In formal ontology, infinite regresses are generally considered a bad sign. One debate where such regresses come into play is the debate about fundamentality. Arguments in favour of some type of fundamentalism are many, but they generally share the idea that infinite chains of ontological dependence must be ruled out. Some motivations for this view are assessed in this article, with the conclusion that such infinite chains may not always be vicious. Indeed, there may even be room (...) for a type of fundamentalism combined with infinite descent as long as this descent is “boring,” that is, the same structure repeats ad infinitum. A start is made in the article towards a systematic account of this type of infinite descent. The philosophical prospects and scientific tenability of the account are briefly evaluated using an example from physics. (shrink)

Once one accepts that certain things metaphysically depend upon, or are metaphysically explained by, other things, it is natural to begin to wonder whether these chains of dependence or explanation must come to an end. This essay surveys the work that has been done on this issue—the issue of grounding and infinite descent. I frame the discussion around two questions: (1) What is infinite descent of ground? and (2) Is infinite descent of ground possible? (...) In addressing the second question, I will consider a number of arguments that have been made for and against the possibility of infinite descent of ground. When relevant, I connect the discussion to two important views about the way reality can be structured by grounding: metaphysical foundationalism and metaphysical infinitism. (shrink)

Physicalism is frequently understood as the thesis that everything depends upon a fundamental physical level. This standard formulation of physicalism has a rarely noted and arguably unacceptable consequence—it makes physicalism come out false in worlds which have no fundamental level, for instance worlds containing things which can infinitely decompose into smaller and smaller parts. If physicalism is false, it should not be for this reason. Thus far, there is only one proposed solution to this problem, and it comes from the (...) so-called via negativa account of physicalism. Via negativa physicalism identifies the physical with the non-mental, such that if everything in the world ultimately depends only on non-mental things, then physicalism is true. To deal with the possibility of worlds without a fundamental level, this account says that physicalism is false in worlds with either a fundamental mental level or an infinite descent of mental levels. Here I argue that there could be a world with an infinite descent of all-mental levels, yet in which physicalism might plausibly be true—thus contradicting the sufficient-for-false condition meant to save physicalism from the threat of infinitely decomposable worlds. This leaves the need for a new dependence-based account of physicalism. (shrink)

This paper contributes to answering the question how physicalism can be defined for a world without fundamental physical phenomena. In a recent paper in this journal, Torin Alter, Sam Coleman, and Robert J. Howell propose a necessary condition on physicalism. They argue that physicalism is true only if there is no infinitely descending chain of mentally constituted phenomena. I argue that this alleged necessary condition faces counterexamples. An infinitely descending chain of mentally constituted phenomena is compatible with physicalism. Afterwards I (...) propose a simpler definition of physicalism that does not fall prey to my counterexample. (shrink)

SummaryFinite, or Fermat arithmetic, as we call it, differs from Peano arithmetic in that it does not involve the existence of an infinite set or Peano's induction postulate. Fermat's method of infinite descent takes the place of bound induction, and we show that a con‐structivist interpretation of logical connectives and quantifiers can account for the predicative finitary nature of Fermat's arithmetic. A non‐set‐theoretic arithemetical logic thus seems best suited to a constructivist‐inspired number theory.

The purpose of this paper may be found in the following quotation. “Whenever an argument can be made to lead to a descending infinitude of natural numbers the hypothesis upon which the argument rests becomes untenable. This method of proof is called the method of infinite descent;.... It would be interesting and valuable to compare this method with the method of mathematical induction.”.

Pierre de Fermat is known as the inventor of modern number theory. He invented–improved many methods useful in this discipline. Fermat often claimed to have proved his most difficult theorems thanks to a method of his own invention: the infinite descent. He wrote of numerous applications of this procedure. Unfortunately, he left only one almost complete demonstration and an outline of another demonstration. The outline concerns the theorem that every prime number of the form 4n + 1 is (...) the sum of two squares. In this paper, we analyse a recent proof of this theorem. It is interesting because: it follows all the elements of which Fermat wrote in his outline; it represents a good introduction to all logical nuances and mathematical variants concerning this method of which Fermat spoke. The assertions by Fermat will also be framed inside their historical context. Therefore, the aims of this paper are related to the history of mathematics and to the logic of proof-methods. (shrink)

How could physicalism be true of a world in which there are no fundamental physical phenomena? A familiar answer, due to Barbara Gail Montero and others, is that physicalism could be true of such a world if that world does not contain an infinite descent of mentality. Christopher Devlin Brown has produced a counterexample to that solution. We show how to modify the solution to accommodate Brown’s example: physicalism could be true of a world without fundamental physical phenomena (...) if that world does not contain an infinite descent of mentally constituted mentality. This solution is independently plausible and is available to physicalists of virtually all significant varieties. (shrink)

Using only uncontentious principles from the logic of ground I construct an infinitely descending chain of ground without a lower bound. I then compare the construction to the constructions due to Dixon and Rabin and Rabern.

