Philosophy of mathematics education

PHILOSOPHY OF MATHEMATICS EDUCATION\ud This thesis supports the view that mathematics teachers should be aware of differing views of the nature of mathematics and of a range of teaching perspectives. The first part of the thesis discusses differing ways in which the subject 'mathematics' can be identified, by relying on existing philosophy of mathematics. The thesis describes three traditionally recognised philosophies of mathematics: logicism, formalism and intuitionism. A fourth philosophy is constructed, the hypothetical, bringing together the ideas of Peirce and of Lakatos, in particular. The second part of the thesis introduces differing ways of teaching mathematics, and identifies the logical and sometimes contingent connections that exist between the philosophies of mathematics discussed in part 1, and the philosophies of mathematics teaching that arise in part 2. Four teaching perspectives are outlined: the teaching of mathematics as aestheticallyorientated, the teaching of mathematics as a game, the teaching of mathematics as a member of the natural sciences, and the teaching of mathematics as technology-orientated. It is argued that a possible fifth perspective, the teaching of mathematics as a language, is not a distinctive approach. A further approach, the Inter-disciplinary perspective, is recognised as a valid alternative within previously identified philosophical constraints. Thus parts 1 and 2 clarify the range of interpretations found in both the philosophy of mathematics and of mathematics teaching and show that they present realistic choices for the mathematics teacher. The foundations are thereby laid for the arguments generated in part 3, that any mathematics teacher ought to appreciate the full range of teaching 4 perspectives which may be chosen and how these link to views of the nature of mathematics. This would hopefully reverse 'the trend at the moment... towards excessively narrow interpretation of the subject' as reported by Her Majesty's Inspectorate (Aspects of Secondary Education in England, 7.6.20, H. M. S. O., 1979). While the thesis does not contain infallible prescriptions it is concluded that the technology-orientated perspective supported by the hypothetical philosophy of mathematics facilitates the aims of those educators who show concern for the recognition of mathematics in the curriculum, both for its intrinsic and extrinsic value. But the main thrust of the thesis is that the training of future mathematics educators must include opportunities for gaining awareness of the diversity of teaching perspectives and the influence on them of philosophies of mathematics

Representation and Reality by Language: How to make a home quantum computer?

A set theory model of reality, representation and language based on the relation of completeness and incompleteness is explored. The problem of completeness of mathematics is linked to its counterpart in quantum mechanics. That model includes two Peano arithmetics or Turing machines independent of each other. The complex Hilbert space underlying quantum mechanics as the base of its mathematical formalism is interpreted as a generalization of Peano arithmetic: It is a doubled infinite set of doubled Peano arithmetics having a remarkable symmetry to the axiom of choice. The quantity of information is interpreted as the number of elementary choices (bits). Quantum information is seen as the generalization of information to infinite sets or series. The equivalence of that model to a quantum computer is demonstrated. The condition for the Turing machines to be independent of each other is reduced to the state of Nash equilibrium between them. Two relative models of language as game in the sense of game theory and as ontology of metaphors (all mappings, which are not one-to-one, i.e. not representations of reality in a formal sense) are deduced

Hilbert's Metamathematical Problems and Their Solutions

This dissertation examines several of the problems that Hilbert discovered in the foundations of mathematics, from a metalogical perspective. The problems manifest themselves in four different aspects of Hilbert’s views: (i) Hilbert’s axiomatic approach to the foundations of mathematics; (ii) His response to criticisms of set theory; (iii) His response to intuitionist criticisms of classical mathematics; (iv) Hilbert’s contribution to the specification of the role of logical inference in mathematical reasoning. This dissertation argues that Hilbert’s axiomatic approach was guided primarily by model theoretical concerns. Accordingly, the ultimate aim of his consistency program was to prove the model-theoretical consistency of mathematical theories. It turns out that for the purpose of carrying out such consistency proofs, a suitable modification of the ordinary first-order logic is needed. To effect this modification, independence-friendly logic is needed as the appropriate conceptual framework. It is then shown how the model theoretical consistency of arithmetic can be proved by using IF logic as its basic logic. Hilbert’s other problems, manifesting themselves as aspects (ii), (iii), and (iv)—most notably the problem of the status of the axiom of choice, the problem of the role of the law of excluded middle, and the problem of giving an elementary account of quantification—can likewise be approached by using the resources of IF logic. It is shown that by means of IF logic one can carry out Hilbertian solutions to all these problems. The two major results concerning aspects (ii), (iii) and (iv) are the following: (a) The axiom of choice is a logical principle; (b) The law of excluded middle divides metamathematical methods into elementary and non-elementary ones. It is argued that these results show that IF logic helps to vindicate Hilbert’s nominalist philosophy of mathematics. On the basis of an elementary approach to logic, which enriches the expressive resources of ordinary first-order logic, this dissertation shows how the different problems that Hilbert discovered in the foundations of mathematics can be solved

Implications of Foundational Crisis in Mathematics: A Case Study in Interdisciplinary Legal Research

As a result of a sequence of so-called foundational crises, mathematicians have come to realize that foundational inquiries are difficult and perhaps never ending. Accounts of the last of these crises have appeared with increasing frequency in the legal literature, and one piece of this Article examines these invocations with a critical eye. The other piece introduces a framework for thinking about law as a discipline. On the one hand, the disciplinary framework helps explain how esoteric mathematical topics made their way into the legal literature. On the other hand, the mathematics can be used to examine some aspects of interdisciplinary legal research

