Banca de DEFESA: DOUGLAS LISBOA SANTOS DE JESUS

Uma banca de DEFESA de DOUTORADO foi cadastrada pelo programa.
STUDENT : DOUGLAS LISBOA SANTOS DE JESUS
DATE: 06/10/2023
TIME: 14:00
LOCAL: PPGF
TITLE:

Between Logic and Rhetoric: the axiomatizations of euclidean geometry in XVI and XVII centuries

KEY WORDS:

Demonstration; Euclidean Geometry; Aristotle; Proclus; Logic; Rhetoric

PAGES: 185
BIG AREA: Ciências Humanas
AREA: Filosofia
SUBÁREA: Epistemologia
SUMMARY:

This thesis aims to present the debate on the nature of Euclidean geometric demonstrations
throughout the 16th and 17th centuries. It starts by examining the reception of the Platonic
philosopher Proclus' comments on Euclid's Elements and the tradition of demonstrative
science presented in Aristotle's Posterior Analytics . At the heart of this debate is a dispute
about the relevance of certain rhetorical concepts in light of the alleged limitations of
syllogistics in accounting for how mathematicians typically demonstrate their propositions.
The mathematical practice inferred from the Elements allows for a coordination between two
types of justifications, one provided by linguistic means and the other by diagrams, primarily
but not exclusively aimed at solving geometric problems. The establishment of a
logical-linguistic definition of demonstration, heavily influenced by David Hilbert, resulted in
the dispensability of diagrams. The axiomatizations of what was understood as Euclidean
geometry in the 16th and 17th centuries, for the most part, were embedded in Aristotle's
doctrine of demonstrative science. From this perspective, the tradition of problem-solving is
secondary. Nevertheless, Aristotle believed that demonstrations should satisfy both logical
and epistemological criteria in equal measure. Unlike 20th-century philosophy of
mathematics, Aristotle's conception of demonstration was not solely concerned with logical
consequence. However, the syllogistic reconstruction of Euclidean demonstrations was not the
primary goal, especially since Proclus raised doubts about the power of Aristotelian logic.
The crisis of the Aristotelian scientific method paved the way for the exploration of
alternative approaches, including rhetoric. Most commentators on the Euclidean text during
this period assumed that the demonstrations in the Elements were intentionally abbreviated,
allowing them to speak of enthymemes, i.e. , rhetorical syllogisms. The inclusion of new
axioms not covered in Proclus' text, many of which anticipated Hilbert's formal axiomatic
method, makes it possible to restore the complete version of the demonstration without
implying that the abbreviated version, i.e. , enthymematic, is considered illegitimate.

COMMITTEE MEMBERS:
Presidente - 1281009 - ABEL LASSALLE CASANAVE
Interno - 3310714 - JORGE ALBERTO MOLINA
Interno - 1615874 - MARCO AURELIO OLIVEIRA DA SILVA
Externo à Instituição - WAGNER DE CAMPOS SANZ - UFG
Externo à Instituição - FRANK THOMAS SAUTTER - UFSM