Finitism--an essay on Hilbert's programme

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Linguistics and Philosophy, 1991.Includes bibliographical references (p. 213-219).by David Watson Galloway.Ph.D

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

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

From mathematics in logic to logic in mathematics : Boole and Frege

This project proceeds from the premise that the historical and logical value of Boole's logical calculus and its connection with Frege's logic remain to be recognised. It begins by discussing Gillies' application of Kuhn's concepts to the history oflogic and proposing the use of the concept of research programme as a methodological tool in the historiography oflogic. Then it analyses'the development of mathematical logic from Boole to Frege in terms of overlapping research programmes whilst discussing especially Boole's logical calculus. Two streams of development run through the project: 1. A discussion and appraisal of Boole's research programme in the context of logical debates and the emergence of symbolical algebra in Britain in the nineteenth century, including the improvements which Venn brings to logic as algebra, and the axiomatisation of 'Boolean algebras', which is due to Huntington and Sheffer. 2. An investigation of the particularity of the Fregean research programme, including an analysis ofthe extent to which certain elements of Begriffsschrift are new; and an account of Frege's discussion of Boole which focuses on the domain common to the two formal languages and shows the logical connection between Boole's logical calculus and Frege's. As a result, it is shown that the progress made in mathematical logic stemmed from two continuous and overlapping research programmes: Boole's introduction ofmathematics in logic and Frege's introduction oflogic in mathematics. In particular, Boole is regarded as the grandfather of metamathematics, and Lowenheim's theorem ofl915 is seen as a revival of his research programme