Hilbert izlencesinin izinde adcılık adına yeni bulgular

Hilbert izlencesinin kanıt kuramsal amacı tarihsel gelişimi içinde özetlendikten sonra arka plandaki model-kuramsal motivasyonu belirtilmektedir. Hilbert'in nihai hedefinin matematiğin temellerine ilişkin tüm epistemolojik ve ontolojik varsayımlardan arındırılmış bir matematik kuramı geliştirmek olduğu savunulmaktadır. Yakın geçmişte mantıktaki bazı gelişmelerin Hilbert izlencesinin yalnızca adcı varsayımlar temelinde sürdürülebileceğine ilişkin yeni bir bakış açısı sağladığı öne sürülmektedir

A pragmatic theory of truth and ontology

At the heart of my pragmatic theory of truth and ontology is a view of the relation between language and reality which I term internal justification: a way of explaining how sentences may have truth-values which we cannot discover without invoking the need for the mystery of a correspondence relation. The epistemology upon which the theory depend~ is fallibilist and holistic (chapter 2); places heavy reliance on modal idioms (chapter 4); and leads to the conclusion that current versions of realism and anti-realism are deficient (chapter 5). Just as my theory avoids the need for an epistemic 'given', it avoids the need for a metaphysical 'given' or 'joints'. I offer a view of the nature of philosophy and what it can properly achieve with respect to ontological questions (chapter 3); since those views lead me to believe that philosophical discussion about what exists should be restricted to 'entities' discussed in non-philosophical contexts, my views on how we should understand claims made about the existence of middle-sized physical objects (chapters 2 and 6), theoretical entities in science (chapter 6), and abstract entities in mathematics (chapter 7), give the thesis a schematic completeness. My theory leads me to a conception of inquiry which defends the cognitive status of moral statements whilst being critical of Kantian and utilitarian approaches to morality (chapter 8). Chapter 1 explores the views of my closest philosophical allies: William James and Nelson Goodman

The Physicist - Philosophers: The Legacy of James Clerk Maxwell and Herrmann von Helmholtz

One of the most effective, and most mysterious, tools of modern theoretical physics is a mathematical method including what is here called “field theory.” The success of this procedure in unraveling the “zoology” of fundamental particles and their behavior is a marvel. The philosophical context of this marvel is the source of endless academic controversy. The core of the method is a blend of mathematics and description created by “physicist-philosophers,” from Maxwell and Helmholtz to Einstein and Schrödinger. This book tries to unravel the mystery, or at least chronicle it.https://digitalcommons.bard.edu/facbooks/1000/thumbnail.jp

Russell, Quine and Wittgenstein in pursuit of truth: A comparative study.

Understanding the intellectual competition facing a philosopher gives a clearer sense of the depth of his work. This thesis is concerned with the reactions of Wittgenstein and Quine to Russell's foundationalism in epistemology. In particular it is concerned with the foundations of mathematics. Wittgenstein's conception of language is the deep source of his philosophy of mathematics. That is why the study of the Wittgensteinian account of mathematical truth goes beyond the limits of reflection on mathematics and extends to the philosophy of language and logic. The claim is that contrary to the framework of thought of both Russell and Quine, there is no language / reality dichotomy. Russell's search for indubitable foundations of knowledge and in particular his attempt to establish the foundations of mathematics in logic is misguided. The very supposition that mathematics needs foundations is an illusion. It is an attempt to transcend the bounds of sense. The epistemological riddles faced by Russell and Quine disappear in the later Wittgensteinian understanding of the matter. They collapse into logical insights. Following modern debates in epistemology, Russell is looking for a proof of the 'external world'. This traditional line of thought continues in Quine's notion of 'The myth of physical objects'. Though Quine's naturalized epistemology is a reaction against foundationalism, the dichotomy in question, still remains. This is finally disposed of, by Wittgenstein's later conception of language. To complete the layout of the discussions; it is demonstrated that the idea of the alleged dichotomy lies behind the arguments of Einstein, Hilbert and all of the logical positivists. Instead of pursuing the source of necessity of a pr/or/propositions in the world or in the mind, we may explore the function of such propositions. Once their role has been properly grasped, the very disturbing epistemological riddles disappear. The absolute certainty of the propositions of logic and mathematics resides in the role that they play in our practice of inference and calculation. According to Russell's account in Principia Mathematica it is a fundamental law of logic that the proposition 'Q' follows from the proposition 'P & (P -- Q)'. But what does this 'following' consist in? There is nothing in reality that provides a foundation for this inference. Logical and mathematical propositions define the techniques of inference and calculation. There is no foundation for our techniques that could justify them from the point of view of a non-participant in the practice. That is why it makes no sense to doubt logical or mathematical propositions. Russell's total loss of the 'objective world' is the inevitable outcome of his understanding of the problem. His scepticism concerning the ordinary empirical judgements is against the mastery of a technique in the practice of describing the world. Without that technique, we would be unable to think or to use language. Our certainty concerning these judgements is a practical certainty that shows how the expressions of our language are used. The function of these judgements makes the question of establishing their ground out of place

Media Ontology and Transcendental Instrumentality

This article takes inspiration from Kittler’s claim that philosophy has neglected the means used for its production. Kittler’s argument for an ontology of media invites us to reflect upon the cybernetic mechanization of logic, which has led practical or instrumental knowledge to challenge the classical division between theory and practice, ideas and demonstrations. This article suggests that procedures, tasks, and functions are part of an instrumental thinking. By drawing on information theory and the mathematical logic of constructivism, the article addresses indeterminacy within automated logic and proposes a re-habilitation of instrumentality whereby the connection between means and ends is articulated away from classical idealism and analytic realism. By following John Dewey’s argument for instrumental reasoning, the article suggests that post-Kantian critique of techne shall be revisited in order to account for a machine philosophy, which has originated from within the practical thinking of machines