Many philosophic arguments concerned with infinite series depend on the mutual inconsistency of statements of the following five forms: (1) something exists which has R to something; (2) R is asymmetric; (3) R is transitive; (4) for any x which has R to something, there is something which has R to x; (5) only finitely many things are related by R. Such arguments are suspect if the two-place relation R in question involves any conceptual vagueness or inexactness. Traditional sorites arguments (...) show that a statement of form (4) can fail to be true even though it has no clear counter-example. Conceptual vagueness allows a finite series not to have any definite first member. I consider the speculative possibilities that there have been only finitely many non-overlapping hours although there has been no first hour and that space and time are only finitely divisible even though there are no smallest spatial or temporal intervals. (shrink)

It amazes children, as they try to count themselves out of numbers, only to discover one day that the hundreds, thousands, and zillions go on forever—to something like infinity. And anyone who has advanced beyond the bounds of basic mathematics has soon marveled at that drunken number eight lying on its side in the pages of their work. Infinity fascinates; it takes the mind beyond its everyday concerns—indeed, beyond everything—to something always more. Infinity makes even the infinite universe seem small; (...) yet it can also be infinitesimal. Infinity thrives on paradox, and it turns the simplest arithmetic on its head, with 1 seeming feasibly to equal 0, after all. Infinity defies common sense. The contemplation of it has relieved at least two great mathematicians of their sanity. Thoroughly readable and entirely accessible, science writer Brian Clegg's lively history explores infinity in its many intriguing facets, from its ancient origins to its place today at the heart of mathematics and science. He examines infinity's paradoxes and profiles the people who first grappled with and then defined and refined them, offering information, mystery, and poetry to conceive the inconceivable and define the indefinable. (shrink)

In this paper two paradoxes of infinity are considered through the lens of counterfactual logic, drawing heavily on a result of Fine (2012b). I will argue that a satisfactory resolution of these paradoxes will have wide ranging implications for the logic of counterfactuals. I then situate these puzzles in the context of the wider role of counterfactuals, connecting them to indicative conditionals, probabilities, rationality and the direction of causation, and compare my own resolution of the paradoxes to alternatives inspired by (...) the theories of Lewis and Fine. Here is a quick overview of the paper. Sections 17.1 and 17.2 introduce two paradoxes of infinity that rest on certain principals concerning the logic of counterfactuals. Section 17.3 considers three possible ways to weaken the counterfactual logic that would resolve these paradoxes, and examines three concrete theories of conditionals which make those weakenings — Lewis (1973), Fine (2012a) and Bacon (2015). With the first two theories found wanting, Sect. 17.4 develops my preferred solution in (Bacon, 2015) a little further, and provides some simple models to show that the paradoxes can indeed be resolved in that framework. (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)

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)

By around the age of 5½, many children in the United States judge that numbers never end, and that it is always possible to add 1 to a set. These same children also generally perform well when asked to label the quantity of a set after one object is added (e.g., judging that a set labeled “five” should now be “six”). These findings suggest that children have implicit knowledge of the “successor function”: Every natural number, n, has a successor, n (...) + 1. Here, we explored how children discover this recursive function, and whether it might be related to discovering productive morphological rules that govern language‐specific counting routines (e.g., the rules in English that represent base‐10 structure). We tested 4‐ and 5‐year‐old children’s knowledge of counting with three tasks, which we then related to (a) children’s belief that 1 can always be added to any number (the successor function) and (b) their belief that numbers never end (infinity). Children who exhibited knowledge of a productive counting rule were significantly more likely to believe that numbers are infinite (i.e., there is no largest number), though such counting knowledge was not directly linked to knowledge of the successor function, per se. Also, our findings suggest that children as young as 4 years of age are able to implement rules defined over their verbal count list to generate number words beyond their spontaneous counting range, an insight which may support reasoning over their acquired verbal count sequence to infer that numbers never end. (shrink)

In preparation for the treatment of the gravitational field at spatial infinity, this paper deals with the electromagnetic field at spatial infinity. The field equations on this three-dimensional(1+2) manifold can be obtained from an action principle, which in turn lends itself to a Hamiltonian formulation. Quantization is formally straightforward, but some thought is given to the physical interpretation of the results.

Jonathan Lear; XII*—Aristotelian Infinity, Proceedings of the Aristotelian Society, Volume 80, Issue 1, 1 June 1980, Pages 187–210, https://doi.org/10.1093/aris.

Es werden zwei Bedeutungen von „Infinitesimal“ unterschieden und zwei Thesen verteidigt: (1) Leibniz glaubte, das Infinitesimale in einer der beiden Bedeutungen sei nicht nur eine nützliche Erdichtung, sondern es sei sogar notwendig fur die Differentialrechnung; (2) die moderne Nichtstand-Analysis rechtfertigt weder Leibniz's Griinde fur die Einführung des Infinitesimalen noch seinen Gebrauch desselben.

Winthrop Parkhurst, W. J. Kingsland, Jr; Infinity and the Infinitesimal, The Monist, Volume 35, Issue 4, 1 October 1925, Pages 633–666, https://doi.org/10.5840/.

The article analyses the explication of the infinity in the philosophical courses taught at Kyiv-Mohyla Academy at the 17th and 18th centuries. It examines 12 philosophical courses – since 1645 (the course by Inokentii Gizel) until 1751 (the course by Georgii Konyskyi). It shows how the infinity was defined and in which kinds it was divided in different courses. In general, all the professors, as well as other scholastic philosophers, agree that categorematic infinity exists only in God, but syncategorematic is (...) present in the created world. Regarding the question whether God, being omnipotent, can create a categorematic infinity in the world, the Mogilyans are divided into several camps: (1) Inokentii Gizel, Stefan Yavorskyi, Inokentii Popovskyi, Sylvestr Pinovskyi, Platon Malynovskyi gave a positive answer to the question; (2) Yosyf Volchanskyi, Ilarion Levytskyi, Amvrosii Dubnevych, Sylvestr Kulabka believed that this kind of infinity is in principle impossible, hence God cannot create it; (3) Teofan Prokopovych and Georgii Konyskyi took a sceptical stand and consider that the human mind as such could not solve this problem. The article analyses which arguments were offered by each camp, and gives suggestions what influenced the position of certain Mohylian professors. Most probably, Mohylian philosophers who supported the possibility of actual infinity in the created world were influenced by nominalistically oriented Jesuit philosophers, like Pedro Hurtando de Mendoza and Rodrigo Arriaga. The other Mohylians backed a more traditional idea, supported by Thomas Aquinas, that only one actual infinity can exist and it is God. (shrink)

The article proposes an interpretation of Kant’s notions of form of, and formal intuition of space to explain and justify the claim that representing space as object requires a synthesis. This involves identifying the transcendental conditions of the analytic unity of consciousness of this formal intuition and distinguishing between it and its content. On this reading which builds upon recent proposals, footnote B160–1n. involves no revision of the Transcendental Aesthetic: space is essentially characterized by non-conceptual features. The article also addresses (...) worries about the infinite magnitude and the unicity of space, by considering the characteristics and requirements of geometric constructions. (shrink)

We describe a variety of sets internal to models of intuitionistic set theory that (1) manifest some of the crucial behaviors of potentially infinite sets as described in the foundational literature going back to Aristotle, and (2) provide models for systems of predicative arithmetic. We close with a brief discussion of Church’s Thesis for predicative arithmetic.

