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.

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.

Propositional logics in general, considered as a set of sentences, can be undecidable even if they have “nice” representations, e.g., are given by a calculus. Even decidable propositional logics can be computationally complex (e.g., already intuitionistic logic is PSPACE-complete). On the other hand, finite-valued logics are computationally relatively simple—at worst NP. Moreover, finite-valued semantics are simple, and general methods for theorem proving exist. This raises the question to what extent and under what circumstances propositional logics represented in various (...) ways can be approximated by finite-valued logics. It is shown that the minimal m-valued logic for which a given calculus is strongly sound can be calculated. It is also investigated under which conditions propositional logics can be characterized as the intersection of (effectively given) sequences of finite-valued logics. (shrink)

A general class of labeled sequent calculi is investigated, and necessary and sufficient conditions are given for when such a calculus is sound and complete for a finite -valued logic if the labels are interpreted as sets of truth values. Furthermore, it is shown that any finite -valued logic can be given an axiomatization by such a labeled calculus using arbitrary "systems of signs," i.e., of sets of truth values, as labels. The number of labels needed is logarithmic (...) in the number of truth values, and it is shown that this bound is tight. (shrink)

The proof theory of many-valued systems has not been investigated to an extent comparable to the work done on axiomatizatbility of many-valued logics. Proof theory requires appropriate formalisms, such as sequent calculus, natural deduction, and tableaux for classical (and intuitionistic) logic. One particular method for systematically obtaining calculi for all finite-valued logics was invented independently by several researchers, with slight variations in design and presentation. The main aim of this report is to develop the proof theory of finite-valued (...) first order logics in a general way, and to present some of the more important results in this area. In Systems covered are the resolution calculus, sequent calculus, tableaux, and natural deduction. This report is actually a template, from which all results can be specialized to particular logics. (shrink)

William James advocated a form of finite theism, motivated by epistemological and moral concerns with scholastic theism and pantheism. In this article, I elaborate James’s case for finite theism and his strategy for dealing with these concerns, which I dub the problems of suffering. I contend that James is at the very least implicitly aware that the problem of suffering is not so much one generic problem but a family of related problems. I argue that one of James’s (...) great contributions to philosophical theism is his advocacy for the view that adequate theistic philosophizing is not so much about cracking this family of problems, but finding a version of the problem to embrace. (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)

I raise three puzzles concerning self-deception: (i) a conceptual paradox, (ii) a dilemma about how to understand human cognitive evolution, and (iii) a tension between the fact of self-deception and Davidson’s interpretive view. I advance solutions to the first two and lay a groundwork for addressing the third. The capacity for self-deception, I argue, is a spandrel, in Gould’s and Lewontin’s sense, of other mental traits, i.e., a structural byproduct. The irony is that the mental traits of which self-deception is (...) a spandrel/byproduct are themselves rational. (shrink)

This paper focuses on order-preserving logics defined from varieties of distributive lattices with negation, and in particular on the problem of whether these can be axiomatized by means Hilbert-style calculi that are finite. On the negative side, we provide a syntactic condition on the equational presentation of a variety that entails failure of finite axiomatizability for the corresponding logic. An application of this result is that the logic of all distributive lattices with negation is not finitely axiomatizable; we (...) likewise establish that the order-preserving logic of the variety of all Ockham algebras is also not finitely axiomatizable. On the positive side, we show that an arbitrary subvariety of semi-De Morgan algebras is axiomatized by a finite number of equations if and only if the corresponding order-preserving logic is axiomatized by a finite Hilbert-style calculus. This equivalence also holds for every subvariety of a Berman variety of Ockham algebras. We obtain, as a corollary, a new proof that the implication-free fragment of intuitionistic logic is finitely axiomatizable, as well as a new corresponding Hilbert-style calculus. Our proofs are constructive in that they allow us to effectively convert an equational presentation of a variety of algebras into a Hilbert-style calculus for the corresponding order-preserving logic, and vice versa. We also consider the assertional logics associated to the above-mentioned varieties, showing in particular that the assertional logics of finitely axiomatizable subvarieties of semi-De Morgan algebras are finitely axiomatizable as well. (shrink)

