I begin with a personal confession. Philosophical discussions of existence have always bored me. When they occur, my eyes glaze over and my attention falters. Basically ontological questions often seem best decided by banging on the table--rocks exist, fairies do not. Argument can appear long-winded and miss the point. Sometimes a quick distinction resolves any apparent difficulty. Does a falling tree in an earless forest make noise, ie does the noise exist? Well, if noise means that an ear must (...) be there to hear it, then the answer to the question is evidently "no." But if noise means that, if there were (counterfactually) someone there, then he would hear it, then just as obviously, the answer becomes "yes.". (shrink)

We examine the arguments on both sides of the recent debate (Hale and Wright v. Field) on the existence, and modal status, of the natural numbers. We formulate precisely, with proper attention to denotational commitments, the analytic conditionals that link talk of numbers with talk of numerosity and with counting. These provide conceptual controls on the concept of number. We argue, against Field, that there is a serious disanalogy between the existence of God and the (...) class='Hi'>existence of numbers. We give stronger reasons than those advanced by Wright for resisting Field's analogy. We argue that the rules governing the basic numerical notions commit us to the natural numbers as necessary existents. We also show that the latest twist in the debate involving 'surdons' leaves both sides in a stalemate. (shrink)

The Frege-Russell view is that existence is a second-order property rather than a property of individuals. One of the most compelling arguments for this view is based on the premise that there is an especially close connection between existence and number. The most promising version of this argument is by C.J.F Williams (1981, 1992). In what follows, I argue that this argument fails. I then defend an account according to which both predications of number and existence attribute (...) properties to individuals. (shrink)

This paper analyzes and explains certain parts of Syrianus’s Commentary on book M of Aristotle’s Metaphysics, which details Syrianus’s response to Aristotle’s attack against the Platonic position of the separate existence of numbers. Syrianus defends the separate existence not only of eidetic but also of mathematical numbers, following a line of argumentation which involves a hylomorphic approach to the latter. He proceeds with an analysis of the mathematical number into matter and form, but his interpretation entails (...) that form is the constituent of number, which has the status and role of a Platonic Form. This solution allows him not only to explain and justify the unity of number, but also to apply the Platonic thesis of the separate existence of numbers, to the mathematical or monadic numbers themselves. It also betrays its tendency to combine theses of the Platonic Ontology with fundamental Aristotelian doctrines. (shrink)

Answers are given to two questions concerning the existence of some sparse subsets of $\mathscr{H} = \{0, 1,..., H - 1\} \subseteq * \mathbb{N}$ , where H is a hyperfinite integer. In § 1, we answer a question of Kanovei by showing that for a given cut U in H, there exists a countably determined set $X \subseteq \mathscr{H}$ which contains exactly one element in each U-monad, if and only if U = a · N for some $a \in (...) \mathscr{H} \backslash \{0\}$ . In §2, we deal with a question of Keisler and Leth in [6]. We show that there is a cut $V \subseteq \mathscr{H}$ such that for any cut U, (i) there exists a U-discrete set $X \subseteq \mathscr{H}$ with X + X = H (mod H) provided $U \subsetneqq V$ , (ii) there does not exist any U-discrete set $X \subseteq \mathscr{H}$ with X + X = H (mod H) provided $\supsetneqq V$ . We obtain some partial results for the case U = V. (shrink)

We investigate properties of universal numberings of finite families of d.c.e. sets. We show different cases of finite families of d.c.e. sets for which there is a universal numbering and for which there is not.

Numbers are concepts whose content, structure, and organization are influenced by the material forms used to represent and manipulate them. Indeed, as argued here, it is the inclusion of multiple forms (distributed objects, fingers, single- and two-dimensional forms like pebbles and abaci, and written notations) that is the mechanism of numerical elaboration. Further, variety in employed forms explains at least part of the synchronic and diachronic variability that exists between and within cultural number systems. Material forms also impart characteristics (...) like linearity that may persist in the form of knowledge and behaviors, ultimately yielding numerical concepts that are irreducible to and functionally independent of any particular form. Material devices used to represent and manipulate numbers also interact with language in ways that reinforce or contrast different aspects of numerical cognition. Not only does this interaction potentially explain some of the unique aspects of numerical language, it suggests that the two are complementary but ultimately distinct means of accessing numerical intuitions and insights. The potential inclusion of materiality in contemporary research in numerical cognition is advocated, both for its explanatory power, as well as its influence on psychological, behavioral, and linguistic aspects of numerical cognition. (shrink)

The capacity of visual attention/STM can be determined by change-detection experiments. Detecting the presence of change leads to an estimate of 4 items, while detecting the absence of change leads to an estimate of 1 item. Thus, there are two magical numbers in vision: 4 and 1. The underlying limits, however, are not necessarily those of central STM.

