Main results are: (i) number e^{e} is transcendental; (ii) the both numbers e+π and e-π are irrational.

Suppose we can save either a larger group of persons or a distinct, smaller group from some harm. Many people think that, all else equal, we ought to save the greater number. This article defends this view (with qualifications). But unlike earlier theories, it does not rely on the idea that several people's interests or claims receive greater aggregate weight. The argument starts from the idea that due to their stakes, the affected people have claims to have a say (...) in the rescue decision. As rescuers, our primary duty is to respect these procedural claims, which we must do by doing what these people would decide, in a process where each is given an equal vote on the matter. So in cases where each votes in their own self-interest, respect for their equal right to decide, or their autonomy, will lead us to save the greater number. The argument is explained in detail, with special attention to the questions of how, exactly, it avoids aggregation, and of why majority rule is superior to lottery procedures. The view has further advantages. Especially, it explains the “partial” relevance of numbers in cases involving unequal harms, and it does so in a way that dissolves the appearance of paradox that besets theories of “partial aggregation.”. (shrink)

David Hilbert's finitistic standpoint is a conception of elementary number theory designed to answer the intuitionist doubts regarding the security and certainty of mathematics. Hilbert was unfortunately not exact in delineating what that viewpoint was, and Hilbert himself changed his usage of the term through the 1920s and 30s. The purpose of this paper is to outline what the main problems are in understanding Hilbert and Bernays on this issue, based on some publications by them which have so (...) far received little attention, and on a number of philosophical reconstructions of the viewpoint (in particular, by Hand, Kitcher, and Tait). (shrink)

With the aid of some results from current linguistic theory I examine a recent anti-Fregean line with respect to hybrid talk of numbers and ordinary things, such as ‘the number of moons of Jupiter is four’. I conclude that the anti-Fregean line with respect to these sentences is indefensible.

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)

Do scientific theories limit human knowledge? In other words, are there physical variables hidden by essence forever? We argue for negative answers and illustrate our point on chaotic classical dynamical systems. We emphasize parallels with quantum theory and conclude that the common real numbers are, de facto, the hidden variables of classical physics. Consequently, real numbers should not be considered as ``physically real" and classical mechanics, like quantum physics, is indeterministic.

Conspiracy theories and conspiracy theorists have been accused of a great many sins, but are the conspiracy theories conspiracy theorists believe epistemically problematic? Well, according to some recent work, yes, they are. Yet a number of other philosophers like Brian L. Keeley, Charles Pigden, Kurtis Hagen, Lee Basham, and the like have argued ‘No!’ I will argue that there are features of certain conspiracy theories which license suspicion of such theories. I will also argue that these features only license (...) a limited suspicion of these conspiracy theories, and thus we need to be careful about generalising from such suspicions to a view of the warrant of conspiracy theories more generally. To understand why, we need to get to the bottom of what exactly makes us suspicious of certain conspiracy theories, and how being suspicious of a conspiracy theory does not always tell us anything about how likely the theory in question is to be false. (shrink)

According to a tradition stemming from Quine and Putnam, we have the same broadly inductive reason for believing in numbers as we have for believing in electrons: certain theories that entail that there are numbers are better, qua explanations of our evidence, than any theories that do not. This paper investigates how modal theories of the form ‘Possibly, the concrete world is just as it in fact is and T’ and ‘Necessarily, if standard mathematics is true and the concrete world (...) is just as it in fact is, then T’ bear on this claim. It concludes that, while analogies with theories that attempt to eliminate unobservable concrete entities provide good reason to regard theories of the former sort as explanatorily bad, this reason does not apply to theories of the latter sort. (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.

I respond to P. McLaughlin and O. Schlaudt’s critique of my analysis of the cross-cultural origins of numbers, noting that my work draws extensively upon number systems as ethnographically attested around the globe, and thus is based only in part on the important Mesopotamian case study. I place the work of Peter Damerow in its historical context, noting its genesis in Piaget’s genetic epistemology and the problems associated with applying Piaget’s developmental theory to societies. While Piaget assumed numeracy (...) involves invariant mental transformations, ongoing research in numerical cognition has been largely unsuccessful in identifying specific brain-bound mechanisms for numerical structure. Accordingly, I suggest the extended mind paradigm from the philosophy of mind may be a more fruitful approach, and detail such an approach using Material Engagement Theory. (shrink)

