The "mathematization" of knowledge is a historically inevitable process governed by two circumstances. In the first place there is the need for the further extension of knowledge in all areas of human activity, whether it be the study of natural phenomena or the theory of taking decisions in the economic or social sphere. Marx pointed out long ago that a science reaches its highest levels only when it succeeds in making use of mathematics. The second circumstance rendering (...) the process of deepening and broadening the mathematization of knowledge not only necessary but possible lies in the constant development both of mathematics itself and of the technical means broadening the sphere of its application. (shrink)

Rather than attempting to characterize a relation of confirmation between evidence and theory, epistemology might better consider which methods of forming conjectures from evidence, or of altering beliefs in the light of evidence, are most reliable for getting to the truth. A logical framework for such a study was constructed in the early 1960s by E. Mark Gold and Hilary Putnam. This essay describes some of the results that have been obtained in that framework and their significance for philosophy of (...) science, artificial intelligence, and for normative epistemology when truth is relative. (shrink)

This book seeks to offer original answers to all the major open questions in epistemology—as indicated by the book’s title. These questions and answers arise organically in the course of a validation of the entire corpus of human knowledge. The book explains how we know what we know, and how well we know it. The author presents a positive theory, motivated and directed at every step not by a need to reply to skeptics or subjectivists, but by the need (...) of a rational individual to know the world. -/- The author draws heavily from Ayn Rand’s theory of concepts as presented in her book, Introduction to Objectivist Epistemology, but departs from her epistemology—especially as explicated by leading exponents of her philosophy—in important ways. Areas of departure include the perception of entities and the role of probability theory in epistemology. -/- A main idea in the book is to begin the validation of the law of causality by identifying that existence causes our consciousness of it. Another key idea is this: The very fact that we perceive entities is the most basic and most extensive evidence that causal forces and causal interactions are regular. -/- Applying Ayn Rand’s principle that characteristics are ranges of measurement, the book offers a philosophical validation of “nonparametric predictive inference,” in turn providing an objective basis for prior probabilities in Bayesian analysis. -/- This book is primarily for a specialized readership—familiar with epistemological ideas and with Ayn Rand’s theory of concepts in particular. The last third of the book uses probability theory and mathematics somewhat beyond basic calculus, but less mathematical explanations are also included to make the general ideas accessible to lay readers. (shrink)

This article examines the role of picturability in mathematical demonstration in the seventeenth and eighteenth centuries and draws attention to the general question of the role that picturability places in cognitive grasp. It suggests that mathematical demonstration is particularly applicable in cognitive grasp it allows the problematic to be identified with some precision. It also discusses infinitesimal analysis and the question of direct proof and evaluates the role of picturability in the analysis of human cognitive capacities.

For millennia, knowledge has eluded a precise definition. The industrialization of knowledge (IoK) and the associated proliferation of the so-called knowledge communities in the last few decades caused this state of affairs to deteriorate, namely by creating a trio composed of data, knowledge, and information (DIK) that is not unlike the aporia of the trinity in philosophy. This calls for a general theory of knowledge (ToK) that can work as a foundation for a science of (...) knowledge (SoK) and additionally distinguishes knowledge from both data and information. In this paper, I attempt to sketch this generality via the establishing of both knowledge structures and knowledge systems that can then be adopted/adapted by the diverse communities for the respective knowledge technologies and practices. This is achieved by means of a formal–indeed, mathematical–approach to epistemological matters a.k.a. formal epistemology. The corresponding application focus is on knowledge systems implementable as computer programs. (shrink)

The Arabic-Islamic tradition is founded on the following new epistemic attitude that reinvents knowledge: to learn from the contributions of previous civilisations through the systematic survey of all extant scientific works; to contribute to the further development of knowledge by linking it, through usefulness, to practice and the practical need of society; to facilitate its learning for younger generations and its transmission to future civilizations since it is conceived not as a finished product but as an ongoing process. (...) The worldwide development of the reinvented knowledge has led to its universalisation and the rapid expansion of mathematics has particularly and led to the complete de-hellinisation of the Greek conception of science and philosophy. (shrink)

