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 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)

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)

Charles Darwin's Origin of Species, published in 1859, was a revolutionary attempt “to overthrow the dogma of separate creations,” a declaration that provoked different reactions among the religious, ranging from mild enthusiasm to anger. Christians sympathetic to Darwin's effort sought to make Darwinism appear compatible with their religious beliefs. Two of Darwin's most prominent defenders in the United States were the Calvinists Asa Gray, a Harvard botanist, and George Frederick Wright, a cleric-geologist. Gray, who long favored a “special origination” in (...) connection with the evolution of humans and questioned whether natural selection can account for the formation of organs, the making of eyes, etc., embraced natural selection as the primary mechanism underlying the production of most species. He also went so far as to invoke divine providence, rather than randomness, to explain the variations on which natural selection acted. This chapter discusses creationism, intelligent design, and modern biology. (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)

This book, in language accessible to the general reader, investigates twelve of the most notorious, most interesting, and most instructive episodes involving the interaction between science and Christianity, aiming to tell each story in its historical specificity and local particularity. Among the events treated in When Science and Christianity Meet are the Galileo affair, the seventeenth-century clockwork universe, Noah's ark and flood in the development of natural history, struggles over Darwinian evolution, debates about the origin of the human species, (...) and the Scopes trial. Readers will be introduced to St. Augustine, Roger Bacon, Pope Urban VIII, Isaac Newton, Pierre-Simon de Laplace, Carl Linnaeus, Charles Darwin, T. H. Huxley, Sigmund Freud, and many other participants in the historical drama of science and Christianity. (shrink)

We explain the general significance of integer-based descriptors for natural shapes and show that the evolution of two such descriptors, called mechanical descriptors (the number _N_(_t_) of static balance points and the Morse–Smale graph associated with the scalar distance function measured from the center of mass) appear to capture (unlike classical geophysical shape descriptors) one of our most fundamental intuitions about natural abrasion: shapes get monotonically _simplified_ in this process. Thus mechanical descriptors help to establish a correlation between (...) subjective and objective descriptors of perceived objects. (shrink)

A common view is that natural language treats numbers as abstract objects, with expressions like the number of planets, eight, as well as the number eight acting as referential terms referring to numbers. In this paper I will argue that this view about reference to numbers in natural language is fundamentally mistaken. A more thorough look at natural language reveals a very different view of the ontological status of natural numbers. On this view, numbers are not (...) primarily treated abstract objects, but rather 'aspects' of pluralities of ordinary objects, namely number tropes, a view that in fact appears to have been the Aristotelian view of numbers. Natural language moreover provides support for another view of the ontological status of numbers, on which natural numbers do not act as entities, but rather have the status of plural properties, the meaning of numerals when acting like adjectives. This view matches contemporary approaches in the philosophy of mathematics of what Dummett called the Adjectival Strategy, the view on which number terms in arithmetical sentences are not terms referring to numbers, but rather make contributions to generalizations about ordinary (and possible) objects. It is only with complex expressions somewhat at the periphery of language such as the number eight that reference to pure numbers is permitted. (shrink)

Constructive set theory started with Myhill's seminal 1975 article [8]. This paper will be concerned with axiomatizations of the natural numbers in constructive set theory discerned in [3], clarifying the deductive relationships between these axiomatizations and the strength of various weak constructive set theories.

CHAPTER Meta-View BRIDGES When I was a child, I lived in an area renowned for its many wooden covered bridges. Sometimes my family would take a Sunday drive ...

One of the most important abilities we have as humans is the ability to think about number. In this chapter, we examine the question of whether there is an essential connection between language and number. We provide a careful examination of two prominent theories according to which concepts of the positive integers are dependent on language. The first of these claims that language creates the positive integers on the basis of an innate capacity to represent real numbers. The second claims (...) that language’s function is to integrate contents from modules that humans share with other animals. We argue that neither model is successful. (shrink)

