This Element, written for researchers and students in philosophy and the behavioral sciences, reviews and critically assesses extant work on number concepts in developmental psychology and cognitive science. It has four main aims. First, it characterizes the core commitments of mainstream number cognition research, including the commitment to representationalism, the hypothesis that there exist certain number-specific cognitive systems, and the key milestones in the development of number cognition. Second, it provides a taxonomy of influential views (...) within mainstream number cognition research, along with the central challenges these views face. Third, it identifies and critically assesses a series of core philosophical assumptions often adopted by number cognition researchers. Finally, the Element articulates and defends a novel version of pluralism about number concepts. (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)

Carey leaves unaddressed an important evolutionary puzzle: In the absence of a numeral list, how could a concept of natural number ever have arisen in the first place? Here we suggest that the initial development of natural number must have bootstrapped on a material culture scaffold of some sort, and illustrate how this might have occurred using strings of beads.

An enduring puzzle in philosophy and developmental psychology is how young children acquire number concepts, in particular the concept of natural number. Most solutions to this problem conceptualize young learners as lone mathematicians who individually reconstruct the successor function and other sophisticated mathematical ideas. In this chapter, I argue for a crucial role of testimony in children’s acquisition of number concepts, both in the transfer of propositional knowledge (e.g., the cardinality concept), and in knowledge-how (e.g., (...) the counting routine). (shrink)

While there has been significant philosophical debate on whether nonlinguistic animals can possess conceptual capabilities, less time has been devoted to considering 'talking' animals, such as parrots. When they are discussed, their capabilities are often downplayed as mere mimicry. The most explicit philosophical example of this can be seen in Brandom's frequent comparisons of parrots and thermostats. Brandom argues that because parrots (like thermostats) cannot grasp the implicit inferential connections between concepts, their vocal articulations do not actually have any (...) conceptual content. In contrast, I argue that Pepperberg's work with Alex (and other African grey parrots) provides evidence that the vocal articulations of at least some parrots have conceptual content. Using Frege's insight that numbers assert something about a concept, I argue that Alex's ability to answer the question "How many?" depended upon a prior grasp of conceptual content. Developing this claim, I argue that Alex's arithmetical abilities show that he was capable of using numbers as both concepts and objects. Frege's theoretical insight and Pepperberg's empirical work provide reason to reconsider the capabilities of parrots, as well as what sorts of tasks provide evidence for conceptual content. (shrink)

I trace changes to Frege's understanding of numbers, arguing in particular that the view of arithmetic based in geometry developed at the end of his life (1924–1925) was not as radical a deviation from his views during the logicist period as some have suggested. Indeed, by looking at his earlier views regarding the connection between numbers and second-level concepts, his understanding of extensions of concepts, and the changes to his views, firstly, in between Grundlagen and Grundgesetze, and, later, (...) after learning of Russell's paradox, this position is a natural position for him to have retreated to, when properly understood. (shrink)

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)

In this paper, two concepts of completing an infinite number of tasks are considered. After discussing supertasks, equisupertasks are introduced. I suggest that equisupertasks are logically possible.

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.

Number Nativism is the view that humans innately represent precise natural numbers. Despite a long and venerable history, it is often considered hopelessly out of touch with the empirical record. I argue that this is a mistake. After clarifying Number Nativism and distancing it from related conjectures, I distinguish three arguments which have been seen to refute the view. I argue that, while popular, two of these arguments miss the mark, and fail to place pressure on Number (...) Nativism. Meanwhile, a third argument is best construed as a challenge: rather than refuting Number Nativism, it challenges its proponents to provide positive evidence for their thesis and show that this can be squared with apparent counterevidence. In response, I introduce psycholinguistic work on The Tolerance Principle (not yet considered in this context), propose that it is hard to make sense of without positing precise and innate representations of natural numbers, and argue that there is no obvious reason why these innate representations couldn’t serve as a basis for mature numeric conception. (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)

In ‘The Train Paradox’, I argued that sequential random selections from the natural numbers would grow through time. I used this claim to present a paradox. In response to this proposed paradox, Jon Pérez Laraudogoitia has argued that random selections from the natural numbers do not grow through time. In this paper, I defend and expand on the argument that random selections from the natural numbers grow through time. I also situate this growth of random selections in the context of (...) my overall work on infinite number, which involves two main claims: 1) infinite numbers, properly understood, are the infinite natural numbers in a nonstandard model of the reals, and 2) ω is potentially infinite. (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)

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.

