In this paper, we explore certain exemplifications of transgression in the history and philosophy of mathematics. We recognize transgressive acts in the transition from a “real” to an “imaginary” world. Further, we suggest the concept of proof-events that transgress traditional concepts of mathematical proof. The theory of proof-events provides us with means to identify transgressive acts in the development of a discovery proof-event. These concern the creative understanding of a purported mathematical proof by (...) reconstructing the meaning conveyed by it, eventually supplementing it, surpassing stylistic hindrances in the narrative formulated by a prover, exceeding the boundaries of the (local) logic of a prover, etc. Finally, we identify transgressive acts with discontinuous points of change in the development of proof-events and suggest formal measures for their recognition. (shrink)

In this paper, we suggest the broader concept of proof-event, introduced by Joseph Goguen, as a fundamental methodological tool for studying proofs in history of mathematics. In this framework, proof is understood not as a purely syntactic object, but as a social process that involves at least two agents; this highlights the communicational aspect of proving. We claim that historians of mathematics essentially study proof-events in their research, since the mathematical proofs they face in the extant (...) sources involve many informal components, often not completely formalizable, and convey some kind of semantic content calling for understanding and verification. We illustrate the application of this methodological approach in some outstanding historical cases, paying particular attention to the process of proof interpretation that makes a proof-event alive. Finally, we suggest a classification of proof-events, according to the conditions imposed upon problem-solving. This enables us to speak about broad classes of proof-events in history of mathematics that share a common characteristic. (shrink)

This paper explores the relationship between informal reasoning, creativity in mathematics, and problem solving. It underscores the importance of environments that promote interaction, hypothesis generation, examination, refutation, derivation of new solutions, drawing conclusions, and reasoning with others, as key factors in enhancing mathematical creativity. Drawing on argumentation logic, the paper proposes a novel approach to uncover specific characteristics in the development of formalized proving using “proof-events.” Argumentation logic can offer reasoning mechanisms that facilitate these environments. This paper proposes (...) how argumentation can be implemented to discover certain characteristics in the development of formalized proving with “proof-events”. The concept of a proof-event was introduced by Goguen who described mathematical proof as a multi-agent social event involving not only “classical” formal proofs, but also other informal proving actions such as deficient or alleged proofs. Argumentation is an integral component of the discovery process for a mathematical proof since a proof necessitates a dialogue between provers and interpreters to clarify and resolve gaps or assumptions. By formalizing proof-events through argumentation, this paper demonstrates how informal reasoning and conflicts arising during the proving process can be effectively simulated. The paper presents an extended version of the proof-events calculus, rooted in argumentation theories, and highlights the intricate relationships among proof, human reasoning, cognitive processes, creativity, and mathematical arguments. (shrink)

The concept of proof can be studied from many different perspectives. Many types of proofs have been developed throughout history such as apodictic, dialectical, formal, constructive and non-constructive proofs, proofs by visualisation, assumption-based proofs, computer-generated proofs, etc. In this paper, we develop Goguen’s general concept of proof-events and the methodology of algebraic semiotics, in order to define the concept of mathematical style, which characterizes the proofs produced by different cultures, schools or scholars. In our view, style can (...) be defined as a semiotic meta-code that depends on the underlying mode of signification (semiosis), the selected code and the underlying semiotic space and determines the individual mode of integration (selection, combination, blending) into a narrative structure (proof). Finally, we examine certain historical types of styles of mathematical proofs, to elucidate our viewpoint. (shrink)

While collaboration has always played an important role in many cases of discovery and creation, recent developments such as the web facilitate and encourage collaboration at scales never seen before, even in areas such as mathematics, where contributions by single individuals have historically been the norm. This new scenario poses a challenge at the theoretical level, as it brings out the importance of various issues which, as of yet, have not been sufficiently central to the study of problem-solving, discovery, and (...) creativity. We analyze the case of collective and web-based proof events in mathematics, which share their temporal and social nature with every case of collective problem-solving. We propose that some ideas from cognitive architectures, in particular, the notion of codelet—understood as an agent engaged in one of a multitude of available tasks—can illuminate our understanding of collective problem-solving and act as a natural bridge from some of the theoretical aspects of collective, web-based discovery to the practical concern of designing cognitively inspired systems to support collective problem-solving. We use the Pythagorean Theorem and its many proofs as a case study to illustrate our approach. (shrink)

String representations of events are applied to Robin Cooper’s proposal that propositions in natural language semantics are types of situations. Links with the higher types of prooftheoretic semantics are forged, deepening type-theoretic interpretations of Discourse Representation Structures to encompass event structures.

On the standard reading of Hume, the belief that the necessity associated with the causal relation is “an entirely mind-independent phenomenon” in the world isunjustified. For example, Jonathan Bennett writes that necessary connections of the sort that Hume allows are not “relations which hold objectively between the ‘objects’ or events which we take to be causally related.” Similarly, Barry Stroud writes that, according to Hume, we believe falsely “that necessity is something that ‘resides’ in the relation between objects or (...) events in the objective world.” In this paper I argue that this reading of Hume is mistaken and that there is a sense of justification (viz., justification qua “proof”) according to which human beings are justified in holding the belief that the necessity associated with the causal relation is “an entirely mind-independent phenomenon” in the world. (shrink)

