We discuss mathematical and physical arguments against continuity and in favor of discreteness, with particular emphasis on the ideas of Émile Borel.

This accessible book gives a new, detailed and elementary explanation of the Gödel incompleteness theorems and presents the Chaitin results and their relation to the da Costa-Doria results, which are given in full, but with no ...

MATHEMATICS is a wonderful, mad subject, full of imagination, fantasy and creativity that is not limited by the petty details of the physical world, but only by the strength of our inner light. Does this sound familiar? Probably not from the mathematics classes you may have attended. But consider the work of three famous earlier mathematicians: Leonhard Euler, Georg Cantor and Srinivasa Ramanujan.

Gregory Chaitin has done seminal work on the foundations of mathematics, especially the meaning of randomness and undecidability. In Meta Maths, he offers his ideas in a new popular version, which has a special interest because it comes directly from their originator. Long associated with the IBM Watson Research Center, Chaitin comes across as a kind of mathematical Richard Feynman, intuitive and high-spirited, irreverent and plain-spoken. Through this jovial persona, he presents many serious ideas in an engagingly (...) dishevelled way, mixing autobiography and reminiscence with exposition and example. He presents his insights so agreeably that it all seems a grand lark. (shrink)

Review of "Exploring Randomness" (200) and "The Unknowable" (1999) by Gregory Chaitin.

This volume presents essays by pioneering thinkers including Tyler Burge, Gregory Chaitin, Daniel Dennett, Barry Mazur, Nicholas Humphrey, John Searle and Ian Stewart. Together they illuminate the Map/Territory Distinction that underlies at the foundation of the scientific method, thought and the very reality itself. It is imperative to distinguish Map from the Territory while analyzing any subject but we often mistake map for the territory. Meaning for the Reference. Computational tool for what it computes. Representations are handy and (...) tempting that we often end up committing the category error of over-marrying the representation with what is represented, so much so that the distinction between the former and the latter is lost. This error that has its roots in the pedagogy often generates a plethora of paradoxes/confusions which hinder the proper understanding of the subject. What are wave functions? Fields? Forces? Numbers? Sets? Classes? Operators? Functions? Alphabets and Sentences? Are they a part of our map? Or do they actually belong to the territory? Researcher, like a cartographer, clothes the reality by stitching multitudes of maps that simultaneously co-exist. A simple apple, for example, can be analyzed from several viewpoints beginning with evolution and biology, all the way down its microscopic quantum mechanical components. Is there a reality out there apart from these maps? How do these various maps interact/intermingle with each other to produce a coherent reality that we interact with? Or do they not? Does our brain uses its own internal maps to facilitate “physicist/mathematician” in us to construct the maps about the external territories in turn? If so, what is the nature of these internal maps? Are there meta-maps? Evolution definitely fences our perception and thereby our ability to construct maps, revealing to us only those aspects beneficial for our survival. But the question is, to what extent? Is there a way out of the metaphorical Platonic cave erected around us by the nature? While “Map is not the territory” as Alfred Korzybski remarked, join us in this journey to know more, while we inquire on the nature and the reality of the maps which try to map the reality out there. The book also includes a foreword by Sir Roger Penrose and an afterword by Dagfinn Follesdal. (shrink)

English summary: Philosophers have sought to apply laws of order, symmetry, and harmony to the observable universe. However, the mathematical work of Gregory Chaitin has shown that complexity, defined as the total absence of symmetry and order, appears everywhere, even in numbers. Complexity thus becomes the unsurpassable limit of reason, reducing the efforts of philosophers and scientists who would propose a system of values that every human should rationally admit. French description: Pour maitriser par la pensee et, - (...) connaitre par le langage l'accablante diversite du reel, les philosophies ont suppose l'existence d'un ordre sous-jacent se manifestant dans la symetrie, l'harmonie et la simplicite. Dans leurs efforts pour elucider et enoncer des lois concises regissant les rapports entre les elements constitutifs du monde observable, les scientifiques leur ont emboite le pas. Or, des recherches recentes en mathematiques de Gregory J. Chaitin, ont fait apparaitre que la complexite, definie comme absence totale d'ordre et de symetrie, se manifeste a tous les niveaux du reel, y compris dans les nombres. Irreductiblement rebelle a l'analyse, la complexite s'erige en limite indepassable de la raison, reduisant les pretentions des scientifiques et rendant irrealisable, comme l'avait deja compris Heidegger, le projet de philosophes qui, comme Platon et Kant, voulaient proposer un systeme de valeurs que tout etre humain devait rationnellement admettre. (shrink)

