‘Mathesis Universalis’ in the Seventeenth Century

Preface. List of Abbreviations and a Note on the Quotation and Translation. Introduction. Rene Descartes and Modern European Mathematics. 1: The Formation of Descartes's Mathematical Thought. 1. Descartes and Jesuit Mathematical Education. 1. Descartes and the Jesuit College of La Fleche. 2. The Curriculum at La Fleche. 3. Mathematical Studies in the Ratio Studiorum 4. Motives for the Teaching of Mathematics in the Jesuit Colleges. 2: The Mathematical Thought of Christoph Clavius. 1. Descartes and Clavius. 2. The Philosophy of Mathematics of Clavius. 3. Pappus in the Works of Clavius. 4. Diophantus in the Works of Clavius. 5. Descartes's Mathematical Background before the Encounter with Beeckman. 3: The First Attempt at Reforming Mathematics. 1. 'An Entirely New Science': The Idea for the Unification of Arithmetic and Geometry. 2. The Mathematics in the Cogitationes privatae. 3. The De Solidorum Elementis. 4. Descartes's Mathematical and Philosophical Dream of 1619. 4: The Mathematical Background of the Regulae Ad Directionem Ingenii. 1. The Old Algebra: The First Fruit of 'An Entirely New Science'. 2. The Mathematics in the Regulae ad Directionem Ingenii. 3. Mathesis Universalis. 5. The Geometrieof 1637. 1. The Pappus Problem. 2. The Composition of the Geometrie. 3. Descartes's Place in the Formative Period of the Modern Analytic Tradition. 4. Beyond Cartesian Mathematics. Interim Consideration. Descartes and the Beginnings ofMathematicism in Modern Thought. II: The Concept of 'Mathesis Universalis' in Historical Perspective. 6. 'Universal Mathematics' In Aristotle. 1. Aristotle's Metaphysics and Posterior Analytics. 2. Greek Commentators: Alexander of Aphrodisias and Asclepius of Tralles. 3. Medieval Commentators: Ibn Rushd (Averroes), Albertus Magnus, Thomas Aquinas, and the Scotist Antonius Andreae. 4. Renaissance Commentators: Agostino Nifo and Pedro da Fonseca. 5. The Status of Mathematics in the Aristotelian Scheme of Learning. 7: 'Mathesis Universalis' in the Sixteenth Century. 1. Proclus Diadochus and Francesco Barozzi. 2. Adriaan van Roomen. 8: 'Mathesis Universalis' in the Seventeenth Century. 1. Reviewing Descartes's Concept of 'Mathesis Universalis' from His Philosophy of Mathematics. 2. The Leibnizian Synthesis. Conclusion: Descartes and the Modern Scheme of Learning. Bibliography. Indices. Name Index. Subject Index. Treatise Index.

Introduction. Beliefs about the nature of mathematics influence a teacherʼs mindset and, as a result, the way he or she teaches in the classroom. Several studies have been conducted in this area, but they have rarely focused on pre-service teachers. The implication is that the instruments used to measure these beliefs must be modified. This study aims to refine and validate a scale to measure pre-service teachersʼ beliefs about the nature of mathematics and to determine the demographic analysis results that influence these beliefs. Materials and Methods. A scale development study was adopted to achieve the objectives of this study. The participants were 410 pre-service teachers from undergraduate programs at one University with A (excellent) accreditation in the capital city of Indonesia. We used factor analysis to obtain a valid and reliable instrument. We also used multiple regression analysis to look at the relationships between pre-service teachersʼ gender, academic major, academic level, and mathematical beliefs. Results. This study established a valid and reliable scale that includes three factors that underlie beliefs about the nature of mathematics. One factor is related to the philosophy of traditional mathematics, namely objective, and the other two factors are related to the philosophy of constructivism mathematics, namely relevant and dynamic. Additionally, we discover that the impact of the academic major variable is more significant than the influence of the other variables (gender and academic level). Discussion and Conclusion. Beliefs about the nature of mathematics are central to the professional development of mathematics teachers because these beliefs have an implicit impact or are related to the beliefs, views, conceptions, or attitudes of teachers about teaching and learning mathematics and, in turn, lead to choices and practices carried out in class. Therefore, the beliefs scale developed in our research will allow researchers and/or interested parties to know the extent to which teachersʼ subjective knowledge of mathematics is used to improve these beliefs and lead to more meaningful mathematics practices.