“The inquiry into the foundations of knowledge is a systematic inquiry into the problem of truth. This problem constitutes one of the three main concerns of philosophical analysis, the others being the problem of beauty and the problem of goodness.” Thus Evangelos P. Papanoutsos, Greece’s leading contemporary philosopher, introduces this third book of his “Trilogy of the Mind.” The first two volumes covered aesthetics and ethics; this one is a major work in epistemology. Combining rigorous analysis with thorough-going scholarship, (...) displaying an intimate acquaintance with the physical and humanistic sciences, and drawing on a deep understanding of philosophical method and the history of philosophy, Professor Papanoutsos is held in high esteem by his European colleagues. This translation of his masterpiece will enhance his reputation and influence among readers of English. The themes of The Foundation of Knowledge range over the topics that have been continually challenging to the modern era of philosophers: being and consciousness, experience and reason, common sense and science, and the domains of knowledge, including the nature of philosophical knowledge. Special attention is paid to the analysis of theoretical consciousness, the problems of categorical thinking, the theory of judgment, mathematics and logic, and the limits of historical understanding. (shrink)

Few philosophers that have been studied as much as Ibn Sīnā have been as much misunderstood. His extraordinary ability to reflect upon and write in a variety of styles about seemingly every topic in every domain has steered his thought from philosophy and theology to mysticism and esoterism. Instead of helping us to learn and understand better Ibn Sīnā than he has previously been understood, the recent surge of Avicennan studies only adds more confusion to the already complex social context (...) which he was living in. (shrink)

This book explains exactly what human knowledge is. The key concepts in this book are structures and algorithms, i.e., what the readers “see” and how they make use of what they see. Thus in comparison with some other books on the philosophy of science, which employ a syntactic approach, the author’s approach is model theoretic or structural. Properly understood, it extends the current art and science of mathematical modeling to all fields of knowledge. The link between structure and (...) algorithms is mathematics. But viewing “mathematics” as such a link is not exactly what readers most likely learned in school; thus, the task of this book is to explain what “mathematics” should actually mean. Chapter 1, an introductory essay, presents a general analysis of structures, algorithms and how they are to be linked. Several examples from the natural and social sciences, and from the history of knowledge, are provided in Chapters 2–6. In turn, Chapters 7 and 8 extend the analysis to include language and the mind. Structures are what the readers see. And, as abstract cultural objects, they can almost always be seen in many different ways. But certain structures, such as natural numbers and the basic theory of grammar, seem to have an absolute character. Any theory of knowledge grounded in human culture must explain how this is possible. The author’s analysis of this cultural invariance, combining insights from evolutionary theory and neuroscience, is presented in the book’s closing chapter. The book will be of interest to researchers, students and those outside academia who seek a deeper understanding of knowledge in our present-day society. (shrink)

The Editors’ vision for this volume is that it should be a selection of essays, contributed by the academics who have worked, studied, collaborated and disagreed with Göran Sundholm; engaging in debated issues and exploring untouched areas maybe only suggested or hinted at in Sundholm’s own work. "Acts of Knowledge" characterizes the papers contained in this volume as bringing something scientifically valuable in their respective fields: all the papers present cutting-edge research in their own style, contributing to very lively (...) debates occurring in the literature in logic, philosophical logic and history of logic. But it also hints at Göran’s constructivist background, which has been an influence or a challenge for many of the contributors. "History, Philosophy and Logic" refers directly to Göran's broad interests into the various aspects of the Philosophy of Logic, Mathematics, and Language, their origins and development, especially with the focus on the Modern History of Logic and the philosophical implications thereof. The readers will find scattered all along this volume pieces of -- and reflections on -- all these themes. (shrink)

Renouvier's place in nineteenth-century French thought -- Renouvier's critique of Comtean positivism -- Renouvier and mathematics -- Renouvier on evolution -- Kant, free will, and the social contract -- Hypothesis and convention in Renouvier's philosophy of science.

