Banca de DEFESA: GUILLERMO NIGRO PUENTE

Uma banca de DEFESA de DOUTORADO foi cadastrada pelo programa.
STUDENT : GUILLERMO NIGRO PUENTE
DATE: 27/03/2026
TIME: 14:00
LOCAL: Web Conferência
TITLE:

The open problem of methodological purity in the Feschrift: Hilbert and the justification of the autonomous foundation of Euclidean geometry

KEY WORDS:

Purity of methods; Axiomatic Method; David Hilbert; Foundations of Geometry

PAGES: 318
BIG AREA: Ciências Humanas
AREA: Filosofia
SUBÁREA: Epistemologia
SUMMARY:

At the end of Festschrift (1899), Hilbert stated that the purity of the method is a ‘subjective’ requirement. However, in his reconstruction of Euclidean geometry, he adopted two clearly purist requirements: (1) to avoid the use of Archimedes’ axiom in elementary propositions. and (2) to demonstrate geometric versions of the Common Notions. These requirements seek an autonomous foundation for geometry and exclude the use of external notions, such as number or general magnitude. What is the epistemic value of these requirements? Can they be justified in any way, or are they merely idiosyncratic? This is the ‘open problem’ of purity in Hilbert. The thesis argues that its justification is essentially dialogical: it depends on philosophical images about the nature of mathematical knowledge and belongs to the realm of rational choices, not to that of ‘technical’ axiomatic analysis. In doing so, the thesis fills two gaps in the current literature: the normative dimension of purity and the conceptualisation of the relationship between purity and the axiomatic method.

The discussion is structured as follows. First, the epistemic function of purity identifies a problem of foundations for which autonomous foundation constitutes a possible (though not the only) solution. At this level, the purist solution is justified by appealing to an ‘architectural’ image of theories and to the adoption of an elementary point of view in reconstruction. Second, the methodological function of purity guides axiomatic research, inducing specific modes of reconstruction. Here, the axiomatic method can show whether a purist requirement is achievable, thus operating as a tool of methodological criticism that enhances the rational evaluation of these requirements. However, this requires the adoption of an image of formal axiomatic theory according to which it maintains an ‘analogy’ with the elementary facts of geometry; otherwise, purist requirements would be directly meaningless. This analogy constitutes the axiomatic expression of the elementary point of view.

Finally, in the absence of a more precise denomination, the concept of deep justification is introduced, understood as that which appeals to an image of the very nature of mathematics. This image corresponds to the ‘conceptual style’ à la Dedekind and the type of methodological formalism that derives from it. From this perspective, purist demands are justified by the requirement that calculation with geometric magnitudes be a result of theory, rather than a prior constitutive element. From an ex ante point of view, this tripartite division can be understood as a progressive effort to offer increasingly substantive reasons, insofar as they appeal to more basic images of knowledge. In this sense, the conceptual style functions as a kind of ultimate justification for purist demands, while the epistemic function represents an “initial motivation”. However,images of knowledge are also justified ex post, fundamentally for reasons of mathematical fertility linked to the generalisation of methods. In this inverse sense, Hilbert conceived of the conceptual style as responsible for the great mathematical advances of the 19th century, with the axiomatic method being the ‘apex’ of this style. Thus, the axiomatic realisation of the conceptual style introduces an architectural image of theories and an analogy between them and geometric intuition. A reconstruction of this type then satisfies the epistemic function: what is justified is the type of solution adopted. The thesis consequently shows the interaction between ex ante and ex post justifications, as well as their expression in Hilbert’s axiomatic practice.

COMMITTEE MEMBERS:
Presidente - 1281009 - ABEL LASSALLE CASANAVE
Interno - 1241476 - HENRIQUE ANTUNES ALMEIDA
Externo à Instituição - EDUARDO NICOLÁS GIOVANNINI
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Externo à Instituição - JOSÉ FERREIRÓS DOMÍNGUEZ