One of the most striking features of mathematics is the fact that we are much more certain about the mathematical knowledge we have than about what mathematical knowledge is knowledge of. Are numbers, sets, functions and groups physical entities of some kind? Are they objectively existing objects in some non-physical, mathematical realm? Are they ideas that are present only in the mind? Or do mathematical truths not involve referents of any kind? It is these kinds of questions that have encouraged philosophers and mathematicians alike to focus their attention on issues in the philosophy of mathematics. Over the centuries, a number of reasonably well-defined positions about the nature of mathematics have been developed and it is these positions (both historical and current) that are surveyed in the current volume. Traditional theories (Platonism, Aristotelianism, and Kantianism), as well as dominant modern theories (logicism, formalism, constructivism, fictionalism, etc.), are all analyzed and evaluated. Leading-edge research in related fields (set theory, computability theory, probability theory, and paraconsistency) is also discussed. The result is a handbook that not only provides a comprehensive overview of recent developments but that also serves as an indispensable resource for anyone wanting to learn about current developments in the philosophy of mathematics. Features of this title include: comprehensive coverage of all main theories in the philosophy of mathematics; clearly written expositions of fundamental ideas and concepts; and, definitive discussions by leading researchers in the field. Summaries of leading-edge research in related fields (set theory, computability theory, probability theory, and paraconsistency) are also included.

The main purpose of this essay is to show that Kant’s account of the nature of mathematical thinking is still highly relevant to the philosophy of mathematics. For Kant has raised important questions which, whether or not one agrees with his answers, must be answered if the nature of pure mathematics and of its application to empirical phenomena is to be understood. The essay begins with a brief summary of Kant’s position, with special emphasis on those of his theses which are still regarded as central by contemporary mathematicians and philosophers (§1). There follows a discussion of some postKantian developments and reactions to Kant’s philosophy of mathematics (§2). The essay ends with an outline of an alternative view of the structure and function of pure and applied mathematics (§3).

Philosophy of mathematics does not coincide as such with the research on the foundations of mathematics. This confluence, however, occurred at the beginning of the twentieth century, in the framework of the efforts spent for overcoming the “crisis” produced by the discovery of the antinomies. Hilbert’s formalism became soon the dominant view in this connection, that had also its philosophical counterpart in the conception of mathematics as a complex of pure formal systems devoid of specific meanings and referents. Agazzi has constantly opposed formalism, relying especially on philosophical reflections about Godel’s theorems, from which he derived the recognition of meanings and contents of many mathematical theories. This has pushed him to revisit the work of Peano and his school (and to stimulate his pupils to investigate their contributions in depth). It turns out that Peano was a pioneer and a champion of that request of logical rigor that animated much of the mathematical community of his time, so that his defense and practice of axiomatizations and his skillful use of mathematical logic as a tool for this critical analysis remained paradigmatic for the foundational investigations. However he never accepted a formalistic conception of mathematics, and this is why he and his school (after having completed their program) remained outside the main stream formalistic outlook of Hilbert’s followers that was dominant in the first half of the twentieth century.

The paper is devoted to the presentation of Chwistek’s philosophical ideas concerning logic and mathematics. The main feature of his philosophy was nominalism, which found full expression in his philosophy of mathematics. He claimed that the object of the deductive sciences, hence in particular of mathematics, is the expression being constructed in them according to accepted rules of construction. He treated geometry, arithmetic, mathematical analysis and other mathematical theories as experimental disciplines, and obtained in this way a nominalistic interpretation of them. The fate of Chwistek’s philosophical conceptions was similar to the fate of his logical conceptions. The system of rational meta-mathematics was not developed by him in detail. He worked on his own ideas without any collaboration with other logicians, mathematicians or philosophers. His investigations were not in the mainstream of the development of logic and philosophy of mathematics.

