Online research in philosophy

This paper is a contribution to the question of how aspects of science have been perceived through history. In particular, I will discuss how the contribution of axiomatics to the development of science and mathematics was viewed in 20th century philosophy of science and philosophy of mathematics. It will turn out that in connection with scientific methodology, in particular regarding its use in the context of discovery, axiomatics has received only very little attention. This is a rather surprising result, since (...) axiomatizations have been employed extensively in mathematics, science, and also by the philosophers themselves. (shrink)

to be some kind of conspiracy against the suitability of Roman numerals for multiplication. This paper aims to dispel..

This article looks at recent work in cognitive science on mathematical cognition from the perspective of history and philosophy of mathematical practice. The discussion is focused on the work of Lakoff and Núñez, because this is the first comprehensive account of mathematical cognition that also addresses advanced mathematics and its history. Building on a distinction between mathematics as it is presented in textbooks and as it presents itself to the researcher, it is argued that the focus of cognitive analyses of (...) historical developments of mathematics has been primarily on the former, even if they claim to be about the latter. (shrink)

A new look at analogical reasoning Content Type Journal Article Pages 1-5 DOI 10.1007/s11016-011-9563-z Authors Dirk Schlimm, Department of Philosophy, McGill University, Montreal, QC H3A 2T7, Canada Journal Metascience Online ISSN 1467-9981 Print ISSN 0815-0796.

Three different ways in which systems of axioms can contribute to the discovery of new notions are presented and they are illustrated by the various ways in which lattices have been introduced in mathematics by Schröder et al. These historical episodes reveal that the axiomatic method is not only a way of systematizing our knowledge, but that it can also be used as a fruitful tool for discovering and introducing new mathematical notions. Looked at it from this perspective, the creative (...) aspect of axiomatics for mathematical practice is brought to the fore. (shrink)

Using four examples of models and computer simulations from the history of psychology, I discuss some of the methodological aspects involved in their construction and use, and I illustrate how the existence of a model can demonstrate the viability of a hypothesis that had previously been deemed impossible on a priori grounds. This shows a new way in which scientists can learn from models that extends the analysis of Morgan (1999), who has identified the construction and manipulation of models as (...) those phases in which learning from models takes place. (shrink)

In 1870 Jordan proved that the composition factors of two composition series of a group are the same. Almost 20 years later Hölder (1889) was able to extend this result by showing that the factor groups, which are quotient groups corresponding to the composition factors, are isomorphic. This result, nowadays called the Jordan-Hölder Theorem, is one of the fundamental theorems in the theory of groups. The fact that Jordan, who was working in the framework of substitution groups, was able to (...) prove only a part of this theorem is often used to emphasize the importance and even the necessity of the abstract conception of groups, which was employed by Hölder. However, as a little-known paper from 1873 reveals, Jordan had all the necessary ingredients to prove the Jordan-Hölder Theorem at his disposal (namely, composition series, quotient groups, and isomorphisms), and he also noted a connection between composition factors and corresponding quotient groups. Thus, I argue that the answer to the question posed in the title is “Yes.” It was not the lack of the abstract notion of groups which prevented Jordan from proving the Jordan-Hölder Theorem, but the fact that he did not ask the right research question that would have led him to this result. In addition, I suggest some reasons why this has been overlooked in the historiography of algebra, and I argue that, by hiding computational and cognitive complexities, abstraction has important pragmatic advantages. (shrink)

The structure-mapping theory has become the de-facto standard account of analogies in cognitive science and philosophy of science. In this paper I propose a distinction between two kinds of domains and I show how the account of analogies based on structure-preserving mappings fails in certain (object-rich) domains, which are very common in mathematics, and how the axiomatic approach to analogies, which is based on a common linguistic description of the analogs in terms of laws or axioms, can be used successfully (...) to explicate analogies of this kind. Thus, the two accounts of analogies should be regarded as complementary, since each of them is adequate for explicating analogies that are drawn between different kinds of domains. In addition, I illustrate how the account of analogies based on axioms has also considerable practical advantages, e. g., for the discovery of new analogies. (shrink)

The main ideas behind Brouwer’s philosophy of Intuitionism are presented. Then some critical remarks against Intuitionism made by William Tait in “Against Intuitionism” [Journal of Philosophical Logic, 12, 173–195] are answered.

Wilfred Sieg and Dirk Schlimm. Dedekind's Analysis of Number: Systems and Axioms.