Randomness, a core concept of gambling, is seen in problem gambling as responsible for the formation of the math-related cognitive distortions, especially the Gambler’s Fallacy. In problem-gambling research, the concept of randomness was traditionally referred to as having a mathematical nature and categorized and approached as such. Randomness is not a mathematical concept, and I argue that its weak mathematical dimension is not decisive at all for the randomness-related issues in gambling and problem gambling, including (...) the correction of the misconceptions and fallacies about probability and statistical concepts applied in gambling. I distinguish between mathematical and nonmathematical dimensions of randomness (the epistemic, the theoretical-methodological, the functional, and the ethical) falling within the general concept, and I argue that both the studies having as object the math-related cognitive distortions among gamblers and the educational programs aiming at correcting them should employ this distinction in their design and content. (shrink)

In light of the observations and research design presented in the previous reports, the current technical report is focused on the relationship between the quality and specificity of the content of the gambling sites and the site’s SEO and marketing policy. This relationship is dependent upon the category of the gambling site and the difference in content quality, and the degree to which the mathematical dimension of gambling is reflected in this content is explained by this dependence.

With the previous technical reports, we presented the main insights from the qualitative analysis of the reflection of the mathematical dimension of gambling in gambling-related sites, in its structural, linguistic, epistemic, and informative aspects. In this report we will focus on the factors responsible for the patterns that the content of the gambling-related sites exhibit and the way that the mathematical dimension of gambling is reflected in this content, namely the factors that influence the content policies (...) of these sites. The report will also point to further directions of research-for both this project and for other investigators-for finding the appropriate balance and relation between the commercial policies on the one hand, and the adequacy and ethics of the content on the other hand. (shrink)

The current technical report of the research project investigating how the mathematical dimension of gambling is reflected in the communication and texts associated with the gambling industry raises the problem of the adequacy of sampling and proposes a new approach in this respect. The qualitative analysis of the reviewed websites is extended to a deeper analysis of language and also to the organization and structure of websites’ content. Although not stated as a goal of the initial project, the (...) research will also try to identify relationships between the specificity of content relative to the mathematical dimension of gambling, and the management and marketing of the website. An overview of the results of the statistical analysis of the current sample is also presented. (shrink)

Games of chance are developed in their physical consumer-ready form on the basis of mathematical models, which stand as the premises of their existence and represent their physical processes. There is a prevalence of statistical and probabilistic models in the interest of all parties involved in the study of gambling – researchers, game producers and operators, and players – while functional models are of interest more to math-inclined players than problem-gambling researchers. In this paper I present a structural (...) analysis of the knowledge attached to mathematical models of games of chance and the act of modeling, arguing that such knowledge holds potential in the prevention and cognitive treatment of excessive gambling, and I propose further research in this direction. (shrink)

The current study evaluates qualitatively how the mathematical dimension of gambling is reflected in the content of gambling websites. A number of gambling websites have been reviewed for their content in that respect. A statistical analysis recorded the presence of the mathematical dimension of gambling and its forms in the content of the participating websites, and a qualitative research study analyzed and assessed the quality of the content with respect to that dimension. The technical reports associated (...) with this study describe the methods used in the processes of review and content analysis, details of the processes, and the specific challenges and difficulties that they posed. (shrink)

The author proposes in this practical guide for both problem and non-problem gamblers a new pragmatic, conceptual approach of gambling mathematics. The primary aim of this guide is the adequate understanding of the essence and complexity of gambling through its mathematical dimension. The author starts from the premise that formal gambling mathematics, which is hardly even digestible for the non-math-inclined gamblers, is ineffective alone in correcting the specific cognitive distortions associated with gambling. By applying (...) the latest research results in this field, the author blends the gambling-mathematics concepts with the epistemology of applied mathematics and cognitive psychology for providing gamblers the knowledge required for rational and safe gambling. (shrink)

On the question of whether gambling behavior can be changed as result of teaching gamblers the mathematics of gambling, past studies have yielded contradictory results, and a clear conclusion has not yet been drawn. In this paper, I bring some criticisms to the empirical studies that tended to answer no to this hypothesis, regarding the sampling and laboratory testing, and I argue that an optimal mathematical scholastic intervention with the objective of preventing problem gambling is possible, (...) by providing the principles that would optimize the structure and content of the teaching module. Given the ethical aspects of the exposure of mathematical facts behind games of chance, and starting from the slots case – where the parametric design is missing, we have to draw a line between ethical and optional information with respect to the mathematical content provided by a scholastic intervention. Arguing for the role of mathematics in problem-gambling prevention and treatment, interdisciplinary research directions are drawn toward implementing an optimal mathematical module in cognitive therapies. (shrink)