In this paper, the author derives the Dedekind-Peano axioms for number theory from a consistent and general metaphysical theory of abstract objects. The derivation makes no appeal to primitive mathematical notions, implicit definitions, or a principle of infinity. The theorems proved constitute an important subset of the numbered propositions found in Frege's *Grundgesetze*. The proofs of the theorems reconstruct Frege's derivations, with the exception of the claim that every number has a successor, which is derived from a modal axiom that (...) (philosophical) logicians implicitly accept. In the final section of the paper, there is a brief philosophical discussion of how the present theory relates to the work of other philosophers attempting to reconstruct Frege's conception of numbers and logical objects. (shrink)

The article develops and extends the theory of Glenn Kessler (Frege, Mill and the foundations of arithmetic, Journal of Philosophy 77, 1980) that a (cardinal) number is a relation between a heap and a unit-making property that structures the heap. For example, the relation between some swan body mass and "being a swan on the lake" could be 4.

How children learn number concepts reflects the conceptual and logical distinction between counting numbers, based on a same-size concept for collections of objects, and natural numbers, constructed as an algebra defined by the Peano axioms for arithmetic. Cross-cultural research illustrates the cultural specificity of counting number systems, and hence the cultural context must be taken into account.

A central part of Frege's logicism is his reconstruction of the natural numbers as equivalence classes of equinumerous concepts or classes. In this paper, I examine the relationship of this reconstruction both to earlier views, from Mill all the way back to Plato, and to later formalist and structuralist views; I thus situate Frege within what may be called the “rise of pure mathematics” in the nineteenth century. Doing so allows us to acknowledge continuities between Frege's and other approaches, (...) but also to understand better the motivation and the significance of his innovations, as well as their limits. (shrink)

During the first months of 1887, while completing the drafts of his Mitteilungen zur Lehre vom Transfiniten, Georg Cantor maintained a continuous correspondence with Benno Kerry. Their exchange essentially concerned two main topics in the philosophy of mathematics, namely, (a) the concept of natural number and (b) the infinitesimals. Cantor's and Kerry's positions turned out to be irreconcilable, mostly because of Kerry's irremediably psychologistic outlook, according to Cantor at least. In this study, I will examine and reconstruct the main (...) points in the discussion around (a) and (b) and stress some interesting aspects of the philosophical and mathematical thought of Benno Kerry. (shrink)

Frege's main contributions to logic and the philosophy of mathematics are, on the one hand, his introduction of modern relational and quantificational logic and, on the other, his analysis of the concept of number. My focus in this paper will be on the latter, although the two are closely related, of course, in ways that will also play a role. More specifically, I will discuss Frege's logicist reconceptualization of the natural numbers with the goal of clarifying two aspects: the (...) motivations for its core ideas; the step-by-step development of these ideas, from Begriffsschrift through Die Grundlagen der Arithmetik and Grundgesetze der Arithmetik to Frege's very last writings, indeed even beyond those, to a number of recent "neo-Fregean" proposals for how to update them. One main development, or break, in Frege's views occurred after he was informed of Russell's antinomy. His attempt to come to terms with this antinomy has found some attention in the literature already. It has seldom been analyzed in connection with earlier changes in his views, however, partly because those changes themselves have been largely ignored. Nor has it been discussed much in connection with Frege's basic motivations, as formed in reaction to earlier positions. Doing both in this paper will not only shed new light on his response to Russell's antinomy, but also on other aspects of his views. In addition, it will provide us with a framework for comparing recent updates of these views, thus for assessing the remaining attraction of Frege's general approach. I will proceed as follows: In the first part of the paper (§1.1 and §1.2), I will consider the relationship of Frege's conception of the natural numbers to earlier conceptions, in particular to what I will call the "pluralities conception", thus bringing into sharper focus his core ideas and their motivations. In the next part (§2.1 and §2.2), I will trace the order in which these ideas come up in Frege's writings, as well as the ways in which his position gets modified along the way, both before and after Russell's antinomy.. (shrink)

Frege discussed Mill’s empiricist ideas and Kant’s rationalist ideas about the nature of mathematics, and employed Set Theory and logico-philosophical notions to develop a new concept for the natural numbers. All this is objectively exposed by this paper.