ABSTRACT: It is argued that the equivalence, which is usually postulated to hold between infinite descent and transfinite induction in the foundations of arithmetic uses the law of excluded middle through the use of a double negation on the infinite set of natural numbers and therefore cannot be admitted in intuitionistic logic and mathematics, and a fortiori in more radical constructivist foundational schemes. Moreover it is shown that the infinite descent used in Dedekind-Peano arithmetic does (...) not correspond to the infinite descent of classical Fermatian arithmetic or number theory. However, from the point of view of classical logic, the principles of complete induction and transfinite induction, the least number principle and infinite descent are all equivalent. We find here a focal point for a foundational critique that aims for a a clarification of philosophical options in the foundations of logic and mathematics. (shrink)

In this paper is presented and discussed the notion of true finite by opposition to the notion of theoretical finite. Examples from mathematics and physics are given. Fermat’s infinite descent principle is challenged.

What Gödel referred to as “outer” consistency is contrasted with the “inner” consistency of arithmetic from a constructivist point of view. In the settheoretic setting of Peano arithmetic, the diagonal procedure leads out of the realm of natural numbers. It is shown that Hilbert’s programme of arithmetization points rather to an “internalisation” of consistency. The programme was continued by Herbrand, Gödel and Tarski. Tarski’s method of quantifier elimination and Gödel’s Dialectica interpretation are part and parcel of Hilbert’s finitist ideal which (...) is achieved by going back to Kronecker’s programme of a general arithmetic of forms or homogeneous polynomials. The paper can be seen as a historical complement to our result on “The Internal Consistency of Arithmetic with Infinite Descent” . An internal consistency proof for arithmetic means that transfinite induction is not needed and that arithmetic can be shown to be consistent within the bounds of arithmetic, that is with the help of Fermat’s infinite descent and Kronecker’s general or polynomial arithmetic, thus returning into arithmetic without the detour of Cantor’s transfinite arithmetic of ideal elements. (shrink)

In this chapter, a generic definition of fundamentality as an ontological minimality thesis is sought and its applicability examined. Most discussions of fundamentality are focused on a mereological understanding of the hierarchical structure of reality, which may be combined with an atomistic, object-oriented metaphysics. But recent work in structuralism, for instance, calls for an alternative understanding and it is not immediately clear that the conception of fundamentality at work in structuralism is commensurable with the mereological conception. However, it is proposed (...) that once we understand fundamentality as an ontological minimality thesis, these two as well as further conceptions of fundamentality can all be treated on a par, including metaphysical infinitism of the ‘boring’ type, where the same structure repeats infinitely. (shrink)

In this paper, I provide an argument for pannormism, the view according to which there are normative properties all the way down. In particular, I argue for what I call the trickling down principle, which says that if there is a metaphysically basic normative property, then, if whatever instantiates it has a ground, that ground instantiates it as well.

Internal logic is the logic of content. The content is here arithmetic and the emphasis is on a constructive logic of arithmetic (arithmetical logic). Kronecker's general arithmetic of forms (polynomials) together with Fermat's infinite descent is put to use in an internal consistency proof. The view is developed in the context of a radical arithmetization of mathematics and logic and covers the many-faceted heritage of Kronecker's work, which includes not only Hilbert, but also Frege, Cantor, Dedekind, Husserl and (...) Brouwer. The book will be of primary interest to logicians, philosophers and mathematicians interested in the foundations of mathematics and the philosophical implications of constructivist mathematics. It may also be of interest to historians, since it covers a fifty-year period, from 1880 to 1930, which has been crucial in the foundational debates and their repercussions on the contemporary scene. (shrink)

A proof of Fermat’s last theorem is demonstrated. It is very brief, simple, elementary, and absolutely arithmetical. The necessary premises for the proof are only: the three definitive properties of the relation of equality (identity, symmetry, and transitivity), modus tollens, axiom of induction, the proof of Fermat’s last theorem in the case of n = 3 as well as the premises necessary for the formulation of the theorem itself. It involves a modification of Fermat’s approach of infinite descent. (...) The infinite descent is linked to induction starting from n = 3 by modus tollens. An inductive series of modus tollens is constructed. The proof of the series by induction is equivalent to Fermat’s last theorem. As far as Fermat had been proved the theorem for n = 4, one can suggest that the proof for n ≥ 4 was accessible to him. An idea for an elementary arithmetical proof of Fermat’s last theorem (FLT) by induction is suggested. It would be accessible to Fermat unlike Wiles’s proof (1995). (shrink)