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)

Iterability, the repetition which alters the idealization it reproduces, is the engine of deconstructive movement. The fact that all experience is transformative-dissimulative in its essence does not, however, mean that the momentum of change is the same for all situations. Derrida adapts Husserl's distinction between a bound and a free ideality to draw up a contrast between mechanical mathematical calculation, whose in-principle infinite enumerability is supposedly meaningless, empty of content, and therefore not in itself subject to alteration through contextual change, (...) and idealities such as spoken or written language which are directly animated by a meaning-to-say and are thus immediately affected by context. Derrida associates the dangers of cultural stagnation, paralysis and irresponsibility with the emptiness of programmatic, mechanical, formulaic thinking. This paper endeavors to show that enumerative calculation is not context-independent in itself but is instead immediately infused with alteration, thereby making incoherent Derrida's claim to distinguish between a free and bound ideality. Along with the presumed formal basis of numeric infinitization, Derrida's non-dialectical distinction between forms of mechanical or programmatic thinking (the Same) and truly inventive experience (the absolute Other) loses its justification. In the place of a distinction between bound and free idealities is proposed a distinction between two poles of novelty; the first form of novel experience would be characterized by affectivites of unintelligibility , confusion and vacuity, and the second by affectivities of anticipatory continuity and intimacy. (shrink)

Humans possess two nonverbal systems capable of representing numbers, both limited in their representational power: the first one represents numbers in an approximate fashion, and the second one conveys information about small numbers only. Conception of exact large numbers has therefore been thought to arise from the manipulation of exact numerical symbols. Here, we focus on two fundamental properties of the exact numbers as prerequisites to the concept of EXACT NUMBERS : the fact that all numbers can be generated by (...) a successor function and the fact that equality between numbers can be defined in an exact fashion. We discuss some recent findings assessing how speakers of Munduruc (an Amazonian language), and young Western children (3-4 years old) understand these fundamental properties of numbers. (shrink)

In graph theory, the concept of domination is essential in a variety of domains. It has broad applications in diverse fields such as coding theory, computer net work models, and school bus routing and facility lo cation problems. If a fuzzy graph fails to obtain acceptable results, neutrosophic sets and neutrosophic graphs can be used to model uncertainty correlated with indeterminate and inconsistent information in arbitrary real-world scenario. In this study, we consider the concept of domination as it relates to (...) single-valued neutrosophic incidence graphs (SVNIGs). Given the importance of domination and its utilization in numerous fields, we propose the application of dominating sets in SVNIG with valid edges. We present some relevant definitions such as those of valid edges, cardinality, and isolated vertices in SVNIG along with some examples. Furthermore, we also show a few significant sets connected to the dominating set in an SVNIG such as independent and irredundant sets. We also investigate a relationship between the concepts of dominating sets and domination numbers as well as irredundant and independence sets in SVNIGs. Finally, a real-life deployment of domination in SVNIGsis investigated in relation to COVID-19 vaccination locations as a practical application. (shrink)

Where do human numerical abilities come from? This article is a commentary on Leibovich et al.’s “From 'sense of number' to 'sense of magnitude' —The role of continuous magnitudes in numerical cognition”. Leibovich et al. argue against nativist views of numerical development by noting limitations in newborns’ vision and limitations regarding newborns’ ability to individuate objects. I argue that these considerations do not undermine competing nativist views and that Leibovich et al.'s model itself presupposes that infant learners have numerical (...) representations. (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)

Radical concept nativism is the thesis that virtually all lexical concepts are innate. Notoriously endorsed by Jerry Fodor (1975, 1981), radical concept nativism has had few supporters. However, it has proven difficult to say exactly what’s wrong with Fodor’s argument. We show that previous responses are inadequate on a number of grounds. Chief among these is that they typically do not achieve sufficient distance from Fodor’s dialectic, and, as a result, they do not illuminate the central question of (...) how new primitive concepts are acquired. To achieve a fully satisfactory response to Fodor’s argument, one has to juxtapose questions about conceptual content with questions about cognitive development. To this end, we formulate a general schema for thinking about how concepts are acquired and then present a detailed illustration. (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)

