We will flip a fair coin infinitely many times. Al calls the first flip, claiming it will land heads. Betty calls every odd numbered flip, claiming they will all land heads. Carl calls every flip bar none, claiming they will all land heads. Pre-theoretically, it seems that Al's claim is infinitely more likely than Betty's, and that Betty's claim is infinitely more likely than Carl's. But standard, real-valued probability theory says that, while Al's claim is infinitely more likely than Betty's, (...) Betty's claim is exactly as likely as Carl's. I use John Conway's surreal numbers to capture these pre-theoretic judgements about the relative probabilities of Al's, Betty's, and Carl's claims. Surreal-valued probabilities also allow us to calculate precise numerical values for those claims. For instance, we will see that Betty's claim has a probability equal to the reciprocal of the product of the square root of cardinality of the continuum and the fourth root of 2. The article provides a 'user's guide' to surreal-valued probabilities and raises questions for further research. (shrink)

Although expected utility theory has proven a fruitful and elegant theory in the finite realm, attempts to generalize it to infinite values have resulted in many paradoxes. In this paper, we argue that the use of John Conway's surreal numbers shall provide a firm mathematical foundation for transfinite decision theory. To that end, we prove a surreal representation theorem and show that our surreal decision theory respects dominance reasoning even in the case of infinite values. We (...) then bring our theory to bear on one of the more venerable decision problems in the literature: Pascal's Wager. Analyzing the wager showcases our theory's virtues and advantages. To that end, we analyze two objections against the wager: Mixed Strategies and Many Gods. After formulating the two objections in the framework of surreal utilities and probabilities, our theory correctly predicts that (1) the pure Pascalian strategy beats all mixed strategies, and (2) what one should do in a Pascalian decision problem depends on what one's credence function is like. Our analysis therefore suggests that although Pascal's Wager is mathematically coherent, it does not deliver what it purports to, a rationally compelling argument that people should lead a religious life regardless of how confident they are in theism and its alternatives. (shrink)

This paper suggests that time could have a much richer mathematical structure than that of the real numbers. Clark & Read (1984) argue that a hypertask (uncountably many tasks done in a finite length of time) cannot be performed. Assuming that time takes values in the real numbers, we give a trivial proof of this. If we instead take the surreal numbers as a model of time, then not only are hypertasks possible but so is an (...) ultratask (a sequence which includes one task done for each ordinal number—thus a proper class of them). We argue that the surreal numbers are in some respects a better model of the temporal continuum than the real numbers as defined in mainstream mathematics, and that surreal time and hypertasks are mathematically possible. (shrink)

In 2001, the algebraico-tree-theoretic simplicity hierarchical structure of J. H. Conway’s ordered field${\mathbf {No}}$of surreal numbers was brought to the fore by the first author and employed to provide necessary and sufficient conditions for an ordered field (ordered$K$-vector space) to be isomorphic to an initial subfield ($K$-subspace) of${\mathbf {No}}$, i.e. a subfield ($K$-subspace) of${\mathbf {No}}$that is an initial subtree of${\mathbf {No}}$. In this sequel, analogous results are established forordered exponential fields, making use of a slight generalization of Schmeling’s (...) conception of atransseries field. It is further shown that a wide range of ordered exponential fields are isomorphic to initial exponential subfields of$({\mathbf {No}}, \exp )$. These include all models of$T({\mathbb R}_W, e^x)$, where${\mathbb R}_W$is the reals expanded by aconvergent Weierstrass system W. Of these, those we calltrigonometric-exponential fieldsare given particular attention. It is shown that the exponential functions on the initial trigonometric-exponential subfields of${\mathbf {No}}$, which includes${\mathbf {No}}$itself, extend tocanonicalexponential functions on theirsurcomplexcounterparts. The image of the canonical map of the ordered exponential field${\mathbb T}^{LE}$oflogarithmic-exponentialtransseries into${\mathbf {No}}$is shown to be initial, as are the ordered exponential fields${\mathbb R}((\omega ))^{EL}$and${\mathbb R}\langle \langle \omega \rangle \rangle $. (shrink)

This paper defines and explores the properties of several effectivizations of the structure of surreal numbers. The construction of one of previously investigated systems, the metadyadics, is shown to be effectively equivalent to the construction of the surreals in L ω 1 CK. This equivalence is used to answer several open questions concerning the metadyadics. Results obtained seem to indicate that the metadyadics best capture the notion of a recursive surreal number.

