This chapter contains sections titled: * Brief History * How We Talk * Science and Infinity * Religion and Infinity * Concluding Remarks * Notes * References * Further Reading.

The epochal transition from the medieval to the modern world has long been thought in terms of the "infinitization" of the world-picture, that is, as the transition from the finite, hierarchically ordered medieval cosmos to the infinite and homogeneous universe of the new astronomy and physics. In this dissertation I argue that this process of "infinitization" must be understood intensively as well as extensively. Nature, in the modern age, is thought not only as infinitely extended in space, but also (...) as exhibiting an infinite richness in all of its parts. Each individual being is grasped as utterly unique and consequently as conceptually inexponable. Thus the infinitization of the real leads to an infinitization of the knowable-- the radical shift in ontology grounds a corresponding shift in epistemology, so that the progress of human knowledge is understood as an unending project infinitely extended over time. ;I take as my point of departure a consideration of Hans Blumenberg's reading of this epochal transition in The Legitimacy of the Modern Age. Blumenberg views the transference of the attribute of infinity from God to the world as characteristic of the modern rejection of the medieval tendency to locate the measure of self- and world-understanding in a transcendent deity. The "worldliness" of the modern age, Blumenberg holds, must be understood as a reaction to the extreme intensification of divine transcendence in late medieval Nominalism. In the face of an absconded God, human "self-assertion" aims at the realization of this-worldly possibilities through the mastery and alteration of reality. I agree with Blumenberg that the primary expression of this self-assertion has been seen in the progress of modern science. The possibility of that progress, however, is grounded in an understanding of nature as a law-like and yet inexhaustible field of investigation directing thought toward an objectivity that is never entirely realized. The origin of such an understanding, I argue, is to be found in late medieval Neoplatonic speculation on the nature of God's omnipresence in the world, in particular in the notion of the immanence of the infinite in the finite. (shrink)

This book clearly explains what an infinite number is, how infinite numbers differ from finite numbers, and how infinite numbers differ from one another. The concept of recursivity is concisely but thoroughly covered, as are the concepts of cardinal and ordinal number. All of Cantor's key proofs are clearly stated, including his epoch-making diagonal proof, whereby he proved that that there are more reals than rationals and, more generally, that there are infinitely large, non-recursive classes. In the final section, (...) Kurt Gödel's celebrated Incompleteness Theorem is shown to be equivalent with a number of the various different theorems proved in this short but profound work. (shrink)

Extending techniques of Dowd and those of Poizat, we study computational complexity of in the case when is a generic oracle, where is a positive integer, and denotes the collection of all -query tautologies with respect to an oracle . We introduce the notion of ceiling-generic oracles, as a generalization of Dowd's notion of -generic oracles to arbitrary finitely testable arithmetical predicates. We study how existence of ceiling-generic oracles affects behavior of a generic oracle, by which we show that is (...) not a subset of is comeager in the Cantor space. Moreover, using ceiling-generic oracles, we present an alternative proof of the fact (Dowd) that the class of all -generic oracles has Lebesgue measure zero. (shrink)

On the question of whether the universe should be infinite or finite, there has been throughout the history of physics a struggle between materialism and idealism, between dialectics and metaphysics. Materialism asserts that the universe is infinite, while idealism advocates finitude. At every stage in the history of physics, these two philosophical lines have engaged in fierce struggle. Although developments in physics always demonstrate the failure of the finite universe doctrine, with every new advance in science the idealists (...) distort and take advantage of the latest results of physics to "prove" with varying sleights of hand that the universe is finite, serving the reactionary rule of the moribund exploiting classes. In the early part of this century after the rise of quantum theory and relativity theory, physics arrived at a new stage of development. After General Relativity was announced in 1916, a lot of people used it and similar theories of gravity to produce all sorts of models of the universe. The "finite universe" point of view became even more fashionable. Lenin pointed out that "That certain schools of the new physics have various dealings with Machism and other variants of modern idealism, is a fact not to be doubted for a moment." It is clear from reading all sorts of foreign literature that the schools of physics promoting a finite universe are linked up with all sorts of idealist philosophy, including theology. (shrink)

This paper addresses the doubts voiced by Wigner about the physical relevance of the concept of geometrical points by exploiting some facts known to all but honored by none: Almost all real numbers are transcendental; the explicit representation of any one will require an infinite amount of physical resources. An instrument devised to measure a continuous real variable will need a continuum of internal states to achieve perfect resolution. Consequently, a laboratory instrument for measuring a continuous variable in a (...) finite time can report only a finite number of values, each of which is constrained to be a rational number. It does not matter whether the variable is classical or quantum-mechanical. Now, in von Neumann’s measurement theory (von Neumann, Mathematical Foundations of Quantum Mechanics, Princeton University Press, Princeton, [1955]), an operator A with a continuous spectrum—which has no eigenvectors—cannot be measured, but it can be approximated by operators with discrete spectra which are measurable. The measurable approximant F(A) is not canonically determined; it has to be chosen by the experimentalist. It is argued that this operator can always be chosen in such a way that Sewell’s results (Sewell in Rep. Math. Phys. 56: 271, [2005]; Sewell, Lecture given at the J.T. Lewis Memorial Conference, Dublin, [2005]) on the measurement of a hermitian operator on a finite-dimensional vector space (described in Sect. 3.2) constitute an adequate resolution of the measurement problem in this theory. From this follows our major conclusion, which is that the notion of a geometrical point is as meaningful in nonrelativistic quantum mechanics as it is in classical physics. It is necessary to be sensitive to the fact that there is a gap between theoretical and experimental physics, which reveals itself tellingly as an error inherent in the measurement of a continuous variable. (shrink)

