2596. How Perspective Shapes Our View of Reality

2343. Science, Mind, and Limits of Understanding

By Noam Chomsky, The Science and Faith Foundation (Vatican), January 2014

René Descartes (1596-1650)

One of the most profound insights into language and mind, I think, was Descartes’s recognition of what we may call “the creative aspect of language use”: the ordinary use of language is typically innovative without bounds, appropriate to circumstances but not caused by them – a crucial distinction – and can engender thoughts in others that they recognize they could have expressed themselves. Given the intimate relation of language and thought, these are properties of human thought as well. This insight is the primary basis for Descartes’s scientific theory of mind and body. There is no sound reason to question its validity, as far as I am aware. Its implications, if valid, are far-reaching, among them what it suggests about the limits of human understanding, as becomes more clear when we consider the place of these reflections in the development of modern science from the earliest days.

It is important to bear in mind that insofar as it was grounded in these terms, Cartesian dualism was a respectable scientific theory, proven wrong (in ways that are often misunderstood), but that is the common fate of respectable theories.

The background is the so-called “mechanical philosophy” – mechanical science in modern terminology. This doctrine, originating with Galileo and his contemporaries, held that the world is a machine, operating by mechanical principles, much like the remarkable devices that were being constructed by skilled artisans of the day and that stimulated the scientific imagination much as computers do today; devices with gears, levers, and other mechanical components, interacting through direct contact with no mysterious forces relating them. The doctrine held that the entire world is similar: it could in principle be constructed by a skilled artisan, and was in fact created by a super-skilled artisan. The doctrine was intended to replace the resort to “occult properties” on the part of the neoscholastics: their appeal to mysterious sympathies and antipathies, to forms flitting through the air as the means of perception, the idea that rocks fall and steam rises because they are moving to their natural place, and similar notions that were mocked by the new science.

The mechanical philosophy provided the very criterion for intelligibility in the sciences. Galileo insisted that theories are intelligible, in his words, only if we can “duplicate [their posits] by means of appropriate artificial devices.” The same conception, which became the reigning orthodoxy, was maintained and developed by the other leading figures of the scientific revolution: Descartes, Leibniz, Huygens, Newton, and others.

Today Descartes is remembered mainly for his philosophical reflections, but he was primarily a working scientist and presumably thought of himself that way, as his contemporaries did. His great achievement, he believed, was to have firmly established the mechanical philosophy, to have shown that the world is indeed a machine, that the phenomena of nature could be accounted for in mechanical terms in the sense of the science of the day. But he discovered phenomena that appeared to escape the reach of mechanical science. Primary among them, for Descartes, was the creative aspect of language use, a capacity unique to humans that cannot be duplicated by machines and does not exist among animals, which in fact were a variety of machines, in his conception.

As a serious and honest scientist, Descartes therefore invoked a new principle to accommodate these non-mechanical phenomena, a kind of creative principle. In the substance philosophy of the day, this was a new substance, res cogitans, which stood alongside of res extensa. This dichotomy constitutes the mind-body theory in its scientific version. Then followed further tasks: to explain how the two substances interact and to devise experimental tests to determine whether some other creature has a mind like ours. These tasks were undertaken by Descartes and his followers, notably Géraud de Cordemoy; and in the domain of language, by the logician-grammarians of Port Royal and the tradition of rational and philosophical grammar that succeeded them, not strictly Cartesian but influenced by Cartesian ideas.

All of this is normal science, and like much normal science, it was soon shown to be incorrect. Newton demonstrated that one of the two substances does not exist: res extensa. The properties of matter, Newton showed, escape the bounds of the mechanical philosophy. To account for them it is necessary to resort to interaction without contact. Not surprisingly, Newton was condemned by the great physicists of the day for invoking the despised occult properties of the neo-scholastics. Newton largely agreed. He regarded action at a distance, in his words, as “so great an Absurdity, that I believe no Man who has in philosophical matters a competent Faculty of thinking, can ever fall into it.” Newton however argued that these ideas, though absurd, were not “occult” in the traditional despised sense. Nevertheless, by invoking this absurdity, we concede that we do not understand the phenomena of the material world. To quote one standard scholarly source, “By `understand’ Newton still meant what his critics meant: `understand in mechanical terms of contact action’.”

It is commonly believed that Newton showed that the world is a machine, following mechanical principles, and that we can therefore dismiss “the ghost in the machine,” the mind, with appropriate ridicule. The facts are the opposite: Newton exorcised the machine, leaving the ghost intact. The mind-body problem in its scientific form did indeed vanish as unformulable, because one of its terms, body, does not exist in any intelligible form. Newton knew this very well, and so did his great contemporaries.

John Locke wrote that we remain in “incurable ignorance of what we desire to know” about matter and its effects, and no “science of bodies [that provides true explanations is] within our reach.” Nevertheless, he continued, he was “convinced by the judicious Mr. Newton’s incomparable book, that it is too bold a presumption to limit God’s power, in this point, by my narrow conceptions.” Though gravitation of matter to matter is “inconceivable to me,” nevertheless, as Newton demonstrated, we must recognize that it is within God’s power “to put into bodies, powers and ways of operations, above what can be derived from our idea of body, or can be explained by what we know of matter.” And thanks to Newton’s work, we know that God “has done so.” The properties of the material world are “inconceivable to us,” but real nevertheless. Newton understood the quandary. For the rest of his life, he sought some way to overcome the absurdity, suggesting various possibilities, but not committing himself to any of them because he could not show how they might work and, as he always insisted, he would not “feign hypotheses” beyond what can be experimentally established.

Replacing the theological with a cognitive framework, David Hume agreed with these conclusions. In his history of England, Hume describes Newton as “the greatest and rarest genius that ever arose for the ornament and instruction of the species.” His most spectacular achievement was that while he “seemed to draw the veil from some of the mysteries of nature, he shewed at the same time the imperfections of the mechanical philosophy; and thereby restored [Nature’s] ultimate secrets to that obscurity, in which they ever did and ever will remain.”

As the import of Newton’s discoveries was gradually assimilated in the sciences, the “absurdity’ recognized by Newton and his great contemporaries became scientific common sense. The properties of the natural world are inconceivable to us, but that does not matter. The goals of scientific inquiry were implicitly restricted: from the kind of conceivability that was a criterion for true understanding in early modern science from Galileo through Newton and beyond, to something much more limited: intelligibility of theories about the world. This seems to me a step of considerable significance in the history of human thought and inquiry, more so than is generally recognized, though it has been understood by historians of science.

