Values, moral values and democratic values are attracting the attention of education researchers in general and mathematics education researchers in particular. Little research has studied pre-service teachers’ perceptions of values in the classroom, their perceptions of the relationship between the different variables of values in the classroom, as well as their relationship with the democratic society. The present research attempts to do so. Twenty-two graduate pre-service teachers who participated in ‘New trends in mathematics (...) education’ course discussed how to cultivated values in the mathematics classroom. Moreover, they answered survey questions related to the cultivation of values in this classroom. We used a combination of deductive and inductive content analysis to characterize the pre-service teachers’ texts. The research results indicate that the pre-service teachers perceived values as encouraging students’ activity in the mathematics classroom. In addition, the pre-service teachers perceived values as encouraging specific categories of values needed as skills for the citizen in a democratic society, as creativity, critical thinking and metacognition. (shrink)

This paper argues that mathematics is imbued with values reflecting its production from human imagination and dialogue. Epistemological, ontological, aesthetic and ethical values are specified, both overt and covert. Within the culture of mathematics, the overt values of truth, beauty, purity, universalism, objectivism, rationalism and utility are identified. In contrast, hidden within mathematics and its culture are the covert values of objectism and ethics, including the specific ethical values of separatism, openness, fairness (...) and democracy. Some of these values emerge from a dialogical view of the nature of mathematics, and certainly from the dialogical nature of culture. It is acknowledged that the choice of certain perspectives, such as absolutism, lead to the appearance of mathematics as largely value-free through hiding value-assumptions behind the initial choice of philosophy. By uncovering the values of mathematics I show that it more closely resembles other components of human culture than is usually recognised. (shrink)

Legal scholarship exploring the nature of evidence and the process of juridical proof has had a complex relationship with formal modeling. As evident in so many fields of knowledge, algorithmic approaches to evidence have the theoretical potential to increase the accuracy of fact finding, a tremendously important goal of the legal system. The hope that knowledge could be formalized within the evidentiary realm generated a spate of articles attempting to put probability theory to this purpose. This literature was both insightful (...) and frustrating. Much light was shed on the legal system, but it also quickly became evident that the tools of probability theory were in many ways ill-constructed for the task. Fundamental incompatibilities between the structure of legal decision making and the extant formal tools were identified, and it became evident that many of the purported explanations of legal phenomena were internally inconsistent. As a consequence, interest in this type of formal modeling declined, and attention was directed toward different kinds of explanations of the phenomena. Perhaps under the influence of a recent trend toward various types of formal modeling in legal scholarship, a recent burst of articles, rather than attempting to explain the macro structure of trials, which was the previous object of interest, attempts to quantify the probative value of various items of evidence in ways consistent with the formal features of various probability theories, and then to study decision making from that perspective. For example, the value of evidence is often purported to be its likelihood ratio, that is, the probability of discovering or receiving the evidence given a hypothesis (e.g., the defendant did it) divided by the probability of discovering or receiving the evidence given the negation of the hypothesis (the defendant didn't do it). Alternatively, the value of evidence is purported (more contextually) to be the information gain it provides, defined as the increase in probability it provides for a hypothesis above the probability of the hypothesis based on the other available evidence. Both conceptions then assume that all of the various probability assessments conform or ought to conform to the dictates of Bayes' theorem (that maintains consistency among such assessments); empirical studies are then done testing the extent to which this is so and proposing how the law can increase the probability that it is so. The general criticisms of using Bayes' theorem as a formal model of juridical proof are well known and were integral to the last wave of interest in formal modeling of the evidentiary process. This paper thus for the most part puts aside that more general issue, and focuses specifically on mathematical modeling of the value of particular items of evidence. The paper demonstrates that formal modeling has only limited value in explaining the value of legal evidence, much more limited than those constructing and discussing the models assume, and thus that the conclusions they draw about the value of evidence are unwarranted. This is done through a discussion of four recent examples that attempt to quantify evidence relating to, respectively, carpet fibers, infidelity, DNA random-match evidence, and character evidence used to impeach a witness. This article thus makes two contributions. First, and most importantly, it is another demonstration of the complex relationship between algorithmic tools and legal decision making. Second, at a minimum it points out serious pitfalls for analytical or empirical studies of juridical proof. (shrink)

The article is devoted to identifying the value of the phenomenon of aesthetic value and beauty of mathematical knowledge and the beauty of mathematical theory of teaching mathematics. The aesthetic potential of mathematical knowledge allows the use of theater technology in the educational process with the active dialogic interaction between teacher and students. The criteria of beauty in mathematical theories are distinguished: the realization of beauty as the unity of the whole, and in the disclosure of the complex through (...) the elementary; methodological interpretation of the beauty in the community of mathematical structures and optimal information content of the meta-language of mathematics; the practical embodiment of beauty in the formalization of the infinite through the finite. The beauty of mathematics is the force that permeates all the ‘layers of knowledge‘ not along, and across, although the effectiveness of mathematical activity due to aesthetic laws, which do not always lend themselves to unambiguous interpretation. In the article it is stated that, depending on the educational goals of communicative impact on the audience, in fact, ‘mathematical lectures theatricality‘ can have different characteristics, the most important of which are teachers artistry and artistic director’s work of a teacher. This cultural phenomenon that includes the theatrical talent, helps create an atmosphere of cooperation needed in varying degrees of activity for pedagogical interaction. The author believes that such approach, developed on the basis of the Stanislavsky system, allows university professors of mathematics significantly improve mathematical lectures. (shrink)

