The Lawless Frontier

Edited by Nancy Cartwright and Keith Ward, Rethinking Order: After the Laws of Nature is a collection of essays about the philosophy of science. Their book is an extended argument, Ward writes, that “the classical Newtonian worldview of a universe made of material particles governed by absolute and unbreakable laws of nature is obsolete.”¹ It is obsolete because there are no absolute and unbreakable laws of nature. Isaac Newton is just there for the fun of it. Cartwright and Ward have made an effort to assign a unity of purpose to these essays, but with only limited success. The book that results contains eleven chapters. The intellectual demands of some chapters have required more than one author for their satisfaction.

In a lecture entitled Der Wissenschaft als Beruf (Science as a Profession), Max Weber assigned to modern science an important role in the disenchantment of the world (enzauberung).² The root of the word enzauberung is zauber, which means magic or charm, but enchantment—verzauberung—does not convey in German quite the sparkle that it does in English.

German forests do not invite loitering.

Magical thinking is, on Weber’s view, an ancient attitude toward nature:

Ye elves of hills, brooks, standing lakes and groves,
And ye, that on the sands with printless foot
Do chase the ebbing Neptune, and do fly him
When he comes back; you demi-puppets that
By moonshine do the green sour ringlets make,
Whereof the ewe not bites.³

It stands opposed to what Weber called Intellektualisierung und Rationalisierung.⁴ These are words that suggest an insistent need to explain things logically, often when the occasion demands of a man that he take off his shoes and go squish the grapes. Whatever they suggest, they mean something else: Sondern sie bedeutet etwas anderes. There are no hidden and incalculable powers playing behind the flux and fleen of things: keine geheimnisvollen unberechenbaren Mächte gebe, die da hineinspielen. The controlling word is berechen—to calculate. What an odd prophetic word! A disenchanted world, Weber goes on to say, is one in which things can be mastered by calculation: daß man vielmehr alle Dinge—im Prinzip—durch Berechnen beherrschen könne. The controlling word is beherrschen, a manly kind of mastery, as the herr in beherrschen might suggest. This is the world of science and bureaucracy. The power that it yields is infinite.

Im Prinzip, natürlich.

It has been fashionable, ever since the appearance of Weber’s lecture, to think of the sciences as if their chief function were hygienic. Only a single affricate separates zauber from sauber. Weber drew no aesthetic or moral conclusions from the cosmic cleanup. It revealed what it did. The universe is bleak, boring, barren, and bloated. Anglo-American philosophers adore it anyway. So do science writers. Physicists have been known to offer a few reservations about the way in which the thing has been described:

When one looks at the succession of blatantly ad hoc moves made in QFT [quantum field theory] (negative-energy sea of electrons, discarding of infinite self energies and vacuum polarizations, local gauge invariance, forcing renormalization in gauge theories, spontaneous symmetry breaking, permanently confined quarks, color, just as examples) and of the picture which emerges of the “vacuum” (aether?), as seething with particle-antiparticle pairs of every description and as responsible for breaking symmetries initially present, one can ask whether or not nature is seriously supposed to be like that [emphasis original].⁵

Physical theories, it would seem, are almost as fairy-filled as German forests. They are positively fey. If magic is magic, what works, works. Fair is fair. The physicist’s variety of magic has, at least, the merit of working. Whether it works for any good reason that we are able to discern is another question.

Nancy Cartwright is a well-known philosopher of science. In her essay, “The Dethronement of Laws in Science,” Cartwright argues in favor of what she calls a dappled worldview.⁶ As a dappler, she is opposed to fundamentalism in the sciences:

A fundamentalist believes in something rather ultimate and mysterious; not God, of course, but something that nevertheless “governs” the whole universe, from top to bottom. What she believes in is the fundamental law(s) of nature. These are what physics has been seeking, and getting closer and closer to actually grasping, since the time of Descartes. They are truths, expressible in mathematical language, that accurately describe the behavior of all things in the physical world, at all times and places. This view has been standard among physicists, and most philosophers of science, for at least a hundred years.⁷

This is very dramatic, and suggests, if it suggests anything at all, a fruity version of Alfred Tennyson’s Ulysses. Striving and seeking are, after all, not too far from getting and grasping. Ballpark, as literary critics like to say. For all that, it is by no means clear what the fundamentalist takes to be fundamental, and for this reason, by no means clear what the denial of fundamentalism entails. The fundamental laws of nature can only govern the universe from top to bottom if the bottom is somehow in charge of the top.