This essay examines some of the ways in which the assumption of the essential finitude of the human mind, in contrast to the infinitude of God’s mind, bears on Leibniz’s and Kant’s accounts of our representational capacities. This examination reveals several underappreciated similarities between their views, but also some notable differences that help us pinpoint where and in what ways Kant departs from his celebrated predecessor. The fruits of this examination are a better understanding of Kant’s conception of the discursivity (...) of our understanding, his account of the difference between concepts and intuitions, and the particular flavor of his idealism. -/- . (shrink)

The article explores the idea that according to Spinoza finite thought and substantial thought represent reality in different ways. It challenges “acosmic” readings of Spinoza's metaphysics, put forth by readers like Hegel, according to which only an infinite, undifferentiated substance genuinely exists, and all representations of finite things are illusory. Such representations essentially involve negation with respect to a more general kind. The article shows that several common responses to the charge of acosmism fail. It then argues that (...) we must distinguish the well-founded ideality of representations of finite things from mere illusoriness, insofar as for Spinoza we can have true knowledge of what is known only abstractly. Finite things can be seen as well-founded beings of reason. The article also proposes that within Spinoza's framework it is possible to represent a finite thing without drawing on representations of mind-dependent entities. (shrink)

It is standard to think that in Spinoza’s system, all things are necessary and in no sense contingent. However, in his classic book, Spinoza’s Metaphysics, published in 1969, Edwin Curley argues based on the proposition 28 of the first part of the Ethics that Spinoza endorses necessitarianism of only a modest kind, according to which when it comes to finite modes, there is a sense in which they are contingent. In this paper, I revisit Curley’s argument. Commentators have responded (...) to Curley’s argument, showing that Spinoza’s remarks on infinite modes entail that finite modes can in no sense be contingent. But this alone falls short of dispelling Curley’s misgivings about the standard interpretation, for it remains unexplained why Curley is wrong in thinking that the proposition 28 supports his moderate necessitarian interpretation. In defense of the standard interpretation, I bolster the usual response to Curley in greater detail than has been done in the literature and explain why, pace Curley, the proposition 28 plays no evidential role in support of Curley’s interpretation. (shrink)

Feynman's sum-over-histories formulation of quantum mechanics has been considered a useful calculational tool in which virtual Feynman histories entering into a coherent quantum superposition cannot be individually measured. Here we show that sequential weak values, inferred by consecutive weak measurements of projectors, allow direct experimental probing of individual virtual Feynman histories, thereby revealing the exact nature of quantum interference of coherently superposed histories. Because the total sum of sequential weak values of multitime projection operators for a complete set of orthogonal (...) quantum histories is unity, complete sets of weak values could be interpreted in agreement with the standard quantum mechanical picture. We also elucidate the relationship between sequential weak values of quantum histories with different coarse graining in time and establish the incompatibility of weak values for nonorthogonal quantum histories in history Hilbert space. Bridging theory and experiment, the presented results may enhance our understanding of both weak values and quantum histories. (shrink)

We investigate risk attitudes when the underlying domain of payoffs is finite and the payoffs are, in general, not numerical. In such cases, the traditional notions of absolute risk attitudes, that are designed for convex domains of numerical payoffs, are not applicable. We introduce comparative notions of weak and strong risk attitudes that remain applicable. We examine how they are characterized within the rank-dependent utility model, thus including expected utility as a special case. In particular, we characterize strong comparative (...) risk aversion under rank-dependent utility. This is our main result. From this and other findings, we draw two novel conclusions. First, under expected utility, weak and strong comparative risk aversion are characterized by the same condition over finite domains. By contrast, such is not the case under non-expected utility. Second, under expected utility, weak comparative risk aversion is characterized by the same condition when the utility functions have finite range and when they have convex range. By contrast, such is not the case under non-expected utility. Thus, considering comparative risk aversion over finite domains leads to a better understanding of the divide between expected and non-expected utility, more generally, the structural properties of the main models of decision-making under risk. (shrink)