continent. 2.3 (2012): 218–223 Vijay Prashad. Arab Spring, Libyan Winter . Oakland: AK Press. 2012. 271pp, pbk. $14.95 ISBN-13: 978-1849351126. Nearly a decade ago, I sat in a class entitled, quite simply, “Corporations,” taught by Vijay Prashad at Trinity College. Over the course of the semester, I was amazed at the extent of Prashad’s knowledge, and the complexity and erudition of his style. He has since authored a number of classic books that have gained recognition throughout the world. The Darker (...) Nations , a peoples’ history of the Third World, sent defibrillating shockwaves through an academic world that had almost forgotten the epic scope and historic dignity of the non-aligned movement and post-colonial struggles. In his recent, award-winning work, Arab Spring, Libyan Winter , Prashad delves into his capacious knowledge of the Third World to excavate the discourses and narratives surrounding the upheavals of 2011. “Revolutions have no specific timetable,” states Prashad in the opening line of this exciting and provocative look at contemporary events. Instantly, an atmosphere of suspense emerges. The reader is alerted to the problem of history. Does the making of history then involve a suspension of historical time, or is it a continuous narrative structure into which events must eventually be integrated? Arab Spring, Libyan Winter shows that the answer to the question, “How is history made?” lies just as easily in the asking. The first half of the book works through the events that transpired to bring about the explosive popular uprisings of Arab Spring: Tunisia and Egypt, Yemen and Bahrain. But the events are not put together in a cohesive, chronological fashion. Using an uncommonly gripping style more akin to the folk-story motif of the djeli than to traditionally Orientalist academia, Prashad suspends a given situation, points out its components, and traces back the characters and genealogies that define each link before resuming a narrative. Every moment is an end to itself, and an origin of something different. Thus, the making of history becomes the suspension of its own progress, it’s catastrophic 'stability,' in a process of differentiation through inclusion. Empirically, one might suggest that history is a condition of time, and by extension, of the subject, but history is also an eminent producer of the Subject and her concept of time through memory and narrative. Hence, history is often overdetermined by a dominant narrative of the sovereign. The apparently chaotic composition of Prashad’s historicity is, then, an interstitial morphology of resistance. It illustrates that the time to act for the revolutionary lies in the gaps within the dominant historical realities. It points out the sorties of signifiers constituting nothing in the obliterated ruins of history. Through such illuminations, Arab Spring, Libyan Winter breaks through “the surface of history” to elaborate revolution’s snapping synapses, exploding interruptions and breaches, connecting flows. Messianic Politics It is tempting to think, 'History is either singular, or it is diffuse. It is either one coherent factual narrative, or it is comprised of the aletheia of the multitude.' Yet the reader finds Prashad examining the structure of history on multiple levels. For Prashad, world history is shaped by geographically defined political movements, such as communism or national liberation, with historical agents like the working class for the former and the nationalist militant for the latter. Religion, however, is another matter: “Religion has an unshakable eschatology which a post-utopian secular politics lacks.” Within the telos of religion, there appears on the horizon the image of Benjamin’s thoughts on messianic time—time as “a small fissure in the continuous catastrophe,” a break with history, may contain an ultimate redemption of the human beyond the political power struggle. Succoring the split between geo-political and utopian-religious (messianic) time in the context of Arab Spring, Prashad indicates that the base of “the deep desire and commitment to some form of democracy” is forged by the affinity between eschatological Islamist politics with the People. To think about the Subject of messianic time in the context of the lack of a “coherent timetable” for emancipatory politics, it is perhaps best to return to the modern tradition of Martin Luther King, Jr., whose prefigurative blending of liberation theology and emancipatory politics became most important during his works of 1963, the climactic year of “The Letter from a Birmingham Jail,” the March on Washington, and the bombing of the 16th Street Baptist Church. In “The Letter,” King states the point bluntly: Frankly, I have yet to engage in a direct-action campaign that was “well timed” in view of those who have not suffered unduly from the disease of segregation. For years now I have heard the word “wait!” It rings in the ear of every Negro with piercing familiarity. This “Wait” has almost always meant ‘Never.” We must come to see, with one of our distinguished jurists, that “justice too long delayed is justice denied.” The timing of rebellious action must not be a part of history. It must change history. The kairos appears spontaneous and untimely in its convulsive, shocking presence, yet it is, deeper still, a path, a longue durée , forged through diligent and rigorous praxis. King put a finer point on the path of historic liberation in his declaration in his 1963 speech at Western Michigan University: “[T]ime is neutral.... Somewhere along the way we must see that time will never solve the problem alone but that we must help time. Somewhere we must see that human progress never rolls in on the wheels on inevitability. It comes through the tireless efforts and the persistent work of dedicated individuals who are willing to be co-workers with God. Without this hard work, time itself becomes an ally of the insurgent and primitive forces of irrational emotionalism and social stagnation. We must always help time and realize that the time is always right to do right.” Time, then, exists in “the neutral,” while action must turn to the kairos , not only of taking time, but taking the right time. By turning our relation to time into kairos , we move time from its context within history to a duration between histories—the longue dure?e of Messianic time. The implementation of neutrality in this context suggests a broad space of time to extend through the valley of positive and negative. In Roland Barthes’s lectures from 1978, the neutral takes place in this minimal distance of non-conflict that exists outside of two extremes. It is, in the words of Maurice Blanchot, “the non-general, the non-generic, as well as the non-specific.” For Blanchot, the neutrality of time permeates life as death, impassive and beyond control; such is the neutrality of messianic time, which permeates the charge of history, changing the "I" into the "one." The neutrality of time therefore implicates history in a case against extremism, showing that the work of the radical is not to tend to any particular side, but to have the time to navigate through the terrible terrain of history by altering its course, its patterns and rhythms of movement, connection, sociality. Hence, one does not simply “make history,” one alters the course of history in accordance to “the right time.” The historical path toward emerging power is transmitted as a gesture to the outside of history, against history; a gesture as simple as reaching out. Thus a counterhistory, to use Foucault’s term, is brought forward as the guide of time past a point of no return, a Rubicon, the point where the normal path of history falls to the past. The subject, who must be the only true agent of history, facilitates time through helpfulness, and both the Subject and her history become decentered in relation to one another. Yet history negates the neutrality of time. As Barthes discloses, although the neutral is the “thought and practice of the nonconflictual, it is nevertheless bound to assertion, to conflict, in order to make itself heard.” Engaged with history, time is charged with a destiny, and becomes an opposition. As Prashad insists, “For Arab lands, the events of early 2011 were not the inauguration of a new history, but the continuation of an unfinished struggle that is a hundred years old.” The emergence of the Subject during Arab Spring did not conceive of a new history, but changed the path of history toward a different destiny. “Historical grievances combined with inflationary pressures now met with the subjective sense that victory might be at hand—this was not simply a protest to scream into the wind, but a protest to actually remove autocrats from their positions of authority. The facts of resistance had given way to the expectation of revolutionary change.” The negativity of this charge toward “revolutionary destiny” rendered control, as a positive force, seriously lacking. The lack of control lies in the problem that the Subject is not totally detemporalized or timeless, but untimely in her presence. The Subject is untimely, because her work is visionary, and it is only through such visionary work that the Rubicon can be crossed. Still, the crossing of this border is haunted by anxiety over an impending disaster that lays in wait. It is only through the form of what Benjamin calls divine violence, captivated not with the justice of the means, but the ends—the transformation of history—that a revelation of history’s traumatic foundations can be liberated, and patterns and rhythms of time developed throughout obscured traditions awakened. In this situation, the Subject appears to be outside of right, but setting the state to rights. Because her position is correct, in-so-far as the rebelling subject rebels due to a lack of recognition, her representation appears outside of the norm, which is mistaken as right. Therefore, such visionary work must be carried out through obscured traditions, underground, away from the surveillance of empire. Explaining his method from the start along the lines of Marx’s metaphor of the mole, Prashad insists, “It is the burrowing that is essential, not simply the emergence onto the surface of history.” Arab Spring, Libyan Winter is a book on uncharted networks of time, spread out over the 252 pages like an elaborate spider’s web of passages intertwining with and overwhelming the machinery of the state. Although untimely, Arab Spring was not a flash in the pan, or a Facebook or Twitter revolution. Prashad notes that the government’s suspension of these tools led to further radicalism by furthering communications through face-to-face encounters. In existential terms, Arab Spring might be thought of as a revolution of Being over techne , an uprising of the unchartable, infinite potential of the Other. The name of this Other is found in Chance. The faith of the revolution lies in the proper decentering of the subject, its giving to the Other of time, for only with respect to time does history actually appear on the horizon of the subject, rather than as an imposition. This time of historical agency appears as a moment when anything can happen—a revolutionary truth event where everything comes into question while being realized in its Otherness as a community of the people begins anew in the streets amidst discourse, friendship, reconfigurations of hegemony, and a becoming of a constituent power. Throwing a wrench into the gears of the “cobwebbed tradition” of Orientalism, the decentering of history in Arab Spring, Libyan Winter is a defining quality of its impassioned revolutionary cry. But perhaps the most paradigmatic points of the book emerge from the unexpected voices. To make the connection between the particular manifestations of revolutionary demands to the general historic terms of revolution, Prashad quotes an anonymous young Egyptian in Tahrir Square, who reminds us, “[T]he French Revolution took a very long time so the people could eventually get their rights.” Far from timelessness, the historicization of Arab Spring must exist precisely within the most uncanny appearance of time, as something that does not appear to come from the natural state of time as we know it, but in arriving has completely transformed the way that we understand time. Strange Monsters Revolution is never as simple as a revolt from below against a rusting structure of elites trying to remain in power. As with the French Revolution, Arab Spring consisted of complex familial ties, outside interests, and religious factions fighting alongside, often in awkward juxtaposition to, liberals, working class parties, farmers, and students. It is perhaps because of this historically difficult and incongruous composition that the Arabic word for revolution is thawra , referring to the image of the bull, or thawr , which has religious significance as a pagan deity for the Ancient Semitic tribes and Cartheginians. If Daniel Guérin was correct in saying, “Anarchism and Marxism drink from the same spring,” then it is quite a different oasis that drove the thirsty beast of revolution through what Prashad calls, “the Libyan labyrinth.” Recalling Bataille’s Acéphale, the intestinal labyrinth gains significance as the mode of metabolism and rumination in which, “(the Acephale) has lost himself, loses me with him, and in which I discover myself as him, in other words, as a monster.” Unraveling the discourse of the revolution, the flows of power and hegemony—from Qaddafi’s nationalist coup in 1969 to the wars with Chad from 1978 to 1987, the neoliberal reforms of the 1990s to, finally, the War on Terror—we find that the metabolic process of Libyan Winter ends in the production of oil. In a process of what Prashad calls “involution,” Qaddafi’s bellicose policies, along with his attempts to nationalize Islam together with oil, mutated the source of his power, turning his house against itself. Already a divided nation, with its two major cities on opposite geographic sides of the country, Libya became split between two opposed political powers—liberal reformists and staunch nationalists—as Qaddafi’s political disengagement manifested through what can only be described in psychoanalytic terms as a recursive passage a l’act (mysteriously calling Wikipedia “Kleenex,” declaring that opponents drank hallucinogens with their instant coffee, and so on). Prashad declares, “The mercurial style that Qaddafi adopted was not about his personality alone, but also a leader’s natural response to a system that relied upon power brokers whose own loyalty… did not have any ideological commitment to the system.” As in the case of other nations during Arab Spring, the true expression of revolt was an outward exposition of what had inwardly been happening in microcosmic societies throughout the realm—from indigenous tribes to political parties, for a century, the people, partly motivated by (and in resistance to) economic impositions of development, had been changing the traditional social compositions and participating in what Ruth Wilson Gilmore calls, “the relative autonomy of the movement from the leaders.” It was this empowerment of political reconfigurations taking place under the context of new democratic assemblages that shocked the elites and evoked such a powerful response. Unlike Qaddafi’s awkward policy choices and rants, real acts of power from the West were clear and decisive. Prashad sets the stages of war and diplomacy, far removed from the deserts, mountains, and cities that forged the backdrop of popular politics. Taking place under the now-classic architecture of the “four pillars” of US interests—oil, the War on Terror, Israel, and the circumvention of Iranian hegemony—we find elites rubbing elbows in “Heliopolist cocktail parties and hushed conferences in Kasr al-Ittihadiya” as well as the Concorde-Lafayette hotel in Paris, which provides the setting for a meeting of anti-Qaddafi figures consolidating their power. These scenes are buttressed with the careful portraiture of key historical actors. As Prashad brings the stage of history to life, we find the diplomats and liberals like Frank Wisner, whose career has brought him from Enron in the late 1990s to the Obama Administration, under the aegis of which he was meeting with Mubarak about military support during Arab Spring. We spy the provocateurs, for instance, gauche cavalier , Bernard Henri-Levy, who telephones Sarkozy from Benghazi about the need for more NATO air strikes. We follow the rebel military establishment as it suffers mysterious deaths and even more mysterious assents (like that of apparent CIA cohort, Khalifa Hifter). In each of these intriguing characters, we find different representations of the security state biopolitique: an oil-injected reification that drives Arab Spring from the resentment of rising food prices to the brink of implosion in Libyan Winter. Reminiscent of the scenes of Cold War soirées represented in old Bond films, the aristocratic fight for oil against democracy that was Libyan Winter presents a harsh truth that the new global crisis is simply the continuation of the old history: the global exploitation of capitalism waged against the imagination of the people. As Benjamin laments, “The labyrinth is the right path for the person who always arrives early enough at his destination. This destination is the marketplace.” Yet, “(t)he labyrinth is the habitat of… a humanity (a class) which does not want to know where its destiny is taking it.” Thus, the labyrinth leads, like the desert, only further into itself. As Blanchot explains, “The desert is even less certain than the world; it is never anything but the approach to the desert.” Only the visionary can take time out of the involution toward oblivion. It is here, in this approach to and escape from history, that we find ourselves within Benjamin’s Golgotha, Blanchot’s desert: A space of indeterminate uncertainty where we become familiar only with our own exile. Blanchot writes, “For the moderated and moderate man, the room, the desert, and the world are strictly determined places. For the man of the desert and the labyrinth, devoted to the error of a journey necessarily a little longer than his life, the same space will be truly infinite, even if he knows that it is not, all the more so since he knows it.” The desert, as allegory, provides an eschatology, a mortality in the destiny of Arab Spring. But any allegory of nature might lead to a labyrinthine eschatology (for example, in Bachelard, the forest presents "a limitless world"); the prescience of the untimely is always an uncanny acceptance of the infinite within the finite—the outside of what is understood. Transgenerational and occupied with the image of the future, the untimely makes otherness its home as it proceeds toward liberation. In the work of helping time navigate this dangerous passage of history, Prashad illustrates that the more violent break with history may have occurred in the peaceable struggles of Tahrir Square, and not in the military clashes of Qaddafi with his former Generals who defected to the CIA and NATO countries. It might be possible to suggest, then, that in the case of Tunisia and Egypt, the historical subject of the people was drawn together in a Dionysian dance of different rhythms in the marvelous realm of the ancients, while we fear that the case of Libya suggests an Apollonian future where time, itself, may lay dying under the machinic hand of history. The properly Nietzschean inversion would follow: time is dead, for history has killed it. Yet, if time is dead, struck down in Golgotha, exposed in the desert, mauled by the thawra , it is only in the present sense that it is rendered impossible outside of the context into which the untimely has thrown us. Thus, if history becomes an art of revolution, life as being-towards-death (death as the provocation of the neutrality of time) becomes what Benjamin calls “the allegory of resurrection” through the glorious ruins of history. Prashad ends his work with a rousing finale: “The time of the impossible has presented itself. In Egypt, where the appetite for the possibilities of the future are greatest, the people continue to assert themselves into Tahrir Square and other places, pushing to reinvigorate a Revolution that must not die… For them the slogan is simple: Down with the Present. Long live the Future. May it be so.&rdquo. (shrink)

Proportionality is a German, and thus continental European, concept in public law that is applied by both the Court of Justice of the European Union (CJEU) and the European Court of Human Rights (ECtHR). The principle specifies that measures adopted by executive authorities should not exceed the limits of what is appropriate and necessary in order to achieve legitimate objectives in the interest of the public. Using a functional comparative approach, this book evaluates the extent to which proportionality has been (...) integrated into the English and Hong Kong judicial systems by comparing case law in these courts with that of the CJEU and the ECtHR. The text also reviews the development of proportionality and presents a topical understanding of why its adoption and application have encountered difficulties, particularly regarding socio-economic rights, in some jurisdictions, such as the United Kingdom and Hong Kong. Written by a scholar with experience from both within the Hong Kong judicial system and from international research, this book is the first all-encompassing reference for legal practitioners worldwide." -- Provided by publisher. (shrink)

