men who met in Gresham College, London, 2 had given notice, in their self-baptism, that the mathematical and experimental approaches were not only compatible but collaborative; even, as it were, one. There is
Some of the great men associated with the formation of modern science.
an important epistemological claim implicit in their stated intention to promote ‘physico-mathematicall experimentall learning’, and the claim was by no means demonstrable in 1660. The thinkers whose work inspired them could be divided into those whose stance was slanted toward the new rationalist understanding of physical explanation – Copernicus, Kepler, Galileo, Descartes – and those who espoused the experimental understanding of physical explanation – Francis Bacon, William Gilbert and William Harvey. This list suggests a geographical divide, with the rationalists on the Continent, the empiricists in England, which makes the ecumenicalism of the sources of inspiration all the more noteworthy.
The temperamental distinction between the mathematical rationalists and experimental empiricists could be, in fact, so marked that we can well wonder how these scientific founders made common cause with one another against the old system. How can such different scientific temperaments, proffering such different answers as to what a scientific explanation ought to look like, have conspired to hammer out the new methodology?
William Gilbert, for example, a luminary of the experimental approach, is acknowledged as the founder of the science of magnetism, and his experiments had been ingenious. He had carved out of a lodestone – a piece of naturally magnetic mineral – a scale model of the Earth he called his terrella, or little Earth, and with it he had been able to explain a phenomenon that had been known for centuries. A freely suspended compass needle pointed North, but later observations had revealed that the direction deviated somewhat from true North, and Robert Norman had published his finding in 1581 that the force on a magnetic needle was not horizontal but slanted into the Earth. Passing a small compass over his terrella, Gilbert demonstrated that a horizontal compass would point toward the magnetic pole, while a dip needle, balanced on a horizontal axis perpendicular to the magnetic one, indicated the proper ‘magnetic inclination’ between the magnetic force and the horizontal direction. The experiments convinced him that the Earth itself was a giant magnet. Galileo, his contemporary, commends his work, but criticises him for not being well-grounded in mathematics, especially geometry.
Galileo, for his part, could be high-handed in regard to experimentation, writing, for example, that it was only the need to convince his ignorant opponents that made him resort to ‘a variety of experiments, though to satisfy his own mind alone he had never felt it necessary to make any’. 3 As one historian of science has written, ‘If this was seriously meant, it was extremely important for the advance of science that Galileo had strong opponents, and in fact there are other passages in his works which show that his confident belief in the mathematical structure of the world emancipated him from the necessity of close dependence on experiment.’4
The two orientations, rationalist and empiricist, were partly defining themselves in opposition to one another, becoming far more adversarial now that the old system was crumbling. That system had blended together both a priori reason and empirical observation, conceiving both as codependently involved in scientific explanation. Aristotle had been a biologist, much given to observing the natural world, and the system that had grown up on Aristotelian foundations had always striven to take account of observable facts. So, for example, as more precise observations of the ‘wandering’ planets were made, a vast complexity of interacting celestial gears, the ever more torturous epicycles and eccentrics, was sketched to accommodate them into the geocentric picture which was an essential part of the old system’s teleology. In Paradise Lost, John Milton speaks of ‘Sphere/With Centric and Eccentric scribbled o’er,/Cycle and Epicycle,
Orb in Orb’. Such complexity was demanded because of ongoing observation. Aristotelians were not given to ignoring the observable facts. Quite the contrary: they observed processes so as to be able to read out of them the narratives of potentiality actualised.
Then again, Aristotle was also a logician, who had laid down the laws of the syllogism. According to Aristotle, logical demonstration, by way of the syllogism, was a necessary component of epistêmê, or scientific knowledge. In his Posterior Analytics, he says that scientific knowledge requires that we know the cause ‘of why the thing is’, and also know that it could not have been otherwise. In other words, scientific knowledge not only must discover causes but demonstrate that they are necessarily the causes, and it is the abstract science of the syllogism that is assigned the latter demonstrative role.
However, both rationalism and empiricism, as they emerged in the seventeenth century, were of an entirely different kind from their counterparts in the old system. The scientific rationalism of Copernicus, Galileo, Kepler, and Descartes had little use for the Aristotelian syllogism, which, so they argued, cannot expand our knowledge but merely rearrange it to set off implicit logical relations. Logic may be perfect, but it is also perfectly inert, incapable of moving substantive discovery forward. For the new scientific rationalists, it is not syllogistic logic but rather mathematics that holds an incomparable active power, capable of generating new knowledge. ‘We do not learn to demonstrate from the manuals of logic,’ Galileo wrote, ‘but from the books which are full of demonstrations, which are mathematical, not logical.’ A priori reason in the form of mathematics provides a methodology for discovery. As Galileo was to put it ringingly in The Assayer:
Philosophy is written in this vast book, which continuously lies upon before our eyes (I mean the universe). But it cannot be understood unless you have first learned to understand the language and recognise the characters in which it is written. It is written in the language of mathematics, and the characters are triangles, circles,
Copernicus with the sphere of the solar system in his hand.
and other geometrical figures. Without such means, it is impossible for us humans to understand a word of it, and to be without them is to wander around in vain through a dark labyrinth.
It was, more than anything else, the new mathematical conception of the physical universe that had hastened the crumbling of the old explanatory system. Copernicus had urged his heliocentric model of the solar system not on the basis of its empirical superiority – both the geocentric and the heliocentric pictures could accommodate the data – but on the basis of its mathematical superiority:
Nor do I doubt that skilled and scholarly mathematicians will agree with me if, what philosophy requires from the beginning, they will examine and judge, not casually but deeply, what I have gathered together in this book to prove these things…Mathematics is written for mathematicians, to whom these my labours, if I am not mistaken, will appear to contribute something.5
Under Galileo, the mathematical conceptualising of nature was radically advanced. He took the concept of motion, agreeing with Aristotle that it is the object of scientific explanation, and he reconfigured it into terms that can be expressed precisely in numbers. Distance travelled is quantifiable, as is time elapsed; and, from Galileo onward, motion is conceived of as a comparison between these two factors, the change of distance and the passing of time. Once motion itself had been reconfigured as a mathematical concept, other concepts, which are functions of motion, can be mathematically defined, so that, by developing the equations between the various functions of mathematical motions, new properties can be uncovered. The mathematical expression of the physical allows for what logic could never accomplish: the generation of new descriptions, going beyond the observable. It is the relations between these mathematical properties which, expressed as equations, remain constant between instances, yielding universal laws of nature. And it is these laws that supplant teleology in the new