Handbook of Mathematical Induction: Theory and Applications shows how to find and write proofs via mathematical induction. This comprehensive book covers the theory, the structure of the written proof, all standard exercises, and hundreds of application examples from nearly every area of mathematics. In the first part of the book, the author discusses different inductive techniques, including well-ordered sets, basic mathematical induction, strong induction, double induction, infinite descent, downward induction, and several variants. He then introduces ordinals and cardinals, (...) transfinite induction, the axiom of choice, Zorn’s lemma, empirical induction, and fallacies and induction. He also explains how to write inductive proofs. The next part contains more than 750 exercises that highlight the levels of difficulty of an inductive proof, the variety of inductive techniques available, and the scope of results provable by mathematical induction. Each self-contained chapter in this section includes the necessary definitions, theory, and notation and covers a range of theorems and problems, from fundamental to very specialized. The final part presents either solutions or hints to the exercises. Slightly longer than what is found in most texts, these solutions provide complete details for every step of the problem-solving process. (shrink)

Book review of 'The Universe As We Find It' (2012, OUP). By John Heil.

In lieu of an abstract, here is a brief excerpt of the content:Five Poems AMIT MAJMUDAR Observing Orpheus I hear the meaning turn back in his throat like Eurydice on the way up from the darkness. Music’s meaning is its making. As for me, I am one more animal in his entourage, learning a new thirst, finding a new south. None of us knew we had this instinct in us. If deserts hide wildflowers until first rain, bright ears are blossoming (...) out of our skulls. He doesn’t have much longer. I know this myth. A God douses the fire with a beehive. The Maenads smear their faces with a warpaint made of stoneground fireflies and pine sap. Hands—hands like his, that drew music through lyre strings into our forest like pieces of bread through a prison fence— are reaching for his body now, his lyre. Their weapons are hands, nothing but hands. They are infinite. They are enough. arion 27.1 spring/summer 2019 Her Metamorphoses First a beetle, body armor forged of chitin, exoskeleton he’ll never bruise, Undiscovered, unapologetically hermetic on her own Galapagos. Next a fruit bat, echolocating tangerines like a blind woman singing up at stars. Then a patio lizard, sunning herself for as long as she needs, A better place for keeping warm than bed with him. Next a sleek water moccasin, quick to make out another woman’s heat signature on his neck, Even quicker to strike. A caterpillar to poke holes in his lying roses And from that destruction to fashion two beautiful wings for herself. A buzzard, to clean up the roadside pile he left of her. A woman in full with no more soft blind pupa stages of her life cycle left, No more forms to force herself to fit for the sake of his pleasure, No more shapeshifts, no more slimming stripes, no more abdominal crunches: Just this body, just this last name, just this apartment, just this bowl of midnight cereal. Just her. 106 five poems Sisyphus Shove hard, shove, shove hard, eyes bloodshot, knees bent Till your boulder goes weightless and races ahead in descent. Force its dark door open inch by grim inch, Shove hard, shove, shove hard, eyes bloodshot, knees bent, Heels dug deep in hell’s steep hillside, teeth clenched. What gravity strips from you, gravity grants you again. Shove hard, shove, shove hard, eyes bloodshot, knees bent Till your body goes weightless and races ahead in ascent. Amit Majmudar 107 A Trinket for Persephone A deathwatch beetle burrows in the pale hall of your marrow, your underground, your tube of bone on which you’re riding home alone. This blackwinged locket with your soul emerges rattling from a hole, a rattle in your emptied throat, the diva’s final, graceless notes, a clicking echoed in the rafters, this bugbear insect coming after to scuttle across the marshy chest, sinkhole navel, and pubic nest out to the yard, below the porch untrackable by any torch until this bug with you inside has burrowed to the other side. It drops into the underworld, as if from some cocoon unfurled, to scale a queenly wrist and rest your brooch of black upon her breast. 