Originally published in 1992. The history of Western philosophy can be seen as a battle between those that insist that the "physical universe" exists and those would claim that there is a much larger "world" which contains atemporal and nonspatial things as well. The central part of this book, and the battle, concerns the existence of universals. Starting with the mediaeval definition of the issue found in Porphry and Boethius, the author then considers modern and contemporary versions of the (...) battle. He concludes that what is at stake between naturalists and ontologists is the existence and nature of a number of important categories, like structures, relations, sets, numbers and so on. (shrink)

Linnebo in 2018 argues that abstract objects like numbers are “thin” because they are only required to be referents of singular terms in abstraction principles, such as Hume's principle. As the specification of existence claims made by analytic truths (the abstraction principles), their existence does not make any substantial demands of the world; however, as Linnebo notes, there is a potential counter-argument concerning infinite regress against introducing objects this way. Against this, he argues that vicious regress is (...) avoided in the account of arithmetic based on Hume's principle because we are specifying numbers in terms of the concept of equinumerosity, or its ordinal equivalent. But far from being only a matter for philosophy, this implies a distinct empirical prediction: in cognitive development, the principle of equinumerosity is primary to number concepts. However, by analysing and expanding on the bootstrapping theory of Carey in 2009, I argue in this paper that there are good reasons to think that the development could be the other way around: possessing numerosity concepts may precede grasping the principle of equinumerosity. I propose that this analysis of early numerical cognition can also help us understand what numbers as thin objects are like, moving away from Platonist interpretations. (shrink)

Challenging the myth that mathematical objects can be defined into existence, Bigelow here employs Armstrong's metaphysical materialism to cast new light on mathematics. He identifies natural, real, and imaginary numbers and sets with specified physical properties and relations and, by so doing, draws mathematics back from its sterile, abstract exile into the midst of the physical world.

A feature of Frege's philosophy of arithmetic that has elicited a great deal of attention in the recent secondary literature is his contention that numbers are ‘self‐subsistent’ objects. The considerable interest in this thesis among the contemporary philosophy of mathematics community stands in marked contrast to Kreisel's folk‐lore observation that the central problem in the philosophy of mathematics is not the existence of mathematical objects, but the objectivity of mathematics. Although Frege was undoubtedly concerned with both questions, a (...) goal of the present paper is to argue that his success in securing the objectivity of arithmetic depends on a less contentious commitment to numbers as objects than either he or his critics have supposed. As such, this paper is an articulation and defense of both Frege's analysis of arithmetic and Kreisel's observation. (shrink)

A feature of Frege's philosophy of arithmetic that has elicited a great deal of attention in the recent secondary literature is his contention that numbers are ‘self‐subsistent’ objects. The considerable interest in this thesis among the contemporary philosophy of mathematics community stands in marked contrast to Kreisel's folk‐lore observation that the central problem in the philosophy of mathematics is not the existence of mathematical objects, but the objectivity of mathematics. Although Frege was undoubtedly concerned with both questions, a (...) goal of the present paper is to argue that his success in securing the objectivity of arithmetic depends on a less contentious commitment to numbers as objects than either he or his critics have supposed. As such, this paper is an articulation and defense of both Frege's analysis of arithmetic and Kreisel's observation. (shrink)

What are the meanings of number expressions, and what can they tell us about questions of central importance to the philosophy of mathematics, specifically 'Do numbers exist?' This Element attempts to shed light on this question by outlining a recent debate between substantivalists and adjectivalists regarding the semantic function of number words in numerical statements. After highlighting their motivations and challenges, I develop a comprehensive polymorphic semantics for number expressions. I argue that accounting for the numerous meanings and how (...) they are related leads to a strengthened argument for realism, one which renders familiar forms of nominalism highly implausible. (shrink)