This is a defense and extension of Stephen Yablo's claim that self-reference is completely inessential to the liar paradox. An infinite sequence of sentences of the form 'None of these subsequent sentences are true' generates the same instability in assigning truth values. I argue Yablo's technique of substituting infinity for self-reference applies to all so-called 'self-referential' paradoxes. A representative sample is provided which includes counterparts of the preface paradox, Pseudo-Scotus's validity paradox, the Knower, and other enigmas of the genre. I (...) rebut objections that Yablo's paradox is not a genuine liar by constructing a sequence of liars that blend into Yablo's paradox. I rebut objections that Yablo's liar has hidden self-reference with a distinction between attributive and referential self-reference and appeals to Gregory Chaitin's algorithmic information theory. The paper concludes with comments on the mystique of self-reference. (shrink)

"A person is not explainable in molecular, field-theoretical, or physiological terms alone." With that declaration, Nobel laureate Gerald M. Edelman goes straight to the heart of Nature's Imagination, a vibrant and important collection of essays by some of the world's foremost scientists. Ever since the Enlightenment, the authors write, science has pursued reductionism: the idea that the whole can be understood by examining and explaining each of its parts. But as this book shows, scientists in every discipline are reaching for (...) a new paradigm that accounts for the whole--from the individual person to the universe itself. Nature's Imagination gathers together the work of thirteen leading mathematicians, astronomers, neuroscientists, and philosophers, as they discuss the revolution sweeping the sciences. Here Roger Penrose, Oliver Sacks, John Barrow, Gregory Chaitin, Maragret Boden, and others explore how and why classic reductionism is falling by the wayside in their own fields. As Freeman Dyson writes in the introduction, science is an art form, not a philosophical method, and it is always in search of new tools. Reductionism has done its work, and scientists are in search of another. Roger Penrose offers a fascinating account of irreducibility in mathematics, starting with the example of an impossible triangle--a drawing of a triangular object twisted so that could not exist in three dimensions. He breaks the triangle into three parts, showing that each corner is physically possible; only in combination is the triangle impossible. Both Penrose and mathematician Gregory Chaitin explore Godel's incompleteness theorem--as does John Barrow, who explains that Chaitin's proof of the theorem shows that, if we ever arrive at a Theory of Everything, there may be a still deeper and simpler unifying theory beyond that. Other contributors discuss the changing thinking in neuroscience, and the limitations of a mechanical view of the mind: as Oliver Sacks writes, "if we are to have a model or theory of mind as this actually occurs in living creatures in the world, it may have to be radically different from anything like a computational one." In addition, this volume includes staunch defenders of the classic scientific approach, such as Peter Atkins ("The omnicompetence of science, and in particular the simplicity its reductionist insight reveals, should be accepted as a working hypothesis until, if ever, it is proved inadequate"). The advance of science has been so startlingly swift in the last century that it has begun to approach limits never dreamed of before. This remarkable volume captures the latest thinking on where we must turn if we are to truly understand oursevles and the universe we live in. (shrink)

La cognition algorithmique joue un rôle central dans le capitalisme contemporain. Depuis la rationalisation du travail industriel et des relations sociales jusqu’à la finance, les algorithmes fondent un nouveau mode de pensée et de contrôle. Dans cette phase du tout-machinique dans l’évolution du capitalisme numérique, il ne suffit plus de se mettre du côté de la théorie critique pour accuser la computation de réduire la pensée humaine à des opérations mécaniques. Comme le théoricien de l’information Gregory Chaitin l’a (...) démontré ; l’incomputabilité et l’aléatoire doivent être conçus comme les conditions de bases de la computation. Si le technocapitalisme est contaminé par l’aléatoire computationnel et le chaos, la critique traditionnelle de la rationalité instrumentale doit elle aussi être remise en question : l’incomputable ne pleut plus être réduit au statut de contraire de la raison. (shrink)