Friedrich Lange, in his classic 19th century history of materialism, observed that we have “so accustomed ourselves to the abstract notion of forces, or rather to a notion hovering in a mystic obscurity between abstraction and concrete comprehension, that we no longer find any difficulty in making one particle of matter act upon another without immediate contact,…through void space without any material link. From such ideas the great mathematicians and physicists of the seventeenth century were far removed. They were all in so far genuine Materialists in the sense of ancient Materialism that they made immediate contact a condition of influence.” This transition over time is “one of the most important turning-points in the whole history of Materialism,” he continued, depriving the doctrine of much significance, if any at all. “What Newton held to be so great an absurdity that no philosophic thinker could light upon it, is prized by posterity as Newton’s great discovery of the harmony of the universe!”

Similar conclusions are commonplace in the history of science. In the mid-twentieth century, Alexander Koyré observed that Newton demonstrated that “a purely materialistic pattern of nature is utterly impossible (and a purely materialistic or mechanistic physics, such as that of Lucretius or of Descartes, is utterly impossible, too)”; his mathematical physics required the “admission into the body of science of incomprehensible and inexplicable `facts’ imposed up on us by empiricism,” by what is observed and our conclusions from these observations.

With the disappearance of the scientific concept of body (material, physical, etc.), what happens to the “second substance,” res cogitans/mind, which was left untouched by Newton’s startling discoveries? A plausible answer was suggested by John Locke, also within the reigning theological framework. He wrote that just as God added to matter such inconceivable properties as gravitational attraction, he might also have “superadded” to matter the capacity of thought. In the years that followed, Locke’s “God” was reinterpreted as “nature,” a move that opened the topic to inquiry. That path was pursued extensively in the years that followed, leading to the conclusion that mental processes are properties of certain kinds of organized matter. Restating the fairly common understanding of the time, Charles Darwin, in his early notebooks, wrote that there is no need to regard thought, “a secretion of the brain,” as “more wonderful than gravity, a property of matter” – all inconceivable to us, but that is not a fact about the external world; rather, about our cognitive limitations.

It is of some interest that all of this has been forgotten, and is now being rediscovered. Nobel laureate Francis Crick, famous for the discovery of DNA, formulated what he called the “astonishing hypothesis” that our mental and emotional states are “in fact no more than the behavior of a vast assembly of nerve cells and their associated molecules.” In the philosophical literature, this rediscovery has sometimes been regarded as a radical new idea in the study of mind. To cite one prominent source, the radical new idea is “the bold assertion that mental phenomena are entirely natural and caused by the neurophysiological activities of the brain.” In fact, the many proposals of this sort reiterate, in virtually the same words, formulations of centuries ago, after the traditional mind-body problem became unformulable with Newton’s demolition of the only coherent notion of body (or physical, material, etc.). For example, 18th century chemist/philosopher Joseph Priestley’s conclusion that properties “termed mental” reduce to “the organical structure of the brain,” stated in different words by Locke, Hume, Darwin, and many others, and almost inescapable, it would seem, after the collapse of the mechanical philosophy that provided the foundations for early modern science, and its criteria of intelligibility.

The last decade of the twentieth century was designated “the Decade of the Brain.” In introducing a collection of essays reviewing its results, neuroscientist Vernon Mountcastle formulated the guiding theme of the volume as the thesis of the new biology that “Things mental, indeed minds, are emergent properties of brains, [though] these emergences are…produced by principles that… we do not yet understand” – again reiterating eighteenth century insights in virtually the same words.

The phrase “we do not yet understand,” however, should strike a note of caution. We might recall Bertrand Russell’s observation in 1927 that chemical laws “cannot at present be reduced to physical laws.” That was true, leading eminent scientists, including Nobel laureates, to regard chemistry as no more than a mode of computation that could predict experimental results, but not real science. Soon after Russell wrote, it was discovered that his observation, though correct, was understated. Chemical laws never would be reducible to physical laws, as physics was then understood. After physics underwent radical changes, with the quantum-theoretic revolution, the new physics was unified with a virtually unchanged chemistry, but there was never reduction in the anticipated sense.

There may be some lessons here for neuroscience and philosophy of mind. Contemporary neuroscience is hardly as well-established as physics was a century ago. There are what seem to me to be cogent critiques of its foundational assumptions, notably recent work by cognitive neuroscientists C.R. Gallistel and Adam Philip King. The common slogan that study of mind is neuroscience at an abstract level might turn out to be just as misleading as comparable statements about chemistry and physics ninety years ago. Unification may take place, but that might require radical rethinking of the neurosciences, perhaps guided by computational theories of cognitive processes, as Gallistel and King suggest.

The development of chemistry after Newton also has lessons for neuroscience and cognitive science. The 18th century chemist Joseph Black recommended that “chemical affinity be received as a first principle, which we cannot explain any more than Newton could explain gravitation, and let us defer accounting for the laws of affinity, till we have established such a body of doctrine as he has established concerning the laws of gravitation.” The course Black outlined is the one that was actually followed as chemistry proceeded to establish a rich body of doctrine. Historian of chemistry Arnold Thackray observes that the “triumphs” of chemistry were “built on no reductionist foundation but rather achieved in isolation from the newly emerging science of physics.” Interestingly, Thackray continues, Newton and his followers did attempt to “pursue the thoroughly Newtonian and reductionist task of uncovering the general mathematical laws which govern all chemical behavior” and to develop a principled science of chemical mechanisms based on physics and its concepts of interactions among “the ultimate permanent particles of matter.” But the Newtonian program was undercut by Dalton’s “astonishingly successful weight-quantification of chemical units,” Thackray continues, shifting “the whole area of philosophical debate among chemists from that of chemical mechanisms (the why? of reaction) to that of chemical units (the what? and how much?),” a theory that “was profoundly antiphysicalist and anti-Newtonian in its rejection of the unity of matter, and its dismissal of short-range forces.” Continuing, Thackray writes that “Dalton’s ideas were chemically successful. Hence they have enjoyed the homage of history, unlike the philosophically more coherent, if less successful, reductionist schemes of the Newtonians.”

Adopting contemporary terminology, we might say that Dalton disregarded the “explanatory gap” between chemistry and physics by ignoring the underlying physics, much as post-Newtonian physicists disregarded the explanatory gap between Newtonian dynamics and the mechanical philosophy by rejecting the latter, and thereby tacitly lowering the goals of science in a highly significant way, as I mentioned.

