Questions like the above are inherently ambiguous and will not have an answer that satisfies everyone because of the difficulty of defining what we mean by the word ‘science’ even with the added qualifier ‘modern’. The latest issue of New Humanist has an interview with David Wootton, professor of history at the University of York and author of the book The Invention of Science, who takes a stab at it and argues that “it happened between 1572 (when astronomer Tycho Brahe saw a new star in the sky) and 1704 (when Isaac Newton drew conclusions about the nature of light, based on experiments).”
It is not my intent to quibble about dates such as these because there is little point. I did think that in the interview Wootton sets up a straw man when he says that he wants to combat the general line of thinking that says that “there is no such thing as the scientific revolution, that nothing important changed, that the new science of Galileo and Newton was no better than the philosophy of Aristotle”. I have read quite a bit about the history and philosophy of science and I think that that is not an accurate characterization of the literature. Most people accept that there has been progress in science from the time of Aristotle. What some of the historians argue is that it is not fair to judge the work of people in the distant past by the standards of today and conclude that what they were doing was not science. As Larry Laudan says in his book Progress and its Problems: Towards a Theory of Scientific Growthi (p. 131)
To ignore the time-specific parameters of rational choice is to put the historian or philosopher in the outrageous position of indicting as irrational some of the major achievements in the history of ideas. Aristotle was not being irrational when he claimed, in the fourth century B.C., that the science of physics should be subordinate to, and legitimated by, metaphysics – even if that same doctrine, at other times and places, might well be characterized as irrational. Thomas Aquinas or Robert Grosseteste were not merely stupid or prejudiced when they espoused the belief that science must be compatible with religious beliefs.
It is true that the word ‘scientist’ was only coined by academic William Whewell in 1834 in an effort to unify the fields of chemistry, mathematics, and physics that he worried were spreading apart. But its introduction was not universally accepted, especially in the UK where the older labels of ‘natural philosopher’ and ‘man of science’ were preferred. The word gained much wider acceptance when the prestigious science journal Nature announced in 1924 that it would allow contributors to use it. The word quickly caught on, with the Royal Society of London, the British Association for the Advancement of Science, the Royal Institution, and the Cambridge University Press adopting it.
But Edward Grant in his book Science and Religion 400 BC-AD 1550: From Aristotle to Copernicus (p. 21) argues that the use of the word ‘science’ as a unifying concept appeared much earlier than the word ‘scientist’.
Thus, the medieval Latin term scientia was used for mathematical astronomy geometric optics, music or harmonics, and mechanics, especially statics, which was known as the “science of weights” in the Middle Ages. When these disciplines are discussed, we are justified in translating scientia as science. Contrary to commonly help opinion, the word science was not first used in the nineteenth century but was first employed, in a limited sense and in its Latin form in the late Middle Ages.
It is possible to identify to origins of specific features of scientific practice more narrowly. For example, the testing of hypotheses seems to have begun around the time of Galileo. But trying to pin down when science itself began is probably a pointless exercise.