This book explores two questions that are integral to the question of the existence of God. The first question concerns the meaning of "existence" and the second concerns the meaning of "God". Regarding the first question, this book motivates, presents and defends the meta-ontology found in Gottlob Frege's writings and defended by Michael Dummett, Crispin Wright and Bob Hale. Frege's approach to questions of existence has mainly found use in connection with abstract objects such as numbers. (...) This is one of the first studies to systematically present Fregean meta-ontology and apply it to theology. Frege's meta-ontology is informed by his context principle. According to this, logico-syntactic notions such as "singular term" and "predicate" are pivotal to questions of what exists. These notions serve to throw light on the second question. Through thorough engagement with Old as well and New Testament texts, the book shows how Frege's logico-syntactic notions are of crucial importance when seeking to understand the meaning and use of "God". To complete the defence of Fregean meta-ontology, the book concludes by pointing to important differences between the otherwise closely associated concepts of an object found in Wittgenstein's Tractatus Logico-Philosophicus and Frege's writings. (shrink)

In the October number of Philosophy x appears a very interesting article by Professor H. H. Price, entitled “Our Evidence for the Existence of Other Minds,” the main object of which is to formulate the grounds on which we may claim logical justification for asserting that other minds exist. No attempt is made to “prove” the existence of other minds—an effort which is regarded as a wild-goose chase. Neither does Prof. Price seek to identify the actual process by (...) which a particular mind accepts the existence of others as a fact; that, presumably, is a matter for the psychologist, and it may well turn out that the phenomena which give the conviction that other minds exist may vary from individual to individual and may often be logically unevidential. The object of the article, in short, is simply to state why it is reasonable to believe that other minds exist. (shrink)

This paper is concerned with an intuitive contrast that arises when we consider sentences containing empty definite descriptions. A sentence like ‘The king of France is bald’ appears neither true nor false, while a sentence like ‘My friend was visited by the king of France’ appears false. Recently, Stephen Yablo has suggested an account of this intuitive contrast. Yablo’s account is particularly interesting, since it has important consequences for the ontological commitments of number sentences like ‘The number of planets is (...) even’. However, the paper argues that Yablo’s account is not convincing and that it can thus not establish these consequences. Further, it develops a Strawsonian account of the intuitive contrast. The developed account allows us to draw important conclusions regarding the correct analysis of definite descriptions and of existence sentences containing definite descriptions. (shrink)

Let X be an infinite set and let and denote the propositions “every filter on X can be extended to an ultrafilter” and “X has a free ultrafilter”, respectively. We denote by the Stone space of the Boolean algebra of all subsets of X. We show: For every well‐ordered cardinal number ℵ, (ℵ) iff (2ℵ). iff “ is a continuous image of ” iff “ has a free open ultrafilter ” iff “every countably infinite subset of has a limit point”. (...) implies “every open filter on extends to an open ultrafilter” implies “has an open ultrafilter” implies It is relatively consistent with that (ω) holds, whereas (ω) fails. In particular, none of the statements given in (2) implies (ω). (shrink)

This paper presents a comprehensive mathematical framework in which a unified treatment of additive and expected utility can be given. For achieving this, elaborate structural assumptions, characterizing a simply ordered, topological semigroup, have to be established in order to construct an isomorphism with the additive group of real numbers.This construction establishes a link between additive and expected utility theory to the extent that the same mathematical considerations leading to the derivation of an additive representation are also valid for proving (...) the expected utility theorem. (shrink)

What is a number? Answering this will answer questions about its philosophical foundations - rational numbers, the complex numbers, imaginary numbers. If we are to write or talk about something, it is helpful to know whether it exists, how it exists, and why it exists, just from a common-sense point of view [Quine, 1948, p. 6]. Generally, there does not seem to be any disagreement among mathematicians, scientists, and logicians about numbers existing in some way, but (...) currently, in the mainstream arena only definitions, descriptions of properties, and effects are presented as evidence. Enough historical description of numbers in history provides an empirical basis of number, although a case can be made that numbers do not exist by themselves empirically. Correspondingly, numbers exist as abstractions. All the while, though, these "descriptions" beg the question of what numbers are ontologically. Advocates for numbers being the ultimate reality have the problem of wrestling with the nature of reality. I start on the road to discovering the ontology of number by looking at where people have talked about numbers as already existing: history. Of course, we need to know not only what ontology is but the problems of identifying one, leading to the selection between metaphysics and provisional approaches. While we seem to be dimensionally limited, at least we can identify a more suitable bootstrapping ontology than mere definitions, leading us to the unity of opposites. The rest of the paper details how this is done and modifies Peano's Postulates. (shrink)

