A general mathematical framework, based on countable partitions of Natural Numbers [1], is presented, that allows to provide a Semantics to propositional languages. It has the particularity of allowing both the valuations and the interpretation Sets for the connectives to discriminate complexity of the formulas. This allows different adequacy criteria to be used to assess formulas associated with the same connective, but that differ in their complexity. The presented method can be adapted potentially infinite number of (...) connectives and truth values, therefore, it can be considered a general framework to provide semantics to several of the known logic systems (eg, LC, L3 LP, FDE). The presented semantics allow to converge to different standard semantics if the separation complexity procedure is annulled. Therefore, it can be understood as a framework that allows greater precision (in complexity terms) with respect to formula satisfaction. Naturally, because of how it is built, it can be incorporated into non-deterministic semantics. The presented procedure also allows generating valuations that grant a different truth value to each formula of propositional language. As a positive side effect, our method allows a constructive proof of the equipotence between N and N^n for all Natural n. (shrink)

Some partitions of Natural Number set are built through recursive processesgenerating in this manner countable examples of countable and disjoint sets whose unionis a set also countable. This process is constructive, so the Axiom of choice is not used.We provide a PC program that generates one of these special partitions and shows howto generate infinite of them. This line of reasoning can have multiple applications in Settheory and Model theory. We proved that the number (...) of ways to make these partitionsof natural numbers is not countable, there are more of these partitions (named doublycountable) than natural numbers. For each natural number greater than 1, we show aneffective procedure that generates these partitions -/- . (shrink)

De Finetti would claim that we can make sense of a draw in which each positive integer has equal probability of winning. This requires a uniform probability distribution over the natural numbers, violating countable additivity. Countable additivity thus appears not to be a fundamental constraint on subjective probability. It does, however, seem mandated by Dutch Book arguments similar to those that support the other axioms of the probability calculus as compulsory for subjective interpretations. These two lines (...) of reasoning can be reconciled through a slight generalization of the Dutch Book framework. Countable additivity may indeed be abandoned for de Finetti's lottery, but this poses no serious threat to its adoption in most applications of subjective probability. Introduction The de Finetti lottery Two objections to equiprobability 3.1 The ‘No random mechanism’ argument 3.2 The Dutch Book argument Equiprobability and relative betting quotients The re-labelling paradox 5.1 The paradox 5.2 Resolution: from symmetry to relative probability Beyond the de Finetti lottery. (shrink)

Number is a major object in mathematics. Mathematics is a discipline which studies the properties of a number. The object is expressible by mathematical language, which has been devised more rigorously than natural language. However, the language is not thoroughly free from natural language. Countability of natural number is also originated from natural language. It is necessary to understand how language leads a number into mathematics, its’ main playground.

A history of the construction of number has been in line with the process of recognition about the properties of geometry. Natural number representing countability is exhibited on a straight line and the completeness of real number is also originated from the continuous property of the number line. Complex number on a plane off the number line is established and thereafter, the whole number system is completed. When the process of constructing a number with geometric features is investigated from (...) different perspectives, it provides a new insight into the fundamental principle of number, which leads to a novel methodology in mathematics. (shrink)

It has been noticed that self-referential, ambiguous definitional formulas are accompanied by complementary self-referential antinomy formulas, which gives rise to contradictions. This made it possible to re-examine ancient antinomies and Cantor’s Diagonal Argument (CDA), as well as the method of nested intervals, which is the basis for evaluating the existence of uncountable sets. Using Georg Cantor’s remark that every real number can be represented as an infinite digital expansion (usually decimal or binary), a simplified system for verifying the definitions of (...) real numbers, subsets, and strings was created - the Cantor Criterion, which allows faulty definitions to be pointed out. For the CDA, the connection of formulas defining objects (real numbers, subsets, strings) from outside the list with supplementary formulas was demonstrated - their indirect and indispensable nature testifies to the lack of unambiguity and gives rise to contradictions for Cantor’s antinomic formulas. Thus, Cantor’s theorem about the higher power of the set of all subsets, using the reductio ad absurdum proof, lost its power and it was indicated that it is necessary to correct the scheme of the Axiom of Specification, which was introduced precisely to exclude antinomies from set theory by excluding from the use of self-referential antinomies and ambiguous supplementary formulas coupled with them identified by the H hypothesis. The method of nested intervals was investigated and it was shown that every real number can be defined by the limit of nested intervals and a countable list of real numbers obtained from a countable pool of all Jules Richard’s texts. (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.