Over the past two decades, gamblers have begun taking mathematics into account more seriously than ever before. While probability theory is the only rigorous theory modeling the uncertainty, even though in idealized conditions, numerical probabilities are viewed not only as mere mathematical information, but also as a decision-making criterion, especially in gambling. This book presents the mathematics underlying the major games of chance and provides a precise account of the odds associated with all gaming events. It begins (...) by explaining in simple terms the meaning of the concept of probability for the layman and goes on to become an enlightening journey through the mathematics of chance, randomness and risk. It then continues with the basics of discrete probability, combinatorics and counting arguments for those interested in the supporting mathematics. These mathematic sections may be skipped by readers who do not have a minimal background in mathematics; these readers can skip directly to the Guide to Numerical Results to pick the odds and recommendations they need for the desired gaming situation. Doing so is possible due to the organization of that chapter, in which the results are listed at the end of each section, mostly in the form of tables. The chapter titled The Mathematics of Games of Chance presents these games not only as a good application field for probability theory, but also in terms of human actions where probability-based strategies can be tried to achieve favorable results. Through suggestive examples, the reader can see what are the experiments, events and probability fields in games of chance and how probability calculus works there. The main portion of this work is a collection of probability results for each type of game. Each game s section is packed with formulas and tables. Each section also contains a description of the game, a classification of the gaming events and the applicable probability calculations. The primary goal of this work is to allow the reader to quickly find the odds for a specific gaming situation, in order to improve his or her betting/gaming decisions. Every type of gaming event is tabulated in a logical, consistent and comprehensive manner. The complete methodology and complete or partial calculations are shown to teach players how to calculate probability for any situation, for every stage of the game for any game. Here, readers can find the real odds, returned by precise mathematical formulas and not by partial simulations that most software uses. Collections of odds are presented, as well as strategic recommendations based on those odds, where necessary, for each type of gaming situation. The book contains much new and original material that has not been published previously and provides great coverage of probabilities for the following games of chance: Dice, Slots, Roulette, Baccarat, Blackjack, Texas Hold em Poker, Lottery and Sport Bets. Most of games of chance are predisposed to probability-based decisions. This is why the approach is not an exclusively statistical one, but analytical: every gaming event is taken as an individual applied probability problem to solve. A special chapter defines the probability-based strategy and mathematically shows why such strategy is theoretically optimal.". (shrink)

The near-miss has been considered an important factor of reinforcement in gambling behavior, and previous research has focused more on its industry-related causes and effects and less on the gaming phenomenon itself. The near-miss has usually been associated with the games of slots and scratch cards, due to the special characteristics of these games, which include the possibility of pre-manipulation of award symbols in order to increase the frequency of these “engineered” near-misses. In this paper, we argue that starting (...) from an elementary mathematical description of the classical (by pure chance) near-miss, generalizable to any game, and focusing equally on the epistemology of its constitutive concepts and their mathematical description, we can identify more precisely the fallacious elements of the near-miss cognitive effects and the inadequate perception and representation of the observational-intentional “I was that close.” This approach further suggests a strategy of using non-standard mathematical knowledge of an epistemological type in problem-gambling prevention and cognitive therapies. (shrink)

This paper outlines a formal recursive wager resolution calculus (WRC) that provides a novel conceptual framework for sentential logic via bridge rules that link wager resolution with truth values. When paired with a traditional truth-centric criterion of logical soundness WRC generates a sentential logic that is broadly truth-conditional but not truth-functional, supports the rules of proof employed in standard mathematics, and is immune to the most vexing features of their traditional implementation. WRC also supports a novel probabilistic criterion of (...) logical soundness, the fair betting probability criterion (FBP). It guarantees that the conclusion of an FBP-valid argument is at least as credible as a conjunction of premises, and also that the conclusion is true if the premises are. In addition, WRC provides a platform for a novel non-probabilistic, computationally simpler criterion of logical soundness – the criterion of Super-validity - that issues the same logical appraisals as FBP, and hence the same guarantees. (shrink)

This eighth book of the author on gambling math presents in accessible terms the cold mathematics behind the sparkling slot machines, either physical or virtual. It contains all the mathematical facts grounding the configuration, functionality, outcome, and profits of the slot games. Therefore, it is not a so-called how-to-win book, but a complete, rigorous mathematical guide for the slot player and also for game producers, being unique in this respect. As it is primarily addressed to the slot player, (...) its goal is to present practical applications of the mathematical models of slot games, in order to provide numerical results that a player can use as criteria for gaming decisions or just as information for any slot game and any predicted winning event. These results are focused on probability and expected value, these being the most important parameters for decisional criteria in slots. The book is packed with plenty of figures, tables, and formulas. The content is organized so that readers can skip the theoretical parts and go picking the practical results (numerical, in tables of values where possible, or ready-to-compute formulas) for the desired situation. The practical results are gathered in the last chapter, titled "Practical Applications and Numerical Results," the largest part of the book, for the most popular categories of slot machines, namely with 3, 5, 9, and 16 reels. Any other category of slot games is covered in the theoretical part of the book, where the general formulas apply not only to existing slot games, but also to possible future slot games of any design and configuration. The author does not just throw the slot mathematics to the audience and run away, but offers an ultimate practical contribution with the chapter "How to estimate the number of stops and the symbol distribution on a reel", a surprise for both players and producers, where one can see that mathematics provides players with some statistical methods as well as methods based on physical measurements for retrieving these missing data. Having these data along with the mathematical results of this book, anyone can generate the PAR sheet of any slot machine. In the last decade, mathematics has been taken more and more seriously into account in gaming, as being the essence that governs the games of chance and the only rigorous tool providing information on optimal play, where possible. For the popular game of slots, mathematics already fulfilled its duty by providing all the data that it can provide and that cannot be found on the display of the slot machines - it is all here in this book. (shrink)