Information Theory, Evolution and The Origin ofLife: The Origin and Evolution of Life as a Digital Message: How Life Resembles a Computer, Second Edition. Hu- bert P. Yockey, 2005, Cambridge University Press, Cambridge: 400 pages, index; hardcover, US $60.00; ISBN: 0-521-80293-8. The reason that there are principles of biology that cannot be derived from the laws of physics and chemistry lies simply in the fact that the genetic information content of the genome for constructing even the simplest organisms is much (...) larger than the information content of these laws. Yockey in his previous book (1992, 335) In this new book, Information Theory, Evolution and The Origin ofLife, Hubert Yockey points out that the digital, segregated, and linear character of the genetic information system has a fundamental significance. If inheritance would blend and not segregate, Darwinian evolution would not occur. If inheritance would be analog, instead of digital, evolution would be also impossible, because it would be impossible to remove the effect of noise. In this way, life is guided by information, and so information is a central concept in molecular biology. The author presents a picture of how the main concepts of the genetic code were developed. He was able to show that despite Francis Crick's belief that the Central Dogma is only a hypothesis, the Central Dogma of Francis Crick is a mathematical consequence of the redundant nature of the genetic code. The redundancy arises from the fact that the DNA and mRNA alphabet is formed by triplets of 4 nucleotides, and so the number of letters (triplets) is 64, whereas the proteome alphabet has only 20 letters (20 amino acids), and so the translation from the larger alphabet to the smaller one is necessarily redundant. Except for Tryptohan and Methionine, all amino acids are coded by more than one triplet, therefore, it is undecidable which source code letter was actually sent from mRNA. This proof has a corollary telling that there are no such mathematical constraints for protein-protein communication. With this clarification, Yockey contributes to diminishing the widespread confusion related to such a central concept like the Central Dogma. Thus the Central Dogma prohibits the origin of life "proteins first." Proteins can not be generated by "self-organization." Understanding this property of the Central Dogma will have a serious impact on research on the origin of life. (shrink)

This book is a state-of-the-art review on the Physics of Emergence. Foreword v Gregory J. Chaitin Preface vii Ignazio Licata Emergence and Computation at the Edge of Classical and Quantum Systems 1 Ignazio Licata Gauge Generalized Principle for Complex Systems 27 Germano Resconi Undoing Quantum Measurement: Novel Twists to the Physical Account of Time 61 Avshalom C. Elitzur and Shahar Dolev Process Physics: Quantum Theories as Models of Complexity 77 Kirsty Kitto A Cross-disciplinary Framework for the Description of (...) Contextually Mediated Change 109 Liane Gabora and Diederik Aerts Quantum-like Probabilistic Models Outside Physics 135 Andrei Khrennikov Phase Transitions in Biological Matter 165 Eliano Pessa Microcosm to Macrocosm via the Notion of a Sheaf (Observers in Terms of t-topos) 229 Goro Kato The Dissipative Quantum Model of Brain and Laboratory Observations 233 Walter J. Freeman and Giuseppe Vitiello Supersymmetric Methods in the Traveling Variable: Inside Neurons and at the Brain Scale 253 H.C. Rosu, O. Cornejo-Perez, and J.E. Perez-Terrazas Turing Systems: A General Model for Complex Patterns in Nature 267 R.A. Barrio Primordial Evolution in the Finitary Process Soup 297 Olof Gornerup and James P. Crutchfield Emergence of Universe from a Quantum Network 313 Paola A. Zizzi Occam's Razor Revisited: Simplicity vs. Complexity in Biology 327 Joseph P. Zbilut Order in the Nothing: Autopoiesis and the Organizational Characterization of the Living 339 Leonardo Bich and Luisa Damiano Anticipation in Biological and Cognitive Systems: The Need for a Physical Theory of Biological Organization 371 Graziano Terenzi How Uncertain is Uncertainty? 389 Tibor Vamos Archetypes, Causal Description and Creativity in Natural World 405 Leonardo Chiatti . (shrink)

The paper concerns basic restrictions (and also simplifications and misinterpretations) which happens when one tries to explain mind processes (especially cognitive ones) by an analogy to formal, algorithmical and anticipated computation processes. The paper puts together the most important reasons why these attempts come to grief. The essence of computative reduction is shown among other things on the basic theorems of modern metamathematics. Especially it gives prominence epistemological consequences Gödel's theorems and recent discovers in metamathematics made by Gregory J. (...) Chaitin. But simplifications of computationisme (and partly of whole cognitivisme) don't mean that the other way is only postmodernism. (shrink)