Ortega y Gasset is known for his philosophy of life and his effort to propose an alternative to both realism and idealism. The goal of this article is to focus on an unfamiliar aspect of his thought. The focus will be given to Ortega’s interpretation of the advancements in modern mathematics in general and Cantor’s theory of transfinite numbers in particular. The main argument is that Ortega acknowledged the historical importance of the Cantor’s Set Theory, analyzed it and (...) articulated a response to it. In his writings he referred many times to the advancements in modern mathematics and argued that mathematics should be based on the intuition of counting. In response to Cantor’s mathematics Ortega presented what he defined as an ‘absolute positivism’. In this theory he did not mean to naturalize cognition or to follow the guidelines of the Comte’s positivism, on the contrary. His aim was to present an alternative to Cantor’s mathematics by claiming that mathematicians are allowed to deal only with objects that are immediately present and observable to intuition. Ortega argued that the infinite set cannot be present to the intuition and therefore there is no use to differentiate between cardinals of different infinite sets. (shrink)

It’s sometimes suggested that we can (in a sense) settle the truth-value of some statements in the language of number theory by stipulation, adopting either φ or ¬φ as an additional axiom. For example, in Clarke-Doane (2020b) and a series of recent APA presentations, Clarke-Doane suggests that any Σ01 sound expansion of our current arithmetical practice would express a truth. In this paper, I’ll argue that (given a certain popular assumption about the model-theoretic representability of languages like ours) (...) we can’t know ourselves to have any such freedom. (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)

I argue that the human mind includes an innate domain-specific system for representing precise small numerical quantities. This theory contrasts with object-tracking theories and with domain-general theories that only make use of mental models. I argue that there is a good amount of evidence for innate representations of small numerical quantities and that such a domain-specific system has explanatory advantages when infants’ poor working memory is taken into account. I also show that the mental models approach requires previously unnoticed (...) domain-specific structure and consequently that there is no domain-general alternative to an innate domain-specific small number system. (shrink)

I follow standard mathematical practice and theory to argue that the natural numbers are the finite von Neumann ordinals. I present the reasons standardly given for identifying the natural numbers with the finite von Neumann's (e.g., recursiveness; well-ordering principles; continuity at transfinite limits; minimality; and identification of n with the set of all numbers less than n). I give a detailed mathematical demonstration that 0 is { } and for every natural number n, n is the set of (...) all natural numbers less than n. Natural numbers are sets. They are the finite von Neumann ordinals. (shrink)

A RELATIVISTIC THEORY OF PHENOMENOLOCICAL CONSTITUTION: A SELF-REFERENTIAL, TRANSCENDENTAL APPROACH TO CONCEPTUAL PATHOLOGY. (Vol. I: French; Vol. II: English) -/- Steven James Bartlett -/- Doctoral dissertation director: Paul Ricoeur, Université de Paris Other doctoral committee members: Jean Ladrière and Alphonse de Waehlens, Université Catholique de Louvain Defended publically at the Université Catholique de Louvain, January, 1971. -/- Universite de Paris X (France), 1971. 797pp. -/- The principal objective of the work is to construct an analytically precise methodology which can (...) serve to identify, eliminate, and avoid a certain widespread _conceptual fault_ or _misconstruction_, called a "projective misconstruction" or "projection" by the author. It is argued that this variety of error in our thinking (i) infects a great number of our everyday, scientific, and philosophical concepts, claims, and theories, (ii) has largely been undetected, and (iii), when remedied, leads to a less controversial and more rigorous elucidation of the transcendental preconditions of human knowledge than has traditionally been possible. The dissertation identifies, perhaps for the first time, a _projective_ variety of self-referential inconsistency, and proposes an innovative, self-reflexive approach to transcendental argument in a logical and phenomenological context. The strength of the approach lies, it is claimed, in the fact that a rejection of the approach is possible only on pain of self-referential inconsistency. The argument is developed in the following stages: A general introduction identifies the central theme of the work, defines the scope of applicability of the results reached, and sketches the direction of the studies that follow. The preliminary discussion culminates in a recognition of the need for a _critique of impure reason_. The body of the work is divided into two parts: Section I seeks to develop a methodology, on a purely formal basis, which is, on the one hand, capable of being used to study the transcendental foundations of the special sciences, including its own proper transcendental foundation. On the other hand, the methodology proposed is intended as a diagnostic and therapeutic tool for dealing with _projective_ uses of concepts. The approach initiates an analysis of concepts from a perspective which views _knowledge as coordination_. Section I describes formal structures that possess the status of preconditions in such a coordinative account of knowledge. Special attention is given to the preconditions of _identifying reference_ to logical particulars. The first section attempts, then, to provide a self-referential, transcendental methodology which is essentially revisionary in that it is motivated by a concern for conceptual error-elimination. Phenomenology, considered in its unique capacity as a self-referential, transcendental discipline, is of special relevance to the study. Section II accordingly examines a group of concepts which come into question in connection with the central theme of _phenomenological constitution_. The "_de-projective methodology_" developed in Section I is applied to these concepts that have a foundational importance in transcendental phenomenology. A translation is, in effect, proposed from the language of consciousness to a language in which preconditions of referring are investigated. The result achieved is the elimination of self-defeating, projective concepts from a rigorous, phenomenological study of the constitutive foundations of science. The dissertation was presented in a two volume, double-language format for the convenience of French and English researchers. Each volume contains an analytical index. (shrink)