In 1980 F. Wattenberg constructed the Dedekind completion

d of the Robinson non-archimedean field

and established basic algebraic properties of

d [6]. In 1985 H. Gonshor established further fundamental properties of

d [7].In [4] important construction of summation of countable sequence of Wattenberg numbers was proposed and corresponding basic properties of such summation were considered. In this paper the important applications of the Dedekind completion

d in transcendental number theory were considered. We dealing using set theory ZFC
(
-model (...) of ZFC).Given an class of analytic functions of one complex variable f  

z, we investigate the arithmetic nature of the values of fz at transcendental points en, n  .Main results are: (i) the both numbers e
 and e   are irrational, (ii) number ee is transcendental. Nontrivial generalization of the Lindemann- Weierstrass theorem is obtained. (shrink)

It is a striking fact from reverse mathematics that almost all theorems of countable and countably representable mathematics are equivalent to just five subsystems of second order arithmetic. The standard view is that the significance of these equivalences lies in the set existence principles that are necessary and sufficient to prove those theorems. In this article I analyse the role of set existence principles in reverse mathematics, and argue that they are best understood as closure conditions on the powerset (...) of the natural numbers. (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)

The nature of time is perceived by intellectuals variedly. An attempt is made in this paper to reconcile such varied views in the light of the Upanishads and related Indian spiritual and philosophical texts. The complex analysis of modern mathematics is used to represent the nature and presentation physical and psychological times so differentiated. Also the relation between time and energy is probed using uncertainty relations, forms of energy and phases of matter.

The systems of arithmetic discussed in this work are non-elementary theories. In this paper, natural numbers are characterized axiomatically in two di erent ways. We begin by recalling the classical set P of axioms of Peano’s arithmetic of natural numbers proposed in 1889 (including such primitive notions as: set of natural numbers, zero, successor of natural number) and compare it with the set W of axioms of this arithmetic (including the primitive notions like: (...) set of natural numbers and relation of inequality) proposed by Witold Wilkosz, a Polish logician, philosopher and mathematician, in 1932. The axioms W are those of ordered sets without largest element, in which every non-empty set has a least element, and every set bounded from above has a greatest element. We show that P and W are equivalent and also that the systems of arithmetic based on W or on P, are categorical and consistent. There follows a set of intuitive axioms PI of integers arithmetic, modelled on P and proposed by B. Iwanuś, as well as a set of axioms WI of this arithmetic, modelled on the W axioms, PI and WI being also equivalent, categorical and consistent. We also discuss the problem of independence of sets of axioms, which were dealt with earlier. (shrink)

Cantor’s proof that the powerset of the set of all natural numbers is uncountable yields a version of Richard’s paradox when restricted to the full definable universe, that is, to the universe containing all objects that can be defined not just in one formal language but by means of the full expressive power of natural language: this universe seems to be countable on one account and uncountable on another. We argue that the claim that definitional contexts (...) impose restrictions on the scope of quantifiers reveals a natural way out. (shrink)

In this short note many conjectures on partitions of integers as summations of prime numbers are presented, which are extension of Goldbach conjecture.

The nature of time is perceived by intellectuals variedly. An attempt is made in this paper to reconcile such varied views in the light of the Upanishads and related Indian spiritual and philosophical texts. The complex analysis of modern mathematics is used to represent the nature and presentation physical and psychological times so differentiated. Also the relation between time and energy is probed using uncertainty relations, forms of energy and phases of matter. Implications to time-dependent Schrodinger wave equation and uncertainty (...) principle are hinted. (shrink)

Roman Suszko said that “Obviously, any multiplication of logical values is a mad idea and, in fact, Łukasiewicz did not actualize it.” The aim of the present paper is to qualify this ‘obvious’ statement through a number of logical and philosophical writings by Professor Jan Woleński, all focusing on the nature of truth-values and their multiple uses in philosophy. It results in a reconstruction of such an abstract object, doing justice to what Suszko held a ‘mad’ project within a generalized (...) logic of judgments. Four main issues raised by Woleński will be considered to test the insightfulness of such generalized truth-values, namely: the principle of bivalence, the logic of scepticism, the coherence theory of truth, and nothingness. (shrink)

This paper introduces a novel object that has less structure than the natural numbers. As such it is a candidate model for the foundation that lies beneath the natural numbers. The implications for the construction of mathematical objects built upon that foundation are discussed.

