by Bill Bryson
Scientific rationalism, then, as it emerged to challenge the old system, placed its hopes not in logic but in mathematics. Whereas the old system’s working hypothesis had been that all physical processes are striving toward an end they seek to accomplish, the working hypothesis of the new rationalists was that all physical processes have a quantitative structure, and it is this abstract structure that distils the laws of nature that provide their explanation. As the über-rationalist Spinoza was to express it:
Thus the prejudice developed into superstitions, and took deep root in the human mind; and for this reason everyone strove most zealously to understand and explain the final causes of things; but in their endeavour to show that nature does nothing in vain, i.e. nothing which is useless to man, they only seem to have demonstrated that nature, the gods, and men are all mad together … Such a doctrine might well have sufficed to conceal the truth from the human race for all eternity if mathematics had not furnished another standard of verity in considering solely the essence and properties of figures without regard to their final causes.7
But what of the new empiricism? How was it in opposition to the old system? Aristotle may not himself have thought much of mathematics, but he was himself an empiricist, who took observation, most especially of biological organisms, very seriously; it was his mathematical-maniacal teacher, Plato, who dismissed sense-data (and many of those in the Copernicus–Kepler–Galileo camp were neo-Platonists). But Aristotle and the grand cathedral of thought that was erected around him advocated a passive form of observation. Nature, working always with its own ends in view, the very ends which provide the explanation in terms of final causes, was not to be interfered with. Teleology trumped technology. The very windingness of the roads of Europe’s medieval cities testifies to the old system’s hands-off approach toward nature. These roads were laid out on paths the rain took as it rolled down inclines. To transpose our own pathways over nature’s choices was a violation of the fundamental assumption of the old system. One must respectfully observe the motions of nature, since their course had been plotted by their implicit end states, and it is in the hands-off observation that the explanation emerges.
The new empiricism, in seeking its non-teleological form of explanation, took an aggressively interventionist attitude toward observation. In doing so it not only asserted its rejection of Aristotelianism, of the teleology that dictated passive observation; its new active observation, in the form of experimentalism, claimed to present a new science, a scientia operativa, that could supplant the old.
The empiricist Bacon, just like the rationalist Galileo, believed that the experience we are presented with does not reflect nature as it is: ‘For the mind of man is far from the nature of a clear and equal glass, wherein the beams of things should reflect according to their true incidence; nay, it is rather like an enchanted glass, full of superstition and imposture, if it be not delivered and reduced. For this purpose, let us consider the false appearances that are imposed upon us by the general nature of the mind …’
Bacon’s solution to how to circumvent these false appearances, which he called the ‘idols of the cave’, lay in his empirical activism. We are not to stand passively by as submissive observers of what nature might offer of itself, but assert ourselves in the gathering of facts through experiment. This assertion is what transforms sense-data, subject to illusion, into facts. The keen but passive gazing that makes sense under the assumptions of teleology made no sense to Francis Bacon.
The Lord Chancellor’s metaphors are telling. Nature should be looked on as an uncooperative witness in a courtroom, who must be interrogated and even tortured in order that the information be extracted. Nature should be treated as a slave who must be ‘constrained’ and ‘moulded’ and compelled to serve man. We must ‘shake her to her foundations’. In short, we force the sense-data to yield up the factual data that nature is actively keeping from us by asserting our own active power over nature in controlled experiments.8 (Although sometimes these experiments end in nature asserting its power over us: the legend is that Francis Bacon died after contracting pneumonia while undertaking some experiments in the dead of winter on the preservation of meat by freezing.)
Thus for both the new rationalists and the new empiricists there was a veil of subjectivity separating the observer from the observed. In this way the two orientations, no matter how distinct their intellectual temperaments, shared a central attitude that went beyond their mere opposition to the old system and explains why they were, even if rivals, also potential allies. Both insisted, against the old system, on more assertiveness. Mathematics, as opposed to inert logic, inserted a generative power into physical description. Experiments, as opposed to passive observation, allow us to wrest the physical facts from illusory experience.
The old system had seen nature as eminently readable by us. The form of explanation spread throughout the cosmos was one which was familiar and natural to us; after all, it was an essentially human form of explanation, taking the sort of explication that applies to human actions and generalising it. The old system saw us as of the universe. There was no reason to suspect our experience, and Aristotle was an unguarded empiricist, an observer who never seemed to worry about what his own mind might be contributing to perception. But not so the post-teleology Baconian empiricist, no more than the post-teleology Galilean rationalist. For both, the experience we have of the world has to be subjected to special treatment in order for reliable information to be extracted.