Contemporary studies of mind are deeply troubled by the “explanatory gap” between the science of mind and neuroscience – in particular, between computational theories of cognition, including language, and neuroscience. I think they would be well-advised to take seriously the history of chemistry. Today’s task is to develop a “body of doctrine” to explain what appear to be the critically significant phenomena of language and mind, much as chemists did. It is of course wise to keep the explanatory gap in mind, to seek ultimate unification, and to pursue what seem to be promising steps towards unification, while nevertheless recognizing that as often in the past, unification may not be reduction, but rather revision of what is regarded as the “fundamental discipline,” the reduction basis, the brain sciences in this case.

Locke and Hume, and many less-remembered figures of the day, understood that much of the nature of the world is “inconceivable” to us. There were actually two different kinds of reasons for this. For Locke and Hume, the reasons were primarily epistemological. Hume in particular developed the idea that we can only be confident of immediate impressions, of “appearances.” Everything else is a mental construction. In particular, and of crucial significance, that is true of identity through time, problems that trace back to the pre-Socratics: the identity of a river or a tree or most importantly a person as they change through time. These are mental constructions; we cannot know whether they are properties of the world, a metaphysical reality. As Hume put the matter, we must maintain “a modest skepticism to a certain degree, and a fair confession of ignorance in subjects, that exceed all human capacity” – which for Hume includes virtually everything beyond appearances. We must “refrain from disquisitions concerning their real nature and operations.” It is the imagination that leads us to believe that we experience external continuing objects, including a mind or self. The imagination, furthermore, is “a kind of magical faculty in the soul, which…is inexplicable by the utmost efforts of human understanding,” so Hume argued.

A different kind of reason why the nature of the world is inconceivable to us was provided by “the judicious Mr. Newton,” who apparently was not interested in the epistemological problems that vexed Locke and Hume. Newton scholar Andrew Janiak concludes that Newton regarded such global skepticism as “irrelevant – he takes the possibility of our knowledge of nature for granted.” For Newton, “the primary epistemic questions confronting us are raised by physical theory itself.” Locke and Hume, as I mentioned, took quite seriously the new science-based skepticism that resulted from Newton’s demolition of the mechanical philosophy, which had provided the very criterion of intelligibility for the scientific revolution. That is why Hume lauded Newton for having “restored [Nature’s] ultimate secrets to that obscurity, in which they ever did and ever will remain.”

For these quite different kinds of reasons, the great figures of the scientific revolution and the Enlightenment believed that there are phenomena that fall beyond human understanding. Their reasoning seems to me substantial, and not easily dismissed. But contemporary doctrine is quite different. The conclusions are regarded as a dangerous heresy. They are derided as “the new mysterianism,” a term coined by philosopher Owen Flanagan, who defined it as “a postmodern position designed to drive a railroad spike through the heart of scientism.” Flanagan is referring specifically to explanation of consciousness, but the same concerns hold of mental processes in general.

The “new mysterianism” is compared today with the “old mysterianism,” Cartesian dualism, its fate typically misunderstood. To repeat, Cartesian dualism was a perfectly respectable scientific doctrine, disproven by Newton, who exorcised the machine, leaving the ghost intact, contrary to what is commonly believed.

The “new mysterianism,” I believe, is misnamed. It should be called “truism” — at least, for anyone who accepts the major findings of modern biology, which regards humans as part of the organic world. If so, then they will be like all other organisms in having a genetic endowment that enables them to grow and develop to their mature form. By simple logic, the endowment that makes this possible also excludes other paths of development. The endowment that yields scope also establishes limits. What enables us to grow legs and arms, and a mammalian visual system, prevents us from growing wings and having an insect visual system.

All of this is indeed truism, and for non-mystics, the same should be expected to hold for cognitive capacities. We understand this well for other organisms. Thus we are not surprised to discover that rats are unable to run prime number mazes no matter how much training they receive; they simply lack the relevant concept in their cognitive repertoire. By the same token, we are not surprised that humans are incapable of the remarkable navigational feats of ants and bees; we simply lack the cognitive capacities, though we can sometimes duplicate their feats with sophisticated instruments. The truisms extend to higher mental faculties. For such reasons, we should, I think, be prepared to join the distinguished company of Newton, Locke, Hume and other dedicated mysterians.

For accuracy, we should qualify the concept of “mysteries” by relativizing it to organisms. Thus what is a mystery for rats might not be a mystery for humans, and what is a mystery for humans is instinctive for ants and bees.

Dismissal of mysterianism seems to me one illustration of a widespread form of dualism, a kind of epistemological and methodological dualism, which tacitly adopts the principle that study of mental aspects of the world should proceed in some fundamentally different way from study of what are considered physical aspects of the world, rejecting what are regarded as truisms outside the domain of mental processes. This new dualism seems to me truly pernicious, unlike Cartesian dualism, which was respectable science. The new methodological dualism, in contrast, seems to me to have nothing to recommend it.

Far from bewailing the existence of mysteries-for-humans, we should be extremely grateful for it. With no limits to growth and development, our cognitive capacities would also have no scope. Similarly, if the genetic endowment imposed no constraints on growth and development of an organism it could become only a shapeless amoeboid creature, reflecting accidents of an unanalyzed environment, each quite unlike the next. Classical aesthetic theory recognized the same relation between scope and limits. Without rules, there can be no genuinely creative activity, even when creative work challenges and revises prevailing rules.

Contemporary rejection of mysterianism – that is, truism – is quite widespread. One recent example that has received considerable attention is an interesting and informative book by physicist David Deutsch. He writes that potential progress is “unbounded” as a result of the achievements of the Enlightenment and early modern science, which directed science to the search for best explanations. As philosopher/physicist David Albert expounds his thesis, “with the introduction of that particular habit of concocting and evaluating new hypotheses, there was a sense in which we could do anything. The capacities of a community that has mastered that method to survive, and to learn, and to remake the world according to its inclinations, are (in the long run) literally, mathematically, infinite.”

The quest for better explanations may well indeed be infinite, but infinite is of course not the same as limitless. English is infinite, but doesn’t include Greek. The integers are an infinite set, but do not include the reals. I cannot discern any argument here that addresses the concerns and conclusions of the great mysterians of the scientific revolution and the Enlightenment.

