Inquiry activities in a classroom: extra-logical processes of illumination vs logical process of deductive and inductive reasoning. A case study

The paper presents results of the research, which was focused on studying students’ inquiry work from a psychological point of view. Inquiry activities of students in a classroom were analysed through the evaluation of the character of these activities within learning process with respect to mathematician’s research practice. A process of learning mathematical discovery was considered in detail as a part of inquiry activities of students in a classroom

From Logical Calculus to Logical Formality—What Kant Did with Euler’s Circles

John Venn has the “uneasy suspicion” that the stagnation in mathematical logic between J. H. Lambert and George Boole was due to Kant’s “disastrous effect on logical method,” namely the “strictest preservation [of logic] from mathematical encroachment.” Kant’s actual position is more nuanced, however. In this chapter, I tease out the nuances by examining his use of Leonhard Euler’s circles and comparing it with Euler’s own use. I do so in light of the developments in logical calculus from G. W. Leibniz to Lambert and Gottfried Ploucquet. While Kant is evidently open to using mathematical tools in logic, his main concern is to clarify what mathematical tools can be used to achieve. For without such clarification, all efforts at introducing mathematical tools into logic would be blind if not complete waste of time. In the end, Kant would stress, the means provided by formal logic at best help us to express and order what we already know in some sense. No matter how much mathematical notations may enhance the precision of this function of formal logic, it does not change the fact that no truths can, strictly speaking, be revealed or established by means of those notations

The logical anti-psychologism of Frege and Husserl

Frege and Husserl are both recognized for their significant contributions to the overthrowing of logical psychologism, at least in its 19th century forms. Between Frege's profound impact on modern logic that extended the influence of his anti-psychologism and Husserl's extensive attempts at the refutation of logical psychologism in the Prolegomena to Logical Investigations, these arguments are generally understood as successful. This paper attempts to account for the development of these two anti-psychologistic conceptions of logical objects and for some of the basic differences between them. It identifies some problems that are common to strongly anti-psychologistic conceptions of logic and compares the extent to which Frege's and Husserl's views are open to these problems. Accordingly, this paper is divided into two parts. Part I develops a conception of the problems of logical psychologism as they are distinctively understood by each philosopher, out of the explicit arguments and criticisms made against the view in the texts. This conception is in each case informed by the overall historical trajectories of each philosopher's philosophical development. Part II examines the two views in light of common problems of anti-psychologism

Mathematical practice, crowdsourcing, and social machines

The highest level of mathematics has traditionally been seen as a solitary endeavour, to produce a proof for review and acceptance by research peers. Mathematics is now at a remarkable inflexion point, with new technology radically extending the power and limits of individuals. Crowdsourcing pulls together diverse experts to solve problems; symbolic computation tackles huge routine calculations; and computers check proofs too long and complicated for humans to comprehend. Mathematical practice is an emerging interdisciplinary field which draws on philosophy and social science to understand how mathematics is produced. Online mathematical activity provides a novel and rich source of data for empirical investigation of mathematical practice - for example the community question answering system {\it mathoverflow} contains around 40,000 mathematical conversations, and {\it polymath} collaborations provide transcripts of the process of discovering proofs. Our preliminary investigations have demonstrated the importance of "soft" aspects such as analogy and creativity, alongside deduction and proof, in the production of mathematics, and have given us new ways to think about the roles of people and machines in creating new mathematical knowledge. We discuss further investigation of these resources and what it might reveal. Crowdsourced mathematical activity is an example of a "social machine", a new paradigm, identified by Berners-Lee, for viewing a combination of people and computers as a single problem-solving entity, and the subject of major international research endeavours. We outline a future research agenda for mathematics social machines, a combination of people, computers, and mathematical archives to create and apply mathematics, with the potential to change the way people do mathematics, and to transform the reach, pace, and impact of mathematics research.Comment: To appear, Springer LNCS, Proceedings of Conferences on Intelligent Computer Mathematics, CICM 2013, July 2013 Bath, U

Kriesel and Wittgenstein

Georg Kreisel (15 September 1923 - 1 March 2015) was a formidable mathematical logician during a formative period when the subject was becoming a sophisticated field at the crossing of mathematics and logic. Both with his technical sophistication for his time and his dialectical engagement with mandates, aspirations and goals, he inspired wide-ranging investigation in the metamathematics of constructivity, proof theory and generalized recursion theory. Kreisel's mathematics and interactions with colleagues and students have been memorably described in Kreiseliana ([Odifreddi, 1996]). At a different level of interpersonal conceptual interaction, Kreisel during his life time had extended engagement with two celebrated logicians, the mathematical Kurt Gödel and the philosophical Ludwig Wittgenstein. About Gödel, with modern mathematical logic palpably emanating from his work, Kreisel has reflected and written over a wide mathematical landscape. About Wittgenstein on the other hand, with an early personal connection established Kreisel would return as if with an anxiety of influence to their ways of thinking about logic and mathematics, ever in a sort of dialectic interplay. In what follows we draw this out through his published essays—and one letter—both to elicit aspects of influence in his own terms and to set out a picture of Kreisel's evolving thinking about logic and mathematics in comparative relief.Accepted manuscrip

The "Artificial Mathematician" Objection: Exploring the (Im)possibility of Automating Mathematical Understanding

Reuben Hersh confided to us that, about forty years ago, the late Paul Cohen predicted to him that at some unspecified point in the future, mathematicians would be replaced by computers. Rather than focus on computers replacing mathematicians, however, our aim is to consider the (im)possibility of human mathematicians being joined by “artificial mathematicians” in the proving practice—not just as a method of inquiry but as a fellow inquirer