Felix Kaufmann: Review of Wilhelm Burkamp’s Begriff und Beziehung (1930)

Felix Kaufmann’s journal book review assesses Wilhelm Burkamp’s Begriff und Beziehung. Studien zur Grundlegung der Logik (1927) as a significant philosophical contribution to the foundations of logic and mathematics. Its argument moves from the historical crisis that motivates Burkamp’s investigations to their organization, their relationship to Frege, and Kaufmann’s differentiated judgment of their results. Kaufmann particularly endorses Burkamp’s criticism of class logic, while questioning his treatment of individuality and qualifying his agreement with the concluding account of logic’s distinctive task.

The opening locates Burkamp’s work within the foundational disturbances produced by Cantorian set theory. Contradictory constructions, including the set of all sets and the set of sets that do not contain themselves, appeared to compromise the consistency of the concept of a set. Their consequences extended beyond a specialized mathematical difficulty: the grounding of natural numbers in sets threatened arithmetic, while the possibility of reformulating the second contradiction without set terminology implicated logic itself. Kaufmann does not reconstruct the competing formalist, intuitionist, and logicist responses. Instead, he uses this background to explain the importance of philosophical engagement with problems that mathematical and logical symbolism had made inaccessible to many philosophers, and that many mathematicians had disregarded while pursuing specialized research.

Burkamp’s five studies form, in Kaufmann’s account, a unified progression: concept and proposition, the individual, existence, class and number in conceptual logic, and finally logic, mathematics, and the a priori. Frege provides the principal point of departure throughout. The book’s dedication to him reflects an intellectual relationship that includes both constructive continuation and opposition to theses in his two major foundational works. Kaufmann identifies two closely connected centers of investigation: existential propositions and individuality, on the one hand, and class logic on the other. His strongest endorsement concerns the latter:

Burkamps Kritik der Klassenlogik scheint mir zum Besten zu gehören, was über dieses Thema geschrieben worden ist, und ich halte seine diesbezüglichen Ergebnisse im wesentlichen für gesichert.

English translation: Burkamp’s critique of class logic seems to me to be among the best things written on this subject, and I regard his results in this respect as essentially established.

This is a judgment of substantive validity, not merely praise for philosophical participation in a technical debate. The review does not, however, reproduce the arguments establishing those results. Its evidential scope is chiefly evaluative: it identifies the contribution Kaufmann considers most secure without supplying a detailed reconstruction of Burkamp’s class logic.

Kaufmann’s reservation about individuality is more specific. He regards the distinction between concrete things or determinate psychophysical individuals and eidetic singularities as logically more consequential than Burkamp allows. A completely determined shade of color exemplifies the latter category. The distinction also bears directly on mathematical foundations because Kaufmann classifies the natural numbers themselves as eidetic singularities. His objection thus connects the analysis of individuality to the book’s foundational project rather than treating it as a peripheral terminological issue.

The closing discussion turns to the fifth study and Burkamp’s attempt to distinguish logic from mathematics through the concept of the pure form of thought. Kaufmann sees the Marburg Neo-Kantian orientation emerging especially clearly here:

Die Logik erhält hiernach ihren spezifischen Sinn nicht durch ihre formale Struktur, sondern durch eine bestimmte Erkenntnisaufgabe, die ihr zufällt.

English translation: According to this account, logic acquires its specific meaning not through its formal structure, but through a particular epistemic task assigned to it.

The decisive move is from formal structure to epistemic function. Logic is distinguished by its task of determining what is not yet known on the basis of what is already known. Burkamp’s transcendental interpretation thereby supplies a reason for singling logic out from the general domain of mathematical forms. Kaufmann reports this position without withdrawing his broader reservations. His final recommendation remains emphatic: despite disagreement with some theses, he regards the book as one of the most important recent contributions to logical theory and commends it to philosophers and mathematicians concerned with foundations. The review’s relevance lies in this discriminating endorsement, which joins recognition of Burkamp’s achievement to a precise objection about the logical status of individuals and numbers.