The Value of Science - Mathematical Physics

Mathematical Physics

In the second part of his book, Poincaré studies the links between physics and mathematics. His approach, at once historical and technical, illustrates the preceding general ideas.

Even though he was rarely an experimenter, Poincaré recognizes and defends the importance of experimentation, which must remain a pillar of the scientific method. According to him, it is not necessary that mathematics incorporate physics into itself, but must develop as an asset unto itself. This asset would be above all a tool: in the words of Poincaré, mathematics is "the only language in which 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. The primary objective of mathematical physics is not invention or discovery, but reformulation. It is an activity of synthesis, which permits one to assure the coherence of theories current at a given time. Poincaré recognized that it is impossible to systematize all of physics of a specific time period into one axiomatic theory. His ideas of a three dimensional space are given significance in this context.

Poincaré states that mathematics (analysis) and physics are in the same spirit, that the two disciplines share a common aesthetic goal and that both can liberate humanity from its simple state. In a more pragmatic way, the interdependence of physics and mathematics is similar to his proposed relationship between intuition and analysis. 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. Mathematics demonstrates the extent of the specific and limited discoveries made by physicists. On the other hand, physics has a key role for the mathematician - a creative role since it presents atypical problems ingrained in reality. In addition, physics offers solutions and reasoning - thus the development of infinitesimal calculus by Isaac Newton within the framework of Newtonian mechanics.

Mathematical physics finds its scientific origins in the study of celestial mechanics. Initially, it was a consolidation of several fields of physics that dominated the 18th century and which had allowed advancements in both the theoretical and experimental fields. However, in conjunction with the development of thermodynamics (at the time disputed), physicists began developing an energy-based physics. In both its mathematics and its fundamental ideas, this new physics seemed to contradict the Newtonian concept of particle interactions. Poincaré terms this the first crisis of mathematical physics.