Philosophy of Mathematics: Sociological Aspects and Mathematical Practice: Unterschied zwischen den Versionen

*** This lecture will approach the question of mathematics’ ‘specialty’ mainly by distinguishing several meanings of ‘specialty' and by describing the roles these aspects play in the contexts of epistemological questions, philosophy of science and philosophy of language. Ideas from Plato, Aristotle and Descartes will be commented upon, and the positions held by Kant and Wittgenstein will be discussed in more detail. Special attention will be paid to the role of mathematics (or `the mathematical')in generalized contexts of teaching/learning of intelligent skills as well as in linguistic meaning[s]. The attempted conclusion of this lecture is that the specialty of mathematics should, above all, be studied within the framework of a philosophical theory of symbols.

** The Indispensability of Mathematical Reasoning

*** It is argued that mathematical reasoning is indispensable in everyday reasoning in both, an analytical or regressive sense, and a synthetical or progressive sense. In the analytical sense mathematical reasoning can be characterized as the transition from given propositions to the conditions necessary for their validity. In the synthetical sense it is the derivation of propositions from given principles. This reading closely connecting mathematics with logic is supported with historical examples from Euclid to Hilbert. In the aspect discussed mathematics is by no means special.

** Is Mathematics Special? - That Depends on the Context.

*** The answer to the question whether mathematics is special depends on what we take mathematics to be. Conceptions of mathematics are marked by the cultural and social contexts (mathematical practice) in which these conceptions are passed on and formed. This presentation will discuss two such contexts: The teaching of mathematics in schools and the context which brought about the modern, formalistic conception of mathematics. In the first part, empirical investigations (in the sense of Löwe & Müller’s “Empirical Philosophy of Mathematics”) will be used to show that pupils acquire a largely empirical understanding of mathematics in school. Mathematical theories serve to describe and explain certain phenomena of reality. This conception of mathematics was also held by mathematicians of past centuries, explicitly for instance by M. Pasch. In the second part of the presentation the historical context of the issue is outlined, which – starting from this conception – has led to the modern Hilbertian conception of mathematics.  The answer to the question “Is mathematics special?” depends on the area the question refers to. Compared to the natural sciences, the mathematics taught in schools is not special; conversely, the modern formalistic conception is. One aim of this presentation is to shed light on the context in which this historical development has taken place. Modern mathematics is not independent of context, much rather it is its context that is ‘special’.