The article examines the effects of non-financial disclosure on corporate social responsibility. We conceptualise trade-offs between two ideal types in relation to CSR. Whereas self-regulation is associated with greater flexibility for businesses to develop best practices, it can also lead to complacency if firms feel no external pressure to engage with CSR. In contrast, government regulation is associated with greater stringency around minimum standards, but can also result in rigidity owing to a ‘one-size-fits-all’ approach. Given these potential trade-offs, we ask (...) how mandatory non-financial disclosure has been shaping CSR practices and examine its potential effectiveness as a regulatory instrument. Our analysis of 24 OECD countries using the Asset4 database shows that firms in countries that require non-financial disclosure adopt significantly more CSR activities. However, we also find that NFD regulation does not lead to lower levels of corporate irresponsibility. Furthermore, our analysis demonstrates that, over time, the variation in CSR activities declines as firms adopt increasingly similar practices. Our study thereby contributes to understanding the impact of government regulation on CSR at firm level. We also discuss the limits of mandatory NFD in addressing regulatory trade-offs between stringency and flexibility in the field of corporate social responsibility. (shrink)

Philosophers have recently argued that self-fulfilling beliefs constitute an important counter-example to the widely accepted theses that we ought not and cannot believe at will. Cases of self-fulfilling belief are thought to constitute a special class where we enjoy the epistemic freedom to permissibly believe for pragmatic reasons, because whatever we choose to believe will end up true. In this paper, I argue that this view fails to distinguish between the aim of acquiring a true belief and the aim of (...) believing what is true. While one cannot usually fail to establish that one will acquire a true belief without establishing the truth of the believed proposition, in the case self-fulfilling belief the two can come apart. I argue that insofar as the aim of belief has to do with determining whether the believed proposition is true, it will be both impossible and impermissible to believe for pragmatic reasons. (shrink)

The field of of imprecise probability has matured, in no small part because of Teddy Seidenfeld’s decades of original scholarship and essential contributions to building and sustaining the ISIPTA community. Although the basic idea behind imprecise probability is (at least) 150 years old, a mature mathematical theory has only taken full form in the last 30 years. Interest in imprecise probability during this period has also grown, but many of the ideas that the mature theory serves can be difficult to (...) apprehend to those new to the subject. Although these fundamental ideas are common knowledge in the ISIPTA community, they are expressed, when they are expressed at all, obliquely, over the course of years with students and colleagues. -/- A single essay cannot convey the store of common knowledge from any research community, let alone the ISIPTA community. But, this essay nevertheless is an attempt to guide those familiar with the basic Bayesian framework to appreciate some of the elegant and powerful ideas that underpin the contemporary theory of lower previsions, which is the theory that most people associate with the term ‘imprecise probabilities’. (shrink)

The possibility of supererogation--doing more than one feels morally obliged to do--is denied by many thinkers.

This is the companion volume to Gregory Vlastos' highly acclaimed work Socrates: Ironist and Moral Philosopher. Four ground-breaking papers which laid the basis for his understanding of Socrates are collected here, in revised form: they examine Socrates' elenctic method of investigative argument, his disavowal of knowledge, his concern for definition, and the complications of his relationship with the Athenian democracy. The fifth chapter is a new and provocative discussion of Socrates' arguments in the Protagoras and Laches. The epilogue 'Socrates (...) and Vietnam' suggests that Socrates was not, as Plato claimed, the most just man of his time. The papers have been prepared for publication by Professor Myles Burnyeat with the minimum of editorial intervention. (shrink)

In this paper I consider the prospects for a skeptical version of infallibilism. For the reasons given above, I think skeptical invariantism has a lot going for it. However, a satisfactory theory of knowledge must account for all of our desiderata, including that our ordinary knowledge attributions are appropriate. This last part will not be easy for the infallibilist invariantist. Indeed, I will argue that it is much more difficult than those sympathetic to skepticism have acknowledged, as there are serious (...) problems with regarding paradigmatic, typical knowledge attributions as loose talk, exaggerations, or otherwise practical uses of language. So, I do not think the pragmatic story that skeptical invariantism needs is one that works without a supplemental error theory of the sort left aside by purely pragmatic accounts of knowledge attributions. In its place, I will offer a compromise pragmatic and error view that I think delivers everything that skeptics can reasonably hope to get. (shrink)