The boundless nature of the natural numbers imposes paradoxically a high formal bound to the use of standard artificial computer programs for solving conceptually challenged problems in number theory. In the context of the new cognitive foundations for mathematics' and physics' program immersed in the setting of artificial mathematical intelligence, we proposed a refined numerical system, called the physical numbers, preserving most of the essential intuitions of the natural numbers. Even more, this new numerical structure additionally possesses the (...) property of being a bounded object allowing us to work with quite similar axioms like the classic Peano axioms, but in a finite environment and with an enriched physical dimension. Finally, we present several enlightening examples and we conclude that the physical numbers provide a natural formal setting for approaching classic problems in number theory from a more hybrid perspective, i.e. with a potential participation in the generation of solutions, not only of working mathematicians but also of more sophisticated artificial interactive (mathematical) intelligent agents (within the context of cognitive-computational metamathematics or artificial mathematical intelligence) and more controlled by physically-inspired principles. Finally, this paper addresses a highly new, bottom-up and paradigm-shifting approach to the foundations of physics: One should refine and improve the conceptual formal setting of the language that we use for understanding physical phenomena (e.g. the mathematical grounding concepts and theories) for being able to understand better more subtle and complex physical phenomena. (shrink)

The new theory of reference has won popularity. However, a number of noted philosophers have also attempted to reply to the critical arguments of Kripke and others, and aimed to vindicate the description theory of reference. Such responses are often based on ingenious novel kinds of descriptions, such as rigidified descriptions, causal descriptions, and metalinguistic descriptions. This prolonged debate raises the doubt whether different parties really have any shared understanding of what the central question of the philosophical (...) theory of reference is: what is the main question to which descriptivism and the causal-historical theory have presented competing answers. One aim of the paper is to clarify this issue. The most influential objections to the new theory of reference are critically reviewed. Special attention is also paid to certain important later advances in the new theory of reference, due to Devitt and others. (shrink)

In his [1937, 1938], Paul Dirac proposed his “Large Number Hypothesis” (LNH), as a speculative law, based upon what we will call the “Large Number Coincidences” (LNC’s), which are essentially “coincidences” in the ratios of about six large dimensionless numbers in physics. Dirac’s LNH postulates that these numerical coincidences reflect a deeper set of law-like relations, pointing to a revolutionary theory of cosmology. This led to substantial work, including the development of Dirac’s later [1969/74] cosmology, and other (...) alternative cosmologies, such as the Brans-Dicke modification of GTR, and to extensive empirical tests. We may refer to the generic hypothesis of “Large Number Relations” (LNR’s), as the proposal that there are lawlike relations of some kind between the dimensionless numbers, not necessarily those proposed in Dirac’s early LNH. Such relations would have a profound effect on our concepts of physics, but they remain shrouded in mystery. Although Dirac’s specific proposals for LNR theories have been largely rejected, the subject retains interest, especially among cosmologists seeking to test possible variations in fundamental constants, and to explain dark energy or the cosmological constant. In the first two sections here we briefly summarize the basic concepts of LNR’s. We then introduce an alternative LNR theory, using a systematic formalism to express variable transformations between conventional measurement variables and the true variables of the theory. We demonstrate some consistency results and review the evidence for changes in the gravitational constant G. The theory adopted in the strongest tests of Ġ/G, by the Lunar Laser Ranging (LLR) experiments, assumes: Ġ/G = 3(dr/dt)/r – 2(dP/dt)/P – (dm/dt)/m, as a fundamental relationship. Experimental measurements show the RHS to be close to zero, so it is inferred that significant changes in G are ruled out. However when the relation is derived in our alternative theory it gives: Ġ/G = 3(dr/dt)/r – 2(dP/dt)/P – (dm/dt)/m – (dR/dt)/R. The extra final term (which is the Hubble constant) is not taken into account in conventional derivations. This means the LLR experiments are consistent with our LNR theory (and others), and they do not really test for a changing value of G at all. This failure to transform predictions of LNR theories correctly is a serious conceptual flaw in current experiment and theory. (shrink)