This article introduces the notion of duplex elements of the finite rings and corresponding neutrosophic rings. The authors establish duplex ring Dup(R) and neutrosophic duplex ring Dup(R)I)) by way of various illustrations. The tables of different duplicities are constructed to reveal the comparison between rings Dup(Zn), Dup(Dup(Zn)) and Dup(Dup(Dup(Zn ))) for the cyclic ring Zn . The proposed duplicity structures have several algebraic systems with dissimilar consequences. Author’s characterize finite rings with R + R is different from the (...) duplex ring Dup(R). However, this characterization supports that R + R = Dup(R) for some well known rings, namely zero rings and finite fields. (shrink)

This paper examines two objections to the infinitist theory of epistemic justification, namely “the finite mind objection” and “the distinction objection.” It criticizes Peter Klein’s response to the distinction objection and offers a more plausible response. It is then argued that this response is incompatible with Klein’s response to the finite mind objection. Infinitists, it would seem, cannot handle both objections when taken together.

EXPANDED EDITION (eBook): -/- Infinity Is Not What It Seems...Infinity is commonly assumed to be a logical concept, reliable for conducting mathematics, describing the Universe, and understanding the divine. Most of us are educated to take for granted that there exist infinite sets of numbers, that lines contain an infinite number of points, that space is infinite in expanse, that time has an infinite succession of events, that possibilities are infinite in quantity, and over half of the world’s population believes (...) in a divine Creator infinite in knowledge, power, and benevolence. -/- According to this treatise, such assumptions are mistaken. In reality, to be is to be finite. The implications of this assessment are profound: the Universe and even God must necessarily be finite. -/- The author makes a compelling case against infinity, refuting its most prominent advocates. Any defense of the infinite will find it challenging to answer the arguments laid out in this book. But regardless of the reader’s position, FOREVER FINITE offers plenty of thought-provoking material for anyone interested in the subject of infinity from the perspectives of philosophy, mathematics, science, and theology. -/- This electronic edition of FOREVER FINITE is offered free online for all and is expanded with content that does not appear in the published edition due to print production page limitations. From the Introduction to the Conclusion, Chapters 1–27 are identical to the published print edition, including the page numbering for the chapters. This electronic edition adds an appendix and associated content—specifically, updates to the book’s back matter (68 new endnotes, 17 new bibliographic references, 3 new figure credits, 14 new glossary terms) and front matter (an updated table of contents and this preface note). (shrink)

The ethical philosophy of Emmanuel Levinas is typically associated with a punishing conception of responsibility rather than freedom. In this chapter, our aim is to explore Levinas’s often overlooked theory of freedom. Specifically, we compare Levinas’s account of freedom to the Kantian (and Fichtean) idea of freedom as autonomy and the Hegelian idea of freedom as relational. Based on these comparisons, we suggest that Levinas offers a distinctive conception of freedom—“finite freedom.” In contrast to Kantian autonomy, finite freedom (...) constitutively involves standing in certain social relations with others. And in contrast to Hegelian relational freedom, the social relations involved in finite freedom are not defined by mutual recognition, but by feelings of separation and even antagonism. Along the way, we promote a reading Levinas’s Totality and Infinity as a Hegel-style story of Bildung (development), and show how, on Levinas’s view, freedom can develop or mature in the life of the individual. (shrink)

A concise exposition of the development of the true infinite is found in Hegel's Encyclopedia Logic (EL92-95). It may be much easier to follow than the one given in the Science of Logic. The following paragraphs are from the Gerates, et al translation of that book, along with some parts of the "Additions" where I felt they were useful. At the end I give my interpretation of the development.

