Felix Kaufmann on Waismann’s Einführung in das mathematische Denken (1937)

Felix Kaufmann’s newspaper book review presents Friedrich Waismann’s introduction to mathematical thinking as a significant pedagogical achievement. Its central claim is that foundational mathematics can become accessible to educated, philosophically interested readers without sacrificing conceptual rigor. Kaufmann evaluates the book through the relationship between accessibility and precision: its success consists in opening a specialist field while preserving what makes its knowledge mathematical.

Denn es ermöglicht durch seine ausgezeichnete Darstellung einem weiteren Kreis von gebildeten und philosophisch interessierten Lesern den Zugang zu einem Erkenntnisgebiet, das bisher fast ausschließlich Fachgelehrten zugänglich war.

English translation: For through its excellent exposition it enables a wider circle of educated and philosophically interested readers to gain access to a field of knowledge that had previously been accessible almost exclusively to specialists.

This widening of access matters because mathematical foundations have helped shape scientific and philosophical thought, and consequently cultural life. Understanding them is therefore an enrichment of education, not simply an acquisition of technical information. Kaufmann connects Waismann’s achievement to his preparation as both researcher and teacher, including his leadership of philosophical seminars at the University of Vienna. The review treats scholarly competence and pedagogical experience as complementary qualifications.

Its decisive criterion, however, is the rigor with which concepts are introduced:

Vor allem wußte er, daß bei der Darstellung der mathematischen Grundlagenprobleme unter keinen Umständen auf Strenge der Begriffseinführung verzichtet werden dürfe; denn in ihr liegt ein Wesensmerkmal der Mathematik.

English translation: Above all, he knew that in presenting the foundational problems of mathematics, rigor in the introduction of concepts must under no circumstances be dispensed with; for it is an essential characteristic of mathematics.

Kaufmann thus distinguishes explanation that conveys genuine knowledge from popular philosophical writing that produces dangerous half-knowledge. Accessibility does not justify loosening definitions: precision is part of the subject itself.

The review then sketches the book’s sixteen short sections. Their center is the rigorous construction of the number system, proceeding through integers, rational numbers, real numbers, and complex numbers. From this center, Waismann develops connections to other problems: the relation between arithmetic and geometry, the character of geometrical knowledge, and differential calculus. Kaufmann particularly hopes that the treatment of calculus will dispel nonmathematicians’ misunderstandings about the infinitely small.

He singles out section nine, on the current state of foundational research, for making disputed questions vividly apparent. Yet his praise stops short of endorsing every position:

Ich kann Dr. Waismanns Thesen in bezug auf jene offenen Probleme nicht durchwegs beipflichten, aber auf diese Meinungsdifferenzen kann hier nicht einmal andeutungsweise eingegangen werden.

English translation: I cannot agree throughout with Dr. Waismann’s theses concerning those open problems, but these differences of opinion cannot be addressed here even in outline.

The review does not specify those disagreements, so it supports no reconstruction of Kaufmann’s alternative views. Its conclusion remains an emphatic recommendation to conscientious readers willing to undertake demanding study. The work’s relevance lies in this qualified advocacy: foundational mathematics deserves a wider readership, and good exposition enables that readership without concealing either conceptual difficulty or unresolved debate.