According to Augustine, abstract objects are ideas in the mind of God. Because numbers are a type of abstract object, it would follow that numbers are ideas in the mind of God. Call such a view the “Augustinian View of Numbers” (AVN). In this paper, I present a formal theory for AVN. The theory stems from the symmetry conception of God as it appears in Studtmann (2021). I show that the theory in Studtmann’s paper can interpret the (...) axioms of Peano Arithmetic minus the induction schema. This fact allows for the development of arithmetic in a natural way. The development eventuates in a theory that can interpret second-order arithmetic. The conception of God that emerges by the end of the discussion is a conception of an infinite, ineffable, self-cause that contains objects that not only serve as numbers but also encode information about each other. (shrink)

Numbers without Science opposes the Quine-Putnam indispensability argument, seeking to undermine the argument and reduce its profound influence. Philosophers rely on indispensability to justify mathematical knowledge using only empiricist epistemology. I argue that we need an independent account of our knowledge of mathematics. The indispensability argument, in broad form, consists of two premises. The major premise alleges that we are committed to mathematical objects if science requires them. The minor premise alleges that science in fact requires mathematical objects. The most (...) common rejection of the argument denies its minor premise by introducing scientific theories which do not refer to mathematical objects. Hartry Field has shown how we can reformulate some physical theories without mathematical commitments. I argue that Field’s preference for intrinsic explanation, which underlies his reformulation, is ill-motivated, and that his resultant fictionalism suffers unacceptable consequences. I attack the major premise instead. I argue that Quine provides a mistaken criterion for ontic commitment. Our uses of mathematics in scientific theory are instrumental and do not commit us to mathematical objects. Furthermore, even if we accept Quine’s criterion for ontic commitment, the indispensability argument justifies only an anemic version of mathematics, and does not yield traditional mathematical objects. The first two chapters of the dissertation develop these results for Quine’s indispensability argument. In the third chapter, I apply my findings to other contemporary indispensabilists, specifically the structuralists Michael Resnik and Stewart Shapiro. In the fourth chapter, I show that indispensability arguments which do not rely on Quine’s holism, like that of Putnam, are even less successful. Also in Chapter 4, I show how Putnam’s work in the philosophy of mathematics is unified around the indispensability argument. In the last chapter of the dissertation, I conclude that we need an account of mathematical knowledge which does not appeal to empirical science and which does not succumb to mysticism and speculation. Briefly, my strategy is to argue that any defensible solution to the demarcation problem of separating good scientific theories from bad ones will find mathematics to be good, if not empirical, science. (shrink)

Our objectives in this paper are, first, to identify several puzzling aspects of the “Trilemma Argument” of Section 6 against the Sense Datum Theory; second, to resolve these puzzles by reconstructing the Trilemma Argument; third to point to a distinction Sellars makes between two versions of the Sense Datum Theory, the “nominalist” version and the “realist” version; fourth, to reconstruct Sellars’s arguments against both; and, finally, to find in an earlier paper, “Is There a Synthetic A Priori?” that (...) his argument against the second version, assumed but not actually given in EPM, is against property realism and depends on taking language expressing propositional perception as fundamental and language expressing non-propositional perception as incomplete and derivative from the former. (shrink)