In this paper we apply social epistemology to mathematical proofs and their role in mathematical knowledge. The most famous modern collaborative mathematical proof effort is the Classification of Finite Simple Groups. The history and sociology of this proof have been well-documented by Alma Steingart (2012), who highlights a number of surprising and unusual features of this collaborative endeavour that set it apart from smaller-scale pieces of mathematics. These features raise a number of interesting philosophical issues, but have received very (...) little attention. In this paper, we will consider the philosophical tensions that Steingart uncovers, and use them to argue that the best account of the epistemic status of the Classification Theorem will be essentially and ineliminably social. This forms part of the broader argument that in order to understand mathematical proofs, we must appreciate their social aspects. (shrink)

The problem of approximating a propositional calculus is to find many-valued logics which are sound for the calculus (i.e., all theorems of the calculus are tautologies) with as few tautologies as possible. This has potential applications for representing (computationally complex) logics used in AI by (computationally easy) many-valued logics. It is investigated how far this method can be carried using (1) one or (2) an infinite sequence of many-valued logics. It is shown that the optimal candidate matrices for (1) can (...) be computed from the calculus. (shrink)

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 are two parts. The notion of an "event" is reinterpreted from being an epistemological state of indefiniteness to being an objective state of indefiniteness. And the mathematical framework of finite probability theory is recast as the quantum probability calculus for QM/sets. The point is (...) not to clarify finite probability theory but to elucidate quantum mechanics itself by seeing some of its quantum features in a classical setting. (shrink)

"The ultimate end goal of the finite I and the not-I, i.e., the end goal of the world," writes Schelling in Of the I as the Principle of Philosophy (1795), "is its annihilation as a world, i.e., as the exemplification of finitude." In this paper, I explicate this statement and its theoretical stakes through a comprehensive re-reading of Schelling's 1795 writings: Of the I and Philosophical Letters on Dogmatism and Criticism, written later in the same year, in relation to (...) what Schelling proclaims to be the central problem of all philosophy: the existence of the world (Dasein der Welt). To that end, I analyze Schelling's 1795 conceptions of synthesis and morality, and the structure of the I's striving in the world that they serve to create and uphold – a structure in which the I is torn between a paradisal past and a striven-for future that it constitutively cannot reach as long as the world is there. Schelling's 1795 metaphysics is ultimately caught, I argue, in the tension between two poles: justifying the world, this world of negativity, division, and endless striving – and annihilating or dissolving it in the bliss of absolute identity and freedom, in which there is no world, and no possibility of or need for a world. (shrink)

Theory of Computation -- Computation by Abstracts Devices.

In this paper, I suggest that infinite numbers are large finite numbers, and that infinite numbers, properly understood, are 1) of the structure omega + (omega* + omega)Ө + omega*, and 2) the part is smaller than the whole. I present an explanation of these claims in terms of epistemic limitations. I then consider the importance, part of which is demonstrating the contradiction that lies at the heart of Cantorian set theory: the natural numbers are too large to be (...) counted by any finite number, but too small to be counted by any infinite number – there is no number of natural numbers. (shrink)

A uniform construction for sequent calculi for finite-valued first-order logics with distribution quantifiers is exhibited. Completeness, cut-elimination and midsequent theorems are established. As an application, an analog of Herbrand’s theorem for the four-valued knowledge-representation logic of Belnap and Ginsberg is presented. It is indicated how this theorem can be used for reasoning about knowledge bases with incomplete and inconsistent information.

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)

The article sketches a view of nature as a movement undermining totality. Awareness of our entanglement in an inherently incomplete movement forces us to rethink or reinvent our relationship to nature. The view provided here has been developed by drawing heavily from the later work of Martin Heidegger, moving with and beyond it in an examination of the limitations of Western philosophy in approaching questions of nature and ecology. Thinking ecologically becomes a question of grammar and the binding (and unbinding) (...) force of language as it pertains to human thought and (or in) its physical environment. (shrink)