Continuing his series of books on the mathematics of gambling, the author shows how a simple-rule game such as roulette is suited to a complex mathematical model whose applications generate improved betting systems that take into account a player's personal playing criteria. The book is both practical and theoretical, but is mainly devoted to the application of theory. About two-thirds of the content is lists of categories and sub-categories of improved betting systems, along with all the parameters that (...) might stand as the main objective criteria in a personal strategy - odds, profits and losses. The work contains new and original material not published before. The mathematical chapter describes complex bets, the profit function, the equivalence between bets and all their properties. All theoretical results are accompanied by suggestive concrete examples and can be followed by anyone with a minimal mathematical background because they involve only basic algebraic skills and set theory basics. The reader may also choose to skip the math and go directly to the sections containing applications, where he or she can pick desired numerical results from tables. The book offers no new so-called winning strategies, although it discusses them from a mathematical point of view. It does, however, offer improved betting systems and helps to organize a player's choices in roulette betting, according to mathematical facts and personal strategies. It is a must-have roulette handbook to be studied before placing your bets on the turn of either a European or American roulette wheel. (shrink)

Book Review Luck, logic, and white lies: the mathematics of games, second edition by Jörg Bewersdorff, New York, Taylor & Francis, CRC Press, 2021, 568 pp., GBP 42.99 (paperback), ISBN 9780367548414, Number of chapters 51.

Slot machines gained a high popularity despite a specific element that could limit their appeal: non-transparency with respect to mathematical parameters. The PAR sheets, exposing the parameters of the design of slot machines and probabilities associated with the winning combinations are kept secret by game producers, and the lack of data regarding the configuration of a machine prevents people from computing probabilities and other mathematical indicators. In this article, I argue that there is no rational justification for this secrecy by (...) giving two reasons, one psychological and the other mathematical. For the latter, I show that mathematics provides us with some statistical methods of retrieving the missing data, which are essential for the numerical probability computations in slots. The slots case raises the problem of the exposure of the parametric configuration and mathematical facts of any game of chance as an ethical obligation. (shrink)

This article explores some open questions related to the problem of verification of theories in the context of empirical sciences by contrasting three epistemological frameworks. Each of these epistemological frameworks is based on a corresponding central metaphor, namely: (a) Neo-empiricism and the gambling metaphor; (b) Popperian falsificationism and the scientific tribunal metaphor; (c) Cognitive constructivism and the object as eigen-solution metaphor. Each of one of these epistemological frameworks has also historically co-evolved with a certain statistical theory and method for (...) testing scientific hypotheses, respectively: (a) Decision theoretic Bayesian statistics and Bayes factors; (b) Frequentist statistics and p-values; (c) Constructive Bayesian statistics and e-values. This article examines with special care the Zero Probability Paradox (ZPP), related to the verification of sharp or precise hypotheses. Finally, this article makes some remarks on Lakatos’ view of mathematics as a quasi-empirical science. (shrink)

In his classic book “the Foundations of Statistics” Savage developed a formal system of rational decision making. The system is based on (i) a set of possible states of the world, (ii) a set of consequences, (iii) a set of acts, which are functions from states to consequences, and (iv) a preference relation over the acts, which represents the preferences of an idealized rational agent. The goal and the culmination of the enterprise is a representation theorem: Any preference relation that (...) satisfies certain arguably acceptable postulates determines a (finitely additive) probability distribution over the states and a utility assignment to the consequences, such that the preferences among acts are determined by their expected utilities. Additional problematic assumptions are however required in Savage's proofs. First, there is a Boolean algebra of events (sets of states) which determines the richness of the set of acts. The probabilities are assigned to members of this algebra. Savage's proof requires that this be a σ-algebra (i.e., closed under infinite countable unions and intersections), which makes for an extremely rich preference relation. On Savage's view we should not require subjective probabilities to be σ-additive. He therefore finds the insistence on a σ-algebra peculiar and is unhappy with it. But he sees no way of avoiding it. Second, the assignment of utilities requires the constant act assumption: for every consequence there is a constant act, which produces that consequence in every state. This assumption is known to be highly counterintuitive. The present work contains two mathematical results. The first, and the more difficult one, shows that the σ-algebra assumption can be dropped. The second states that, as long as utilities are assigned to finite gambles only, the constant act assumption can be replaced by the more plausible and much weaker assumption that there are at least two non-equivalent constant acts. The second result also employs a novel way of deriving utilities in Savage-style systems -- without appealing to von Neumann-Morgenstern lotteries. The paper discusses the notion of “idealized agent" that underlies Savage's approach, and argues that the simplified system, which is adequate for all the actual purposes for which the system is designed, involves a more realistic notion of an idealized agent. (shrink)

The article addresses the necessity of increasing the role of mathematics in the psychological intervention in problem gambling, including cognitive therapies. It also calls for interdisciplinary research with the direct contribution of mathematics. The current contributions and limitations of the role of mathematics are analysed with an eye toward the professional profiles of the researchers. An enhanced collaboration between these two disciplines is suggested and predicted.

