Online research in philosophy

I re-examine Kuhn’s account of scientific revolutions. I argue that the sorts of events Kuhn regards as scientific revolutions are a diverse lot, differing in significant ways. But, I also argue that Kuhn does provide us with a principled way to distinguish revolutionary changes from non-revolutionary changes in science. Scientific revolutions are those changes in science that (1) involve taxonomic changes, (2) are precipitated by disappointment with existing practices, and (3) cannot be resolved by appealing to shared standards. I argue that an important and often overlooked dimension of the Kuhnian account of scientific change is the shift in focus from theories to research communities. Failing to make this shift in perspective might lead one to think that when individual scientists change theories a scientific revolution has occurred. But, according to Kuhn, it is research communities that undergo revolutionary changes, not individual scientists. I show that the change in early modern astronomy is aptly characterized as a Kuhnian revolution.

The aim of this paper is to provide a unified account of scientific and mathematical change in a thoroughly empiricist setting. After providing a formal modelling in terms of embedding, and criticising it for being too restrictive, a second modelling is advanced. It generalises the first, providing a more open-ended pattern of theory development, and is articulated in terms of da Costa and French's partial structures approach. The crucial component of scientific and mathematical change is spelled out in terms of partial embeddings. Finally, an application of this second pattern is made, with the examination of the early formulation of set theory (in particular, the works of Cantor, Zermelo and Skolem).

This book argues for the need to put into practice a profound and comprehensive intellectual revolution, affecting to a greater or lesser extent all branches of scientific and technological research, scholarship and education. This intellectual revolution differs, however, from the now familiar kind of scientific revolution described by Kuhn. It does not primarily involve a radical change in what we take to be knowledge about some aspect of the world, a change of paradigm. Rather it involves a radical change in the fundamental, overall intellectual aims and methods of inquiry. At present inquiry is devoted to the enhancement of knowledge. This needs to be transformed into a kind of rational inquiry having as its basic aim to enhance personal and social wisdom. This new kind of inquiry gives intellectual priority to the personal and social problems we encounter in our lives as we strive to realize what is desirable and of value – problems of knowledge and technology being intellectually subordinate and secondary. For this new kind of inquiry, it is what we do and what we are that ultimately matters: our knowledge is but an aspect of our life and being.

In this paper I explore possibilities of bringing post-positivist philosophies of empirical science to bear on the dynamics of mathematical development. This is done by way of a convergent accommodation of a mathematical version of Lakatos's methodology of research programmes, and a version of Kuhn's account of scientific change that is made applicable to mathematics by cleansing it of all references to the psychology of perception. The resulting view is argued in the light of two case histories of radical conceptual innovations.

In this paper it is argued, firstly, that Kuhnian revolutions in mathematics are logically possible, in the sense of not being inconsistent with the nature of mathematics; and, secondly, that Kuhnian revolutions are actually possible, in the sense that a Kuhnian paradigm for mathematics can be exhibited which would, if accepted by the mathematical community, produce a full Kuhnian revolution. These two arguments depend on first proving that a shift from a classical conception of mathematics to an intuitionist conception would be incommensurable, that is, that some classical statements, possessing meanings which cannot be preserved in the intuitionist language, would become unintelligible. The vague but intriguing thesis is then tentatively advanced that Kuhnian revolutions are even historically possible, in the sense that only what we might call 'accidental' historical factors may have prevented mathematics from undergoing just such a Kuhnian revolution in the early years of the twentieth century.

It is shown how the historiographic purport of Lakatosian methodology of mathematics is structured on the theme of analysis and synthesis. This theme is explored and extended to the revolutionary phase around 1800. On the basis of this historical investigation it is argued that major innovations, crucial to the appraisal of mathematical progress, defy reconstruction as irreducibly rational processes and should instead essentially be understood as processes of social-cognitive interaction. A model of conceptual change is developed whose essential ingredients are the variability of rational responses to new intellectual and practical challenges arising in the cultural environment of mathematics, and the shifting selective pressure of society. The resulting view of mathematical development is compared with Kuhn's theory of scientific paradigms in the light of some personal communications.

In this paper I propose a position in the ontology of mathematics which is inspired mainly by a case study in the mathematical discipline if-theory. The main theses of this position are that mathematical objects are introduced by mathematicians and that after mathematical objects have been introduced, they exist as objectively accessible abstract objects.

In this paper, I introduce and examine the notion of “mathematical engineering” and its impact on mathematical change. Mathematical engineering is an important part of contemporary mathematics and it roughly consists of the “construction” and development of various machines, probes and instruments used in numerous mathematical fields. As an example of such constructions, I briefly present the basic steps and properties of homology theory. I then try to show that this aspect of contemporary mathematics has important consequences on our conception of mathematical knowledge, in particular mathematical growth.

Social revolutions--that is critical periods of decisive, qualitative change--are a commonly acknowledged historical fact. But can the idea of revolutionary upheaval be extended to the world of ideas and theoretical debate? The publication of Kuhn's The Structure of Scientific Revolutions in 1962 led to an exciting discussion of revolutions in the natural sciences. A fascinating, but little known, off-shoot of this was a debate which began in the United States in the mid-1970's as to whether the concept of revolution could be applied to mathematics as well as science. Michael Grove declared that revolutions never occur in mathematics, while Joseph Dauben argued that there have been mathematical revolutions and gave some examples. This book is the first comprehensive examination of the question. It reprints the original papers of Grove, Dauben, and Mehrtens, together with additional chapters giving their current views. To this are added new contributions from nine further experts in the history of mathematics, who each discuss an important episode and consider whether it was a revolution. The whole question of mathematical revolutions is thus examined comprehensively and from a variety of perspectives. This thought-provoking volume will interest mathematicians, philosophers, and historians alike.