Proof requires a dialogue between agents to clarify obscure inference steps, fill gaps, or reveal implicit assumptions in a purported proof. Hence, argumentation is an integral component of the discovery process for mathematical proofs. This work presents how argumentation theories can be applied to describe specific informal features in the development of proof-events. The concept of proof-event was coined by Goguen who described mathematical proof as a public social event that takes place in space (...) and time. This new meta-methodological concept is designed to cover not only “traditional” formal proofs but all kinds of proofs and inference steps, including incomplete or purported proofs. Our approach attempts to make proof-events more comprehensive to express the complete trajectory of a mathematical proof-event until the ultimate validation of the proving outcome. Thus, we advance an extended version of proof-event calculus which is built on argumentation theories designed to capture the internal and external structure of collaborative mathematical practice and highlight the relationship between proof, human reasoning, and cognitive processes. In addition, another area in which argumentation can make a significant contribution is dealing with the defeasible knowledge of the Web which is a product of its open and ubiquitous nature. This approach seems to be sufficient for the presentation of Web-based proving processes as manifested in the case of the Mini-Polymath 4 project. (shrink)

A proof is given that there does not exist an event, that is not already in the past for some possible distant observer at the (our) moment that the latter is "now" for us. Such event is as "legally" past for that distant observer as is the moment five minutes ago on the sun for us (irrespective of the circumstance that the light of the sun cannot reach us in a period of five minutes). Only an extreme positivism: "that (...) which cannot yet be observed does not yet exist", can possibly withstand the conclusion concerned. Therefore, there is determinism, also in micro-physics. (shrink)

God Exists! 50 Profound Proofs By: Matthew Armstrong Some of the most important questions in one’s life are: Where will I live for all eternity? Is God real? Will I live in His kingdom? Is it possible to prove the existence of God? God Exists! explores fifty proofs that God absolutely exists. Prophecy demonstrates the existence of God. Events taking place in the world today reveal that God exists and knows what will happen well in advance. The universe displays (...) the handiwork of God. (shrink)

We construct an event-based computer simulation model of the Einstein-Podolsky-Rosen-Bohm experiments with photons. The algorithm is a one-to-one copy of the data gathering and analysis procedures used in real laboratory experiments. We consider two types of experiments, those with a source emitting photons with opposite but otherwise unpredictable polarization and those with a source emitting photons with fixed polarization. In the simulation, the choice of the direction of polarization measurement for each detection event is arbitrary. We use three different procedures (...) to identify pairs of photons and compute the frequency of coincidences by analyzing experimental data and simulation data. The model strictly satisfies Einstein’s criteria of local causality, does not rely on any concept of quantum theory and reproduces the results of quantum theory for both types of experiments. We give a rigorous proof that the probabilistic description of the simulation model yields the quantum theoretical expressions for the single- and two-particle expectation values. (shrink)

On the standard reading of Hume, the belief that the necessity associated with the causal relation is “an entirely mind-independent phenomenon” in the world isunjustified. For example, Jonathan Bennett writes that necessary connections of the sort that Hume allows are not “relations which hold objectively between the ‘objects’ or events which we take to be causally related.” Similarly, Barry Stroud writes that, according to Hume, we believe falsely “that necessity is something that ‘resides’ in the relation between objects or (...) events in the objective world.” In this paper I argue that this reading of Hume is mistaken and that there is a sense of justification (viz., justification qua “proof”) according to which human beings are justified in holding the belief that the necessity associated with the causal relation is “an entirely mind-independent phenomenon” in the world. (shrink)

Writing strategic documents is a major practice of many actors striving to see their educational ideas realised in the curriculum. In these documents, arguments are systematically developed to create the legitimacy of a new educational goal and competence to make claims about it. Through a qualitative analysis of the writing strategies used in these texts, I show how two of the main actors in the Czech educational discourse have developed a proof that a new educational goal is needed. I (...) draw on the connection of the relational approach in the sociology of education with Lyotard’s analytical semantics of instances in the event. The comparison of the writing strategies in the two documents reveals differences in the formation of a particular pattern of justification. In one case the texts function as a herald of pure reality, and in the other case as a messenger of other witnesses. This reveals different regimens of proof, although both of them were written as prescriptive directives – normative models of the educational world. (shrink)

The aim of this essay is to develop a coherence theory for the justification of evidentiary judgments in law. The main claim of the coherence theory proposed in this article is that a belief about the events being litigated is justified if and only if it is a belief that an epistemically responsible fact finder might hold by virtue of its coherence in like circumstances. The article argues that this coherentist approach to evidence and legal proof has the (...) resources to meet some of the main objections that may be addressed against attempts to analyze the justification of evidentiary judgments in law in coherentist terms. It concludes by exploring some implications of the proposed version of legal coherentism for a jurisprudence of evidence. (shrink)

In an attempt to improve upon Alexander Pruss’s work (The principle of sufficient reason: A reassessment, pp. 240–248, 2006), I (Weaver, Synthese 184(3):299–317, 2012) have argued that if all purely contingent events could be caused and something like a Lewisian analysis of causation is true (per, Lewis’s, Causation as influence, reprinted in: Collins, Hall and paul. Causation and counterfactuals, 2004), then all purely contingent events have causes. I dubbed the derivation of the universality of causation the “Lewisian argument”. (...) The Lewisian argument assumed not a few controversial metaphysical theses, particularly essentialism, an incommunicable-property view of essences (per Plantinga’s, Actualism and possible worlds, reprinted in: Davidson (ed.) Essays in the metaphysics of modality, 2003), and the idea that counterfactual dependence is necessary for causation. There are, of course, substantial objections to such theses. While I think a fight against objections to the Lewisian argument can be won, I develop, in what follows, a much more intuitive argument for the universality of causation which takes as its inspiration a result from Frederic B. Fitch’s work (J Symb Logic 28(2):135–142, 1963) [with credit to who we now know was Alonzo church’s, Referee Reports on Fitch’s Definition of value, in: (Salerno (ed.), New essays on the knowability paradox, 2009)] that if all truths are such that they are knowable, then (counter-intuitively) all truths are known. The resulting Church–Fitch proof for the universality of causation is preferable to the Lewisian argument since it rests upon far weaker formal and metaphysical assumptions than those of the Lewisian argument. (shrink)

I EXISTENTIAL PROOFS INTRODUCTION Metaphysical problems, like all philosophical problems, arise from a sense of puzzlement. What is puzzling is that the ...