“Dutch Book” arguments and references to gambling theorems are typical in the debate between Bayesians and scientists committed to “classical” statistical methods. These arguments have rarely convinced non-Bayesian scientists to abandon certain conventional practices, partially because many scientists feel that gambling theorems have little relevance to their research activities. In other words, scientists “don’t bet.” This article examines one attempt, by Schervish, Seidenfeld, and Kadane, to progress beyond such apparent stalemates by connecting “Dutch Book”–type mathematical results with principles (...) actually endorsed by practicing experimentalists. (shrink)

The field of of imprecise probability has matured, in no small part because of Teddy Seidenfeld’s decades of original scholarship and essential contributions to building and sustaining the ISIPTA community. Although the basic idea behind imprecise probability is (at least) 150 years old, a mature mathematical theory has only taken full form in the last 30 years. Interest in imprecise probability during this period has also grown, but many of the ideas that the mature theory serves can be difficult to (...) apprehend to those new to the subject. Although these fundamental ideas are common knowledge in the ISIPTA community, they are expressed, when they are expressed at all, obliquely, over the course of years with students and colleagues. -/- A single essay cannot convey the store of common knowledge from any research community, let alone the ISIPTA community. But, this essay nevertheless is an attempt to guide those familiar with the basic Bayesian framework to appreciate some of the elegant and powerful ideas that underpin the contemporary theory of lower previsions, which is the theory that most people associate with the term ‘imprecise probabilities’. (shrink)

In mid-2019, the controversy regarding South African runner Caster Semenya’s eligibility to participate in competitions against other female runners culminated in a Court of Arbitration for Sport judgement. Semenya possessed high endogenous testosterone levels (arguably a performance advantage), secondary to a disorder of sexual development. In this commentary, Aristotelean teleology is used to defend the existence of ‘male’ and ‘female’ as discrete categories. It is argued that once the athlete’s sex is established, they should be allowed to compete in the (...) category of their sex without obligatory medical treatment. Indeed, other athletes who possess advantageous genetic or phenotypic traits that fall outside of the human norm have been allowed to compete as humans without restraint. In both cases, if an athlete possesses the essential attributes of being a human or being male or female they should be permitted to compete in those respective categories; athletes’ eligibilities should not be based upon accidental attributes. (shrink)

Nền kinh tế nước ta đang chuyển biến mạnh mẽ từ nền kinh tế bao cấp sang kinh tế thị trường, nhất là từ khi nước ta gia nhập WTO. Đảng và chính phủ đã đề ra rất nhiều các chính sách để cải tiến các thể chế quản lý nền kinh tế và tài chính. Thị trường chứng khoán Việt Nam đã ra đời và đang đóng một vai trò quan trọng trong việc huy động vốn phục vụ cho (...) việc phát triển nền kinh tế quốc dân. Việc phân tích tác động của các chính sách đến nền kinh tế vĩ mô và nền tài chính, những đối sách của chính phủ ta và chính phủ các nước đối với cuộc khủng khoảng tài chính toàn cầu là các vấn đề rất thời sự đang thu hút sự chú ý của nhiều nhà kinh tế và toán học. (shrink)

Don’t form beliefs on the basis of coin flips or random guesses. More generally, don’t take belief gambles: if a proposition is no more likely to be true than false given your total body of evidence, don’t go ahead and believe that proposition. Few would deny this seemingly innocuous piece of epistemic advice. But what, exactly, is wrong with taking belief gambles? Philosophers have debated versions of this question at least since the classic dispute between William Clifford and William James (...) near the end of the nineteenth century. Here I reassess the normative standing of belief gambles from the perspective of epistemic decision theory. The main lesson of the paper is a negative one: it turns out that we need to make some surprisingly strong and hard-to-motivate assumptions to establish a general norm against belief gambles within a decision-theoretic framework. I take this to pose a dilemma for epistemic decision theory: it forces us to either make seemingly unmotivated assumptions to secure a norm against belief gambles, or concede that belief gambles can be rational after all. (shrink)

The city of Kruja dates back to its existence in the 5th and 6th centuries. In the inner city are preserved great historical, cultural, and architectural values that are inherited from generation to generation. In the city interact and coexist three different typologies of dwellings: historic buildings that belong to the XIII, XIV, XV, XIII, XIX centuries (built using the foundations of previous buildings); socialist buildings dating back to the Second World War until 1990; and modern buildings which were built (...) from 1990 onwards. According to the questionnaires, the creation of mathematical models applied to each category will result in contradictory attitudes but also fairy ones based on different percentages. There are underlined 5 quality of life indicators(questions) from a total of 30 questions of the questionnaire, which are involved in mathematical regression and are statistically significant with a significance level p<5% (with a reliability of 95%), which interact for all three categories of buildings. The quality of life indicators: dwelling area, heating mode during winter, level of dwelling improvement, time spent in the dwelling, and monthly electricity payments, are the main actors who will be compared in order to draw conclusions. According to the statistical calculations trend, it is noted that the socialist buildings category does not directly participate in the debate between historic and modern buildings by means of the quality of life indicators (questions). (shrink)

Pluralist mathematical realism, the view that there exists more than one mathematical universe, has become an influential position in the philosophy of mathematics. I argue that, if mathematical pluralism is true (and we have good reason to believe that it is), then mathematical realism cannot (easily) be justified by arguments from the indispensability of mathematics to science. This is because any justificatory chain of inferences from mathematical applications in science to the total body of mathematical theorems can cover (...) at most one mathematical universe. Indispensability arguments may thus lose their central role in the debate about mathematical ontology. (shrink)

