Logical Inferences and Visualization
Horacio Faas
Universidad Nacional de Córdoba - Argentina
horaciofaas@gmail.com
As inferred from the number of publications and academic meetings on the
subject, interest on visualization is greatly renewed nowadays in mathematical
and logic circles. Although visualization is usually understood in different
ways it is surely closely related with intuition, and it is well known that in
the beginnings of past century this one suffered rough attacks as misguiding
and not trustworthy. From then on, the reliable inferences were solely those
formalized in a language. But even after it was widely accepted the position
against intuition there were several opinions remarking its enormous
contribution to advances in precise knowledge. ("Einstein thought in images",
said Hadamard).
It cannot be refused that in daily life we make inferences without appealing
to linguistic intermediation. Even more, it seems that there exists some type
of inferences made by animal species other than human. As said above,
visualization is coming back accompanied sometimes by rigorous presentations
to arrive at, in some cases, to accept it like genuine part of demonstrations.
In this paper I show a few different approaches to non linguistic inferences,
aiming at highlighting the possibility of including diagrammatic and
heterogeneous reasoning in proofs. Examples of it should be how to justify newo
geometrical knowledge based on pictures (perhaps a "kantian" synthetic a priori
knowledge, to be criticized), and the way inferences understood as information
extraction by means of information flux presentation would be useful in proofs.
I also make some comments about a bit of the history of calculus on this
subject, going from Leibnizian diagrams to Cauchy's proof of the fundamental
theorem.