Cantor’s theory of cardinal numbers offers a way to generalize arithmetic from finite sets to infinite sets using the notion of one-to-one association between two sets. As is well known, all countable infinite sets have the same ‘size’ in this account, namely that of the cardinality of the natural numbers. However, throughout the history of reflections on infinity another powerful intuition has played a major role: if a collectionAis properly included in a collectionBthen the ‘size’ ofAshould be less than (...) the ‘size’ ofB(part–whole principle). This second intuition was not developed mathematically in a satisfactory way until quite recently. In this article I begin by reviewing the contributions of some thinkers who argued in favor of the assignment of different sizes to infinite collections of natural numbers (Thabit ibn Qurra, Grosseteste, Maignan, Bolzano). Then, I review some recent mathematical developments that generalize the part–whole principle to infinite sets in a coherent fashion (Katz, Benci, Di Nasso, Forti). Finally, I show how these new developments are important for a proper evaluation of a number of positions in philosophy of mathematics which argue either for the inevitability of the Cantorian notion of infinite number (Gödel) or for the rational nature of the Cantorian generalization as opposed to that, based on the part–whole principle, envisaged by Bolzano (Kitcher). (shrink)

LetMbe a countably infinite ω-categorical structure. Consider Aut(M) as a complete metric space by definingd(g, h) = Ω{2−n:g(xn) ≠h(xn) org−1(xn) ≠h−1(xn)} where {xn:n∈ ω} is an enumeration ofMAn automorphism α ∈ Aut(M) is generic if its conjugacy class is comeagre. J. Truss has shown in [11] that if the set P of all finite partial isomorphisms contains a co-final subset P1closed under conjugacy and having the amalgamation property and the joint embedding property then there is a generic automorphism. In (...) the present paper we give a weaker condition of this kind which is equivalent to the existence of generic automorphisms. Really we give more: a characterization of the existence of generic expansions (defined in an appropriate way) of an ω-categorical structure. We also show that Truss' condition guarantees the existence of a countable structure consisting of automorphisms ofMwhich can be considered as an atomic model of some theory naturally associated toM. We do it in a general context of weak models for second-order quantifiers.The author thanks Ludomir Newelski for pointing out a mistake in the first version of Theorem 1.2 and for interesting discussions. Also, the author is grateful to the referee for very helpful remarks. (shrink)

Mate Meršić (Merchich, 1850-1928) sees the origin of Zeno’s paradox ‘Achilles’ in the ambiguities of the concept of the infinity. According to him (and to the tradition started by Gregory St. Vincent), those ambiguities are resolved by the concept of convergent geometric series. In this connection, Meršić proposes a general ontological theory with the priority of the finite over the infinite, and, proceeding from Newton’s concept of fluxion, he develops a modal interpretation of differential calculus.

The first two sections of this paper investigate what Newton could have meant in a now famous passage from “De Graviatione” (hereafter “DeGrav”) that “space is as it were an emanative effect of God.” First it offers a careful examination of the four key passages within DeGrav that bear on this. The paper shows that the internal logic of Newton’s argument permits several interpretations. In doing so, the paper calls attention to a Spinozistic strain in Newton’s thought. Second it sketches (...) four interpretive options: (i) one approach is generic neo-Platonic; (ii) another approach is associated with the Cambridge Platonist, Henry More; a variant on this (ii*) emphasizes that Newton mixes Platonist and Epicurean themes; (iii) a necessitarian approach; (iv) an approach connected with Bacon’s efforts to reformulate a useful notion of form and laws of nature. Hitherto only the second and third options have received scholarly attention in scholarship on DeGrav. The paper offers new arguments to treat Newtonian emanation as a species of Baconian formal causation as articulated, especially, in the first few aphorisms of part two of Bacon’s New Organon. If we treat Newtonian emanation as a species of formal causation then the necessitarian reading can be combined with most of the Platonist elements that others have discerned in DeGrav, especially Newton’s commitment to doctrines of different degrees of reality as well as the manner in which the first existing being ‘transfers’ its qualities to space (as a kind of causa-sui). This can clarify the conceptual relationship between space and its formal cause in Newton as well as Newton’s commitment to the spatial extended-ness of all existing beings. While the first two sections of this paper engage with existing scholarly controversies, in the final section the paper argues that the recent focus on emanation has obscured the importance of Newton’s very interesting claims about existence and measurement in “DeGrav”. The paper argues that according to Newton God and other entities have the same kind of quantities of existence; Newton is concerned with how measurement clarifies the way of being of entities. Newton is not claiming that measurement reveals all aspects of an entity. But if we measure something then it exists as a magnitude in space and as a magnitude in time. This is why in DeGrav Newton’s conception of existence really helps to “lay truer foundations of the mechanical sciences.”. (shrink)