Legal reasoning on the requirements and application of law has been studied for centuries, but in this subject area the legal profession maintains predominantly the same stance it did in the time of the Ancient Greeks. There is a gap between the standards of proof, one which has been always demonstrated by percentages and in terms of the evaluation of these standards by percentages by mathematical or statistical methods. One method to fill the gap is Bayes theorem that describes (...) an event’s probability based on conditions that might be related to an event. Bayes theorem can help to establish or confirm a relation between facts and rules if there is sufficient other evidence that connect a party in a procedure with a considered legal action. (shrink)

Recently there have emerged an assortment of theorems relating to the ‘absoluteness of emerged events,’ and these results have sometimes been used to argue that quantum mechanics may involve some kind of metaphysically radical non-absoluteness, such as relationalism or perspectivalism. However, in our view a close examination of these theorems fails to convincingly support such possibilities. In this paper we argue that the Wigner’s friend paradox, the theorem of Bong et al and the theorem of Lawrence et al are (...) all best understood as demonstrating that if quantum mechanics is universal, and if certain auxiliary assumptions hold, then the world inevitably includes various forms of ‘disaccord,’ but this need not be interpreted in a metaphysically radical way; meanwhile, the theorem of Ormrod and Barrett is best understood either as an argument for an interpretation allowing multiple outcomes per observer, such as the Everett approach, or as a proof that quantum mechanics cannot be universal in the sense relevant for this theorem. We also argue that these theorems taken together suggest interesting possibilities for a different kind of relational approach in which _interaction_ states are relativized whilst observed events are absolute, and we show that although something like ‘retrocausality’ might be needed to make such an approach work, this would be a very special kind of retrocausality which would evade a number of common objections against retrocausality. We conclude that the non-absoluteness theorems may have a significant role to play in helping converge towards an acceptable solution to the measurement problem. (shrink)

The semantics of progressive sentences presents a challenge to linguists and philosophers alike. According to a widely accepted view, the truth-conditions of progressive sentences rely essentially on a notion of inertia. Dowty suggested inertia worlds to implement this “inertia idea” in a formal semantic theory of the progressive. The main thesis of the paper is that the notion of inertia went through a subtle, but crucial change when worlds were replaced by events in Landman and Portner :760–787, 1998), and (...) that this new, event-related concept of inertia results in a possibility-based theory of the progressive. An important case in point in the paper is a proof that, despite its surface structure, the theory presented in Portner does not implement the notion of inertia in Dowty ; rather, it belongs together with Dowty’s earlier, 1977 theory according to which the progressive is a possibility operator. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Reviewed by:Kant’s Transcendental Proof of RealismBrandon C. LookKenneth R. Westphal. Kant’s Transcendental Proof of Realism. New York: Cambridge University Press, 2004. Pp. x + 299. Cloth, $80.00.Westphal's book is a rich and exciting contribution to the field of Kant studies. Its claims run counter to much contemporary discussion of Kant's theoretical philosophy and indeed challenge some of Kant's fundamental doctrines, but the arguments are very compelling and (...) therefore likely to force us to rethink both the Critique of Pure Reason [CPR] and Kant's relevance to contemporary epistemology.The central point of this work is that, if we follow Kant's injunction to engage in reflection on our cognitive capacities, then we will be led not to Kantian transcendental idealism but to an unqualified realism about objects and events in space and time. The book's first three chapters are devoted to analyzing Kant's methodology and the much-overlooked idea of transcendental affinity. Westphal focuses on the Kantian duty of engaging in "transcendental reflection," which is "the action through which I make the comparison of representations in general with the cognitive power in which they are situated, and through which I distinguish whether they are to be compared with one another as belonging to pure understanding or to sensible intuition" (CPR, A 261/B 317). And he adds to this the more general methodological obligation of "epistemic reflection," or the duty to determine precisely our cognitive capacities and incapacities. If, then, we reflect on our representations, their origins in sensibility and the understanding, and on our powers to know things of ourselves and the world in general, then certain puzzles arise—puzzles that were confronted by Kant (and Hume and Hegel) but that have not received the attention they have deserved. Chief among these puzzles is our ability to identify single objects and events and track them in space through time. But, as Westphal is keen to point out, identifying any particular over time requires that we be able to re-identify that particular. Now, while this puzzle is addressed by Kant in the Transcendental Deduction and Analogies of Experience, it is important to see that there is a material condition necessary for the possibility of self-conscious experience of the world that is not fleshed out in great detail by Kant and seems to be completely overlooked in the literature on Kant, namely, "the presupposition of diversity in nature" and the homogeneity of its objects among themselves (CPR, A 657/B 685). In other words, the fact that we are able to associate our various sensations with each other at all (and thereby identify and re-identify objects) is due to their "transcendental affinity." And it is Westphal's contention that "on Kant's own principles, the transcendental affinity of the manifold of sensory intuition can only be reconstructed, but not constructed, by intellectual syntheses of the understanding" (91). Therefore, pace Kant, who held that only his transcendental idealism could explain the possibility of experience, Westphal argues that transcendental idealism runs afoul of many of Kant's own arguments and observations in the Transcendental Aesthetic and Transcendental Analytic and, more important, that transcendental affinity provides grounds for "realism sans phrase" (the author's phrase) about objects and events in space and time, and for the view that our mental states are determined by external objects. As Westphal puts it in his conclusion: "Kant's analysis of transcendental affinity is in fact a transcendental argument for mental content externalism" (266). [End Page 665]The second part of Westphal's argument revolves around a detailed examination of Kant's theses about transeunt causality and the nature of matter as expressed not only in the Critique of Pure Reason, but also in the Metaphysical Foundations of Natural Science and the Opus postumum. Roughly the problem is this: in order to have an adequately grounded metaphysics of nature, Kant needs not only the results of the Analogies of Experience, the transcendental causal thesis that every event has a cause, but also the metaphysical causal thesis that every physical event has an external physical cause. But, according to Westphal, in the Metaphysical Foundations... (shrink)