Theories of number concepts often suppose that the natural numbers are acquired as children learn to count and as they draw an induction based on their interpretation of the first few count words. In a bold critique of this general approach, Rips, Asmuth, Bloomfield [Rips, L., Asmuth, J. & Bloomfield, A.. Giving the boot to the bootstrap: How not to learn the natural numbers. Cognition, 101, B51–B60.] argue that such an inductive inference is consistent with a representational system (...) that clearly does not express the natural numbers and that possession of the natural numbers requires further principles that make the inductive inference superfluous. We argue that their critique is unsuccessful. Provided that children have access to a suitable initial system of representation, the sort of inductive inference that Rips et al. call into question can in fact facilitate the acquisition of larger integer concepts without the addition of any further principles. (shrink)

Until the second half of the 19th century the natural numbers were regarded as given and not further analysable. The concept of a class as defined by mathematicians of the time, Seeming more fundamental, Was then used to define the natural numbers. Their definitions of a class are unsuitable because of paradoxes and other difficulties. In this paper a new definition of a class is stated, And from this the natural numbers are defined.

The paper is concerned with the problem of individuation in Spinoza. Spinoza's account of individuation leads to the apparent contradiction between, on the one hand, the view that substance (God or Nature) is simple, eternal, and infinite, and on the other, the claim that substance contains infinite differentiation - determinate and finite modes, i.e. individuals. A reconstruction of Spinoza's argument is offered which accepts the reality of the contradiction and sees it as a consequence of Spinoza's way of posing the (...) problem of individuation: it is argued that Spinoza's ontology is constructed on the basis of his methodology, rather than conversely. The contradiction appears in its acutest form in Spinoza's discussion of number. The paper explains the philosophical motivation behind Spinoza's Problematik and reflects on the historical context of the contradiction to which that Problematik gives rise. (shrink)

It is known (see [1, 3.1.5]) that the order type of the nonstandard natural number system * N has the form ω + (ω * + ω) θ, where θ is a dense order type without first or last element and ω is the order type of N. Concerning this, Zakon [2] examined * N more closely and investigated the nonstandard real number system * R, as an ordered set, as an additive group and as a uniform space. He (...) raised five questions which remained unsolved. These questions are concerned with the cofinality and coinitiality of θ (which depend on the underlying nonstandard universe * U). In this paper, we shall treat nonstandard models where the cofinality and coinitiality of θ coincide with some appropriated cardinals. Using these nonstandard models, we shall give answers to three of these questions and partial answers to the other to questions in [2]. (shrink)

The conceptual building blocks suggested by developmental psychologists may yet play a role in how the human learner arrives at an understanding of natural number. The proposal of Rips et al. faces a challenge, yet to be met, faced by all developmental proposals: to describe the logical space in which learners ever acquire new concepts.

Theorems of arithmetic are used, perhaps essentially, to reach conclusions about the natural world. This applicability can be explained in a natural way by analogy with the applicability of statements of law to the world. ;In order to carry out an ontological argument for my thesis, I assume the existence of universals as a working hypothesis. I motivate a theory of laws according to which statements of law are singular statements about scientific properties. Such statements entail generalizations about (...) instances of those properties. My task, then, is to show that theorems of arithmetic, in application, are similar to statements of law in this respect. On the basis of my working hypothesis, I argue that there are inclining reasons to think that cardinal numbers are scientific properties whose instances occur in the natural world. The works of Mill, Frege and Benacerraf play central roles in these arguments. ;In general, I urge that there are no conclusive epistemological or ontological factors which preclude arithmetical truths from being laws of nature. Consequently, I think we can extend any semantic theory which is adequate for statements of law to theorems of arithmetic. This would apply even to a theory of predication which does not involve commitment to universals. So, to the degree that my argument succeeds with the working hypothesis, it also supports the unconditioned thesis that arithmetical truths are laws of nature. (shrink)