Writing, the exigency of writing: no longer the writing that has always (through a necessity in no way avoidable) been in the service of the speech or thought that is called idealist (that is to say, moralizing), but rather the writing that through its own slowly liberated force (the aleatory force of absence) seems to devote itself solely to itself as something that remains without identity, and little by little brings forth possibilities that are entirely other: an anonymous, distracted, deferred, (...) and dispersed way of being in relation, by which everything is brought into question – and first of all the idea of God, of the Self, of the Subject, then of Truth and the One, then finally the idea of the Book and the Work so that this writing (understood in its enigmatic rigor), far from having the Book as its goal rather signals its end: a writing that could be said to be outside discourse, outside language. – Maurice Blanchot ( The Infinite Conversation , xi) The Compendium * Beginning with the idea and practice of writing, and moving to the subject or self, then to truth and the One, and arriving at the Book and the Work, Blanchot’s words reflect the trajectory of the following work of writing. True to the embeddedness of the subject in the work of writing, the term “Compendium” has been a metonym for me over the past few years as I have thought about the concept of totality alongside its expression in figures such as the One, the Whole, or the All. In the following I aim to share some of the content of this metonym, and to enrich it by making some distinctions. 1 Here the term “Compendium” will refer to a concept of totality that is ontological (pertaining to being, and the copula) and also textual (pertaining to the symbolic, and the signifier-signified relationship). The Compendium is a figure for thinking the world as a Book, in the broadest sense—an approach to reality that is by no means new, but one that is due for renewal. 2 In partial answer to the question of the Compendium we will say that the word ‘Compendium’ stands in for expressions which seek to totalize, or concepts which seek to approach the concept of ‘everything,’ particularly when these expressions are bound up in the question of the Book (both the book as a physical object and the Book as a metaphysical figure: a way of thinking about the work in and of writing). The question of the Compendium is strongly associated with the physical and metaphysical form of the Book, like the mysterious and symbolic books which are opened in the Biblical book of Revelation, the book of life and the book of death, which together constitute an important couplet. 3 Parataxis & Hypotaxis There are two helpful distinctions that will bring us closer to an answer to the question “What is a Compendium?” The first distinction is between two figures in and for writing: parataxis and hypotaxis . Typically paired in contrast to one another as literary techniques, with parataxis indicating a side-by-side placement of textual elements and hypotaxis referring to subordinate arrangements of textual elements, the two figures can be understood as having a philosophical significance in addition to their practical function as devices for writers. For us these two terms will remain between philosophical theory and writing practice, and serve as a perspective for our writing and creation of texts. Here I take texts to refer to anything from the concrete written words in a book, to the ontological text of the world that we experience. Parataxis , as a figural way of thinking about the ontological structure of texts, refers to texts which are tightly woven and interdependent – texts within which each sentence bears the weight of the entire work. Parataxis often involves repetition, great density, and fragility. One of the best examples of this sort of text is Theodor Adorno’s posthumous magnum opus , Aesthetic Theory . In addition to the text itself, the translation history of the book may also help us to better understand parataxis and its relation to hypotaxis . The final published version of Aesthetic Theory , in German, was a text that Adorno intended to revise and rewrite, but this intention was never realized because of his untimely death. Where the German text of Aesthetic Theory certainly exemplifies the concept of parataxis , the first English edition took this densely woven text and carved Adorno’s lengthy paragraphs and lengthy sentences into manageable ‘bite-sized’ pieces of English text. The first translator took further liberty and inserted headings and new paragraph breaks where there were none in the original. Aesthetic Theory was eventually retranslated by Robert Hullot-Kentor and is now available in a form that is much more faithful to the original work. 4 The pertinent idea that this translation history points to is the distinction between parataxis and hypotaxis . Where parataxis describes texts which are repetitive, densely woven, and often fragile, hypotaxis describes texts which are hierarchical and in which the primary relation is linear – the latter of which is very similar to the first English translation of Aesthetic Theory . On the other hand, the original German text that Adorno wrote was very dense, often repetitive, and contained long sections of text unbroken by paragraphs or headings. This example of parataxis was then turned towards hypotaxis through the initial translation which took a text that was, in many ways, nonhierarchical and nonlinear, and artificially subjected it to a hierarchy that was not its own (and here I take the word of Fredric Jameson who comments on the two translations in a note at the beginning of his book Late Marxism ). 5 The point here is not about translation, but rather the distinction between parataxis and hypotaxis precisely as they are figures for discourse, written or otherwise. The concept of parataxis looks far more postmodern and rhizomic than the concept of hypotaxis which remains very modern and arborescent. This is a more figural way of talking about the distinction. However, if we wanted to take a more precise and analytical approach we could say that on the level of form parataxis involves a relationship between sentences and paragraphs in which each part bears an equally crushing responsibility to present the whole content of the text. As well, on the level of content, parataxis requires that each concept in a work take on the full conceptual weight of the total work. Continuing the analysis, we could say that parataxis describes texts in which there is no (or very little) linear or causal move from antecedent to consequent, whether in content or form. Instead, parataxis describes texts which weave together concepts via conjugations or associations. The concept of hypotaxis , on the other hand, requires that texts submit themselves to hierarchy, one example of which is logical argumentation. The contingency of the conclusion upon premises in hypotaxis is certainly distinct from the repetition, re-presentation, and conjugation of concepts in parataxis , and I think that this is evident in the difference between the writing styles prevalent in contemporary Analytic philosophy and Continental philosophy. It is not the case that the distinction between parataxis and hypotaxis absolutely corresponds with the writing styles of Continental and Analytic philosophy (respectively). There are thinkers who take exception to this pattern, such as Badiou’s more formal approach in the discourse of Continental philosophy to give one example. However, when Analytic philosophy takes its writing lessons from the sciences, and seeks to do philosophy via the clean linear move from antecedent to consequent, then I think that Analytic philosophy writes with hypotaxis in mind. On the other hand, when Continental (particularly German and French) philosophy takes its writing lessons from narrative or poetry then Continental philosophy writes with parataxis in mind. It is not fair to say that all those writing Continental philosophy are looking to narrative and poetry for stylistic direction, just as it is not the case that all those writing Analytic philosophy have mathematics and science as their writing format, however there are some striking ways in which the differing epistemologies of Continental and Analytic philosophy encourage writing with parataxis and hypotaxis in mind (respectively). For some, writing with parataxis or hypotaxis in mind is a conscious decision and a product of stylistic self-consciousness, and for others it is an unconscious discursive and epistemic requirement that must be met in order to be involved in a particular discourse. Before moving on to our second distinction, and then an examination of the question of the Book, it is important to point out that both totality and figurations like the One, the Whole, and the All, come out of very human desires to totalize or to actualize our being-towards-totality. The relationship between identity and totality then can be construed, with Heidegger in mind, as striving toward wholeness, completion, or fulfillment, each of which is a quality of compendia. 6 Compilation & Selection The first distinction between parataxis and hypotaxis pertains to both form and content, and to both the mechanics of writing and the ideas to which writing refers. The second distinction pertains to both as well. In the process of writing and in the process of perception as well, there is a tension between compilation and selection . To begin with, in the work of theory-writing there is compilation, and this is because writing theory requires a broad perspective and certain measure of totalization. On the other hand, writing theory requires that the writer select an idea or a combination of ideas to individuate out of a radical and infinite multiplicity of identities and combinations. The concern here is not primarily for the writing of a theory about a specific idea, like a secondary work or a reference text. But rather the concern is with the writing of grand theories: Marx and Marxism, Derrida and Deconstruction, Husserl and Phenomenology, Sartre and Existentialism, Saussure and Structuralism —each of which strives along a trajectory towards being all-encompassing, whether it intends to or not. Today, we could even say that Speculative Realism has embarked on this journey, and perhaps now we could say that the discourse of Speculative Realism has reached the inevitable point after which a grand theory leaves the hands of its writer or writers and becomes a possession of the collective consciousness of the academy, or (perhaps now) the mass consciousness of the blogosphere—and this is a true pharmakon , a poison and a cure, a blessing and a curse. 7 When we write theory, whether the scope is that of a grand theory or a particular theory, we write in the tension between compilation (making good on the desire to be all encompassing) and selection (being required to decide and discern and to judge what is included in the work and what is deleted or appended or abridged or given over to ellipses). Generally speaking, the work of theory-writing often comes out of a desire to totalize, that is, to develop a theory of everything that is able to apprehend new experiences and ideas while still remaining whole. I grant that this is not a universal desire, and that not everyone who sits down to write a work (or book) of theory does so because of their will-to-totality, but it remains that this drive to totalize does condition a great many writers of theory (especially those who seek to develop grand theories like those mentioned previously). At the beginning of his book of interviews, Between Existentialism and Marxism , Sartre is quoted as saying, after completing the first volume of his Critique of Dialectical Reason : “I no longer feel the need to make long digressions in my books, as if I were forever chasing after my own philosophy. It will now be deposited in little coffins, and I will feel completely emptied and at peace—as I felt after Being and Nothingness . A feeling of emptiness: a writer is fortunate if he can attain such a state. For when one has nothing to say, one can say everything .” 8 This is where our two initial distinctions come to bear on the being-towards-totality as it is expressed in the concrete practice of writing: the first being between parataxis and hypotaxis , and the second being between compilation and selection. Where this second distinction is concerned there is a certain paradox at play given that selection is inescapable, and compilation is unachievable. Selection is inescapable (with the figure of the Compendium in mind) because, even when the imperative to compile is followed to excruciating lengths, selection still has the last word. This is why the Compendium is a figure rather than some material thing that can actually be accomplished or actualized. No matter how far one goes along with the will-to-archive and the will-to-compile, it is only ever a question of minimizing selection and never eliminating it. This leads to the second point, which is that total compilation is unachievable. The closest thing to total compilation that we have before us is the ontological and textual fabric of the world. Even in the case of the world, compilation cannot be enacted in a total fashion because of the need to include both the sphere of the actual and the sphere of the possible or potential. This as another way in which compilation is always-already selection, and further evidence that total compilation is practically impossible. Discursive Figuration and Total Writing Rather than figuration referring to a figure of speech or an image, here the term points to a way in which to think about the work of writing, both the work put into writing, and the work that is the result of writing: the finished yet incomplete final piece. As figures or figurations, the Book and the Compendium are ways of thinking about the work of writing and the human desire for the wholeness, completion, and fulfillment that are made manifest in the completed form of the book (whether a published or printed or saved document put to rest by the author). These two figures bear more strongly upon works that seek to be all-encompassing, and works that strive towards expressions of totality such as the One, the Whole, or the All. Moving on to the topic of the subtitle, and on a more prescriptive note, I would say that there is more hope for theory-writing to be found in parataxis than hypotaxis . Part of the reason for this claim is the poststructuralist critique of hierarchies, and yet another part is the compelling line of thinking called “weak thought” (in theology by John D. Caputo, and in philosophy by Gianni Vattimo, among others). 9 The consequences of these two convictions are such that if one is to strongly assert the truth of a grand theory without leaving the realm of hierarchy-critique and weak thought, then one must write paradoxically with parataxis in mind, and in so doing write weakly and non-hierarchically. This does not mean avoiding a sort of topography or topology when writing theory, rather it means avoiding both subjugation and oppression in thought by pursuing a nonviolent sort of ontology. In order to do this, theory writing does not present itself as an exercise in logical analysis where one mechanically moves from a set of premises (via contingency) to the inevitable conclusion (via necessity). Instead, theory approaches the idea of a Compendium. Given that the figural Compendium places parataxis and compilation above hypotaxis and selection, then the question becomes: is placing one part of a binary term before another not just another way of selecting, or another expression of hypotaxis ? Counter to the urge to resign oneself to the reign of selection, with some qualification one can nonetheless write without deciding whether selection and hypotaxis should be subservient to compilation and parataxis (which is to say that writing-without-decision is somewhere between and beyond possibility and impossibility). This dilemma can be disarmed by stating that theory writing is not about primacy, and not about having-decided-beforehand, both stylistically and also where content is concerned. This is the paradox of writing: always striving along a trajectory ( telos ) towards totality via parataxis and compilation, but always being drawn back to finitude and making selections, and placing one idea before another with hypotaxis in mind. In light of this paradox, the question of the Compendium as a figure for discourse and a figural Book has resonance with Jacques Derrida’s essay on Edmund Jabès’ Book of Questions , found in his Writing and Difference . The Question of the Book “Little by little the book will finish me.” 10 Derrida quotes Jabès, and proceeds to outline the reflexive relationship between the author of the book and the book itself, each of which are subjected to the other through a sort of chiasmus, both ontological and textual (not that the two are entirely separable). The Book, for Derrida like Blanchot in the introductory quotation, “infinitely reflects itself” and “develops as a painful questioning of its own possibility”, and in light of this we can draw an association with the painful tension between the practical reality of selection and the ideal trajectory of compilation. 11 This tension in writing, between the will to total compilation and the necessity of abridgment, is an ontological tension between part and whole just as it is a textual tension between signifier and signified. The ontological tension in writing that Derrida expresses in his essay on Jabès is between everything and nothing – a contradiction which Derrida finds in Jabès’ Book of Questions and also in the divine, in God. When one writes between compilation and selection one writes towards totality, and here we can follow Derrida who states that the lapse in signification, presumably in the signifier-signified discrepancy, is a “ rupture with totality itself” and furthermore that this lapse cannot be rectified through deductive reason or even philosophical discourse. 12 This rupture with totality, found in the troubled relation between signifier and signified, can also tell us a lot about the Book and about writing. Given an understanding of writing as an ontological act that strives towards (but never accomplishes) totality, we can see the written book as the manifested and given body of that striving. Perhaps in other disciplines or even other schools of philosophy there is a cultural climate within which the article is prized above the book, but I am fairly confident, especially when referring to the grand theories of Continental thought, that there is a respect for the book as a uniquely meaningful object capable of apprehending totality. Derrida writes, Between the too warm flesh of the literal event and the cold skin of the concept runs meaning. This is how it enters into the book. Everything enters into, transpires in the book. This is why the book is never finite. It always remains suffering and vigilant. 13 Here the vitality of the literal event is compromised by the deadening weight of the concept, and the figure of the Book assists in this lapse of meaning. The Book is a totalizing object, much like the figure of the Compendium, and yet this totalization lacks its final object of completed totality both because the figural Book is necessarily incomplete, and because the physical book is always-already a product of selection. Instead of achieving its end of being a place where everything takes place, it suffers from the lapse of writing, the discrepancy between event and concept. Derrida continues his exposition on Jabès by bringing to light yet another reflexive relation, a reversal of the relationship between the Book and the world. Derrida writes that, for Jabès, “the book is not in the world, but the world is in the book.” 14 In addition to what was stated before, I take figuration to mean that the concepts employed, such as the Book or the Compendium, are not subject to the supposed rigor of analysis that is so prized by Enlightenment or Capitalist realisms. The figure of the Book and the figure of the Compendium cannot be held accountable to standards imported from scientific method, given an understanding that these standards require a concept to be replicable, consistent, falsifiable, measurable, and so on… This is an important point to make, especially in the present atmosphere within which theory must justify itself under conditions that are not its own. Instead of being held to these standards, figuration serves as a way in which to think about writing that leaves thought open to idealistic speculation, imperfect analogies, and most importantly the ever-present gain and loss that occur in the relationship between thought and being. By extension the figural way of addressing writing indulges in enough generalization to accommodate the excess/lack relationship between the ideal form of the Book or the Compendium, and the manifested and given body of a text (as it is practically completed and closed). Here we speak against the hostile atmosphere which would have theory submit itself to hierarchical rigor, rather than Blanchot’s “enigmatic rigor”, by being explicit about the discursive conditions and epistemic conditions under which theory operates. To write with parataxis in mind is, to a certain degree, to write with weakness as one’s methodology (with weak thought being in opposition to hypotaxis and hierarchy as much as weak thought can be in opposition to anything). Rather than asserting the strength of hierarchical theory, and rather than employing antagonistic argumentation with the goal of refutation different discursive and epistemic conditions for theory writing must be cultivated (and these are by no means new). First the critical and theoretical spirit reveals itself as being concerned with the task of complication—especially the complication and critique of binaries, dichotomies, dualities, polarities, paradoxes, parallaxes, hybridities, and especially antinomies. Second, theory positions itself as a sort of showing or revealing, rather than being ultimately focused on coming to full agreement or disagreement. This is where figuration can help theory-writing, both by placing emphasis on teleologies and trajectories (like parataxis and compilation), and by shifting focus away from pure primacy, power, or absolute origin. Figuration then, assists theory writing by placing the concern of theory outside of the concerns of the hard or soft sciences, and into the realm of thought or the idea. To speak of ‘the’ Compendium or ‘the’ Book, here, is to generalize not regarding the perfect form of the Compendium or Book, but to engage speculatively with an abstract idea which remains singular and yet complicated by multiplicity. The question of the Book that Derrida asks through Jabès is a question of everything and nothing, totality and nonbeing, and this question is a concern for writing and a concern for the figure of the Compendium so defined by the tension between compilation and selection, and parataxis and hypotaxis . On the note of nonbeing, we can look to the final page of Derrida’s first essay on Jabès in Writing and Difference and notice the introduction of his neologism, différance (with an ‘a’). Derrida writes: Life negates itself in literature only so that it may survive better. So that it may be better. It does not negate itself any more than it affirms itself: it differs from itself, defers itself, and writes itself as différance . Books are always books of life (the archetype would be the Book of Life kept by the God of the Jews) or of afterlife (the archetype would be the Books of the Dead kept by the Egyptians). 15 This introduction of différance into the equation of the Book, alongside ontological affirmation and negation, hearkens back to the discussion of the symbolic lapse earlier in his essay. Differing and deferring, the Book is always a Book of Life and a figure for the intersection of the vital (life) and the total in writing. In a way, the writer of the Book may be someone who lives life, and in another way the writer of the book-as-object may be someone who engages in the act of writing, both practically (by inscribing words on paper, or typing script on a computer) and ontologically or symbolically (by investing their existence and inexistence into a work worthy of the figural Book). In addition to the writer as writer, the writer also serves as an editor, and the editorial role alongside the idea of the Compendium shows the editor to be one who collects and compiles, while being restrained by eventual selection and decision. Beyond this the editor of works and texts, in both the Book and the world, is engaged in a process of inscribing notation—of annotating (on) the Compendium. Eternal commentary is a feature of the textual Compendium, just as open-ended totality describes the ontological Compendium. The writer as writer does not simply write texts, but rather engages in a profoundly ontological act of inscribing their existence into the world, and the writer as editor does not merely annotate or alter texts as they are, but rather comments upon the text of the world. In/Conclusion So as we create texts, and as we write and create theory, let us be and remain attentive to the figure of the Compendium and the figure of the Book. These two concurrent metonymies are essential for total writing (grand theories, etc.), and may be forgettable for those concerned with fragmentary or hierarchical writing (which are valid in their own right). From the ontological and symbolic text of the world and phenomenological experience, to concrete texts such as books or mixed media, the figure of the Compendium and the figure of the Book are important for the actualization of the human will to be all-encompassing. The figure of the Compendium teaches writers of theory that the repetition, density, and fragility of parataxis offers a strong sort of weakness which does not become yet another variety of oppressive and hegemonic thought. Rather than write with hypotaxis in mind, subjugating one thought to another via cause and effect or antecedent and consequent, I would hope that theory-writing could guiltlessly indulge in the dialectical and contradictory conjugation of ideas, and take this as a legitimate methodology. Rather than being ashamed of showing resonances or giving way to abridgment or ellipses, it is my own authorial (but not authoritative) conviction that one need not give up on totalizing and constructing grand theories because of the fact that complete totalization is impossible, and one need not give up on totalizing and constructing grand theories because of the worry of violence, for the Book and the Compendium lack their completed object and are ever incomplete trajectories. NOTES * UPDATED 7/19/13: we've updated the HTML version to reflect the (correct) PDF version. Our apologies to the author—eds. 1. I would like extend thanks to Dr. Peter Schwenger of the University of Western Ontario for his comments on this paper and his hospitality as it was presented as a Theory Session at the Centre for Theory and Criticism on October 26th 2012. I would also like to thank Andrew Weiss for conversation and critique. 2. I should clarify my use of the term ‘figure’ at the outset. Given the use of the term by Jean-Francois Lyotard in Discourse, Figure (and also Gilles Deleuze in Francis Bacon ), I should state clearly that my use of the term will not correspond to the term ‘figure’ understood as the representation of an object. Instead, my use of the term ‘figure’ and its variations (‘figural’, ‘figurative’, ‘figuration’) will refer to the illustrative or metaphorical use of the Compendium or the Book as models or ways of thinking about writing and discourse. 3. Cf. Revelations 20:12-15. 4. Theodor Adorno, Aesthetic Theory , Trans. Robert Hullot-Kentor (London & New York: Continuum Press, 1997). 5. Fredric Jameson, Late Marxism: Adorno, or the Persistence of the Dialectic (London & New York: Verso, 1990), ix-x. 6. I have written about this concept of being-towards-totality elsewhere in two pieces of writing: the first is called Notes on the Compendium (an unfinished draft) which addresses some very wide theoretical concerns, being structured as a sort of itinerary or archive, and the second is Dialectics Unbound (Punctum Books, 2013) which outlines a concept of totality without the violence of totalization. While these works are more concerned with ontological totality, here I would like to focus on the idea of the Compendium as a textual totality. 7. Cf. Louis Morelle, “Speculative Realism: After Finitude and Beyond, A vade mecum” in Speculations: Journal of Speculative Realism . Issue 3 (Brooklyn, New York: Punctum Books, 2012), 241-272. 8. Jean-Paul Sartre, Between Existentialism and Marxism . Trans. John Matthews (London & New York: Verso, 1974), 9. 9. Cf. John D. Caputo, The Weakness of God (Indiana: Indiana University Press, 2006). 10. Jacques Derrida, “Edmond Jabès and the Question of the Book” in Writing and Difference . Trans. Alan Bass (Chicago: University of Chicago Press, 1978), 65. 11. Ibid, 65. 12. Ibid, 71. 13. Ibid, 75. 14. Ibid, 76. 15. Ibid, 78. (shrink)