Many argued (Accardi and Fedullo, Pitowsky) that Kolmogorov's axioms of classical probability theory are incompatible with quantum probabilities, and that this is the reason for the violation of Bell's inequalities. Szabó showed that, in fact, these inequalities are not violated by the experimentally observed frequencies if we consider the real, “effective” frequencies. We prove in this work a theorem which generalizes this results: “effective” frequencies associated to quantum events always admit a Kolmogorovian representation, when these events are collected (...) through different experimental setups, the choice of which obeys a classical distribution. (shrink)

This paper outlines a logical representation of certain aspects of the process of mathematical proving that are important from the point of view of Artificial Intelligence. Our starting point is the concept of proof-event or proving, introduced by Goguen, instead of the traditional concept of mathematical proof. The reason behind this choice is that in contrast to the traditional static concept of mathematical proof, proof-events are understood as processes, which enables their use in Artificial Intelligence (...) in such contexts in which problem-solving procedures and strategies are studied. We represent proof-events as problem-centred spatio-temporal processes by means of the language of the calculus of events, which adequately captures certain temporal aspects of proof-events (i.e. that they have history and form sequences of proof-events evolving in time). Further, we suggest a “loose” semantics for the proof events by means of Kolmogorov’s calculus of problems. Finally, we expose the intented interpretations for our logical model from the fields of automated theorem-proving and Web-based collective proving. (shrink)

In this paper the class of minimal models C ZI for Kiczuk's system of physical change ZI is provided and soundness and completeness proofs of ZI with respect to these models are given. ZI logic consists of propositional logic von Wright's And Then and six specific axioms characterizing the meaning of unary propositional operator "Zm", read "there is a change in the fact that". ZI is intended to be a logic which provides a formal account for describing two kinds of (...) process change: the change from one state of the process to its other state (e.g., transmitting or absorbing energy with greater or less than the usual intensity) and the perishing of the process (e.g., cessation of the energetic activity of the sun). (shrink)

The heuristics and strategies presented in Lakatos' Proofs and Refutations are well-known. However they hardly present the whole story as many authors have shown. In this paper a recent, rather spectacular, event in the history of mathematics is examined to gather evidence for two new strategies. The first heuristic concerns the expectations mathematicians have that a statement will be proved using given methods. The second heuristic tries to make sense of the mathematicians' notion of the quality of a proof.