Brouwer's philosophy of mathematics is usually regarded as an intra-subjective, even solipsistic approach, an approach that also underlies his mathematical intuitionism, as he strived to create a mathematics that develops out of something inner and a-linguistic. Thus, points of connection between Brouwer's mathematical views and his views about and the social world seem improbable and are rarely mentioned in the literature. The current paper aims to challenge and change that. The paper employs a socially oriented prism to examine (...) Brouwer's views on the construction, use, and practice of mathematics. It focuses on Brouwer's views on language, his social interactions, and the importance of group context as they appear in the significs dialogues. It does so by exploring the establishment and dissolution of the significs movement, focusing on Gerrit Mannoury's influence and relationship with Brouwer and analyzing several fragments from the significs dialogues while emphasizing the role Brouwer ascribed to groups in forming and sharing new ideas. The paper concludes by raising two questions that challenge common historical and philosophical readings of intuitionism. (shrink)

The desirable gambles framework offers the most comprehensive foundations for the theory of lower pre- visions, which in turn affords the most general ac- count of imprecise probabilities. Nevertheless, for all its generality, the theory of lower previsions rests on the notion of linear utility. This commitment to linearity is clearest in the coherence axioms for sets of desirable gambles. This paper considers two routes to relaxing this commitment. The first preserves the additive structure of the desirable gambles framework and (...) the machinery for coherent inference but detaches the interpretation of desirability from the multiplicative scale invariance axiom. The second strays from the additive combination axiom to accommodate repeated gambles that return rewards by a non-stationary processes that is not necessarily additive. Unlike the first approach, which is a conservative amendment to the desirable gambles framework, the second is a rad- ical departure. Yet, common to both is a method for describing rewards called discounted utility. (shrink)

An inquiry on the training needs in Mathematics was conducted to Laura Vicuña Center - Palawan (LVC-P) learners. Specifically, this aimed to determine their level of performance in numbers, measurement, geometry, algebra, and statistics, identify the difficulties they encountered in solving word problems and enumerate topics where they needed coaching. -/- To identify specific training needs, the study employed a descriptive research design where 36 participants were sampled purposively. The data were gathered through a problem set test and focus (...) group discussion. Findings revealed that LVC-P learners had an unsatisfactory performance in numbers, measurement, and statistics while alarmingly poor in geometry and algebra. They also faced difficulties in remembering, understanding, applying, and analyzing mathematical concepts when solving problems. Further, the learners were able to name certain topics subjected to tutorials. -/- The above facts and observations suggest that learners of LVC-P have urgent training needs in Mathematics. It is recommended that the Western Philippines University - College of Education RDE Unit continue its research, development, and extension services to LVC-P. More importantly, the results of this inquiry regarding the training needs of the learners will serve as bases for conducting an extension project and development program for LVC-P. Series of tutorial sessions is deemed necessary to address the needs and difficulties of the learners. (shrink)

There is a major debate as to whether there are non-causal mathematical explanations of physical facts that show how the facts under question arise from a degree of mathematical necessity considered stronger than that of contingent causal laws. We focus on Marc Lange’s account of distinctively mathematical explanations to argue that purported mathematical explanations are essentially causal explanations in disguise and are no different from ordinary applications of mathematics. This is because these explanations work not by appealing to what (...) the world must be like as a matter of mathematical necessity but by appealing to various contingent causal facts. (shrink)

The paper discusses Hilbert mathematics, a kind of Pythagorean mathematics, to which the physical world is a particular case. The parameter of the “distance between finiteness and infinity” is crucial. Any nonzero finite value of it features the particular case in the frameworks of Hilbert mathematics where the physical world appears “ex nihilo” by virtue of an only mathematical necessity or quantum information conservation physically. One does not need the mythical Big Bang which serves to concentrate all (...) the violations of energy conservation in a “safe”, maximally remote point in the alleged “beginning of the universe”. On the contrary, an omnipresent and omnitemporal medium obeying quantum information conservation rather than energy conservation permanently generates action and thus the physical world. The utilization of that creation “ex nihilo” is accessible to humankind, at least theoretically, as long as one observes the physical laws, which admit it in their new and wider generalization. One can oppose Hilbert mathematics to Gödel mathematics, which can be identified as all the standard mathematics until now featureable by the Gödel dichotomy of arithmetic to set theory: and then, “dialectic”, “intuitionistic”, and “Gödelian” mathematics within the former, according to a negative, positive, or zero value of the distance between finiteness and infinity. A mapping of Hilbert mathematics into pseudo-Riemannian space corresponds, therefore allowing for gravitation to be interpreted purely mathematically and ontologically in a Pythagorean sense. Information and quantum information can be involved in the foundations of mathematics and linked to the axiom of choice or alternatively, to the field of all rational numbers, from which the pair of both dual and anti-isometric Peano arithmetics featuring Hilbert arithmetic are immediately inferable. Noether’s theorems (1918) imply quantum information conservation as the maximally possible generalization of the pair of the conservation of a physical quantity and the corresponding Lie group of its conjugate. Hilbert mathematics can be interpreted from their viewpoint also after an algebraic generalization of them. Following the ideas of Noether’s theorem (1918), locality and nonlocality can be realized both physically and mathematically. The “light phase of the universe” can be linked to the gap of mathematics and physics in the Cartesian organization of cognition in Modernity and then opposed to its “dark phase”, in which physics and mathematics are merged. All physical quantities can be deduced from only mathematical premises by the mediation of the most fundamental physical constants such as the speed of light in a vacuum, the Planck and gravitational constants once they have been interpreted by the relation of locality and nonlocality. (shrink)