Introduction 1 -- Ancient Greece -- Reason and the irrational : Sophocles' Oedipus tyrannus -- Psuchê : literature and moral psychology from Homer to Sophocles -- Aristotle's poetics : Oedipus and the problem of tragedy -- Psuchê redux : philosophy and the new psychology -- Psychologizing Oedipus : reason and unreason in Aristotle's ethics -- Golden age denmark -- Kierkegaard's retrieval of Greek tragedy -- Tragedy as historical idea : either/or ancient drama reflected in the modern -- Stages on life's (...) way : Hamartia in the wake of modernity -- Fear and trembling : tragedy, comedy, and the heroism of Abraham -- The concept of anxiety : fate and the tragic logos of a second ethics -- Beyond eudaimonism : tragic virtue and the practice of eternity -- Moral psychology in the pseudonyms, search for a method ethicscontra ethics : Climacus on eternal happiness and tragic virtue -- Kierkegaard and the tragedy of authorship. (shrink)

Light speed, c = 3 × 108 meters per second, is the ultimate speed limit of the universe. The welltested physics orthodoxy of special relativity tells us that nothing can go faster than c. When any massive object with rest mass M (taken to be in energy units) has velocity v=c (or relativistic velocity ß = v/c = 1), the object's mass-energy becomes infinite. This is because the relativistic..

This chapter contains sections titled: * The Fundamental Question * 1 Whatever Begins to Exist Has a Cause * 2 The Universe Began To Exist * 3 The Cause of the Universe * Notes.

Hegel’s very first acknowledged publication was, among other things, an attack on Fichte.1 In 1801, Hegel was still laboring in almost complete obscurity, while Fichte was an international sensation, though already somewhat past the peak of his meteoric career. In the 1801 Differenzschrift, Hegel cut his teeth by criticizing Fichte’s already widely-criticized Wissenschaftslehre, and by demonstrating that Schelling’s philosophical system was not simply to be equated with it. Fichte himself never bothered to respond to Hegel’s criticisms; indeed he never publicly (...) acknowledged their existence. This was not because he was unconcerned with criticisms of his views; quite the contrary. But at the time he had bigger fish to fry. He responded to Jacobi’s criticisms, and to Schelling’s; he replied in great detail to critical questions raised by Reinhold, and with vituperative intensity to objections raised by skeptics and purportedly loyal Kantians. But Hegel’s Differenzschrift was left without a Fichtean rebuttal. This is a pity, both because of the missed opportunity to illuminate by controversy central issues at stake in the post-Kantian period, but also because it made it easier for Hegel simply to reiterate his youthful criticism as if it were the last word. And reiterate it he did: in one form or another Hegel’s early criticisms of Fichte reappear at every subsequent stage of his career: in the Phenomenology, in the Science of Logic, in the Encyclopaedia, as the final chapter in Hegel’s History of Philosophy, and in countless other minor works and documents from the Nachlass and correspondence. (shrink)

Modal logic provides an elegant way to understand the notion of potential infinity. This raises the question of what the right modal logic is for reasoning about potential infinity. In this article I identify a choice point in determining the right modal logic: Can a potentially infinite collection ever be expanded in two mutually incompatible ways? If not, then the possible expansions are convergent; if so, then the possible expansions are branching. When possible expansions are convergent, the right modal logic (...) is S4.2, and a mirroring theorem due to Linnebo allows for a natural potentialist interpretation of mathematical discourse. When the possible expansions are branching, the right modal logic is S4. However, the usual box and diamond do not suffice to express everything the potentialist wants to express. I argue that the potentialist also needs an operator expressing that something will eventually happen in every possible expansion. I prove that the result of adding this operator to S4 makes the set of validities Pi-1-1 hard. This result makes it unlikely that there is any natural translation of ordinary mathematical discourse into the potentialist framework in the context of branching possibilities. (shrink)

Modal logic provides an elegant way to understand the notion of potential infinity. This raises the question of what the right modal logic is for reasoning about potential infinity. In this article I identify a choice point in determining the right modal logic: Can a potentially infinite collection ever be expanded in two mutually incompatible ways? If not, then the possible expansions are convergent; if so, then the possible expansions are branching. When possible expansions are convergent, the right modal logic (...) is S4.2, and a mirroring theorem due to Linnebo allows for a natural potentialist interpretation of mathematical discourse. When the possible expansions are branching, the right modal logic is S4. However, the usual box and diamond do not suffice to express everything the potentialist wants to express. I argue that the potentialist also needs an operator expressing that something will eventually happen in every possible expansion. I prove that the result of adding this operator to S4 makes the set of validities Pi-1-1 hard. This result makes it unlikely that there is any natural translation of ordinary mathematical discourse into the potentialist framework in the context of branching possibilities. (shrink)

In his late reflections on values and forms of life from the 1920s and 1930s, Husserl develops the concept of personal value and argues that these values open two kinds of infinities in our lives. On the one hand personal values disclose infinite emotive depths in human individuals while on the other hand they connect human individuals in continuous and progressive chains of care. In order to get at the core of the concept, I will explicate Husserl’s discussion of personal (...) values of love by distinguishing between five related features. I demonstrate that values of love (1) are rooted in egoic depts and define who we are as persons, (2) differ from objective values in being absolute and non-comparative, (3) ground vocational lives as organizing principles, (4) are endlessly self-disclosing and self-intensifying, and (5) establish transitive relations of care between human beings. On the basis of my five-partite distinction, I argue that Husserl’s concepts of love and value of love reveal the dynamic character of human subjectivity and intersubjectivity. (shrink)

We measure, in the presence of the axiom of infinity, the proof-theoretic strength of the axioms of set theory which make the theory look really like a “theory of sets”, namely, the axiom of extensionality Ext, separation axioms and the axiom of regularity Reg . We first introduce a weak weak set theory as a base over which to clarify the strength of these axioms. We then prove the following results about proof-theoretic ordinals:1. and ,2. and . We also show (...) that neither Reg nor affects the proof-theoretic strength, i.e., where T is Basic plus any combination of Ext and. (shrink)

Some, of course, would go further and claim that Jesus was the very content of what he preached, the ontological embodiment of his message, or as Origin put it centuries ago, the kingdom-of-God-in-person, ho autobasileia.1 This affirmation in fact lies at the heart of the Christian tradition, and if the guardians of that orthodoxy were to answer the question we are posing today, they would say: What the Christ of faith will be is the same as what the Jesus of (...) history was: the incarnate presence of the self-communicating God. (shrink)

Here, we continue the discussion in [1], about infinities in Physics. Our goal is to create a Mathematical system to give a probable explanation for infinities in QED, based on Fuzzy time. This Mathematical system should be sufficiently satisfactory and Simple. In general, our goal of these series, is to provide more reasons to consider time as a fuzzy concept in a way that is explained in [4], [5], [6].

As is well known, Epicurus and his followers held that the universe was infinite and f that its two primary components, void and atoms, were each infinite. The void was infinite in extension, the atoms were infinite in number and their total was infinite also in extension. The chief Epicurean proofs of these infinities are found in Epicurus, Ad Herod. 41–2, and in Lucretius 1.951–1020. As far as I can see, both the commentators to these works and writers on Epicurean (...) physics in general have neglected to take into account some material pertinent to these proofs, material found in Aristotle and especially in his commentators Alexander of Aphrodisias, Themistius, Simplicius, and Philoponus. In this article I wish to compare this neglected information with the proofs of infinity found in Epicurus and Lucretius and to discuss their authorship. (shrink)

Related Works: Original Paper: M. Randall Holmes. Strong Axioms of Infinity in NFU. J. Symbolic Logic, Volume 66, Issue 1 , 87--116. Project Euclid: euclid.jsl/1183746361.

Related Works: Original Paper: M. Randall Holmes. Strong Axioms of Infinity in NFU. J. Symbolic Logic, Volume 66, Issue 1, 87--116. Project Euclid: euclid.jsl/1183746361.

