Science is continually faced with describing that which is beyond. This book, through contributions from nine prominent scholars, tackles that challenge.

Spinoza’s bold claim that there exists only a single infinite substance entails that finite things pose a deep challenge: How can Spinoza account for their finitude and their plurality? Taking finite bodies as a test case for finite modes in general I articulate the necessary conditions for the existence of finite things. The key to my argument is the recognition that Spinoza’s account of finite bodies reflects both Cartesian and Hobbesian influences. This recognition leads to the surprising realization there must (...) be more to finite bodies than their finitude, a claim that goes well beyond the basic substance-monism claim, namely, that anything that is, is in God. This leads to the conclusion, which may seem paradoxical, that finite bodies have both an infinite as well as a finite aspect to them. Finite bodies, I argue, both actively partially determine all the other finite bodies, thereby partially causing their existence insofar as they are finite, as well as are determined by the totality of other bodies. I articulate precisely what this infinite aspect is and how it is distinct from the general substance-monism dictum. (shrink)

Written by a highly respected scholar of Thomas Aquinas's writings, this volume offers a comprehensive presentation of Aquinas's metaphysical thought. It is based on a thorough examination of his texts organized according to the philosophical order as he himself describes it rather than according to the theological order. -/- In the introduction and opening chapter, John F. Wippel examines Aquinas's view on the nature of metaphysics as a philosophical science and the relationship of its subject to divine being. Part One (...) is devoted to his metaphysical analysis of finite being. It considers his views on the problem of the One and the Many in the order of being, and includes his debt to Parmenides in formulating this problem and his application of analogy to finite being. Subsequent chapters are devoted to participation in being, the composition of essence and esse in finite beings, and his appeal to a kind of relative nonbeing in resolving the problem of the One and the Many. Part Two concentrates on Aquinas's views on the essential structure of finite being, and treats substance-accident composition and related issues, including, among others, the relationship between the soul and its powers and unicity of substantial form. It then considers his understanding of matter-form composition of corporeal beings and their individuation. Part Three explores Aquinas's philosophical discussion of divine being, his denial that God's existence is self-evident, and his presentation of arguments for the existence of God, first in earlier writings and then in the "Five Ways" of his Summa theologiae. A separate chapter is devoted to his views on quidditative and analogical knowledge of God. The concluding chapter revisits certain issues concerning finite being under the assumption that God's existence has now been established. -/- John F. Wippel, professor of philosophy at The Catholic University of America, was recently awarded the prestigious Aquinas Medal by the American Catholic Philosophical Association. In addition to numerous articles and papers, Wippel has coauthored or edited several other works, including Metaphysical Themes in Thomas Aquinas and The Metaphysical Thought of Godfrey of Fontaines, both published by CUA Press. (shrink)

When attempting to generalize recursion theory to admissible ordinals, it may seem as if all classical priority constructions can be lifted to any admissible ordinal satisfying a sufficiently strong fragment of the replacement scheme. We show, however, that this is not always the case. In fact, there are some constructions which make an essential use of the notion of finiteness which cannot be replaced by the generalized notion of $\alpha$-finiteness. As examples we discuss bothcodings of models of arithmetic into the (...) recursively enumerable degrees, and non-distributive lattice embeddings into these degrees. We show that if an admissible ordinal $\alpha$ is effectively close to $\omega$ (where this closeness can be measured by size or by cofinality) then such constructions maybe performed in the $\alpha$-r.e. degrees, but otherwise they fail. The results of these constructions can be expressed in the first-order language of partially ordered sets, and so these results also show that there are natu. (shrink)

The Bayes Blind Spot of a Bayesian Agent is the set of probability measures on a Boolean algebra that are absolutely continuous with respect to the background probability measure of a Bayesian Agent on the algebra and which the Bayesian Agent cannot learn by conditionalizing no matter what evidence he has about the elements in the Boolean algebra. It is shown that if the Boolean algebra is finite, then the Bayes Blind Spot is a very large set: it has the (...) same cardinality as the set of all probability measures ; it has the same measure as the measure of the set of all probability measures ; and is a ``fat'' set in topological sense in the set of all probability measures taken with its natural topology. (shrink)