This article offers an argument for a form of panentheism in which the divine is conceived as both ‘God the World’ and ‘God the Good’. ‘God the World’ captures the notion that the totality of everything which exists is ‘in’ God, while acknowledging that, given evil and suffering, not everything is ‘of’ God. ‘God the Good’ encompasses the idea that God is also the universal concept of Goodness, akin to Plato’s Form of the Good as developed by Iris Murdoch, which (...) is inextricably conjoined with God the World because it is the nature of the world which determines the nature of perfect Goodness. This form of ‘conjoined’ panentheism yields a concept of divine personhood which includes both divine agency and human/divine engagement. God the Good is an agent of change by providing human persons with a standard of Goodness against which to measure the goodness of their own actions, while God the World provides the physical embodiment through which God acts. Human engagement with the divine may take a number of forms and may lead to moral action, the means by which the divine acts upon the world and changes it for the better. (shrink)

This study examines a number of different answers to the question: wheredoes Avicenna demonstrate the existence of God within the Metaphysics of the Healing? Many interpreters have contended that there is an argument for God’s existence in Metaphysics of the Healing I.6–7. In this study I show that such views are incorrect and that the only argument for God’s existence in the Metaphysics of the Healing is found in VIII.1–3. My own interpretation relies upon a careful consideration (...) of the scientific order and first principles of the Metaphysics of the Healing, paying attention to Avicenna’s own explicit statements concerning the goals andintentions of different books and chapters, and a close analysis of the structure ofthe different arguments found in the relevant texts of the Metaphysics of the Healing. I conclude that Avicenna’s explicit goal in I.6–7 is to establish the properties that belong to necessary existence and possible existence, which consists, not in ademonstration of God’s existence, but in a dialectical treatment of the first principlesof metaphysics. (shrink)

In ordinary life everyone assumes that he has a great deal of knowledge about other minds or persons. This assumption has naturally aroused the curiosity of philosophers; though perhaps they have not been as curious about it as they ought to have been, for they have devoted many volumes to our consciousness of the material world, but very few to our consciousness of one another. It was thought at one time that each of us derives his knowledge of other minds (...) from the observation of other human organisms. I observe that there are a number of bodies which resemble my own fairly closely in their shape, size, and manner of movement; I conclude by analogy that each of these bodies is animated by a. (shrink)

Recent years have witnessed an enormous increase in behavioral and neuroimaging studies of numerical cognition. Particular interest has been devoted toward unraveling properties of the representational medium on which numbers are thought to be represented. We have argued that a correct inference concerning these properties requires distinguishing between different input modalities and different decision/output structures. To back up this claim, we have trained computational models with either symbolic or nonsymbolic input and with different task requirements, and showed that this (...) allowed for an integration of the existing data in a consistent manner. In later studies, predictions from the models were derived and tested with behavioral and neuroimaging methods. Here we present an integrative review of this work. (shrink)