In this work, we derive the standard formalism of quantum theory by analyzing the behavior of single-variable systems under measurements. These systems, with minimal information capacity, exhibit indeterministic behavior in independent measurements while yielding probabilistically predictable outcomes in dependent measurements. Enforcing probability conservation in the probability transformations leads to the derivation of the Born rule, which subsequently gives rise to the Hilbert space structure and the Schrödinger equation. Additionally, we show that preparing physical systems in coherent states —crucial for (...) observing quantum phenomena— effectively reduces the number of independent variables to one. This first-principles derivation of quantum theory from probability conservation in single-variable systems offers new insights into the physical meaning of quantum theory and clarifies its domain of applicability. (shrink)

This paper develops and explores a new framework for theorizing about the measurement and aggregation of well-being. It is a qualitative variation on the framework of social welfare functionals developed by Amartya Sen. In Sen’s framework, a social or overall betterness ordering is assigned to each profile of real-valued utility functions. In the qualitative framework developed here, numerical utilities are replaced by the properties they are supposed to represent. This makes it possible to characterize the measurability and interpersonal comparability of (...) well-being directly, without the use of invariance conditions, and to distinguish between real changes in well-being and merely representational changes in the unit of measurement. The qualitative framework is shown to have important implications for a range of issues in axiology and social choice theory, including the characterization of welfarism, axiomatic derivations of utilitarianism, the meaningfulness of prioritarianism, the informational requirements of variable-population ethics, the impossibility theorems of Arrow and others, and the metaphysics of value. (shrink)

In this paper intuitionistic set theory INC# in infinitary set theoretical language is considered. External induction principle in nonstandard intuitionistic arithmetic were derived. Non trivial application in number theory is considered.

There are three main moral theories: virtue ethics, the deontological approach and utilitarianism. The concern here is how they interrelate, why they come into focus at different times and places, and how they are configured in their application to a modern democratic society. Person-oriented virtue ethics was the dominant understanding in Ancient Greece but within the Western tradition this was later subordinated to the monotheism of Ancient Judaism as modified by Christianity. Of growing importance by the eighteenth century was rights (...) theory which was often still situated religiously. Kant’s principle of the categorical imperative has been highly influential but was challenged by the emerging nature of industrial and capitalist society. Utilitarianism, within which the moral rightness of activity resides in its tendency to promote happiness or unhappiness, represented the decisive move from the transcendental to the immanent approach. Although all three approaches to moral theory continue to be relevant to identifiable situations and aspects of modern society, there has been a substantial turn towards a heavily modified utilitarianism associated with parliamentary democracy and market economies founded on property ownership. The root cause of this is the ability of utilitarianism, as opposed to the other approaches, to handle considerations of number and probability. The concept of utility is fundamental in economics but the idea has evolved away from its origins to mean “preference”. There is a sense in which the straightforward appeal of basic utilitarianism has been “leased out” in modified form to a set of institutional arrangements. Certain “pressure points” in a modern society are noted which pose particular problems pertinent to moral theory. Bernard Williams argues persuasively for an appropriately modified form of virtue ethics. (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)

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)

Discussing theories at length, including their origin, development, and replacement by other theories, can help students in understanding of both objective and subjective aspects of the scientific process. Presenting theories in the form of- models helps in this undertaking, and the history of science provides a number of suitable models. The paper describes specific examples that have been used in in-service courses for science teachers.

In his highly perceptive, if underappreciated introduction to Wittgenstein’s Tractatus, Russell identifies a “lacuna” within Wittgenstein’s theory of number, relating specifically to the topic of transfinite number. The goal of this paper is two-fold. The first is to show that Russell’s concerns cannot be dismissed on the grounds that they are external to the Tractarian project, deriving, perhaps, from logicist ambitions harbored by Russell but not shared by Wittgenstein. The extensibility of Wittgenstein’s theory of number (...) to the case of transfinite cardinalities is, I shall argue, a desideratum generated by concerns intrinsic, and internal to Wittgenstein’s logical and semantic framework. Second, I aim to show that Wittgenstein’s theory of number as espoused in the Tractatus is consistent with Russell’s assessment, in that Wittgenstein meant to leave open the possibility that transfinite numbers could be generated within his system, but did not show explicitly how to construct them. To that end, I show how one could construct a transfinite number line using ingredients inherent in Wittgenstein’s system, and in accordance with his more general theories of number and of operations. (shrink)