If we are material beings living in a material world -- and all the scientific evidence suggests that we are -- then we must find existential meaning, if there is such a thing, in this physical world. We must cast our lot with the natural rather than the supernatural. Many Westerners with spiritual inclinations are attracted to Buddhism -- almost as a kind of moral-mental hygiene. But, as Owen Flanagan points out in The Bodhisattva's Brain, Buddhism is hardly naturalistic. In (...) _The Bodhisattva's Brain_, Flanagan argues that it is possible to discover in Buddhism a rich, empirically responsible philosophy that could point us to one path of human flourishing. Some claim that neuroscience is in the process of validating Buddhism empirically, but Flanagan's naturalized Buddhism does not reduce itself to a brain scan showing happiness patterns. "Buddhism naturalized," as Flanagan constructs it, offers instead a fully naturalistic and comprehensive philosophy, compatible with the rest of knowledge -- a way of conceiving of the human predicament, of thinking about meaning for finite material beings living in a material world. (shrink)

We show that the variety of residuated lattices is generated by its finite simple members, improving upon a finite model property result of Okada and Terui. The reasoning is a blend of proof-theoretic and algebraic arguments.

Finite Transcendence: Existential Exile and the Myth of Home introduces and situates “existential exile” as an experience of the fundamental finitude of human existence and demonstrates how a particular way of responding in faith may enable one to find home in exile. Using the literary and philosophical oeuvre of Albert Camus as a model, this book demonstrates the manner in which mythic literature can both present and engage the condition of exile toward its possible transcendence.

Hilary Putnam has argued that computational functionalism cannot serve as a foundation for the study of the mind, as every ordinary open physical system implements every finite-state automaton. I argue that Putnam's argument fails, but that it points out the need for a better understanding of the bridge between the theory of computation and the theory of physical systems: the relation of implementation. It also raises questions about the class of automata that can serve as a basis for understanding the (...) mind. I develop an account of implementation, linked to an appropriate class of automata, such that the requirement that a system implement a given automaton places a very strong constraint on the system. This clears the way for computation to play a central role in the analysis of mind. (shrink)

The aim here is to describe how to complete the constructive logicist program, in the author’s book Anti-Realism and Logic, of deriving all the Peano-Dedekind postulates for arithmetic within a theory of natural numbers that also accounts for their applicability in counting finite collections of objects. The axioms still to be derived are those for addition and multiplication. Frege did not derive them in a fully explicit, conceptually illuminating way. Nor has any neo-Fregean done so.

This book is a rich collection of philosophical essays radically interrogating key notions and preoccupations of the phenomenological tradition. While using Heidegger’s Being and Time as its permanent point of reference and dispute, this collection also confronts other important philosophers, such as Kant, Nietzsche, and Derrida. The projects of these pivotal thinkers of finitude are relentlessly pushed to their extreme, with respect both to their unexpected horizons and to their as yet unexplored analytical potential. A Finite Thinking shows that, paradoxically, (...) where the thought of finitude comes into its own it frees itself, not only to reaffirm a certain transformed and transformative presence, but also for a non-religious reconsideration and reaffirmation of certain theologemes, as well as of the body, heart, and love. This book shows the literary dimension of philosophical discourse, providing important enabling ideas for scholars of literature, cultural theory, and philosophy. (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.