The present paper deals with the ontological status of numbers and considers Frege ́s proposal in Grundlagen upon the background of the Post-Kantian semantic turn in analytical philosophy. Through a more systematic study of his philosophical premises, it comes to unearth a first level paradox that would unset earlier still than it was exposed by Russell. It then studies an alternative path, that departin1g from Frege’s initial premises, drives to a conception of numbers as synthetic a priori in a more (...) Kantian sense. On this basis, it tentatively explores a possible derivation of basic logical rules on their behalf, suggesting a more rudimentary basis to inferential thinking, which supports reconsidering the difference between logical thinking and AI. Finally, it reflects upon the contributions of this approach to the problem of the a priori. (shrink)

The mapping of numbers onto space is fundamental to measurement and to mathematics. Is this mapping a cultural invention or a universal intuition shared by all humans regardless of culture and education? We probed number-space mappings in the Mundurucu, an Amazonian indigene group with a reduced numerical lexicon and little or no formal education. At all ages, the Mundurucu mapped symbolic and nonsymbolic numbers onto a logarithmic scale, whereas Western adults used linear mapping with small or symbolic numbers and (...) logarithmic mapping when numbers were presented nonsymbolically under conditions that discouraged counting. This indicates that the mapping of numbers onto space is a universal intuition and that this initial intuition of number is logarithmic. The concept of a linear number line appears to be a cultural invention that fails to develop in the absence of formal education. (shrink)

Which role do concepts play in a person's actions? Do concepts underwrite the very idea of agency in somebody's acting? Or is the appeal to concepts in action a problematic form of over-intellectualization which obstructs a proper picture of genuine agency? Within the large and complicated terrain of these questions, the debate about know-how has been of special interest in recent years. In this paper, I shall try to spell out what know-how can tell us about the (...) role of concepts in action. I will argue that the fact that people possess and exercise know-how is indeed suited to deliver a number of important insights as to where and how concepts play a role in action. But I shall argue that these insights are limited in that they fail to cover the whole realm of relevant phenomena. (shrink)

The conceptualisation of numbers is culturally bound. This may seem like a counterintuitive claim, but one illustration thereof is the limitations of the resemblance of the ancient Greek concept of number to that in modern mathematics.

What are numbers, and where do they come from? A novel answer to these timeless questions is proposed by cognitive archaeologist Karenleigh A. Overmann, based on her groundbreaking study of material devices used for counting in the Ancient Near East—fingers, tallies, tokens, and numerical notations—as interpreted through the latest neuropsychological insights into human numeracy and literacy. The result, a unique synthesis of interdisciplinary data, outlines how number concepts would have been realized in a pristine original condition to develop (...) into one of the ancient world’s greatest mathematical traditions, a foundation for mathematical thinking today. In this view, numbers are abstract from their inception and materially bound at their most elaborated. The research updates historical work on Neolithic tokens and interpretations of Mesopotamian numbers, challenging several longstanding assumptions about numbers in the process. The insights generated are also applied to the role of materiality in human cognition more generally, including how concepts become distributed across and independent of the material forms used for their representation and manipulation; how societies comprised of average individuals use material structures to create elaborated systems of numeracy and literacy; and the differences between thinking through and thinking about materiality. (shrink)

In this chapter we argue that our concept of time is a functional concept. We argue that our concept of time is such that time is whatever it is that plays the time role, and we spell out what we take the time role to consist in. We evaluate this proposal against a number of other analyses of our concept of time, and argue that it better explains various features of our dispositions as speakers and our practices as agents.

