Seeing Further: The Story of Science and the Royal Society

Gödel was a strong mathematical Platonist who thought in a serious way about the notion that the entities that are the subject matter of mathematics really exist, though not in our physical universe, and that when we do mathematics we in some sense perceive those entities. An almost painfully meticulous scholar, he was well aware of Kant’s objections to Leibniz’s metaphysics, and understood that those objections would have to be dealt with in order for him to make any progress. According to his friend and biographer Hao Wang, Gödel discovered the works of Edmund Husserl (1859–1938) in the late 1950s and devoted much of the remainder of his life to studying them. He felt that Husserl had solved many, if not all, of the metaphysical problems that Gödel had set for himself, including doing away with Kant’s objections to Leibniz’s work. Husserl is prolix, prolific and infamously difficult to read (even Gödel complained of this) and so a reader of sub-Gödel IQ, eyeing a heap of Husserl translations on a table, might despair of ever putting his finger on the passages that Gödel is thinking of. Fortunately, Hao Wang did us the favour of listing the specific Husserl books that Gödel most admired. One of them is Cartesian Meditations, based on a series of lectures that Husserl delivered late in his career. In the fifth and last of these, Husserl gets around to mentioning, in an approving way, Leibniz and monads. Husserl has come round to Leibniz’s way of thinking, but he has got there by taking a different route, pioneered by Husserl, through phenomenology – the premises and development of which I’ll spare the reader. Since Gödel’s death, mathematical Platonism has come in for serious study both by philosophers such as Edward N. Zalta, a metaphysician at Stanford University, and scientists such as Max Tegmark, an MIT cosmologist. Zalta and Te gmark (like Deutsch) have been influenced by David Lewis’ work on modal realism. Beginning from different premises, they have arrived at markedly similar approaches.

None of these latter-day echoes of Leibnizian thinking has generated traceable, exact results in the same way that, for example, Newtonian mechanics was able to predict the orbit of the Moon. If such a thing happens in the future – if, for example, the practitioners of loop quantum gravity use their theory to make predictions that are verified by experiment – then credit will have to go to them and not to Leibniz, who could never have imagined such a science. It’s not the point of this chapter, in other words, to argue that Leibniz was right, much less that Newton was wrong. Leibniz was not even doing science as we now define the term. My conclusions are two. First of all, that the infamous duel between Newton and Leibniz – which was only superficially about who had invented the calculus – came back from the dead a hundred years ago to exert remarkable influence over the course of modern science. Secondly, that Leibniz’s most fundamental assumption, namely that the universe makes sense and that the human has the power to make sense of it and that, consequently, pure metaphysics is no waste of time, remains perhaps the central question of all science. In 1960, Eugene Wigner wrote a paper, The Unreasonable Effectiveness of Mathematics in the Physical Sciences, in which he addressed the nearly miraculous way in which pure mathematics – seemingly a product of human cognition, and nothing else – predicts the behaviour of the physical world. The examples cited by Wigner would have made sense to Leibniz. Leibniz, however, would have been baffled by Wigner’s use of the adjective ‘unreasonable’ in the title of his paper. Wigner was a modern: a product of a sceptical age. He was uneasy (or felt obliged to pretend to be uneasy) with the philosophical implications of the way in which the physical world answered to mathematics. This unease could not have been more alien to Leibni z, who, during his long philosophical career, questioned many things that would have been easier to leave alone, but believed, with a kind of medieval serenity, in the reasonableness of Creation.

Notes

* (#ulink_553b9565-ee29-52cd-86cc-4883e37dbea0) This perennial theological chestnut seems to have occasioned some soul-searching for Newton as well, since he risked serious trouble by semi-openly espousing the Arian heresy, which denies the Trinity.

5 REBECCA NEWBERGER GOLDSTEIN (#ulink_47b9d0e5-6fdb-50a5-bc62-fcb9a92606f8)

WHAT’S IN A NAME? RIVALRIES AND THE BIRTH OF MODERN SCIENCE

Rebecca Newberger Goldstein is a philosopher and a novelist. Her work, both literary and scholarly, has received numerous prizes, including a MacArthur ‘genius’ award. Her non-fiction works are Incompleteness: The Proof and Paradox of Kurt Godel and Betraying Spinoza: The Renegade Jew Who Gave Us Modernity. Her fiction includes the novels Properties of Light: A Novel of Love, Betrayal and Quantum Physics, Mazel, The Dark Sister and The Mind-Body Problem, as well as a volume of stories, Strange Attractors. Her latest book is 36 Arguments for the Existence of God: A Work of Fiction.

A MID THE DIVISIONS IN THOUGHT WHICH MARKED THE SCIENTIFIC REVOLUTION, THE FOUNDERS OF THE ROYAL SOCIETY INSISTED ON BINDING TOGETHER TWO CONTENDERS FOR THE BASIS OF NATURAL EXPLANATION. AS REBECCA GOLDSTEIN EXPLAINS, THERE WERE DEEP COMMITMENTS TO THE PRIMACY OF EXPERIMENTAL RESULTS, AND TO RECOGNISING UNDERLYING MATHEMATICAL PATTERNS. BUT THE REALLY POWERFUL TRICK, THEN AS NOW, LAY IN FINDING HOW TO BRING THEM TOGETHER.

After a lecture given by Christopher Wren, then the Gresham College Professor of Astronomy, twelve prominent gentlemen, deciding that they would meet weekly to discuss science and perform experiments, recorded their intention to form a ‘Colledge for the Promoting of Physico-Mathematicall Experimentall Learning’.