The notion of an ideal reasoner has several uses in epistemology. Often, ideal reasoners are used as a parameter of (maximum) rationality for finite reasoners (e.g. humans). However, the notion of an ideal reasoner is normally construed in such a high degree of idealization (e.g. infinite/unbounded memory) that this use is unadvised. In this dissertation, I investigate the conditions under which an ideal reasoner may be used as a parameter of rationality for finite reasoners. In addition, I present (...) and justify the research program of computational epistemology, which investigates the parameter of maximum rationality for finite reasoners using computer simulations. (shrink)

Kant’s account of the sublime makes frequent appeals to infinity, appeals which have been extensively criticised by commentators such as Budd and Crowther. This paper examines the costs and benefits of reconstructing the account in finitist terms. On the one hand, drawing on a detailed comparison of the first and third Critiques, I argue that the underlying logic of Kant’s position is essentially finitist. I defend the approach against longstanding objections, as well as addressing recent infinitist work by Moore and (...) Smith. On the other hand, however, I argue that finitism faces distinctive problems of its own: whilst the resultant theory is a coherent and interesting one, it is unclear in what sense it remains an analysis of the sublime. I illustrate the worry by juxtaposing the finitist reading with analytical cubism. -/- . (shrink)

This book offers a theoretical investigation into the general problem of reality as a multiplicity of ‘finite provinces of meaning’, as developed in the work of Alfred Schutz. A critical introduction to Schutz’s sociology of multiple realities as well as a sympathetic re-reading and reconstruction of his project, Experiencing Multiple Realities traces the genesis and implications of this concept in Schutz’s writings before presenting an analysis of various ways in which it can shed light on major sociological problems, such (...) as social action, social time, social space, identity, or narrativity. (shrink)

Darwin famously held that his use of the term "chance" in evolutionary theory merely "serves to acknowledge plainly our ignorance of the causes of each particular variation". Is this a tenable view today? Or should we revise our thinking about chance in evolution in light of the more advanced, quantitative models of Neo-Darwinian theory, which make substantial use of statistical reasoning and the concept of probability? Is determinism still a viable metaphysical doctrine about biological reality after the quantum revolution in (...) physics, or dowe have to abandon it in favor of an objective indeterminism? In light of such reflections, what is the relevant interpretation of probability in evolutionary theory? Do biologists use the concept of probability because they are finite cognitive agents or because the evolutionary process is fundamentally probabilistic? In this paper, I will show that we do not yet fully understand the nature of chance in evolution. (shrink)

The prospects and limitations of defining truth in a finite model in the same language whose truth one is considering are thoroughly examined. It is shown that in contradistinction to Tarski's undefinability theorem for arithmetic, it is in a definite sense possible in this case to define truth in the very language whose truth is in question.

On the rise over the past 20 years has been ‘moderate supernaturalism’, the view that while a meaningful life is possible in a world without God or a soul, a much greater meaning would be possible only in a world with them. William Lane Craig can be read as providing an important argument for a version of this view, according to which only with God and a soul could our lives have an eternal, as opposed to temporally limited, significance, by (...) virtue of our moral choices then making an ultimate difference. I present two major objections to this position. On the one hand, I contend that if God existed and we had souls that lived forever, then, in fact, all our lives would turn out the same. On the other hand, I maintain that, if this objection is wrong, so that our moral choices would indeed make an ultimate difference and thereby confer an eternal significance on our lives (only) in a supernatural realm, then Craig could not capture the view, aptly held by moderate supernaturalists, that a meaningful life is possible in a purely natural world. An eternal significance would be ‘too big’, reducing any meaning possible during an earthly life to nothing by comparison. I conclude that moderate supernaturalists would be wise to avoid appealing to eternal (or infinite) meaning when spelling out the way God and a soul could alone impart a great meaning to our lives; some other notion of greatness should be considered, perhaps one involving a temporary afterlife. (shrink)