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)

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)

The author tells us that this book was originally intended to be part of a larger work, with the provisional title God and Infinity, but that he opted instead for a separate and independent treatment of the notion of infinity in philosophy and related areas. The original purpose is very well-hidden, showing itself clearly only in the Preface and a few other places. The book begins with a chapter describing some known alleged difficulties having to do with the infinitely large (...) and infinitely small . This is followed by a chapter on some basic mathematical notions and facts, including sketchy references to Zermelo-Fraenkel set theory , ordinal and cardinal numbers, Cantor's diagonal method, some special set-theoretic statements , analysis, non-standard analysis, model theory, internal set theory , Conway numbers, and the ‘finite mathematics’ of Mycielski and Lavine. Of course, readers will obtain very little solid information about these matters, but at least they will be introduced to some of the terminology. Chapter 3 returns to the problems and conundrums introduced in Chapter 1 and tries to resolve them or show that they are illusory. In some cases, the trouble arose because an infinite sequence of experiments was imagined which would be physically impossible . Other apparent difficulties arose only because of unfamiliarity with the …. (shrink)

Cosmological arguments attempt to prove the existence of God by appeal to the necessity of a first cause. Schematically, a cosmological argument will thus appear as: (1) All contingent beings have a cause of existence. (2) There can be no infinite causal chains. (3) Therefore, there must be some non-contingent First Cause. Cosmological arguments come in two species, depending on their justification of the second premiss. Non-temporal cosmological arguments, such as those of Aristotle and Aquinas, view causation as requiring explanatory (...) or conceptual priority, and thus insist that there can be no infinite regresses in such priority. Temporal cosmological arguments, also called kalam cosmological arguments due to their historical roots in Islamic kalam philosophers such as Abu Yusuf Ya'qub b. Ishaq al-Kindi and Abu Ali al-Hussain ibn Sina, view causation as requiring temporal priority, and thus insist that there can be no infinite temporal regresses.1 The kalam cosmological argument thus requires some supporting argument showing the incoherence of an infinite temporal regress of causally related events. William Lane Craig, in "The Finitude of the Past and the Existence of God"2, attempts to provide such an argument: (4) An actual infinite cannot exist. (5) An infinite temporal regress of events is an actual infinite. (6) Therefore an infinite temporal regress of events cannot exist. (9) I will not be concerned here with the general status of cosmological arguments, kalam or otherwise, or with contesting Craig's assumption that an infinite past would (unlike an infinite future) constitute a problematic actual infinity. I am rather concerned with Craig's general working principle, embodied in (4) above, that actual infinities are impossible. Craig, of course, is not alone in denying the possibility of the actually infinite. Resistance to such infinities is at least as old as Aristotle (Physics 3.5.204b1 – 206a8), and, as Craig rightly points out, persists through much of modern (i.e., post-scholastic, pre-twentieth-century) philosophy.. (shrink)

We study the first-order consequences of Ramsey’s Theorem fork-colourings ofn-tuples, for fixed$n, k \ge 2$, over the relatively weak second-order arithmetic theory$\mathrm {RCA}^*_0$. Using the Chong–Mourad coding lemma, we show that in a model of$\mathrm {RCA}^*_0$that does not satisfy$\Sigma ^0_1$induction,$\mathrm {RT}^n_k$is equivalent to its relativization to any proper$\Sigma ^0_1$-definable cut, so its truth value remains unchanged in all extensions of the model with the same first-order universe.We give a complete axiomatization of the first-order consequences of$\mathrm {RCA}^*_0 + \mathrm {RT}^n_k$for$n \ge (...) 3$. We show that they form a non-finitely axiomatizable subtheory of$\mathrm {PA}$whose$\Pi _3$fragment coincides with$\mathrm {B} \Sigma _1 + \exp $and whose$\Pi _{\ell +3}$fragment for$\ell \ge 1$lies between$\mathrm {I} \Sigma _\ell \Rightarrow \mathrm {B} \Sigma _{\ell +1}$and$\mathrm {B} \Sigma _{\ell +1}$. We also give a complete axiomatization of the first-order consequences of$\mathrm {RCA}^*_0 + \mathrm {RT}^2_k + \neg \mathrm {I} \Sigma _1$. In general, we show that the first-order consequences of$\mathrm {RCA}^*_0 + \mathrm {RT}^2_k$form a subtheory of$\mathrm {I} \Sigma _2$whose$\Pi _3$fragment coincides with$\mathrm {B} \Sigma _1 + \exp $and whose$\Pi _4$fragment is strictly weaker than$\mathrm {B} \Sigma _2$but not contained in$\mathrm {I} \Sigma _1$.Additionally, we consider a principle$\Delta ^0_2$-$\mathrm {RT}^2_2$which is defined like$\mathrm {RT}^2_2$but with both the$2$-colourings and the solutions allowed to be$\Delta ^0_2$-sets rather than just sets. We show that the behaviour of$\Delta ^0_2$-$\mathrm {RT}^2_2$over$\mathrm {RCA}_0 + \mathrm {B}\Sigma ^0_2$is in many ways analogous to that of$\mathrm {RT}^2_2$over$\mathrm {RCA}^*_0$, and that$\mathrm {RCA}_0 + \mathrm {B} \Sigma ^0_2 + \Delta ^0_2$-$\mathrm {RT}^2_2$is$\Pi _4$- but not$\Pi _5$-conservative over$\mathrm {B} \Sigma _2$. However, the statement we use to witness failure of$\Pi _5$-conservativity is not provable in$\mathrm {RCA}_0 +\mathrm {RT}^2_2$. (shrink)

1. If A is strongly amorphous , then its power set P is dually Dedekind infinite, i. e., every function from P onto P is injective. 2. The class of “inexhaustible” sets is not closed under supersets unless AC holds.

Familiar proof theoretical and especially automated deduction methods sometimes accept infinity where, in fact, it can be omitted. Our first example deals with the infinite supply of individual variables admitted in 1-order deductions, the second one deals with infinite-branching rules in sequent calculi with number-theoretical induction. The contents of Section 1 summarize and extend basic ideas and results published elsewhere, whereas basic ideas and results of Section 2 are exposed for the first time in the present paper. We consider classical (...) formalisms only, to which constructive formalisms can be reduced by Shanin-style interpretations. (shrink)

The subject of this paper is a characterization of the $\Sigma_1$ -definable set functions of Kripke-Platek set theory with infinity and a uniform version of axiom of choice: $KP\omega+(uniform\;AC)$ . This class of functions is shown to coincide with the collection of set functionals of type 1 primitive recursive in a given choice functional and $x\mapsto\omega$ . This goal is achieved by a Gödel Dialectica-style functional interpretation of $KP\omega+(uniform\;AC)$ and a computability proof for the involved functionals.

The Bachmann-Howard structure, that is the segment of ordinal numbers below the proof theoretic ordinal of Kripke-Platek set theory with infinity, is fully characterized in terms of CARLSON’s approach to ordinal notation systems based on the notion of Σ1-elementarity.

From the perspective of Christian theology, divine freedom is the paradigm of human freedom, but it is also completely unlike ours in its infinity. This is the paradox of the analogy of being: in its infinity, the Archetype of our being is also completely other. In contrast, likeness between contingent beings is limited in that each being is individuated yet similar to those of like species. No matter how alike beings are, “unlikeness” increases with generic distance. At the asymptotic limit, (...) the Archetype is infinitely unlike us, but remains the ultimate blueprint for each being. If as Archetype, God’s infinite freedom is qualitative of infinite being, then all finite beings must possess freedom to some finite degree. Herein lies the problem treated in this paper: how can bothanimate and inanimate being possess freedom? To answer this question, the author draws upon the first volume of part two of Hans Urs von Balthasar’s sixteen volume trilogy, Theo-logic: The Truth of the World. (shrink)