We are left with a serious and challenging scientific inquiry: to determine the innate components of our cognitive nature in language, perception, concept formation, reflection, inference, theory construction, artistic creation, and all other domains of life, including the most ordinary ones. By pursuing this task we may hope to determine the scope and limits of human understanding, while recognizing that some differently structured intelligence might regard human mysteries as simple problems and wonder that we cannot find the answers, much as we can observe the inability of rats to run prime number mazes because of the very design of their cognitive nature.

There is no contradiction in supposing that we might be able to probe the limits of human understanding and try to sharpen the boundary between problems that fall within our cognitive range and mysteries that do not. There are possible experimental inquiries. Another approach would be to take seriously the concerns of the great figures of the early scientific revolution and the Enlightenment: to pay attention to what they found “inconceivable,” and particularly their reasons. The “mechanical philosophy” itself has a claim to be an approximation to common sense understanding of the world, a suggestion that might be clarified by experimental inquiry. Despite much sophisticated commentary, it is also hard to escape the force of Descartes’s conviction that free will is “the noblest thing” we have, that “there is nothing we comprehend more evidently and more perfectly” and that “it would be absurd” to doubt something that “we comprehend intimately, and experience within ourselves” merely because it is “by its nature incomprehensible to us,” if indeed we do not “have intelligence enough” to understand the workings of mind, as he speculated. Concepts of determinacy and randomness fall within our intellectual grasp. But it might turn out that “free actions of men” cannot be accommodated in these terms, including the creative aspect of language and thought. If so, that might be a matter of cognitive limitations – which would not preclude an intelligible theory of such actions, far as this is from today’s scientific understanding.

Honesty should lead us to concede, I think, that we understand little more today about these matters than the Spanish physician-philosopher Juan Huarte did 500 years ago when he distinguished the kind of intelligence humans shared with animals from the higher grade that humans alone possess and is illustrated in the creative use of language, and proceeding beyond that, from the still higher grade illustrated in true artistic and scientific creativity. Nor do we even know whether these are questions that lie within the scope of human understanding, or whether they fall among what Hume took to be Nature’s ultimate secrets, consigned to “that obscurity in which they ever did and ever will remain.”

2342. Within Any Possible Universe, No Intellect Can Ever Know It All

By Graham P. Collins, Scientific American, August 1, 2009

Deep in the deluge of knowledge that poured forth from science in the 20th century were found ironclad limits on what we can know. Werner Heisenberg discovered that improved precision regarding, say, an object’s position inevitably degraded the level of certainty of its momentum. Kurt Gödel showed that within any formal mathematical system advanced enough to be useful, it is impossible to use the system to prove every true statement that it contains. And Alan Turing demonstrated that one cannot, in general, determine if a computer algorithm is going to halt.

David H. Wolpert, a physics-trained computer scientist at the NASA Ames Research Center, has chimed in with his version of a knowledge limit. Because of it, he concludes, the universe lies beyond the grasp of any intellect, no matter how powerful, that could exist within the universe. Specifically, during the past two years, he has been refining a proof that no matter what laws of physics govern a universe, there are inevitably facts about the universe that its inhabitants cannot learn by experiment or predict with a computation. Philippe M. Binder, a physicist at the University of Hawaii at Hilo, suggests that the theory implies researchers seeking unified laws cannot hope for anything better than a “theory of almost everything.”

Wolpert’s work is an effort to create a formal rigorous description of processes such as measuring a quantity, observing a phenomenon, predicting a system’s future state or remembering past information—a description that is general enough to be independent of the laws of physics. He observes that all those processes share a common basic structure: something must be configured (whether it be an experimental apparatus or a computer to run a simulation); a question about the universe must be specified; and an answer (right or wrong) must be supplied. He models that general structure by defining a class of mathematical entities that he calls inference devices.

The inference devices act on a set of possible universes. For instance, our universe, meaning the entire world line of our universe over all time and space, could be a member of the set of all possible such universes permitted by the same rules that govern ours. Nothing needs to be specified about those rules in Wolpert’s analysis. All that matters is that the various possible inference devices supply answers to questions in each universe. In a universe similar to ours, an inference device may involve a set of digital scales that you will stand on at noon tomorrow and the question relate to your mass at that time. People may also be inference devices or parts of one.

Wolpert proves that in any such system of universes, quantities exist that cannot be ascertained by any inference device inside the system. Thus, the “demon” hypothesized by Pierre-Simon Laplace in the early 1800s (give the demon the exact positions and velocities of every particle in the universe, and it will compute the future state of the universe) is stymied if the demon must be a part of the universe.

Researchers have proved results about the incomputability of specific physical systems before. Wolpert points out that his result is far more general, in that it makes virtually no assumptions about the laws of physics and it requires no limits on the computational power of the inference device other than it must exist within the universe in question. In addition, the result applies not only to predictions of a physical system’s future state but also to observations of a present state and examining a record of a past state.

The theorem’s proof, similar to the results of Gödel’s incompleteness theorem and Turing’s halting problem, relies on a variant of the liar’s paradox—ask Laplace’s demon to predict the following yes/no fact about the future state of the universe: “Will the universe not be one in which your answer to this question is yes?” For the demon, seeking a true yes/no answer is like trying to determine the truth of “This statement is false.” Knowing the exact current state of the entire universe, knowing all the laws governing the universe and having unlimited computing power is no help to the demon in saying truthfully what its answer will be.

In a sense, however, the existence of such a paradox is not exactly earth-shattering. As Scott Aaronson, a computer scientist at the Massachusetts Institute of Technology, puts it: “That your predictions about the universe are fundamentally constrained by you yourself being part of the universe you’re predicting, always seemed pretty obvious to me—and I doubt Laplace himself would say otherwise if we could ask him.” Aaronson does allow, though, that it is “often a useful exercise to spell out all the assumptions behind an idea, recast everything in formal notation and think through the implications in detail,” as Wolpert has done. After all, the devil, or demon, is in the details.

2026. Video: Richard Levins on "The Two Faces of Science"

By Richard Levins, HealthRoots, October 17, 2012

1813. The Unreasonable Effectiveness of Mathematics in the Natural Sciences

By Eugene Wigner, Communications in Pure and Applied Mathematics, February 1960

"The Unreasonable Effectiveness of Mathematics in the Natural Sciences," in Communications in Pure and Applied Mathematics, vol. 13, No. I (February 1960). New York: John Wiley & Sons, Inc. Copyright © 1960 by John Wiley & Sons, Inc.

Mathematics, rightly viewed, possesses not only truth, but supreme beautya beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show. The true spirit of delight, the exaltation, the sense of being more than Man, which is the touchstone of the highest excellence, is to be found in mathematics as surely as in poetry.