Catholic Social Teaching takes a rather cautious view toward the value of the ideas of Adam Smith, due to his emphasis on negative political and economic liberty. Detractors of Smith within CST point to what they consider to be deficiencies within his works: an impoverished moral anthropology, a lack of concern for the common good, and markets untethered to human needs. Defenders of Smith within CST tend to emphasize the material benefits that derive from Smithian institutions, such as economic growth, (...) improvements in standards of living, and the new opportunities that arise from cultures focused on innovation. This paper argues that Smith’s ideas have real value for CST. However, this value primarily lies in his moral psychology. While not denying the dangers of business activity for our moral lives, Smith also helps us to understand the possibility of the moral corruption of those who wield political power, thus providing an indirect defense of political and economic liberty that coheres with important Christian ethical notions found in CST. This is the case even though CST does not consistently recognize concerns surrounding the dangers of wielding political power—and thus Smith’s arguments offer real challenges to CST as well. This paper further suggests that these basic insights are relevant regardless of one’s favored institutional arrangements. (shrink)

That a philosophical thesis entails a vicious regress is commonly taken to be decisive evidence that the thesis is false. In this paper, I argue that the existence of a vicious regress is insufficient to reject a proposed analysis provided that certain constraints on the analysis are met. When a vicious regress is present, some further consequence of the thesis must be established that, together with the presence of the vicious regress, shows the thesis to be false. The argument is (...) provided largely through the examination of Michael Bergmann's vicious regress argument against strong awareness internalism and a partial defense of that thesis against Bergmann. (shrink)

This paper considers the relationship between evidence and self-fulfilling beliefs. Following Grice (1971), many philosophers hold that adopting a self-fulfilling belief would involve an impermissible form of bootstrapping. I argue that such objections gets their force from a popular but problematic model of theoretical deliberation which pictures deliberation as a function, treating the deliberation’s inputs as given, fixed prior to and independently from the deliberation. Though such a picture may seem plausible, attending to the case of self-fulfilling beliefs can help (...) us see how such a function-model is inadequate for a general understanding of theoretical deliberation. Because it is unable to accommodate the possibility that the outcomes of our deliberation can sometimes be evidence for the proposition we are deliberating about, the function model is unable to account for cases where what we believe affects the truth of what is believed. (shrink)

This essay presents results about a deviation from independence measure called focused correlation . This measure explicates the formal relationship between probabilistic dependence of an evidence set and the incremental confirmation of a hypothesis, resolves a basic question underlying Peter Klein and Ted Warfield's ‘truth-conduciveness’ problem for Bayesian coherentism, and provides a qualified rebuttal to Erik Olsson's claim that there is no informative link between correlation and confirmation. The generality of the result is compared to recent programs in Bayesian epistemology (...) that attempt to link correlation and confirmation by utilizing a conditional evidential independence condition. Several properties of focused correlation are also highlighted. Introduction Correlation Measures 2.1 Standard covariance and correlation measures 2.2 The Wayne–Shogenji measure 2.3 Interpreting correlation measures 2.4 Correlation and evidential independence Focused Correlation Conclusion Appendix CiteULike Connotea Del.icio.us What's this? (shrink)

Many philosophers think that part of what makes an event lucky concerns how probable that event is. In this paper, I argue that an epistemic probability account of luck successfully resists recent arguments that all theories of luck, including probability theories, are subject to counterexample (Hales 2016). I argue that an event is lucky if and only if it is significant and sufficiently improbable. An event is significant when, given some reflection, the subject would regard the event as significant, and (...) the event’s epistemic probability, determined by the subject’s evidence, is the only kind of probability that directly bears on whether or not the event is lucky. I conclude with some lessons that are applicable to probability theorists of luck generally, including those defending non-epistemic probability theories. (shrink)

This paper considers the relationship between evidence and self-fulfilling beliefs—beliefs whose propositional contents will be true just in case—and because—an agent believes them. Following Grice, many philosophers hold that believing such propositions would involve an impermissible form of bootstrapping. This paper argues that such objections get their force from a popular but problematic function-model of theoretical deliberation, and that attending to the case of self-fulfilling belief can help us see why such a model is mistaken. The paper shows that on (...) a revised model of theoretical deliberation our evidence will problematically underdetermine any appropriate doxastic attitude: when belief in a proposition is self-fulfilling, our evidence is insufficient to support belief, disbelief, or even suspended judgment toward that proposition. (shrink)

A half-billion years ago, a few species of single-celled protozoa stumbled irreversibly from loose social interaction into a tight, specialized interdependence. They became multi-celled metazoa, and human beings are one sort. Metazoa greatly transcend their constituent cells in lifetime, abilities, experiences and even materials (like bone). New kind of beings emerged out of the interactions of the old.