The Web may critically transform the way we understand the activity of proving. The Web as a collaborative medium allows the active participation of people with different backgrounds, interests, viewpoints, and styles. Mathematical formal proofs are inadequate for capturing Web‐based proofs. This article claims that Web provings can be studied as a particular type of Goguen's proof‐events. Web‐based proof‐events have a social component, communication medium, prover‐interpreter interaction, interpretation process, understanding and validation, historical component, and styles. To (...) demonstrate its claim, the article discusses the Kumo and Polymath projects, both of which employ Web‐based communication as part of proving. Web proving is a novel type of proving activity that may have a serious impact on the change in mathematical practices, despite the fact that it is not currently a universally acceptable methodology. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Reviewed by:Kant’s Transcendental Proof of RealismBrandon C. LookKenneth R. Westphal. Kant’s Transcendental Proof of Realism. New York: Cambridge University Press, 2004. Pp. x + 299. Cloth, $80.00.Westphal's book is a rich and exciting contribution to the field of Kant studies. Its claims run counter to much contemporary discussion of Kant's theoretical philosophy and indeed challenge some of Kant's fundamental doctrines, but the arguments are very compelling and (...) therefore likely to force us to rethink both the Critique of Pure Reason [CPR] and Kant's relevance to contemporary epistemology.The central point of this work is that, if we follow Kant's injunction to engage in reflection on our cognitive capacities, then we will be led not to Kantian transcendental idealism but to an unqualified realism about objects and events in space and time. The book's first three chapters are devoted to analyzing Kant's methodology and the much-overlooked idea of transcendental affinity. Westphal focuses on the Kantian duty of engaging in "transcendental reflection," which is "the action through which I make the comparison of representations in general with the cognitive power in which they are situated, and through which I distinguish whether they are to be compared with one another as belonging to pure understanding or to sensible intuition" (CPR, A 261/B 317). And he adds to this the more general methodological obligation of "epistemic reflection," or the duty to determine precisely our cognitive capacities and incapacities. If, then, we reflect on our representations, their origins in sensibility and the understanding, and on our powers to know things of ourselves and the world in general, then certain puzzles arise—puzzles that were confronted by Kant (and Hume and Hegel) but that have not received the attention they have deserved. Chief among these puzzles is our ability to identify single objects and events and track them in space through time. But, as Westphal is keen to point out, identifying any particular over time requires that we be able to re-identify that particular. Now, while this puzzle is addressed by Kant in the Transcendental Deduction and Analogies of Experience, it is important to see that there is a material condition necessary for the possibility of self-conscious experience of the world that is not fleshed out in great detail by Kant and seems to be completely overlooked in the literature on Kant, namely, "the presupposition of diversity in nature" and the homogeneity of its objects among themselves (CPR, A 657/B 685). In other words, the fact that we are able to associate our various sensations with each other at all (and thereby identify and re-identify objects) is due to their "transcendental affinity." And it is Westphal's contention that "on Kant's own principles, the transcendental affinity of the manifold of sensory intuition can only be reconstructed, but not constructed, by intellectual syntheses of the understanding" (91). Therefore, pace Kant, who held that only his transcendental idealism could explain the possibility of experience, Westphal argues that transcendental idealism runs afoul of many of Kant's own arguments and observations in the Transcendental Aesthetic and Transcendental Analytic and, more important, that transcendental affinity provides grounds for "realism sans phrase" (the author's phrase) about objects and events in space and time, and for the view that our mental states are determined by external objects. As Westphal puts it in his conclusion: "Kant's analysis of transcendental affinity is in fact a transcendental argument for mental content externalism" (266). [End Page 665]The second part of Westphal's argument revolves around a detailed examination of Kant's theses about transeunt causality and the nature of matter as expressed not only in the Critique of Pure Reason, but also in the Metaphysical Foundations of Natural Science and the Opus postumum. Roughly the problem is this: in order to have an adequately grounded metaphysics of nature, Kant needs not only the results of the Analogies of Experience, the transcendental causal thesis that every event has a cause, but also the metaphysical causal thesis that every physical event has an external physical cause. But, according to Westphal, in the Metaphysical Foundations... (shrink)

What has become known as the ‘faith/history’ problem for historical religions like Christianity centres on the attempt to combine the ontological decisiveness, for faith, of an historical event characterized as an actual Incarnation of God with the epistemological indifference, or irrelevance, of historical information about that event which is decisive for faith. Without the former there is nothing to be related to or personally appropriated; without the latter faith is rendered vulnerable to the vagaries of historical research. Soren Kierkegaard's Climacus (...) writings – the Philosophical Fragments and the Concluding Unscientific Postscript – provide a classic formulation of the inherent tension. Twentieth-century variations on the same theme are found in the writings of Paul Tillich and Rudolf Bultmann, among others. In what follows I want to highlight an element in Kierkegaard's Climacus account which crucially distinguishes his attempt to deal with the faith/history relation from other attempts to deal with it, and thus offers what I suggest is a novel approach to overcoming the tension. This element – effectively a kind of a priori or conceptual proof – has been neglected, and whatever one's final judgement on its contribution to the problem as a whole, a consideration of it and its implications is, at the very least, integral to a full understanding of the thought-experiment in the Fragments . Moreover, in the process, useful light can be shed indirectly on twentieth-century attempts to deal with the faith/history problem. (shrink)

In an earlier paper the present author proved that de Finetti coherence is preserved under taking products of coherent books on two finite sets of independent events. Conversely, in this note it is proved that product is the only coherence preserving operation on coherent books. Our proof shows that the traditional definition of stochastically independent classes of events actually follows from the combination of two more basic notions: boolean algebraic independence and de Finetti coherent betting system.

A classicist, philosopher, and poet, Poul Martin Møller was an important figure in the Danish Golden Age. The traumatic event of the death of his wife led him to think more profoundly about the question of the immortality of the soul. In 1837 he published his most important philosophical treatise, "Thoughts on the Possibility of Proofs of Human Immortality," presented here in English for the first time. It was read and commented upon by the leading figures of the Golden Age, (...) such as Søren Kierkegaard. It proved to be the last important work that Møller wrote before his death in March of 1838 at the age of 43. (shrink)

Suppose you attend a seminar where a mathematician presents a proof to some of his colleagues. Suppose further that what he is proving is an important mathematical statement Now the following happens: as the mathematician proceeds, his audience is amazed at first, then becomes angry and finally ends up disturbing the lecture (some walk out, some laugh, …). If in addition, you see that the proof he is presenting is formally speaking (nearly) correct, would you say you are (...) witnessing an extraordinary event in urgent need of explanation? Surely, your answer would be yes. But do events of this type actually occur? The matter of the fact is, yes, they do. This paper presents the details of such an event, suggests a possible explanation and examines its implications for our understanding of mathematical practice. (shrink)

The Web may critically transform the way we understand the activity of proving. The Web as a collaborative medium allows the active participation of people with different backgrounds, interests, viewpoints, and styles. Mathematical formal proofs are inadequate for capturing Web-based proofs. This article claims that Web provings can be studied as a particular type of Goguen's proof-events. Web-based proof-events have a social component, communication medium, prover-interpreter interaction, interpretation process, understanding and validation, historical component, and styles. To (...) demonstrate its claim, the article discusses the Kumo and Polymath projects, both of which employ Web-based communication as part of proving. Web proving is a novel type of proving activity that may have a serious impact on the change in mathematical practices, despite the fact that it is not currently a universally acceptable methodology. (shrink)