Bottoms up.

No physical theory allows this conclusion to be deduced, nor is it an experimental fact.

An appeal to the fundamental laws of nature might suggest that those laws are fundamental in virtue of their place in some logical structure. Accepted as axioms, everything follows from them while they follow from nothing. Kurt Gödel’s incompleteness theorem rules this out. There are no such laws.⁸ If the fundamental laws of physics contain the arithmetic of the natural numbers, they cannot be fundamental, and if they are fundamental, they cannot contain the arithmetic of the natural numbers. This leaves open the possibility that nature is simpler than arithmetic. In a moment of fantasy, Richard Feynman once entertained this possibility.

No one is betting on it.

Cartwright’s animadversions with respect to fundamentalism are contingent on the argument that the fundamental laws of nature are false. She has, indeed, written a book entitled How the Laws of Physics Lie. This is not the happiest of titles. Nor does it convey the happiest of arguments. That Burmese women are short does not imply that no women are tall. Neither physicists nor logicians, it is quite true, have provided a coherent account of quantum field theories. They remain insusceptible to precise logical analysis. Axiomatic versions are inadequate, and adequate versions are not axiomatic. Hardly anything is what it seems. Those skeptical about fundamentalism for logical reasons may repair to un buffet à volonté. Cartwright has no intention of breaking bread with them. Her argument is, on the contrary, a masterpiece of primitive simplicity. The model in which the fundamental laws of nature are satisfied, she argues, is not real, and in the real world, they are not satisfied.

Let me see. Newtonian mechanics treats of a system of points, with masses m_i and radius vectors ${\mathbf{r}}_i\in E^3$. The equations of motion are of the form $m_i{\mathbf{r}}_i={\mathbf{F}}_i$, for i = 1, 2, 3, …, n. A Newtonian equation $m_i{\mathbf{r}}_i={\mathbf{F}}_i$ is true in some model M only if there is a force in M corresponding to ${\mathbf{F}}_i$. This is existential generalization in action. In a system of n interacting Newtonian particles open to external forces, ${\mathbf{F}}_i=\sum {\mathbf{F}}_{i,j}+{\mathbf{F}}_i^e$. The resulting force ${\mathbf{F}}_i$ is the vector sum of the internal forces, $\sum {\mathbf{F}}_{i,j}$, acting on the system and its external forces, ${\mathbf{F}}_i^e$.

Cartwright disputes none of this. And for every good reason; it is a textbook trot. Nor does she doubt that the existence of ${\mathbf{F}}_i$ follows inexorably from $m_i{\mathbf{r}}_i={\mathbf{F}}_i$. If three plus seven equals ten, there is some number, bold as brass, such that it is the sum of three and seven. Cartwright has divided her scruples so that they encompass the components, but not the resultant, of vector addition. One and the same object may be controlled both by Newton and Coulomb’s law, she observes, but while the resultant force, ${\mathbf{F}}_i$, is real enough, the component forces $\sum {\mathbf{F}}_{i,j}$ and ${\mathbf{F}}_i^e$ are imaginary. “Nature does not perform vector addition.” If nature does not perform vector addition, she could not be much inclined to scalar addition either. Of two fat men weighed simultaneously, only their sum is real, circumstances that suggest that radical weight loss is possible to those prepared to endure the embarrassment of simultaneous measurement. In this way lies madness. Existential generalization is polyvalent. If a + b = c there are then numbers x, y and z such that x + y = z. Cartwright has offered the reader no reason to suppose that this logical law fails in the case of component forces.

And, indeed, it doesn’t.

“If you wish to kill the King,” Omar Little argued in The Wire, “you’d best not miss.”

Eric Watkins is a professor of philosophy at the University of California in San Diego. He opens his essay, “The Rise and Fall of Laws of Nature,” with a strophic appreciation of order:

It is a basic fact, if not about the world itself, then at least about us, that order is crucial. Whatever our specific commitments may be, and they obviously vary considerable, we are deeply committed to some notion of order. Whether that is a natural order, a moral order, or a divine order, we firmly hold onto the idea that reality is, must or, at the very least, ought to be ordered in some way. The contrast concept—disorder or, if taken to the extreme, chaos—has undeniable negative connotations for us, as is clear from the dismay caused by the inconvenient placement of an “out of order” sign. This kind of negative attitude toward the lack of order is common, not only in the mass of papers strewn across my desk, but also in the fundamental structure of our social lives, since chaos and disorder are destructive of our most basic abilities and opportunities, and the corresponding demand for “law and order” is undeniable even for the most liberal-minded person.⁹