This work is a critical introduction to Alfred Schutz’s sociology of the multiple reality and an enterprise that seeks to reassess and reconstruct the Schutzian project. In the first part of the study, I inquire into Schutz’s biographical con- text that surrounds the germination of this conception and I analyse the main texts of Schutz where he has dealt directly with ‘finite provinces of meaning.’ On the basis of this analysis, I suggest and discuss, in Part II, several solutions (...) to the shortcomings of the theoretical system that Schutz drew upon the sociological problem of multiple reality. Specifically, I discuss problems related to the struc- ture, the dynamics, and the interrelationing of finite provinces of meaning as well as the way they relate to the questions of narrativity, experience, space, time, and identity. (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)

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)

In this article, I consider the importance of philosophy in the dialogue between religious believers and non-believers. I begin by arguing that a new epistemology of epistemic peer disagreement is required if the dialogue is to progress. Rather than viewing the differences between the positions as due to a deficit of understanding, I argue that differences result from the existential anchoring of such enquiries in life projects and the under-determination of interpretations by experience. I then explore a central issue which (...) is often implicit in these dialogues, namely the ontological status of God-world relations. Drawing on the reflections of Hegel on the infinite and the finite, I argue that his version of panentheism provides an insightful way to conceptualise God-world relations that avoids both dualistic and monistic approaches and helps to explicate a holistic ontology of transcendence from within the world of experience. (shrink)

In 'Forever Finite: The Case Against Infinity' (Rond Books, 2023), the author argues that, despite its cultural popularity, infinity is not a logical concept and consequently cannot be a property of anything that exists in the real world. This article summarizes the main points in 'Forever Finite', including its overview of what debunking infinity entails for conceptual thought in philosophy, mathematics, science, cosmology, and theology.

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)

This dissertation examines Fichte's critical idealism in an effort to formulate a compelling model of how we can be said to be free, despite our subjection to both rational and nonrational constraints. ;Fichte grounds idealism in a "drive to freedom" that involves two disparate strands of thought: the standpoint of idealism is said to be both the result of an absolutely free adoption of the principle of self-determination and conditioned by reason, to which the finite I is necessarily subject. (...) However, if there is no reason prior to the choice of a first principle, the choice of idealism over dogmatism cannot be motivated, and so is not attributable to a responsible subject. And if there is a prior basis for the election of idealism, the constraints of reason already necessitate the I's attempt to consider itself to be independent and are, therefore, not self-legislated. Fichte's argument strategy thus yields the following question: How can there be an absolute election of the philosophical principle of one's life, when the adoption of such a standpoint is already a manifestation of who one is? Fichte confronts this issue by conceiving of the I as necessarily subject to an original limit and by radically subjectivizing the world in terms of the I's ethical and conceptual demands. ;By locating the tension between these positions, this dissertation reveals how philosophers indebted to Fichte as well as interpreters of the Wissenschaftslehre have mischaracterized Fichte's treatment of rational self-constraint. Fichte's synthesis of freedom and necessity constitutes a model of subjectivity according to which the normatively constrained structure of experience is impossible without my relating that experience to my practical activity, and the ways in which I am able to characterize that experience are answerable to what is practically required of me by that experience. Fichte shows the interdetermination of the givenness of the given and the autonomy of the I by qualifying each in terms of the other. As a description of this interdetermination, Fichte's striving doctrine establishes the tension involved in this mutually constitutive relation as a positive conception of human freedom. (shrink)

If there truly is a proof that shows that no universal halt decider exists on the basis that certain tuples: (H, Wm, W) are undecidable, then this very same proof (implemented as a Turing machine) could be used by H to reject some of its inputs. When-so-ever the hypothetical halt decider cannot derive a formal proof from its input strings and initial state to final states corresponding the mathematical logic functions of Halts(Wm, W) or Loops(Wm, W), halting undecidability has been (...) decided. (shrink)