The mathematician Georg Cantor strongly believed in the existence of actually infinite numbers and sets. Cantor’s “actualism” went against the Aristotelian tradition in metaphysics and mathematics. Under the pressures to defend his theory, his metaphysics changed from Spinozistic monism to Leibnizian voluntarist dualism. The factor motivating this change was two-fold: the desire to avoid antinomies associated with the notion of a universal collection and the desire to avoid the heresy of necessitarian pantheism. We document the changes in Cantor’s thought (...) with reference to his main philosophical-mathematical treatise, the Grundlagen (1883) as well as with reference to his article, “Über die verschiedenen Standpunkte in bezug auf das aktuelle Unendliche” (“Concerning Various Perspectives on the Actual Infinite”) (1885). (shrink)

Philosophy of science is seen by most as a meta-discipline – one that takes science as its subject matter, and seeks to acquire knowledge and understanding about science without in any way affecting, or contributing to, science itself. Karl Popper’s approach is very different. His first love is natural philosophy or, as he would put it, cosmology. This intermingles cosmology and the rest of natural science with epistemology, methodology and metaphysics. Paradoxically, however, one of his best known contributions, (...) his proposed solution to the problem of demarcation, helps to maintain the gulf that separates science from metaphysics, thus fragmenting cosmology into falsifiable science on the one hand, untestable philosophy on the other. This has damaging repercussions for a number of issues Popper tackles, from the problem of induction to simplicity of theory and quantum theory. But his proposed solution to the demarcation problem is untenable. Metaphysical assumptions are an integral part of scientific knowledge, inherent in the persistent acceptance of unified theories against the evidence. Once this is appreciated, it becomes obvious that natural philosophy, a synthesis of science and philosophy, is both more rigorous and of greater intellectual value than the two dissociated components we have today. What Popper sought for could come to full fruition. Problems that Popper tackled, from the problem of induction, to the problem of unity of theory, problems of quantum theory, and problems concerning the scope and limits of physics, all receive more adequate resolution within the new, fully-fledged natural philosophy. (shrink)

Over the past 50 years, there has been a great deal of philosophical interest in laws of nature, perhaps because of the essential role that laws play in the formulation of, and proposed solutions to, a number of perennial philosophical problems. For example, many have thought that a satisfactory account of laws could be used to resolve thorny issues concerning explanation, causation, free-will, probability, and counterfactual truth. Moreover, interest in laws of nature is not constrained to metaphysics or philosophy of (...) science; claims about laws play essential roles in areas as diverse as the philosophy of religion (e.g., in the argument from design) and the philosophy of mind (e.g., in the formulation of Davidson’s anomalous monism). In my dissertation, I consider and reject the widely-held thesis that the facts concerning laws can be reduced to the facts concerning the particular entities that the laws “govern,” and that the laws thus have no independent existence. I instead defend a version of nomic primitivism, according to which the facts about laws cannot be reduced to facts that are themselves non-nomic – i.e., to facts that do not fundamentally involve laws, counterfactuals, causes, etc. Insofar as the truth or falsity of reductionism about laws has implications for many of the problems mentioned above, I think that this result should be of interest even to those who who do not work in metaphysics or the philosophy of science. My methodology, which I lay out and defend in Chapter One, is a version of Carnapian explication. This method emphasizes the importance of articulating and maintaining clear distinctions between (1) the vague concept (or concepts) law of nature inherent in ordinary language and scientific practice and (2) the precise analyses of “law of nature” that philosophers have proposed as potential replacements for this concept. I argue that metaphysics-as-explication has clear advantages over rival conceptions of metaphysical methodology; in particular, it allows us to formulate evaluative criteria for metaphysical claims. In Chapter Two, I offer an example of how careful attention to concepts already in use can help resolve philosophical debate. Specifically, I argue that much recent literature has mistakenly assumed that there is only one concept of “law of nature” in use, while there are in fact at least two. Strong laws are the principles pursued by fundamental physics: they are true, objective, and bear distinctive relationships to counterfactuals and explanation. Weak laws, by contrast, lack at least one of these distinctive characteristics but play central roles in both the “special sciences” and in everyday life. In Chapters Three and Four, I offer extended arguments against the two most prominent versions of reductionism about laws – Humeanism and law necessitarianism. According to philosophical Humeans, the laws of nature supervene upon the non-modal, non-nomic facts concerning the behavior of particular things at particular times and places. Law necessitarians, by contrast, argue that the laws are in fact metaphysically necessary, and that which laws there are is determined by a class of primitive, modally loaded facts concerning the essences, natures, or dispositions. I argue that both of these views are mistaken insofar as they disagree with well-entrenched scientific practices and those in favor of reductionism have failed to provided sufficient reason for thinking that these practices should be revised. Much of my argument is focused on the role played by a number of supposed methodological principles, including appeals to intuition, parsimony, and methodological naturalism. While the conclusions of this dissertation are explicitly constrained to laws, many of the arguments should be of interest to those who are concerned about philosophical methodology (especially in the role of intuition in philosophical argument) or the appropriate relation between metaphysics, science, and the philosophy of science. (shrink)