Call an explanation in which a non-mathematical fact is explained—in part or in whole—by mathematical facts: an extra-mathematical explanation. Such explanations have attracted a great deal of interest recently in arguments over mathematical realism. In this article, a theory of extra-mathematical explanation is developed. The theory is modelled on a deductive-nomological theory of scientific explanation. A basic DN account of extra-mathematical explanation is proposed and then redeveloped in the light of two difficulties that the basic theory faces. The final view (...) appeals to relevance logic and uses resources in information theory to understand the explanatory relationship between mathematical and physical facts. 1Introduction2Anchoring3The Basic Deductive-Mathematical Account4The Genuineness Problem5Irrelevance6Relevance and Information7Objections and Replies 7.1Against relevance logic7.2Too epistemic7.3Informational containment8Conclusion. (shrink)

This monograph offers a fresh perspective on the applicability of mathematics in science. It explores what mathematics must be so that its applications to the empirical world do not constitute a mystery. In the process, readers are presented with a new version of mathematical structuralism. The author details a philosophy of mathematics in which the problem of its applicability, particularly in physics, in all its forms can be explained and justified. Chapters cover: mathematics as a formal (...) science, mathematical ontology: what does it mean to exist, mathematical structures: what are they and how do we know them, how different layers of mathematical structuring relate to each other and to perceptual structures, and how to use mathematics to find out how the world is. The book simultaneously develops along two lines, both inspired and enlightened by Edmund Husserl’s phenomenological philosophy. One line leads to the establishment of a particular version of mathematical structuralism, free of “naturalist” and empiricist bias. The other leads to a logical-epistemological explanation and justification of the applicability of mathematics carried out within a unique structuralist perspective. This second line points to the “unreasonable” effectiveness of mathematics in physics as a means of representation, a tool, and a source of not always logically justified but useful and effective heuristic strategies. (shrink)

We demonstrate how real progress can be made in the debate surrounding the enhanced indispensability argument. Drawing on a counterfactual theory of explanation, well-motivated independently of the debate, we provide a novel analysis of ‘explanatory generality’ and how mathematics is involved in its procurement. On our analysis, mathematics’ sole explanatory contribution to the procurement of explanatory generality is to make counterfactual information about physical dependencies easier to grasp and reason with for creatures like us. This gives precise content (...) to key intuitions traded in the debate, regarding mathematics’ procurement of explanatory generality, and adjudicates unambiguously in favour of the nominalist, at least as far as explanatory generality is concerned. (shrink)

We know very little about mathematical skepticism in modem times. Imre Lakatos once remarked that “in discussing modem efforts to establish foundations for mathematical knowledge one tends to forget that these are but a chapter in the great effort to overcome skepticism by establishing foundations for knowledge in general." And in a sense he was clearly right: modem thought — with its new discoveries in mathematical sciences, the mathematization of physics, the spreading of Pyrrhonist doctrines, the centrality of epistemological foundationalism (...) and the diffusion of the geometrical method in philosophy — was the most natural arena in which skepticism and mathematics could confront each other. The problem remains, however, that no investigation of the whole topic has yet been attempted. Thus, as far as we know, mathematical certainties should have clashed with skeptical doubts, but whether and to what extent there was indeed a historical debate on mathematical skepticism in modern thought remains to be ascertained. (shrink)

Some authors have begun to appeal directly to studies of argumentation in their analyses of mathematical practice. These include researchers from an impressively diverse range of disciplines: not only philosophy of mathematics and argumentation theory, but also psychology, education, and computer science. This introduction provides some background to their work.

In “Models and Reality” (1980), Putnam sketched a version of his internal realism as it might arise in the philosophy of mathematics. Here, I will develop that sketch. By combining Putnam’s model-theoretic arguments with Dummett’s reflections on Gödelian incompleteness, we arrive at (what I call) the Skolem-Gödel Antinomy. In brief: our mathematical concepts are perfectly precise; however, these perfectly precise mathematical concepts are manifested and acquired via a formal theory, which is understood in terms of a computable system of (...) proof, and hence is incomplete. Whilst this might initially seem strange, I show how internal categoricity results for arithmetic and set theory allow us to face up to this Antinomy. This also allows us to understand why “Models are not lost noumenal waifs looking for someone to name them,” but “constructions within our theory itself,” with “names from birth.”. (shrink)

I argue that certain species of belief, such as mathematical, logical, and normative beliefs, are insulated from a form of Harman-style debunking argument whereas moral beliefs, the primary target of such arguments, are not. Harman-style arguments have been misunderstood as attempts to directly undermine our moral beliefs. They are rather best given as burden-shifting arguments, concluding that we need additional reasons to maintain our moral beliefs. If we understand them this way, then we can see why moral beliefs are vulnerable (...) to such arguments while mathematical, logical, and normative beliefs are not—the very construction of Harman-style skeptical arguments requires the truth of significant fragments of our mathematical, logical, and normative beliefs, but requires no such thing of our moral beliefs. Given this property, Harman-style skeptical arguments against logical, mathematical, and normative beliefs are self-effacing; doubting these beliefs on the basis of such arguments results in the loss of our reasons for doubt. But we can cleanly doubt the truth of morality. (shrink)

