bošković's work during his Roman period. The two constructed theories of Bošković, the theory of forces and the theory of geometric transformations, directly influenced the idea for the construction of his third theory. These written theories refer to understanding and effective application of continuity and infinity in natural philosophy and geometry, and this task, according to Bošković, requires methodological support from the theory of infinitesimals. (shrink)

This paper offers a new formulation of the “Hilbert's Hotel Argument” (HHA) which is superior to existing formulations because it (1) demonstrates that HH is logically impossible in the concrete world, (2) takes into account the need to consider the assumptions of HHA, and (3) offers a reply to an important objection concerning the validity of HHA. In addition, this paper contributes to the discussion by using the new HHA to defend a relevant difference between the past and the future (...) by demonstrating that the HHA applies to the former but not to the latter. Finally, this paper demonstrates the significance of metaphysical arguments such as the HHA for physical cosmology. (shrink)

This paper offers a new formulation of the “Hilbert's Hotel Argument” (HHA) which is superior to existing formulations because it (1) demonstrates that HH is logically impossible in the concrete world, (2) takes into account the need to consider the assumptions of HHA, and (3) offers a reply to an important objection concerning the validity of HHA. In addition, this paper contributes to the discussion by using the new HHA to defend a relevant difference between the past and the future (...) by demonstrating that the HHA applies to the former but not to the latter. Finally, this paper demonstrates the significance of metaphysical arguments such as the HHA for physical cosmology. (shrink)

_Abstract_: The overarching aim of this paper is to provide a comprehensive account of melancholic depression from the perspective of hermeneutic phenomenology. More specifically, we propose that this condition should be interpreted as an alteration in the intentional arc that affects corporeality, temporality, and spatiality, rather than as a mood disorder. In fact, classifying melancholic depression as a mood disorder seems a particularly poor choice; the mood disorder is not a cause but a consequence of a primary disturbance in operative (...) intentionality. Our paper comprises four sections: (1) We provide a methodological and conceptual framework for our theoretical work, adopting a hermeneutic phenomenological perspective on melancholic depression, and underlining the main differences between melancholia and depression. (2) We offer a detailed reconstruction of the different interpretations of melancholic depression and their relationships to the issue of temporality. (3) We analyze the disruption of intentionality in melancholic depression and how it affects corporeality and directionality. (4) We focus on the disruption of moods and affects in melancholic depression, observing how they shape subjective experience. The goal of this theoretical contribution is to underscore the role of operative intentionality and its disruptions in melancholic depression. We argue that taking elements such as intentionality, embodiment, and spatiality into consideration could help to direct therapies along more promising paths. _Keywords_: Melancholic Depression; Operative Intentionality; Corporeality; Directionality; Temporality; Mood Disorders _La depressione melancolica. Un’interpretazione fenomenologico-ermeneutica_ _Riassunto_: L’intento generale di questo contributo è di fornire un’interpretazione della depressione melancolica attraverso la fenomenologia ermeneutica. Più specificatamente, la nostra ipotesi è che questa condizione potrebbe essere interpretata come un’alterazione dell’arco intenzionale la quale impatta la corporeità, la temporalità e la spazialità, e non come un disordine dell’umore. Infatti, classificare la depressione melancolica come un disordine dell’umore ci sembra una scelta particolarmente limitata; il disordine dell’umore non è la causa ma la conseguenza di un disturbo a livello primario dell’intenzionalità operativa. Il nostro contributo si articola in quattro sezioni: nella prima, forniamo una cornice metodologica e concettuale finalizzata a chiarificare le fondamenta teoretiche del nostro lavoro, ad illustrare l’utilizzo della fenomenologia ermeneutica in riferimento alla depressione melancolica, e a sottolineare le differenze principali tra melancolia e depressione. Nella seconda sezione, ricostruiamo in modo dettagliato le diverse interpretazioni della depressione melancolica e la loro relazione con la temporalità. Nella terza sezione, analizziamo l’alterazione dell’intenzionalità nella depressione melancolica e come essa impatti sia la corporeità che la direzionalità. Infine, nell’ultima sezione ci concentriamo sull’alterazione degli stati umorali e della componente affettiva, osservando come questi elementi vadano a caratterizzare l’esperienza soggettiva. Il fine di questo contributo teoretico è di evidenziare il ruolo dell’intenzionalità operativa e delle sue alterazioni nella depressione melancolica. La nostra tesi è che prendere in considerazione alcuni elementi, come l’intenzionalità, l’embodiment e la spazialità, possa contribuire a delineare interventi terapeutici in modo più promettente. _Parole chiave_: Depressione melancolica; Intenzionalità operativa; Corporeità; Direzionalità; Temporalità; Disturbi dell’umore. (shrink)

