Thèse dirigée par : Aldo Brigaglia & Marco Panza.
- Aldo Brigaglia, Università di Palermo (directeur)
- Marco Panza, Université Panthéon-Sorbonne (codirecteur)
- Cinzia Cerroni, Università di Palermo
- Gaetana Restuccia, Università di Messina
- Gilles Aldon, École normale supérieure de Lyon
- Jean-Jacques Szczeciniarz, Université Paris-Diderot
- Massimo Galuzzi, Università di Milano (rapporteur)
- Dominique Tournès, Université de la Réunion (rapporteur)
Résumé :
In La Géométrie, Descartes
proposed a “balance” between geometric constructions and symbolic
manipulation with the introduction of suitable ideal machines. In
particular, Cartesian tools were polynomial algebra (analysis) and
a class of diagrammatic constructions (synthesis). This setting provided
a classification of curves, according to which only the algebraic ones
were considered “purely geometrical.” This limit was overcome with a
general method by Newton and Leibniz introducing the infinity in the
analytical part, whereas the synthetic perspective gradually lost
importance with respect to the analytical one — geometry became a mean
of visualization, no longer of construction.
Descartes’s
foundational approach (analysis without infinitary objects
and synthesis with diagrammatic constructions) has, however, been
extended beyond algebraic limits, albeit in two different periods. In
the late 17th century, the synthetic aspect was extended by “tractional
motion” (construction of transcendental curves with idealized machines).
In the first half of the 20th century, the analytical part was extended
by “differential algebra,” now a branch of computer algebra.
This
thesis seeks to prove that it is possible to obtain a new balance
between these synthetic and analytical extensions of Cartesian tools for
a class of transcendental problems. In other words, there is a
possibility of a new convergence of machines, algebra, and geometry that
gives scope for a foundation of (a part of) infinitesimal calculus
without the conceptual need of infinity.
The
peculiarity of this work lies in the attention to the constructive
role of geometry as idealization of machines for foundational purposes.
This approach, after the “de-geometrization” of mathematics, is far
removed from the mainstream discussions of mathematics, especially
regarding foundations. However, though forgotten these days, the problem
of defining appropriate canons of construction was very important in
the early modern era, and had a lot of influence on the definition of
mathematical objects and methods. According to the definition of Bos
[2001], these are “exactness problems” for geometry.
Such
problems about exactness involve philosophical and psychological
interpretations, which is why they are usually considered external to
mathematics. However, even though lacking any final answer, I propose in
conclusion a very primitive algorithmic approach to such problems,
which I hope to explore further in future research.
From
a cognitive perspective, this approach to calculus does not
require infinity and, thanks to idealized machines, can be set with
suitable “grounding metaphors” (according to the terminology of Lakoff
and Nuñez [2000]). This concreteness can have useful fallouts for math
education, thanks to the use of both physical and digital artifacts
(this part will be treated only marginally).