The paper discusses a few tensions “crucifying” the works and even personality of the great Georgian philosopher Merab Mamardashvili: East and West; human being and thought, symbol and consciousness, infinity and finiteness, similarity and differences. The observer can be involved as the correlative counterpart of the totality: An observer opposed to the totality externalizes an internal part outside. Thus the phenomena of an observer and the totality turn out to converge to each other or to be one and the same. (...) In other words, the phenomenon of an observer includes the singularity of the solipsistic Self, which (or “who”) is the same as that of the totality. Furthermore, observation can be thought as that primary and initial action underlain by the phenomenon of an observer. That action of observation consists in the externalization of the solipsistic Self outside as some external reality. It is both a zero action and the singularity of the phenomenon of action. The main conclusions are: Mamardashvili’s philosophy can be thought both as the suffering effort to be a human being again and again as well as the philosophical reflection on the genesis of thought from itself by the same effort. Thus it can be recognized as a powerful tension between signs and symbol, between conscious structures and consciousness, between the syncretism of the East and the discursiveness of the West crucifying spiritually Georgia. (shrink)

A well known and referenced global result is the nilpotent characterisation of the finite p-groups. This un doubtedly transends into neutrosophy. Hence, this fact of the neutrosophic nilpotent p-groups is worth critical studying and comprehensive analysis. The nilpotent characterisation depicts that there exists a derived series (Lower Central) which must terminate at {ϵ} (an identity), after a finite number of steps. Now, Suppose that G(I) is a neutrosophic p-group of class at least m ≥ 3. We show in (...) this paper that Lm−1(G(I)) is abelian and hence G(I) possesses a characteristic abelian neutrosophic subgroup which is not supposed to be contained in Z(G(I)). Furthermore, If L3(G(I)) = 1 such that pm is the highest order of an element of G(I)/L2(G(I)) (where G(I) is any neutrosophic p-group) then no element of L2(G(I)) has an order higher than pm. (shrink)

The Koch snowflake is one of the first fractals that were mathematically described. It is interesting because it has an infinite perimeter in the limit but its limit area is finite. In this paper, a recently proposed computational methodology allowing one to execute numerical computations with infinities and infinitesimals is applied to study the Koch snowflake at infinity. Numerical computations with actual infinite and infinitesimal numbers can be executed on the Infinity Computer being a new supercomputer patented in USA (...) and EU. It is revealed in the paper that at infinity the snowflake is not unique, i.e., different snowflakes can be distinguished for different infinite numbers of steps executed during the process of their generation. It is then shown that for any given infinite number n of steps it becomes possible to calculate the exact infinite number, Nn, of sides of the snowflake, the exact infinitesimal length, Ln, of each side and the exact infinite perimeter, Pn, of the Koch snowflake as the result of multiplication of the infinite Nn by the infinitesimal Ln. It is established that for different infinite n and k the infinite perimeters Pn and Pk are also different and the difference can be infinite. It is shown that the finite areas An and Ak of the snowflakes can be also calculated exactly (up to infinitesimals) for different infinite n and k and the difference An − Ak results to be infinitesimal. Finally, snowflakes constructed starting from different initial conditions are also studied and their quantitative characteristics at infinity are computed. (shrink)

The goal of this paper consists of developing a new (more physical and numerical in comparison with standard and non-standard analysis approaches) point of view on Calculus with functions assuming infinite and infinitesimal values. It uses recently introduced infinite and infinitesimal numbers being in accordance with the principle ‘The part is less than the whole’ observed in the physical world around us. These numbers have a strong practical advantage with respect to traditional approaches: they are representable at a new kind (...) of a computer – the Infinity Computer – able to work numerically with all of them. An introduction to the theory of physical and mathematical continuity and differentiation (including subdifferentials) for functions assuming finite, infinite, and infinitesimal values over finite, infinite, and infinitesimal domains is developed in the paper. This theory allows one to work with derivatives that can assume not only finite but infinite and infinitesimal values, as well. It is emphasized that the newly introduced notion of the physical continuity allows one to see the same mathematical object as a continuous or a discrete one, in dependence on the wish of the researcher, i.e., as it happens in the physical world where the same object can be viewed as a continuous or a discrete in dependence on the instrument of the observation used by the researcher. Connections between pure mathematical concepts and their computational realizations are continuously emphasized through the text. Numerous examples are given. (shrink)