In this paper, I pursue such a logical foundation for arithmetic in a variant of Zermelo set theory that has axioms of subset separation only for quantifier-free formulae, and according to which all sets are Dedekind finite. In section 2, I describe this variant theory, which I call ZFin0. And in section 3, I sketch foundations for arithmetic in ZFin0 and prove that certain foundational propositions that are theorems of the standard Zermelian foundation for arithmetic are independent of ZFin0.<br><br>An equivalent (...) theory of sets and an equivalent foundation for arithmetic was introduced by John Mayberry and developed by the current author in his doctoral thesis. In that thesis, the independence results mentioned above are proved using proof-theoretic methods. In this paper, I offer model-theoretic proofs of the central independence results using the technique of cumulation models, which was introduced by Steve Popham, a doctoral student of Mayberry<br>from the early 1980s. (shrink)

The Polish logician Roman Suszko has extensively pleaded in the 1970s for a restatement of the notion of many-valuedness. According to him, as he would often repeat, “there are but two logical values, true and false.” As a matter of fact, a result by W´ojcicki-Lindenbaum shows that any tarskian logic has a many-valued semantics, and results by Suszko-da Costa-Scott show that any many-valued semantics can be reduced to a two-valued one. So, why should one even consider using logics with more (...) than two values? Because, we argue, one has to decide how to deal with bivalence and settle down the tradeoff between logical 2-valuedness and truth-functionality, from a pragmatical standpoint. -/- This paper will illustrate the ups and downs of a two-valued reduction of logic. Suszko’s reductive result is quite non-constructive.We will exhibit here a way of effectively constructing the two-valued semantics of any logic that has a truth-functional finite-valued semantics and a sufficiently expressive language. From there, as we will indicate, one can easily go on to provide those logics with adequate canonical systems of sequents or tableaux. The algorithmic methods developed here can be generalized so as to apply to many non-finitely valued logics as well —or at least to those that admit of computable quasi tabular two-valued semantics, the so-called dyadic semantics. (shrink)

This paper, which in its subject matter goes back to works on strongly nite logics , is concerned with the following problems: Let Cn1; Cn2 be two strongly nite logics over the same propositional language. Is the supremum of Cn1 and Cn2 also a strongly nite operation? Is any nite matrix axiomatizable by a nite set of standard rules? The rst question can be found in [9] . The second conjec- ture was formulated by Wolfgang Rautenberg, but investigations into this (...) problem had been carried out earlier in works of many logicians . Moreover, Stephen Bloom [1] posed a conjecture stronger than that: the consequence de- termined by a nite matrix is nitely based, i.e. it is the consequence generated by a nite set of standard rules. This hypothesis was, however, disproved by Andrzej Wronski [10] . In the present paper it is shown that neither nor holds true. The negative answer to can be viewed as a generalization of the result given by Andrzej Wronski [10]. (shrink)

Forcing extensions yield models of ZFC in which a long sequence of club subsets of ω 1 has the following property: every subsequence of size ℵ 1 has a finite intersection.

Much of the philosophical discussion of explanations has centered around two broad conceptions of what sorts of ‘things’ explanations are – namely, the descriptive and ontic conceptions. Defenders of each argue that scientific psychology has at best little to contribute to the study of explanations. These anti-psychologistic arguments come in two main varieties, the metaphysical and the epistemic. Both varieties trace back to Hempel and recur in the more recent writings of prominent mechanists. The metaphysical arguments attempt to combat psychologism (...) about explanation by doubly dissociating explanations themselves from the commonly associated phenomenology of explanation. The epistemic arguments attempt to combat psychologism by doubly dissociating the good-making features of explanations from the CAPE. I focus on the metaphysical arguments here, for only they directly concern the nature of, as opposed to the evaluation conditions for, explanations. The metaphysical arguments are all shown to equivocate over ‘explanation’, and thus they are no threat to the proposal that sometimes explanations just are psychological events. Such arguments do, nonetheless, gesture towards something important – namely, that the CAPE is neither necessary nor sufficient for explanation in any sense. Similar considerations lead another latter-day mechanist to conclude that no phenomenology of any sort is essential for explanation. What all of these thinkers overlook, however, is that there is another important phenomenological correlate of explanation – namely, finding a happening intelligible or understanding how or why-possibly – and it may well be of the very essence of explanation, at least in the psychological sense. I propose a psychomechanistic theory of intelligibility that accounts for how we can, through finite means, make boundless, endlessly qualified inferences about the implications of our explanations. In closing I note that, although finding a happening intelligible does sometimes engender an unjustified confidence in correctness, we should not overlook how attaining this state does at least put us in a kind of epistemic base camp. I also hint at how intelligibility could turn out to be the ultimate source of the descriptive and ontic conceptions of explanation. (shrink)

