Reviews elsewhere on the web:
Mathematical Association of America
Philosophica Mathematica
Edward Nelson, Princeton (pdf)

Dennis Hesseling

Gnomes in the fog

This book looks at the development of Brouwer's Intuitionism, and its reception by the mathematics community. Is mathematics just axioms, or is there more to it? How are we to think of infinite sets? The book will certainly be of interest to students of the history of mathematics. It might also be of interest to those wanting to know about the culture in the 1920's - how the war had introduced uncertainty everywhere. It certainly requires some knowledge of university level mathematics, but was more readable than I expected. Readers who topped up what they learned at school with some reading about mathematical axiomatics would probably be able to follow it, in particular chapters 1, 2, and 6. It would also be of use to those considering research into the foundations of mathematics - possibly as a warning of the quagmire that this can become.

Hesseling puts the resolution of the debate around 1928, when Intutionism was axiomatised, but I'm not convinced - axiomatisation somewhat defeats the point. I would say that the results of Gödel and Skolem at the start of the 1930's played a significant rôle. Having definite results showed the shallowness of the philosophical posturing which preceded them.

The title comes from Kundera's observation that hindsight makes history seem clearer than it really was. Personally, I think that patches of the fog are still around.

Amazon.com info
Hardcover 448 pages  
ISBN: 3764365366
Salesrank: 7846097
Weight:2.2 lbs
Published: 2004 Birkhäuser
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Hardcover 476 pages  
ISBN: 3764365366
Salesrank: 3097595
Weight:2.2 lbs
Published: 2003 Birkhäuser
Amazon price £108.00
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Hardcover 448 pages  
ISBN: 3764365366
Salesrank: 3995694
Weight:2.2 lbs
Published: 2004 Birkhäuser
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Product Description

The significance of foundational debate in mathematics that took place in the 1920s seems to have been recognized only in circles of mathematicians and philosophers. A period in the history of mathematics when mathematics and philosophy, usually so far away from each other, seemed to meet. The foundational debate is presented with all its brilliant contributions and its shortcomings, its new ideas and its misunderstandings.