(3) A compatibilist needs to explain how free will can co-exist with determinism, paradigmatically by offering an analysis of ‘free’ action that is demonstrably compatible with determinism. (Here is the late Roderick Chisholm, in defense of irreducible or libertarian agent-causation: ‘Now if you can analyze such statements as “Jones killed his uncle” into event-causation statements, then you may have earned the right to make jokes about the agent as cause. But if you haven’t done this, and if all the same (...) you do believe such things as that I raised my arm and that Jolns [sic] killed his uncle, and if moreover you still think it’s a joke to talk about the agent as cause, then, I’m afraid, the joke is entirely on you.’). (shrink)

The Web may critically transform the way we understand the activity of proving. The Web as a collaborative medium allows the active participation of people with different backgrounds, interests, viewpoints, and styles. Mathematical formal proofs are inadequate for capturing Web‐based proofs. This chapter claims that Web provings can be studied as a particular type of Goguen's proof‐events. Web‐based proof‐events have a social component, communication medium, prover‐interpreter interaction, interpretation process, understanding and validation, historical component, and styles. To (...) demonstrate its claim, the chapter discusses the Kumo and Polymath projects, both of which employ Web‐based communication as part of proving. Web proving is a novel type of proving activity that may have a serious impact on the change in mathematical practices, despite the fact that it is not currently a universally acceptable methodology. The Web may critically transform the way we understand the activity of proving. The Web as a collaborative medium allows the active participation of people with different backgrounds, interests, viewpoints, and styles. Mathematical formal proofs are inadequate for capturing Web‐based proofs. This chapter claims that Web provings can be studied as a particular type of Goguen's proof‐events. Web‐based proof‐events have a social component, communication medium, prover‐interpreter interaction, interpretation process, understanding and validation, historical component, and styles. To demonstrate its claim, the chapter discusses the Kumo and Polymath projects, both of which employ Web‐based communication as part of proving. Web proving is a novel type of proving activity that may have a serious impact on the change in mathematical practices, despite the fact that it is not currently a universally acceptable methodology. (shrink)

Starting from a connected, partially ordered set of events, it is shown that results of the measurement of time are elements of a partially ordered and filtering field, as used in a previous paper. Moreover, some relations between physical formulas and properties of the field are proved. Finally, some open problems and suggestions are pointed out. For the convenience of the reader not acquainted with elementary algebraic methods, proofs are given in detail.

This article deals with the events that preceded the U.S. House Resolution No. 269 of June 11, 2002, acknowledging the primacy of Antonio Meucci in the invention of the telephone and that were decisive to the passing of the same. Among them are the author’s lecture at the University of NewYork of October 10, 2000, and Resolution No. 1566 of the New York City Council urging the U.S. Congress to recognize the priority of Antonio Meucci in the invention of (...) the telephone, which followed 2 days after. This lecture is reported here as it was newly delivered in Canada during the author’s visit in April 2003. (shrink)

Mc Taggart's celebrated proof of the unreality of time is a chain of implications whose final step asserts that the A-series (i.e. the classification of events as past, present or future) is intrinsically contradictory. This is widely believed to be the heart of the argument, and it is where most attempted refutations have been addressed; yet, it is also the only part of the proof which may be generalised to other contexts, since none of the notions involved (...) in it is specifically temporal. In fact, as I show in the first part of the paper, McTaggart's refutation of the A-series can be easily interpreted in mathematical terms; subsequently, in order to strengthen my claim, I apply the same framework by analogy to the cases of space, modality, and personal identity. Therefore, either McTaggart's proof as a whole may be extended to each of these notions, or it must embed some distinctly temporal element in one of the steps leading up to the contradiction of the A-series. I conclude by suggesting where this element might lay, and by hinting at what I believe to be the true logical fallacy of the proof. (shrink)

Kant, Hume, and the Interruption of Dogmatic Slumber offers an interpretation of Kant’s “confession,” in the Prolegomena, that “it was the objection of David Hume that first, many years ago, interrupted my dogmatic slumber.” It argues that Hume roused Kant not, as has often been thought, by challenging the principle “every event has a cause” that governs experience, but by attacking the principle of sufficient reason, the basis of rationalist metaphysics and of the cosmological proof of the existence of (...) God. This proposal makes it possible to reconcile Kant’s declaration about Hume with his later assertion that it was the Antinomy of pure reason that first woke him from dogmatic slumber, because the Antinomy, like Hume’s challenge, is directed against the dogmatic use of the principle of sufficient reason. The proposal put forward here also makes it possible to understand why Kant speaks of “the objection of David Hume” after mentioning Hume’s attack on metaphysics; for the “objection” that Kant has in mind, it is argued here, is a challenge to metaphysics, rather than to the foundations of empirical knowledge. (shrink)

Transmission of justification across inference is a valuable and indeed ubiquitous epistemic phenomenon in everyday life and science. It is thanks to the phenomenon of epistemic transmission that inferential reasoning is a means for substantiating predictions of future events and, more generally, for expanding the sphere of our justified beliefs or reinforcing the justification of beliefs that we already entertain. However, transmission of justification is not without exceptions. As a few epistemologists have come to realise, more or less trivial (...) forms of circularity can prevent justification from transmitting from p to q even if one has justification for p and one is aware of the inferential link from p to q. In interesting cases this happens because one can acquire justification for p only if one has independent justification for q. In this case the justification for q cannot depend on the justification for p and the inferential link from p to q, as genuine transmission would require. The phenomenon of transmission failure seems to shed light on philosophical puzzles, such as Moore's proof of a material world and McKinsey's paradox, and it plays a central role in various philosophical debates. For this reason it is being granted continued and increasing attention. (shrink)