We introduce the $\Sigma _1$ -definable universal finite sequence and prove that it exhibits the universal extension property amongst the countable models of set theory under end-extension. That is, the sequence is $\Sigma _1$ -definable and provably finite; the sequence is empty in transitive models; and if M is a countable model of set theory in which the sequence is s and t is any finite extension of s in this model, then there is an end-extension of M to a (...) model in which the sequence is t. Our proof method grows out of a new infinitary-logic-free proof of the Barwise extension theorem, by which any countable model of set theory is end-extended to a model of $V=L$ or indeed any theory true in a suitable submodel of the original model. The main theorem settles the modal logic of end-extensional potentialism, showing that the potentialist validities of the models of set theory under end-extensions are exactly the assertions of S4. Finally, we introduce the end-extensional maximality principle, which asserts that every possibly necessary sentence is already true, and show that every countable model extends to a model satisfying it. (shrink)

A far-reaching reinterpretation of Plato’s Timaeus and its engagement with time, eternity, body, and soul that in its original French edition profoundly influenced Derrida The Tomb of the Artisan God provides a radical rereading of Timaeus, Plato’s metaphysical text on time, eternity, and the relationship between soul and body. First published in French in 1995, the original edition of Serge Margel’s book included an extensive introductory essay by Jacques Derrida, who drew on Margel’s insights in developing his own concepts of (...) time, the promise, the world, and khōra. Now available in English with a new preface by Margel, this engagement with Platonic thought proceeds from two questions that span the history of philosophy: What is time? What is the body? Margel’s twinned interrogation centers around Plato’s concept of the demiurge (divine artisan or craftsman): its body, its anthropomorphic attributes, its productive capacities and regulatory functions in the ordering/organization/assembling of the world. He posits that this paradoxical figure is not merely a cosmological metaphor for the living body but also the site of its destruction, dissolution, and disappearance. Torn between the finite and the infinite, being and becoming, the concept of demiurge also poses metaphysical questions about time, time before time, and the end of time. The ontological status of the demiurge’s body, Margel argues, would become increasingly decisive in the history of philosophy, particularly in Christianity and the dogma of incarnation. (shrink)

Arrhenius’s impossibility theorems purport to demonstrate that no population axiology can satisfy each of a small number of intuitively compelling adequacy conditions. However, it has recently been pointed out that each theorem depends on a dubious assumption: Finite Fine-Grainedness. This assumption states that there exists a finite sequence of slight welfare differences between any two welfare levels. Denying Finite Fine-Grainedness makes room for a lexical population axiology which satisfies all of the compelling adequacy conditions in each theorem. Therefore, Arrhenius’s theorems (...) fail to prove that there is no satisfactory population axiology. In this paper, I argue that Arrhenius’s theorems can be repurposed. Since all of our population-affecting actions have a non-zero probability of bringing about more than one distinct population, it is population prospect axiologies that are of practical relevance, and amended versions of Arrhenius’s theorems demonstrate that there is no satisfactory population prospect axiology. These impossibility theorems do not depend on Finite Fine-Grainedness, so lexical views do not escape them. (shrink)