108 five poems Four Ways of Looking at Argus 1. His Vigil These hundred eyes, these hundred balloons Tied to the boy-small wrist of my mind Are down to ninety-nine. I lost one to a bee sting late last June— I never thought of them as fragile Until that pinprick broke my vigil. I’m slowing down. I didn’t blink in time. I have no inkling why I keep This vigil—surely not to watch, Panoptic from this mountaintop, A heifer drifting off to sleep? I’m here, I must be here, to see A signal fire from the east That warns of Olympus under siege And Ares gory from a rout, Athena’s owl’s eyes gouged out, The naiads naked in a pen, The whole world given up to men. The Gods will need a monster then. 2. After Martial Hermes was in a fix: He needed lullabies To shutter Argus’s one hundred eyes And get the tethered heifer—coal-eyed Io—free Of Hera’s... (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Eclipsing Art: Method and Metaphysics in Coleridge’s Biographia Literaria *Tim MilnesColeridge’s PredicamentIn his self-addressed “letter” which precipitates the abrupt end to the thirteenth chapter (and with it, the first volume) of the Biographia Literaria, Coleridge likens the current state of his argument to “the fragments of the winding steps of an old ruined tower.” 1 The suggestion of intellectual ascent in this is revealing and is echoed a few (...) years later by his explanation of the pedagogical function of the “Landing-Places” between the essays in The Friend (1818). 2 The underlying idea, as usual with Coleridge, is akin to that of an exercise in intellectual and spiritual mountaineering. The thinker, having established a safe base camp in fixed first principles, gradually ascends to higher truths, which are at the same time taken to be foundational and which form the ground of the whole process. The logical return here is what Coleridge elsewhere explains as the “seeming argumentum in circulo, incident to all spiritual Truths.” 3 In the Biographia, however, this process of simultaneous levitation and descent remains notoriously incomplete, left over for the projected but never completed “great book on the constructive philosophy.” 4 Thus, the procedure initiated in the philosophical [End Page 125] theses of chapter twelve having been arrested, the project of chapter thirteen, “in which the results [of the theses] will be applied to the deduction of the imagination, and with it the principles of production and genial criticism in the fine arts,” 5 is left as a fragmentary exposition in the hugely influential but attenuated and gnomic sequence of distinctions between primary and secondary imagination, and fancy.Consequently, students of Coleridge have been left to puzzle over the “missing” argument for themselves and to explain why Coleridge seems to give up at this point. At least since Thomas McFarland’s Coleridge and the Pantheist Tradition it has been accepted that a significant part of the problem lies with the “counter-pull” 6 exerted upon Coleridge’s thought at this point in time between Kant and Schelling (or, as McFarland sees it, between Kant and Schelling / Spinoza). In other words, it is the pull between a system in which free will is preserved at the price of being declared noumenal and one in which free will is imperilled but set against the goal of a system of total and undivided philosophy at ease with the infinite. The difference between the methodologies of Kant and Schelling, as Gian Orsini noted, is the product of their contrasting first postulates, or philosophical starting-points. While Kant sets out from the epistemological problem of to what extent objects can be said to conform to human knowledge, Schelling asks, “do we deduce Mind from Nature or Nature from Mind?” 7 Each of these presumes a completely different kind of enquiry from the other, and thus a different method; a fact Coleridge appears to have fully appreciated only during the composition of the Biographia.The Biographia’s problems have provoked a diverse range of critical responses over the years. 8 Among these, however, there is a more or less settled [End Page 126] opinion that Coleridge’s failure at this stage to come to terms with the Kant / Fichte / Schelling debate merely added to the difficulty of a work which, having begun its life as a reply to and a rebuttal of Wordsworth’s empirical and associative definition of imagination as set out in the Preface to his Poems (1815), had already far exceeded its initial plan both in scope and size. As the statement of his poetic principles and critique of Wordsworth had turned into a literary life, and this into a broader exploration of his developing views on epistemology, religion, and the history of language, so the “Preface” to the Sibylline Leaves (as it had originally been conceived) now dwarfed that work. Coleridge’s printer’s decision to produce the Biographia in two volumes only served to emphasize the break between its “philosophical” and literary-critical planes and the failure of Coleridge’s declared intention to apply the rules of art, “deduced from philosophical principles, to poetry and criticism.” 9Yet there was always something... (shrink)

