Jeremy Gray is Professor of the History of Mathematics and Director of the Centre for the History of the Mathematical Sciences at the Open University, and a regular Visiting Professor at the University of Warwick. He works on the history of mathematics in the 19th and 20th Centuries, and also on issues in the philosophy and social significance of mathematics. He wrote 'The Hilbert Challenge', Oxford University Press, 2000, and he was the editor of and a contributor to 'The Symbolic Universe: geometry and physics, 1890-1930', Oxford University Press, 1999. His book 'Janos Bolyai, non-Euclidean geometry, and Nature of Space', a Burndy Library publication, MIT, came out in 2004.

José Ferreirós is 'profesor titular' for Logic and Philosophy of Science at the University of Seville. He is the author of 'Labyrinth of Thought: A history of set theory and its role in modern mathematics', Basel, Birkhäuser, 1999, 'Science Networks' no. 23. He has edited in Spanish 'Riemanniana Selecta' (Madrid, CSIC, 2000, bilingual edition) and Dedekind's foundational works (Madrid, Alianza Editorial, 1997), and has published papers on the history and philosophy of logic, mathematics, and general science in journals such as 'The Bulletin of Symbolic Logic', 'Historia Mathematica', 'Archive for History of Exact Sciences', 'Science in Context', 'Theoria' and 'Critíca'. Among his forthcoming papers are contributions to 'Landmarks in Western Mathematics' (Elsevier), 'The Shaping of Arithmetic' (Springer), 'Filosofia de las ciencias matemáticas, naturales y sociales' (Trotta), and 'Philosophical insights into logic and mathematics' (Kluwer).

• Introduction


• Reinterpretations in the history and philosophy of foundations


• Frege and the role of historical elucidation: Methodology and the foundations of mathematics


• Riemann's Habilitationsvortrag at the crossroads of mathematics, physics and philosophy


• The Riemannian background to Frege's philosophy


• Axiomatics, Empiricism, and Anschauung in Hilbert's conception of geometry: Between arithmetic and general relativity


• Explorations into the emergence of modern mathematics


• Methodology and metaphysics in the development of Dedekind's theory of ideals


• Emmy Noether's "set-theoretic" topology: from Dedekind to the rise of functors


• Tarski on models and logical consequence


• A path to the epistemology of mathematics: Homotopy theory


• Alternative views and programs in the philosophy of mathematics


• Felix Hausdorff's 'considered empiricism'


• Practice-related symbolic realism in H. Weyl's mature view of mathematical knowledge


• From Kant to Hilbert: French philosophy of concepts in the beginning of the 20th Century.


• Relative consistency and accessible domains, (Synthese, vol. 84, 1990)


• Coda


• Modern mathematics as a cultural phenomenon