In this paper, I suggest that infinite numbers are large finite numbers, and that infinite numbers, properly understood, are 1) of the structure omega + (omega* + omega)Ө + omega*, and 2) the part is smaller than the whole. I present an explanation of these claims in terms of epistemic limitations. I then consider the importance, part of which is demonstrating the contradiction that lies at the heart of Cantorian set theory: the natural numbers are too large to be (...) counted by any finite number, but too small to be counted by any infinite number – there is no number of natural numbers. (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)

One of Derek Parfit’s greatest legacies was the search for Theory X, a theory of population ethics that avoided all the implausible conclusions and paradoxes that have dogged the field since its inception: the Absurd Conclusion, the Repugnant Conclusion, the Non-Identity Problem, and the Mere Addition Paradox. In recent years, it has been argued that this search is doomed to failure and no satisfactory population axiology is possible. This chapter reviews Parfit’s life’s work in the field and argues (...) that he provided all the necessary components necessary for a Theory X. It then shows how these components can be combined together and applied to the global challenges Parfit argued matter most: preventing human extinction, managing catastrophic risks, and eradicating global poverty and suffering. Finally, it identifies a number of challenges facing his theory and suggests how these may be overcome. (shrink)

Saul Kripke once noted that there is a tight connection between computation and de re knowledge of whatever the computation acts upon. For example, the Euclidean algorithm can produce knowledge of which number is the greatest common divisor of two numbers. Arguably, algorithms operate directly on syntactic items, such as strings, and on numbers and the like only via how the numbers are represented. So we broach matters of notation. The purpose of this article is to explore the relationship (...) between the notations acceptable for computation, the usual idealizations involved in theories of computability, flowing from Alan Turing’s monumental work, and de re propositional attitudes toward numbers and other mathematical objects. (shrink)

This paper presents a sequent calculus and a dual domain semantics for a theory of definite descriptions in which these expressions are formalised in the context of complete sentences by a binary quantifier I. I forms a formula from two formulas. Ix[F, G] means ‘The F is G’. This approach has the advantage of incorporating scope distinctions directly into the notation. Cut elimination is proved for a system of classical positive free logic with I and it is shown to (...) be sound and complete for the semantics. The system has a number of novel features and is briefly compared to the usual approach of formalising ‘the F ’ by a term forming operator. It does not coincide with Hintikka’s and Lambert’s preferred theories, but the divergence is well-motivated and attractive. (shrink)

It is usual to identify initial conditions of classical dynamical systems with mathematical real numbers. However, almost all real numbers contain an infinite amount of information. I argue that a finite volume of space can’t contain more than a finite amount of information, hence that the mathematical real numbers are not physically relevant. Moreover, a better terminology for the so-called real numbers is “random numbers”, as their series of bits are truly random. I propose an alternative classical mechanics, which is (...) empirically equivalent to classical mechanics, but uses only finite-information numbers. This alternative classical mechanics is non-deterministic, despite the use of deterministic equations, in a way similar to quantum theory. Interestingly, both alternative classical mechanics and quantum theories can be supplemented by additional variables in such a way that the supplemented theory is deterministic. Most physicists straightforwardly supplement classical theory with real numbers to which they attribute physical existence, while most physicists reject Bohmian mechanics as supplemented quantum theory, arguing that Bohmian positions have no physical reality. (shrink)

Evolutionary theory is a paradigmatic example of a well-supported scientific theory. In this chapter we consider a number of objections to evolutionary theory, and show how responding to these objections reveals aspects of the way in which scientific theories are supported by evidence. Teaching these objections can therefore serve two pedagogical aims: students can learn the right way to respond to some popular arguments against evolutionary theory, and they can learn some basic features of the (...) structure of scientific theories and evidence. (shrink)

Hofweber (Ontology and the ambitions of metaphysics, Oxford University Press, 2016) argues for a thesis he calls “internalism” with respect to natural number discourse: no expressions purporting to refer to natural numbers in fact refer, and no apparent quantification over natural numbers actually involves quantification over natural numbers as objects. He argues that while internalism leaves open the question of whether other kinds of abstracta exist, it precludes the existence of natural numbers, thus establishing what he calls “restricted nominalism” (...) about natural numbers. We argue that Hofweber’s internalism fails to establish restricted nominalism. Not only is his primary argument for restricted nominalism invalid, the analysis of quantification proposed threatens to collapse internalism into either a traditional form of error theory or realism. (shrink)

In the first month of 2024, there was a significant increase in the number of downloads for the Bayesian stats / MCMC computing program, bayesvl, developed by AISDL running on R and Stan. The following RDocumentation (CRAN) graph illustrates the noticeable leap in data for January 2024.