It might not have been the most elegant of designations, but it did, in its very wordiness, portend great things. It gave notice to the hope – because it was still, in 1660, only a hope – that two distinct orientations, one mathematical, the other experimental, would be pounded together into one coherent scientific method. The hope paid off, and it was from

A colour engraving of Gresham College, home of the Royal Society from 1660 to 1710.

within the ranks of the Royal Society that the new compound emerged. Two cognitive stances that had seemed to have little to do with one another, except in their opposition to the system of natural philosophy dominant for centuries, were rendered equally necessary in the explanation of physical phenomena.

It was a time of epistemological urgency. A grandly unifying cathedral of thought was crumbling.

(#litres_trial_promo) The all-inclusive view of the cosmos, laid down by Aristotle and buttressed by the medical theories of Galen, the astronomy of Ptolemy, and the theology of Christianity, had offered a way of explaining…absolutely everything. From the falling of objects to the rising of smoke; from generation and decay to the four basic personality types; from the relation between body and soul to the pathways of the planets; the supposed nature and reason for every aspect of the world could be extracted from an interlocking system that employed a homogeneous form of explanation throughout.

The form of explanation had been purpose-driven, or teleological, and its scaffolding was the metaphor of human action. We explain human actions by citing the end state that the agent has in mind in undertaking it. The old system took this familiar model of explanation and expanded it to apply to the world at large. ‘To be ignorant of motion is to be ignorant of nature,’ Aristotle had written, but by motions he meant not just displacements of bodies but such processes as becoming a parent, gaining knowledge, growing older. All were subsumed under the same conception: a striving to actualise an end state that was implicit in the motion and provided the explanation, the final cause, for the course that the motion took. The explanatory logic of human actions – based on intentions – was one with the explanatory logic of the cosmos.

The working hypothesis behind teleology was, of course, that all natural phenomena and processes do in fact have goals, allowing them to be viewed as potentialities on the way to being actualised. But every form of explanation makes use of some working hypothesis or other, ascribing to nature the features that allow such explanation to work. The mode of physical explanation that was to supplant teleology, making essential use of mathematics, also staked its claim on the world’s being a certain way.

We are today understandably prepared to believe that the only reasons anyone might have had to cling to the old crumbling teleological cathedral, in light of the superior science battering it, were speciously theological; and, in fact, such reasons probably did motivate most of those who clung to the old system. Still, there was nothing a priori fallacious about the old system’s assumptions about reality, just as there was nothing a priori true about the assumptions that would replace them.

The grand old system was crumbling, and it made for a capacious space into which genius could expand. When foundations fall, everything can and must be rethought. The exhilaration on display in the writings of the new scientists bears witness to how bracingly liberating such possibilities can be, at least for those with the intellectual imagination and bravado to take advantage of them. ‘You cannot help it, Signor Sarsi,’ Galileo exults in The Assayer, written in the form of a letter to a friend, ‘that it was granted to me alone to discover all the new phenomena in the sky and nothing to anybody else.’

EXPLANATION RE-EXPLAINED

And what question is more foundational than the question of what counts as a good explanation? All the great men whom we now associate with the formation of modern science – Copernicus (1473–1543), William Gilbert (1544–1603), Francis Bacon (1561–1626), Galileo Galilei (1564–1642), Johannes Kepler (1571–1630), William Harvey (1578–1657), René Descartes (1596–1650), Robert Boyle (1627–1691), John Locke (1632–1704) and Isaac Newton (1643–1727) – were intensely involved with the question of what form explanation ought to take, if teleology was truly to be abandoned, and there was by no means a consensus among them. Two different orientations emerged: one rationalist, stressing abstract reason, the other empiricist, stressing experience.

In some sense, this cognitive split was nothing new. It had made itself felt in the ancient world, in the distinction between the Platonists and the Aristotelians. It is probably as old as thought itself, shadowing two distinct intellectual temperaments. But the new rationalist and empiricist orientations were not like the old. The rationalist orientation looked to mathematics to provide the new mode of explanation. The empiricists saw the new scientific method as emerging out of experimentation. In responding to the need for a new mode of explanation to take the place of teleology, they became epistemological rivals, offering competing models to take the place of the old system’s final causes.

The men who met in Gresham College, London,

(#litres_trial_promo) 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’.

(#litres_trial_promo) 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.’

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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.

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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 conception of explanation.

A priori mathematics, according to Galileo, does not entirely obviate the need for observation (only the most extreme of rationalists, Spinoza and Leibniz, were to argue the expendability, at least in principle, of all empirical knowledge, claiming that all could be a priori deduced from first principles

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Of course, this meant that not all of the processes conceived of as motions by Aristotle were Galilean motions. Only motions susceptible to mathematical translation came under the purview of science; the rest were expelled from the possibility of physical explanation. Even more than this, Galileo, and those who followed him, defined physical nature itself in terms of mathematics. It was Galileo who first drew the distinction between primary and secondary qualities. If all aspects of physical reality are mathematically expressible, and if not all aspects of our experience are susceptible to mathematical treatment, the implication is that not all aspects of our experience are physically real. Our minds contribute to what we seem to see out there in the world. Our experience is not transparent; there is a gauzy veil of subjectivity hung between us and the objective physical world of mathematical bodies, compounded out of mathematically arranged mathematical constituents, mathematically moving through mathematical space over the course of mathematical time. All those aspects of our experience that can be rendered in mathematical language are ‘primary’ and correlated with what is out there; the rest are ‘secondary’ qualities, features of our subjective experience, caused by the interaction between the primary qualities out there and our own sensory organs. This distinction was widely accepted, not only by rationalists like Galileo and Descartes, but empiricists like John Locke. The portions of res cogitans lurking in our cerebral hemispheres provide a sanctuary for the otherwise inexplicable flotsam and jetsam of perception.

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.

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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…’