According to Hegel, the true infinite is the fundamental concept of philosophy. Yet despite this fact, there is absence of consensus concerning its meaning and significance. The true infinite challenges the currently dominant non-metaphysical interpretations of Hegel, as it challenged the dominance of the Kantian framework in its own day, specifically Kant’s attack on theology and his treatment of theology as a postulate of moralit y. Kant admits that the God-postulate has only subjective necessity and validity, and is an expression (...) of moral faith. Hegel both accepts Kant’s approach to the God-question through freedom and practical reason, but he rejects Kant’s philosophy of the postulates as incoherent, burdened with finitude and antithesis. The ought is only the beginning of the transcendence of finitude, but also essentially clings to finitude. This is the spurious infinite. In contrast to the traditional view of abstract transcendence, Hegel shows that the very attempt to separate the infinite from the finite only renders the infinite finite and levels it. The consciousness of limit (finitude) implies a transcendence of limit. The true infinite is an onto-theological principle, a social infinite that overcomes the limits imposed by abstract transcendence and the dualisms imposed by the Kantian frame. It is of vital importance for Hegel’s philosophy of religion, as both a doctrine of divine presence and absolute spirit in its community. (shrink)

According to Hegel, the true infinite is the fundamental concept of philosophy. Yet despite this fact, there is absence of consensus concerning its meaning and significance. The true infinite challenges the currently dominant non-metaphysical interpretations of Hegel, as it challenged the dominance of the Kantian framework in its own day, specifically Kant’s attack on theology and his treatment of theology as a postulate of moralit y. Kant admits that the God-postulate has only subjective necessity and validity, and is an expression (...) of moral faith. Hegel both accepts Kant’s approach to the God-question through freedom and practical reason, but he rejects Kant’s philosophy of the postulates as incoherent, burdened with finitude and antithesis. The ought is only the beginning of the transcendence of finitude, but also essentially clings to finitude. This is the spurious infinite. In contrast to the traditional view of abstract transcendence, Hegel shows that the very attempt to separate the infinite from the finite only renders the infinite finite and levels it. The consciousness of limit (finitude) implies a transcendence of limit. The true infinite is an onto-theological principle, a social infinite that overcomes the limits imposed by abstract transcendence and the dualisms imposed by the Kantian frame. It is of vital importance for Hegel’s philosophy of religion, as both a doctrine of divine presence and absolute spirit in its community. (shrink)

Daniel Isaacson has advanced an epistemic notion of arithmetical truth according to which the latter is the set of truths that we grasp on the basis of our understanding of the structure of natural numbers alone. Isaacson’s thesis is then the claim that Peano Arithmetic (PA) is the theory of finite mathematics, in the sense that it proves all and only arithmetical truths thus understood. In this paper, we raise a challenge for the thesis and show how it can (...) be overcome. We introduce the concept of purity for theories of arithmetic: a theory of arithmetic is pure when it only proves arithmetical truths. Then, we argue that, under Isaacson’s thesis, some PA-provable truths—including transfinite induction claims for infinite ordinals and some consistency statements—are seemingly not arithmetical in Isaacson’s sense, and hence that Isaacson’s thesis might entail the impurity of PA. Nonetheless, we conjecture that the advocate of Isaacson’s thesis can avoid this undesirable consequence: the arithmetical nature, as understood by Isaacson, of all contentious PA-provable statements can be justified. As a case study, we explore how this is done for transfinite induction claims with infinite ordinals below ε0. To this end, we show that the PA-proof of such claims employs exclusively resources from finite mathematics, and that ordinals below ε0 are finitary objects despite being infinite. (shrink)

This article concerns the split between syntactic and semantic approaches to scientific theories. It aims at showing that an axiomatic representation of a scientific theory is a precondition of comprehending if the models of contain infinite entities. This result is established on the basis of the proposition that the human mind—which is finitely bounded for all we know—is not capable of directly grasping infinite entities. In view of this cognitive limitation, an indirect and finite representation of possibly infinite components (...) of the models of a scientific theory proves to be indispensable. Sets of axioms and sets of axiom schemes provide such a representation. These considerations will be cast into an argument for an axiomatic conception of scientific theories. The article concludes with a case study of the ideal gas model. (shrink)

The concept of formal transcendentalism is utilized. The fundamental and definitive property of the totality suggests for “the totality to be all”, thus, its externality (unlike any other entity) is contained within it. This generates a fundamental (or philosophical) “doubling” of anything being referred to the totality, i.e. considered philosophically. Thus, that doubling as well as transcendentalism underlying it can be interpreted formally as an elementary choice such as a bit of information and a quantity corresponding to the number of (...) elementary choices to be defined. This is the quantity of information defined both transcendentally and formally and thus, philosophically and mathematically. If one defines information specifically, as an elementary choice between finiteness (or mathematically, as any natural number of Peano arithmetic) and infinity (i.e. an actually infinite set in the meaning of set theory), the quantity of quantum information is defined. One can demonstrate that the so-defined quantum information and quantum information standardly defined by quantum mechanics are equivalent to each other. The equivalence of the axiom of choice and the well-ordering “theorem” is involved. It can be justified transcendentally as well, in virtue of transcendental equivalence implied by the totality. Thus, all can be considered as temporal as far anything possesses such a temporal counterpart necessarily. Formally defined, the frontier of time is the current choice now, a bit of information, furthermore interpretable as a qubit of quantum information. (shrink)

According to Hegel, the true infinite is the fundamental concept of philosophy. Yet despite this fact, there is absence of consensus concerning its meaning and significance. The true infinite challenges the currently dominant non-metaphysical interpretations of Hegel, as it challenged the dominance of the Kantian framework in its own day, specifically Kant’s attack on theology and his treatment of theology as a postulate of moralit y. Kant admits that the God-postulate has only subjective necessity and validity, and is an expression (...) of moral faith. Hegel both accepts Kant’s approach to the God-question through freedom and practical reason, but he rejects Kant’s philosophy of the postulates as incoherent, burdened with finitude and antithesis. The ought is only the beginning of the transcendence of finitude, but also essentially clings to finitude. This is the spurious infinite. In contrast to the traditional view of abstract transcendence, Hegel shows that the very attempt to separate the infinite from the finite only renders the infinite finite and levels it. The consciousness of limit (finitude) implies a transcendence of limit. The true infinite is an onto-theological principle, a social infinite that overcomes the limits imposed by abstract transcendence and the dualisms imposed by the Kantian frame. It is of vital importance for Hegel’s philosophy of religion, as both a doctrine of divine presence and absolute spirit in its community. (shrink)