Proofs are central, and unique, to mathematics. They establish the truth of theorems and provide us with the most secure knowledge we can possess. It is thus perhaps unsurprising that philosophers once thought that the only value proofs have lies in establishing the truth of theorems. However, such a view is inconsistent with mathematical practice. If a proof’s only value is to show a theorem is true, then mathematicians would have no reason to reprove the same theorem in different (...) ways, yet this is a practice they frequently engage in. This suggests that proofs can have a wide variety of values. My purpose in this chapter is to provide a survey of some of them. I will discuss how proofs can have practical value, for example, by helping mathematicians to make new discoveries or learn new mathematical tools, and how they can have abstract value, for example, by being beautiful or explanatory. I will also explore the relationships between different proof values. (shrink)

At the beginning of the present century, a series of paradoxes were discovered within mathematics which suggested a fundamental unclarity in traditional mathemati­cal methods. These methods rested on the assumption of a realm of mathematical idealities existing independently of our thinking activity, and in order to arrive at a firmly grounded mathematics different attempts were made to formulate a conception of mathematical objects as purely human constructions. It was, however, realised that such formulations necessarily result in a (...) class='Hi'>mathematics which lacks the richness and power of the old ‘platonistic’ methods, and the latter are still defended, in various modified forms, as embodying truths about self-existent mathematical entities. Thus there is an idealism-realism dispute in the philosophy of mathematics in some respects parallel to the controversy over the existence of the experiential world to the settle­ment of which lngarden devoted his life. The present paper is an attempt to apply Ingarden’s methods to the sphere of mathematical existence. This exercise will reveal new modes of being applicable to non-real objects, and we shall put forward arguments to suggest that these modes of being have an importance outside mathematics, especially in the areas of value theory and the ontology of art. (shrink)

We argue that it is neither necessary nor sufficient for a mathematical proof to have epistemic value that it be “correct”, in the sense of formalizable in a formal proof system. We then present a view on the relationship between mathematics and logic that clarifies the role of formal correctness in mathematics. Finally, we discuss the significance of these arguments for recent discussions about automated theorem provers and applications of AI to mathematics.

This article surveys recent literature by Parsons, McGee, Shapiro and others on the significance of categoricity arguments in the philosophy of mathematics. After discussing whether categoricity arguments are sufficient to secure reference to mathematical structures up to isomorphism, we assess what exactly is achieved by recent ‘internal’ renditions of the famous categoricity arguments for arithmetic and set theory.

The main item in the present volume was published in 1930 under the title Das Unendliche in der Mathematik und seine Ausschaltung. It was at that time the fullest systematic account from the standpoint of Husserl's phenomenology of what is known as 'finitism' (also as 'intuitionism' and 'constructivism') in mathematics. Since then, important changes have been required in philosophies of mathematics, in part because of Kurt Godel's epoch-making paper of 1931 which established the essential in completeness of arithmetic. (...) In the light of that finding, a number of the claims made in the book (and in the accompanying articles) are demon strably mistaken. Nevertheless, as a whole it retains much of its original interest and value. It presents the issues in the foundations of mathematics that were under debate when it was written (and in some cases still are);, and it offers one alternative to the currently dominant set-theoretical definitions of the cardinal numbers and other arithmetical concepts. While still a student at the University of Vienna, Felix Kaufmann was greatly impressed by the early philosophical writings (especially by the Logische Untersuchungen) of Edmund Husser!' He was never an uncritical disciple of Husserl, and he integrated into his mature philosophy ideas from a wide assortment of intellectual sources. But he thought of himself as a phenomenologist, and made frequent use in all his major publications of many of Husserl's logical and epistemological theses. (shrink)

Do multi-level selection explanations of the evolution of social traits deepen the understanding provided by single-level explanations? Central to the former is a mathematical theorem, the multi-level Price decomposition. I build a framework through which to understand the explanatory role of such non-empirical decompositions in scientific practice. Applying this general framework to the present case places two tasks on the agenda. The first task is to distinguish the various ways of suppressing within-collective variation in fitness, and moreover to evaluate their (...) biological interest. I distinguish four such ways: increasing retaliatory capacity, homogenising assortment, and collapsing either fitness structure or character distribution to a mean value. The second task is to discover whether the third term of the Price decomposition measures the effect of any of these hypothetical interventions. On this basis I argue that the multi-level Price decomposition has explanatory value primarily when the sharing-out of collective resources is `subtractable'. Thus its value is more circumscribed than its champions Sober and Wilson (1998) suppose. (shrink)