William Oliver Martin published "The Order and Integration of Knowledge" in 1957 to address the problem of the nature and the order of various kinds of knowledge; in particular, the theoretical problem of how one kind of knowledge is related to another kind. Martin characterizes kinds of knowledge as being either autonomous or synthetic. The latter are reducible to two or more of the autonomous (or irreducible) kinds of knowledge, viz., history (H), metaphysics (Meta), theology (...) (T), formal logic (FL), mathematics (Math), and generalizations of experimental science (G). Metaphysics and theology constitute the two domains of the ontological context while history and experimental science are the two domains of the phenomenological context. The relation of one kind of knowledge to another may be instrumental, constitutive, and/or regulative. For instance, historical propositions are constitutive of G, metaphysical propositions are regulative of G, and propositions in formal logic and mathematics are instrumental to G. Theological propositions are not related to G and so there is no conflict between science and theology. Martin's work sheds light on the possible areas of incompatibility between science and religion. (shrink)

On knowledge and practices: a manifesto -- The web of practices -- Agents and frameworks -- Complementarity in mathematics -- Ancient Greek mathematics: a role for diagrams -- Advanced math: the hypothetical conception -- Arithmetic certainty -- Mathematics developed: the case of the reals -- Objectivity in mathematical knowledge -- The problem of conceptual understanding.

This book collects eight essays — written over multiple decades, for a general audience — that address Fenstad’s thoughts on the topics of what there is and how.

This book draws its inspiration from Hilbert, Wittgenstein, Cavaillès and Lakatos and is designed to reconfigure contemporary philosophy of mathematics by making the growth of knowledge rather than its foundations central to the study of mathematical rationality, and by analyzing the notion of growth in historical as well as logical terms. Not a mere compendium of opinions, it is organised in dialogical forms, with each philosophical thesis answered by one or more historical case studies designed to support, complicate or (...) question it. The first part of the book examines the role of scientific theory and empirical fact in the growth of mathematical knowledge. The second examines the role of abstraction, analysis and axiomatization. The third raises the question of whether the growth of mathematical knowledge constitutes progress, and how progress may be understood. Readership: Students and scholars concerned with the history and philosophy of mathematics and the formal sciences. (shrink)

In his classic discussion of liberal education and the nature of knowledge, Professor Hirst argues for a liberal education which is “directly concerned with the development of mind and rational knowledge.”1He sets out clear conditions which any activity must satisfy if it is to be a form of knowledge and suggests that there are seven distinct forms which satisfy these conditions:“mathematics, physical sciences, human sciences, history, religion, literature and the fine arts, philosophy”2The first argument of this paper (...) is that literary criticism exhibits all the features which Hirst demands of a form of knowledge. Second, it is argued that ‘literature’ does not exhibit the characteristics of a form of knowledge. The third argument is that Hirst's acceptance of ‘literature and the fine arts’ as a ‘form of knowledge’ reflects a fundamental deficiency in his list of the distinguishing features of a ‘form of knowledge’, which is that he omits any explicit reference to the necessity for a form of knowledge to exhibit criteria for distinguishing truth from falsity within its domain. (shrink)

In this paper we apply social epistemology to mathematical proofs and their role in mathematical knowledge. The most famous modern collaborative mathematical proof effort is the Classification of Finite Simple Groups. The history and sociology of this proof have been well-documented by Alma Steingart (2012), who highlights a number of surprising and unusual features of this collaborative endeavour that set it apart from smaller-scale pieces of mathematics. These features raise a number of interesting philosophical issues, but have received very (...) little attention. In this paper, we will consider the philosophical tensions that Steingart uncovers, and use them to argue that the best account of the epistemic status of the Classification Theorem will be essentially and ineliminably social. This forms part of the broader argument that in order to understand mathematical proofs, we must appreciate their social aspects. (shrink)