This is a doctoral thesis in the Department of Philosophy of the Madras University. Central to Indian philosophical thought is Mukti or liberation from the present state of ignorance and bondage. But the positive meaning of this liberation is not conceived in the same way by all Indian schools. In the first part of the book the author examines the opinions of various Indian schools other than Vedanta, including Buddhism and Jainism. The second part explains the point of Advaita Vedanta, (...) the chief exponent of which is Sankara. For Advaita or Non-Dualism Brahman is the unique, infinite and immutable reality; Finite beings are maya or unreal beside the absolute reality of Brahman. They have only a relative and practical reality for human life until Mukti or liberation is attained in the realization that Brahman is the real self of all and the world of finite beings ceases to have any further meaning. All religious exercises are only means towards this final realization. On the whole the book is a synthesis of metaphysics from the Indian, especially non-dualistic, point of view. Twelve pages of Errata at the end of the book do little honor to the printers and publishers of the Madras University Philosophical Series.—J. K. (shrink)

It is a received view that Kant’s formal logic (or what he calls “pure general logic”) is thoroughly intensional. On this view, even the notion of logical extension must be understood solely in terms of the concepts that are subordinate to a given concept. I grant that the subordination relation among concepts is an important theme in Kant’s logical doctrine of concepts. But I argue that it is both possible and important to ascribe to Kant an objectual notion of logical (...) extension according to which the extension of a concept is the multitude of objects falling under it. I begin by defending this ascription in response to three reasons that are commonly invoked against it. First, I explain that this ascription is compatible with Kant’s philosophical reflections on the nature and boundary of a formal logic. Second, I show that the objectual notion of extension I ascribe to Kant can be traced back to many of the early modern works of logic with which he was more or less familiar. Third, I argue that such a notion of extension makes perfect sense of a pivotal principle in Kant’s logic, namely the principle that the quantity of a concept’s extension is inversely proportional to that of its intension. In the process, I tease out two important features of the Kantian objectual notion of logical extension in terms of which it markedly differs from the modern one. First, on the modern notion the extension of a concept is the sum of the objects actually falling under it; on the Kantian notion, by contrast, the extension of a concept consists of the multitude of possible objects—not in the metaphysical sense of possibility, though—to which a concept applies in virtue of being a general representation. While the quantity of the former extension is finite, that of the latter is infinite—as is reflected in Kant’s use of a plane-geometrical figure (e.g., circle, square), which is continuum as opposed to discretum, to represent the extension in question. Second, on the modern notion of extension, a concept that signifies exactly one object has a one-member extension; on the Kantian notion, however, such a concept has no extension at all—for a concept is taken to have extension only if it signifies a multitude of things. This feature of logical extension is manifested in Kant’s claim that a singular concept (or a concept in its singular use) can, for lack of extension, be figuratively represented only by a point—as opposed to an extended figure like circle, which is reserved for a general concept (or a concept in its general use). Precisely on account of these two features, the Kantian objectual extension proves vital to Kant’s theory of logical quantification (in universal, particular and singular judgments, respectively) and to his view regarding the formal truth of analytic judgments. (shrink)

First published in 1990, this is a reissue of Professor Hilary Putnam’s dissertation thesis, written in 1951, which concerns itself with The Meaning of the Concept of Probability in Application to Finite Sequences and the problems of the deductive justification for induction. Written under the direction of Putnam’s mentor, Hans Reichenbach, the book considers Reichenbach’s idealization of very long finite sequences as infinite sequences and the bearing this has upon Reichenbach’s pragmatic vindication of induction.

Despite the importance of the variational principles of physics, there have been relatively few attempts to consider them for a realistic framework. In addition to the old teleological question, this paper continues the recent discussion regarding the modal involvement of the principle of least action and its relations with the Humean view of the laws of nature. The reality of possible paths in the principle of least action is examined from the perspectives of the contemporary metaphysics of modality and Leibniz's (...) concept of essences or possibles striving for existence. I elaborate a modal interpretation of the principle of least action that replaces a classical representation of a system's motion along a single history in the actual modality by simultaneous motions along an infinite set of all possible histories in the possible modality. This model is based on an intuition that deep ontological connections exist between the possible paths in the principle of least action and possible quantum histories in the Feynman path integral. I interpret the action as a physical measure of the essence of every possible history. Therefore only one actual history has the highest degree of the essence and minimal action. To address the issue of necessity, I assume that the principle of least action has a general physical necessity and lies between the laws of motion with a limited physical necessity and certain laws with a metaphysical necessity. (shrink)

We answer a question raised in [9], that is whether the infinite weight of the generic type of the free group is witnessed in F ω . We also prove that the set of primitive elements in finite rank free groups is not uniformly definable. As a corollary, we observe that the generic type over the empty set is not isolated. Finally, we show that uncountable free groups are not N₁-homogeneous.

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

The article analyzes the features of understanding of the history of philosophical process by the leading representative of French Neo-Kantianism. The binary scheme, thoroughly substantiated by Renоuvier in the “Sketch of the Systematic Classification of Philosophical Doctrines”, was a certain result of his previous creative way. In the “Textbook of Philosophy of the New Age”, he, highlighting the pantheism–idealism dilemma, expresses his sympathies for eclecticism. In the “Textbook of Ancient Philosophy”, he advocates giving philosophy a scientific character by combining the (...) laws of reason and history. In “Experiences of General Criticism” he builds a system of neo-criticism, identifying phenomena with reality. Finally, the “Sketch” presents an attempt at an objective consideration of the history of philosophy. Renоuvier identifies six oppositions passing through the entire history of thought (thing–consciousness, infinite–finite, evolution–creation, necessity–freedom, happiness–duty, evidence–belief). The strength of his method is the desire to make a choice between opposing trends, taking into account intellectual motives, passions, practical inducements of varying degrees of community. But both elements of reductionism and unfounded optimism about the lack of influence of all kinds of systems on his own constructions are present in his approach. (shrink)

The wrong version of my article ‘Aspects of the Infinite in Kant’ was printed in the last issue of Mind (pp. 205–23). I should like to correct an error that thereby appeared on page 207. In A430–2/B458–60 of the Critique of Pure Reason Kant does not deny that what is (mathematically) infinite should be what I called an actual measurable totality—if, by its measure, we mean ‘the multiplicity of given units which it contains’. His point is simply that what (...) makes it infinite cannot be the fact that its measure is the greatest possible; for there is no such thing. What he has in mind here can be illustrated as follows. Take the infinite multiplicity of hours that have elapsed up to a given moment; then the multiplicity of minutes that have elapsed, and indeed the multiplicity of hours that will have elapsed an hour later, are both greater. No multiplicity is so great that it cannot be increased in this way. (Of course, standard contemporary formal work on the infinite has superseded Kant here. A modern mathematician would want either to quarrel with this or at least to refine it.) He also refuses to allow that an infinite multiplicity is a number (cf. BIII and A526–7/B554–5). In the end, he thinks, there is no saying what it is for something to be (mathematically) infinite without falling back on ‘the true transcendental concept of infinitude,’ namely ‘that the successive synthesis of units required for the enumeration of a quantum can never be completed’. (shrink)

This paper proposes a reassessment of Hegel’s critical reading of Spinoza and of the charge of acosmism, for which this reading is known. We argue that this charge is actually the consequence of a more fundamental criticism, namely Spinoza’s presumable inability to conceive the unity of the finite and the infinite. According to Hegel, the infinite and the finite remain two poles apart in Spinoza’s metaphysics, which thus fails to be a true monism, insofar as it contains an (...) irreducible duality. Against this reading, we argue that Spinoza’s conception of the causal co-determination of finite modes entails the acknowledgment of their essentially infinite nature. The study of this particular instance of coincidentia oppositorum enables to counter some of Hegel’s criticisms as well as to illuminate a fairly unexplored aspect of Spinoza’s substance monism. (shrink)

This paper proposes a reassessment of Hegel’s critical reading of Spinoza and of the charge of acosmism, for which this reading is known. We argue that this charge is actually the consequence of a more fundamental criticism, namely Spinoza’s presumable inability to conceive the unity of the finite and the infinite. According to Hegel, the infinite and the finite remain two poles apart in Spinoza’s metaphysics, which thus fails to be a true monism, insofar as it contains an (...) irreducible duality. Against this reading, we argue that Spinoza’s conception of the causal co-determination of finite modes entails the acknowledgment of their essentially infinite nature. The study of this particular instance of coincidentia oppositorum enables to counter some of Hegel’s criticisms as well as to illuminate a fairly unexplored aspect of Spinoza’s substance monism. (shrink)

Abstract Robert Stern's Understanding Moral Obligation is a remarkable achievement, representing an original reading of Kant's contribution to modern moral philosophy and the legacy he bequeathed to his later-eighteenth- and early-nineteenth-century successors in the German tradition. On Stern's interpretation, it was not the threat to autonomy posed by value realism, but the threat to autonomy posed by the obligatory nature of morality that led Kant to develop his critical moral theory grounded in the concept of the self-legislating moral agent. Accordingly, (...) Stern contends that Kant was a moral realist of sorts, holding certain substantive views that are best characterized as realist commitments about value. In this paper, I raise two central objections to Stern's reading of Kant. The first objection concerns what Stern identifies as Kant's solution to the problem of moral obligation. Whereas Stern sees the distinction between the infinite will and the finite will as resolving the problem of moral obligation, I argue that this distinction merely explains why moral obligations necessarily take the form of imperatives for us imperfect human beings, but does not solve the deeper problem concerning the obligatory nature of morality: why we should take moral norms to be supremely authoritative laws that override all other norms based on our non-moral interests. The second objection addresses Stern's claim that Kantian autonomy is compatible with value realism. Although this is an idea with which many contemporary readers will be sympathetic, I suggest that the textual evidence actually weighs in favor of constructivism. (shrink)

This paper proposes an extension of Chaitin's halting probability Ω to a measurement operator in an infinite dimensional quantum system. Chaitin's Ω is defined as the probability that the universal self-delimiting Turing machine U halts, and plays a central role in the development of algorithmic information theory. In the theory, there are two equivalent ways to define the program-size complexity H of a given finite binary string s. In the standard way, H is defined as the length of the (...) shortest input string for U to output s. In the other way, the so-called universal probability m is introduced first, and then H is defined as –log2m without reference to the concept of program-size.Mathematically, the statistics of outcomes in a quantum measurement are described by a positive operator-valued measure in the most general setting. Based on the theory of computability structures on a Banach space developed by Pour-El and Richards, we extend the universal probability to an analogue of POVM in an infinite dimensional quantum system, called a universal semi-POVM. We also give another characterization of Chaitin's Ω numbers by universal probabilities. Then, based on this characterization, we propose to define an extension of Ω as a sum of the POVM elements of a universal semi-POVM. The validity of this definition is discussed.In what follows, we introduce an operator version equation image of H in a Hilbert space of infinite dimension using a universal semi-POVM, and study its properties. (shrink)

The research focuses on the problem of interpretation of such a concept as the origin, which is ultimate for philosophical thought and bases existentially oriented constructions of Franz Rosenzweig and Martin Heidegger. It is argues in this paper the fundamental difference between the interpretations of the origin in “The Star of Redemption» and in «Being and Time”, despite all points of intersection and coincidences that bring closer together dialogical and existential-historical thinking. Such differentiation of positions is determined by the necessity (...) both to neutralize romantic connotations, which expose the origin ( Ursprung ) as abyss ( Abgrund ), and to methodologically clarify the possibility not so much to think as to practically assert a limited and unclosed within itself integral being, including ontological and ethical dimensions. The origin of oblivion and renewal, imposed by Heidegger’s hermeneutics of facticity, is contrasted with Rosenzweig's origin of the “eternal overworld”, reconstructed on the basis of Herman Cohen's religio-philosophical intuitions and his analysis of the infinitesimal as a principle of reality. Rosenzweig’s position, which aims at a correlative consideration of the purely logical content of the infinitesimal quantity (almost-Nothing) and the existential experience of the finiteness of everything, is favorably distinguished by the absence of a one-sided focus on recalling, where fore-running although is declared, but it is immediately restrained by being-towards-death. As a result, Rosenzweig’s dialogism turns out to be free from overestimating what Heidegger’s interpretation calls resoluteness, which in fact is not much different from paralysis in the clearance of being. Cohen’s problem of an infinite task's realization, being actualized in Rosenzweig’s conception, allows to go beyond being-towards-death in juxtaposing the trajectories of initial elements into a true gestalt. (shrink)

Combining historical fact with a retelling of ancient myths and legends, Alistair Wilson shows how mathematics arose out of the problems of everyday life. He introduces concepts such as geometry, prime numbers, and trigonometry in a way that will totally disarm the reader who fears mathematics.