In his monograph On Numbers and Games, J. H. Conway introduced a real-closed field containing the reals and the ordinals as well as a great many less familiar numbers including $-\omega, \,\omega/2, \,1/\omega, \sqrt{\omega}$ and $\omega-\pi$ to name only a few. Indeed, this particular real-closed field, which Conway calls No, is so remarkably inclusive that, subject to the proviso that numbers—construed here as members of ordered fields—be individually definable in terms of sets of NBG, it may be (...) said to contain “All Numbers Great and Small.” In this respect, No bears much the same relation to ordered fields that the system ℝ of real numbers bears to Archimedean ordered fields. In Part I of the present paper, we suggest that whereas $\mathbb{R}$should merely be regarded as constituting an arithmetic continuum, No may be regarded as a sort of absolute arithmetic continuum, and in Part II we draw attention to the unifying framework No provides not only for the reals and the ordinals but also for an array of non-Archimedean ordered number systems that have arisen in connection with the theories of non-Archimedean ordered algebraic and geometric systems, the theory of the rate of growth of real functions and nonstandard analysis. In addition to its inclusive structure as an ordered field, the system No of surreal numbers has a rich algebraico-tree-theoretic structure—a simplicity hierarchical structure—that emerges from the recursive clauses in terms of which it is defined. In the development of No outlined in the present paper, in which the surreals emerge vis-à-vis a generalization of the von Neumann ordinal construction, the simplicity hierarchical features of No are brought to the fore and play central roles in the aforementioned unification of systems of numbers great and small and in some of the more revealing characterizations of No as an absolute continuum. (shrink)

Skolem studied the germs at infinity of the smallest class of real valued functions on the positive real line containing the constant $1$, the identity function ${\mathbf {x}}$, and such that whenever f and g are in the set, $f+g,fg$ and $f^g$ are in the set. This set of germs is well ordered and Skolem conjectured that its order type is epsilon-zero. Van den Dries and Levitz computed the order type of the fragment below $2^{2^{\mathbf {x}}}$. Here we prove that (...) the set of asymptotic classes within any Archimedean class of Skolem functions has order type $\omega $. As a consequence we obtain, for each positive integer n, an upper bound for the fragment below $2^{n^{\mathbf {x}}}$. We deduce an epsilon-zero upper bound for the fragment below $2^{{\mathbf {x}}^{\mathbf {x}}}$, improving the previous epsilon-omega bound by Levitz. A novel feature of our approach is the use of Conway’s surreal number for asymptotic calculations. (shrink)

On a very natural conception of sets, every set has an absolute complement. The ordinary cumulative hierarchy dismisses this idea outright. But we can rectify this, whilst retaining classical logic. Indeed, we can develop a boolean algebra of sets arranged in well-ordered levels. I show this by presenting Boolean Level Theory, which fuses ordinary Level Theory (from Part 1) with ideas due to Thomas Forster, Alonzo Church, and Urs Oswald. BLT neatly implement Conway’s games and surreal numbers; and (...) a natural extension of BLT is definitionally equivalent with ZF. (shrink)

In a number of works, we have suggested that whereas the ordered field R of real numbers should merely be regarded as constituting an arithmetic continuum (modulo the Archimedean axiom), the ordered field No of surreal numbers may be regarded as a sort of absolute arithmetic continuum (modulo NBG). In the present chapter, as part of a more general exposition of the absolute arithmetic continuum, we will outline some of the properties of the system of surreal (...) numbers that we believe lend credence to this mathematico-philosophical thesis. We will also provide an overview of No’s rich structure as a simplicity-hierarchical (or s-hierarchical) ordered field that recursively emerges from the interplay between its structure as an ordered field and its structure as a lexicographically ordered full binary tree. Finally, we will draw attention to how properties of the system of surreal numbers considered as an s-hierarchical ordered algebraic structure can be appealed to in conjunction with classical relations between ordered algebraic and geometric systems to help resolve, for the surreal case, one of the purported difficulties that lies at the heart of attempts to bridge the gap between the domains of number and of geometrical magnitude. In particular, we will explain how it is possible that despite the fact that every surreal number, considered as a member of an s-hierarchical ordered field, differs from every other in characteristic individual properties, the absolute (Euclidean) geometrical continuum, which is modeled by the Cartesian space over the ordered field No of surreal numbers, appears as an amorphous pulp of points that display little individuality. (shrink)