"The goal of this volume is to revise Hans Urs von Balthasar's theology of divine suffering, that is, his disputed discourse on the descent of Christ into hell and its implications for the Triune God, according to a robust contemporary Catholic theology. In order to accomplish such an appropriation, I have recourse not only to twentieth-century Thomistic theology, but also to the thought of Pope Emeritus Benedict XVI and Pope St. John Paul II. I seek to engage the best (...) of the vast relevant secondary literature on Balthasar and to offer a balanced assessment of his work on the topic of divine suffering, both critical and appreciative in different respects. I argue for a peculiar interpretation of Balthasar's take on Holy Saturday and for a more refined use of language in theological discourse than is typical of his sometimes-hyperbolic style. Recognizing his laudable attempt to integrate mystical spirituality and systematic theology, I seek to distinguish valid insights from confused mixtures of metaphorical, meta-symbolic, and philosophical discourse on God, particularly, with respect to the classical problem of how the Creator who willed to become incarnate may be said to suffer. Truly, "One of the Trinity has suffered," and yet this is mystery of faith that ought to be carefully explained and understood in conformity with sustained Catholic reflection on divine immutability and simplicity, the dual nature and unique personhood of Christ, the Trinity of divine subsistent relations, the freedom of God in creating and becoming man, the analogy of being, the problem of evil, and the immensity and infinite value of Christ's redemptive suffering"--. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:The Petrification of Cleopatra in Nineteenth Century Art MARGARET MALAMUD MARTHA MALAMUD What did Cleopatra look like? Was she a Roman, a Ptolemaic Greek, an Egyptian, an African? Was she a precocious child, a devastatingly beautiful seductress, an astute practitioner of imperial politics, a murderess, a longnosed blue-stocking? [Figure 1] Cleopatra is dead, but “Cleopatra ” exists in the eye of the beholder. What other human being has been (...) portrayed in such “infinite variety”? In her life, and even more in her afterlife, the protean queen of the Nile has, like a shaft of sunlight focused through a magnifying glass, clarified and set ablaze underlying ideologies and overt assertions of race and gender, manifested differently in disparate times and places. Here we consider two case studies of nineteenth century representations of Cleopatra in stone: William Wetmore Story’s Cleopatra (1860) and Edmonia Lewis’s The Death of Cleopatra (1876). [Figarion 28.1 spring/summer 2020 Figure 1: Bust of Cleopatra VII, 40–30 bce, Altes Museum, Berlin. 32 the petrification of cleopatra ure 2] Story, a wealthy and well-connected white expatriate living in Rome, gave the famous queen features that deviated from classicizing portrayals of the queen and struck critics as startling and original. American artist Edmonia Lewis, of Black West Indies / Ojibwa Native American descent, lived and worked in Rome, but as a poor, racially-mixed female artist she moved in quite different social circles than Story. Her decision to sculpt the Egyptian queen as a white woman Figure 2: William Wetmore Story, Cleopatra, 1858; this carving 1869, Metropolitan Museum of Art, New York. Margaret Malamud, Martha Malamud 33 dying in agony during the last moments of her suicide shocked and fascinated critics. Both works display a very different understanding of women, the Orient, the Classical past, and the exotic Black other. We understand and contextualize these strikingly different representations of Cleopatra by looking at the comments of contemporary writers and critics. Lewis and her Death of Cleopatra inspired the twenty-first century Black poet Tyehimba Jess, with whose poetic evocation of Lewis and her work we conclude. In the middle of the nineteenth century, when most portrayals were of Cleopatra as a classical Greek beauty, William Wetmore Story boldly gave the famous queen African features. Story, an American emigré artist and “cultural influencer,” was living in Rome on the eve of the American Civil War. While there, he sculpted Cleopatra, which he began in 1858, completed in 1860, and exhibited to great acclaim at the 1862 World Exposition in London. Story was from Boston, where abolitionist sentiments were prominent, and was well aware of racial politics in the United States. Concerned by British support for the American South, he wrote a series of letters from Rome to the British Daily News in December of 1861 in which he denounced the institution of slavery, explained why he was devoted to the Union, and urged British neutrality in the Civil War. His Cleopatra was a provocative abolitionist statement. From the late eighteenth century on, free African Americans faced two pressing tasks: to refute charges that they were racially inferior to white people and to insert themselves into the historical record. ManyAfrican Americans and their supporters argued that there was a clear connection between ancient Africa (Ethiopia and Egypt) and the classical world based on their reading of classical texts and late eighteenth and early nineteenth century world histories. Educated African Americans documented the debt of Greece and Rome to Egyptian civilization, and repeatedly pointed out that Egypt was a major source of the cultural achievements of Greco-Roman an- 34 the petrification of cleopatra tiquity.It was conventional wisdom that Egyptian civilization influenced Greece. What was new and controversial was the assertion of a racial connection between ancient Egyptians and modern African Americans. Abolitionists used this constructed racial history to argue that the magnificent civilization of ancient Egypt and its influences on Greek and Roman culture were proof that African Americans were not racially inferior and therefore fully worthy to be citizens with equal rights in the American Republic.1 By the 1860s it had become a commonplace in American abolitionist circles that antiquity ’s most famous queen was... (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Reviewed by:Deep Mysteries: God, Christ, and Ourselves by Aidan NicholsGerard T. MundyDeep Mysteries: God, Christ, and Ourselves by Aidan Nichols, O.P. (Lanham, MD: Lexington, 2020), vii + 133 pp.Basic Catholic teaching declares that God's will must be trusted and that perfect knowledge of all that is resides in the Creator. An implication of this claim is that all of God's work within time and history—in man's linearly conception of (...) time and history—is meaningful. For, as perfect order himself God's work is neither random nor coincidental. God's perfect understanding means that he is able, colloquially speaking, "to connect the dots" perfectly. Thus, the historical particulars of the New Testament events are not random and coincidental, but they connect perfectly within God's plan.In terms of biblical events, then, what are the significances of the time periods during which, and places at which, they occurred? "Why was Jesus born on a particular date, at a particular time, and in a particular manger in Bethlehem? Why was the womb of Mary, Mater Dei, chosen at that exact time, and at that exact place? Why did Jesus stand on that particular