One of the most frequently asked question is whether Hegel’s idea of God is immanent or transcendent. In Transcending Subjects: Augustine, Hegel, and Theology, Geoffrey Holsclaw attempts to solve this puzzle by contrasting the political theologies of Hegel and Augustine. He argues that Hegel produces a political theology of ‘self-transcending immanence’ while Augustine produces a political theology of ‘self-immanentizing transcendence’. The primary problem with Holsclaw’s dialectical procedure results from its uncritical appeal to a transcendent source for the supersession of opposites. (...) Rather than critically annulling each opposed position, he often resorts to simply describing the inadequacy of positions in light of the privileged ideal of Augustinian transcendence. Holsclaw’s dialectic has not been directly derived from Augustine but from the more recent writings of William Desmond. He translates Desmond’s opposition between the second and third types of transcendence into his own opposition between Hegelian self-transcending immanence and Augustinian self-immanentizing transcendence. Hegel’s apparent ambiguity on transcendence seems to expose him to Holsclaw’s critique. Yet Hegel may, arguably, have never needed to answer this objection because the opposition between immanence and transcendence has already been resolved by Hegel’s dialectic of infinity: for if the opposition between the categories of immanence and transcendence that appears at the level of Spirit merely replicates the vanishing opposition between the finite and the infinite at the level of Logic, then the supersession of this negative opposition between the infinite and the finite into the true and speculative infinity can be virtually replicated for religious consciousness in the supersession of transcendence and immanence into an equally speculative transcendence. Where Holsclaw seeks to supersede immanence into transcendence, Hegel seeks to preserve both transcendence and immanence in the true ‘speculative transcendence’ that corresponds to the true ‘speculative infinity’. The decisive disagreement between Hegel and Holsclaw thus concerns, not merely any relation between transcendence and immanence, but – more fundamentally – the very possibility of superseding the spurious transcendence into a genuinely speculative notion of infinity and transcendence. (shrink)

Clauses (1) and (2) guarantee the inclusion of all 'intuitive' natural numbers, and (3) guarantees the exclusion of all other objects. Thus, in particular, no nonstandard numbers, which would follow after the intuitive ones are admitted (nonstandard numbers are found in nonstandard models of Peano arithmetic, in which the standard natural numbers are followed by one or more 'copies' of integers running from minus infinity to infinity)1. What is problematic about this delimitation? I suspect that its hypothetical proponent would see (...) its weakest point in the unexplained concept of successor. However, we logicians know better (or at least some of us are convinced that we do): it is clause (3) which harbours the neuralgic spot, by dint of resisting any reasonable logical formalization to the point of appearing utterly void! There is, of course, no problem with regimenting (1) - we only need an individual constant 0 and a unary predicate constant N (the one whose meaning we are interested in) and we postulate.. (shrink)

Linnebo and Shapiro have recently given an analysis of potential infinity using modal logic. A key technical component of their account is to show that under a suitable translation ◊ of nonmodal language into modal language, nonmodal sentences ϕ 1, …, ϕ n entail ψ just in case ϕ 1 ◊, …, ϕ n ◊ entail ψ ◊ in the modal logic S4.2. Linnebo and Shapiro establish this result in nonfree logic. In this note I argue that their analysis of (...) potential infinity should be carried out in a free logic. I then extend their key theorems to the setting of negative free logic. (shrink)

It is commonly assumed that Aristotle denies any real existence to infinity. Nothing is actually infinite. If, in order to resolve Zeno’s paradoxes, Aristotle must talk of infinity, it is only in the sense of a potentiality that can never be actualized. Aristotle’s solution has been both praised for its subtlety and blamed for entailing a limitation of mathematic. His understanding of the infinite as simply indefinite (the “bad infinite” that fails to reach its accomplishment), his conception of the cosmos (...) and even its prime mover as finite (in the sense of autarchic/self-contained) have been contrasted with the subsequent claim of God as ens infinitum (understood as a “positive” infinity). The goal of this essay is to reexamine the major texts (notably De caelo) and to demonstrate that (1) Aristotle’s claim according to which there is no actual infinite concerns only substances, not processes. (2) That Aristotle does not deny an actual infinite as such. (3) That when considering time and God (qua eternal) Aristotle acknowledges an actual infinite. (shrink)

1. Approximations of arbitrarily large but finite systems are often mistaken for infinite idealizations in statistical and thermal physics. The problem is illustrated by thermodynamically reversible processes. They are approximations of processes requiring arbitrarily long, but finite times to complete, not processes requiring an actual infinity of time.2. The present debate over whether phase transitions comprise a failure of reduction is confounded by a confusion of two senses of “level”: the molecular versus the thermodynamic level and the few component versus (...) the many component level. (shrink)

In his Lectures on the History of Philosophy, Hegel offers the following verdict on Spinoza’s ontology: “According to Spinoza what is, is God, and God alone. Therefore, the allegations of those who accuse Spinoza of atheism are the direct opposite of the truth; with him there is too much God” (Hegel 1995, vol. 3, 281-2). It is not easy to dismiss Hegel’s grand pronouncement, since Spinoza indeed clearly affirms: “whatever is, is in God” (E1p15). Crocodiles, porcupines (and your thoughts about (...) crocodiles and porcupines) are all in God. There is nothing that is not in Spinoza’s God. Spinoza defines God at the very opening page of the Ethics, and the definition (and its explication) unfold in three successively elucidatory steps: First, Spinoza characterizes God as “a being absolutely infinite [ens absolute infinitum]. Then, he spells out this characterization: “i.e., a substance consisting of an infinity of attributes, of which each one expresses an eternal and infinite essence” (E1def6). Finally, Spinoza adds an explicatio to the entire definition: “I say absolutely infinite, not infinite in its own kind; for if something is only infinite in its own kind, we can deny infinite attributes of it; but if something is absolutely infinite, whatever expresses essence and involves no negation pertains to its essence.” Throughout his philosophical career Spinoza plays with the precise formulation of this definition; its core remains pretty stable, though the small variations might indicate attempts to explore alternative nuances (see, for example, KV II.1; I/19; Ep. 2; IV/7; Ep. 83; IV/335). (shrink)

The article develops the conception that Leibniz has of organisms as machines of a particular type, differing from artificial machines because 1. all the parts of an organic machine are in turn composed by smaller machines and thus to infinity; and 2. the maintenance of the individual identity in living machines is provided by the fact that they have folds going to infinity which can unfold and fold back, thus allowing infinite transformations of the body. The author then discusses these (...) two theses. With respect to thesis 1 the question is how the idea that the artificial machines have parts which are not in their turn machines is compatible with the mechanistic worldview hold by Leibniz. With respect to thesis 2 the author emphasizes the fact that Leibniz's conception of the infinite convolution of the organic machines seems to anticipate several points in the mathematics of fractal curves. (shrink)