Central to Levinas’ “phenomenological” approach to ethics is his identification of an “infinite signification” in the human face. This insistence on the appearance of an infinitely signifying phenomenon has led many, notably Dominique Janicaud, to decry Levinas’ work as anti-phenomenological: little more than a novel approach to metaphysics. A significant element of the phenomenological revolution, Janicaud insists, referencing Husserl and the early Heidegger for support, is grounded in the recognition that phenomena arise in and are circumscribed by finitude. Any reference (...) to an infinite phenomenon which breaks with apperception, he therefore concludes, necessarily breaks as well with the phenomenological tradition, signaling a reversion to idealism, metaphysics, and ultimately theology. As this paper will argue however, a close reading of Husserl reveals not only the recognition of infinite phenomena; but, more significantly, the possibility that all phenomena are potentially infinite. This means that in many ways Levinas is not only more faithful to Husserl’s project than is often admitted; but moreover, that Levinas’ work serves as a lens through which a hitherto under-emphasized element of Husserl’s work can be examined – an element which challenges the traditional critique of Levinas. What’s more, the possibility of infinite phenomena opened up in Husserl’s work, when read through Levinas’ analysis of the ethical significance of the infinite, creates the possibility of recognizing the potential ethical value of all objects, and not merely one object in particular: the human face. This possibility, as I will show, contributes to further blurring the already ambiguous distinction within Levinas between “good” and “bad” infinities – a blurring which, as I will argue, serves as a powerful heuristic in interpreting certain human pathologies, from obsessive compulsive disorder and certain forms of schizophrenia, to the temptation of evil. In this way, an analysis of the status of the infinite in Husserl and Levinas provides a purely phenomenological ground for exploring issues traditionally seen as the exclusive domain of metaphysics and theology. (shrink)

John Tomarchio - Aquinas's Concept of Infinity - Journal of the History of Philosophy 40:2 Journal of the History of Philosophy 40.2 163-187 Aquinas's Concept of Infinity John Tomarchio MUCH HAS BEEN WRITTEN of late about Aquinas's concept of divine infinity, but the attention given to his other metaphysical uses of the term 'infinite' has been peripheral -- sometimes to ill effect in the interpretation of his concept of divine infinity. The intent of this article is to offer an explication (...) of Aquinas's analogical concept of infinity. What Aquinas means when he calls beings or their principles infinite is sufficiently complex that it needs to be analyzed in its own right before his arguments for things' actually being infinite can be evaluated. There is no one place in his writings where Aquinas gives a full treatment of the various metaphysical senses of infinity, and so his view has to be reconstructed from the analysis of sundry texts in juxtaposition. As my intent is to reconstruct his overall theory, I will take up the relevant texts in a logical rather than historical order. I will first address the kind of relative infinity Aquinas predicates of certain principles of things, namely, the matter, form, and existence of finite beings, and then the kind of actual infinity he predicates of certain things, namely angelic substances and God. In the course of doing so, I will identify four global distinctions that run through all these.. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:diacritics 34.2 (2006) 14-36 [Access article in PDF] "The Possibility of the Poetic Said" in Otherwise than Being (Allusion, or Blanchot in Lévinas) Gabriel Riera Language would exceed the limits of what is thought, by suggesting, letting be understood without ever making understandable [en laissant sous-entendre, sans jamais faire entendre] an implication of meaning distinct from that which comes to signs from the simultaneity of systems or the logical (...) definition of concepts. This possibility [vertu] is laid bare in the poetic said. [...] It is shown in the prophetic said. —Emmanuel Lévinas, Otherwise than Being [my emphasis] Writing, without placing itself above art, supposes that one not prefer art, but efface art as writing effaces itself. —Maurice Blanchot, Writing of the DisasterIn an effort to elucidate a more originary difference than Heidegger's ontological difference between Being and entities, as well as an ethicity of thinking able to respond to the trace of transcendence of the other's face, Lévinas's philosophy redefines language radically and does so in uneasy proximity to poetic language. In Otherwise than Being or Beyond Essence (1974), Lévinas deploys a constellation of terms—discourse, expression, communication, face, sincerity, witnessing, prophecy—in order to elucidate the ethical dimension of language, the ethicity of philosophical discourse. This constellation revolves around the crucial notion of signification and the two operative concepts of the said (le dit) and saying (le dire). By the former Lévinas understands the semantic content of an utterance or the giving of signs by a sender to an addressee, but also the different modalities by which a subject masters the world by assimilating it to the measure of consciousness (discourse, narration, history, manifestation, representation). Saying (le dire) refers to the way of signifying primarily that I am for the other, the expression of being-for-the-other that subjectivity "is" before positioned as the source of signs or comprehension: subjectivity as exposition. While the said expresses a content, the saying is expression without content.Language for Lévinas is the "structure" of both the retraction and the inscription of infinity (l'infini), and this "structure" is what Lévinas calls ethics. It is thus possible to affirm that in Lévinas there is no ethical turn without a certain linguistic turn, one in which the a priori of language is interrupted by an anteriority whose a priori character is enigmatic. But Lévinas's linguistic turn is turned inside out: for him language contains the other's call and thus harbors a form of exteriority within itself. This is because language is elevated to a metaphysical category (in the sense of Totality and Infinity [1961], where it means the breaching or interruption of totality) that exceeds the simple [End Page 14] "giving out of signs" [délivrance de signes] [AE 81/OB 48] or communication. Lévinas claims that "saying is communication, to be sure, but as a condition for all communication, as exposure [exposition]" [AE 82/OB 48]. Signification precedes and defines language as proximity and contact with the other and not as the "vicissitude [peripetia] of the thematizing intentionality" [AE 216/OB 137] of a subject whose main attribute is its ability to synchronize questions and answers, being and its manifestations, and by so doing leaves no margin for being exposed to the other. Further, signification also names an enigmatic pre-diction that challenges the primacy that language, understood as a system of signs, enjoys in other philosophies (structuralism and analytic philosophy, for example).Most critics agree that Lévinas's philosophy is inhospitable to aesthetic phenomena and ambivalent toward poetic language. His severe condemnation of the work of art in "Reality and Its Shadow" (1948) is still present in Totality and Infinity, Lévinas's first major work, where he establishes a relation with the other as face-to-face in discourse. And although in Otherwise than Being Lévinas finally acknowledges "poetic" language's ability to suggest significations that exceed the order of discourse and even endows it with a (quasi)-ethical force... (shrink)

Power, according to Spinoza, is God's essence. Hence understanding Spinoza's thoughts about power will help us understand Spinoza's God. Since Spinoza's metaphysics is the foundation for his ethics, this understanding will provide insights into the latter as well. ;I begin by examining Spinoza's interpretation of Descartes. This has God outside the universe recreating it each moment; otherwise it would cease to exist. Spinoza concludes that all events exist solely through God's power; neither minds nor bodies have power of their own. (...) In other words, Spinoza's Descartes is an occasionalist. ;I then argue that the primary difference between Spinoza's God and Descartes' God is that Spinoza places God's power inside the universe, rather than outside. Spinoza's God is an immanent creative power: a power in all things which continually creates them, keeping the universe itself from disappearing. ;Next I explain why Spinoza thinks the existence of motion in the universe follows from God's power. Spinoza believes modes in extension are only differentiated by their motion. God's power necessarily creates infinitely many modes. Therefore, there must be motion. ;Next I argue that God's attributes include infinite powers of thought and of extension. Individual beings are finite modifications of these powers. This metaphysics is extended into Spinoza's ethics, showing that it is a system which describes how we share in God's power and how virtuous behavior increases our power, thereby increasing our joy. Then a discussion of power in Spinoza's epistemology shows that ideas are not conceived by him as inert, but as acts of understanding, which strive to maintain themselves in our minds and cause other ideas to follow from them. This leads to an examination of ways in which an individual person, by increasing knowledge of the passions that afflict him or her, is supposed to gain power over them. The concluding chapter speculatively ties the earlier discoveries to other important aspects of Spinoza's metaphysics, striving toward a more comprehensive understanding of Spinoza's God. (shrink)

Aggregative consequentialism and several other popular moral theories are threatened with paralysis: when coupled with some plausible assumptions, they seem to imply that it is always ethically indifferent what you do. Modern cosmology teaches that the world might well contain an infinite number of happy and sad people and other candidate value-bearing locations. Aggregative ethics implies that such a world contains an infinite amount of positive value and an infinite amount of negative value. You can affect only a finite (...) amount of good or bad. In standard cardinal arithmetic, an infinite quantity is unchanged by the addition or subtraction of any finite quantity. So it appears you cannot change the value of the world. Modifications of aggregationism aimed at resolving the paralysis are only partially effective and cause severe side effects, including problems of “fanaticism”, “distortion”, and erosion of the intuitions that originally motivated the theory. Is the infinitarian challenge fatal? (shrink)

The concept of indistinguishable particles in quantum theory is fundamental to questions of ontology. All ordinary matter is made of electrons, protons, neutrons, and photons and they are all indistinguishable particles. Yet the concept itself has proved elusive, in part because of the interpretational difficulties that afflict quantum theory quite generally, and in part because the concept was so central to the discovery of the quantum itself, by Planck in 1900; it came encumbered with revolution. I offer a deflationary reading (...) of the concept ‘indistinguishable’ that is identical to Gibbs’ concept ‘generic phase’, save that it is defined for state spaces with only finitely-many states of bounded volume and energy. That, and that alone, makes for the difference between the quantum and Gibbs concepts of indistinguishability. This claim is heretical on several counts, but here we consider only the content of the claim itself, and its bearing on the early history of quantum theory rather than in relation to contemporary debates about particle indistinguishability and permutation symmetry. It powerfully illuminates that history. (shrink)

