The Value of Science - Intuition and Logic

Intuition and Logic

The first part of the book deals exclusively with the mathematical sciences, and particularly, the relationship between intuition and logic in mathematics. It first examines which parts of science correspond to each of these two categories of scientific thought, and outlines a few principles:

• What we define as intuition changes with the course of time (Classical philosophers were seen as logicians in their time, but today we might think of them as using intuition) – it is therefore the ideas that change, in the evolution of scientific thought;
• This evolution began with the arithmetization of analysis, and ended with the revival of intuitive ideas in an axiomatic system, by the first (true) logicians.

This historic intuition is therefore mathematical intuition. For Poincaré, it is a result of the principle of least effort, that is, of a link to scientific convention based on experimentation. Convention, thus given a context, permits one to consider different theories of the same problem, and subsequently make a choice based on the degree of simplicity and usefulness of explanations advanced by each of these theories (see also Occam's razor). The example chosen by Poincaré is that of three-dimensional space. He shows how the representation of this space is only one possibility, chosen for its usefulness among many models that the mind could create. His demonstration rests on the theory of The Mathematical Continuum (1893), one of Poincaré's earlier publications.

Finally, Poincaré advances the idea of a fundamental relationship between the sciences of geometry and analysis. According to him, intuition has two major roles: to permit one to choose which route to follow in search of scientific truth, and to allow one to comprehend logical developments:

Moreover, this relation seems to him inseparable from scientific advancement, which he presents as an enlargement of the framework of science – new theories incorporating previous ones, even while breaking old patterns of thought.