There are two ways of thinking about the natural numbers: as ordinal numbers or as cardinal numbers. It is, moreover, well-known that the cardinal numbers can be defined in terms of the ordinal numbers. Some philosophies of mathematics have taken this as a reason to hold the ordinal numbers as (metaphysically) fundamental. By discussing structuralism and neo-logicism we argue that one can empirically distinguish between accounts that endorse this fundamentality claim and those that do not. In particular, we argue (...) that if the ordinal numbers are metaphysically fundamental then it follows that one cannot acquire cardinal number concepts without appeal to ordinal notions. On the other hand, without this fundamentality thesis that would be possible. This allows for an empirical test to see which account best describes our actual mathematical practices. We then, finally, discuss some empirical data that suggests that we can acquire cardinal number concepts without using ordinal notions. However, there are some important gaps left open by this data that we point to as areas for future empirical research. (shrink)

Number concepts must support arithmetic inference. Using this principle, it can be argued that the integer concept of exactly ONE is a necessary part of the psychological foundations of number, as is the notion of the exact equality - that is, perfect substitutability. The inability to support reasoning involving exact equality is a shortcoming in current theories about the development of numerical reasoning. A simple innate basis for the natural number concepts can be proposed that embodies the arithmetic principle, (...) supports exact equality and also enables computational compatibility with real- or rational-valued mental magnitudes. (shrink)

The purpose of this article is to analyse the mathematical practices leading to Rafael Bombelli’s L’algebra (1572). The context for the analysis is the Italian algebra practiced by abbacus masters and Renaissance mathematicians of the fourteenth to sixteenth centuries. We will focus here on the semiotic aspects of algebraic practices and on the organisation of knowledge. Our purpose is to show how symbols that stand for underdetermined meanings combine with shifting principles of organisation to change the character of algebra.

How do we get out knowledge of the natural numbers? Various philosophical accounts exist, but there has been comparatively little attention to psychological data on how the learning process actually takes place. I work through the psychological literature on number acquisition with the aim of characterising the acquisition stages in formal terms. In doing so, I argue that we need a combination of current neologicist accounts and accounts such as that of Parsons. In particular, I argue that we learn (...) the initial segment of the natural numbers on the basis of the Fregean definitions, but do not learn the natural number structure as a whole on the basis of Hume's principle. Therefore, we need to account for some of the consistency of our number concepts with the Dedekind-Peano axioms in other terms. (shrink)

It is sometimes suggested that criteria of identity should play a central role in an account of our most fundamental ways of referring to objects. The view is nicely illustrated by an example due to (Quine, 1950). Suppose you are standing at the bank of a river, watching the water that floats by. What is required for you to refer to the river, as opposed to a particular segment of it, or the totality of its water, or the current temporal (...) part of this water? According to Quine, you must at least implicitly be operating with some criterion of identity that informs you when two sightings of water count as sightings of the same referent. For unless you have at least an implicit grasp of what is required for your intended referent to be identical with another object with which you are presented, you have not succeeded in singling out a unique object for reference. (shrink)

We propose a reformulation of the ideal $\mathcal {N}$ of Lebesgue measure zero sets of reals modulo an ideal J on $\omega $, which we denote by $\mathcal {N}_J$. In the same way, we reformulate the ideal $\mathcal {E}$ generated by $F_\sigma $ measure zero sets of reals modulo J, which we denote by $\mathcal {N}^*_J$. We show that these are $\sigma $ -ideals and that $\mathcal {N}_J=\mathcal {N}$ iff J has the Baire property, which in turn is equivalent to (...) $\mathcal {N}^*_J=\mathcal {E}$. Moreover, we prove that $\mathcal {N}_J$ does not contain co-meager sets and $\mathcal {N}^*_J$ contains non-meager sets when J does not have the Baire property. We also prove a deep connection between these ideals modulo J and the notion of nearly coherence of filters (or ideals). We also study the cardinal characteristics associated with $\mathcal {N}_J$ and $\mathcal {N}^*_J$. We show their position with respect to Cichoń’s diagram and prove consistency results in connection with other very classical cardinal characteristics of the continuum, leaving just very few open questions. To achieve this, we discovered a new characterization of $\mathrm {add}(\mathcal {N})$ and $\mathrm {cof}(\mathcal {N})$. We also show that, in Cohen model, we can obtain many different values to the cardinal characteristics associated with our new ideals. (shrink)