World news can be discouraging these days. In order to counteract the effects of fake news and corruption, scientists have a duty to present the truth and propose ethical solutions acceptable to the world at large. -/- By starting from scratch, we can lay down the scientific principles underlying our very existence, and reach reasonable conclusions on all major topics including quantum physics, infinity, timelessness, free will, mathematical Platonism, happiness, ethics and religion, all the way to creation and a special (...) type of multiverse. -/- This article amounts to a summary of my personal Theory of Everything. -/- DOI: 10.13140/RG.2.2.36046.31049. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:REFLECTIONS ON PANNENBERG'S SYSTEMATIC THEOLOGY 1 PAUL D. MOLNAR St. John's University Jamaica, New York RADING PANNENBERG leaves no doubt that one is encountering an intellectual giant. His thought is clear, systematic, comprehensive, and fact-filled. In many respects this book is exciting; topics are introduced and developed with details from scripture, from obscure and renowned Protestant theologians, from Aquinas, Augustine, Origen, Duns Scotus, Barth, Jiingel, Moltmann, Rahner, and (...) many others until Pannenberg 's own position is clarified and fully developed. Its six chapters explore : 1) how truth is the foundation of systematic theology; 2) how the concept of God relates to this truth (natural theology) ; 3) how the reality of God is understood in relation to other religions; 4) how to understand revelation; 5) the trinity; and 6) the unity and attributes of God. This last lengthy chapter (337-448) presents a very challenging description of God's attributes arguing that " the trinitarian life of God in his economy of salvation proves to be the true infinity of his omnipotence " (415). Hence the doctrine of the trinity makes it possible to speak correctly about God's immanence and transcendence without falling into pantheism, dualism, modalism, or subordinationism. This book is essential reading for any serious student of theology. Because of its complexity and difficulty, however, we shall focus on the relation of theology and philosophy and briefly explore the roles of experience and faith, natural theology, the 1 Wolfhart Pannenberg, Systematic Theology, Volume 1, trans. Geoffrey W. Bromiley (Grand Rapids : William B. Eerdmans Publishing Company, 1991). 501 502 PAUL D. MOLNAR trinity, revelation, God's being and act, and the person and work of Jesus, with a view toward seeing how Pannenberg's philosophy affects his theology. In spite of his view that Rahner may succumb to " subjectivism " through a consensus theory of truth ( 14, n.22), many of Pannenberg's ideas, e.g., that "Nonthematic knowledge of God" is real and " part of our original situation " ( 115), and that, in the context of modern secularism, the word " God " still reminds us that "the wholeness of human existence has become an unanswered question" (71), would resonate positively with a Rahnerian method. And while he believes Barth did not succeed in grounding the doctrine of the trinity in revelation (327, 296), he contends that Rahner made progress with his thesis of identity, even though by allowing the Father to remain transcendent, he did not go far enough and should have seen that the Father " has made himself dependent upon the course of history " (329, emphasis mine). While Pannenberg deliberately intends to emphasize God's freedom and to avoid subjectivism (36ff., 127ff., 376-7, 391, 419), as when he argues that dogmaticians do not decide what is true, but only God decides (56), fundamental problems do affiict his approach to theology from the beginning. For example: The natural theology of the philosophers had formulated a criterion for judging whether any God could be seriously considered as the author of the whole cosmos, and Christian theology had to meet this criterion if its claim could be taken seriously that the God who redeems us in Jesus Christ is the Creator of heaven and earth and thus the one true God of all peoples (79). The obvious question which arises here is whether it is philosophy or God the creator, known through faith, who determines the meaning of creation and redemption; problems connected with this assertion arise within each chapter. Even though Pannenberg recognizes the problem of establishing our knowledge of God by negating our limited experience ( 184£., 406), he actually attempts to understand God's being in terms of the limits of creaturely experience when he argues that finite knowledge " is grounded especially in the time-bound nature PANNENBERG'S SYSTEMATIC THEOLOGY 503 of the knowledge" (54). Thus all of our statements about God, who is not an object in the world, "rest on anticipations of the totality of the world and therefore on the as yet nonexistent future of its uncompleted history " (55, emphasis mine) ; following Descartes, he writes " We attain a sense of the infinite only by negation of the limit of the finite "; while our direct unthematic... (shrink)