Mathematicians do not claim to know a proposition unless they think they possess a proof of it. For all their confidence in the truth of a proposition with weighty non-deductive support, they maintain that, strictly speaking, the proposition remains unknown until such time as someone has proved it. This article challenges this conception of knowledge, which is quasi-universal within mathematics. We present four arguments to the effect that non-deductive evidence can yield knowledge of a mathematical proposition. We also (...) show that some of what mathematicians take to be deductive knowledge is in fact non-deductive. 1 Introduction2 Why It Might Matter3 Two Further Examples and Preliminaries4 An Exclusive Epistemic Virtue of Proof?5 Analyses of Knowledge6 The Inductive Basis of Deduction7 Physical to Mathematical Linkages8 Conclusion. (shrink)

Many books explain what is known about the universe. This book investigates what cannot be known. Rather than exploring the amazing facts that science, mathematics, and reason have revealed to us, this work studies what science, mathematics, and reason tell us cannot be revealed. In The Outer Limits of Reason, Noson Yanofsky considers what cannot be predicted, described, or known, and what will never be understood. He discusses the limitations of computers, physics, logic, and our own thought processes. Yanofsky describes (...) simple tasks that would take computers trillions of centuries to complete and other problems that computers can never solve; perfectly formed English sentences that make no sense; different levels of infinity; the bizarre world of the quantum; the relevance of relativity theory; the causes of chaos theory; math problems that cannot be solved by normal means; and statements that are true but cannot be proven. He explains the limitations of our intuitions about the world -- our ideas about space, time, and motion, and the complex relationship between the knower and the known. Moving from the concrete to the abstract, from problems of everyday language to straightforward philosophical questions to the formalities of physics and mathematics, Yanofsky demonstrates a myriad of unsolvable problems and paradoxes. Exploring the various limitations of our knowledge, he shows that many of these limitations have a similar pattern and that by investigating these patterns, we can better understand the structure and limitations of reason itself. Yanofsky even attempts to look beyond the borders of reason to see what, if anything, is out there. (shrink)

Menachem Butler. Bibliography: Mark Steiner’s main works, pp. 3–7.

In this paper, we argue that Williamson’s arguments against luminosity and the KK principle do not work, at least in a scientific context. Both of these arguments are based on the presence of a so-called “buffer zone” between situations in which one is in a position to know p and situations in which one is in a position to know ¬p. In those positions belonging to the buffer zone ¬p holds, but one is not in a position to know ¬p. (...) The presence of this buffer zone triggers two types of sorites arguments. We show that this kind of argument does not hold in a scientific context, where the buffer zone is controlled by a quantitative measurement of the experimental error. (shrink)

This book explores a theory of human knowledge through a model of rationality combined with some fundamental logical, mathematical, physical and neuroscientific considerations. Its ultimate goal is to present a philosophical system of integrated knowledge, in which the different domains of human understanding are unified by common conceptual structures, such that traditional metaphysical and epistemological questions may be addressed in light of these categories. Philosophy thus becomes a that may reproduce and even expand the conceptual chain followed by (...) nature and thought, in an effort to organize the results of the different branches of knowledge by inserting them in a broader framework. (shrink)

The nominalistic ontology of Kotarbinski, Slupecki and Tarski does not provide any direct interpretations of the sets of higher types which play important roles in type theory and in set theory. For this and other reasons I will interpret those theories as descriptions of some finite structures which are actually constructed in human imaginations and stored in their memories. Those structures will be described in this lecture. They are hinted by the idea of Skolem functions and Hilbert's -symbols, and they (...) constitute a finitistic modification of Tarski's concept of a model. They suggest also a form of the evolutionary process which leads to the development of human intelligence and language. (shrink)