mountain and not upon another? There are perfect answers to all of those questions and more, for, as Proverbs declares in this vein: "The LORD has made everything for a purpose" (16:4).In Deep Mysteries: God, Christ, and Ourselves, Father Aidan Nichols, O.P., meditates on questions related to the potential reasons for some of the historical particulars of "the principal Christological mysteries." Fr. Nichols defines the mysteries as these: "the Annunciation, the Nativity, the Baptism, the Transfiguration, the Passion and Death, the Descent into Hell, the Resurrection, the Ascension, Pentecost, and the Second Coming." (41).Attempting to understand how all of the parts of the New Testament fit together perfectly in an historical sense—as God's mind would see it—is a fascinating intellectual endeavor. The primary difference between the innocent inquirer and the vain one, however, is that the former understands that God's reason is infinite and that only the First Mover has the ability to see and to put together all particulars. Thomas Aquinas elucidates the essence of the endeavor to connect the particulars when, in reference to man's ability to understand "the truth of faith," he argues: "Yet it is useful for the human reason to exercise itself in such arguments, however weak they may be, provided only that there be present no presumption to comprehend or to demonstrate. For to be able to see something of the loftiest realities, however thin and weak the sight may be, is... a cause of the greatest joy" [Summa contra gentiles I, ch. 8).Posing the question of what the particulars of history might mean brings immediately another question. As Fr. Nichols asks: "How can the Eternal [End Page 386] and time be conjoined in a phrase, and the truth of such conjoining made credible?" (29). Indeed, if God transcends the man-conceived conception of time, how can man even understand God's movements if man can see time only in linear fashion? Fr. Nichols contends that Christ's "mystery-events are not just historical occurrences that, like all such events, belong to the human past" (31). Rather, argues Fr. Nichols, the mysteries "are permanently able to affect the intended beneficiary of the mysteries: namely, the human race" (31). In other words, contends Fr. Nichols, the mysteries are past, present, and future, in the human conception of time.Attempting "to connect the dots" in Christological historical particulars is a subjective endeavor that requires a great deal of individual deciphering, with reliance especially on the interpretation of perceived symbolism. Fr. Nichols's broad theme-based meditations can provide starting points for further study on more concentrated topics.The most exceptional part of the book is Fr. Nichols's discussion on how Jesus's mysteries exist, simultaneously, in the past, in the present, and in the future. This point is of great significance for both theology and philosophy. For, until the Second Coming, God, not being tied to man's limited experience of time, simultaneously, and... (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Journal of the History of Philosophy 41.3 (2003) 417-418 [Access article in PDF] Heidi M. Ravven and Lenn E. Goodman, editors. Jewish Themes in Spinoza's Philosophy. Albany: The State University of New York Press, 2002. Pp. ix + 290. Cloth, $78.50. Paper, $26.95.The current anthology presents an important contribution to the study of Spinoza's relation to Jewish philosophy as well as to contemporary scholarship of Spinoza's metaphysics and political (...) theory.In the opening essay, Lenn Goodman takes upon himself the ambitious task of evaluating Spinoza's positions as to several central disputes throughout the history of philosophy. In this rich and extensive essay Goodman argues that Spinoza's radical rationalism makes him pursue syntheses between traditionally opposed poles. Lee Rice's article, "Love of God in Spinoza," carefully analyzes Spinoza's concept of love and suggests the existence of three kinds of love parallel to the three kinds of knowledge. Warren Montag addresses the unresolved issue of Spinoza's relation to the Kabbalah. Montag sides with Deleuze against the association of Spinozism with the Neo-Platonic theory of emanation, arguing that unlike the Neo-Platonists and Kabbalists, Spinoza's view of God rules out any hierarchy, and does not assume a descent from primal simplicity into complexity. Edwin Curley contributes a beautiful reading of the story of Job and of Maimonides' interpretation of the story. Curley follows Maimonides' discussion of the various positions regarding the question of divine providence and his attempt to identify the speeches of the characters of the book of Job with each of these positions. Following a sensitive consideration of various attempts to [End Page 417] recover Maimonides' own rather concealed view about providence, Curley concludes that "[W]hatever you think Maimonides' final word on Job is, our examination of him certainly supports [Spinoza's] claim that [Maimonides] reads Scripture in terms of Platonic or Aristotelian speculations" (170).Warren Zev Harvey's essay addresses Spinoza's most neglected work, the Compendium of the Grammar of the Hebrew Language. Apart from providing an outline of Spinoza's work as a Hebrew grammarian ("a noun-intoxicated grammarian") and pointing out some of Spinoza's more eccentric views about the nature of the Hebrew language, Harvey makes a compelling case for the importance of the book for the study of Spinoza's metaphysics. He points out a fascinating analogy which Spinoza draws between the parts of speech: (proper) noun, adjectives, and participles, and Spinoza's key metaphysical terms: substance, attributes, and modes. (Think about the weighty implications of this analogy for Curley's interpretation of the substance-mode relation.) According to Harvey, Spinoza's claim that all Hebrew words are derived from nouns is a linguistic parallel to Spinoza's pivotal metaphysical doctrine which makes all things be in the substance (or God). Kenneth Seeskin's article discusses the views of Maimonides and Spinoza on the question of creation and God's relation to the world. Seeskin reconstructs an interesting critical dialogue between these two philosophers. According to Seeskin, both philosophers held that an effect must be similar to its cause. Maimonides' commitment to Negative Theology and to the denial of any common measure between God and finite things forced him to deny causal or explanatory relation between God and the world. What Maimonides can suggest instead is rather to prove that the emergence of the world ex nihilo and according to the divine will, is possible, and even likely given certain astronomical facts of medieval science. Thus, Maimonides would hold that "although we have grounds for believing that creation occurred, we will never be in a position to say how" (120). For Spinoza, such a view constitutes nothing but a "sanctuary of ignorance." Thus, according to Seeskin, Spinoza affirms a clear causal relation between God and the world of finite things insofar as God is said to explain the world. While I tend to agree with most aspects of this analysis, it is important to note that Spinoza also claims that "between the finite and infinite there is no relation" (Letter 54). This... (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)

