Mathematical intuition. Phenomenology and mathematical knowledge.

*(English)*Zbl 0686.03003
Synthese Library, 203. Dordrecht etc.: Kluwer Academic Publishers Group. xiv, 209 p. Dfl. 130.00; £39.50; $ 69.00 (1989).

This book aims to give a detailed reconstruction of Edmund Husserl’s philosophy of mathematics as developed especially in “Philosophie der Arithmetik”. Ch. 1 gives a general overview of different conceptions of intuition in mathematics. Ch. 2 describes Husserl’s peculiar phenomenological view of intuition which is compared in Ch. 3 with his theory of perception. Ch. 4 deals with some objections that might be raised against this approach.

The main body of the work is to be found in Chs. 5-7, where the author tries to show that there are certain analogies between mathematical and perceptual intuition. Mathematical intuition has to do with genuinely mathematical (abstract) objects, not with perceptual objects like signs (sign-tokens) that may be taken to represent natural numbers or sets. Notwithstanding the ontological differences between mathematical and perceptual objects, the analogy between both kinds of intuition is seen “insofar as in each case we are concerned with certain processes that produce evidence for M’s beliefs”. In both cases the objects are said to “transcend the actual intuition”. In the case of mathematical objects, this roughly means that when we perceive certain perceptual entities (e.g., sign-tokens) we do not see them as perceptual objects “but rather as elements of a set, where the particular nature of the individuals is irrelevant to the set”. (p. 175).

The main body of the work is to be found in Chs. 5-7, where the author tries to show that there are certain analogies between mathematical and perceptual intuition. Mathematical intuition has to do with genuinely mathematical (abstract) objects, not with perceptual objects like signs (sign-tokens) that may be taken to represent natural numbers or sets. Notwithstanding the ontological differences between mathematical and perceptual objects, the analogy between both kinds of intuition is seen “insofar as in each case we are concerned with certain processes that produce evidence for M’s beliefs”. In both cases the objects are said to “transcend the actual intuition”. In the case of mathematical objects, this roughly means that when we perceive certain perceptual entities (e.g., sign-tokens) we do not see them as perceptual objects “but rather as elements of a set, where the particular nature of the individuals is irrelevant to the set”. (p. 175).

Reviewer: W.Lenzen

##### MSC:

03A05 | Philosophical and critical aspects of logic and foundations |

00A35 | Methodology of mathematics |

03-02 | Research exposition (monographs, survey articles) pertaining to mathematical logic and foundations |

00-02 | Research exposition (monographs, survey articles) pertaining to mathematics in general |