An exponential $\exp $ on an ordered field $$. The structure$$ is then called an ordered exponential field. A linearly ordered structure $$is called o-minimal if every parametrically definable subset of M is a finite union of points and open intervals of M.The main subject of this thesis is the algebraic and model theoretic examination of o-minimal exponential fields$$ whose exponential satisfies the differential equation $\exp ' = \exp $ with initial condition $\exp = 1$. This study (...) is mainly motivated by the Transfer Conjecture, which states as follows:Any o-minimal exponential field $$ whose exponential satisfies the differential equation $\exp ' = \exp $ with initial condition $\exp =1$ is elementarily equivalent to $\mathbb {R}_{\exp }$.Here, $\mathbb {R}_{\exp }$ denotes the real exponential field $$, where $\exp $ denotes the standard exponential $x \mapsto \mathrm {e}^x$ on $\mathbb {R}$. Moreover, elementary equivalence means that any first-order sentence in the language $\mathcal {L}_{\exp } = \{+,-,\cdot,0,1, <,\exp \}$ holds for $$ if and only if it holds for $\mathbb {R}_{\exp }$.The Transfer Conjecture, and thus the study of o-minimal exponential fields, is of particular interest in the light of the decidability of $\mathbb {R}_{\exp }$. To the date, it is not known if $\mathbb {R}_{\exp }$ is decidable, i.e., whether there exists a procedure determining for a given first-order $\mathcal {L}_{\exp }$ -sentence whether it is true or false in $\mathbb {R}_{\exp }$. However, under the assumption of Schanuel’s Conjecture—a famous open conjecture from Transcendental Number Theory—a decision procedure for $\mathbb {R}_{\exp }$ exists. Also a positive answer to the Transfer Conjecture would result in the decidability of $\mathbb {R}_{\exp }$. Thus, we study o-minimal exponential fields with regard to the Transfer Conjecture, Schanuel’s Conjecture, and the decidability question of $\mathbb {R}_{\exp }$.Overall, we shed light on the valuation theoretic invariants of o-minimal exponential fields—the residue field and the value group—with additional induced structure. Moreover, we explore elementary substructures and extensions of o-minimal exponential fields to the maximal ends—the smallest elementary substructures being prime models and the maximal elementary extensions being contained in the surreal numbers. Further, we draw connections to models of Peano Arithmetic, integer parts, density in real closure, definable Henselian valuations, and strongly NIP ordered fields.Parts of this thesis were published in [2–5].Abstract prepared by Lothar Sebastian KrappE-mail: sebastian.krapp@uni-konstanz.deURL: https://d-nb.info/1202012558/34. (shrink)

I argue for the possibility of hypergunk: that is, it is possible that there exists an x such that every part of x has a proper part and, for any set S of parts of x, there is a set S′ of parts of...

Since Blood Simple, the first film they wrote and directed together, the Coen Brothers have been working their way up in the film world and, in spite of their outside-the-mainstream taste for the noir and the surreal, have earned a number of prestigious prizes. After Fargo, one of their most critically acclaimed films, expectations were high, and when the Brothers released their next bizarre venture, most critics rushed to measure it against Fargo’s success. Consequently, The Big Lebowski, the Coens’ (...) 1998 neo-noir detective comedy, was considered an incoherent, “unsatisfactory” medley of genres and styles and a box office bomb, and nothing hinted that this unorthodox story of mistaken identity, featuring a pot-smoking, unemployed character named the Dude as its “hee-ro,” would gain a following. Yet, since its 1998 DVD release, The Big Lebowski has been hailed as the first cult film of the Internet, continuously inspiring versatile cultural phenomena as nonconformist in their nature as the movie itself. This essay examines particular factors which initially might have been responsible for alienating the audience only to help The BigLebowski become a peculiar cultural event in later years. It looks at TheBig Lebowski’s characters, the historical time and place of the film’s action as well as at various external historical events, phenomena, places and people such as, for example, the Port Huron Statement, the Reagan-Bush era, Los Angeles and its immigration issues, racial minorities, civil rights activists, the Western genre and, last but not least, Arnold Schwarzenegger. Reflecting the film’s oddities, this bag of cultural idiosyncrasies appears to provide some plausible explanations for The Big Lebowski’s unexpected, against-all-odds rise from the marginal position of a critical and commercial failure to the status of a cult classic and cultural landmark. (shrink)

