Das Unendliche in der Mathematik und seine Ausschaltung: Eine Untersuchung über die Grundlagen der Mathematik

Felix Kaufmann’s Das Unendliche in der Mathematik und seine Ausschaltung is a foundational study of the role of infinity in mathematics. Its purpose is to clarify the assumptions by which actual infinity, and especially the uncountable infinite, enters mathematical theory. Kaufmann does not present this as a destructive attack on mathematics, but as an attempt to preserve mathematical validity by removing illegitimate metaphysical interpretations of infinite totalities.

Die vorliegende Arbeit behandelt die Grundlagenprobleme der Mathematik, die größtenteils mit dem Begriffe des Unendlichen verknüpft sind, und versucht über die wichtigsten umstrittenen Fragen zu klarer Entscheidung zu gelangen.

English translation: The present work deals with the foundational problems of mathematics, which are for the most part bound up with the concept of the infinite, and attempts to reach clear decisions on the most important disputed questions.

The argument proceeds through an analysis of meaning, assertion, and justification. Kaufmann treats foundational disputes not merely as technical problems within set theory, but as consequences of confusions about the logical force of universal and existential claims. In particular, he argues that statements about “all” objects or the “existence” of objects function differently depending on whether they are empirical or non-empirical.

Wir werden im folgenden erkennen, daß die mangelnde Trennung von empirischen und nicht-empirischen All-Aussagen bzw. Existenzaussagen eine der Hauptquellen der Schwierigkeiten ist, die sich bei der Grundlegung der Mengenlehre ergeben.

English translation: We shall recognize in what follows that the failure to distinguish between empirical and non-empirical universal or existential propositions is one of the chief sources of the difficulties that arise in the foundation of set theory.

This distinction allows Kaufmann to redirect the problem of infinity. The question is not whether the mind can somehow grasp an already given infinite realm, but whether the supposed realm has been introduced with a legitimate mathematical meaning. His critique therefore concerns the conditions under which symbols, predicates, and domains can function as meaningful parts of mathematical discourse.

Kaufmann’s philosophy of symbolism is central to this project. He rejects the idea that formal signs can be treated as empty marks whose subsequent manipulation alone secures mathematical sense. A sign is already embedded in the possibility of understanding; otherwise it is not a sign in the relevant logical sense. This supports his suspicion of formal systems that appear to name or range over infinite totalities without having justified the meaning of those totalities.

Ein „sinnloses Zeichen“ ist eine contradictio in adjecto, denn in der Behauptung, daß visuelle oder akustische Phänomene „Zeichen“ sind, liegt diejenige eingeschlossen, daß man durch sie etwas verstehen, mit ihrer Hilfe fremde Gedanken erfassen kann.

English translation: A "meaningless sign" is a contradictio in adjecto, for the assertion that visual or acoustic phenomena are "signs" already contains the assertion that through them one can understand something, that with their help one can grasp the thoughts of others.

A major part of the book’s eliminative strategy depends on distinguishing kinds of generality. Kaufmann argues that many appeals to infinity arise from confusing specific generality with an imagined survey of infinitely many individual cases. Once this confusion is removed, arithmetic does not require the completion of an infinite traversal. The alleged need to pass through infinitely many possibilities is, for him, a misleading picture rather than a genuine requirement of proof.

Unterscheidet man aber in korrekter Weise zwischen spezifischer Allgemeinheit und individueller Allgemeinheit, so fällt der Ungedanke des Durchlaufens von unendlich vielen Möglichkeiten und damit das Hauptmotiv für die irrige Annahme der Unentscheidbarkeit arithmetischer Probleme fort.

English translation: But if one correctly distinguishes between specific generality and individual generality, then the pseudo-thought of running through infinitely many possibilities falls away, and with it the chief motive for the erroneous assumption that arithmetical problems are undecidable.

The same line of reasoning governs Kaufmann’s treatment of uncountable domains, above all the supposed totality of real numbers. He does not simply claim that all propositions about such domains are decidable. Instead, he argues that the domains themselves have not been logically warranted in the way classical set theory assumes. Apparent limits of decision are therefore not evidence for a hidden mathematical being beyond knowledge, but symptoms of an unjustified formation of the range of discourse.

In this sense, the “elimination” of infinity is not the rejection of mathematical rigor, but a critique of the ontological surplus attached to mathematical symbolism. Kaufmann seeks to show that mathematics can retain its legitimate content without appeal to completed infinite totalities. His monograph thus combines logical analysis, philosophy of language, and critique of set-theoretic realism in order to secure mathematics against the confusions generated by actual infinity.