continent. 1.1 : 3-13. / 0/ – Introduction I want to propose, as a trajectory into the philosophically weird, an absurd theoretical claim and pursue it, or perhaps more accurately, construct it as I point to it, collecting the ground work behind me like the Perpetual Train from China Mieville's Iron Council which puts down track as it moves reclaiming it along the way. The strange trajectory is the following: Kant's critical philosophy and much of continental philosophy which has followed, (...) has been a defense against horror and madness. Kant's prohibition on speculative metaphysics such as dogmatic metaphysics and transcendental realism, on thinking beyond the imposition of transcendental and moral constraints, has been challenged by numerous figures proceeding him. One of the more interesting critiques of Kant comes from the mad black Deleuzianism of Nick Land stating, “Kant’s critical philosophy is the most elaborate fit of panic in the history of the Earth.” And while Alain Badiou would certainly be opposed to the libidinal investments of Land's Deleuzo-Guattarian thought, he is likewise critical of Kant's normative thought-bureaucracies: Kant is the one author for whom I cannot feel any kinship. Everything in him exasperates me, above all his legalism—always asking Quid Juris? Or ‘Haven’t you crossed the limit?’—combined, as in today’s United States, with a religiosity that is all the more dismal in that it is both omnipresent and vague. The critical machinery he set up has enduringly poisoned philosophy, while giving great succour to the academy, which loves nothing more than to rap the knuckles of the overambitious [….] That is how I understand the truth of Monique David-Menard’s reflections on the properly psychotic origins of Kantianism. I am persuaded that the whole of the critical enterprise is set up to to shield against the tempting symptom represented by the seer Swedenborg, or against ‘diseases of the head’, as Kant puts it. An entire nexus of the limits of reason and philosophy are set up here, namely that the critical philosophy not only defends thought from madness, philosophy from madness, and philosophy from itself, but that philosophy following the advent of the critical enterprise philosophy becomes auto-vampiric; feeding on itself to support the academy. Following Francois Laruelle's non-philosophical indictment of philosophy, we could go one step further and say that philosophy operates on the material of what is philosophizable and not the material of the external world. [1] Beyond this, the Kantian scheme of nestling human thinking between our limited empirical powers and transcendental guarantees of categorical coherence, forms of thinking which stretch beyond either appear illegitimate, thereby liquefying both pre-critical metaphysics and the ravings of the mad in the same critical acid. In rejecting the Kantian apparatus we are left with two entities – an unsure relation of thought to reality where thought is susceptible to internal and external breakdown and a reality with an uncertain sense of stability. These two strands will be pursued, against the sane-seal of post-Kantian philosophy by engaging the work of weird fiction authors H.P. Lovecraft and Thomas Ligotti. The absolute inhumanism of the formers universe will be used to describe a Shoggothic Materialism while the dream worlds of the latter will articulate the mad speculation of a Ventriloquil Idealism. But first we must address the relation of philosophy to madness as well as philosophy to weird fiction. /1/ – Philosophy and Madness There is nothing that the madness of men invents which is not either nature made manifest or nature restored. Michel Foucault. Madness and Civilization. The moment I doubt whether an event that I recall actually took place, I bring the suspicion of madness upon myself: unless I am uncertain as to whether it was not a mere dream. Arthur Schopenhauer. The World as Will and Idea, Vol. 3. Madness is commonly thought of as moving through several well known cultural-historical shifts from madness as a demonic or otherwise theological force, to rationalization, to medicalization psychiatric and otherwise. Foucault's Madness and Civilization is well known for orientating madness as a form of exclusionary social control which operated by demarcating madness from reason. Yet Foucault points to the possibility of madness as the necessity of nature at least prior to the crushing weight of the church. Kant’s philosophy as a response to madness is grounded by his humanizing of madness itself. As Adrian Johnston points out in the early pages of Time Driven pre-Kantian madness meant humans were seized by demonic or angelic forces whereas Kantian madness became one of being too human. Madness becomes internalized, the external demonic forces become flaws of the individual mind. Foucault argues that, while madness is de-demonized it is also dehumanized during the Renaissance, as madmen become creatures neither diabolic nor totally human reduced to the zero degree of humanity. It is immediately clear why for Kant, speculative metaphysics must be curbed – with the problem of internal madness and without the external safeguards of transcendental conditions, there is nothing to formally separate the speculative capacities for metaphysical diagnosis from the mad ramblings of the insane mind – both equally fall outside the realm of practicality and quotidian experience. David-Menard's work is particularly useful in diagnosing the relation of thought and madness in Kant's texts. David-Menard argues that in Kant's relatively unknown “An Essay on the Maladies of the Mind” as well as his later discussion of the Seer of Swedenborg, that Kant formulates madness primarily in terms of sensory upheaval or other hallucinatory theaters. She writes: “madness is an organization of thought. It is made possible by the ambiguity of the normal relation between the imaginary and the perceived, whether this pertains to the order of sensation or to the relations between our ideas” Kant's fascination with the Seer forces Kant between the pincers of “aesthetic reconciliation” – namely melancholic withdrawal – and “a philosophical invention” – namely the critical project. Deleuze and Guattari's schizoanalysis is a combination and reversal of Kant's split, where an aesthetic over engagement with the world entails prolific conceptual invention. Their embrace of madness, however, is of course itself conceptual despite all their rhizomatic maneuvers. Though they move with the energy of madness, Deleuze and Guattari save the capacity of thought from the fangs of insanity by imbuing materiality itself with the capacity for thought. Or, as Ray Brassier puts it, “Deleuze insists, it is necessary to absolutize the immanence of this world in such a way as to dissolve the transcendent disjunction between things as we know them and as they are in themselves”. That is, whereas Kant relied on the faculty of judgment to divide representation from objectivity Deleuze attempts to flatten the whole economy beneath the juggernaut of ontological univocity. Speculation, as a particularly useful form of madness, might fall close to Deleuze and Guattari’s shaping of philosophy into a concept producing machine but is different in that it is potentially self destructive – less reliant on the stability of its own concepts and more adherent to exposing a particular horrifying swath of reality. Speculative madness is always a potential disaster in that it acknowledges little more than its own speculative power with the hope that the gibbering of at least a handful of hysterical brains will be useful. Pre-critical metaphysics amounts to madness, though this may be because the world itself is mad while new attempts at speculative metaphysics, at post-Kantian pre-critical metaphysics, are well aware of our own madness. Without the sobriety of the principle of sufficient reason we have a world of neon madness: “we would have to conceive what our life would be if all the movements of the earth, all the noises of the earth, all the smells, the tastes, all the light – of the earth and elsewhere, came to us in a moment, in an instant – like an atrocious screaming tumult of things”. Speculative thought may be participatory in the screaming tumult of the world or, worse yet, may produce its spectral double. Against theology or reason or simply commonsense, the speculative becomes heretical. Speculation, as the cognitive extension of the horrorific sublime should be met with melancholic detachment. Whereas Kant's theoretical invention, or productivity of thought, is self -sabotaging, since the advent of the critical project is a productivity of thought which then delimits the engine of thought at large either in dogmatic gestures or non-systematizable empirical wondrousness. The former is celebrated by the fiction of Thomas Ligotti whereas the latter is espoused by the tales of H.P. Lovecraft. /2/ – Weird Fiction and Philosophy. Supernatural horror, in all its eerie constructions, enables a reader to taste treats inconsistent with his personal welfare. Thomas Ligotti Songs of a Dead Dreamer. I choose weird stories because they suit my inclination best—one of my strongest and most persistent wishes being to achieve,momentarily, the illusion of some strange suspension or violation of the galling limitations of time, space, and natural law which forever imprison us and frustrate our curiosity about the infinite cosmic spaces beyond the radius of our sight and analysis H.P. Lovecraft. “Notes on Writing Weird Fiction” Lovecraft states that his creation of a story is to suspend natural law yet, at the same time, he indexes the tenuousness of such laws, suggesting the vast possibilities of the cosmic. The tension that Lovecraft sets up between his own fictions and the universe or nature is reproduced within his fictions in the common theme of the unreliable narrator; unreliable precisely because they are either mad or what they have witnessed questions the bounds of material reality. In “The Call of Cthulhu” Lovecraft writes: The most merciful thing in the world, I think, is the inability of the human mind to correlate all its contents. We live on a placid island of ignorance in the midst of black seas of infinity, and it was not meant that we should voyage far. The sciences, each straining in its own direction, have hitherto harmed us little; but some day the piecing together of dissociated knowledge will open up such terrifying vistas of reality, and of our frightful position therein, that we shall either go mad from the revelation or flee from the deadly light into the peace and safety of a new dark age. Despite Lovecraft's invocations of illusion, he is not claiming that his fantastic creations such as the Old Ones are supernatural but, following Joshi, are only ever supernormal. One can immediately see that instead of nullifying realism Lovecraft in fact opens up the real to an unbearable degree. In various letters and non-fictional statements Lovecraft espoused strictly materialist tenets, ones which he borrowed from Hugh Elliot namely the uniformity of law, the denial of teleology and the denial of non-material existence. Lovecraft seeks to explore the possibilities of such a universe by piling horror upon horror until the fragile brain which attempts to grasp it fractures. This may be why philosophy has largely ignored weird fiction – while Deleuze and Guattari mark the turn towards weird fiction