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

MARGARET LYNN SCHABAS (Toronto, 1954) is professor of philosophy at the University of British Columbia in Vancouver and served as the head of the Philosophy Department from 2004-2009. She has held professoriate positions at the University of Wisconsin-Madison and at York University, and has also taught as a visiting professor at Michigan State University, University of Colorado-Boulder, Harvard, CalTech, the Sorbonne, and the École Normale de Cachan. As the recipient of several fellowships, she has enjoyed visiting terms at Stanford, Duke, (...) MIT, Cambridge, the LSE, and the MPI-Berlin. In addition to her doctorate in the history and philosophy of science and technology (Toronto 1983), she holds a bachelor of science in music (oboe) and the philosophy of science (Indiana 1976), a master’s degree in the history and philosophy of science (Indiana 1977), and a master’s degree in economics (Michigan1985). -/- She has published four books and over forty articles or book chapters in science studies. Some of the journals in which her articles can be found are Isis, Monist, History of Political Economy, Public Affairs Quarterly, Daedaelus, Journal of Economic Perspectives, and Studies in the History and Philosophy of Science. Her first book, A world ruled by number (1990) examines the emergence of mathematical economics in the second half of the nineteenth century. Her second book, The natural origins of economics (2005), traces the transformation of economics from a natural to a social science. She also has two co-edited collections, Oeconomies in the age of Newton (2003), with Neil De Marchi, and David Hume’s political economy (2008), with Carl Wennerlind. She is currently writing a monograph on Hume’s economics, as well as articles on the history and philosophy of bioeconomics. She is currently president of the History of Economics Society. -/- EJPE interviewed Margaret Schabas at the University of British Columbia in March 2013. In this interview, she recounts her earliest foray into the history and philosophy of economics, the conceptual trade between economics and natural science, and her most recent undertaking: the history and philosophy of bioeconomics. (shrink)

In this paper I present an in-class game designed to simulate the dynamics of the state of nature. I first explain the mechanics of the game, and how to administer it in the classroom. Then I address how the game can help introduce students to a number of important topics in political philosophy. In broad terms, the game serves to generate discussion regarding to main questions. (1) How does civil society come about? (2) Is the state of nature and the (...) arrangement which arises from it fair? In so doing I suggest how the game can further student understanding of figures such as Aristotle, Hobbes, Locke, Hume, Rousseau, Marx, and Rawls. (shrink)

I criticise a paper by Peter Forrest in which he argues that a principle of unrestricted countable fusion has paradoxical consequences. I argue that the paradoxical consequences that he exhibits may be due to his Whiteheadean assumptions about the nature of spacetime rather than to the principle of unrestricted countable fusion.