The efforts to prove God's existence in the history of thought have been one of the fundamental problems of philosophy and theology, and even the most important one. The evidences put furword to prove the existence of God constitute the center of philosophy of religion’s problems not only philosophy of religion, but also the disciplines such as theology-kalam and Islamic philosophy are also seriously concerned. When we look at the history of philosophy, it is clear that almost all (...) philosophers are interested in the evidence in favour of and against the existence of God. Especially in recent times, religious experience has often been regarded as evidence of the existence of God. Studies in the field of religious and mystical experience have put forward many arguments concerning the use of religious experience as evidence. However, this study has been limited to experience of sense and pure consciousness claims rather than examining all of these arguments. In this study firstly, the analogy between experience of sense and religious experience will be deal with, and then claims that conciousness can be emptied by a number of methods which are defended by mystics. In addition, it will be tried to defend the opinion that the mystical experiences which are the common point of the two approaches are valid. The main emphasis of the article will be that these arguments can be used as evidence for the existence of God.Summary: Religious experience - which has not been considered as one of the classical evidences regarding the existence of God before being popular by recent studies – is started to be called one of the arguments. The most important reason for not arguing religious experience among the evidences of God’s existence is that it is a personal experience and therefore contains subjective perspectives. However, many researchers, who refuse to accept the criticism raised against religious experience with restricting to subjectivity, emphasized the importance of religious experience by highlighting the objective aspects of such experiences. This emphasis was not only on its significance, but also on the fact that it had a cognitive feature and thus it could provide evidence for the existence of God. The argument of religious experience is generally defined as an effort for demonstrating to existence of God with reference to several feelings such as spiritual experiences of human in face of creator and sacred Being, encounter with God, uniting, and annihilating selfness in Him. The emergence of such efforts, especially after the XVIIIth century, indicates that the history of the evidence does not go back much earlier. However, even if notas evidence, it is possible to extend the traces of religious and mystical experience to the history of religions such as Buddhism and Hinduism. Considering a great amount of the previous literature focused on religious experience in the West, It is easily seen that such research is still new. The main purpose in addressing such a controversial issue is to make the argument on the religious experience more current and to show whether the religious experience would be able to use as evidence for the existence of God.First title of this study comprises many different claims such as Walter T. Stace’s The Argument from Unanimity Richard Swinburne’s The Principle of Credulity, William P. Alston’s Perceiving God, John Hick’s Experience-as. However, instead of briefly giving out very different claims, the study chooses to focus on two arguments which are less discussed and can be given in more detail. The first is the argument based on the analogy between the religious experience and the sense experience. The fundamental point of this argument is sense experience. With reference to the similarities between the sense experience and the religious experience, it can be argued that religious experience is also cognitive and may function as an evidence for the existence of God. In other words, being perceptual and noetic, namely cognitive, of sense experience is addressed as a similar side of religious experience. Whether it is imaginary or real, eventually, it is necessary for an experience to have an object and being experience of something. Therefore, the religious experience includes the perception of a Transcendent Being that is beyond and in excess of the individual who experiences. Perception of the sense experience will definitely be different from the perception of religious experience. Researchers who think that the five sense organs do not run the perception of Sacred Beings, identified broader and various meanings for the perception. Thereby, in this approach, where the semantic field of perception is expanded, it was contended that human beings were able to perceive the transcendent, eternal, and unlimited God, and this can be explained in the epistemological sense. In order to use religious experiences as an evidence for the existence of God, it is necessary to determine whether such experiences are caused by God rather than something else. This raises the issue of the validity of religious experience. The researchers in this area have propounded various criteria to identify whether religious experience is existed or not. Experiences, which do not meet the criteria, are not considered existed even if they claim to be religious experience.The second is the argument that discusses whether it is possible to suggest an evidence for the existence of God by utilizing pure consciousness events. In general, this argument claims that the consciousness gets clear, and thus, mystic or Sufi experiences the Divine One with pure consciousness by discharging consciousness from all sensual contents and removing it from secular desire and wishes. The argument, which claims the experience of Divine One is direct, therefore uninterpretable, and experienced by pure consciousness, believes clearing the consciousness of mystics from sensual qualities such as seeing, hearing, and tasting. It is stated that it is possible for mystics to empty their sensory images and get rid of them, and thus, their desires and wishes could be lost. Especially, the thinkers who are connected with the philosophy of perennialism and defend this approach, thought that people, who have distinct religion, culture, and beliefs in different parts of the world and in various time periods, had the same experience and considered such experiences were universal by suggesting that this similarity could be called as common essence.Even though the claim which offers that the events of pure consciousness are the same everywhere have been criticized in many respects, the advocates of this approach have argued that the experience has the common essence, but the interpretation is different. This relationship, which was tried to be established between experience and commentary, caused a lot of work for both the thinkers who put forward the claims of pure consciousness and the researchers who opposed it. However, examining the relationship between experience and interpretation requires serious studies in socio-psychological, cultural and religious fields. Therefore, in order not to exceed the limits of the article, the relationship between experience and interpretation is briefly mentioned.While discussing two arguments, the main claims of the arguments were addressed, and the criticisms directed to this argument were not discussed as much as possible. However, some fundamental criticisms have been mentioned shortly for a better understanding of the discussion. Since such western-oriented claims mostly were deployed in a western context, the study excluded the comparison of the claims across different cultures. (shrink)

The present essay examines and critically discusses Paul Benacerraf's antiplatonist argument of "What Numbers Could Not Be." In the course of defending platonism against Benacerraf's semantic skepticism, I develop a novel platonist analysis of the content of arithmetic on the basis of which the necessary existence of the natural numbers and the nature of numerical reference are explained.

