 
Summary: Gnomes in the Fog: The Reception of Brouwer's Intuitionism in the 1920s
by Dennis E. Hesseling
Basel, Birkhšauser, 2003
xxiii+448 pages. US $139.00. ISBN 3764365366
Reviewed by Jeremy Avigad
The early twentieth century was a lively time for the foundations of math
ematics. This ensuing debates were, in large part, a reaction to the set
theoretic and nonconstructive methods that had begun making their way
into mathematical practice around the turn of the twentieth century. The
controversy was exacerbated by the discovery that overly našive formulations
of the fundamental principles governing the use of sets could result in contra
dictions. Many of the leading mathematicians of the day, including Hilbert,
Henri PoincarŽe, ŽEmile Borel, and Henri Lebesgue, weighed in with strong
views on the role that the new methods should play in mathematics. Dennis
E. Hesseling's book, Gnomes in the Fog, documents reactions to the "crisis
of foundations" that was inaugurated by the attempts of L.E.J. Brouwer's
to refound modern mathematics on "intuitionistic" principles.
Brouwer's interests in foundational issues extend at least as far back
as his 1907 doctoral dissertation at the University of Amsterdam. In that
work, he tried to ground mathematics, especially the mathematics of the