The purpose of this paper is to analyse possibilities of including and conveying a variety of values in teaching mathematics through tasks in terms of their openness. In the first part of the introduction we present the theoretical ideas about values in mathematics education by Bishop and Lim & Ernest. The second part of introduction sets out the reasons why an emphasis on values seems advisable. Mathematics is not commonly associated with a variety of (...) values. However, for mathematical education an orientation towards societal demands appear relevant. Scholars like Skovsmose and Fischer alike demanded the development of a reflecting ability. Some national state curricula depict some of the societal demands by calling for an orientation towards general values for all school subjects, but in this we see discrepancies to practiced teaching. The section “Tasks and Values” relates to justifying why tasks appear to be a useful means of influencing mathematics education. Several scholars have already investigated the possibilities of different task-types for teaching values. Here we connect this purpose to tasks characterized by their varying openness. As teachers are essential to unfold the potential of tasks, the present analysis is intendent to give them concrete ideas for mediating values in mathematics classes. In the main part we analyse traditional tasks from ancient textbooks, with experiences in Armenian classes, as well as an open task with experiences of Austrian students in partner work. The discussed tasks provide opportunities for mathematical values like control, openness and for epistemological values as well as for more general values like self-confidence, mutual respect, justice and mathematics educational values like accuracy and consistency. In conclusion, the encouragement towards discussions among and with the students during the solving process of the task appears to be essential. Our experiences show that students mainly react positively towards such discussing activities. This could be a motivation for teachers to take up this issue besides of idealistic reasons. (shrink)

Scientific explanations are widely recognized to have instrumental value by helping scientists make predictions and control their environment. In this paper I raise, and provide a first analysis of, the question whether explanatory proofs in mathematics have analogous instrumental value. I first identify an important goal in mathematical practice: reusing resources from existing proofs to solve new problems. I then consider the more specific question: do explanatory proofs have instrumental value by promoting reuse of the resources they contain? In (...) general, I argue that the answer to this question is “no” and demonstrate this in detail for the theory of mathematical explanation developed by Marc Lange. (shrink)

Philosophers of mathematics agree that the only interpretation of arithmetic that takes that discourse at 'face value' is one on which the expressions 'N', '0', '1', '+', and 'x' are treated as proper names. I argue that the interpretation on which these expressions are treated as akin to free variables has an equal claim to be the default interpretation of arithmetic. I show that no purely syntactic test can distinguish proper names from free variables, and I observe that any (...) semantic test that can must beg the question. I draw the same conclusion concerning areas of mathematics beyond arithmetic. This paper is a greatly extended version of my response to Stewart Shapiro's paper in the conference 'Structuralism in physics and mathematics' held in Bristol on 2–3 December, 2006. (shrink)

Definitions are traditionally seen as abbreviations, as tools for notational convenience that do not increase inferential power. From a Philosophy of Mathematical Practice point of view, however, there is much more to definitions. For example, definitions can play a role in problem solving, definitions can contribute to understanding, sometimes equivalent definitions are appreciated differently, and so on. This chapter reviews the literature on definitions and (to a certain extent) concepts in mathematical practice. It is structured according to four themes through (...) which definitions (and concepts) in mathematical practice have been studied. These themes concern (1) the nature of definitions, (2) whether and how concepts evolve, (3) definitions and concepts from a communal perspective, and (4) different values relating to definitions and concepts. (shrink)

2nd edition. Many-valued logics are those logics that have more than the two classical truth values, to wit, true and false; in fact, they can have from three to infinitely many truth values. This property, together with truth-functionality, provides a powerful formalism to reason in settings where classical logic—as well as other non-classical logics—is of no avail. Indeed, originally motivated by philosophical concerns, these logics soon proved relevant for a plethora of applications ranging from switching theory to cognitive (...) modeling, and they are today in more demand than ever, due to the realization that inconsistency and vagueness in knowledge bases and information processes are not only inevitable and acceptable, but also perhaps welcome. The main modern applications of (any) logic are to be found in the digital computer, and we thus require the practical knowledge how to computerize—which also means automate—decisions (i.e. reasoning) in many-valued logics. This, in turn, necessitates a mathematical foundation for these logics. This book provides both these mathematical foundation and practical knowledge in a rigorous, yet accessible, text, while at the same time situating these logics in the context of the satisfiability problem (SAT) and automated deduction. The main text is complemented with a large selection of exercises, a plus for the reader wishing to not only learn about, but also do something with, many-valued logics. (shrink)

INTRODUCTION In ita original form the present paper was presented to the American Mathematical Society, April 2k,, as a companion piece to the writer's ...