Joan Weiner has argued that Frege’s definitions of numbers constitute linguistic stipulations that carry no ontological commitment: they don’t present numbers as pre-existing objects. This paper offers a critical discussion of this view, showing that it is vitiated by serious exegetical errors and that it saddles Frege’s project with insuperable substantive difficulties. It is first demonstrated that Weiner misrepresents the Fregean notions of so-called Foundations-content, and of sense, reference, and truth. The discussion then focuses on the role of (...) definitions in Frege’s work, demonstrating that they cannot be understood as mere linguistic stipulations, since they have an ontological aim. The paper concludes with stressing both the epistemological and the ontological aspects of Frege’s project, and their crucial interdependence. (shrink)

In the first experiment, I exhibit unreliable judgment about the primeness or divisibility of four-digit numbers, in contrast to a seeming Excel program. In the second experiment, I exhibit an imperfect memory for arbitrary-seeming three-digit number and letter combinations, in contrast to my seeming collaborator with seemingly hidden notes. In the third experiment, I seem to suffer repeated defeats at chess. In all three experiments, the most straightforward interpretation of the experiential evidence is that something exists in the universe (...) that is superior in the relevant respects – theoretical reasoning (about primes), memorial retention (for digits and letters), or practical reasoning (at chess) – to my own solipsistically-conceived self. (shrink)

In this paper, I study the kind of questions we can ask about the existence of numbers. In addition to asking whether numbers exist, and how, I argue that there is also a third relevant question: why numbers exist. In platonist and nominalist accounts this question may not make sense, but in the psychologist account I develop, it is as well-placed as the other two questions. In fact, there are two such why-questions: the causal why-question asks (...) what causes numbers to exist and the teleological why-question asks for what purpose numbers exist. I argue that in a psychologist constructivist account, in which numbers are understood to exist as referents of a particular type of culturally shared concepts, both why-questions can get plausible answers. (shrink)

This paper argues that Carnap both did not view and should not have viewed Frege's project in the foundations of mathematics as misguided metaphysics. The reason for this is that Frege's project was to give an explication of number in a very Carnapian sense — something that was not lost on Carnap. Furthermore, Frege gives pragmatic justification for the basic features of his system, especially where there are ontological considerations. It will be argued that even on the question of the (...) independent existence of abstract objects, Frege and Carnap held remarkably similar views. I close with a discussion of why, despite all this, Frege would not accept the principle of tolerance. (shrink)

It is probably true to say that the majority of philosophers have considered the universe to be mental. If the universe is really mental, it follows that matter cannot be quite real, and many philosophers have in fact brought forward cogent reasons for regarding matter as in some sense illusory. Those who hold this view are called Idealists. Idealism has historically assumed a number of different forms, between some of which there is little in common, but all forms of Idealism (...) agree in maintaining that objects, such as chairs and tables, of which common sense believes the external world to exist, are not material. (shrink)

It is frequently claimed that the human mind is organized in a modular fashion, a hypothesis linked historically, though not inevitably, to the claim that many aspects of the human mind are innately specified. A specific instance of this line of thought is the proposal of an innately specified geometric module for human reorientation. From a massive modularity position, the reorientation module would be one of a large number that organized the mind. From the core knowledge position, the reorientation module (...) is one of five innate and encapsulated modules that can later be supplemented by use of human language. In this paper, we marshall five lines of evidence that cast doubt on the geometric module hypothesis, unfolded in a series of reasons: (1) Language does not play a necessary role in the integration of feature and geometric cues, although it can be helpful. (2) A model of reorientation requires flexibility to explain variable phenomena. (3) Experience matters over short and long periods. (4) Features are used for true reorientation. (5) The nature of geometric information is not as yet clearly specified. In the final section, we review recent theoretical approaches to the known reorientation phenomena. (shrink)

The existence of natural laws, whether deterministic or indeterministic, and whether exceptionless or ceteris paribus, seems puzzling because it implies that mindless bits of matter behave in a consistent and co-ordinated way. I explain this puzzle by showing that a number of attempted solutions fail. The puzzle could be resolved if it were assumed that natural laws are a manifestation of God’s activity. This argument from natural law to God’s existence differs from its traditional counterparts in that, whereas (...) the latter seek to explain the fact of natural laws, the former seeks to explain their possibility. The customary objections to the traditional arguments cannot be successfully adapted to counter this new argument, with one exception which has only limited effect. I rebut four claims that the theistic solution to the puzzle about natural laws is paradoxical, though I concede that one of these claims has merit. I consider four objections to the new argument but find three of them more or less unsatisfactory. The fourth, if successful, would undermine our claims to know the truth about the world. (shrink)