The inclusion of an out of order sign in this list is evidence, if any were needed, that only a sibilant separates the cosmic from the comic side of things. Watkins is, nevertheless, a writer able inexpertly to express a certain unease. “Creation itself,” James Parker observed in writing about Beowulf, “and the fact that it makes sense, is unbearable to Grendel.”¹⁰ Grendel is the poem’s alarmingly human monster. What may well be unbearable to Grendel is the fact that anyone should think optimistically to deny the senselessness of creation. I do not know the interpretative strategies Watkins may have followed in meditating about order. His specialty is Kant. There is something in his paragraph that nonetheless suggests that, like the Christian faith in ninth-century Denmark, fundamentalism in science commands allegiance in proportion to the anxieties it allays.

Or defers.

The idea of a unique set of physical laws governing the universe from top to bottom is curiously divided. Logicians very naturally think of the laws of nature in their own familiar terms. Theories are sets of sentences, and sets of sentences are true in the structures that satisfy them. Satisfaction does not endow a theory with agency. Cayley’s theorem discloses the connection between groups and permutations. That is all that it does. The laws of nature seem remarkably pressed by their administrative obligations. No sooner have they accelerated an errant astronaut toward the center of the earth than they are concerned to run up the tides in Kuala Lumpur. What a life! If they are busy governing the universe, the universe is busy being governed. Submission is obligatory. Governance and submission are not normally counted among the scientific concepts.

How might a set of sentences have acquired a universal whip?

The more that the logician gains the right to say what the laws of nature are, the less they seem capable of running anything. “What breathes fire into the laws of nature?” Stephen Hawking once asked, in an unexpected access of elegance. Something must, otherwise they would just sit there on the printed page—a consent decree, at best, but one without power of enforcement. With the advent of quantum mechanics, Watkins observes, an element of chance enters into various calculations. If God is not busy throwing the dice, He is doing little to interrupt their use. This is not quite what Weber envisaged under the aspect of scientific sanitation. What remains of a disenchanted universe if things in the world are, as Watkins implies, more hit and less miss, or the reverse? The question is sociological, not scientific, but it cannot easily be dismissed. That we live in a disenchanted world is a contemporary article of faith. No miracles. Nothing beyond the evidence. Nothing fishy. Science commentators in their thousands have expressed their commitment to this view. Their inane gabble fills the internet. The more virile of them are ill-disposed to take their own grandmothers on faith. Or on anything else.

From what madrassa are their replacements apt to come?

Robert Bishop and Roman Frigg, in their contribution to this volume, identify order with systems, like Newtonian mechanics, that are both deterministic and predictable. One-stop shopping. We know what makes a system predictable—it is a prediction. What about deterministic? “A law is deterministic,” they argue, “if it uniquely determines the evolution of a system.”¹¹ If this is intended as a definition, it is circular, and if it is intended as an aside, it is inconclusive. The complex function $f\left(z\right)=\sqrt{z}$ is deterministic, but not unique. Bishop and Frigg are beyond the niceties of delicate definitions. They have a big picture in mind. Laplace’s Demon offers them a cinemascopic perspective:

Nous devons donc envisager l'état présent de l'universe comme l'effet de son état antérieur, et comme la cause de celui qui va suivre. Une intelligence qui pour un instant donné connaîtrait toutes les forces dont la nature est animée et la situation respective des êtres qui la composent, si d'ailleurs elle était assez vaste pour soumettre ces données à l'analyse, embrasserait dans la même formule les mouvements des plus grands corps de l'universe et ceux du plus léger atome; rien ne serait incertain pour elle, et l'avenir comme le passé serait présent a ses yeux.¹²

Like so many other philosophers, Bishop and Frigg find themselves unable to sit through the screening without fidgeting. They discount the competence of une vaste intelligence because we cannot, after all, know the initial condition of Newton’s laws all that precisely. This is attacking Laplace on the wrong grounds. His demon is like Brunhilde: she is not impeded by trifles. It would be better to ask what model of inference might be adequate to her task? There are $2^{{\aleph }_0}$ initial states for Brunhilde to consider in a Newtonian system; and as many inferences into the future for her to conduct. No finite formalized language contains $2^{{\aleph }_0}$ sentences.