Traditional classifications of the main schools in the philosophy of mathematics are based upon two questionable presuppositions. First, it is assumed that a realist, who wishes to defend objective truth values of mathematical statements, has to be either a Platonist or a physicalist. Secondly, a constructivist, who regards mathematical entities as human constructs rather than pre-existing objects, has to be either a subjective mentalist or an objective idealist. In contrast to these alternatives and their many variants, this paper (...) argues that it is possible to be a constructivist and a realist at the same time. Mathematical entities are public creations in the human-made domain of abstract artifacts; this domain Karl Popper called World 3. They are not completely transparent to their makers, so that mathematical constructs are full of open problems for us to study. The applicability of mathematics to physical, mental, and social reality can be explained by the Theory of Measurement, which also helps to show what is wrong and what is right with the empiricist approaches to mathematical knowledge. (shrink)

What I suggest we can see in this brief overview of the literature is an extensive interpenetration on both sides of these debates between scientific, political, and social values. Important shifts in political and social values were of course occurring over the same period, some of them in parallel with, and perhaps even contributing to, these transitions I have been speaking of in evolutionary discourse. The developments that I think of as at least suggestive of possible parallels include (...) the progressive encroachment of public values into the private domain of post-World War II American life, the cold war, the rise of consumerism, and the flowering of what Christopher Lasch calls a “narcissistic individualism.”35 In popular language, the 1960s gave birth to the “me” generation. Perhaps the most tantalizing analogue is suggested by Barbara Ehrenreich's argument for the emergence of a new meaning of masculinity — an ideal of masculinity measured not by commitment, responsibility, or success as family provider, but precisely by the strength of a man's autonomy in the private sphere, his resistance to the demands of a hampering female.36 It is tempting to speculate about possible connections between changes in scientific discourse and developments in the social and political spheres, but such connections, however suggestive, would clearly have to be demonstrated.For now, however, I want to focus on another kind of change —a transformation not so much in the social or political sphere as in the scientific sphere. I make this turn, or return, in support of a more complex account of scientific change that incorporates reverberations within the scientific communty along with social and political changes.In the 1960s, all of biology was undergoing a major transformation in direct response to the dramatic successes of molecular biology. These successes seemed to completely vindicate the values on which the molecular revolution was premised — namely, simplicity and mechanism. Following the victory of Watson and Crick, and of others after them, the fever of that endeavor swept through biology leaving in its wake a new standard of science, and of scientific discourse — one predicated on clarity, simplicity, and analyzability; on the definition of legitimate questions as those capable of clear and unambiguous answers. Every biological discipline felt it — even evolutionary biology, which in some respects was at the furthest pole. Perhaps precisely because it seemed conceptually so remote, evolutionary biology may have felt it most of all. Lewontin inadvertently provides us with some direct support for this view. Indeed, he begins his introduction to Population Biology and Evolution with the following remarks: The twenty years since World War II have seen a vindication in biology of our faith in the Cartesian method as a way of doing science. Some of the most fundamental and interesting problems of biology have been solved or are very nearly solved by an analytic technique that is now loosely called “molecular biology.” But it is not specifically the “molecular” aspect of biology of the last twenty years that has led to its success. It is, rather, the analytic aspect, the belief that by breaking systems down into their component parts, by simplifying them or using simpler organisms, one can learn about more complex systems. As it happens, the problems that were attacked and are being attacked by this method lead to answers in terms of molecules and cell organelles.... There is a host of problems in biology, however, that has been much neglected in these twenty exciting years, because the answers to them cannot be meaningfully framed in molecular and cellular terms.37Lewontin is referring, of course, to problems in evolution. The remainder of his remarks is devoted to an argument for the applicability of the method, if not the content, of molecular biology to these problems. He writes, “It is not the case that molecular biology is Cartesian and analytic while population biology is holistic. Population biology is properly analytic and operates, within the framework of its own problems, by the process of simplification, analysis, and resynthesis.”38 With these remarks, he leads into the criticism of the “holists” who have “held up progress.”This new ethic of simplicity, clarity, and mechanism — embodying the very virtues lauded by Williams — was explicitly carried into evolutionary biology in the name of scientific progress. As it happened, the values implied also fit conveniently well with other values — each set of values providing crucial support for the other.However substantive the scientific gains may have been in some respects, the net effect of this ethic has also been a systematic “perceptual bias” — a bias with profound practical consequences for the entire program of methodological individualism in evolutionary biology, if not elsewhere as well. It may well be that the whole is equivalent to the reconstituted aggregate of its parts, if, in the process of aggregation or summation, all possible interactions among the parts are included. But if certain kinds of interactions are systematically excluded, our confidence in that program necessarily founders. My claim here is that such systematic exclusion does occur, and that it occurs on a number of different levels. To briefly review the interlocking kinds of “bias” that I see occurring in practice, I suggest the following schematic listing:On the most general level: The ethic of simplicity — the privileging of certain values, even certain methodologies, as having an a priori superior claim to scientific credibility.Only slightly less general, and crucially related, is the equation of “scientific” with “tractible”: Given the techniques of analysis available, the equation of science with what we can do inevitably leads to a systematic technical bias favoring simplicity. That is, because we don't know how to model complex dynamics, nonlinear interactions are systematically biased against because of the limitations of our technical know-how. The consequences of this equation of the scientific with the tractible are greatly compounded by the additional equation between what we can do and what is — that is, by our temptation to confuse tractibility with reality.Finally, and also closely related, a further kind of elision occurs even within the confines of tractibility. This kind of elision — taking the form almost of inferring tractibility from one's prior assumptions of what is real — is exemplified by the history of a mathematical ecology of mutualism. Even when mutualism can be introduced into the same technical machinery, it is still not pursued. The basic assumption is that competition is what is real, not because it is easier to model, but because it is what we expect. When the actual difficulties of modeling competition are then in turn suppressed, as in the Robert May story, what we have, given the temptation to equate the tractible with the real, is the possibility of a truly self-fulfilling prophecy. (shrink)