In his classic discussion of liberal education and the nature of knowledge, Professor Hirst argues for a liberal education which is “directly concerned with the development of mind and rational knowledge.”1He sets out clear conditions which any activity must satisfy if it is to be a form of knowledge and suggests that there are seven distinct forms which satisfy these conditions:“mathematics, physical sciences, human sciences, history, religion, literature and the fine arts, philosophy”2The first argument of this paper (...) is that literary criticism exhibits all the features which Hirst demands of a form of knowledge. Second, it is argued that ‘literature’ does not exhibit the characteristics of a form of knowledge. The third argument is that Hirst's acceptance of ‘literature and the fine arts’ as a ‘form of knowledge’ reflects a fundamental deficiency in his list of the distinguishing features of a ‘form of knowledge’, which is that he omits any explicit reference to the necessity for a form of knowledge to exhibit criteria for distinguishing truth from falsity within its domain. (shrink)

In this paper we present a paradigmatic example of the use in knowledge management of techniques from other fields, namely mathematical analysis. We also highlight that the Jacobi method presented here takes precedence over the better known Hungarian method. Finally, we signify that the Jacobi method represents the first known or recognized case of serendipity in both knowledge management and operational research. This paper thus demonstrates the intersection between knowledge management, mathematical analysis and operational research and how (...) taking historical perspectives are important for recognising future applications of knowledge management. This results in a better understanding of knowledge management and how it can be applied in the future. It has been shown how knowledge management relates to historical mathematical principles. (shrink)

This book argues against the view that mathematical knowledge is a priori,contending that mathematics is an empirical science and develops historically,just as ...

This book describes in detail the relationship between symbolic representations of knowledge and abstract states of knowledge, exploring along the way the foundations of knowledge, knowledge bases, knowledge-based systems, and knowledge representation and reasoning. The idea of knowledge bases lies at the heart of symbolic, or "traditional," artificial intelligence. A knowledge-based system decides how to act by running formal reasoning procedures over a body of explicitly represented knowledge—a knowledge base. The (...) system is not programmed for specific tasks; rather, it is told what it needs to know and expected to infer the rest. This book is about the logic of such knowledge bases. It describes in detail the relationship between symbolic representations of knowledge and abstract states of knowledge, exploring along the way the foundations of knowledge, knowledge bases, knowledge-based systems, and knowledge representation and reasoning. Assuming some familiarity with first-order predicate logic, the book offers a new mathematical model of knowledge that is general and expressive yet more workable in practice than previous models. The book presents a style of semantic argument and formal analysis that would be cumbersome or completely impractical with other approaches. It also shows how to treat a knowledge base as an abstract data type, completely specified in an abstract way by the knowledge-level operations defined over it. (shrink)

: In his autobiographical account, the Munqidh min al-Dalāl, al-Ghazālī reflects on his conversion from skepticism to faith. Previous scholarship has interpreted this text as an anticipation of Cartesian positions regarding epistemic certainty. Although the existing similarities between al-Ghazālī and Descartes are striking, the focus of the present essay lies on the different philosophical aims pursued by the two thinkers. It is thus argued that al-Ghazālī operates with a broader notion of the Self than Descartes, because it is inclusive of (...) the body. And it is shown that the two philosophers use completely diverging paradigms. While Descartes models his notion of evidence after mathematical certainty, al-Ghazālī draws his famous 'ilm al-yaqīnī (certain knowledge) from a religious context. (shrink)

This chapter tries to answer the following question: How should we conceive of the method of mathematics, if we take a naturalist stance? The problem arises since mathematical knowledge is regarded as the paradigm of certain knowledge, because mathematics is based on the axiomatic method. Moreover, natural science is deeply mathematized, and science is crucial for any naturalist perspective. But mathematics seems to provide a counterexample both to methodological and ontological naturalism. To face this problem, some authors tried (...) to naturalize mathematics by relying on evolutionism. But several difficulties arise when we try to do this. This chapter suggests that, in order to naturalize mathematics, it is better to take the method of mathematics to be the analytic method, rather than the axiomatic method, and thus conceive of mathematical knowledge as plausible knowledge. (shrink)