Kant's concept of time forms an integral part of his mature system of transcendental idealism. That system is a critical response to his predecessors' treatments of time and related issues. Hence, a proper assessment of Kant's understanding of time requires an elaboration of its distinctive historical and systematic matrix. The aim of the dissertation is to examine critically Kant's mature conception of time in light of both the historical factors that shaped it and the role it plays in Kant's doctrine (...) of transcendental idealism. ;The most important discussion shaping Kant's thought on time is the Leibniz-Clarke Correspondence . The relevance of the Correspondence to Kant's mature view of time has, however, been overlooked by many commentators. The reason for this neglect can be traced in part to a misinterpretation of the question underlying the first antinomy in Kant's Critique of Pure Reason . According to that misinterpretation, the first antinomy revolves around the question whether time itself is finite or infinite. The correction of this mistaken, but common interpretation paves the way for a main task of this dissertation, namely, to reconstruct with historical and systematic accuracy the relation between the issue in the Correspondence underlying the first antinomy and Kant's concept of time. ;The dissertation is divided into four chapters. Chapter One examines the particular issue discussed in the Correspondence which had the greatest impact on Kant's notion of time, namely, the question whether time is absolute, as Clarke and the Newtonians held, or whether it is relational, as Leibniz held. Chapter Two explores how the concept of time in Kant's "Inaugural Dissertation" is influenced by this dispute between Leibniz and Clarke. Chapter Three considers the concept of time and its role in Kant's fully developed transcendental idealism, as articulated in the Critique of Pure Reason. Particular attention is paid to the "Transcendental Aesthetic" and the first antinomy. Finally, Chapter Four critically examines a central assumption of Kant's concept of time that he shares with his predecessors: the infinity of time. It is argued that this assumption overlooks the metaphysical significance of the "moving" present. (shrink)

The kind of unknowability I will discuss concerns the count of certain natural finite sets of objects. Even the situation with regard to our present strong formal systems is rather unclear. One can just profitably focus on that, putting aside issues of general unknowability.

The philosophy of superdeterminism is based on a single scientific fact about the universe, namely that cause and effect in physics are not real. In 2020, accomplished Swedish theoretical physicist, Dr. Johan Hansson published a physics proof using Albert Einstein’s Theory of Special Relativity that our universe is superdeterministic meaning a predetermined static block universe without cause and effect in physics. In the absence of cause and effect in physics, there can be no actual energy in our universe but only (...) the illusion of energy. This is consistent with the zero energy universe theory, which says that the positive matter energy is exactly balanced by the negative gravitational energy, so that the total energy of the universe is zero. An infinite universe would have an infinite amount of positive matter energy exactly cancelling out an infinite amount of negative gravitational energy. However, mathematically infinity minus infinity is not equal to zero, but rather is undefined. Consequently, an infinite universe would be inconsistent with the perfect cancelling balance of positive matter energy and negative gravitational energy under the zero energy universe theory. As a result, only a finite universe is consistent with the zero energy universe theory and the philosophy of superdeterminism. (shrink)

Quasi-set theory Q is an alternative set-theory designed to deal mathematically with collections of indistinguishable objects. The intended interpretation for those objects is the indistinguishable particles of non-relativistic quantum mechanics, under one specific interpretation of that theory. The notion of cardinal of a collection in Q is treated by the concept of quasi-cardinal, which in the usual formulations of the theory is introduced as a primitive symbol, since the usual means of cardinal definition fail for collections of indistinguishable objects. In (...) a recent work, Domenech and Holik have proposed a definition of quasi-cardinality in Q. They claimed their definition of quasi-cardinal not only avoids the introduction of that notion as a primitive one, but also that it may be seen as a first step in the search for a version of Q that allows for a greater representative power. According to them, some physical systems can not be represented in the usual formulations of the theory, when the quasi-cardinal is considered as primitive. In this paper, we discuss their proposal and aims, and also, it is presented a modification from their definition we believe is simpler and more general. (shrink)

We investigate properties of universal numberings of finite families of d.c.e. sets. We show different cases of finite families of d.c.e. sets for which there is a universal numbering and for which there is not.