This article reports the macro-organizational structure of research articles in mathematics, based on an analysis of 30 published pure and applied mathematics articles. Math RAs eschew the Introduction-Methods-Results-Discussion structure for an Introduction-Results model that enables researchers to present new knowledge as clearly and succinctly as possible. Notable omissions from the mathematics RA structure are Method and Discussion sections, which mathematicians do not need because of the well-established methodology used in the field and the relative absence of extended (...) discussion required to interpret research findings. We contextualize the macrostructure of RAs in mathematics within the discourse conventions and disciplinary assumptions about knowledge in the field to suggest the value of such a strategy to teachers and students of academic writing. (shrink)

The conflict between Platonic realism and Constructivism marks a watershed in philosophy of mathematics. Among other things, the controversy over the Axiom of Choice is typical of the conflict. Platonists accept the Axiom of Choice, which allows a set consisting of the members resulting from infinitely many arbitrary choices, while Constructivists reject the Axiom of Choice and confine themselves to sets consisting of effectively specifiable members. Indeed there are seemingly unpleasant consequences of the Axiom of Choice. The non-constructive nature (...) of the Axiom of Choice leads to the existence of non-Lebesgue measurable sets, which in turn yields the Banach-Tarski Paradox. But the Banach-Tarski Paradox is so called in the sense that it is a counter-intuitive theorem. To corroborate my view that mathematical truths are of non-constructive nature, I shall draw upon Gödel’s Incompleteness Theorems. This also shows the limitations inherent in formal methods. Indeed the Löwenheim-Skolem Theorem and the Skolem Paradox seem to pose a threat to Platonists. In this light, Quine/Putnam’s arguments come to take on a clear meaning. According to the model-theoretic arguments, the Axiom of Choice depends for its truth-value upon the model in which it is placed. In my view, however, this is another limitation inherent in formal methods, not a defect for Platonists. To see this, we shall examine how mathematical models have been developed in the actual practice of mathematics. I argue that most mathematicians accept the Axiom of Choice because the existence of non-Lebesgue measurable sets and the Well-Ordering of reals open the possibility of more fruitful mathematics. Finally, after responding to Benacerraf’s challenge to Platonism, I conclude that in mathematics, as distinct from natural sciences, there is a close connection between essence and existence. Actual mathematical theories are the parts of the maximally logically consistent theory that describes mathematical reality. (shrink)

The goal of the article is to substantiate that despite the criticism the paradigm in economics will not change because of the axiomatic assumptions of value-free economics. How these assumptions work is demonstrated on the example of Gary Becker’s economic approach which is analyzed from the perspective of scientific research programme. The author indicates hard core of economic approach and the protective belt which makes hard core immune from any criticism. This immunity leads economists to believe that they are objective (...) scientists and, consequently, it results in epistemological hubris. Due to its tautological nature, economic approach is considered to be a degenerative programme. This conclusion is extended on value-free economics. In spite of these problems, many economists still believe in positive economics and they dismiss normative approaches. It has a negative influence on people. The conclusion of the article is that thanks to axiomatic assumptions economists do not have objective and ironclad methodology and they should accept normative values in their research. (shrink)

This paper explores the relationship of time and value in the history of economics, using the contributions of Girard, Achterhuis, Kula and Mirowski. In the ‘anthropometric stage’ time and value are intertwined: value and time are not abstract concepts, but they express a concrete process which incorporates the social positions of individuals. In the ‘lineamentric stage’ the concepts of time and value remain cyclical, but they receive an abstract character. The economy reproduces itself cyclically, because the origin of value – (...) human labour – reproduces itself in each production period. In the ‘syndetic stage’ the cyclical conception of the economy is abandoned through the perception of a non-reproductive system. Time and value are detached from each other, and linear time enters economic theory as an analytical tool. The labour theory of value is replaced by external denominants of value: utility and external scarcity. (shrink)

In an attempt to justify research efforts in various branches of science, scholars have tried to capture the essence of the relevant subject-matters in a definition, or at least have declared these subject-matters to exist. Otherwise the study of topics in the branches would be of questionable value, to say the least. For example, when dealing with numbers, their ontological status somehow has to be declared.