In his groundbreaking book, Against Coherence (2005), Erik Olsson presents an ingenious impossibility theorem that appears to show that there is no informative relationship between probabilistic measures of coherence and higher likelihood of truth. Although Olsson's result provides an important insight into probabilistic models of epistemological coherence, the scope of his negative result is more limited than generally appreciated. The key issue is the role conditional independence conditions play within the witness testimony model Olsson uses to establish his result. Olsson (...) maintains that his witness model yields charitable ceteris paribus conditions for any theory of probabilistic coherence. Not so. In fact, Olsson's model, like Bayesian witness models in general, selects a peculiar class of models that are in no way representative of the range of options available to coherence theorists. Recent positive results suggest that there is a way to develop a formal theory of coherence after all. Further, although Bayesian witness models are not conducive to the truth, they are conducive to reliability. (shrink)

In this chapter we draw connections between two seemingly opposing approaches to probability and statistics: evidential probability on the one hand and objective Bayesian epistemology on the other.

In this paper, I argue for the ethics of business based on the way that business activity may embody a vocation to partake in “the Good.” Following a Platonist framework for ethics and recent work on vocations by Robert M. Adams, I argue that understanding the ethics of vocations allows us to avoid the charges that business persons have to do something more for others—often couched in terms of social responsibility, sustainability, or consideration of stakeholders—in order to legitimize their careers (...) ethically. Rather, I claim, the ethics of a business vocation, as in any vocation, rests first and foremost in the way that a person pursues projects that answer an invitation to partake in the good things of this world. Thus, the promotion of the well-being of others, while ethically admirable and constitutive of some vocations, is not fundamental for understanding the ethics of vocations themselves, even in business. There are important implications for the ethics of markets, and I also consider a recent challenge to my approach that claims that true business vocations demand a more direct promotion of the well-being of others. (shrink)

In the age of big data and a machine epistemology that can anticipate, predict, and intervene on events in our lives, the problem once again is that a few individuals possess the knowledge of how to regulate these activities. But the question we face now is not how to share such knowledge more widely, but rather of how to enjoy the public benefits bestowed by this knowledge without freely sharing it. It is not merely personal privacy that is at stake (...) but a range of unsung benefits that come from ignorance and forgetting, traits that are inherently human and integral to the functioning of our society. (shrink)

In this paper, I explain the curious role played by the Argument from Absolute Terms in Peter Unger's book Ignorance, I provide a critical presentation of the argument, and I consider some outstanding issues and the argument’s contemporary significance.

Since 1995, students, researchers, and professionals have turned to The Craft of Research for clear and helpful guidance on how to conduct research and report it effectively. Now, master teachers Wayne C. Booth, Gregory G. Colomb, and Joseph M. Williams have completely revised and updated their classic handbook. The new edition will continue to help thousands of students and writers plan, carry out, and report on research to produce effective term papers, dissertations, articles, or books -- in any field, (...) at any level. (shrink)

We examine the notion of conditionals and the role of conditionals in inductive logics and arguments. We identify three mistakes commonly made in the study of, or motivation for, non-classical logics. A nonmonotonic consequence relation based on evidential probability is formulated. With respect to this acceptance relation some rules of inference of System P are unsound, and we propose refinements that hold in our framework.

The desirable gambles framework offers the most comprehensive foundations for the theory of lower pre- visions, which in turn affords the most general ac- count of imprecise probabilities. Nevertheless, for all its generality, the theory of lower previsions rests on the notion of linear utility. This commitment to linearity is clearest in the coherence axioms for sets of desirable gambles. This paper considers two routes to relaxing this commitment. The first preserves the additive structure of the desirable gambles framework and (...) the machinery for coherent inference but detaches the interpretation of desirability from the multiplicative scale invariance axiom. The second strays from the additive combination axiom to accommodate repeated gambles that return rewards by a non-stationary processes that is not necessarily additive. Unlike the first approach, which is a conservative amendment to the desirable gambles framework, the second is a rad- ical departure. Yet, common to both is a method for describing rewards called discounted utility. (shrink)

Spengler’s work is typically represented as speculative philosophy of history. However, I argue that there is good reason to consider much of his thought as preoccupied with existential and phenomenological questions about the nature and ends of human existence, rather than with history per se. In this paper I consider Spengler’s work in comparison with Heidegger’s history of Being and analysis of technological modernity. I argue that Spengler’s considerable proximity to much of Heidegger’s thought compels us to reconsider the nature (...) and scope of Spengler’s philosophical project. (shrink)