--BERTRAND RUSSELL, Study of Mathematics

THERE IS A story about two friends, who were classmates in high school, talking about their jobs. One of them became a statistician and was working on population trends. He showed a reprint to his former classmate. The reprint started, as usual, with the Gaussian distribution and the statistician explained to his former classmate the meaning of the symbols for the actual population, for the average population, and so on. His classmate was a bit incredulous and was not quite sure whether the statistician was pulling his leg. "How can you know that?" was his query. "And what is this symbol here?" "Oh," said the statistician, "this is pi." "What is that?" "The ratio of the circumference of the circle to its diameter." "Well, now you are pushing your joke too far," said the classmate, "surely the population has nothing to do with the circumference of the circle.”

Naturally, we are inclined to smile about the simplicity of the classmate's approach. Nevertheless, when I heard this story, I had to admit to an eerie feeling because, surely, the reaction of the classmate betrayed only plain common sense. I was even more confused when, not many days later, someone came to me and expressed his bewilderment [1 The remark to be quoted was made by F. Werner when he was a student in Princeton.] with the fact that we make a rather narrow selection when choosing the data on which we test our theories. "How do we know that, if we made a theory which focuses its attention on phenomena we disregard and disregards some of the phenomena now commanding our attention, that we could not build another theory which has little in common with the present one but which, nevertheless, explains just as many phenomena as the present theory?" It has to be admitted that we have no definite evidence that there is no such theory.

The preceding two stories illustrate the two main points which are the subjects of the present discourse. The first point is that mathematical concepts turn up in entirely unexpected connections. Moreover, they often permit an unexpectedly close and accurate description of the phenomena in these connections. Secondly, just because of this circumstance, and because we do not understand the reasons of their usefulness, we cannot know whether a theory formulated in terms of mathematical concepts is uniquely appropriate. We are in a position similar to that of a man who was provided with a bunch of keys and who, having to open several doors in succession, always hit on the right key on the first or second trial. He became skeptical concerning the uniqueness of the coordination between keys and doors.

Most of what will be said on these questions will not be new; it has probably occurred to most scientists in one form or another. My principal aim is to illuminate it from several sides. The first point is that the enormous usefulness of mathematics in the natural sciences is something bordering on the mysterious and that there is no rational explanation for it. Second, it is just this uncanny usefulness of mathematical concepts that raises the question of the uniqueness of our physical theories. In order to establish the first point, that mathematics plays an unreasonably important role in physics, it will be useful to say a few words on the question, "What is mathematics?", then, "What is physics?", then, how mathematics enters physical theories, and last, why the success of mathematics in its role in physics appears so baffling. Much less will be said on the second point: the uniqueness of the theories of physics. A proper answer to this question would require elaborate experimental and theoretical work which has not been undertaken to date.

WHAT IS MATHEMATICS?

Somebody once said that philosophy is the misuse of a terminology which was invented just for this purpose. [2 This statement is quoted here from W. Dubislav's Die Philosophie der Mathematik in der Gegenwart (Berlin: Junker and Dunnhaupt Verlag, 1932), p. 1.] In the same vein, I would say that mathematics is the science of skillful operations with concepts and rules invented just for this purpose. The principal emphasis is on the invention of concepts. Mathematics would soon run out of interesting theorems if these had to be formulated in terms of the concepts which already appear in the axioms. Furthermore, whereas it is unquestionably true that the concepts of elementary mathematics and particularly elementary geometry were formulated to describe entities which are directly suggested by the actual world, the same does not seem to be true of the more advanced concepts, in particular the concepts which play such an important role in physics. Thus, the rules for operations with pairs of numbers are obviously designed to give the same results as the operations with fractions which we first learned without reference to "pairs of numbers." The rules for the operations with sequences, that is, with irrational numbers, still belong to the category of rules which were determined so as to reproduce rules for the operations with quantities which were already known to us. Most more advanced mathematical concepts, such as complex numbers, algebras, linear operators, Borel setsãand this list could be continued almost indefinitelyãwere so devised that they are apt subjects on which the mathematician can demonstrate his ingenuity and sense of formal beauty. In fact, the definition of these concepts, with a realization that interesting and ingenious considerations could be applied to them, is the first demonstration of the ingeniousness of the mathematician who defines them. The depth of thought which goes into the formulation of the mathematical concepts is later justified by the skill with which these concepts are used. The great mathematician fully, almost ruthlessly, exploits the domain of permissible reasoning and skirts the impermissible. That his recklessness does not lead him into a morass of contradictions is a miracle in itself: certainly it is hard to believe that our reasoning power was brought, by Darwin's process of natural selection, to the perfection which it seems to possess. However, this is not our present subject. The principal point which will have to be recalled later is that the mathematician could formulate only a handful of interesting theorems without defining concepts beyond those contained in the axioms and that the concepts outside those contained in the axioms are defined with a view of permitting ingenious logical operations which appeal to our aesthetic sense both as operations and also in their results of great generality and simplicity. [3 M. Polanyi, in his Personal Knowledge (Chicago: University of Chicago Press, 1958), says: "All these difficulties are but consequences of our refusal to see that mathematics cannot be defined without acknowledging its most obvious feature: namely, that it is interesting" (p 188).]

The complex numbers provide a particularly striking example for the foregoing. Certainly, nothing in our experience suggests the introduction of these quantities. Indeed, if a mathematician is asked to justify his interest in complex numbers, he will point, with some indignation, to the many beautiful theorems in the theory of equations, of power series, and of analytic functions in general, which owe their origin to the introduction of complex numbers. The mathematician is not willing to give up his interest in these most beautiful accomplishments of his genius. [4 The reader may be interested, in this connection, in Hilbert's rather testy remarks about intuitionism which "seeks to break up and to disfigure mathematics," Abh. Math. Sem., Univ. Hamburg, 157 (1922), or Gesammelte Werke (Berlin: Springer, 1935), p. 188.]

WHAT IS PHYSICS?

The physicist is interested in discovering the laws of inanimate nature. In order to understand this statement, it is necessary to analyze the concept, "law of nature."

The world around us is of baffling complexity and the most obvious fact about it is that we cannot predict the future. Although the joke attributes only to the optimist the view that the future is uncertain, the optimist is right in this case: the future is unpredictable. It is, as Schrodinger has remarked, a miracle that in spite of the baffling complexity of the world, certain regularities in the events could be discovered. One such regularity, discovered by Galileo, is that two rocks, dropped at the same time from the same height, reach the ground at the same time. The laws of nature are concerned with such regularities. Galileo's regularity is a prototype of a large class of regularities. It is a surprising regularity for three reasons.