It is no pure and simple truism to say that the philosophy of mathematics has always existed in Western culture, at least since philosophy itself came into being. In fact, not only were most of the early Greek philos ophers mathematicians as well, but it is also well known that speculation within the Greek systems of thought was very often based on problems of a mathematical nature. Soon mathematics, besides being a source of philosophical problems and of their solutions, became also a direct object of philosophical enquiry, to such an extent that the early forms of philosophy of science were in fact forms of the philosophy of mathematics. This sort of privileged link between mathematics and philosophy has never been interrupted in Western thought, and even today there are philosophies that are in various ways influenced by mathematical thought (from the philosophy of Russell to that of Husserl, from Wittgenstein to Carnap and logical empiricism), while a large amount of research is being directed to the study of the 'philosophy of mathematics' proper. Underneath this undeniable continuity, however, considerable differ ences emerge between the ancient and modern conceptions of the philos ophy of mathematics, and some problems, once considered of primary importance, seem now to have almost completely lost their value. To mention only a few of them, let us recall certain classical questions of the old philosophy of mathematics: What are numbers? What kind of relationship exists between mathematics and the structure of the world? How can the property of universality and necessity of mathematical truths be explained? In what sense is mathematical infinity to be con ceived? Is mathematics man's invention or man's discovery? By attributing such questions to the past, we do not mean to confine them to ages far distant from our own. Even Dedekind, for example, entitled one of his most celebrated essays: "What are numbers and what must they be?" ; and Frege devoted the whole of his profound logico-math ematical research to the effort of pointing out the essence of the natural number to mention only these two great authors. However, the theories

This study aimed to investigate the espoused beliefs held by secondary-level mathematics teachers about mathematics and its teaching and learning processes. The research employed a quantitative survey design, focusing on four districts (Chitwan, Makawanpur, Gorkha, and Parsa) as representatives of Nepal. The selection of 168 mathematics teachers from these districts utilized proportional allocation of stratified random sampling. A questionnaire, employing a Likert five-point scale, was designed to collect data on teachers' espoused beliefs. The results revealed a prevalent alignment of secondary-level mathematics teachers' espoused beliefs with the fallibilist philosophy of mathematics. Additionally, the findings indicated a strong inclination towards the use of the constructivist method in teaching mathematics. This research contributes valuable insights into the prevailing beliefs among mathematics teachers at the secondary level, providing a basis for understanding their perspectives on the nature of mathematics and preferred teaching methodologies.

What is philosophy of mathematics and what is it about? The most popular answer, I suppose, to this question would be that philosophers should provide a justification for our presently most cherished mathematical theories and for the most important tool to develop such theories, namely logico-mathematical proof. In fact, it does cover a large part of the activity of philosophers that think about mathematics. Discussions about the merits and faults of classical logic versus one or other ‘deviant’ logics as the logical basis for mathematical theories, ranging from intuitionist over modal logic to paraconsistent logic, typically belong to this area, as do debates about the natural-number concept, its ‘nature’, its properties, especially its uniqueness, and so on. No doubt sociologists of knowledge could explain why (the community of) philosophers of mathematics came to select these particular problems and deal with them the way they do. What it does imply,...

The core idea of mathematical structuralism is that mathematical theories, always or at least in many central cases, are meant to characterize abstract structures (as opposed to more concrete, individual objects). As such, structuralism is a general position about the subject matter of mathematics, namely abstract structures; but it also includes, or is intimately connected with, views about its methodology, since studying such structures involves distinctive tools and procedures. The goal of the present collection of essays is to discuss mathematical structuralism with respect to both aspects. This is done by examining contributions by a number of mathematicians and philosophers of mathematics from the second half of the 19th and the early 20th centuries.

It was Frege who first forced both philosophers and mathematicians to acknowledge the lack of any satisfactory philosophical account of the nature and epistemological basis of mathematics. He himself constructed a complete system of philosophy of mathematics; and, in the early part of the present century, others, most notably Hilbert and Brouwer, constructed alternative systems from quite different philosophical standpoints. Of the various attempts made in that period to create over-all philosophies of mathematics providing, simultaneously, solutions to all the fundamental philosophical problems concerning mathematics, only the intuitionist system originated by Brouwer survives today as a viable theory to which, as a whole, anyone now could declare himself an adherent. The two main rival systems, those of Frege and of Hilbert, contributed much of lasting value to the foundations of mathematics; but neither remains as an integral structure of doctrines to which allegiance could still be given.

In essence, the position of science is to facilitate human life. Likewise with mathematics which is classified as a science. The birth of mathematics is none other than to provide convenience and overcome various life problems. Philosophy is a science that seeks the truth value of the essence of a problem. Philosophy and mathematics have no doubt that from the past until now these two fields of knowledge are very closely related. Philosophy of mathematics is a branch of philosophy that examines the philosophical assumptions, foundations and effects of mathematics. The aim of the philosophy of mathematics is to provide a record of the nature and methodology of mathematics and to understand the place of mathematics in human life. The position and relevance of philosophy in learning mathematics can be seen in epistemology, mathematical ontology, mathematical methodology, and mathematical logic.

