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: Logic, which can only give certainties, is the instrument of demonstration; intuition is that of invention 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.
This asset would be above all a tool: in the words of Poincaré, mathematics is "the only language in which [physicists] could speak" to understand each other and to make themselves heard.
This language of numbers seems elsewhere to reveal a unity hidden in the natural world, when there may well be only one part of mathematics that applies to theoretical physics.
Poincaré recognized that it is impossible to systematize all of physics of a specific time period into one axiomatic theory.
The language of mathematics not only permits one to express scientific advancements, but also to take a step back to comprehend the broader world of nature.
In addition, physics offers solutions and reasoning - thus the development of infinitesimal calculus by Isaac Newton within the framework of Newtonian mechanics.
Poincaré himself remained relatively optimistic regarding the evolution of physics with respect to these severe experimental difficulties.
He had little confidence in the nature of principles: they were constructed by physicists because they accommodate and take into account a large number of laws.
On the contrary, if the principles can no longer sustain laws adequately, in accordance with experimental observation, they lose their utility and are rejected, without even having been contradicted.
He said that it was the responsibility of mathematical physics to reconstitute those principles, or to find a replacement for them (the greater goal being to return the field to unity), given that it had played the main role in questioning them only after consolidating them to begin with.
Poincaré argued that the advancement of the physical sciences would have to consider a new kind of determinism, giving a new place to chance.
In 1905, the same year as the publication of The Value of Science, Albert Einstein published a decisive article on the photoelectric effect, which he based on the work of Planck.
To this teleological problem, Poincaré responds by taking the opposite position from that of Édouard Le Roy, philosopher and mathematician, who argued in a 1905 article (Sur la logique de l'invention, "On the logic of invention") that science is intrinsically anti-intellectual (in the sense of Henri Bergson) and nominalistic.
He explains that the notion that science is anti-intellectual is self-contradictory, and that the accusation of nominalism can be strongly criticized, because it rests on confusions of thoughts and definitions.
He wishes rather to demonstrate that objectivity in science comes precisely from the fact that the scientist does no more than translate raw facts into a particular language: "(...) tout ce que crée le savant dans un fait, c'est le langage dans lequel il l'énonce".