The first reason that it is surprising is that it is true not only in Pisa, and in Galileo's time, it is true everywhere on the Earth, was always true, and will always be true. This property of the regularity is a recognized invariance property and, as I had occasion to point out some time ago, without invariance principles similar to those implied in the preceding generalization of Galileo's observation, physics would not be possible. The second surprising feature is that the regularity which we are discussing is independent of so many conditions which could have an effect on it. It is valid no matter whether it rains or not, whether the experiment is carried out in a room or from the Leaning Tower, no matter whether the person who drops the rocks is a man or a woman. It is valid even if the two rocks are dropped, simultaneously and from the same height, by two different people. There are, obviously, innumerable other conditions which are all immaterial from the point of view of the validity of Galileo's regularity. The irrelevancy of so many circumstances which could play a role in the phenomenon observed has also been called an invariance. However, this invariance is of a different character from the preceding one since it cannot be formulated as a general principle. The exploration of the conditions which do, and which do not, influence a phenomenon is part of the early experimental exploration of a field. It is the skill and ingenuity of the experimenter which show him phenomena which depend on a relatively narrow set of relatively easily realizable and reproducible conditions. [5 See, in this connection, the graphic essay of M. Deutsch, Daedalus 87, 86 (1958). A. Shimony has called my attention to a similar passage in C. S. Peirce's Essays in the Philosophy of Science (New York: The Liberal Arts Press, 1957), p. 237.] In the present case, Galileo's restriction of his observations to relatively heavy bodies was the most important step in this regard. Again, it is true that if there were no phenomena which are independent of all but a manageably small set of conditions, physics would be impossible.

The preceding two points, though highly significant from the point of view of the philosopher, are not the ones which surprised Galileo most, nor do they contain a specific law of nature. The law of nature is contained in the statement that the length of time which it takes for a heavy object to fall from a given height is independent of the size, material, and shape of the body which drops. In the framework of Newton's second "law," this amounts to the statement that the gravitational force which acts on the falling body is proportional to its mass but independent of the size, material, and shape of the body which falls.

The preceding discussion is intended to remind us, first, that it is not at all natural that "laws of nature" exist, much less that man is able to discover them. [6 E. Schrodinger, in his What Is Life? (Cambridge: Cambridge University Press, 1945), p. 31, says that this second miracle may well be beyond human understanding.] The present writer had occasion, some time ago, to call attention to the succession of layers of "laws of nature," each layer containing more general and more encompassing laws than the previous one and its discovery constituting a deeper penetration into the structure of the universe than the layers recognized before. However, the point which is most significant in the present context is that all these laws of nature contain, in even their remotest consequences, only a small part of our knowledge of the inanimate world. All the laws of nature are conditional statements which permit a prediction of some future events on the basis of the knowledge of the present, except that some aspects of the present state of the world, in practice the overwhelming majority of the determinants of the present state of the world, are irrelevant from the point of view of the prediction. The irrelevancy is meant in the sense of the second point in the discussion of Galileo's theorem. [7 The writer feels sure that it is unnecessary to mention that Galileo's theorem, as given in the text, does not exhaust the content of Galileo's observations in connection with the laws of freely falling bodies.]

As regards the present state of the world, such as the existence of the earth on which we live and on which Galileo's experiments were performed, the existence of the sun and of all our surroundings, the laws of nature are entirely silent. It is in consonance with this, first, that the laws of nature can be used to predict future events only under exceptional circumstancesãwhen all the relevant determinants of the present state of the world are known. It is also in consonance with this that the construction of machines, the functioning of which he can foresee, constitutes the most spectacular accomplishment of the physicist. In these machines, the physicist creates a situation in which all the relevant coordinates are known so that the behavior of the machine can be predicted. Radars and nuclear reactors are examples of such machines.

The principal purpose of the preceding discussion is to point out that the laws of nature are all conditional statements and they relate only to a very small part of our knowledge of the world. Thus, classical mechanics, which is the best known prototype of a physical theory, gives the second derivatives of the positional coordinates of all bodies, on the basis of the knowledge of the positions, etc., of these bodies. It gives no information on the existence, the present positions, or velocities of these bodies. It should be mentioned, for the sake of accuracy, that we discovered about thirty years ago that even the conditional statements cannot be entirely precise: that the conditional statements are probability laws which enable us only to place intelligent bets on future properties of the inanimate world, based on the knowledge of the present state. They do not allow us to make categorical statements, not even categorical statements conditional on the present state of the world. The probabilistic nature of the "laws of nature" manifests itself in the case of machines also, and can be verified, at least in the case of nuclear reactors, if one runs them at very low power. However, the additional limitation of the scope of the laws of nature which follows from their probabilistic nature will play no role in the rest of the discussion.

THE ROLE OF MATHEMATICS IN PHYSICAL THEORIES

Having refreshed our minds as to the essence of mathematics and physics, we should be in a better position to review the role of mathematics in physical theories.

Naturally, we do use mathematics in everyday physics to evaluate the results of the laws of nature, to apply the conditional statements to the particular conditions which happen to prevail or happen to interest us. In order that this be possible, the laws of nature must already be formulated in mathematical language. However, the role of evaluating the consequences of already established theories is not the most important role of mathematics in physics. Mathematics, or, rather, applied mathematics, is not so much the master of the situation in this function: it is merely serving as a tool.

Mathematics does play, however, also a more sovereign role in physics. This was already implied in the statement, made when discussing the role of applied mathematics, that the laws of nature must have been formulated in the language of mathematics to be an object for the use of applied mathematics. The statement that the laws of nature are written in the language of mathematics was properly made three hundred years ago;[8 It is attributed to Galileo] it is now more true than ever before. In order to show the importance which mathematical concepts possess in the formulation of the laws of physics, let us recall, as an example, the axioms of quantum mechanics as formulated, explicitly, by the great physicist, Dirac. There are two basic concepts in quantum mechanics: states and observables. The states are vectors in Hilbert space, the observables self-adjoint operators on these vectors. The possible values of the observations are the characteristic values of the operatorsãbut we had better stop here lest we engage in a listing of the mathematical concepts developed in the theory of linear operators.