This is the first interdisciplinary exploration of the philosophical foundations of the Web, a new area of inquiry that has important implications across a range of domains. - Contains twelve essays that bridge the fields of philosophy, cognitive science, and phenomenology. - Tackles questions such as the impact of Google on intelligence and epistemology, the philosophical status of digital objects, ethics on the Web, semantic and ontological changes caused by the Web, and the potential of the Web to serve as (...) a genuine cognitive extension. - Brings together insightful new scholarship from well-known analytic and continental philosophers, such as Andy Clark and Bernard Stiegler, as well as rising scholars in “digital native” philosophy and engineering. - Includes an interview with Tim Berners-Lee, the inventor of the Web. (shrink)

Ten years ago, researchers using event-related brain potentials to study language comprehension were puzzled by what looked like a Semantic Illusion: Semantically anomalous, but structurally well-formed sentences did not affect the N400 component—traditionally taken to reflect semantic integration—but instead produced a P600 effect, which is generally linked to syntactic processing. This finding led to a considerable amount of debate, and a number of complex processing models have been proposed as an explanation. What these models have in common is that they (...) postulate two or more separate processing streams, in order to reconcile the Semantic Illusion and other semantically induced P600 effects with the traditional interpretations of the N400 and the P600. Recently, however, these multi-stream models have been called into question, and a simpler single-stream model has been proposed. According to this alternative model, the N400 component reflects the retrieval of word meaning from semantic memory, and the P600 component indexes the integration of this meaning into the unfolding utterance interpretation. In the present paper, we provide support for this “Retrieval–Integration ” account by instantiating it as a neurocomputational model. This neurocomputational model is the first to successfully simulate the N400 and P600 amplitude in language comprehension, and simulations with this model provide a proof of concept of the single-stream RI account of semantically induced patterns of N400 and P600 modulations. (shrink)

In this article, I present a novel account of a priori warrant, which I then use to examine the relationship between a priori and a posteriori warrant in mathematics. According to this account of a priori warrant, the reason that a posteriori warrant is subordinate to a priori warrant in mathematics is because processes that produce a priori warrant are reliable independent of the contexts in which they are used, whereas this is not true for processes that produce a posteriori (...) warrant. Following this, I show how this difference explains why a priori warranting processes, such as proof (and mathematical intuition) can, and a posteriori warranting processes cannot, definitively establish mathematical results. I then show why, in the event of a conflict between a priori and a posteriori warranting processes, the former undermine, and cannot be undermined by, the latter. Finally, I explain the connection between apriority and mathematical necessity, but do so in a way that allows for the existence of contingent a priori knowledge. (shrink)

This book is an examination of the foundations and applications of the program of truth-theoretic semantics for natural languages introduced in 1967 by Donald Davidson in his classic paper “Truth and Meaning.” This is the second of two books on Donald Davidson’s central philosophical project. The first, Donald Davidson: Meaning, Truth, Language and Reality (Oxford: Oxford University Press, 2005), dealt with the basic framework of Davidson’s truth-theoretic approach to providing a meaning theory for a natural language, and then with his (...) development of his general project in the philosophy of language and theory of meaning by way of the analysis of the empirical confirmation of a meaning theory for a speaker, from the stance of the radial interpreter, that is, the interpreter who starts out without any detailed knowledge of the speakers’ meanings or attitudes. This book turns to the development of details of Davidson’s semantic program, specifically, the pursuit of the project of providing a compositional semantic theory for natural languages by making strategic use of an appropriately adapted Tarski-style axiomatic truth theory. Such a theory aims to provide a specification of the truth conditions of any felicitous utterance of a natural language sentence which serves to interpret it, and exhibits in the proof of the theorem that gives the truth conditions the compositional structure of the sentence and how its truth depends on its context of utterance. The book develops the basic framework, applies it to quantifiers, proper names, indexicals, simple and complex demonstratives, quotation, adjectives and adverbs, simple tenses of state and event verbs, temporal adverbs and quantifiers, tense in sentential complements and perfect tenses, opaque contexts, especially indirect discourse and attitude sentences, non-declarative sentences, semantic structure and logical form, and discusses the relation of the framework the metaphysics of truth. (shrink)