_Abstract_: In this introduction to the thematic issue on _the future of the cognitive science(s)_, we examine how challenges and uncertainties surrounding the past and present of this discipline make it difficult to chart its future. We focus on two main questions. The first is whether cognitive science is a single unified field or inherently pluralistic. This question can be asked at various levels: First, with respect to the disciplines that should be included in the cognitive hexagon and their reciprocal (...) relationships: should we speak of cognitive science or of the cognitive sciences? Second, with regard to the conceptual and methodological changes (turns or revolutions) that have taken place within the cognitive project from its inception to the present day. Third, it pertains to cognitive psychology as a discipline. Before the emergence of cognitive science psychology was a fragmented discipline characterized by different traditions and approaches: has cognitive science been able to stem this fragmentation? Finally, we can question the unity of the cognitive architecture itself: is cognition produced by homogeneous or heterogenous mechanisms for information processing? We show that the issue of unity is addressed by several of the papers included in this thematic issue. In the second part of this introduction, we query the role that each component discipline should play in the cognitive project and in particular which should lead the project going forward, and why. Again, we show how this issue has been tackled by several articles featured in this collection. _Keywords_: Future of Cognitive Science; Cognitive Psychology; Pluralism and Cognitive Science; Philosophy and Cognitive Science; Fragmentation of Psychology _Quale futuro per la scienza cognitiva?_ _Riassunto_: In questa introduzione al fascicolo tematico dedicato al _futuro della scienza cognitiva_ prendiamo in considerazione sfide e incertezze che caratterizzano il passato e il presente di questa disciplina, rendendo difficile prevedere il suo futuro. Due le questioni principali su cui concentriamo l’attenzione. La prima: la scienza cognitiva è un ambito unitario o intrinsecamente pluralistico? Questo problema si manifesta a diversi livelli. In primo luogo, riguarda le discipline che dovrebbero comporre l’esagono cognitivo e le loro relazioni reciproche: dovremmo parlare di scienza cognitiva o di scienze cognitive? In secondo luogo, riguarda le trasformazioni (svolte o rivoluzioni) concettuali e metodologiche avvenute all’interno di questo progetto dalla sua nascita fino ai giorni nostri. In terzo luogo, riguarda la psicologia cognitiva. Prima della nascita della scienza cognitiva la psicologia era una disciplina frammentaria caratterizzata da diverse tradizioni e approcci: possiamo dire che la scienza cognitiva abbia posto rimedio a tale frammentazione? Infine, il problema dell’unità sorge anche in relazione all’architettura cognitiva stessa, considerando che la cognizione potrebbe o potrebbe non essere prodotta da meccanismi omogenei di elaborazione delle informazioni. Nella presentazione che segue si cerca di mostrare come l’unità, in tutte queste varianti, sia una questione affrontata da diversi articoli presenti in questo fascicolo tematico. Nella seconda parte di questa introduzione prendiamo in considerazione il problema del ruolo che ciascuna disciplina componente dovrebbe svolgere nel progetto cognitivo e, in particolare, quale dovrebbe guidare il progetto e perché, ponendo in evidenza come anche questa sia una questione centrale, tematizzata da diversi lavori presentati in questo fascicolo. _Parole chiave_: Futuro della scienza cognitiva; Psicologia cognitiva; Pluralismo e scienza cognitiva; Filosofia e scienza cognitiva; Frammentazione della psicologia. (shrink)

ABSTRACT I suggest how a broadly Kantian critique of classical logic might spring from reflections on constructibility conditions. According to Kant, mathematics is concerned with objects that are given through ‘arbitrary synthesis,’ in the form of ‘constructions of concepts’ in the medium of ‘pure intuition.’ Logic, by contrast, is narrowly constrained – it has no objects of its own and is fixed by the very forms of thought. That is why there is not much room for developments within logic, (...) as compared to the progress in mathematics. Kant’s view of logic remains critical, though – through considerations that are effectively on the scope and limits of classical logic and which play a part in his transcendental idealism. The most important ones are to be found in his critique of the use of reductio ad absurdum proofs in metaphysics and his solutions to the ‘antinomies of pure reason.’ Arguably, these considerations carry over to mathematics as well – by way of ‘analogues’ to the antinomies – in particular in resolving infinity paradoxes. (shrink)