It is true, of course, that physics chooses certain mathematical concepts for the formulation of the laws of nature, and surely only a fraction of all mathematical concepts is used in physics. It is true also that the concepts which were chosen were not selected arbitrarily from a listing of mathematical terms but were developed, in many if not most cases, independently by the physicist and recognized then as having been conceived before by the mathematician. It is not true, however, as is so often stated, that this had to happen because mathematics uses the simplest possible concepts and these were bound to occur in any formalism. As we saw before, the concepts of mathematics are not chosen for their conceptual simplicityeven sequences of pairs of numbers are far from being the simplest conceptsbut for their amenability to clever manipulations and to striking, brilliant arguments. Let us not forget that the Hilbert space of quantum mechanics is the complex Hilbert space, with a Hermitean scalar product. Surely to the unpreoccupied mind, complex numbers are far from natural or simple and they cannot be suggested by physical observations. Furthermore, the use of complex numbers is in this case not a calculational trick of applied mathematics but comes close to being a necessity in the formulation of the laws of quantum mechanics. Finally, it now begins to appear that not only complex numbers but so-called analytic functions are destined to play a decisive role in the formulation of quantum theory. I am referring to the rapidly developing theory of dispersion relations.

It is difficult to avoid the impression that a miracle confronts us here, quite comparable in its striking nature to the miracle that the human mind can string a thousand arguments together without getting itself into contradictions, or to the two miracles of the existence of laws of nature and of the human mind's capacity to divine them. The observation which comes closest to an explanation for the mathematical concepts' cropping up in physics which I know is Einstein's statement that the only physical theories which we are willing to accept are the beautiful ones. It stands to argue that the concepts of mathematics, which invite the exercise of so much wit, have the quality of beauty. However, Einstein's observation can at best explain properties of theories which we are willing to believe and has no reference to the intrinsic accuracy of the theory. We shall, therefore, turn to this latter question.

IS THE SUCCESS OF PHYSICAL THEORIES TRULY SURPRISING?

A possible explanation of the physicist's use of mathematics to formulate his laws of nature is that he is a somewhat irresponsible person. As a result, when he finds a connection between two quantities which resembles a connection well-known from mathematics, he will jump at the conclusion that the connection is that discussed in mathematics simply because he does not know of any other similar connection. It is not the intention of the present discussion to refute the charge that the physicist is a somewhat irresponsible person. Perhaps he is. However, it is important to point out that the mathematical formulation of the physicist's often crude experience leads in an uncanny number of cases to an amazingly accurate description of a large class of phenomena. This shows that the mathematical language has more to commend it than being the only language which we can speak; it shows that it is, in a very real sense, the correct language. Let us consider a few examples.

The first example is the oft-quoted one of planetary motion. The laws of falling bodies became rather well established as a result of experiments carried out principally in Italy. These experiments could not be very accurate in the sense in which we understand accuracy today partly because of the effect of air resistance and partly because of the impossibility, at that time, to measure short time intervals. Nevertheless, it is not surprising that, as a result of their studies, the Italian natural scientists acquired a familiarity with the ways in which objects travel through the atmosphere. It was Newton who then brought the law of freely falling objects into relation with the motion of the moon, noted that the parabola of the thrown rock's path on the earth and the circle of the moon's path in the sky are particular cases of the same mathematical object of an ellipse, and postulated the universal law of gravitation on the basis of a single, and at that time very approximate, numerical coincidence. Philosophically, the law of gravitation as formulated by Newton was repugnant to his time and to himself. Empirically, it was based on very scanty observations. The mathematical language in which it was formulated contained the concept of a second derivative and those of us who have tried to draw an osculating circle to a curve know that the second derivative is not a very immediate concept. The law of gravity which Newton reluctantly established and which he could verify with an accuracy of about 4% has proved to be accurate to less than a ten thousandth of a per cent and became so closely associated with the idea of absolute accuracy that only recently did physicists become again bold enough to inquire into the limitations of its accuracy. [9 See, for instance, R. H. Dicke, Am. Sci., 25 (1959).] Certainly, the example of Newton's law, quoted over and over again, must be mentioned first as a monumental example of a law, formulated in terms which appear simple to the mathematician, which has proved accurate beyond all reasonable expectations. Let us just recapitulate our thesis on this example: first, the law, particularly since a second derivative appears in it, is simple only to the mathematician, not to common sense or to non-mathematically-minded freshmen; second, it is a conditional law of very limited scope. It explains nothing about the earth which attracts Galileo's rocks, or about the circular form of the moon's orbit, or about the planets of the sun. The explanation of these initial conditions is left to the geologist and the astronomer, and they have a hard time with them.

The second example is that of ordinary, elementary quantum mechanics. This originated when Max Born noticed that some rules of computation, given by Heisenberg, were formally identical with the rules of computation with matrices, established a long time before by mathematicians. Born, Jordan, and Heisenberg then proposed to replace by matrices the position and momentum variables of the equations of classical mechanics. They applied the rules of matrix mechanics to a few highly idealized problems and the results were quite satisfactory. However, there was, at that time, no rational evidence that their matrix mechanics would prove correct under more realistic conditions. Indeed, they say "if the mechanics as here proposed should already be correct in its essential traits." As a matter of fact, the first application of their mechanics to a realistic problem, that of the hydrogen atom, was given several months later, by Pauli. This application gave results in agreement with experience. This was satisfactory but still understandable because Heisenberg's rules of calculation were abstracted from problems which included the old theory of the hydrogen atom. The miracle occurred only when matrix mechanics, or a mathematically equivalent theory, was applied to problems for which Heisenberg's calculating rules were meaningless. Heisenberg's rules presupposed that the classical equations of motion had solutions with certain periodicity properties; and the equations of motion of the two electrons of the helium atom, or of the even greater number of electrons of heavier atoms, simply do not have these properties, so that Heisenberg's rules cannot be applied to these cases. Nevertheless, the calculation of the lowest energy level of helium, as carried out a few months ago by Kinoshita at Cornell and by Bazley at the Bureau of Standards, agrees with the experimental data within the accuracy of the observations, which is one part in ten million. Surely in this case we "got something out" of the equations that we did not put in.