and Lovecraft in particular, with the precursors to speculative realism as well as contemporary related thinkers have begun to view Lovecraft as making philosophical contributions. Lovecraft's own relation to philosophy is largely critical while celebrating Nietzsche and Schopenhauer. This relationship of Lovecraft to philosophy and philosophy to Lovecraft is coupled with Lovecraft's habit of mercilessly destroying the philosopher and the figure of the academic more generally in his work, a destruction which is both an epistemological destruction and an ontological destruction. Thomas Ligotti's weird fiction which he has designated as a kind of “confrontational escapism” might be best described in the following quote from one of his shortstories, “The human phenomenon is but the sum of densely coiled layers of illusion each of which winds itself on the supreme insanity. That there are persons of any kind when all there can be is mindless mirrors laughing and screaming as they parade about in an endless dream”. Whereas Lovecraft's weirdness draws predominantly from the abyssal depths of the uncharted universe, Ligotti's existential horror focuses on the awful proliferation of meaningless surfaces that is, the banal and every day function of representation. In an interview, Ligotti states: We don't even know what the world is like except through our sense organs, which are provably inadequate. It's no less the case with our brains. Our whole lives are motored along by forces we cannot know and perceptions that are faulty. We sometimes hear people say that they're not feeling themselves. Well, who or what do they feel like then? This is not to say that Ligotti sees nothing beneath the surface but that there is only darkness or blackness behind it, whether that surface is on the cosmological level or the personal. By addressing the implicit and explicit philosophical issues in Ligotti's work we will see that his nightmarish take on reality is a form of malevolent idealism, an idealism which is grounded in a real, albeit dark and obscure materiality. If Ligotti's horrors ultimately circle around mad perceptions which degrade the subject, it takes aim at the vast majority of the focus of continental philosophy. While Lovecraft's acidic materialism clearly assaults any romantic concept of being from the outside, Ligotti attacks consciousness from the inside: Just a little doubt slipped into the mind, a little trickle of suspicion in the bloodstream, and all those eyes of ours, one by one, open up to the world and see its horror [...] Not even the solar brilliance of a summer day will harbor you from horror. For horror eats the light and digests it into darkness. Clearly, the weird fiction of Lovecraft and Ligotti amount to a anti-anthrocentric onslaught against the ramparts of correlationist continental philosophy. /3/ – Shoggothic Materialism or the Formless Formless protoplasm able to mock and reflect all forms and organs and processes—viscous agglutinations of bubbling cells—rubbery fifteen-foot spheroids infinitely plastic and ductile—slaves of suggestion, builders of cities—more and more sullen, more and more intelligent, more and more amphibious, more and more imitative—Great God! What madness made even those blasphemous Old Ones willing to use and to carve such things? H.P. Lovecraft. “At the Mountains of Madness” On the other hand, affirming that the universe resembles nothing and is only formless amounts to saying that the universe is something like a spider or spit. Georges Bataille. “Formless”. The Shoggoths feature most prominently in H.P. Lovecraft's shortstory “At the Mountains of Madness” where they are described in the following manner: It was a terrible, indescribable thing vaster than any subway train – a shapeless congeries of protoplasmic bubbles, faintly self -luminous, and with myriads of temporary eyes forming and un-forming as pustules of greenish light all over the tunnel-filling front that bore down upon us, crushing the frantic penguins and slithering over the glistening floor that it and its kind had swept so evilly free of all litter. The term is a litmus test for materialism itself as the Shoggoth is an amorphous creature. The Shoggoths were living digging machines bio engineered by the Elder Things, and their protoplasmic bodies being formed into various tools by their hypnotic powers. The Shoggoths eventually became self aware and rose up against their masters in an ultimately failed rebellion. After the Elder Ones retreated into the oceans leaving the Shoggoths to roam the frozen wastes of the Antarctic. The onto-genesis of the Shoggoths and their gross materiality, index the horrifyingly deep time of the earth a concept near and dear to Lovecraft's formulation of horror as well as the fear of intelligences far beyond, and far before, the ascent of humankind on earth and elsewhere. The sickly amorphous nature of the Shoggoths invade materialism at large, where while materiality is unmistakably real ie not discursive, psychological, or otherwise overly subjectivist, it questions the relation of materialism to life. As Eugene Thacker writes: The Shoggoths or Elder Things do not even share the same reality with the human beings who encounter them—and yet this encounter takes place, though in a strange no-place that is neither quite that of the phenomenal world of the human subject or the noumenal world of an external reality. Amorphous yet definitively material beings are a constant in Lovecraft's tales. In his tale “The Dream-Quest of Unknown Kadatth” Lovecraft describes Azathoth as, “that shocking final peril which gibbers unmentionably outside the ordered universe,” that, “last amorphous blight of nethermost confusion which blashphemes and bubbles at the centre of all infinity,” who, “gnaws hungrily in inconceivable, unlighted chambers beyond time”. Azathoth's name may have multiple origins but the most striking is the alchemy term azoth which is both a cohesive agent and a acidic creation pointing back to the generative and the decayed. The indistinction of generation and degradation materially mirrors the blur between the natural and the unnatural as well as life and non-life. Lovecraft speaks of the tension between the natural and the unnatural is his short story “The Unnameable.” He writes, “if the psychic emanations of human creatures be grotesque distortions, what coherent representation could express or portray so gibbous and infamous a nebulousity as the spectre of a malign, chaotic perversion, itself a morbid blasphemy against Nature?”. Lovecraft explores exactly the tension outlined at the beginning of this chapter, between life and thought. At the end of his short tale Lovecraft compounds the problem as the unnameable is described as “a gelatin—a slime—yet it had shapes, a thousand shapes of horror beyond all memory”. Deleuze suggests that becoming-animal is operative throughout Lovecraft's work, where narrators feel themselves reeling at their becoming non-human or of being the anomalous or of becoming atomized. Following Eugene Thacker however, it may be far more accurate to say that Lovecraft's tales exhibit not a becoming-animal but a becoming-creature. Where the monstrous breaks the purportedly fixed laws of nature, the creature is far more ontologically ambiguous. The nameless thing is an altogether different horizon for thought. The creature is either less than animal or more than animal – its becoming is too strange for animal categories and indexes the slow march of thought towards the bizarre. This strangeness is, as aways, some indefinite swirling in the category of immanence and becoming. Bataille begins “The Labyrinth” with the assertion that being, to continue to be, is becoming. More becoming means more being hence the assertion that Bataille's barking dog is more than the sponge. This would mean that the Shoggotth is altogether too much being, too much material in the materialism. Bataille suggests that there is an immanence between the eater and the eaten, across the species and never within them. That is, despite the chaotic storm of immanence there must remain some capacity to distinguish the gradients of becoming without reliance upon, or at least total dependence upon, the powers of intellection to parse the universe into recognizable bits, properly digestible factoids. That is, if we undo Deleuze's aforementioned valorization of sense which, for his variation of materialism, performed the work of the transcendental, but refuse to reinstate Kant's transcendental disjunction between thing and appearance, then it must be a quality of becoming-as-being itself which can account for the discernible nature of things by sense. In an interview with Peter Gratton, Jane Bennett formulates the problem thusly: What is this strange systematicity proper to a world of Becoming? What, for example, initiates this congealing that will undo itself? Is it possible to identify phases within this formativity, plateaus of differentiation? If so, do the phases/plateaus follow a temporal sequence? Or, does the process of formation inside Becoming require us to theorize a non-chronological kind of time? I think that your student’s question: “How can we account for something like iterable structures in an assemblage theory?” is exactly the right question. Philosophy has erred too far on the side of the subject in the subject-object relation and has furthermore, lost the very weirdness of the non-human. Beyond this, the madness of thought need not override. /4/ - Ventriloquial Idealism or the Externality of Thought My aim is the opposite of Lovecraft's. He had an appreciation for natural scenery on earth and wanted to reach beyond the visible in the universe. I have no appreciation for natural scenery and want the objective universe to be a reflection of a character. Thomas Ligotti. “Devotees of Decay and Desolation.” Unless life is a dream, nothing makes sense. For as a reality, it is a rank failure [….] Horror is more real than we are. Thomas Ligotti. “Professor Nobody's Little Lectures on Supernatural Horror”. Thomas Ligotti's tales are rife with mannequins, puppets, and other brainless entities which of replace the valorized subject of philosophy – that of the free thinking human being. His tales such as “The Dream of the Manikin” aim to destroy the rootedness of consciousness. James Trafford has connected the anti-egoism of Ligotti to Thomas Metzinger – where the self is at best an illusion and we plead desperately for someone else to acknowledge that we are real. Trafford has stated it thus, “Life is played out as an inescapable puppet show, an endless dream in which the puppets are generally unaware that they are trapped within a mesmeric dance of whose mechanisms they know nothing and over which they have no control”. An absolute materialism, for Ligotti, implies an alienation of the idea which leads to a ventriloquil idealism. As Ligotti notes in an interview, “the fiasco and nightmare of existence, the particular fiasco and nightmare of human existence, the sense that people are puppets of powers they cannot comprehend, etc.” And then further elaborates that,“[a]ssuming that anything has to exist, my perfect world would be one in which everyone has experienced the annulment of his or her ego. That is, our consciousness of ourselves as unique individuals would entirely disappear”. The externality of the idea leads to the unfortunate consequence of consciousness eating at itself through horror