In his Contra Academicos, Augustine offers one of the most detailed responses to scepticism to have come down to us from antiquity. In this paper, I examine Augustine’s defence of the existence of infallible knowledge in Contra Academicos 3. I challenge a number of established views (including those of Myles Burnyeat, Gareth Matthews, and Christopher Kirwan) concerning the nature and merit of Augustine’s defence of knowledge and propose a new understanding of Augustine’s response to scepticism (including his semantic (...) response to external world scepticism) and several important elements of Augustine’s thought concerning signification, cognition, and object-directed thought. I argue that once we properly understand Augustine’s views about these issues, his arguments in defence of knowledge are more interesting and more successful than usually thought. (shrink)

The philosophy of mathematical practice sometimes investigates the social constitution of mathematics but does not always make explicit the philosophical-normative framework that guides the discussion. This chapter investigates some recent proposals in the philosophy of mathematical practice that compare social facts and mathematical objects, discussing similarities and differences. An attempt will be made to identify, through a comparison with three different perspectives in social ontology, the kind of objectivity attributed to mathematical knowledge, the type of representational or non-representational semantics (...) adopted, and the justificatory or coordinative role entrusted to axiomatics. After a brief introduction to key issues in social ontology, Sect. 3 of the chapter offers a survey of contributions by Feferman, Ferreirós, and Cole, highlighting a difference between approaches based on rules, practices, and intentional states, respectively. Section 4.1 also discusses results by Carter, Collin, Giardino, and Pantsar, focusing on differences between the objectivity of knowledge and objectivity of objects and showing that many authors combine a realist, structuralist, and pragmatist perspective, as well as the idea that objectivity comes in degrees. Section 4.2 focuses on the kind of semantics that is adopted in socially oriented approaches. A representational semantics is preferred by authors grounding their views on mental states, whereas a non-representational semantics best fits with views based on practices. Section 5 considers how axiomatics can be understood in a social perspective. Axiomatics generally plays a justificatory role also in theories that aim to explain the social constitution of mathematical knowledge. Yet, if one shifts from approaches based on mental states to approaches anchored on rules or habits, axiomatics can be viewed as an institution based on obligations, functions, coordination problems, and agent’s actions and roles, thus playing an organizational and coordination role. (shrink)

This extensive work is both a systematization of past developments, and an extension to new areas, of the application of mathematical apparatus to the study of logical systems; it does not aim to include all such metamathematical devices, Gödel-numbering for example, but to emphasize algebraic and topological ones. The first part surveys required algebraic and topological notions; in the second part they are applied to classical logic—propositional and predicate calculi; in the final section, modal and intuitionistic, non-classical logics come under (...) scrutiny. The reader should come prepared with a working knowledge of the forementioned areas of mathematics; but the pace is leisurely, and the book tries to be self-contained. It succeeds admirably and may become the standard examination of the topic.—P. J. M. (shrink)

