Felix Kaufmann · 1978
Kaufmann’s essay reads the foundational crises of modern mathematics not as a refutation of logic, but as a demand to clarify what logical principles, symbolic systems, and mathematical objects are. The rise of set theory, formal logic, and non-Euclidean geometry has unsettled inherited doctrines, yet Kaufmann argues that the resulting paradoxes expose confusions in language and abstraction more than defects in reason itself.
The development of mathematics in the last hundred years and the theoretical reflections to which this has given rise have produced results that seem apt to bring about a complete overturning of traditional conceptions of logic.
The essay’s central move is linguistic and phenomenological. Language is not a stock of sounds or inscriptions, nor is mathematics merely a calculus of marks. Meaning arises through rules that coordinate signs with intended objects and states of affairs. For Kaufmann, therefore, “speaking about language” is not a retreat from mathematics into psychology; it is the clarification of the conditions under which symbolic expressions can signify at all.
We have emphasized that language is not a system of acoustic complexes and their configurations, but a system of co-ordinating rules between these and thoughts of objects and facts in the world.
This view lets Kaufmann criticize both naïve realism about mathematical entities and crude formalism. Symbols need not resemble what they symbolize, but a successful symbolism must still preserve rule-governed relations that make operations meaningful. Mathematical signs are not self-interpreting; their legitimacy depends on the disciplined connection between sign, intention, fulfillment, and possible object.
On the other hand there will have to be certain similarity connections between symbol and symbolized, if a symbolism is to satisfy a certain condition (to be specified presently) and so to 'achieve' something.
Kaufmann applies this analysis to the antinomies. Russell’s paradox and related difficulties do not show that logic has collapsed. They show what happens when predicates, classes, names, functions, and objects are treated as though they belonged to one undifferentiated domain. Type restrictions and formal devices may be technically useful, but the philosophical task is deeper: to distinguish legitimate symbolic construction from the manufacture of pseudo-objects.
The same point governs his treatment of abstraction. Mathematical objects are not empirical things abstracted by simply omitting sensible qualities, nor are they mental images. Geometry, number, and formal systems depend on idealizing acts whose sense must be specified by their rules of construction and use. Kaufmann’s warning is that ordinary grammar tempts us to speak as though figures, colors, sounds, classes, or numbers were objects in the same sense.
However, this ‘explanation’ is apt to mislead us because it conceals the radical insight that ‘about’ geometrical figures, colours and sounds as such we can predicate nothing at all.
The discussion of arithmetic sharpens the issue. Kaufmann rejects psychologistic accounts of number as inner temporal succession and objectivist accounts that treat numbers as peculiar properties. Counting is grounded in ordered distinction, and arithmetic articulates the formal structure of the number series. Induction is not an empirical generalization but an expression of the completeness of this ordered form.
His engagement with intuitionism is accordingly sympathetic but limited. Brouwer’s critique of classical existence proofs reveals genuine ambiguities in mathematical language, especially where proof, construction, and assertion are confused. Yet Kaufmann does not conclude that a new logic must replace the old. Once the meaning of existential and universal claims is clarified through their rules of fulfillment, the dispute becomes less a battle between rival logics than a demand for stricter interpretation.
The essay ultimately mediates among logicism, formalism, intuitionism, and phenomenology. Mathematics is neither empirical discovery nor empty manipulation. It is a disciplined development of symbolic forms whose possibility rests on prior logical and intentional structures. The foundational crises of mathematics therefore call not for abandoning logic, but for a more exact account of how language, abstraction, and symbolic construction make mathematical thought possible.
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