which, for Ligotti, is more real than reality and goes beyond horror-as-affect. Beyond this, taking together with the unreality of life and the ventriloquizing of subjectivity, Ligotti's thought becomes an idealism in which thought itself is alien and ultimately horrifying. The role of human thought and the relation of non-relation of horror to thought is not completely clear in Ligotti's The Conspiracy Against the Human Race. Ligotti argues in his The Conspiracy Against the Human Race,that the advent of thought is a mistake of nature and that horror is being in the sense that horror results from knowing too much. Yet, at the same time, Ligotti seems to suggest that thought separates us from nature whereas, for Lovecraft, thought is far less privileged – mind is just another manifestation of the vital principal, it is just another materialization of energy. In his brilliant “Prospects for Post-Copernican Dogmatism” Iain Grant rallies against the negative definition of dogmatism and the transcendental, and suggests that negatively defining both over-focuses on conditions of access and subjectivism at the expense of the real or nature. With Schelling, who is Grant's champion against the subjectivist bastions of both Fichte and Kant, Ligotti's idealism could be taken as a transcendental realism following from an ontological realism. Yet the transcendental status of Ligotti's thought move towards a treatment of the transcendental which may threaten to leave beyond its realist ground. Ligotti states: Belief in the supernatural is only superstition. That said, a sense of the supernatural, as Conrad evidenced in Heart of Darkness, must be admitted if one's inclination is to go to the limits of horror. It is the sense of what should not be- the sense of being ravaged by the impossible. Phenomenally speaking, the super-natural may be regarded as the metaphysical counterpart of insanity, a transcendental correlative of a mind that has been driven mad. Again, Ligotti equates madness with thought, qualifying both as supernatural while remaining less emphatic about the metaphysical dimensions of horror. The question becomes one of how exactly the hallucinatory realm of the ideal relates to the black churning matter of Lovecraft's chaos of elementary particles. In his tale “I Have a Special Plan for This World” Ligotti formulates thus: A: There is no grand scheme of things. B: If there were a grand scheme of things, the fact – the fact – that we are not equipped to perceive it, either by natural or supernatural means, is a nightmarish obscenity. C: The very notion of a grand scheme of things is a nightmarish obscenity. Here Ligotti is not discounting metaphysics but implying that if it does exist the fact that we are phenomenologically ill-equipped to perceive that it is nightmarish. For Ligotti, nightmare and horror occur within the circuit of consciousness whereas for Lovecraft the relation between reality and mind is less productive on the side of mind. It is easier to ascertain how the Kantian philosophy is a defense against the diseases of the head as Kant armors his critical enterprise from too much of the world and too much of the mind. The weird fiction of both Lovecraft and Ligotti demonstrates that there is too much of both feeding into one another in a way that corrodes the Kantian schema throughly, breaking it down into a dead but still ontologically potentiated nigredo. The haunting, terrifying fact of Ligotti's idealism is that the transcendental motion which brought thought to matter, while throughly material and naturalized, brings with it the horror that thought cannot be undone without ending the material that bears it either locally or completely. Thought comes from an elsewhere and an elsewhen being-in-thought. The unthinkable outside thought is as maddening as the unthought engine of thought itself within thought which doesn't exist except for the mind, the rotting décor of the brain. /5/ - Hyperstitional Transcendental Paranoia or Self -Expelled Thought Weird fiction has been given some direct treatment in philosophy in the mad black Deleuzianism of Nick Land. Nick Land along with others in the 1990s created the Cyber Culture Research Unit as well as the research group Hyperstition. The now defunct hyperstitional website, an outgrowth of the Cyber Culture Research Unit, defined hyperstition in the following fourfold: 1-Element of effective culture that makes itself real. 2-Fictional quantity functional as a time-traveling device. 3-Coincidence intensifier. 4-Call to the Old Ones. The distinctively Lovecraftian character of hyperstition is hard to miss as is its Deleuzo-Guattarian roots. In the opening pages of A Thousand Plateaus Deleuze and Guattari write, “We have been criticized for over-quoting literary authors. But when one writes, the only question is which other machine the literary machine can be plugged into”. The indisinction of literature and philosophy mirrors the mess of being and knowing as post-correlationist philosophy, where philosophy tries to make itself real where literature, especially the weird, aims itself at the brain-circuit of horror. The texts of both Lovecraft and Ligotti work through horror as epistemological plasticity meeting with proximity as well as the deep time of Lovecraft and the glacially slow time of paranoia in Ligotti. Against Deleuze, and following Brassier, we cannot allow the time of consciousness, the Bergsonian time of the duree, to override natural time, but instead acknowledge that it is an unfortunate fact of existence as a thinking being. Horror-time, the time of consciousness, with all its punctuated moments and drawn out terrors, cannot compare to the deep time of non-existence both in the unreachable past and the unknown future. The crystalline cogs of Kant's account of experience as the leading light for the possibility of metaphysics must be throughly obliterated. His gloss of experience in Prolegomena to any Future Metaphysics could not be more sterile: Experience consists of intuitions, which belong to the sensibility, and of judgments, which are entirely a work of the understanding. But the judgments which the understanding makes entirely out of sensuous intuitions are far from being judgments of experience. For in the one case the judgment connects only the perceptions as they are given in sensuous intuition [....] Experience consists in the synthetic connection of appearances in consciousness, so far as this connection is necessary. Here it is difficult to dismiss the queasiness that Kant's legalism induces upon sight for both Badiou and David-Menard. Kant's thought becomes, as Foucault says when reflecting on Sade's text in relation to nature, “the savage abolition of itself”. For Badiou, Kant's philosophy simply closes off too much of the outside, freezing the world of thought in an all too limited formalism. Critical philosophy is simply the systematized quarantine on future thinking, on thinking which would threaten the formalism which artificially grants thought its own coherency in the face of madness. Even the becoming-mad of Deleuze, while escaping the rumbling ground, makes grounds for itself, mad grounds but grounds which are thinkable in their affect. The field of effects allows for Deleuze's aesthetic and radical empiricism, in which effects and/or occasions make up the material of the world to be thought as a chaosmosis of simulacra. Given a critique of an empiricism of aesthetics, of the image, it may be difficult to justify an attack on Kantian formalism with the madness of literature, which does not aim to make itself real but which we may attempt to make real. That is, how do Lovecraft's and Ligotti's materials, as materials for philosophy to work on, differ from either the operative formalisms of Kant or the implicitly formalized images of Deleuzian empiricism? It is simply that such texts do not aim to make themselves real, and make claims to the real which are more alien to us than familiar, which is why their horror is immediately more trustworthy. This is the madness which Blanchot discusses in The Infinite Conversation through Cervantes and his knight – the madness of book-life, of the perverse unity of literature and life a discussion which culminates in the discussion of one of the weird's masters, that of Kafka. The text is the knowing of madness, since madness, in its moment of becoming-more-mad, cannot be frozen in place but by the solidifications of externalizing production. This is why Foucault ends his famous study with works of art. Furthermore extilligence, the ability to export the products of our maligned brains, is the companion of the attempts to export, or discover the possibility of intelligences outside of our heads, in order for philosophy to survive the solar catastrophe. To borrow again from Deleuze, writing is inseparable from becoming. The mistake is to believe that madness is reabsorbed by extilligence, by great works, or that it could be exorcised by the expelling of thought into the inorganic or differently organic. Going out of our heads does not guarantee we will no longer mean we cannot still go out of our minds. This is simply because of the outside, of matter, or force, or energy, or thing-in-itself, or Schopenhauerian Will. In Lovecraft’s “The Music of Erich Zahn” an “impoverished student of metaphysics” becomes intrigued by strange viol music coming from above his room. After meeting the musician the student discovers that each night he plays frantic music at a window in order to keep some horridness at bay, some “impenetrable darkness with chaos and pandemonium”. The aesthetic defenses provided by the well trained brain can bear the hex of matter for so long, the specter of unalterability within it which too many minds obliterate, collapsing everything before the thought of thought as thinkable or at least noetically mutable on our own terms. Transcendental paranoia is the concurrent nightmare and promise of Paul Humphrey's work, of being literally out of our minds. It is the gothic counterpart of thinking non-conceptually but also of thinking never belonging to any instance of purportedly solid being. As Bataille stated, “At the boundary of that which escapes cohesion, he who reflects within cohesion realizes there is no longer any room for him” Thought is immaterial only to the degree that it is inhuman, it is a power that tries, always with failure, to ascertain its own genesis. Philosophy, if it can truly return to the great outdoors, if it can leave behind the dead loop of the human skull, must recognize not only the non-priority of human thought, but that thought never belongs to the brain that thinks it, thought comes from somewhere else. To return to the train image from the beginning “a locomotive rolling on the surface of the earth is the image of continuous metamorphosis” this is the problem of thought, and of thinking thought, of being no longer able to isolate thought, with only a thought-formed structure. [1] One of the central tenets of Francois Laruelle's non-philosophy is that philosophy has traditionally operated on material already presupposed as thinkable instead of trying to think the real in itself. Philosophy, according to Laruelle, remains fixated on transcendental synthesis which shatters immanence into an empirical datum and an a prori factum which are then fused by a third thing such as the ego. For a critical account of Laruelle's non-philosophy see Ray Brassier's Nihil Unbound. (shrink)

