Mathematics used to be portrayed as a deductive science. Stemming from Polya, however, is a philosophical movement which broadens the concept of mathematical reasoning to include inductive or quasi-empirical methods. Interest in inductive methods is a welcome turn from foundationalism toward a philosophy grounded in mathematical practice. Regrettably, though, the conception of mathematical reasoning embraced by quasi-empiricists is still too narrow to include the sort of thought-experiment which Mueller describes as traditional mathematical proof and which Lakatos examines in Proofs and (...) refutations. This paper extends the concept of mathematical reasoning along two further dimensions to accommodate thought-experiment.Keywords: Thought-experiment; Informal proof; Mathematical reasoning. (shrink)

Recent accounts of the role of diagrams in mathematical reasoning take a Platonic line, according to which the proof depends on the similarity between the perceived shape of the diagram and the shape of the abstract object. This approach is unable to explain proofs which share the same diagram in spite of drawing conclusions about different figures. Saccheri’s use of the bi-rectangular isosceles quadrilateral in Euclides Vindicatus provides three such proofs. By forsaking abstract objects it is possible to give a (...) natural explanation of Saccheri’s proofs as well as standard geometric proofs and even number-theoretic proofs. (shrink)

The “origins” of (geometric) space is examined from the perspective of the so-called “conceptual space” or “semantic space”. Semantic space is characterized by its fundamental “locality” that generates an “implicit” mode of geometrizing. This view is examined from within three perspectives. First, the role that various diagrammatic entities play in the everyday life and pragmatic activities of selected ethnic groups is illustrated. Secondly, it is shown how conceptual spaces are fundamentally linked to the meaning effects of particular natural languages and (...) these are very different from the global and universal aspects of Euclidean spaces. Thirdly, it is contended that these modes of creating body and culture-based spatial frameworks and related cosmogonies and cosmologies can be described as forms of “latent geometry” that initially appear unexplainable in any rational way. Nonetheless, and thanks to the deep mathematical reflections provided by René Thom, it is illustrated how the various ways of generating space can be further analyzed as distortions of mainstream spatialization furnished by Euclidean geometry that established the dominant universality of the ideas of space (and time). (shrink)

The development of ancient civilizations and their achievements in sciences such as mathematics and astronomy are well researched for script-using civilizations. On the basis of oral tradition and mnemonic artifacts illiterate ancient civilizations were able to attain an adequate level of knowledge. The Neolithic and Bronze Age earthworks and circles are such mnemonic artifacts. Explanatory models are given for the shape of the stone formations and the ditch of Stonehenge reflecting the circular and specific non-circular shapes of these structures. The (...) basic mathematical concepts are Pythagorean triangles, thus adopting the construction procedures of the Neolithic circular ditches of Central Europe in the fifth Millennium bc and later earthworks and stone circles in Britain and Brittany. This knowledge was extended with new elliptical concepts. Approximations for the values of π and the square root of 2 are encoded in the henge. All constructions were performed using a standardized “Babylonian” metrology that shows a remarkable consistency and comprehensible development over some 14 centuries. (shrink)