This contribution considers whether or not it is possible to devise a coherent form of external skepticism about the normative if we ‘relax’ about normative ontology by regarding claims about the existence of normative truths and properties themselves as normative. I answer this question in the positive: A coherent form of non-normative error-theories can be developed even against a relaxed background. However, this form no longer makes any reference to the alleged falsity of normative judgments, nor the non-existence of normative (...) properties. Instead, it concerns a specifically inferentialist construal of error-theories which suggests that error-theorists should abstain from any claims about normative ontology to focus exclusively on claims about the inferential role of normative vocabulary. As I will show, this suggestion affords a number of important advantages. However, it also comes at a cost, in that it might not only change the letter, but also the spirit of traditional error-theories. (shrink)

A number of cognitive and behavioral variables influence the performance in tasks of theory of mind (ToM). Since two of the most important variables, memory and explicit expression, are impaired in schizophrenic patients, the ToM appears inconsistent in these patients. An ideal instrument of ToM should therefore account for deficient memory and impaired ability of these patients to explicitly express intentions. If such an instrument is developed, it should provide information that can be used not only to understand (...) the pathophysiology but also to monitor patients. (shrink)

In this paper, I present the case of the discovery of complex numbers by Girolamo Cardano. Cardano acquires the concepts of (specific) complex numbers, complex addition, and complex multiplication. His understanding of these concepts is incomplete. I show that his acquisition of these concepts cannot be explained on the basis of Christopher Peacocke’s Conceptual Role Theory of concept possession. I argue that Strong Conceptual Role Theories that are committed to specifying a set of transitions that is both necessary and (...) sufficient for possession of mathematical concepts will always face counterexamples of the kind illustrated by Cardano. I close by suggesting that we should rely more heavily on resources of Anti-Individualism as a framework for understanding the acquisition and possession of concepts of abstract subject matters. (shrink)

An excerpt of "The SignalGlyph Project and Prime Numbers" (a chapter of the book THE IMAGE OF LANGUAGE) that attempts to illustrate how dimensional limitations of mathematical language have obscured recognition of the system of patterning in the distribution of prime numbers.

I here defend a theory consisting of four claims about ‘property’ and properties, and argue that they form a coherent whole that can solve various serious problems. The claims are (1): ‘property’ is defined by the principles (PR): ‘F-ness/Being F/etc. is a property of x iff F’ and (PA): ‘F-ness/Being F/etc. is a property’; (2) the function of ‘property’ is to increase the expressive power of English, roughly by mimicking quantification into predicate position; (3) property talk should be understood (...) at face value: apparent commitments are real and our apparently literal use of ‘property’ is really literal; (4) there are no properties. In virtue of (1)–(2), this is a deflationist theory and in virtue of (3)–(4), it is an error theory. (1) is fleshed out as a claim about understanding conditions, and it is argued at length, and by going through a number of examples, that it satisfies a crucial constraint on meaning claims: all facts about ‘property’ can be explained, together with auxiliary facts, on its basis. Once claim (1) has been expanded upon, I argue that the combination of (1)–(3) provides the means for handling several problems: they help giving a happy-face solution to what I call the paradox of abstraction , they form part of a plausible account of the correctness of committive sentences, and, most importantly, they help respond to various indispensability arguments against nominalism. (shrink)

A Commonplace of recent philosophy of mind is that intentional states are relations between thinkers and propositions. This thesis-call it the 'Relational Thesis'-does not depend on any specific theory of propositions. One can hold it whether one believes that propositions are Fregean Thoughts, ordered n-tuples of objects and properties or sets of possible worlds. An assumption that all these theories of propositions share is that propositions are abstract objects, without location in space or time...