A WIDE variety of symbolic devices are employed in sociology for the representation of facts and ideas. These range all the way from descriptive devices such as maps, pictures, and charts to more conceptual devices like words, statistics, and mathematics. Of all these, one of the less exploited and yet most promising types of symbolism is that of mathematics. Particularly in the field of social theory, where emphasis is upon the relationships among data rather than upon the data themselves, does mathematics offer us a useful and exact language. The truth of this becomes evident upon consideration of the nature of theory itself. Essentially any theory is a system of abstract statements. Each one of these statements describes a general mode of relationship between the terms which comprise the statement. Each of these terms, in turn, refers to a general class of data, i.e., to a universal as distinct from a particular. It follows that all of the statements comprising the theory are themselves generalizations. That is to say, they specify certain common attributes, certain regularities in the phenomenon described. Theory is not primarily concerned with unique relationships; rather it is concerned with invariant relationships. Consider now the nature of mathematics. It too is characterized by abstractness. Its statements, like those of theory, specify general modes of relationship between the component terms of the system. Again, these terms are by their very nature universals rather than particulars. But here a new feature appears, one which endows mathematics with certain properties that are not characteristic of theory in general. The terms which form the elements of a mathematical system are variables. As such these terms have the special property of denoting any one of a class of elements each of which may possess a different value. The statements which are composed of variables-i.e., of terms having this property-are known as propositional functions, or formulas. By itself, of course, a formula or propositional function has no empirical reference whatever. But when specific values are put in the place of the variables which comprise the formula, a proposition is expressed which can be empirically tested. A set of variables thus related to one another by a formula is the most general kind of statement that can possibly be made about a phenomenon. A system of such formulas has a flexibility and a rigor which perhaps no other theoretical system can equal. By means of it a complexity and subtlety of relationships can be indicated between facts and ideas which wholly escape a verbal symbolism. Mathematics, it may be suggested, is the most elegant language in which a theory can be expressed. In the development of social theory there have been numerous attempts to employ a mathematical symbolism, some of them being of remarkable antiquity. As long ago as the ninth century the Arab mathematician Al-Khowarizmi evolved an algebra of property inheritance by which he endeavored to show the logic of Arab property laws.' Spinoza, as is well known, used a geometrical language in his Ethics. Early in the seventeen hundreds Ceva outlined the logical requirements for a mathematical theory of value.2 At about the same time Weigel sought to formulate a universal theory of moral quantities. 3 During the last half century there have appeared numerous sociological systems which in one way or another have employed mathematical symbolism, among them being the works of Haret, Barcelo, and Pareto,4 and most recently the contributions of Menger, Dodd, Chapple, Rashevsky, and Von Neumann and Morgenstern.6 A study of these mathematical theories of society

THIS conference on "Mathematics in the Medical Sciences" was called because medical scientists have begun to ask questions that entail the development of new mathematical theory and the application of more complex mathematical reasoning than has been the case in the past. Also, the effective use of new instruments now becoming available to medicine demands an understanding of their underlying physical theory and its mathematical application. The meeting was concerned with both the nature of mathematics and the technics for its application that have been useful to biology and medicine in the past or may be applicable in the near . . .

Computer science provides an in-depth understanding of technical aspects of programming concepts, but if we want to understand how programming concepts evolve, how programmers think and talk about them and how they are used in practice, we need to consider a broader perspective that includes historical, philosophical and cognitive aspects. In this paper, we develop such broader understanding of monads, a programming concept that has an infamous formal definition, syntactic support in several programming languages and a reputation for being elegant and powerful, but also intimidating and difficult to grasp. This paper is not a monad tutorial. It will not tell you what a monad is. Instead, it helps you understand how computer scientists and programmers talk about monads and why they do so. To answer these questions, we review the history of monads in the context of programming and study the development through the perspectives of philosophy of science, philosophy of mathematics and cognitive sciences. More generally, we present a framework for understanding programming concepts that considers them at three levels: formal, metaphorical and implementation. We base such observations on established results about the scientific method and mathematical entities -- cognitive sciences suggest that the metaphors used when thinking about monads are more important than widely accepted, while philosophy of science explains how the research paradigm from which monads originate influences and restricts their use. Finally, we provide evidence for why a broader philosophical, sociological look at programming concepts should be of interest for programmers. It lets us understand programming concepts better and, fundamentally, choose more appropriate abstractions as illustrated in number of case studies that conclude the paper.