This paper traces the genealogy of a long-enduring controversy in Western philosophy viz, whether philosophic and mathematical methodologies are equal but separate and distinct approaches to rational inquiry, or whether one is superior to the other from the standpoint of epistemology, and, ultimately, a pedagogy which supports and promotes conceptual and critical thinking. With the Socratic teacher in mind, philosophic methodology, viewed by Plato as a dialectical process of free-ranging inquiry, compelled him to distinguish the work of philosophy from that (...) of mathematics, since philosophic methodology could not function unless freed from the constraints contained in a system built on axioms, propositions, and images. In his dialogue Meno, Socrates’ pedagogic demonstration, supported by the unique capacity of mathematics to provide epistemic closure, foreshadowed a slow-growing rift between philosophy and mathematics, one which came to a head in the 17th century rationalism of Descartes. In Descartes’ work on methodology, mathematics, by overriding the seemingly inconclusive meanderings of philosophy, became the supreme pathway to knowledge. Philosophy, thereafter, had to fight for its life, while pedagogy, questing more and more for “completeness” through formalization, ironically moved toward a fragmentation and reductionism which a mathematized Cartesian methodology supported. The curriculum divisions which resulted were presaged in Descartes’ writing on methodology; the isolation of the individual learner from the larger community of discourse was supported by Descartes’ “cogito.” Suggested here is that a pedagogic approach to the teaching of mathematics, which extends beyond the limitations of the “single solution” methodology utilized in Meno, is possible—an approach which allows pedagogy to spread the Socratic use of dialectical thinking even into that sphere of apodicticity. This suggests that an alternative to the monistic rationalism of Descartes’ mathematized methodology is possible for pedagogy in general, now bringing back into play what Descartes found expendable—viz. dialectical philosophy itself, dialogical inquiry, history, literature, and the arts. Had Socrates, who came to philosophy primarily as a teacher fascinated with intellectual paradoxes, explored, in Meno, the pedagogic possibility of a dialectical approach to mathematics, his larger work in dialectical pedagogy might have been less teacher-centered—notwithstanding his ironic “ignorance”; and given his historic significance as the most celebrated teacher in Western culture, modern pedagogy might have been less overwhelmed by the Cartesian drive toward mechanized and reductionist “single-solution,” non-communal teaching and thinking. (shrink)

The paper is divided into two parts. In the first one, I set forth a hypothesis to explain the failure of Husserl’s project presented in the Philosophie der Arithmetik based on the principle that the entire mathematical science is grounded in the concept of cardinal number. It is argued that Husserl’s analysis of the nature of the symbols used in the decadal system forces the rejection of this principle. In the second part, I take into account Husserl’s explanation of why, (...) albeit independent of natural numbers, the system is nonetheless correct. It is shown that its justification involves, on the one hand, a new conception of symbols and symbolic thinking, and on the other, the recognition of the question of “the formal” and formalization as pivotal to understand “the mathematical” overall. (shrink)

Integration Area C. Nature, sources, and limits of human knowledge; roles of perception, reason, testimony, and intuition in acquiring rational beliefs; e.g. science, mathematics, values, the arts, religion, social issues, and psychological states. G.E. Integration IC.

There are four types of information an agent can have concerning the state of the universe: information acquired via observation, via control, via prediction, or via retrodiction, i.e., memory. Each of these four types of information appear to rely on a different kind of physical device. However it turns out that there is some mathematical structure that is common to those four types of devices. Any device that possesses that structure is known as an “inference device”. Here I review some (...) of the properties of IDs, including their relation with Turing machines, and quantum mechanics. I also review the bounds on the joint information about the physical universe that can be held by any set of IDs. These bounds constrain the possible mathematical structure of any universe that contains agents with information concerning that universe in which they are embedded. (shrink)

The purpose of this cross-sectional study was to examine pre-service teachers’ perceptions of their mathematical pedagogical content knowledge and to determine effect of demographic variables (Gender, year of study) on their mathematical pedagogical content knowledge. A Likert scale questionnaire was used to collect data from 104 pre-service teachers. Descriptive statistics and Mann-Whitney U-test were used to examine pre-service teachers’ perceived knowledge of teaching strategies, mathematical language and symbols, misconceptions, curriculum, and their perceived knowledge of learners. Results (...) show that in general, majority pre-service teachers have strong perceptions of their mathematical pedagogical content knowledge. They consider lesson planning, purpose of the mathematics curriculum and assessment tools used in a mathematics classroom as important aspects of their perceived knowledge of the curriculum. It was also revealed that their perceived knowledge of misconception would enable them to know and anticipate their learners’ misconceptions. However, results also indicated that there was no statistically significant difference in pre-service teachers’ mathematical pedagogical content knowledge based on gender and year of study. Study implications are also discussed. (shrink)