Significant scientific and technological advances in intensive care have been made. However, patients in the intensive care unit may experience discomfort, loss of control, and surreal experiences. This has generated relevant debates about how to humanize the intensive care units and whether humanization is necessary at all. This paper aimed to explore how humanizing intensive care is described in the literature. A scoping review was performed. Studies published between 01.01.1999 and 02.03.2020 were identified in the CINAHL, Embase, PubMed, and (...) Scopus databases. After removing 185 duplicates, 363 papers were screened by title and abstract. Full-text screening of 116 papers led to the inclusion of 68 papers in the review based on the inclusion criteria; these papers mentioned humanizing or dehumanizing intensive care in the title or abstract. Humanizing care was defined as holistic care, as a general attitude of professionals toward patients and relatives and an organizational ideal encompassing all subjects of the healthcare system. Technology was considered an integral component of intensive care that must be balanced with caring for the patient as a whole and autonomous person. This holistic view of patients and relatives could ameliorate the negative effects of technology. There were geographical differences and the large number of studies from Spain and Brazil reflect the growing interest in humanizing intensive care in these particular countries. In conclusion, a more holistic approach with a greater emphasis on the individual patient, relatives, and social context is the foundation for humanizing intensive care, as reflected in the attitudes of nurses and other healthcare professionals. Demands for mastering technology may dominate nurses’ attention toward patients and relatives; therefore, humanized intensive care requires a holistic attitude from health professionals and organizations toward patients and relatives. Healthcare organizations, society, and regulatory frameworks demanding humanized intensive care may enforce humanized intensive care. (shrink)

Originally published in 1995, The Selected Works of George McCready Price is the seventh volume in the series, Creationism in Twentieth Century America, reissued in 2019. The volume brings together the original writings and pamphlets of George McCready Price, a leading creationist of the early antievolution crusade of the 1920s. McCready Price labelled himself the 'principal scientific authority of the Fundamentalists' and as a self-taught scientist he enjoyed more scientific repute amongst fundamentalists of the time. This interesting and unique collection (...) of original source material includes five of his writings between 1906 and 1924, challenging the new Darwinian theory of evolution and natural selection through his writings on the natural sciences. His literature covers the topics of evolution and biology and critiques biological arguments for evolution. He also wrote widely on geology offering his own alternative argument of 'flood geography' in opposition to the Darwinian theory concerning palaeontology and geology. This volume will be of interest to historians of natural history and the creationism movement, as well as scholars of religion and American history. (shrink)

Originally published in 1995, The Antievolution Works of Arthur I. Brown is the third volume in the series, Creationism in Twentieth Century America. The volume brings together original sources from the prominent surgeon and creationist Arthur I. Brown. Brown discredited evolution as it was contrary to the 'clear statements of scripture' which he believed infallible, stating evolution instead to be both a hoax and 'a weapon of Satan'. The works included focus on Brown's polemic through his early twentieth century writings. (...) The essays focus on his scientific investigations and provide a negative commentary upon Darwin's theory of evolution instead focusing on biblical explanations for evolution. As a scientist Brown's unique view of evolution from a creationist and scientific viewpoint provides a fascinating lens through which to view the historical debates surrounding evolution and provides a unique insight into how Darwinian theory affected both the scientific and religious communities. This book will be of interest to natural historians, and theologians as well as academics of philosophy and history. (shrink)