Hegel's Science of Logic makes the just not low claim to be an absolute, ultimate-grounded knowledge. This project, which could not be more ambitious, has no good press in our post-metaphysical age. However: That absolute knowledge absolutely cannot exist, cannot be claimed without self-contradiction. On the other hand, there can be no doubt about the fundamental finiteness of knowledge. But can absolute knowledge be finite knowledge? This leads to the problem of a self-explication of logic (in the sense of Hegel) (...) and further, as will be shown, to a new definition of the dialectical procedure. The stringency of which results from the fact that always exactly that implicit content is explicated that was generated by the preceding explication step itself and is thus concretely comprehensible. At the same time, a new implicit content is generated by this act of explication, which requires a new explication step, and so forth. In the dialectical procedure reinterpreted in this way, dialectical arguments are not beheld, guessed at or even surreptitiously obtained, but are methodically accountable. Thereby dialectics is understood as a self-explication of logic by logical means and thus as a proof of the possibility of ultimate-grounding in the form of absolute and nevertheless finite – and thus also fallible – knowledge. (shrink)

The neutrosophic automorphisms of a neutrosophic groups G (I) , denoted by Aut(G (I)) is a neu-trosophic group under the usual mapping composition. It is a permutation of G (I) which is also a neutrosophic homomorphism. Moreover, suppose that X1 = X(G (I)) is the neutrosophic group of inner neutrosophic auto-morphisms of a neutrosophic group G (I) and Xn the neutrosophic group of inner neutrosophic automorphisms of Xn-1. In this paper, we show that if any neutrosophic group of the sequence (...) G (I), X1, X2, … is the identity, then G (I) is nilpotent. (shrink)

We study model theory of actions of finite groups on substructures of a stable structure. We give an abstract description of existentially closed actions as above in terms of invariants and PAC structures. We show that if the corresponding PAC property is first order, then the theory of such actions has a model companion. Then, we analyze some particular theories of interest (mostly various theories of fields of positive characteristic) and show that in all the cases considered the PAC property (...) is first order. (shrink)

Take any putative ordinary object which is divisible into a finite number of small units and tolerant to the loss of one of them. We can remove these units one at a time, and since our object definitely doesn’t exist when there are zero units, and since we cannot pinpoint which removal brings about this destruction, the Sorites Puzzle threatens common sense. We can rescue ordinary objects from its grip, but since independently motivated linguistic explanations of vagueness depend on there (...) being multiple candidate contents of vague terms, these efforts succeed only if there are multiple candidates that can be meant by ordinary object terms. Thus, many more objects than common sense accepts have to exist. The 'arguments from vagueness' also offer soritical reasons for object proliferation. Problematising the categories composite object and persisting thing (rather than specific object concepts), these show that either there are none of these (except simples) or every candidate for making up one, does. The latter, less revisionary alternative is plenitudinism. I defend Modally Full Plenitude, because it is three-dimensionalist and non-reductivist about de re modality – a distinct persisting object in any region, and for every modal profile satisfied by the matter in that region. (shrink)

Given a sequence of sets An⊆{0,…,n−1}, the Furstenberg correspondence principle provides a shift-invariant measure on2N that encodes combinatorial information about infinitely many of the An's. Here it is shown that this process can be inverted, so that for any such measure, ergodic or not, there are finite sets whose combinatorial properties approximate it arbitarily well. The finite approximations are obtained from the measure by an explicit construction, with an explicit upper bound on how large n has to be to yield (...) a sufficiently good approximation. We draw conclusions for computable measure theory, and show, in particular, that given any computable shift-invariant measure on 2N, there is a computable element of 2N that is generic for the measure. We also consider a generalization of the correspondence principle to countable discrete amenable groups, and once again provide an effective inverse. (shrink)

§ I. A FINITE, RATIONAL BEING CAN NOT POSIT ITSELF WITHOUT ASCRIBING TO ITSELF A FREE CAUSALITY. PROOF. A. If a rational being is to posit itself as such, ...