This article is about the special, subjective concepts we apply to experience, called “phenomenal concepts”. They are of special interest in a number of ways. First, they refer to phenomenal experiences, and the qualitative character of those experiences whose metaphysical status is hotly debated. Conscious experience strike many philosophers as philosophically problematic and difficult to accommodate within a physicalistic metaphysics. Second, PCs are widely thought to be special and unique among concepts. The sense that there is (...) something special about PCs is very closely tied up with features of the epistemic access they afford to qualia. When we deploy phenomenal concepts introspectively to some phenomenally conscious experience as it occurs, we are said to be acquainted with our own conscious experiences. Accounts of PCs either have to explain the acquaintance relation, or acquaintance with our phenomenal experiences has to be denied. PCs have received much attention in recent philosophy of mind mainly because they figure in arguments for dualism and in physicalist responses to these arguments. The main topic of this paper is to explore different accounts of phenomenal concepts and their role in recent debates over the metaphysical status of phenomenal consciousness. (shrink)

A concept is traditionally defined via the necessary and sufficient conditions that clearly determine its extension. By contrast, cognitive views of concepts intend to account for empirical data that show that categorisation under a concept presents typicality effects and a certain degree of indeterminacy. We propose a formal language to compactly represent concepts by leveraging on weighted logical formulas. In this way, we can model the possible synergies among the qualities that are relevant for categorising an object under (...) a concept. We show that our proposal can account for a number of views of concepts such as the prototype theory and the exemplar theory. Moreover, we show how the proposed model can overcome some limitations of cognitive views. (shrink)

Drawing on the material culture of the Ancient Near East as interpreted through Material Engagement Theory, the journey of how material number becomes a conceptual number is traced to address questions of how a particular material form might generate a concept and how concepts might ultimately encompass multiple material forms so that they include but are irreducible to all of them together. Material forms incorporated into the cognitive system affect the content and structure of concepts through (...) their agency and affordances, the capabilities and constraints they provide as the material component of the extended, enactive mind. Material forms give concepts the tangibility that enables them to be literally grasped and manipulated. As they are distributed over multiple material forms, concepts effectively become independent of any of them, yielding the abstract irreducibility that makes a concept like number what it is. Finally, social aspects of material use—collaboration, ordinariness, and time—have important effects on the generation and distribution of concepts. (shrink)

There are a number of different senses of the term “evil.” We examine in this paper the term “evil” when it is used to say things such as: “what Hitler did was not merely wrong, it was evil”, and “Hitler was not merely a bad person, he was an evil person”. Failing to keep a promise or telling a white lie may be morally wrong, but unlike genocide or sadistic torture, it is not evil in this sense. In this (...) paper we analyze the specific moral difference between “evil” and “mere wrongdoing”. In so doing we shall defend a specific conception of what acts and which persons should count as evil. On the view defended in this paper it is a necessary feature of an evil act that the victim of that act suffer what would at least normally be a life-wrecking or life-ending harm. (shrink)

This article is about the special, subjective concepts we apply to experience, called “phenomenal concepts”. They are of special interest in a number of ways. First, they refer to phenomenal experiences, and the qualitative character of those experiences whose metaphysical status is hotly debated. Conscious experience strike many philosophers as philosophically problematic and difficult to accommodate within a physicalistic metaphysics. Second, PCs are widely thought to be special and unique among concepts. The sense that there is (...) something special about PCs is very closely tied up with features of the epistemic access they afford to qualia. When we deploy phenomenal concepts introspectively to some phenomenally conscious experience as it occurs, we are said to be acquainted with our own conscious experiences. Accounts of PCs either have to explain the acquaintance relation, or acquaintance with our phenomenal experiences has to be denied. PCs have received much attention in recent philosophy of mind mainly because they figure in arguments for dualism and in physicalist responses to these arguments. The main topic of this article is to explore different accounts of phenomenal concepts and their role in recent debates over the metaphysical status of phenomenal consciousness. (shrink)

A number of recent writers have expressed scepticism about the viability of a specifically moral concept of obligation, and some of the considerations offered have been interesting and persuasive. This is a scepticism that has its roots in Nietzsche, even if he is mentioned only rather rarely in the debate. More proximately, the scepticism in question receives seminal expression in Elizabeth Anscombe's 1958 essay, ‘Modern Moral Philosophy’, a piece that is often paid lip-service to, but—like Nietzsche's work—has only rarely (...) been taken seriously by those wishing to defend the conception of obligation under attack. This is regrettable. Anscombe's essay is powerful and direct, and it makes a forthright case for the claim that, in the absence of a divine law conception of ethics, any specifically moral concept of obligation must be redundant, and that the best that can be hoped for in a secular age is some sort of neo-Aristotelianism. Anscombe is right about this, we think. And, among those who disagree, one of the very few to have taken her on at all explicitly is Christine Korsgaard, whose Kantianism of course commits her to the view that the concept of moral obligation is central, with or without God. Here, we try to show that Korsgaard loses the argument. (shrink)