It is well known that the set of algebraic numbers (let us call it A) is countable. In this paper, instead of the usage of the classical terminology of cardinals proposed by Cantor, a recently introduced methodology using ①-based infinite numbers is applied to measure the set A (where the number ① is called grossone). Our interest to this methodology is explained by the fact that in certain cases where cardinals allow one to say only whether a (...) set is countable or it has the cardinality of the continuum, the ①-based methodology can provide a more accurate measurement of infinite sets. In this article, lower and upper estimates of the number of elements of A are obtained. Both estimates are expressed in ①-based numbers. (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)

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)

This book pursues the question of how and whether natural language allows for reference to abstract objects in a fully systematic way. By making full use of contemporary linguistic semantics, it presents a much greater range of linguistic generalizations than has previously been taken into consideration in philosophical discussions, and it argues for an ontological picture is very different from that generally taken for granted by philosophers and semanticists alike. Reference to abstract objects such as properties, numbers, propositions, (...) and degrees is considerably more marginal than generally held. (shrink)

Abstract: The paper argues that mainstream economics and mainstream philosophy of natural science had much in common during the period 1945-1965. It examines seven common features of the two fields and suggests a number of historical developments that might help explain these similarities. The historical developments include: the Vienna Circle connection, the Samuelson-Harvard-Foundations connection, and the Cold War operations research connection.

Abstract: This article explores the medical theories of the Dutch philosopher and physician Henricus Regius (1598-1679), who sought to provide clearer notions of medicine than the traditional theories of Jean Fernel, Daniel Sennert and Vopiscus Plempius. To achieve this, Regius overtly built upon the natural philosophy of René Descartes, in particular his theories of mechanical physiology and the corpuscular nature of matter. First, I show that Regius envisaged a novel partitioning of medicine, intended to make it independent in exposition (...) but conceptually grounded in natural philosophy. This served his overall purpose of making medicine a ‘clearer’ discipline. To this end Regius detaches the general notion of physiology as the study of the human body (which pertains to natural philosophy) from a medical physiology that concerns only health. Secondly, I show that Regius’s notions of health, disease, temperament and medicaments were the product of a Cartesian interpretation of traditional concepts, which was influenced by Santorio Santorio’s project of practicing medicine without occult elements. As a consequence, Regius’s classification of these notions is largely traditional. In conclusion, I show that Regius’s method of investigation in the area of natural philosophy consisted of a problem-solving procedure based on sensorial ideas. This approach can be traced back to Descartes, but was grounded on a purely empirical theory of knowledge, by which he rejected Descartes’s metaphysics. Regius’s order of exposition in natural philosophy and medicine, on the other hand, proceeds by definitions and divisions, and omits proofs. The reasons why Regius adopted this order were: 1) his rejection of Descartes’s metaphysical interpreta-tion of physics, and 2) his wish to emphasize the conceptual clarity and continuity of natural philosophy and medicine. (shrink)

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)

The term ‘human nature’ can refer to different things in the world and fulfil different epistemic roles. Human nature can refer to a classificatory nature (classificatory criteria that determine the boundaries of, and membership in, a biological or social group called ‘human’), a descriptive nature (a bundle of properties describing the respective group’s life form), or an explanatory nature (a set of factors explaining that life form). This chapter will first introduce these three kinds of ‘human nature’, together with seven (...) reasons why we disagree about human nature. In the main, this chapter focuses on the explanatory concept of human nature, which is related to one of the seven reasons for disagreement, namely, the scientific authority inherent in the term ‘nature’. I will examine why, in a number of historical contexts, it was attractive to refer to ‘nature’ as an explanatory category, and why this usage has led to the continual contestation of the term within the sciences. The claim is that even if the contents of talk about ‘nature’ varied historically, the term’s pragmatic function of demarcation stayed the same. The term ‘nature’ conveys scientific authority over a territory; ‘human nature’ is a concept used to divide causes, as well as experts, and thereby conquer others who threaten to invade one’s epistemic territory. Analysing this demarcation, which has social as well as epistemic aspects, will help us to understand why the explanatory role has been important and why it is unlikely that people will ever agree on either the meaning or the importance of ‘human nature’ as an explanatory category. (shrink)

Euclid's classic proof about the infinitude of prime numbers has been a standard model of reasoning in student textbooks and books of elementary number theory. It has withstood scrutiny for over 2000 years but we shall prove that despite the deceptive appearance of its analytical reasoning it is tautological in nature. We shall argue that the proof is more of an observation about the general property of a prime numbers than an expository style of natural deduction of (...) the proof of their infinitude. (shrink)