The physical cosmology of Alfred North Whitehead

Throughout the history of philosophy, cosmological theories have always deservedly enjoyed a position of special prominence. Of all recent cosmologies, or phi - losophies of Nature, perhaps the most comprehensive and satisfactory is that offered. by Alfred North Whitehead. Whitehead, always both mathematician and philosopher, enjoyed a full career as mathematician at Cambridge and London Universities before answering an invitation from Harvard University to a chair in philosophy there. His interests invariably carried him to the forefront of the advance, and his more technical mathematical works bore the imprint of a philosopher. His philosophy carried the marks of its birth in mathematics and the physical sciences.Although his Treatise on Universal Algebra (1898) won him an enviable reputation, it was his collaboration with Bertrand Russell in the first decade of the twentieth century on Principia Nathematica which proved his pioneering genius. In the middle of this decade, Whitehead offered to the Royal Society of London a memoir entitled "On Mathematical Concepts of the Material World." This memoir, which fell into oblivion, employed the symbolic technique of Principia Nathematica in solving the fundamental problem of importance to cosmological theory. Given a set of entities and a relation between those entities, Whitehead attempted to show the whole of Euclidean geometry to be an expression of the properties of the field of that relation. Certain extraneous relations served to associate the axioms with the material world of the physicists, of which Whitehead offered seven alternative concepts.The first three volumes of Princiaá Mathematica had been published, and Whitehead had begun his work on the fourth, which was to have been concerned with the application of symbolic reasoning to the foundations of geometry and the problem of space. But by this time the scientific world had been captivated by the publication of the special and general theories of relativity by Einstein. These novelties naturally attracted Whitehead, who wrote several essays on the presuppositions of relativity. Whitehead was convinced that the principle and the method introduced by Einstein constituted a revolution in physical science, but found his explanation faulty.A series of three important "Nature" volumes introduced the philosophy of "Nature" as conceived by Whitehead, using his own interpretation of the meaning of the new relativity. A powerful method of analysis, called the Method of Extensive Abstraction and having as its purpose the definition of spatial and temporal entities so as to avoid a circularity of reasoning was born at this period. The third of the volumes was devoted entirely to the development of his own theory of relativity, to which the philosophically more satisfactory interpretation of relativity could be readily applied. From his original presuppositions Whitehead offered four alternative relativity theories, one of which coincided with Einstein's, and two of which were attempts at a unified field theory. The fourth, a theory of gravitation, used a physical element, the "impetus," instead of an infinitesimal metric element, as Einstein had done. This theory proved to be empirically less satisfactory than that of Einstein. But Professor George Temple generalized this fourth theory by using a space -time of positive uniform curvature, and results more satisfactory empirically than those of Einstein followed. The philosophical advantages of Whitehead's relativity were retained. This result seems to invite a more careful consideration of Temple's generalization of ;Whitehead's relativity than has been obtained at present.But by this time Whitehead's speculations, which took as their restricted field the area of nature in which mind was irrelevant, began to concentrate on the enlarged field of cosmological theory in its points of contact with metaphysics. The most important discovery he believed he had made was that in this enlarged area, all the more special physical and extensive properties of nature were dependent for their existence upon process.Now in his sixties, Whitehead accepted Harvard's invitation to a chair in philosophy. Within a very few years he returned to the United Kingdom to deliver the Gifford Lectures at the University of Edinburgh, in which the implications of adopting process as the central principle in the universe were systematically presented.One outstanding; feature of these lectures has been unfortunately ignored; it is a major and original suggestion of this thesis that the categoreal scheme of Process and Reality is really the axiomatic scheme of "On Mathematical Concepts of the Material World" generalized on the metaphysical level. An attempt at the application of the symbolic method to the axioms (categories of explanation and obligation) is made here. Thus the generalized problem in Process and Reality becomes, "Given a set of onto - logical existents and the operation of creativity, what axioms regarding the operation of creativity will have as their result that the more specialized discoveries of the humanities and the sciences follow from the properties of those entities forming the field of creativity?"These lectures, although they offered a comprehensive metaphysical system justifying the operation of physical field theories, suffered under the misfa' tune that they were given at just the time when the quantum mechanics revolution was precipitated in the physical sciences. From the point of view of quantum mechanics, therefore, the philosophy of organism does not supply a satisfactory cosmology within which it can operate. This is especially unfortunate in view of his possibly superior physical theory of relativity; possible points of expansion to allow for quantum mechanics are indicated, although they do violence to the base of the philosophy of organism.As the chief exemplification of the metaphysical principles, Whitehead postulated a brilliantly conceived metaphysical God who was important in physical cosmology. It is suggested that this metaphysical God is, nevertheless, inadequate to satisfy the demands of the religious conscience.Despite the originality of most of the elements introduced by Whitehead, a full understanding of his meaning and an appreciation of his novelties is possible only by referring his writings to their proper settings. Thus, the philosophy of organism is explained against the background of the process philosophies of Bergson, Alexander, and Horgan. Because of its many similarities in respect to the setting of the cosmological problem and the essentials of the solution to the Timaeus, a special chapter is devoted to the correspondence between the two. Whitehead's relativity and philosophy of Nature requires an understanding of the development of the theory of relativity, the world- models of the relativistic cosmologies, and the attempts at a unified field theory. Similarly, the memoir of 1905 is described in a more general back ground setting forth a broad picture of the state of geometry, physical science, and philosophy at the turn of the century.As a final reflection, certain presuppositions at the base of Whitehead's philosophy of organism are investigated and evaluated. The points believed by the present writer to be especially vulnerable in the philosophy of organism are exposed. An experiment in suggesting the prospectus of an alternative system which might avoid the difficulties, and incorporate the advantages of, the philosophy of organism, is made with the warning that it is no more than a suggestion.Throughout the thesis, certain dominant strains of "Ihitehead's thinking can be detected: the importance in his mind of the axiomatic -deductive method in the sciences; the realization that prevalent habits of thinking need to be altered by new discoveries, but are resisted; the conviction that the sciences must be ontologically centered; the faith in field theories; and the conviction that cosmology must be the search for the forms in the facts; to designate the more outstanding convictions

Second-order logic is logic

"Second-order logic" is the name given to a formal system. Some claim that the formal system is a logical system. Others claim that it is a mathematical system. In the thesis, I examine these claims in the light of some philosophical criteria which first motivated Frege in his logicist project. The criteria are that a logic should be universal, it should reflect our intuitive notion of logical validity, and it should be analytic. The analysis is interesting in two respects. One is conceptual: it gives us a purchase on where and how to draw a distinction between logic and other sciences. The other interest is historical: showing that second-order logic is a logical system according to the philosophical criteria mentioned above goes some way towards vindicating Frege's logicist project in a contemporary context

Mathematics and general education

Democracy and General Education My purpose in Part I is to develop a model of general mathematical education: that is, to identify aims appropriate to a course of mathematical education which forms part of a programme of general education. To do so presumes, of course, that it is possible to justify both the inclusion of mathematics-related aims and content in the curriculum, and their organisation around a unit entitled 'mathematics'. I will offer arguments for both these presuppositions, as well as for my model of general mathematical education