Generality is a key value in scientific discourses and practices. Throughout history, it has received a variety of meanings and of uses. This collection of original essays aims to inquire into this diversity. Through case studies taken from the history of mathematics, physics and the life sciences, the book provides evidence of different ways of understanding the general in various contexts. It aims at showing how individuals have valued generality and how they have worked with specific types of "general" (...) entities, procedures, and arguments. (shrink)

In this paper we consider the major development of mathematical analysis during the mid-nineteenth century. On the basis of Jahnke’s (Hist Math 20(3):265–284, 1993 ) distinction between considering mathematics as an empirical science based on time and space and considering mathematics as a purely conceptual science we discuss the Swedish nineteenth century mathematician E.G. Björling’s general view of real- and complexvalued functions. We argue that Björling had a tendency to sometimes consider mathematical objects in a naturalistic way. One (...) example is how Björling interprets Cauchy’s definition of the logarithm function with respect to complex variables, which is investigated in the paper. Furthermore, in view of an article written by Björling (Kongl Vetens Akad Förh Stockholm 166–228, 1852 ) we consider Cauchy’s theorem on power series expansions of complex valued functions. We investigate Björling’s, Cauchy’s and the Belgian mathematician Lamarle’s different conditions for expanding a complex function of a complex variable in a power series. We argue that one reason why Cauchy’s theorem was controversial could be the ambiguities of fundamental concepts in analysis that existed during the mid-nineteenth century. This problem is demonstrated with examples from Björling, Cauchy and Lamarle. (shrink)

It is the purpose of this paper to present a theory of probability derived from two-valued logic—the logic of which an aspect is given in Part I, Section A, of Principia Mathematica. The symbolic system of Mr. Keynes, given in his Treatise on Probability, will be shown to be a part of our system. We have, however, little if anything in common with his philosophical analysis; a definition of Keynes’ fundamental probability relation, free from psychological or material reference, will be (...) given, enabling us to offset some of the objections to his theory. (shrink)

The present book of proceedings includes chapters related to the areas of pure, applied and inter-disciplinary mathematics reflecting the potential applications in the domains of sciences and engineering. The main areas include algebra and its applications, analysis and approximation theory, cryptography, computational fluid dynamics, continuum mechanics and vibrations, differential equations and applications, graph theory, fuzzy mathematics and logic, numerical analysis, optimization and its applications, wave propagation, etc. The scientists, engineers, academicians and researchers working in the proposed areas of (...) coding and information theory, computational fluid dynamics, differential equations, fuzzy sets and systems, numerical analysis, optimization, vibrations, etc. looking for new insight and ideas in these areas will be benefitted by the contents of this book. In fact, it will be useful to a large class of readers interested in recent findings related to mathematical sciences and their applications to the diverse domains of knowledge. The book will be a fruitful contribution to all knowledge seekers and researchers in the concerned areas who are looking for new insight and ideas. As it consists of updated research articles on emerging areas of mathematical sciences and their applications, it will be a valued addition to the learning resource centres of various universities, industries, and research and development organizations across the globe. (shrink)

This article focuses on defective conditionals ? namely indicative conditionals whose antecedents are false and whose truth-values therefore cannot be determined. The problem is to decide which formal connective can adequately represent this usage. Classical logic renders defective conditionals true whereas traditional mathematics dismisses them as irrelevant. This difference in treatment entails that, at the propositional level, classical logic validates some sentences that are intuitively false in plane geometry. With two proofs, I show that the same flaw is (...) shared by a family of trivalent logics. I go on to examine the strict conditional and its derivatives. This family is the only one to avoid the faulty inference but it does so without addressing the status of the truth-value assigned to defective conditionals. (shrink)

The article is based upon the following starting position. In this post-modern time, it seems that no scholar in Europe supports what is called “Enlightenment Project” with its naïve objectivism and Correspondence Theory of Truth1, - though not being really hostile, just strongly skeptical about it. No old-fasioned “classical” academical texts; only His Majesty Discourse as chain of interpretations and reinterpretations. What was called objectivity “proved to be” intersubjectivity; what was called Object (in Latin and German and Russian tradition) now (...) is related to as a phenomenon; what was called Subject either is looked upon as Cartesian “cogito” or disappeares at all; what was called Truth turned to be either method of demonstration (by positivists) or the author’s sincerity (by existentialists); what was called Essence is now merely a joke. Alternatively, we speak about Sense, Meaning, and Value. (shrink)

Aesthetics in philosophy of mathematics is too narrowly construed. Beauty is not the only feature in mathematics that is arguably aesthetic. While not the highest aesthetic value, being interesting is a sine qua non for publishability. Of the many ways to be interesting, being explanatory has recently been discussed. The motivational power of what is interesting is important for both directing research and stimulating education. The scientific satisfaction of curiosity and the artistic desire for beautiful results are complementary (...) but both aesthetic. (shrink)

During his lifetime Peirce was appreciated for his erudite and accurate thinking by experts in a few borderline philosophical subjects—in particular the history and philosophy of physics, semeiotics, mathematical logic, the theory of probability—and was suspected by a few prescient souls to be a general philosopher of outstanding genius; but by academic and other publishers he was consistently regarded as a bad bet. These facts largely explain the history of Peircean publication in the last few decades. It was only in (...) 1958, nearly fifty years since Peirce stopped writing, that the wealthiest university in the wealthiest nation in the world was able to complete its eight volume edition of his Collected Papers; but in the meanwhile at least three useful smaller anthologies of Peirce’s writings had been put before the public. Professor Wiener’s volume makes a fourth. And the obvious initial question is: was this additional anthology really necessary? (shrink)