Modern categorical logic as well as the Kripke and topological models of intuitionistic logic suggest that the interpretation of ordinary “propositional” logic should in general be the logic of subsets of a given universe set. Partitions on a set are dual to subsets of a set in the sense of the category-theoretic duality of epimorphisms and monomorphisms—which is reflected in the duality between quotient objects and subobjects throughout algebra. If “propositional” logic is thus seen as the logic of subsets (...) of a universe set, then the question naturally arises of a dual logic of partitions on a universe set. This paper is an introduction to that logic of partitions dual to classical subset logic. The paper goes from basic concepts up through the correctness and completeness theorems for a tableau system of partition logic. (shrink)

Over the last thirty years there have been a number of attempts to analyse the distinction between intrinsic and extrinsic properties in terms of the facts about naturalness. This article discusses the three most influential of these attempts, each of which involve David Lewis. These are Lewis's 1983 analysis, his 1986 analysis, and his joint 1998 analysis with Rae Langton.

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)

Standard methods in experimental philosophy have sought to measure folk intuitions using experiments, but certain limitations are inherent in experimental methods. Accordingly, we have designed the Free-Will Intuitions Scale to empirically measure folk intuitions relevant to free-will debates using a different method. This method reveals what folk intuitions are like prior to participants' being put in forced-choice experiments. Our results suggest that a central debate in the experimental philosophy of free will—the “natural” compatibilism debate—is mistaken in assuming that folk (...) intuitions are exclusively either compatibilist or incompatibilist. They also identify a number of important new issues in the empirical study of free-will intuitions. (shrink)

A phenomenological exploration of the meta-physics of categories, relations, and signs as encountered in physics and the natural sciences.

Academic discussions of the count/mass distinction in Chinese feature three general problems, upon which this essay critically reflects: 1) Most studies focus either on modern or on classical Chinese thus representing parallel discussions that never intersect; 2) studies on count/mass grammar are often detached from reflections on count/mass semantics, which results in serious theoretical and terminological flaws; 3) approaches to Chinese often crucially depend on observations of English grammar and semantics, as, e.g., many/much vs. few/little patterns, the use of plural (...) inflections, etc., which is seldom justified. The article investigates the relevant discourse on the count/mass issue in classical and modern Chinese and concludes with exploring two distinct areas related to countability: the semantics of singular in contexts in which objects are introduced as referential-indefinite and the semantics of number and countability as revealed in diangu. (shrink)

On a now orthodox view, humans and many other animals possess a “number sense,” or approximate number system, that represents number. Recently, this orthodox view has been subject to numerous critiques that question whether the ANS genuinely represents number. We distinguish three lines of critique – the arguments from congruency, confounds, and imprecision – and show that none succeed. We then provide positive reasons to think that the ANS genuinely represents numbers, and not just non-numerical confounds or exotic substitutes (...) for number, such as “numerosities” or “quanticals,” as critics propose. In so doing, we raise a neglected question: numbers of what kind? Proponents of the orthodox view have been remarkably coy on this issue. But this is unsatisfactory since the predictions of the orthodox view, including the situations in which the ANS is expected to succeed or fail, turn on the kind of number being represented. In response, we propose that the ANS represents not only natural numbers, but also non-natural rational numbers. It does not represent irrational numbers, however, and thereby fails to represent the real numbers more generally. This distances our proposal from existing conjectures, refines our understanding of the ANS, and paves the way for future research. (shrink)

The question whether numbers are objects is a central question in the philosophy of mathematics. Frege made use of a syntactic criterion for objethood: numbers are objects because there are singular terms that stand for them, and not just singular terms in some formal language, but in natural language in particular. In particular, Frege (1884) thought that both noun phrases like the number of planets and simple numerals like eight as in (1) are singular terms referring to (...) numbers as abstract objects. (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.

Considering the set of natural numbers ℕ, then in the context of Peano axioms, starting from inequalities between finite sets, we find a fundamental contradiction, about the existence of ℕ, from a not-finitist point of view.

A number of authors in favor of a unitary account of singular descriptions have alleged that the unitary account can be extrapolated to account for plural definite descriptions. In this paper I take a closer look at this suggestion. I argue that while the unitary account is clearly onto something right, it is in the end empirically inadequate. At the end of the paper I offer a new partitive account of plural definite descriptions that avoids the problems with both the (...) unitary account and standard Russellian analyses. (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)

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).