Robert Pargetter has recently argued that, even if the theist cannot produce plausible explanations for the evil we experience, the atheologian has no justifiable basis for claiming that evil can in any sense count as strong evidence against God's existence. His strategy is to challenge as question-begging (1) the atheologian's assumption that a prima facie conflict between God and evil exists and (2) the atheologian's claim that God's nonexistence is a more plausible explanation for unresolved (unexplained) evil than a (...) number of theistic options. I argue that Pargetter is unsuccessful, mainly because he (1) fails to understand clearly the conditions under which a prima facie moral conflict exists and (2) fails to distinguish clearly between 'plausibility' and 'possibility' as these terms are applicable to explanatory hypotheses. (shrink)

Since the 1960s an increasing number of philosophers have endorsed the thesis that there can be no such thing as “the best possible world.” In this paper I examine the main arguments for this thesis as put forth by George Schlesinger, Alvin Plantinga, Bruce Reichenbach, Peter Forrest, and Richard Swinburne. I argue that none of these arguments succeed in establishing the thesis and that the logical possibility of the best possible world is as yet an open question.

In a number a passages Descartes appears to insist that "I am, I exist" and its variants are wholly indubitable. These passages present an intractable problem of interpretation in the face of passages in which Descartes allows that any result is dubitable, "I am, I exist" included. Here I pull together a number of elements of Descartes' system to show how all of these passages hang together. If my analysis is correct, it tells us something about the perspective that Descartes (...) himself thinks we should take in reading the "Meditations". (shrink)

We show how to determine the k-th bit of Chaitin’s algorithmically random real number Ω by solving k instances of the halting problem. From this we then reduce the problem of determining the k-th bit of Ω to determining whether a certain Diophantine equation with two parameters, k and N , has solutions for an odd or an even number of values of N . We also demonstrate two further examples of Ω in number theory: an exponential Diophantine equation with (...) a parameter k which has an odd number of solutions iff the k-th bit of Ω is 1, and a polynomial of positive integer variables and a parameter k that takes on an odd number of positive values iff the k-th bit of Ω is 1. (shrink)

Hindman’s Theorem says that every finite coloring of the positive natural numbers has a monochromatic set of finite sums. Ramsey algebras, recently introduced, are structures that satisfy an analogue of Hindman’s Theorem. It is an open problem posed by Carlson whether every Ramsey algebra has an idempotent ultrafilter. This paper develops a general framework to study idempotent ultrafilters. Under certain countable setting, the main result roughly says that every nondegenerate Ramsey algebra has a nonprincipal idempotent ultrafilter in some nontrivial (...) countable field of sets. This amounts to a positive result that addresses Carlson’s question in some way. (shrink)

This book is about universals. "Mathematics," we learn, "is the theory of universals". Natural numbers turn out to be universals, as do real numbers, complex numbers, and sets. It is natural, therefore, that the reader demand some evidence that universals abound sufficiently to supply models of canonical mathematical theories. The author devotes nearly a third of his book to one potential source of such evidence: the Truthmaker axiom. In the case of some propositions P, Truthmaker entails that (...) if P is true, then there are things whose joint existence strictly implies P's truth. These things are truthmakers for P. Might Truthmaker, then, transform evidence for the truth of propositions into evidence for the existence of truthmaking universals? And might we thus be guaranteed a rich supply of numbers and sets? The author thinks not: "universals must not be construed as truthmakers". Something is a universal only if it can be instantiated by individuals x and y such that x's being an instance does not strictly imply y's being an instance. Since this characteristic of universals is incompatible with natural assumptions about truthmakers, the author's disheartening conclusion follows: "universals should not be expected to play any distinctive role in truthmaking". So Truthmaker is not a wellspring of universals. Why should we, then, believe that universals abound? One must understand, first, that the author embraces David Armstrong's a posteriori realism: "Everything there is is physical.... Hence universals, too, are physical. That is to say, the universals which exist are all real physical properties and relations among physical things". Universals are reasonably posited, then, only if their existence is entailed by our best accounts of the physical universe: "In cutting universals away from truthmakers, I am making their existence a matter, not for a priori proof from logic alone, but for total science". Sentences not interpretable in optimal physical theory will, on this view, not be modeled by universals and, hence, will be either bad mathematics or no part of mathematics at all. (shrink)