Benardete paradoxes involve a beginningless set each member of which satisfies some predicate just in case no earlier member satisfies it. Such paradoxes have been wielded on behalf of arguments for the impossibility of an infinite past. These arguments often deploy patchwork principles in support of their key linking premise. Here I argue that patchwork principles fail to justify this key premise.

Traditional act utilitarianism judges an action permissible just in case it produces as much aggregate utility as any alternative. It is often supposed that utilitarianism faces a serious problem if the future is infinitely long. For in that case, actions may produce an infinite amount of utility. And if that is so for most actions, then utilitarianism, it appears, loses most of its power to discriminate among actions. For, if most actions produce an infinite amount of utility, then (...) few actions produce non-maximal utility, and so most actions are permissible.1 I will argue that potentially infinite futures create no major problems for utilitarianism. Utilitarianism has, I argue, the resources to distinguish among actions all of which produce infinite amounts of utility -- judging some permissible and some impermissible. For brevity of expression I will focus on act utilitarianism, but all the points apply equally well to many other traditional forms of utilitarianism. (shrink)

By “Brentanian inner consciousness” I mean the conception of inner consciousness developed by Franz Brentano. The aim of this paper is threefold: first, to present Brentano’s account of inner consciousness; second, to discuss this account in light of the mereology outlined by Brentano himself; and third, to decide whether this account incurs an infinite regress. In this regard, I distinguish two kinds of infinite regress: external infinite regress and internal infinite regress. I contend that the most (...) plausible reading of Brentano’s account is the so-called fusion thesis, and I argue that internal infinite regress turns out to be inherent to Brentanian inner consciousness. (shrink)

Throughout history, almost all mathematicians, physicists and philosophers have been of the opinion that space and time are infinitely divisible. That is, it is usually believed that space and time do not consist of atoms, but that any piece of space and time of non-zero size, however small, can itself be divided into still smaller parts. This assumption is included in geometry, as in Euclid, and also in the Euclidean and non- Euclidean geometries used in modern physics. Of the few (...) who have denied that space and time are infinitely divisible, the most notable are the ancient atomists, and Berkeley and Hume. All of these assert not only that space and time might be atomic, but that they must be. Infinite divisibility is, they say, impossible on purely conceptual grounds. (shrink)

In the Transcendental Ideal Kant discusses the principle of complete determination: for every object and every predicate A, the object is either determinately A or not-A. He claims this principle is synthetic, but it appears to follow from the principle of excluded middle, which is analytic. He also makes a puzzling claim in support of its syntheticity: that it represents individual objects as deriving their possibility from the whole of possibility. This raises a puzzle about why Kant regarded it as (...) synthetic, and what his explanatory claim means. I argue that the principle of complete determination does not follow from the principle of excluded middle because the externally negated or ?negative? judgement ?Not (S is P)? does not entail the internally negated or ?infinite? judgement ?S is not-P.? Kant's puzzling explanatory claim means that empirical objects are determined by the content of the totality of experience. This entails that empirical objects are completely determinate if and only if the totality of experience has a completely determinate content. I argue that it is not a priori whether experience has such a completely determinate content and thus not analytic that objects obey the principle of complete determination. (shrink)

An infinite lottery machine is used as a foil for testing the reach of inductive inference, since inferences concerning it require novel extensions of probability. Its use is defensible if there is some sense in which the lottery is physically possible, even if exotic physics is needed. I argue that exotic physics is needed and describe several proposals that fail and at least one that succeeds well enough.