As past president of both the History of Science Society and the American Society of Church History, Ronald L. Numbers is uniquely qualified to assess the historical relations between science and Christianity. In this collection of his most recent essays, he moves beyond the clichés of conflict and harmony to explore the tangled web of historical interactions involving scientific and religious beliefs. In his lead essay he offers an unprecedented overview of the history of science and Christianity from the (...) perspective of the ordinary people who filled the pews of churchesor loitered around outside. Unlike the elite scientists and theologians on whom most historians have focused, these vulgar Christians cared little about the discoveries of Copernicus, Newton, and Einstein. Instead, they worried about the causes of the diseases and disasters that directly affected their lives and about scientists preposterous attempts to trace human ancestry back to apes. Far from dismissing opinion-makers in the pulpit, Numbers closely looks at two the most influential Protestant theologians in nineteenth-century America: Charles Hodge and William Henry Green. Hodge, after decades of struggling to harmonize Gods two revelationsin nature and in the Biblein the end famously described Darwinism as atheism. Green, on the basis of his careful biblical studies, concluded that Ussher's chronology was unreliable, thus opening the door for Christian anthropologists to accommodate the subsequent discovery of human antiquity. In Science without God Numbers traces the millennia-long history of so-called methodological naturalism, the commitment to explaining the natural world without appeals to the supernatural. By the early nineteenth century this practice was becoming the defining characteristic of science; in the late twentieth century it became the central point of attack in the audacious attempt of intelligent designers to redefine science. Numbers ends his reassessment by arguing that although science has markedly changed the world we live in, it has contributed less to secularizing it than many have claimed. Taken together, these accessible and authoritative essays form a perfect introduction to Christian attitudes towards science since the 17th century. (shrink)

Belief in the divine origin of the universe began to wane most markedly in the nineteenth century, when scientific accounts of creation by natural law arose to challenge traditional religious doctrines. Most of the credit - or blame - for the victory of naturalism has generally gone to Charles Darwin and the biologists who formulated theories of organic evolution. Darwinism undoubtedly played the major role, but the supporting parts played by naturalistic cosmogonies should also be acknowledged. Chief among these was (...) the nebular hypothesis proposed by Pierre Simon Laplace in 1796, which explained the origin of the solar system as a natural development over extended periods of time. Ronald Numbers focuses on Laplace's theory as it affected American scientific thought. he first traces the history of Laplace's cosmogony chronologically, from its European inception to its demise about 1900. the last three chapters explore some of the theological and scientific consequences resulting from the acceptance of this cosmogony. Most significant was the change in the status of supernatural doctrine. When the nebular hypothesis lost credence at the end of the nineteenth century, those who had before tried to accommodate natural theory with supernatural doctrine no longer felt compelled to do so when faced with succeeding theories. The nebular hypothesis, it seems, had established natural law in the heavens. (shrink)

As the crusade to outlaw the teaching of evolution changed to a battle for equal time for creationism, the ideological defenses of that doctrine also shifted from primarily biblical to more scientific grounds. This essay describes the historical development of “scientific creationism” from a variety of late–nineteenth– and early–twentieth–century creationist reactions to Charles Darwin's theory of evolution, through the Scopes trial and the 1960s revival of creationism, to the current spread of strict creationism around the world.

Originally published in 1995, Creation-Evolution Debates is the second volume in the series, Creationism in Twentieth Century America, reissued in 2021. The volume comprises eight debates from the early 1920s and 1930s between prominent evolutionists and creationists of the time. The original sources detail debates that took place either orally or in print, as well as active debates between creationists over the true meaning of Genesis I. The essays in this volume feature prominent discussions between the likes of Edwin Grant (...) Conklin, Henry Fairfield Osbourne and William Jennings Bryan, John Roach Francis and Charles Francis Potter, George McCready Price and Joseph McCabe and William Bell Riley versus Charles Smith, amongst many others. The collection will be of especial interest to natural historians, and theologians as well as academics of philosophy, and history. (shrink)

Over the course of human history, the sciences, and biology in particular, have often been manipulated to cause immense human suffering. For example, biology has been used to justify eugenic programs, forced sterilization, human experimentation, and death camps—all in an attempt to support notions of racial superiority. By investigating the past, the contributors to _Biology and Ideology from Descartes to Dawkins_ hope to better prepare us to discern ideological abuse of science when it occurs in the future. Denis R. Alexander (...) and Ronald L. Numbers bring together fourteen experts to examine the varied ways science has been used and abused for nonscientific purposes from the fifteenth century to the present day. Featuring an essay on eugenics from Edward J. Larson and an examination of the progress of evolution by Michael J. Ruse, _Biology and Ideology_ examines uses both benign and sinister, ultimately reminding us that ideological extrapolation continues today. An accessible survey, this collection will enlighten historians of science, their students, practicing scientists, and anyone interested in the relationship between science and culture. (shrink)