Gnomes in the Fog: The Reception of Brouwer's Intuitionism in the 1920s (Science Networks. Historical Studies) [Hardcover]

This is a unique, thorough, and excellent treatment of the reception of intuitionist mathematics and philosophy of mathematics. It presents the views of the semi-intuitionists such as Kronecker and Poincare. It extensively discusses the various foundational debates and opinions of intutionism by various mathematicians,including several less well known ones, but it goes far beyond that. For instance, it includes parallels to similar currents such as constructivism in art to the popularity of appeals to "constructivism" by not only the mathematical intuitionists, but by the formalist Hilbert and the logicist Carnap. Also mentioned is the case of a high school text of the period that included some discussion of intuitionist mathematics. It also discusses parallels between Weyl's defense of intuitionism and Spengler's "Decline of the West" and the influences of Brouwer on Wittgenstein. This is all and all a fascinating book. It doesn't develop the conceptual points as far as it might, but it lays out numerous suggestions and alternative accounts of the relation of mathematical intuitionism to lebensphilosophie, Weimar culture and other cultural currents of the time.

This book is a revised version of the author's 1999 dissertation. It gives only a sketch Brouwer's views on the epistemology and ontology, and thus on the nature and boundaries, of mathematics. Hesseling writes: "I am here dealing with the genesis of his intuitionism as far as it is relevant to its reception. Since most reactions did not go deeply into intuitionistic mathematics, neither shall I. Instead I restrict myself to giving an impression of how Brouwer developed intuitionism by treating the main ideas of some of his papers." (67) In the last numbered chapter, Hesseling wanders off in a fanciful quest for cultural undertones to the debate.

Besides the lack of mathematical detail, the philosophical basis of Brouwer's views on mathematics is not presented with any depth. Brouwer's views on the criteria that certify what is legitimate mathematics arise from his philosophy. His philosophical claims require justification if they are to support a critique of mathematics. Furthermore, Kant's epistemology, certainly the most influential philosophical background to the debate, although for some participants filtered through Husserl's phenomenology, is given little attention. Hesseling hasn't gone into the intellectual depths of the controversy, and the result is a superficial presentation of ideas.

The debate resolved predominantly into the differences between Brouwer's Intuitionism, supported by Weyl, and Hilbert's Formalism, supported by Bernays. The Logicism of Frege, and of Russell and Whitehead is not discussed. Hesseling sees Weyl's 1921 essay "On the New Foundational Crisis of Mathematics" as a pivotal text in the debate.

Hesseling has surveyed the entire corpus of written, public reactions to Brouwer's Intuitionism from 1909 to 1933, and lists these texts by year in an appendix. Another appendix gives a chronology of the debate from 1897 to 1932. Brouwer lived until 1966, but Hesseling looks at the debate only up to 1933. He writes: "After 1933, Brouwer published some 30 more papers on intuitionism, some which were technical contributions to intuitionistic mathematics, others of a more expository nature. Compared to his ideas before 1930, no spectacular new insights were presented in the later papers." (86)

In 1928, Hilbert and Ackermann published their logic text. In 1930, Gödel announced his incompleteness proof and Heyting published a formalization of Intuitionistic logic. In 1932, Gödel began publishing on Intuitionistic logic. In 1934, Hilbert and Bernays published the first volume of their text on the foundations of mathematics. The features of the debate were changed.

Brouwer's position arises from the assertion that the ontology of mathematics is determined solely by epistemology, and thus a statement which has not been determined to be true or to be false cannot be determined to refer to a state of affairs or to anything which exists. If we cannot find it, then we cannot claim it exists and we also cannot claim it does not exist. Moreover, we cannot claim it is certainly either one or the other, because the limits of our epistemology are the limits of ontological fact. Moreover, knowledge is temporal and a process; and it is pre-linguistic. The negation of what is known is absurdity.

Brouwer claimed that mathematics is not in any sense linguistic and that mathematics precedes logic. Logic arises from mathematics via language. As Hesseling writes: "Mathematical language follows upon mathematical activity, and logic consists of looking at that language in a mathematical way." (39) Likewise: "Logical principles only hold for words that have mathematical meaning." (44)

Since mathematical meaning derives from emergent, epistemologically generated ontology, it follows that logic itself is not a formative and central structure of cognition, but rather is itself an emergent, structural overlay. It then follows that where mathematical meaning is indeterminate, so is logic.

The reductio ad absurdum method of proof involves the law of excluded middle. The dichotomy of either S or not-S (at least one and not both) ensures that reductio ad absurdum is valid. If the dichotomy fails, so does the justification for reductio ad absurdum. This is important to Brouwer because reductio ad absurdum allows for non-constructive proofs, which circumvent, overrun, and thwart his ontology.

In place of the law of excluded middle, in Intuitionism one has a logic where every proposition is either determined to be true or to be false or it is undetermined. One could then instate a logic, not of truth values, but of values of determination.

For a collection of fundamental texts from the debate, see Paulo Mancosu's [1998] From Brouwer to Hilbert: The Debate on the Foundations of Mathematics in the 1920s.

For a study devoted strictly to Brouwer's views on Intuitionism, see Walter P. van Stigt's [1990] Brouwer's Intuitionism.