Abstract In this article, I attempt to trace Emmanuel Levinas's notion of transcendence and its relation to infinity to his Talmudic lectures to offer both a philosophical diagnosis as well as a counter to the essentialist logic of what Levinas considers the traditional or ?metaphysical? concept of time. This opens my speculative argument up to two levels of interpretation as it requires an historical investigation into the cultural context that conditioned Levinas's particular understanding of transcendence and infinity in relation (...) to his turn toward Talmud and sacred Jewish texts in the years following World War Two. But it also requires that we take seriously the critique of history as inadequate to such a task that is inherent in Levinas's presentation of these concepts. All of this hinges on the role and place of ?time? in Levinas. I consider Levinas's category of ?time immemorial? as a counter to the essentialist logic of what Levinas considers the traditional or ?metaphysical? concept of time which I claim is at the core of the ?realist? attitude toward the practice of history founded upon the categories of agency, experience, memory, testimony, and most recently the importance of ?presence?. This in turn leads me to propose two possible alternatives, through Freud and Derrida, that conserve Levinas's critique of the essentialist understanding of time uncoupled from any transcendent or theological mechanism. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Transitions to a Modern Cosmology: Meister Eckhart and Nicholas of Cusa on the Intensive InfiniteElizabeth BrientThe Epochal Transition from the late medieval to the early modern world has long been thought in terms of the gradual “infinitization” of the cosmos. Traditionally this process has been studied by focusing on the pre-history and the aftermath of the Copernican revolution, that is, by describing the transition from the finite, hierarchically (...) ordered medieval cosmos to the infinite and homogeneous universe of the new astronomy. I would like to situate this paper within the context of a more general reading of the process of the infinitization of the medieval cosmos. That is, I want to argue that this process of “infinitization” must be understood intensively as well as extensively. Nature, in the modern age, is thought of not only as infinitely extended in space, but also as exhibiting an infinite richness in all of its parts. Each particular, each individual being is grasped as utterly unique, as infinitely rich and consequently as conceptually inexponable. Thus the infinitization of the real leads to an infinitization of the knowable—the radical shift in ontology grounds a corresponding shift in epistemology, so that the progress of human knowledge is understood as an unending project infinitely extended over time. In order to underscore both aspects of the infinitization of the medieval cosmos (intensive as well as extensive) I will focus on the way in which a single metaphor (that of the infinite sphere) is unfolded historically in the thought of Meister Eckhart and Nicholas of Cusa.1 This paper may thus be viewed as a case study of [End Page 575] the way in which the modern notion of “intensive infinity” may be traced back to late medieval speculation on the nature of God’s omnipresence in the world.1. EARLY SOURCES OF THE INFINITE SPHERE METAPHORThe metaphor of the infinite sphere makes its first appearance in the pseudo-Hermetic Liber XXIV philosophorum, a small book in the literary form of an “opus propositionum” which, in twenty-four propositions, describes the ineffable essence of God in the mystical language of Neoplatonic theology. It is in the second proposition of this book that we find for the first time the formulation, “God is an infinite sphere, whose center is everywhere, whose circumference is nowhere.”2 This formulation represents a significant development in the history of the Neoplatonic conception of intensive unity, and in the Christian understanding of divine infinity. The Neoplatonic One is here identified with God as infinite being. As a result the relationship between the One and the many, between God and the world, comes to be viewed in a new light, through the lens of “infinite unity.” The metaphor of the infinite sphere is a paradoxical formulation which pictures the coincidence of divine immanence with divine transcendence. That is, it expresses a double infinity: both intensive and extensive. God’s center is located at every point, as it were, indicating the plenitude and infinite richness of God’s being, while there is no limit, no boundary to the extent of God’s all-encompassing reach. The infinite sphere figures the divine nature as an intensive plenitude without end or bounds. The third and next proposition in this same work goes on to strengthen this sense of divine intensity, by asserting that God in his entirety is present in every part (Deus est totus in quolibet sui). The divine nature is not spread out compositionally, as it were, but is wholly present throughout. Proposition 18 describes God as a sphere which has just as many circumferences as points (Deus est sphaera cuius tot sunt circumferentiae, quot sunt puncta), once again, an exploding image which can only make sense in the case of an infinite sphere. These three propositions taken together provide us with a springboard for considering the nature of infinite unity. The exploding paradox of the infinite sphere [End Page 576] provides a rubric for thinking the coincidence of transcendence and immanence, a way of conceptualizing infinite unity, which cannot be “grasped” with finite concepts or through discursive reasoning. The metaphor of the infinite sphere finds its meaning precisely in the way... (shrink)