The characterization of early token-based accounting using a concrete concept of number, later numerical notations an abstract one, has become well entrenched in the literature. After reviewing its history and assumptions, this article challenges the abstract–concrete distinction, presenting an alternative view of change in Ancient Near Eastern number concepts, wherein numbers are abstract from their inception and materially bound when most elaborated. The alternative draws on the chronological sequence of material counting technologies used in the Ancient Near (...) East—fingers, tallies, tokens, and numerical notations—as reconstructed through archaeological and textual evidence and as interpreted through Material Engagement Theory, an extended-mind framework in which materiality plays an active role (Malafouris 2013). (shrink)

In his "Two concepts of possible worlds", Peter Van Inwagen explores two kinds of views about the nature of possible worlds : abstractionism and concretism. The latter is the view defended by David Lewis who claims that possible worlds are concrete spatio-temporal universes, very much like our own, causally and spatio-temporally disconnected from each other. The former is the view of the majority who claims that possible worlds are some kind of abstract objects – such as propositions, properties, states (...) of affairs, or sets of numbers. In this paper, I will develop this view in an 'extreme abstractionist' way, appealing to a 'modal bundle theory', and I will try to show that it is preferable to the standard abstractionist ones. Finally, I will compare this kind of abstractionism to concretism, only to find that the difference between the two is minimal. (shrink)

In recent years, academics and educators have begun to use software mapping tools for a number of education-related purposes. Typically, the tools are used to help impart critical and analytical skills to students, to enable students to see relationships between concepts, and also as a method of assessment. The common feature of all these tools is the use of diagrammatic relationships of various kinds in preference to written or verbal descriptions. Pictures and structured diagrams are thought to be (...) more comprehensible than just words, and a clearer way to illustrate understanding of complex topics. Variants of these tools are available under different names: “concept mapping”, “mind mapping” and “argument mapping”. Sometimes these terms are used synonymously. However, as this paper will demonstrate, there are clear differences in each of these mapping tools. This paper offers an outline of the various types of tool available and their advantages and disadvantages. It argues that the choice of mapping tool largely depends on the purpose or aim for which the tool is used and that the tools may well be converging to offer educators as yet unrealised and potentially complementary functions. (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)

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)

A multiverse is comprised of many universes, which quickly leads to the question: How many universes? There are either finitely many or infinitely many universes. The purpose of this paper is to discuss two conceptions of infinite number and their relationship to multiverses. The first conception is the standard Cantorian view. But recent work has suggested a second conception of infinite number, on which infinite numbers behave very much like finite numbers. I will argue that that this second (...) conception of infinite number is the correct one, and analyze what this means for multiverses. (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)

Beck presents an outline of the procedure of bootstrapping of integer concepts, with the purpose of explicating the account of Carey. According to that theory, integer concepts are acquired through a process of inductive and analogous reasoning based on the object tracking system, which allows individuating objects in a parallel fashion. Discussing the bootstrapping theory, Beck dismisses what he calls the "deviant-interpretation challenge"—the possibility that the bootstrapped integer sequence does not follow a linear progression after some point—as being (...) general to any account of inductive learning. While the account of Carey and Beck focuses on the OTS, in this paper I want to reconsider the importance of another empirically well-established cognitive core system for treating numerosities, namely the approximate number system. Since the ANS-based account offers a potential alternative for integer concept acquisition, I show that it provides a good reason to revisit the deviant-interpretation challenge. Finally, I will present a hybrid OTS-ANS model as the foundation of integer concept acquisition and the framework of enculturation as a solution to the challenge. (shrink)

There are at least two traditional conceptions of numerical degree of similarity. According to the first, the degree of dissimilarity between two particulars is their distance apart in a metric space. According to the second, the degree of similarity between two particulars is a function of the number of (sparse) properties they have in common and not in common. This paper argues that these two conceptions are logically independent, but philosophically inconsonant.