The publication of Frege’s Begriffsschrift in 1879 forever altered the landscape for many Western philosophers. Here, Sebastian Rödl traces how the Fregean influence, written all over the development and present state of analytic philosophy, led into an unholy alliance of an empiricist conception of sensibility with an inferentialist conception of thought. -/- According to Rödl, Wittgenstein responded to the implosion of Frege’s principle that the nature of thought consists in its inferential order, but his Philosophical Investigations shied away (...) from offering an alternative. Rödl takes up the challenge by turning to Kant and Aristotle as ancestors of this tradition, and in doing so identifies its unacknowledged question: the relation of judgment and truth to time. Rödl finds in the thought of these two men the answer he urges us to consider: the temporal and the sensible, and the atemporal and the intelligible, are aspects of one reality and cannot be understood independently of one another. In demonstrating that an investigation into the categories of the temporal can be undertaken as a contribution to logic, Rödl seeks to transform simultaneously our philosophical understanding of both logic and time. (shrink)

The main aim of this essay is to show that, for Stevens, the concept of reality is very fluctuating. The essay begins with addressing the relationship between poetry and philosophy. I argue, contra Critchley, that Stevens’ poetic work can elucidate, or at least help us to understand better, the ideas of philosophers that are usually considered obscure. The main “obscure” philosophical work introduced in and discussed throughout the essay is Schelling’s System of Transcendental Idealism. Both a (shellingian) philosopher and a (...) (stevensian) poet search for reality. In order to understand Stevens’ poetry better, I distingush several concepts of reality: initial reality (the external world of the common sense), imagined reality (a fiction, a product of one’s mind), final reality (the object of a philosopher’s and a poet’s search) and total reality (the sum of all realities, Being). These determinations are fixed by reason (in the present essay), whereas in Stevens’ poetic works, they are made fluid by the imagination. This fluidity leads the concept of reality from its initial stage through the imagined stage to its final stage. Throughout this process, imagined reality must be distinguished from both a mere fancy and its products. Final reality is, however, nothing transcendent. It is rather a general transpersonal order of reality created by poetry/the imagination. The main peculiarity of final reality is that it is a dynamic order. It is provisional at each moment. Stevens (and Schelling too) characterizes this order as that of a work of art which is a finite object, but has an infinite meaning. Stevens calls this order “the central poem” or the “endlessly elaborating poem”. If ultimate reality is a poem created by the imagination, one may ask who is the imagining subject. I argue that this agent is best to be thought as total reality, that is, as Being. Stevens, however, maintains that if there were such an agency, it would be an inhuman agency, “an inhuman meditation”. The essay concludes, in a Derridian manner, with the claim that this agency cannot have any name; it is the “unnamed creator of an unknown sphere, / Unknown as yet, unknowable, / Uncertain certainty” (OP: 127). It is best thought as an X, as an unknown variable. Being has no name. (shrink)

RESUMEN El presente artículo intenta desarrollar los elementos de una concepción del conocimiento que resulte operativa para una versión epistémica de la democracia deliberativa. Se intenta superar algunas limitaciones de los modelos epistémico-deliberativos actuales y profundizar la relación entre conocimiento, deliberación y legitimidad. ABSTRACT The article develops the elements of a conception of knowledge that turns out to be operative for an epistemic version of deliberative democracy. It attempts to overcome some of the limitations of current deliberative-epistemic models and (...) carry out an in-depth study of the relation among knowledge, deliberation, and legitimacy. (shrink)

Since both berkeley and hume are committed to the view that a line is composed of finitely many fundamental parts, They must find responses to the standard geometrical proofs of infinite divisibility. They both repeat traditional arguments intended to show that infinite divisibility leads to absurdities, E.G., That all lines would be infinite in length, That all lines would have the same length, Etc. In each case, Their arguments rest upon a misunderstanding of the concept of a limit, And thus (...) are not successful. Berkeley, However, Adds a further ingenious argument to the effect that the standard geometrical proofs of infinite divisibility misread the unlimited representational capacity of geometrical diagrams as a substantive feature of the objects that these diagrams represent. The article concludes that berkeley is right on this matter, And that the traditional proofs of infinite divisibility do not show what they are intended to show. (shrink)

Discusses Frege's formal definitions and characterizations of infinite and finite sets. Speculates that Frege might have discovered the "oddity" in Dedekind's famous proof that all infinite sets are Dedekind infinite and, in doing so, stumbled across an axiom of countable choice.

As we know, the whole system of Hegel 's thought is based totally on the self-development process of the Geist. In other words, according to dialectical scheme of neoplatonism which Proclus systematized as a triad, μονή-πρόοδος-ἐπιστροφή, Hegel says that the Geist remains in himself, comes out from himself, and then returns to himself. With this process of self-development of the Geist, Hegel tries to explain the realities in general such as nature, history, art, religion, and philosophy. This process of the (...) development of the Geist makes a circle mode with the triadic way of an sich - für sich - an und für sich, but itself as the final destination of the Geist after finishing the whole process, is not a mere itself at the beginning, but the completed itself filled with the self-consciousness and freedom in full bloom. The last moment when the Geist returns to itself, Hegel says it is ‘Ruhe in Gott’. This is the final moment which the Geist as reality of the world and the history arrives and also the conclusion of his philosophy. Then, what happens to this final stage of the return of the Geist to itself, and what does it mean? Most of all, Hegel saw that, as the goal of world history is the achievement of self-consciousness of the Geist on its freedom and the realization of the freedom, this goal was achieved in the self-return of the Geist. And he has seen that the finite and the infinite, which showed constant opposition in the history of Western thought, are united here. According to Hegel, the infinite and finite in the bad concept that came out of the intellect(Verstand) are confronted with each other, and existence and non-existence are substantially separated. However, in the true concept of the infinite derived from reason(Vernunft), is formed the unified moment, in which the confrontations of both parties has been resolved and everything had been included in itself. The transcendence that he advocates is not a transmundane transcendence that is separated absolutely from the finite sensory world in existence and then unreachable absolutely, but a ‘right transcendence’ in which the finite and the infinite are really inseparable and remain as moments of the process of the Geist. In his view, the former which Platonism represents in Western philosophy, is an 'empty transcendence,' but the transcendence that he speaks is a 'full transcendence' that encompasses all moments of the finite in itself. The return of the absolute idea to itself means not only the sublation (Aufhebung) of the finite but also the the sublation of the reality, of the other, of the outwardness, of the abandonment of history, and secures the opposites to those such as ideality, self-identity, inwardness, and trans-historicity. Hegel's Geist as returning to itself has self-consciousness recognizing that it negates and rules the world and the history at the same time. In other words, it is honored and then the glorifying of God has been achieved in the consciousness of itself as the Absolute that it has been the only subject and the sole ruler of world history. According to Hegel, this glorifying of God is the most valuable object of the Geist and history. The Geist that has completed the process of self-development in the return to the itself enjoys the serenity, which is not a 'vacant serenity' but a ‘full serenity’ integrating the finite and the infinite into its inner simplicity. Hegel regarded the transition from the other to being self-internal as a transition from necessity to freedom, and in this serenity the Geist enjoys the freedom that it has in its own nature, as possible as it can. That is to say, in this serenity, the Geist becomes absolutely free and became itself, negating and sublating all other-being(Anderssein). In his view, the "Serenity in God," where the Geist becomes itself and its absolute freedom is obtained, is the ultimate goal of the Geist, and the omega point of the history. (shrink)

The first goal of this paper is to develop a theoretical and practical framework which can help to measure the emotional value generated by organizations in quantitative terms. Its second goal is to use data obtained from the UCAN in Spain as a case study to illustrate the quantification of the emotional value generated, with a view to factoring that value into a social accounting system. Ever greater recognition of the social role of organizations in recent years has led (...) to a need for a consistent definition of the concept of socio-emotional value, and for instruments that can be used to measure that value in terms of generic social accounting. Taking the current lack of standardization in such models and instruments, especially those that deal with emotional value as its starting point, the paper proposes a new instrument for measuring that value quantitatively in such a way as to overcome some of the limitations of earlier proposals. The underlying perspective is that the monetary values identified in market and non-market transactions do not accurately account for all the value generated for different stakeholders, and that adjustments are required through a correction factor applied to the value variables identified. The quantification of the socio-emotional value generated by an organization is seen as a more comprehensive indicator of its performance, given that it provides more information and takes into account the value generated for stakeholders as a whole in all dimensions. (shrink)

BackgroundVulnerability is a key concept in traditional and contemporary bioethics. In the philosophical literature, vulnerability is understood not only to be an ontological condition of humanity, but also to be a consequence of contingent factors. Within bioethics debates, vulnerable populations are defined in relation to compromised capacity to consent, increased susceptibility to harm, and/or exploitation. Although vulnerability has historically been associated with older adults, to date, no comprehensive or systematic work exists on the meaning of their vulnerability. To fill this (...) gap, we analysed the literature on aged care for the meaning, foundations, and uses of vulnerability as an ethical concept.MethodsUsing PRISMA guidelines, we conducted a systematic review of argument-based ethics literature in four major databases: PubMed, Embase®, Web of Science™, and Philosopher’s Index. These covered biomedical, philosophy, bioethical, and anthropological literature. Titles, abstracts, and full texts of identified papers were screened for relevance. The snowball technique and citation tracking were used to identify relevant publications. Data analysis and synthesis followed the preparatory steps of the coding process detailed in the QUAGOL methodology.ResultsThirty-eight publications met our criteria and were included. Publication dates ranged from 1984 to 2020, with 17 publications appearing between 2015 and 2020. Publications originated from all five major continents, as indicated by the affiliation of the first author. Our analyses revealed that the concept of vulnerability could be distinguished in terms of basic human and situational vulnerability. Six dimensions of older adults’ vulnerability were identified: physical; psychological; relational/interpersonal; moral; sociocultural, political, and economic; and existential/spiritual. This analysis suggested three ways to relate to older adults’ vulnerability: understanding older adults’ vulnerability, taking care of vulnerable older adults, and intervening through socio-political-economic measures.ConclusionsThe way in which vulnerability was conceptualised in the included publications overlaps with distinctions used within contemporary bioethics literature. Dimensions of aged care vulnerability map onto defining features of humans, giving weight to the claim that vulnerability represents an inherent characteristic of humans. Vulnerability is mostly a value-laden concept, endowed with positive and negative connotations. Most publications focused on and promoted aged care, strengthening the idea that care is a defining practice of being human. (shrink)

The basic thesis is that the problem of infinity underlies the current dilemma in modern theoretical physics. The traditional and set-theoretic conceptions of infinity are considered. It is demonstrated that standard mathematical analysis is dependent on the complete relativity of the infinite. In examining the domains of modern physics, infinity is found to lose its entirely relative character and, therefore, to be less amenable to classical analysis. Complementary aspects of microworld infinity are identified and are associated with the equivalent features (...) (inertial and gravitational mass) of Einstein's macroworld theory. The persisting effort to treat essentially non-classical phenomena in classical terms is critically discussed. A new attitude toward the infinite is recommended, one that might lead to establishing a second principle of the relativity of the infinite. The prospect for implementing the suggested approach through a "trans-analytic" meta-theory of dimensional generation is briefly entertained. (shrink)

Kennedy discusses, through dialogue, old concepts in philosophy with children regarding the finite and infinite parts of the Earth and galaxy.