Indispensablists argue that when our belief system conflicts with our experiences, we can negate a mathematical belief but we do not because if we do, we would have to make an excessive revision of our belief system. Thus, we retain a mathematical belief not because we have good evidence for it but because it is convenient to do so. I call this view ‘ mathematical convenientism.’ I argue that mathematical convenientism commits the consequential fallacy and that it demolishes the Quine-Putnam (...) indispensability argument and Baker’s enhanced indispensability argument. (shrink)

Mathematics has always been a core part of western education, from the medieval quadrivium to the large amount of arithmetic and algebra still compulsory in high schools. It is an essential part. Its commitment to exactitude and to rigid demonstration balances humanist subjects devoted to appreciation and rhetoric as well as giving the lie to postmodernist insinuations that all “truths” are subject to political negotiation. In recent decades, the character of mathematics has changed – or rather broadened: it (...) has become the enabling science behind the complexity of contemporary knowledge, from gene interpretation to bank risk. Mathematical understanding is all the more necessary for future jobs, as well as remaining, as ever, a prophylactic against the more corrosive philosophical views emanating from the humanities. (shrink)

Socialist buildings in the city of Kruja (Albania) date back after the Second World War between the years 1945-1990. These buildings were built during the time of the socialist Albanian dictatorship and the totalitarian communist regime. A questionnaire with 30 questions was conducted and 14 people were interviewed. The interviewed residents belong to a certain area of the city of Kruja. Based on the results obtained, diagrams have been conceived and mathematical regression models have been developed which will serve as (...) a base for drawing conclusions. The purpose of this study is to find a relationship that specifies the role of the inhabitants, the role of the dwellings, the physical-mechanical characteristics, and their energy efficiency, in order to improve the living conditions of the inhabitants. The independent variables interact with each other assuming positive or negative values depending on the probabilistic mathematical regression, giving different results based on the calculated data. The variables that co-exist among them are living space, quality of life, time spent at home, social interaction of residents, dwelling humidity level, dwelling orientation, satisfaction level of the inhabitants, dwelling improvements demand, and monthly electricity bills. (shrink)

In this handbook, I put into practice my philosophical views on children's mathematics. The handbook contains brief instructions and examples of mathematical activities. In the INSTRUCTIONS section, instructions are given on how, and in part why that way, to help preschool children in their mathematical development. In the ACTIVITIES section, there are examples of activities through which the child develops her mathematical abilities.

The published works of scientists often conceal the cognitive processes that led to their results. Scholars of mathematical practice must therefore seek out less obvious sources. This article analyzes a widely circulated mathematical joke, comprising a list of spurious proof types. An account is proposed in terms of argumentation schemes: stereotypical patterns of reasoning, which may be accompanied by critical questions itemizing possible lines of defeat. It is argued that humor is associated with risky forms of inference, which are essential (...) to creative mathematics. The components of the joke are explicated by argumentation schemes devised for application to topic-neutral reasoning. These in turn are classified under seven headings: retroduction, citation, intuition, meta-argument, closure, generalization, and definition. Finally, the wider significance of this account for the cognitive science of mathematics is discussed. (shrink)

Recent experimental evidence from developmental psychology and cognitive neuroscience indicates that humans are equipped with unlearned elementary mathematical skills. However, formal mathematics has properties that cannot be reduced to these elementary cognitive capacities. The question then arises how human beings cognitively deal with more advanced mathematical ideas. This paper draws on the extended mind thesis to suggest that mathematical symbols enable us to delegate some mathematical operations to the external environment. In this view, mathematical symbols are not only used (...) to express mathematical concepts—they are constitutive of the mathematical concepts themselves. Mathematical symbols are epistemic actions, because they enable us to represent concepts that are literally unthinkable with our bare brains. Using case-studies from the history of mathematics and from educational psychology, we argue for an intimate relationship between mathematical symbols and mathematical cognition. (shrink)

Analysing several characteristic mathematical models: natural and real numbers, Euclidean geometry, group theory, and set theory, I argue that a mathematical model in its final form is a junction of a set of axioms and an internal partial interpretation of the corresponding language. It follows from the analysis that (i) mathematical objects do not exist in the external world: they are our internally imagined objects, some of which, at least approximately, we can realize or represent; (ii) mathematical truths are not (...) truths about the external world but specifications (formulations) of mathematical conceptions; (iii) mathematics is first and foremost our imagined tool by which, with certain assumptions about its applicability, we explore nature and synthesize our rational cognition of it. (shrink)

For over thirty years I have argued that all branches of science and scholarship would have both their intellectual and humanitarian value enhanced if pursued in accordance with the edicts of wisdom-inquiry rather than knowledge-inquiry. I argue that this is true of mathematics. Viewed from the perspective of knowledge-inquiry, mathematics confronts us with two fundamental problems. (1) How can mathematics be held to be a branch of knowledge, in view of the difficulties that view engenders? What could (...) mathematics be knowledge about? (2) How do we distinguish significant from insignificant mathematics? This is a fundamental philosophical problem concerning the nature of mathematics. But it is also a practical problem concerning mathematics itself. In the absence of the solution to the problem, there is the danger that genuinely significant mathematics will be lost among the unchecked growth of a mass of insignificant mathematics. This second problem cannot, it would seem, be solved granted knowledge-inquiry. For, in order to solve the problem, mathematics needs to be related to values, but this is, it seems, prohibited by knowledge-inquiry because it could only lead to the subversion of mathematical rigour. Both problems are solved, however, when mathematics is viewed from the perspective of wisdom-inquiry. (1) Mathematics is not a branch of knowledge. It is a body of systematized, unified and inter-connected problem-solving methods, a body of problematic possibilities. (2) A piece of mathematics is significant if (a) it links up to the interconnected body of existing mathematics, ideally in such a way that some problems difficult to solve in other branches become much easier to solve when translated into the piece of mathematics in question; (b) it has fruitful applications for (other) worthwhile human endeavours. If ever the revolution from knowledge to wisdom occurs, I would hope wisdom mathematics would flourish, the nature of mathematics would become much more transparent, more pupils and students would come to appreciate the fascination of mathematics, and it would be easier to discern what is genuinely significant in mathematics (something that baffled even Einstein). As a result of clarifying what should count as significant, the pursuit of wisdom mathematics might even lead to the development of significant new mathematics. (shrink)