The book aims to examine how a Trinitarian Theism can be formulated through the elaboration of a Relational Ontology and a Trinitarian Metaphysics, in the context of a hyperphatic epistemology. This metaphysics has been proposed by some supporters of the so-called Open Theism as a solution to the numerous dilemmas of Classical Theism. The hypothesis they support is that the Trinitarian nature of God, reflected in a world of multiplicity, relationality, substance and relations, demands that we think of God as (...) dynamic, internally multiple and relational. However, if the expression «God is love» – understood as the formula of the Trinity – is the key to a new theism, it leaves a problem open: how can it be translated philosophically? The research develops on two different levels: first, it assesses whether the expression should be translated into the Trinitarian paradigm, and the aporias it generates; secondly, it tries to assess whether this paradigm (eminently relational) has a correspondence in ontology: is there a satisfactory Relational Metaphysics already available? The suspicion is that such Trinitarian-Relational Ontology, invoked by many as a solution to the incongruities of classical theism, has yet to be formulated in a satisfactorily manner, despite the existence of various attempts to formulate Relational Ontologies. In order to provide an evaluation of all these attempts and to outline some possible new perspectives, the thesis consists of five chapters. It is the aim of the present dissertation to evaluate such attempts, and to outline some possible new solutions. In the chapters some points have been established: 1. there must be at least one irreducible real relation (non-reductionist realism); 2. the relation must be equally fundamental to the substance: this means that it is both external and internal; 3. this relation must be able to account for contingent causality; 4. holism is a plausible position; 5. the Trinity is a good theistic model of the divine, apophatic but rational. The last chapter then returns to ontological questions: several Relational Ontologies are examined – including ontic structural realism and process (or eventist) ontology – together with their applications in philosophy of religion (e.g.: the Relational God, Process Trinitarianism, the Entangled God). This assessment shows how all these ontologies postulate real transcendental relations (subsistent relations), inside the substances or inside the powers, inside the tropes or the structures, describing them as monads or processes (or actual occasions). These relations are the same we need to describe the Trinity. Therefore, they are either possible for both domains – ontology and theology – or they are both impossible. It has been opted for the second conclusion. But they are both impossible and inevitable: the fundamental entities of the world and their interactions (causation) must be described as real transcendental relations because each ontology transforms entities or relations into real transcendental relationships at some point. Thus, neither relationalism nor substantialism are convincing: from the fall of these two dogmas (or, rather, from the fall of their naive interpretations) we can realise that the fundamental reality is something that lies between processism (relationalism) and substantialism. It is impossible to completely substitute the notion of substance with the notion of event, process, structure or relation, both in speaking of God and in speaking about the entities of fundamental ontology. Neither monism nor pluralism can be affirmed in their pure forms. The hypothesis proposed, then, is that the notion of gunk-junk is the only one that can translate the relationality hitherto sought in an ontological model. The central part of the chapter describes the merits and defects of an eventist-infinitive ontology (EIO) based on the concept of gunk, and its potentiality to generate new theistic accounts. Through the notion of infinityings it seems to be possible to find some solutions for the questions left open by the causal relation, and therefore to defend at least the existence of one relation (external and internal). Each fundamental entity is described, in this ontology, through four transcendentals: ‘entity’, ‘relation’, ‘unity’ and ‘multiplicity’. The co-primarity between substance and relations (borrowed from the notion of relatio subsistens) differentiates EIO from the process philosophies precisely because it does not pretend to eliminate the substantial principle, or the category of substance, but wants to think of it with the transcendental of relationality. Of course, EIO poses a challenge to the role, the method and the explicative capacity of metaphysics, because what we can state of the fundamental reality is antinomic. EIO tries to assume the antinomy as a result, to make it the basis of a kind of apophatic ontology. If our best ontology is gunky, then it is possible to confirm what has been said in chap. 4: the ‘how’ of individual substances is unknown to us at least as much as the ‘how’ of God. Even in the world we have the mystery of a distinction that is not division. However, the convergence of these antinomies can find an ultimate explanation in a theistic metaphysics (EIM): God creates inside of Himself and his Trinitarian substance is “contracted” into the worldly entities. They are made contingent and ontologically different by this self-limitation, but it is still the infinity of God that makes space in Himself for something new, even though He is totally present in every entity. It is not absurd to think that the substances of the world keep track of the divine nature, even in His “contraction”. Among the characteristics of the divine nature that each created entity maintains we have the subsistent relationality, the pericoretic indwelling (the infinite gunk-junk), the fact of being always one-and-many. If EIO and EIM are accepted as a good compromise between relationalism and substantialism, even in their apophaticity, then, on this basis, a Trinitarian Philosophy is possible. The picture of reality that emerges represents the world as multiple, substantial, contingent and intrinsically relational, forcing us to postulate the transcendental of relationality and multiplicity. Such transcendental leads us to think of the world and God (and their relations) in a Trinitarian way – With the necessary acknowledgement that this is a reasonable but apophatic discourse. (shrink)