OF ENDS AND MMEANS
The activist empiricism of Bacon was correlated with a practical stance toward scientific knowledge, which blazed forth into utopian zeal:
I humbly pray … that knowledge being now discharged of that venom which the serpent infused into it, and which makes the mind of man to swell, we may not be wise above measure and sobriety, but cultivate truth in charity … Lastly, I would address one general admonition to all; that they consider what are the true ends of knowledge, and that they seek it not either for pleasure of the mind, or for contention, or for superiority to others, or for profit, or fame, or power, or any of these inferior things; but for the benefit and use of life; and that they perfect and govern it in charity. For it was from the lust of power that the angels fell, from lust of knowledge that man fell; but of charity there can be no excess, neither did angel or man ever come in danger by it.9
Here, too, on this question of the ‘true end of knowledge’ a temperamental difference parts the new rationalists and empiricists. A Galileo or Descartes would not have been as inclined to archly dismiss ‘pleasure of the mind’ or ‘lust of knowledge’ as Bacon had been. Though the scientific rationalists and scientific empiricists might share the belief that experience must be subjected to special treatment to be rendered profitable for science, they had differing views on the profit of science. The experimental/empiricists (Gilbert, Harvey) tended to agree with Bacon’s practical goals. As men must experimentally assert their power over nature, so, too, the value of possessing nature’s secrets was that they be utilised for the practical improvement of men’s lives. For the mathematical/rationalists the knowledge was sufficient unto itself, a thing deserving to be desired, whether it yielded practical improvements or not.
By 1660, the mathematical understanding of physical explanation could not be ignored, not with the work of people like Copernicus, Galileo and Descartes; and the men who came together to form a Colledge for the Promoting of Physico-Mathematicall Experimentall Learning acknowledged the mathematical conception of the physical in their self-designation. Nevertheless by temperament these early men of the Royal Society were more allied with Bacon, Gilbert and Harvey than with Galileo and Descartes. It was the ‘experimentall learning’ that most engaged them, and so, too, they were inclined to embrace the practical humanitarian goals of science that Bacon had linked with his experimentalism.
Christopher Wren gave the inaugural lecture at Gresham College, after the Royal Society had been officially forme
d in 1662, and in his address he spoke passionately of the manner in which the new thinking had thrown off the tyranny of the old system of thought, bringing in its stead the freedom of scientific investigation. In the course of his celebratory advocacy he extolled William Gilbert (chastised by Galileo for his lack of geometry) as the very embodiment of the new science:
Among the honourable Assertors of this Liberty, I must reckon Gilbert, who having found an admirable Correspondence between his Terrela, and the great Magnet of the Earth, thought, this Way, to determine this great Question, and spent his studies and Estate upon this Enquiry; by which obiter, he found out many admirable magnetical Experiments: This Man would I have adored, not only as the sole Inventor of Magneticks, a new Science to be added to the Bulk of Learning, but as the Father of the new Philosophy.
But if any thinker hovered as a guiding spirit over the group it was the thoroughly empiricist Francis Bacon. Bacon had dreamed of a science that would operate in the way of a collaboration, a ‘Fellowship’ to take the place of individual geniuses working in isolation; it was all of a piece with his utopian ambitions for the new knowledge, and the members of the Royal Society called themselves ‘Fellows’ in homage to the Lord Chancellor’s vision.
And yet intimations of a union between the ‘physico-mathematicall’ and ‘experimentall’ there had no doubt been. It is in the chemist Robert Boyle, the most important scientist among the twelve original Fellows, that we can see the two approaches groping somewhat dazedly toward one another. Boyle was certainly, in many ways, a disciple of Bacon – but not in all ways. He preserved an interest in the practical control of nature through knowledge of cases, which had been such a prominent feature in Francis Bacon, and which both men regarded as closely related to the empirical method; and yet he also had been touched by the Galilean spirit. Though not himself a profound mathematician, Boyle was keenly aware that astronomy and mechanics had outstripped chemistry. He was eager to carry chemistry forward by allying it with an atomistic interpretation of matter, and he recognised that mathematics was integral to the atomistic interpretation of physical phenomena.
But he also contended that chemistry, in its vigorous experimentalism, had something to teach the fields of astronomy and mechanics that had been so transformed by its mathematical reconfiguration. These latter endeavours ‘have hitherto presented us rather a mathematical hypothesis of the universe than a physical, having been careful to show us the magnitudes, situations, and motions of the great globes, without being solicitous to declare what simpler bodies, and what compounded ones, the terrestrial globe we inhabit does or may consist in’.10
Boyle’s suggestion is that the new science, as understood by Galileo et al., is all very well and good, but that, in its overly abstract mathematical demonstrations and idealised formulations, it had travelled too far in the direction of apriorism. Robert Boyle is proposing that chemistry, though lagging behind on the theoretical side, might yet have something to offer the fledgling methodology in the way of getting one’s hands stained with the stuff of ‘the terrestrial globe we inhabit’. His distinction between mathematical and physical hypotheses is important, and we shall see it again. It reveals Boyle’s intuition that there was still something missing in the systems of Galileo and Descartes, no matter how impressive they were.
It is relevant that Boyle was a chemist. The example of the alchemists, though they strayed too near to mysticism and magic for Boyle’s taste, was not purely negative, for they had defied the old system’s passivity toward nature. (Bacon, too, had praised alchemy as a scientia operativa.)