Let the sceptics scoff. Newton and Brunhilde were plainly meant for one another. Newtonian mechanics is another story, a system that is not what it seems. Newton’s laws of motion permit a beam of light to be infinitely accelerated along a trajectory T. Who’s to stop it? No matter the boundary D of the observable universe, it follows that there is some velocity v such that D < v × t.

So much for going out. What of coming in? If at a given time, a beam of light has crossed the observable universe and entered into the bosom of infinity, then at the very same time, its negative doppelgänger may be observed entering the universe from nowhere, uncaused, undetermined, unexplained. The philosopher of science John Earman has called such visitors “space invaders.”¹³

Nice.

Together with the other authors in this volume, Bishop and Frigg write under the impression that chaotic systems have come to undermine the intellectual serenity afforded by Newtonian mechanics. For a time, systems sensitive to their initial conditions enjoyed a precarious perch in the popular imagination. A butterfly in Brazil, it was reported, might by flapping its translucent wings cause a hurricane in Bermuda. Entomologists were briefly regarded with anxious admiration. At least, it was hoped, they retained access to the DDT. Bishop and Frigg assimilate chaos to the broad and noble class of non-linear systems. A secret must now be imparted. No matter its claims to deterministic excellence, Newtonian mechanics is itself a non-linear system, as a glance at Newton’s law of universal gravitation would at once reveal. Nor do systems flamboyantly exhibiting a sensitive dependence on their initial conditions seem particularly wonderful. The exponential function f(x) = a^x is no one’s idea of a mathematical butterfly; its sensitivity is positively morbid.

Don’t get me wrong. Practical matters of Newtonian astronomy are fine. The spaceships are going where they are aimed. Unsupported objects do fall toward the center of the earth.

All is well.

Quantum mechanics does represent a sharp and an ineradicable break with classical physics. Schrödinger’s wave function is entirely deterministic, but its interpretation is a matter of the prevailing odds. Curiously enough, quantum mechanics exhibits nothing like sensitivity to its initial conditions. Quantum systems are as sensitive as a collocation of plumbers. How chaotic systems might emerge from them is not well understood.

In his essay, “Freedom and the Causal Order,” Tim Mawson observes that “it is very difficult to reconcile libertarian free will with the universal rule of laws of nature.”¹⁴ Quite so. It is my free will or the wallpaper—one or the other will have to go. Neither seems to be going anywhere, I must say. Like Bishop and Friggs, Mawson is persuaded that indeterminism has been rehabilitated by quantum mechanics. He, for one, welcomes his new quantum overlords: “[a]nd thus our understanding of ourselves as free in the sense we’re thinking about now fits with, rather than clashes with, what science is telling us.”¹⁵ In a dramatic, but temporary, moment of lucidity, Mawson adds that “it is not obvious quite how our universe being indeterministic can really help us be free in the relevant sense.”¹⁶

Very well. What is that relevant sense? When it comes to freedom of the will, we could all use a little help. As long as no one is looking, Schrödinger’s wave moves seamlessly across the face of time, as determined in its progression as a drum roll. Quantum indeterminism emerges on the level of the look-see, when a measurement is made. The collapse of the wave function is enough to give anyone the heebie-jeebies. But quantum indeterminism is every bit as conceptually constraining as Newtonian determinism. In any random experiment, the law of large numbers establishes that the empirical frequency of events converges in the limit to its theoretical frequency. What can occur must occur, and what might occur cannot occur.

Chaotic indeterminism is no improvement, if only because chaotic systems are not indeterminate. No philosopher is prepared to argue for freedom of will on the grounds that, as usual, she has just missed stepping into the right initial conditions. Freedom of will is a property of human and animal life that is impossible to explain, or to justify, whether the universe is determinate, indeterminate, or something in between. It is a mystery. Fundamentalism assigns to various theories the power to run the universe. As we are ruled, we are not free. The dismissal of fundamentalism rules out rulers, a good thing, I suppose. Theories go to where theories should be, and that is within the logician’s embrace. Some are true, others not. Mortuary science is less fundamental than quantum electrodynamics (QED), but less fundamental only in a relative sense. If the morticians were put in charge of the philosophy of science, who knows where QED would wind up? Theories assigning freedom of will to human or animal agents might well be in conflict with theories in physics or biology. What of it? No one cares. Our various theories are not complete. Why should they be consistent? With nothing once again in charge of the universe, the universe loses its hygienic claims, and we are returned to world that would be familiar to any thirteenth-century scholastic.

The theories are nutty; the world is what it is; and we have no real idea why it is what it is.

That seems about right.