ABSTRACTThe paper deals with a problem posed by Mathieu Vidal to provide a formal representation for defective conditional in mathematics Vidal, M. [. The defective conditional in mathematics. Journal of Applied Non-Classical Logics, 24, 169–179]. The key feature of defective conditional is that its truth-value is indeterminate if its antecedent is false. In particular, we are interested in two explanations given by Vidal with the use of trivalent logics. By analysing a simple argument from plane geometry, where defective (...) conditional is in use, he gives two trivalent formal explanations for it. For both explanations, Vidal rigorously shows that trivalent logics cannot adequately represent defective conditional. Preserving Vidal's criteria of defective conditional ad max, we indicate some arguable points in his explanations and present an alternative explanation containing the original conjunction and disjunction in order to show that there are trivalent logics that might be an adequate formal explanation for defective conditional. (shrink)

This paper contains a defense against anti-realism in mathematics in the light both of incompleteness and of the fact that mathematics is a ‘cultural artifact.’. Anti-realism (here) is the view that theorems, say, of aritltmetic cannot be taken at face value to express true propositions about the system of numbers but must be reconstrued to be about somctliiiig else or about nothing at all. A ‘bite-the-bullet’ aspect of the defease is that, adopting new axioms, liitherto independent, is not. (...) a matter of recognizing trutlis wliich had previoasly been unrecognized, but of extending the domain of what is true. (shrink)

We introduce the notion of a convex tree. We show that the binary expansion for real numbers in the unit interval ) is equivalent to weak König lemma ) for trees having at most two nodes at each level, and we prove that the intermediate value theorem is equivalent to \ for convex trees, in the framework of constructive reverse mathematics.

Despite progress, gender gaps persist in mathematical and language-related fields, and gender stereotypes likely play a role. The current study examines the relations between parents’ gender-related beliefs and their adolescent child’s motivation and career aspirations through a survey of 172 parent-child dyads. Parents reported their gendered beliefs about ability in mathematics and language arts, as well as their prescriptive gender role beliefs. Students reported their expectancies and values in these two domains, as well as their career aspirations The (...) results of path models suggested that parents’ ability stereotypes about language boosted girls’ motivation for language arts, thereby nudging them away from STEM pathways. Girls’ career aspirations stemmed not only from their valuation of the corresponding domain, but also from their valuation of competing domains. Such findings highlight the need to consider multiple domains simultaneously in order to better capture the complexity of girls’ career decisions. For boys, parents’ language ability stereotypes were directly related to mathematical career aspirations. These results suggest that stereotypes that language arts is not for boys push them instead toward mathematics. Our study also highlighted the unique role of parental beliefs in traditional gender roles for boys’ motivation and career aspirations. Specifically, parents’ gender role stereotypes directly related to less interest in language arts only among boys. This highlights that research into gender gaps in female-dominated fields should consider stereotypes related to appropriate behavior and social roles for boys. (shrink)

A clearly written and uncomplicated text, suitable for use with elementary and high school students as well as in college classes. It presents, in thorough detail, the techniques for making deductions, testing for validity, etc., in the logic of sentences and of universal quantification. The exposition rests upon the basic notion of inference according to rules; some fourteen rules of inference are presented and explained. Truth values and truth tables are discussed as means for determining important properties of inferences, (...) e.g., testing for validity of the inference, consistency of the premisses, etc. These techniques are then applied in formulating a simple mathematical system, a set of axioms for addition. The system is then used to illustrate the deduction of theorems with universal quantification.—K. P. F. (shrink)

This study examined individual differences in mathematics learning by combining antecedent (A), opportunity (O), and propensity (P) indicators within the Opportunity-Propensity model. Although there is already some evidence for this model based on secondary datasets, there currently is no primary data available that simultaneously takes into account A,O and P factors in children with and without Mathematical Learning Disabilities (MLD). Therefore the mathematical abilities of 114 school-aged children (grade 3 till 6) with and without MLD were analyzed and combined (...) with information retrieved from standardized tests and questionnaires. Results indicated significant differences in personality, motivation, temperament, subjective well-being, self-esteem, self-perceived competence and parental aspirations when comparing children with and without MLD. In addition, A, O and P factors were found to underlie mathematical abilities and disabilities. For the A factors, parental aspirations explained about half of the variance in fact retrieval speed in children without MLD, and SES was especially involved in the prediction of procedural accuracy in general. Teachers’ experience contributed as O factor and explained about 6% of the variance in mathematical abilities. P indicators explained between 52 and 69% of the variance, with especially intelligence as overall significant predictor. Indirect effects pointed towards the interrelatedness of the predictors and the value of including A, O and P indicators in a comprehensive model. The role parental aspirations played in fact retrieval speed was partially mediated through the self-perceived competence of the children, whereas the effect of SES on procedural accuracy was partially mediated through intelligence in children of both groups and through working memory capacity in children with MLD. Moreover, in line with the componential structure of mathematics, our findings were dependent on the math task used. Different A, O and P indicators seemed to be important for fact retrieval speed compared to procedural accuracy. Also, mathematical development type (MLD or typical development) mattered since some A, O and P factors were predictive for MLD only and the other way around. Practical implications of these findings and recommendations for future research on MLD and on individual differences in mathematical abilities are provided. (shrink)