Abstract. In this article we developed scientific and applied foundations of commercialization of the nature-resource potential of anthropogenic objects, on the example of exhausted mines. It is determined that the category of “anthropogenic object” can be considered in a narrow-applied sense, as specific anthropogenic objects to ensure the target needs, and in a broad theoretical sense, meaning everything that is created and changed by human influence, that is the objects of both artificial and natural origin. It was determined that (...) problems of commercialization of the natural-resource potential of anthropogenic objects are most often considered by researchers for specific objects, without having complex methodological coverage from the point of view of combining environmental, technical, economic and managerial components. When studying the substantiation of the scientific base, the authors confirmed the feasibility of the commercialization of natural-resource potential of anthropogenic objects on the example of a number of theoretical scientific studies in reclamation, reconstruction, recreation, remediation, restoration of biological productivity and economic value of land disturbed by economic activity. The considered examples of exhausted mines in the 21st century in the USA, Canada, Germany, Romania, and Poland indicate a wide range of opportunities for their commercialization. The study of the potential for commercialization of exhausted mines in the post-Soviet countries testified to the underused reserves for the commercialization of their nature-resource potential and their high potential for further development. The authors proposed the identification of anthropogenic objects on the basic livelihood spheres of society. There were identified the main system (natural, biological, technical, economic, social, managerial) and structural (subjects, trends, threats, risks, problems, challenges) factors of diagnosing the state of an anthropogenic object. A set of measures has been developed for commercialization of an anthropogenic object in functional and production activities, product policy, financial and investment spheres, pricing and sales policies, promotion, management and determination of property rights. Recommendations were provided on optimizing the management decision-making process based on a set of positivistic development principles, methods, and management functions. The study allows international organizations, state and local authorities, territorial communities, owners and potential investors to see new opportunities and make mutually beneficial decisions on the rational use of the nature-resource potential of anthropogenic objects. (shrink)

This is a book about numbers– what they are as concepts and how and why they originate–as viewed through the material devices used to represent and manipulate them. Fingers, tallies, tokens, and written notations, invented in both ancestral and contemporary societies, explain what numbers are, why they are the way they are, and how we get them. Cognitive archaeologist Karenleigh A. Overmann is the first to explore how material devices contribute to numerical thinking, initially by helping us to (...) visualize and manipulate the perceptual experience of quantity that we share with other species. She explores how and why numbers are conceptualized and then elaborated, as well as the central role that material objects play in both processes. Overmann’s volume thus offers a view of numerical cognition that is based on an alternative set of assumptions about numbers, their material component, and the nature of the human mind and thinking. Karim Zahidi (“Radicalizing numerical cognition,” Synthese, 2021, p. 530) called this reconstruction, as presented in journal articles, a “naturalistically plausible account of the emergence of the modern natural number concept," while César Dos Santos ("Review of: 'The Materiality of Numbers'," Qeios, 2023, p. 1) called it "a 'Copernican Revolution' in the way we understand the relationship between numbers and the material devices we use to record and manipulate them." . (shrink)

The approximate number system (ANS) enables organisms to represent the approximate number of items in an observed collection, quickly and independently of natural language. Recently, it has been proposed that the ANS goes beyond representing natural numbers by extracting and representing rational numbers (Clarke & Beck, 2021a). Prior work demonstrates that adults and children discriminate ratios in an approximate and ratio-dependent manner, consistent with the hallmarks of the ANS. Here, we use a well-known “connectedness illusion” to (...) provide evidence that these ratio-dependent ratio discriminations are (a) based on the perceived number of items in seen displays (and not just non-numerical confounds), (b) are not dependent on verbal working memory, or explicit counting routines, and (c) involve representations with a part-whole (or subset-superset) format, like a fraction, rather than a part-part format, like a ratio. These results vindicate key predictions of the hypothesis that the ANS represents rational numbers. (shrink)

There are mathematical structures with elements that cannot be distinguished by the properties they have within that structure. For instance within the field of complex numbers the two square roots of −1, i and −i, have the same algebraic properties in that field. So how do we distinguish between them? Imbedding the complex numbers in a bigger structure, the quaternions, allows us to algebraically tell them apart. But a similar problem appears for this larger structure. There seems to (...) be always a background and a context that we rely upon. Thus mathematicians naturally make use of Kantian intuition and references fixed by names and denotations. I argue that such features cannot be avoided. (shrink)