Schutz’s references to literature and arts in his theoretical works are manifold. But literature and theory are both a certain kind of a finite province of meaning, that means they are not easily accessible from the paramount reality of everyday life. Now there is another kind of referring to literature: metaphorizing it. Using it, as may be said with Lakoff and Johnson, to understand and to experience one kind of thing in terms of another. Literally metapherein means “to carry over”. (...) Metaphorizing in this view is then a specific kind of border-crossing between different provinces of meaning. That poses two questions: 1. What means finiteness of those provinces of meaning, what kind of border crossings are possible? What is the ground for metaphorizing meaning? 2. Could this concept used for founding a theory of the constitution of the societal and of society, that overcomes the dichotomy of structure/agency? These questions will be answered with one example in view: Schutz’ report to Kaufmann of his first visit of Husserl describing his experience as feeling like Wilhelm Meister at the Society of the Tower. In a first step this metaphor is presented together with some crumbs of metaphor theory. In a second step these crumbs will be connected to Husserl’s concept of experience. After developing a short overview over Schutz’ “finite provinces of meaning,” the relation of experience, metaphors to the intersubjectivity of these provinces in their dependence from writing and printing is discussed. (shrink)

What are the assumptions and tasks hidden in contemporary calls to "overcome" the metaphysical tradition? Reflecting upon the internal contradictions of the notions of "tradition" and "finiteness," Dennis J. Schmidt offers novel insights into how philosophy must relate to its traditions if it is to retain a vital sense of the plurality of "edges" that constitute its finiteness. He does this through a close examination of issues found in the work of Hegel and Heidegger, two philosophers who made the ideas (...) of both tradition and finiteness the center of their concern.Schmidt begins by asking how Heidegger can claim to have destroyed metaphysics despite Hegel's claim to have perfected its possibilities. Systematically following the development of Heidegger's critique of Hegel, Schmidt generates a dialogue between them. The topic of that dialogue is the nature of finiteness as it is articulated in time, nothing, the dialectical and hermeneutical circles, and in the notions of experience, work, technology, history, and preSocratic thought.Beginning with Heidegger's critique of Hegel in Being and Time, Schmidt's strategy is to disclose the complexities of philosophical discourse about the finite by drawing out the proximities between Hegel and Heidegger. The dialogue that results presents novel portraits of both philosophers. It also reveals that Heidegger's early, unacknowledged failure to separate himself from the Hegelian dialectic is the motive behind many of the turns and decisions of his later career.In concluding, Schmidt offers an interpretation of the wider significance of the results of that dialogue, and connects his study to other contemporary discussions of postmodernism. He expands upon the idea of the plurality of edges opened by finiteness, arguing that philosophy only understands its own past and future once it recognizes the meaning of its own finiteness.Dennis J. Schmidt is Associate Professor of Philosophy at the State University of New York at Binghamton. The Ubiquity of the Finite is included in the series Studies in Contemporary German Social Thought, edited by Thomas McCarthy. (shrink)

In this paper, the properties of the super quantum measures are studied. Firstly, the products of Dirac measures are discussed; Secondly, based on the properties of Dirac measures, the structures of super quantum measures are characterized; At last, we prove that any super quantum measure can determine a unique diagonally positive strongly symmetric signed measure. This result verifies the conjecture which was proposed by Gudder.

This paper shows how the classical finite probability theory (with equiprobable outcomes) can be reinterpreted and recast as the quantum probability calculus of a pedagogical or toy model of quantum mechanics over sets (QM/sets). There have been several previous attempts to develop a quantum-like model with the base field of ℂ replaced by ℤ₂. Since there are no inner products on vector spaces over finite fields, the problem is to define the Dirac brackets and the probability calculus. The previous attempts (...) all required the brackets to take values in ℤ₂. But the usual QM brackets <ψ|ϕ> give the "overlap" between states ψ and ϕ, so for subsets S,T⊆U, the natural definition is <S|T>=|S∩T| (taking values in the natural numbers). This allows QM/sets to be developed with a full probability calculus that turns out to be a non-commutative extension of classical Laplace-Boole finite probability theory. The pedagogical model is illustrated by giving simple treatments of the indeterminacy principle, the double-slit experiment, Bell's Theorem, and identical particles in QM/Sets. A more technical appendix explains the mathematics behind carrying some vector space structures between QM over ℂ and QM/Sets over ℤ₂. (shrink)

In this paper a model for barrecursion is presented. It has as a novelty that it contains discontinuous functionals. The model is based on a concept called strong majorizability. This concept is a modification of Howard's majorizability notion; see [T, p. 456].