In this article, we evaluate various responses to a noteworthy objection, namely, the infinite God objection to the kalām cosmological argument. As regards this objection, the proponents of the kalām argument face a dilemma—either an actual infinite cannot exist or God cannot be infinite. More precisely, this objection claims that God’s omniscience entails the existence of an actual infinite with God knowing an actually infinite number of future events or abstract objects, such as mathematical truths. (...) We argue, however, that the infinite God objection is based on two questionable assumptions, namely, that it is possible for an omniscient being to know an actually infinite number of things and that there exist an actually infinite number of abstract objects for God to know. (shrink)

In this work we propose a labelled tableau method for ukasiewicz infinite-valued logic L . The method is based on the Kripke semantics of this logic developed by Urquhart [25] and Scott [24]. On the one hand, our method falls under the general paradigm of labelled deduction [8] and it is rather close to the tableau systems for sub-structural logics proposed in [4]. On the other hand, it provides a CoNP decision procedure for L validity by reducing the check (...) of branch closure to linear programming. (shrink)

The creation-of-matter hypothesis of the Bondi-Gold-Hoyle steady-state cosmology requires that in an infinite time to which the first transfinite number may be assigned the number of atoms of matter produced would be equal to the cardinal number of the set of mathematical points in the continuum. The existence of a set of finite atoms with that cardinal number is physically unacceptable. The argument for the production of a non-denumerable set of atoms, in infinite time, is given in terms (...) of a model which is shown to be isomorphic with the original Cantor "diagonal" proof for the existence of a non-denumerable infinity. An alternative model which meets the requirements of the steady-state theory is presented; in this model, the number of atoms is explicitly no greater than countably infinite, and remains countably infinite as long as the past time of the universe is restricted to the unlimited set of finite unit-time intervals. If the origin of the steady-state universe is taken as being within that infinite set, expressed by the negative natural numbers, the contradiction of an atom at every mathematical point does not arise. The contradiction does arise if the origin is not within the set of finite numbers, and accordingly there is a restriction as to which concept of infinite past may properly be maintained in the steady-state theory. (shrink)

Ramsey's device [2], for eliminating theoretical terms from a theory assumes that the theory has only finitely many axioms. Paul Berent [1], has recently suggested a modification of Ramsey's method that is designed to cover the infinite case. But, his modified method is quite unlike Ramsey's in that it involves an ascent from the original theory to a suitable metatheory. Thus, it is perhaps of some interest to note that there is an extension of Ramsey's method that covers the (...) infinite case and at the same time is much closer in spirit to his original suggestion. (shrink)

How are inferences to design affected when one makes the (plausible) assumption that the universe is spatially infinite? I will show that arguments for the existence of God based on the improbable development of life don’t go through. I will also show that the model of design inferences promulgated by William Dembski is flawed. My model for design inferences has the (desirable) consequence that there are circumstances where a seeming miracle can count as evidence for the existence of God, (...) even if one would expect that type of event to naturalistically occur in a spatially infinite universe. (shrink)

The following four theses all have some intuitive appeal: (I) There are valid norms. (II) A norm is valid only if justified by a valid norm. (III) Justification, on the class of norms, has an irreflexive proper ancestral. (IV) There is no infinite sequence of valid norms each of which is justified by its successor. However, at least one must be false, for (I)--(III) together entail the denial of (IV). There is thus a conflict between intuition and logical possibility. (...) This paper, after distinguishing various conceptions of a norm, of validity and of justification, argues for the following position. (I) is true. (II) is false for legislative justification and true for epistemic justification. (III) is true for legislative and false for epistemic justification. (IV) is true for legislative justification; for epistemic justification (IV) is true or false depending on the conception taken of a norm. Our intuition in favour of (II) must therefore be abandoned where justification is conceived legislatively. Our intuition in favour of (III) must be abandoned, and our intuition in favour of (IV) qualified, where justification is conceived epistemically. (shrink)

In this paper we shall give a characterization of D-algebras in terms of lattice ordered abelian groups. To make this paper self-contained we shall recall some notations from [4]. The symbols !; ^; _; serve as implication, conjunction, disjunction, and negation, respectively. By D we mean a set of connectives from the list above containing the implication connective !. By a D-formula we mean a formula built up in a usual way from an innite set of the propositional variables and (...) connectives from D. (shrink)

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

: It is possible and necessary to compare stretches of human life with other goods, such as the good of conserving resources for others. A minute of human life is not of infinite value; all else being equal, a minute of life is less valuable than 10 years of the same life. Nevertheless, this ability to evaluate human life does not necessarily lead to total commodification of human life.