In Book I, part iv, section 2 of the Treatise, "Of scepticism with regard to the senses," Hume presents two different answers to the question of how we come to believe in the continued existence of unperceived objects. He rejects his first answer shortly after its formulation, and the remainder of the section articulates an alternative account of the development of the belief. The account that Hume adopts, however, is susceptible to a number of insurmountable objections, which motivates a (...) reassessment of his original proposal. This paper defends a version of Hume's initial explanation of the belief in continued existence and examines some of its philosophical implications. (shrink)

‘Number sense’ is a short‐hand for our ability to quickly understand, approximate, and manipulate numerical quantities. My hypothesis is that number sense rests on cerebral circuits that have evolved specifically for the purpose of representing basic arithmetic knowledge. Four lines of evidence suggesting that number sense constitutes a domain‐specific, biologically‐determined ability are reviewed: the presence of evolutionary precursors of arithmetic in animals; the early emergence of arithmetic competence in infants independently of other abilities, including language; the existence of a (...) homology between the animal, infant, and human adult abilities for number processing; and the existence of a dedicated cerebral substrate. In adults of all cultures, lesions to the inferior parietal region can specifically impair number sense while leaving the knowledge of other cognitive domains intact. Furthermore, this region is demonstrably activated during number processing. I postulate that higher–level cultural devel‐opments in arithmetic emerge through the establishment of linkages between this core analogical representation and other verbal and visual representations of number notations. The neural and cognitive organization of those representations can explain why some mathematical concepts are intuitive, while others are so difficult to grasp. Thus, the ultimate foundations of mathematics rests on core representations that have been internalized in our brains through evolution. (shrink)

Kim :1099–1112, 2013) defends a logicist theory of numbers. According to him, numbers are adverbial entities, similar to those denoted by “frequently” and “at 100 mph”. He even introduces new adverbs for numbers: “1-wise”, “2-wise”, and so on. For example, “Fs exist 2-wise” means that there are two Fs. Kim claims that, because we can derive Dedekind–Peano axioms from his definition of numbers as adverbial entities, it is a new form of logicism. In this paper, I (...) will, however, argue that his theory is vulnerable to an analogue of the so-called Bad Company objection to neo-Fregeanism. This means that we cannot be sure that numbers are actually given to us by Kim’s definition; for, we don’t know whether it is indeed a good definition. So, unless Kim, or somebody else, provides a demarcation criterion between good and bad adverbial definitions, Kim’s theory will remain incomplete. (shrink)

The 4th volume of “Manuscripts and Speeches” by the prominent contemporary thinker Paul Ricoeur (1913‑2005) contains works discussing one of his central themes — the problem of a common existence of men considered from the point of view of politics, economics, power, law, culture, morality, and ethics. At the same time, the French thinker specifically highlights and discusses such burning problems of modern life as mutual recognition, the fragility of human existence and earthly civilization in general, tolerance, care, (...) justice, responsibility for the future of humanity, for the very idea of man. In a number of works, Ricoeur expresses his views on the possible future of Europe — on its new ethos overcoming the boundaries and structures of national states, on the combination of their “identity” and “otherness” at the global level. Translation from one language to another as a kind of a priori communication, exchange of memory as a preliminary transfer to a different cultural environment, forgiveness that stems from the suffering of others and raises the question of the burden of debt, — all this is being considered as a models of integration. Originally presented at philosophical congresses and symposia, as well as qua journal publications, the highlighted problems were fundamentally studied by the philosopher in his works “History and Truth” (1955), “Oneself as Another” (1990), “Ethics and Responsibility” (1994), “The Just” (vols. 1, 2 — 1995, 2001), “Ideology and Utopia” (1997), “Memory, History, Oblivion” (2000), “The Way of Recognition” (2004). The book “Paul Ricoeur. Politics, Economics and Society. Manuscripts and Speeches 4” is addressed to philosophers, cultural and political scientists and scholars representing different branches of humanitarian studies, as well as to readers interested in contemporary philosophy. (shrink)

The dissertation is an inquiry into the ontology and epistemology of numbers. As regards the former, the Fregean conception of numbers as objects and the Russellian conception of numbers as higher-level entities are both critically examined. A conception of numbers as modes of existence , that is, ways or manners in which things exist, is introduced and defended instead. As regards the latter, the basic concepts of arithmetic are explicated in terms of pure logic alone, (...) and all the truths of arithmetic are shown to follow from those explications solely by logical means. A new version of logicism in the philosophy of arithmetic is thereby established. (shrink)