Remarks on the Controversy about the Foundations of Logic and Mathematics

Felix Kaufmann’s 1931 essay intervenes in the controversy over the foundations of logic and mathematics by treating it chiefly as a problem of sense. Foundational disputes arise when symbolic forms, abstractions, or abbreviations are mistaken for independent objects.

The central problem in the theory of a science lies in clearly grasping the sense of the relevant propositions.

For Kaufmann, a science is understood through the objects its propositions intend and through the procedures by which they are verified. Clarification does not create meaning from nothing; it raises already operative meanings to a higher level of determinacy. The foundational philosopher must therefore examine how mathematical and logical propositions function before asking what entities they allegedly describe.

The first major target is the received theory of abstraction. Kaufmann denies that abstraction produces separate objects by stripping away features. Rather, abstraction holds some aspects fixed while leaving others undetermined within a connected unity.

The foregoing considerations already contain the basic ideas of the analyses that follow; they start from a critique of the current view as to the nature of abstraction.

This account allows him to oppose both conceptual realism and overly extensional logic. A number, color, judgment, or function is not clarified by attaching to it accidental associations or by treating it as a thing-like bearer of further properties. What matters is the invariant structure intended in a proposition.

Abstraction thus here rests on the invariance of certain aspects within that connected unity with respect to changes in other aspects.

Kaufmann applies this point to “properties of properties,” “judgments about judgments,” “concepts of concepts,” and “functions of functions.” Such expressions may be legitimate as abbreviations, but confusion begins when they are read as names for higher-order entities. Logical symbolism is useful only if its abbreviating role remains explicit; otherwise it generates sham problems by disguising rules of translation as ontology.

The essay then distinguishes sharply between empirical and non-empirical judgments.

Here we have to distinguish between empirical and non-empirical judgments.

This distinction is crucial for Kaufmann’s criticism of quantification and extension. In empirical propositions, universality and existence depend on an indicated field of individuation; without such a field, a universal empirical proposition remains incomplete. In non-empirical propositions, by contrast, generality belongs to the structure of the meaning itself. Thus “all” and “there is” do not have one uniform sense across logic, mathematics, and empirical science.

Kaufmann’s treatment of number follows from this semantic analysis. Natural number is not a class of equivalent classes, nor a property of collections, but an abstraction from the ordered activity of counting. Counting assigns signs according to a rule; cardinal number emerges when the order of counted objects is allowed to vary while the relevant invariant remains fixed. Temporal features of the act are then abstracted away, leaving the logical structure of the counting process.

On this basis Kaufmann reinterprets Peano’s axioms as an explicit definition of the natural-number structure. Complete induction is not an obscure infinitary power but part of the determination of that structure: it excludes further, unlicensed determinations. Hence undecidability, where it appears, is not a deep metaphysical feature of arithmetic but a sign that the presuppositions have not fully determined the intended object.

The final sections extend the critique to impredicative definitions, higher-level functions, and set-theoretic infinities. Negative, rational, irrational, and complex numbers can be understood as symbolic extensions whose content is secured by translation back into propositions about natural numbers. Difficulty arises when mathematical forms are confused with completed totalities of values satisfying them. A sequence is governed by a law or general term; it is not first given as an unordered completed set.

Kaufmann’s importance lies in the essay’s combination of phenomenological analysis, verificationist semantics, and constructivist restraint. He does not merely reject classical mathematics; he argues that its valid content survives once misleading higher-order and set-theoretic formulations are translated into clearer forms. The foundational controversy is therefore recast as a critique of ambiguous signs, illicit reifications, and failures to distinguish logical meaning from empirical multiplicity.