There is no uniquely standard concept of an effectively decidable set of real numbers or real n-tuples. Here we consider three notions: decidability up to measure zero [M.W. Parker, Undecidability in Rn: Riddled basins, the KAM tori, and the stability of the solar system, Phil. Sci. 70(2) (2003) 359–382], which we abbreviate d.m.z.; recursive approximability [or r.a.; K.-I. Ko, Complexity Theory of Real Functions, Birkhäuser, Boston, 1991]; and decidability ignoring boundaries [d.i.b.; W.C. Myrvold, The decision problem for entanglement, in: R.S. (...) Cohen et al. (Eds.), Potentiality, Entanglement, and Passion-at-a-Distance: Quantum Mechanical Studies fo Abner Shimony, Vol. 2, Kluwer Academic Publishers, Great Britain, 1997, pp. 177–190]. Unlike some others in the literature, these notions apply not only to certain nice sets, but to general sets in Rn and other appropriate spaces. We consider some motivations for these concepts and the logical relations between them. It has been argued that d.m.z. is especially appropriate for physical applications, and on Rn with the standard measure, it is strictly stronger than r.a. [M.W. Parker, Undecidability in Rn: Riddled basins, the KAM tori, and the stability of the solar system, Phil. Sci. 70(2) (2003) 359–382]. Here we show that this is the only implication that holds among our three decidabilities in that setting. Under arbitrary measures, even this implication fails. Yet for intervals of non-zero length, and more generally, convex sets of non-zero measure, the three concepts are equivalent. (shrink)

The concept of impetus denoted the transmission of a power from the mover to the object moved. Many authors resorted to this concept to explain why a projectile keeps on moving when no longer in contact with its initial mover. But its application went further, as impetus was also appealed to in attempts to explain the acceleration of falling bodies or the motion of the heavens. It was widely applied in Renaissance natural philosophy, but it also raised a number (...) of ontological questions concerning its precise nature. (shrink)

Many theorists hold that there is, among value concepts, a fundamental distinction between thin ones and thick ones. Among thin ones are concepts like good and right. Among concepts that have been regarded as thick are discretion, caution, enterprise, industry, assiduity, frugality, economy, good sense, prudence, discernment, treachery, promise, brutality, courage, coward, lie, gratitude, lewd, perverted, rude, glorious, graceful, exploited, and, of course, many others. Roughly speaking, thick concepts are value concepts with significant descriptive content. (...) I will discuss a number of problems having to do with how best to understand the notion of a thick concept. Thick concepts have been widely discussed in the .. (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)

Frege’s rejection of singular reference to concepts is centrally implicated in his notorious paradox of the concept horse. I distinguish a number of claims in which that rejection might consist and detail the dialectical difficulties confronting the defense of several such claims. Arguably the least problematic such claim—that it is simply nonsense to say that a concept can be referred to with a singular term—has recently received a novel defense due to Robert Trueman. I set out Trueman’s argument (...) for this claim, identifying and remedying some omissions and errors of formulation therein. I then develop a response to the argument by showing, pace Trueman, that it is possible—and how it is possible—to express identities between objects and concepts. (shrink)

Commercial lawyers working across borders know that globalization has changed commercial law. To think of commercial law as only the law of states is to have an inadequate understanding of the norms governing commercial transactions. Some have argued for a transnational conception of commercial law, but their grounds of justification have been unpersuasive, often grounded on claims about the common content among national legal systems. Legal positivism is a rich literature on the concept of a legal system and the validity (...) conditions for rules in legal systems, but it has not been used to understand legal order outside or beyond the state. This article aims to use legal positivism to conceptualize a transnational commercial law order. Prevailing positivist accounts at least implicitly condition legal order on state sovereignty. The article offers a cosmopolitan conception of legal positivism, in which the state is no longer an enabling condition for law. The cosmopolitan conception provides the means by which to adequately describe a transnational commercial law order. There are limits to the conceptual analysis this article provides, one of which is that it does not purport to evaluate the justice or morality of transnational legal order. But the cosmopolitan conception of legal positivism elucidated in this article stands on its own as a way of understanding a number of transnational legal orders other than commercial law. The attractiveness of the account is that it describes law as a human social practice even when it is not solely the product of the state, so that we do not have to rely on natural law theories to understand legal rules that states do not maintain. (shrink)