Miracles are extraordinary events that occur in the hands of those who claim to be prophets and which cannot be repeated by others. By these miracles, the prophets prove to society that the truths they convey are of divine origin. The miracles bestowed upon prophets vary according to the scope of the message they deliver and the interests and relevance of the societies with which they deal. Accordingly, Islamic scholars have classified miracles into three groups. The miraculous events (...) that appeal to people’s senses and lead them to the truths they convey by captivating their astonishment are called emotional miracles. Such miracles are events that occur without cause by overcoming the laws of nature, which are in the realm of causes, through the direct intervention of Allāh. On the other hand, spiritual miracles are truths that arouse the interlocutor’s interest in moral conduct and rational conclusions rather than in his sensory perceptions. Since these truths are attained through contemplation, they are not confined to a particular time and place, as in the case of emotional miracles. Some Kalām scholars have also considered the Prophets’ revelational knowledge about events that have occurred in the past or will occur in the future as miracles. They have studied these events that a person could not bear in the past or could not predict for the future under the title of "The news miracle" Among these three types of miracles, scholars have considered spiritual miracles superior to the others both in terms of the breadth of their influence and their continuity. That Prophet Muhammad (PBUH) is the last of the prophets, and that the teachings he received through revelation remain valid until the Day of Judgment necessitates that some of the miracles bestowed upon him must be of a quality that attracts the attention of all people. For this reason, Islamic scholars have placed the permanent miracle of Prophet Muhammad in the category of spiritual miracles. Scholars agree that his greatest miracle that has astonished all humankind is the Qurʾān. The clearest proof of this are the verses in the Qurʾān that challenge the opponents to bring a similar one. Allāh commands the disbelievers first to bring a similitude of the Qurʾān (al-Isrāʾ 17:88), then the equivalent of ten surahs (Hūd 11:13), and finally any surah of the Qurʾān (al-Baqara 2:23). Thus, the deniers are made aware of their inability to meet this challenge under gradually mitigating circumstances. Although scholars agree on the Qurʾānic miracle’s reality, they have expressed different views on its nature and essence. Thus, some scholars have argued that the miracle of the Qurʾān does not arise from some properties special to it but from the direct intervention of Allāh in human who have the ability to bring something like it. Most Islamic scholars, on the other hand, pointed out that the Qurʾān has aspects that exceed human powers in various fields, and they explained the inability to reply in the same vein to the Qurʾān due to its uniquenesses. Accordingly, some scholars have argued that miracles are perceptible, but it is never possible to define them. Other scholars, however, viewed miracles as an identifiable phenomenon and stated that the Qurʾān is an unusual and unique book in terms of verses. In this article, the questions Why was the Quran sent down as a miraculous Kalam with an attribute not found in other divine books and in what way did the Qurʾān make people weak to have a similar of it? were scrutinized, thus, some fundamental questions regarding the nature and character of iʿdjāz al-Qurʾān are elucidated. Since the topic deals with history and literature, scanning sources and data analysis were mainly used as methods in the research. The results can be summerized as follows: Although the Prophet was supported by both emotional and mental miracles in proving the Prophetic case, his greatest miracle is the Quran. The miracle of the Qurʾān is different from the other miracles, based on both the challenging verses and the Prophet's explanations of the miracles. The reason for this difference lies in the fact that the miracle of the Qurʾān is based on reason, unlike the other miracles which are emotional. Most Islamic scholars who deal with the miracle of the Qurʾān emphasize the issues of rhetoric, verse, style, and content. At this point, they emphasized that the Qurʾān has a unique expression and content that cannot be surpassed. (shrink)

Epiphenomenalism holds that mental events are caused by physical events while not causing any physical effects whatsoever. The self-stultification objection is a venerable argument against epiphenomenalism according to which, if epiphenomenalism were true, we would not have knowledge of our own sensations. For the past three decades, W.S. Robinson has called into question the soundness of this objection, offering several arguments against it. Many of his arguments attempt to shift the burden of proof onto the opponents of (...) epiphenomenalism, hoping to show that epiphenomenalism is no less stultifying than its contenders, such as dualism, functionalism, or identity theory. In the current paper I attempt to shift the burden of proof back to Robinson, and thus to defend the self-stultification objection, by offering two counterarguments against one of Robinson's objections to one of the key premises of the self-stultification objection. (shrink)

Brian Leftow offers a theist theory of necessity and possibility, and a new sort of argument for God's existence. He argues that necessities of logic and mathematics are determined by God's nature, but that it is events in God's mind - his imagination and choice - that account for necessary truths about concrete creatures.

The disjunctive theory of perception claims that we should understand statements about how things appear to a perceiver to be equivalent to statements of a disjunction that either one is perceiving such and such or one is suffering an illusion (or hallucination); and that such statements are not to be viewed as introducing a report of a distinctive mental event or state common to these various disjoint situations. When Michael Hinton first introduced the idea, he suggested that the burden of (...) proof or disproof lay with his opponent, that what was needed was to show that our talk of how things look or appear to one.. (shrink)

Not all scientific explanations work by describing causal connections between events or the world's overall causal structure. In addition, mathematicians regard some proofs as explaining why the theorems being proved do in fact hold. This book proposes new philosophical accounts of many kinds of non-causal explanations in science and mathematics.

Does God exist? What are the various arguments that seek to prove the existence of God? Can atheists refute these arguments? The Existence of God: A Philosophical Introduction assesses classical and contemporary arguments concerning the existence of God: the ontological argument, introducing the nature of existence, possible worlds, parody objections, and the evolutionary origin of the concept of God the cosmological argument, discussing metaphysical paradoxes of infinity, scientific models of the universe, and philosophers’ discussions about ultimate reality and the meaning (...) of life the design argument, addressing Aquinas’s Fifth Way, Darwin’s theory of evolution, the concept of irreducible complexity, and the current controversy over intelligent design and school education. Bringing the subject fully up to date, Yujin Nagasawa explains these arguments in relation to recent research in cognitive science, the mathematics of infinity, big bang cosmology, and debates about ethics and morality in light of contemporary political and social events. The book also includes fascinating insights into the passions, beliefs and struggles of the philosophers and scientists who have tackled the challenge of proving the existence of God, including Thomas Aquinas, and Kurt Gödel - who at the end of his career as a famous mathematician worked on a secret project to prove the existence of God. The Existence of God: A Philosophical Introduction is an ideal gateway to the philosophy of religion and an excellent starting point for anyone interested in arguments about the existence of God. (shrink)