The existence of fundamental moral disagreements is a central problem for moral realism and has often been contrasted with an alleged absence of disagreement in mathematics. However, mathematicians do in fact disagree on fundamental questions, for example on which set-theoretic axioms are true, and some philosophers have argued that this increases the plausibility of moral vis-à-vis mathematical realism. I argue that the analogy between mathematical and moral disagreement is not as straightforward as those arguments present it. In particular, I (...) argue that pluralist accounts of mathematics render fundamental mathematical disagreements compatible with mathematical realism in a way in which moral disagreements and moral realism are not. 11. (shrink)

It cannot be overstated that in recent times the gravity and complexity of irregular maritime migration have triggered concerns and debates in the academic domains of International Relations and International Law. The article examines how states’ compliance with, and enforcement of, international law and legal norms would tackle the challenges of irregular migration in the Indian Ocean region. The legislative and policy challenges regarding irregular migration can be analyzed under two segments. First, there is a lack of comprehensive discussion about (...) the sending and the receiving states’ commitments (regarding irregular migrants) to international law and legal norms. Secondly, deficit of comparative analysis to show how the littoral states comply with international law norms on migration and refugee influx. Such analysis from multiple perspectives would be helpful to get insights and make policy recommendations about how effectively international law norms could be enforced through national legislation and policy framework. (shrink)

There is a wide range of realist but non-Platonist philosophies of mathematics—naturalist or Aristotelian realisms. Held by Aristotle and Mill, they played little part in twentieth century philosophy of mathematics but have been revived recently. They assimilate mathematics to the rest of science. They hold that mathematics is the science of X, where X is some observable feature of the (physical or other non-abstract) world. Choices for X include quantity, structure, pattern, complexity, relations. The article lays (...) out and compares these options, including their accounts of what X is, the examples supporting each theory, and the reasons for identifying the science of X with (most or all of) mathematics. Some comparison of the options is undertaken, but the main aim is to display the spectrum of viable alternatives to Platonism and nominalism. It is explained how these views answer Frege’s widely accepted argument that arithmetic cannot be about real features of the physical world, and arguments that such mathematical objects as large infinities and perfect geometrical figures cannot be physically realized. (shrink)

The triumph of scientific materialism in the Seventeenth Century not only bifurcated nature into matter and mind and primary and secondary qualities, as Alfred North Whitehead pointed out in Science and the Modern World. It divided science and the humanities. The core of science is the effort to comprehend the cosmos through mathematics. The core of the humanities is the effort to comprehend history and human nature through narratives. The life sciences can be seen as the zone in which (...) the conflict between these two very different ways of comprehending the world collide. Evolutionary theory as defended by Schelling developed out of natural history, but efforts have been made to formulate neo-Darwinism through mathematical models. However, it is impossible to eliminate stories from biology. As Stuart Kauffman argued, mathematical models attempt to pre-state all possibilities, but in evolution there can be adjacent possibles that can be embraced by organisms but cannot be pre-stated. To account for such actions it is necessary to tell stories. Mathematics provides analytic precision allowing long chains of deductions, but tends to deny temporal becoming and cannot do justice to the openness of the future, while narratives focus on processes and events, but lack exactitude that would provide precise deductions and predictions. In advancing mathematics adequate to life, Robert Rosen argued that living beings as anticipatory systems must have models of themselves, and strove to develop a form of mathematics able to model life itself. It has been convincingly argued that narratives are central to human self-creation and they are lived out before being explicitly told. Their models of themselves are first and foremost, stories or narratives. If this is the case, might not living beings as biological entities be characterized by proto-stories or narratives in their models of themselves? Biosemiotics, largely inspired by C.S. Peirce, provides a bridge between mathematical and narrative comprehension, conceiving them as different forms of semiosis. The study of life through biosemiotics could reveal how mathematics and narratives can be understood in relation to each other. This could have implications for how we understand science and the humanities and their relationship to each other. In this paper I will examine work in theoretical biology that might advance these efforts. (shrink)

In the first part of this article we survey general similarities and differences between biological and social macroevolution. In the second (and main) part, we consider a concrete mathematical model capable of describing important features of both biological and social macroevolution. In mathematical models of historical macrodynamics, a hyperbolic pattern of world population growth arises from non-linear, second-order positive feedback between demographic growth and technological development. Based on diverse paleontological data and an analogy with macrosociological models, we suggest that the (...) hyperbolic character of biodiversity growth can be similarly accounted for by non-linear, second-order positive feedback between diversity growth and the complexity of community structure. We discuss how such positive feedback mechanisms can be modelled mathematically. (shrink)