But though Boyle seemed to have sensed the presence of a unified methodology binding together the activist approaches of the new rationalism and new empiricism, he does not manage to bring it forth, perhaps because he himself lacked mathematical muscle.11 The best that he can offer is a reconciliation wrought by relativism: if what one is after is knowledge of nature then quantitative deductions on the model of Galileo and Cartesianism will yield satisfaction; but if one’s aim is control of nature in the interest of particular ends, the necessary relations can often be discovered between qualities immediately experienced or drawn forth from experiments. It all depends on what one wants out of one’s science, he writes, although the implication is that true knowledge, if that’s what one wants, will require something more deductive than experimental.
The true blending of the two rivals for replacing the teleological understanding of explanation finally arrived in a work whose very title is telling: Philosophi$$ Naturalis Principia Mathematica, The Mathematical Principles of Natural Philosophy. With Isaac Newton, a scientist who saw mathematics as essential to physical understanding had entered the ranks of the Royal Society. And yet the experimental aspect is also of fundamental importance to his methodology.
Newton observes in his preface to the Principia that ‘all the difficulty of philosophy seems to consist in this – from the phenomena of motions to investigate the forces of nature, and then from these forces to demonstrate the other phenomena’. The phrase to ‘demonstrate the other phenomena’ reiterates the message of the work’s title: the fundamental place of mathematics in Newton’s method:
We offer this work as mathematical principles of philosophy. By the propositions mathematically demonstrated in the first book, we then derive from the celestial phenomena the forces of gravity with which bodies tend to the sun and the several planets. Then, from these forces, by other propositions which are also mathematical, we deduce the motions of the planets, the comets, the moon, and the sea.
As it was for Aristotle, so it was for Newton: to investigate nature is to investigate motions. Only, of course, Newton has inherited Galileo’s transformed conception of motion, reconfigured by, and restricted to, mathematical expression. The mathematical imagination of Newton, surpassing that of Galileo or Descartes, made possible the mathematical absorption of far vaster reaches of physical phenomena. The language of the Book of Nature is not confined to geometry, as it had been according to Galileo and Descartes; rather it is analysis that becomes the more important means of expressing what is physically relevant. His invention of fluxional calculus afforded him a powerful tool whose operations could not be fully represented geometrically. On the question of mathematical type, Newton is pragmatically flexible, writing in his preface to the Principia, ‘For you may assume any quantities by the help whereof it is possible to come to equations; only taking this care, that you obtain as many equations from them as you assume quantities really unknown.’
But Newton follows as much in the footsteps of Bacon, Gilbert and Harvey, as in those of Copernicus, Kepler, Galileo and Descartes. This is most sharply brought home by his reiterated denunciation of ‘hypotheses’. By hypothesis, Newton means empirically unattached claims about reality, and by his emphatic rejection of ‘hypotheses’, he is emphasising the necessity of tying scientific statements down to experience. Unlike Galileo or Descartes, Newton distinguishes between mathematical truth and physical truth (echoing the intuition in Boyle’s complaint against the rationalists). That the resistance of bodies is in the ratio of the velocity, ‘is more a mathematical hypothesis than a physical one’, he says in Principia II, 9, and makes similar statements in connection with his discussion of fluids (Principia, II, 62). A mathematical truth that has not been made manifest in experience has not advanced to a physical truth. And experience must be experimentally manipulated in order for the mathematical truth to be made manifest in it. Galileo and Descartes were right that the mathematical structure that is latent in physical processes provides their explanation; but Bacon, too, had been right that nature requires prodding by way of experimentation in order for the mathematical and the physical to be rendered one.
In fact – and here is where the two anti-Aristotelian strains are finally brought together – it is precisely because ultimate explanation is mathematical, and this mathematical structure is not immediately given up in passively observed nature, that experimentation i
s necessary. The explanation of the motion is to be found in uncovering the mathematical structure within it; but experience as such does not readily give up the latent mathematical structure. Experiments are necessary to tease out the implicit mathematics, whose consequences can then be mathematically drawn, leading to further mathematical conclusions that must again be tied down to experience by way of experiment.
Newton’s work on optics is as instructive as his mechanics, demonstrating both the fundamental place of mathematics and the necessity for experiment. His eagerness to reduce yet another sphere of phenomena to mathematical formulae results in a science of colours. And yet mere observation could not have given Newton the phenomena that would yield to mathematical formulae. His famous interventions – for example, placing two prisms within the path of a light beam, one that would split white light into the spectrum, the other that would reconstruct white light out of the spectrum – were as essential to the science as the resultant mathematical equations. To paraphrase Immanuel Kant (who was three years old when Newton died in 1727): Experimentation without mathematical explanation is blind; mathematical explanation without experimentation is empty.
UNREASONABLE EFFECTIVENESS
Looking back now, there seems something almost accidental about the emergence of both the new rationalism and the new empiricism as coevals, each offering a rival substitute for the disputed teleology of the old system, each appealing to different sorts of intellects, tending toward divergent opinions as regards the ultimate worth and purpose of knowledge. All these centuries later, the methodological amalgamation can still call forth our wonder – most memorably expressed by the late physicist and Nobel laureate Eugene Wigner, in the phrase ‘the unreasonable effectiveness of mathematics in the physical sciences’.
