Mathematics in science
The main event usually viewed as marking the beginning of the modern mathematical approach to science was the publication of Isaac Newton's 1687 book Mathematical Principles of Natural Philosophy (the Principia). The idea that mathematics might be relevant to science nevertheless had long precursors in both practical and philosophical traditions. Before 500 BC the Babylonians were using arithmetic to describe and predict astronomical data. And by 500 BC the Pythagoreans had come to believe that all natural phenomena should somehow be reducible to relationships between numbers. Many Greek philosophers then discussed the general concept that nature should be amenable to abstract reasoning of the kind used in mathematics. And at a more practical level, the results and methodology of Euclid's work on geometry from around 300 BC became the basis for studies in astronomy, optics and mechanics, notably by Archimedes and Ptolemy. In medieval times there were some doubts about the utility of mathematics in science, and in the late 1200s, for example, Albertus Magnus made the statement that "many of the geometer's figures are not found in natural bodies, and many natural figures, particularly those of animals and plants, are not determinable by the art of geometry". Roger Bacon nevertheless wrote in 1267 that "mathematics is the door and key to the sciences", and by the 1500s it was often believed that for science to be meaningful it must somehow follow the systematic character of mathematics. (Typical of the time was the statement of Leonardo da Vinci that "no human inquiry can be called science unless it pursues its path through mathematical exposition and demonstration".) Around the end of the 1500s Galileo began to develop more explicit connections between concepts in mathematics and in physics, and concluded that the universe could be understood only in the "language of mathematics", whose "characters are triangles, circles and other geometric figures". What Isaac Newton then did was in effect to suggest that natural systems are at some fundamental level actually governed by purely abstract laws that can be specified in terms of mathematical equations. This idea has met with its greatest success in physics, where for the past three centuries essentially every major theory has been formulated in terms of mathematical equations. Starting in the mid-1800s, it has also had increasing success in chemistry. And in the past century, it has had a few scattered successes in dealing with simpler phenomena in fields like biology and economics. But despite the vast range of phenomena in nature that have never successfully been described in mathematical terms, it has become quite universally assumed that, as David Hilbert put it in 1900, "mathematics is the foundation of all exact knowledge of natural phenomena". There continue to be theories in science that are not explicitly mathematical—examples being continental drift and evolution by natural selection—but, as for example Alfred Whitehead stated in 1911, it is generally believed that "all science as it grows toward perfection becomes mathematical in its ideas".