According to finite frequentism, the probability of an attribute A in a finite reference class B is the relative frequency of actual occurrences of A within B. I present fifteen arguments against this position.

The small sample properties of estimators and tests are frequently too complex to be useful or are unknown. Much econometric theory is therefore developed for very large or asymptotic samples where it is assumed that the behaviour of estimators and tests will adequately represent their properties in small samples. Refined asymptotic methods adopt an intermediate position by providing improved approximations to small sample behaviour using asymptotic expansions. Dedicated to the memory of Michael Magdalinos, whose work is a major contribution to (...) this area, this book contains chapters directly concerned with refined asymptotic methods. In addition, there are chapters focusing on new asymptotic results; the exploration through simulation of the small sample behaviour of estimators and tests in panel data models; and improvements in methodology. With contributions from leading econometricians, this collection will be essential reading for researchers and graduate students concerned with the use of asymptotic methods in econometric analysis. (shrink)

The essay compares Ricœur’s and Levinas’s conceptions of the constitution of conscience in the sense of German Gewissen. Beginning with Ricœur’s basic distinction between “identity-idem” and “identity-ipse” it shows the proper place of conscience in his conception. For Ricœur conscience is a finite category of otherness as related to the self, i.e. its most interior, intrinsic otherness. For Levinas, on the other hand, conscience – the other in the same – is an initiatory category, which is described in terms of (...) absolute passivity, persecution and substitution. In conclusion Ricœur´s critique of Levinas’ category of “the Other” is examined on the background of Levinas’ conception of subjectivity as a vocation for Good. (shrink)

Formal learning theory constitutes an attempt to describe and explain the phenomenon of learning, in particular of language acquisition. The considerations in this domain are also applicable in philosophy of science, where it can be interpreted as a description of the process of scientific inquiry. The theory focuses on various properties of the process of hypothesis change over time. Treating conjectures as informational states, we link the process of conjecture-change to epistemic update. We reconstruct and analyze the temporal aspect of (...) learning in the context of dynamic and temporal logics of epistemic change. We first introduce the basic formal notions of learning theory and basic epistemic logic. We provide a translation of the components of learning scenarios into the domain of epistemic logic. Then, we propose a characterization of finite identifiability in an epistemic temporal language. In the end we discuss consequences and possible extensions of our work. (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.

The Kansas-Nebraska Act of 1854 officially opened the Great Plains to westward settlement, and the public survey of 1855 by Charles A. Manners and Joseph Ledlie along the Sixth Principal Meridian established the grid by which the uncharted expanse of the Great Plains was brought into scale. The mechanical act performed by land surveyors is believed by photographer Joe Deal to be powerfully similar to the artistic act of making a photograph.To Deal, both acts are about establishing a frame around (...) a vast scene that suggests no definite boundaries of its own. Thus, when approaching his own photographs of the Great Plains, Deal viewed his photography as a form of reenactment, a method of understanding how it felt to contain the Great Plains in smaller, more measurable units. In West and West, Deal, who was born and raised in Kansas, revisited the Kansas-Nebraska territory and applied his photographic understanding of the landscape grid and horizon line to illuminate the sense of infinite space that transcends the reality of the survey. As Deal writes in his concluding essay: “If the square, as employed in the surveys of public lands, could function like a telescope, framing smaller and smaller sections of the plains down to a transect, it can also be used as a window, equilaterally divided by the horizon, that begins with a finite section of the earth and sky and restores them in the imagination to the vastness that now exists as an idea: the landscape that is contained within the perfect symmetry of the square implies infinity.” The stunning photographs in West and West present the Great Plains from a rare perspective. From this vantage point, Deal is able to distill and contemplate its expanse. (shrink)