In an attempt to justify research efforts in various branches of science, scholars have tried to capture the essence of the relevant subject-matters in a definition, or at least have declared these subject-matters to exist. Otherwise the study of topics in the branches would be of questionable value, to say the least. For example, when dealing with numbers, their ontological status somehow has to be declared.

The main thesis defended in this paper is that, interpreted in the light of reflections of Peirce and Poincaré, one can found in mathematical reasoning a non-logical symptom that may be aesthetic in Goodman’s sense. This symptom is called exemplification and serves to distinguish between only logically correct and even explanatory proofs. It broadens the scope of aesthetics to include all activities involving symbolic systems and blurs the boundaries between logic and aesthetics in mathematics. It gives a better understanding (...) of Poincaré’s thesis that to affect aesthetic value to certain properties is not simply an added value, a bonus that somehow rewards the mathematician’s mechanical labor, but on the contrary, taking the aesthetic value into account can be helpful to mathematical practice. As an example, three proofs of the irrationality of √2 are compared for their aesthetic functioning. (shrink)

Many philosophers have claimed that extensive or additive measurement is incompatible with the existence of "higher values", any amount of which is better than any amount of some other value. In this paper, it is shown that higher values can be incorporated in a non-standard model of extensive measurement, with values represented by sets of ordered pairs of real numbers, rather than by single reals. The suggested model is mathematically fairly simple, and it applies to structures including (...) negative as well as positive values. (shrink)

Journal of Mathematical Logic, Volume 24, Issue 03, December 2024. When k is a finite field, [J. Becker, J. Denef and L. Lipshitz, Further remarks on the elementary theory of formal power series rings, in Model Theory of Algebra and Arithmetic, Proceedings Karpacz, Poland, Lecture Notes in Mathematics, Vol. 834 (Springer, Berlin, 1979)] observed that the total residue map [math], which picks out the constant term of the Laurent series, is definable in the language of rings with a parameter (...) for t. Driven by this observation, we study the theory [math] of valued fields equipped with a linear form [math] which restricts to the residue map on the valuation ring. We prove that [math] does not admit a model companion. In addition, we show that [math] is undecidable whenever k is an infinite field. As a consequence, we get that [math] is undecidable, where [math] maps f to its complex residue at 0. (shrink)

In recent years, philosophical work directly concerned with the practice of mathematics has intensified, giving rise to a movement known as the philosophy of mathematical practice . In this paper we offer a survey of this movement aimed at mathematics educators. We first describe the core questions philosophers of mathematical practice investigate as well as the philosophical methods they use to tackle them. We then provide a selective overview of work in the philosophy of mathematical practice covering topics (...) including the distinction between formal and informal proofs, visualization and artefacts, mathematical explanation and understanding, value judgments, and mathematical design. We conclude with some remarks on the potential connections between the philosophy of mathematical practice and mathematics education. (shrink)

An international group of distinguished scholars brings a variety of resources to bear on the major issues in the study and teaching of mathematics, and on the problem of understanding mathematics as a cultural and social phenomenon. All are guided by the notion that our understanding of mathematical knowledge must be grounded in and reflect the realities of mathematical practice. Chapters on the philosophy of mathematics illustrate the growing influence of a pragmatic view in a field traditionally (...) dominated by platonic perspectives. In a section on mathematics, politics, and pedagogy, the emphasis is on politics and values in mathematics education. Issues addressed include gender and mathematics, applied mathematics and social concerns, and the reflective and dialogical nature of mathematical knowledge. The concluding section deals with the history and sociology of mathematics, and with mathematics and social change. Contributors include Philip J. Davis, Helga Jungwirth, Nel Noddings, Yehuda Rav, Michael D. Resnik, Ole Skovsmose, and Thomas Tymoczko. (shrink)

Purity has been recognized as an ideal of proof. In this paper, I consider whether purity continues to have value in contemporary mathematics. The topics (e.g., algebraic topology, algebraic geometry, category theory) and methods of contemporary mathematics often favor unification and generality, values that are more often associated with impurity rather than purity. I will demonstrate this by discussing several examples of methods and proofs that highlight the epistemic significance of unification and generality. First, I discuss the (...) examples of algebraic invariants and of considering a mathematical object from several different perspectives to illustrate that the methods used in contemporary mathematics favor impurity. Then I consider an example from category theory which demonstrates how unification and generality are related to impurity and that impure solutions can be explanatory. In light of this discussion, we see that purity only has marginal value within contemporary mathematics which instead prioritizes the epistemic values associated with impurity. (shrink)