The primary aim of this paper is to critically evaluate the deductive model of ethical applications, which is based on normative ethical theories like deontology and consequentialism, and to show why a number of models have failed to furnish appropriate resolutions to practical moral problems. Here, for the deductive model, I want to call it a “Linear Mechanical Model” because the basic assumption of this model is that if a normative theory is sacrosanct, then the case is as (...) it is. The conclusion derived from the case will also be correct, true and acceptable. However, traditional ethicists used to apply their ethical theories, but they did not know which moral theory was effective on the ground level of reality. The study will show readers how ethical theories are in conflict with each other in the case of euthanasia. In more precise words, “which ethical theories are said to be applied, meta-ethical or normative, or both for the resolution of ethical problems? If normative theories are said to be applied, how the application can take place when it is contrary to our experience, that (then) in a situation of moral crises, no one really applies a theory?” For that, my argument is the linear model has failed because it is rigid, often ignores the agents’ intrinsic values, and has no space to amend it, no matter how bizarre the consequence is. Its alternative is the Inductive model. For that, the paper will take three moral principles (autonomy, beneficence including maleficence, and justice) of Beauchamp & Childress. This suggests us for resolving value-laden moral problems, we should consider some steps such as a) recognising moral issues to start with; b) developing the moral imagination; c) sharpening analytical/critical skills; d) testing out disagreements; e) effecting decisions and behavior; and f) implementation, closure, and process are of vital importance, in other words, it starts with the free and informed consensus of all interested parties, but this model also has been failed because the model could not give a systematic organization to their way of resolution. Here, my argument is that the inductive model provides resolution of the practical problem but ignores what is ethically obligatory, permissible, or wrong in that situation, and there are no appropriate suggestions in the case of a moral crisis. (shrink)

Starting from the typical case of utilitarianism, I distinguish three ways a moral theory may be deemed (over-)demanding: practical, epistemic, and cognitive. I focus on the latter, whose specific nature has been overlooked. Taking animal ethics as a case study, I argue that knowledge of human cognition is critical to spelling out moral theories (including their implications) that are accessible and acceptable to the greatest number of agents. In a nutshell: knowing more about our cognitive apparatus with a (...) view to play better with it. This meta-theoretical suggestion, however, differs from a classical objection drawn from the intuition that a given theory demands too much. -/- //// A partir du cas typique de l’utilitarisme, je distingue trois façons dont une théorie morale pourrait être jugée (trop) exigeante : pratique, épistémique et cognitive. La spécificité de cette dernière – qui fait l’objet de cet article – a été négligée. Je soutiens, en prenant l’exemple de l’éthique animale, que la connaissance de la cognition humaine est essentielle à la formulation de théories morales (et de leurs implications) qui soient accessibles à, et acceptables par, le plus grand nombre d’agents possible. En d’autres termes : comment mieux connaître notre machine cognitive pour mieux jouer avec elle. Cette recommandation méta-théorique se distingue cependant d’une objection classique tirée de l’intuition selon laquelle une théorie exigerait trop de nous. (shrink)

Donors to global health programs and policymakers within national health systems have to make difficult decisions about how to allocate scarce health care resources. Principled ways to make these decisions all make some use of summary measures of health, which provide a common measure of the value (or disvalue) of morbidity and mortality. They thereby allow comparisons between health interventions with different effects on the patterns of death and ill health within a population. The construction of a summary measure of (...) health requires that a number be assigned to the harm of death. But the harm of death is currently a matter of debate: different philosophical theories assign very different values to the harm of death at different ages. This chapter considers how we should assign numbers to the harm of deaths at different ages in the face of uncertainty and disagreement. (shrink)

John D. Norton is responsible for a number of influential views in contemporary philosophy of science. This paper will discuss two of them. The material theory of induction claims that inductive arguments are ultimately justified by their material features, not their formal features. Thus, while a deductive argument can be valid irrespective of the content of the propositions that make up the argument, an inductive argument about, say, apples, will be justified (or not) depending on facts about apples. (...) The argument view of thought experiments claims that thought experiments are arguments, and that they function epistemically however arguments do. These two views have generated a great deal of discussion, although there hasn’t been much written about their combination. I argue that despite some interesting harmonies, there is a serious tension between them. I consider several options for easing this tension, before suggesting a set of changes to the argument view that I take to be consistent with Norton’s fundamental philosophical commitments, and which retain what seems intuitively correct about the argument view. These changes require that we move away from a unitary epistemology of thought experiments and towards a more pluralist position. (shrink)