The same is true of the qualitative characteristics of the "complex spectra," that is, the spectra of heavier atoms. I wish to recall a conversation with Jordan, who told me, when the qualitative features of the spectra were derived, that a disagreement of the rules derived from quantum mechanical theory and the rules established by empirical research would have provided the last opportunity to make a change in the framework of matrix mechanics. In other words, Jordan felt that we would have been, at least temporarily, helpless had an unexpected disagreement occurred in the theory of the helium atom. This was, at that time, developed by Kellner and by Hilleraas. The mathematical formalism was too dear and unchangeable so that, had the miracle of helium which was mentioned before not occurred, a true crisis would have arisen. Surely, physics would have overcome that crisis in one way or another. It is true, on the other hand, that physics as we know it today would not be possible without a constant recurrence of miracles similar to the one of the helium atom, which is perhaps the most striking miracle that has occurred in the course of the development of elementary quantum mechanics, but by far not the only one. In fact, the number of analogous miracles is limited, in our view, only by our willingness to go after more similar ones. Quantum mechanics had, nevertheless, many almost equally striking successes which gave us the firm conviction that it is, what we call, correct.

The last example is that of quantum electrodynamics, or the theory of the Lamb shift. Whereas Newton's theory of gravitation still had obvious connections with experience, experience entered the formulation of matrix mechanics only in the refined or sublimated form of Heisenberg's prescriptions. The quantum theory of the Lamb shift, as conceived by Bethe and established by Schwinger, is a purely mathematical theory and the only direct contribution of experiment was to show the existence of a measurable effect. The agreement with calculation is better than one part in a thousand.

The preceding three examples, which could be multiplied almost indefinitely, should illustrate the appropriateness and accuracy of the mathematical formulation of the laws of nature in terms of concepts chosen for their manipulability, the "laws of nature" being of almost fantastic accuracy but of strictly limited scope. I propose to refer to the observation which these examples illustrate as the empirical law of epistemology. 

Together with the laws of invariance of physical theories, it is an indispensable foundation of these theories. Without the laws of invariance the physical theories could have been given no foundation of fact; if the empirical law of epistemology were not correct, we would lack the encouragement and reassurance which are emotional necessities, without which the "laws of nature" could not have been successfully explored. Dr. R. G. Sachs, with whom I discussed the empirical law of epistemology, called it an article of faith of the theoretical physicist, and it is surely that. However, what he called our article of faith can be well supported by actual examplesãmany examples in addition to the three which have been mentioned.

THE UNIQUENESS OF THE THEORIES OF PHYSICS

The empirical nature of the preceding observation seems to me to be self-evident. It surely is not a "necessity of thought" and it should not be necessary, in order to prove this, to point to the fact that it applies only to a very small part of our knowledge of the inanimate world. It is absurd to believe that the existence of mathematically simple expressions for the second derivative of the position is self-evident, when no similar expressions for the position itself or for the velocity exist. It is therefore surprising how readily the wonderful gift contained in the empirical law of epistemology was taken for granted. The ability of the human mind to form a string of 1000 conclusions and still remain "right," which was mentioned before, is a similar gift.

Every empirical law has the disquieting quality that one does not know its limitations. We have seen that there are regularities in the events in the world around us which can be formulated in terms of mathematical concepts with an uncanny accuracy. There are, on the other hand, aspects of the world concerning which we do not believe in the existence of any accurate regularities. We call these initial conditions. The question which presents itself is whether the different regularities, that is, the various laws of nature which will be discovered, will fuse into a single consistent unit, or at least asymptotically approach such a fusion. Alternatively, it is possible that there always will be some laws of nature which have nothing in common with each other. At present, this is true, for instance, of the laws of heredity and of physics. It is even possible that some of the laws of nature will be in conflict with each other in their implications, but each convincing enough in its own domain so that we may not be willing to abandon any of them. We may resign ourselves to such a state of affairs or our interest in clearing up the conflict between the various theories may fade out. We may lose interest in the "ultimate truth," that is, in a picture which is a consistent fusion into a single unit of the little pictures, formed on the various aspects of nature.

It may be useful to illustrate the alternatives by an example. We now have, in physics, two theories of great power and interest: the theory of quantum phenomena and the theory of relativity. These two theories have their roots in mutually exclusive groups of phenomena. Relativity theory applies to macroscopic bodies, such as stars. The event of coincidence, that is, in ultimate analysis of collision, is the primitive event in the theory of relativity and defines a point in space-time, or at least would define a point if the colliding panicles were infinitely small. Quantum theory has its roots in the microscopic world and, from its point of view, the event of coincidence, or of collision, even if it takes place between particles of no spatial extent, is not primitive and not at all sharply isolated in space-time. The two theories operate with different mathematical conceptsãthe four dimensional Riemann space and the infinite dimensional Hilbert space, respectively. So far, the two theories could not be united, that is, no mathematical formulation exists to which both of these theories are approximations. All physicists believe that a union of the two theories is inherently possible and that we shall find it. Nevertheless, it is possible also to imagine that no union of the two theories can be found. This example illustrates the two possibilities, of union and of conflict, mentioned before, both of which are conceivable.

In order to obtain an indication as to which alternative to expect ultimately, we can pretend to be a little more ignorant than we are and place ourselves at a lower level of knowledge than we actually possess. If we can find a fusion of our theories on this lower level of intelligence, we can confidently expect that we will find a fusion of our theories also at our real level of intelligence. On the other hand, if we would arrive at mutually contradictory theories at a somewhat lower level of knowledge, the possibility of the permanence of conflicting theories cannot be excluded for ourselves either. The level of knowledge and ingenuity is a continuous variable and it is unlikely that a relatively small variation of this continuous variable changes the attainable picture of the world from inconsistent to consistent. [10 This passage was written after a great deal of hesitation. The writer is convinced that it is useful, in epistemological discussions, to abandon the idealization that the level of human intelligence has a singular position on an absolute scale. In some cases it may even be useful to consider the attainment which is possible at the level of the intelligence of some other species. However, the writer also realizes that his thinking along the lines indicated in the text was too brief and not subject to sufficient critical appraisal to be reliable.] Considered from this point of view, the fact that some of the theories which we know to be false give such amazingly accurate results is an adverse factor. Had we somewhat less knowledge, the group of phenomena which these "false" theories explain would appear to us to be large enough to "prove" these theories. However, these theories are considered to be "false" by us just for the reason that they are, in ultimate analysis, incompatible with more encompassing pictures and, if sufficiently many such false theories are discovered, they are bound to prove also to be in conflict with each other. Similarly, it is possible that the theories, which we consider to be "proved" by a number of numerical agreements which appears to be large enough for us, are false because they are in conflict with a possible more encompassing theory which is beyond our means of discovery. If this were true, we would have to expect conflicts between our theories as soon as their number grows beyond a certain point and as soon as they cover a sufficiently large number of groups of phenomena. In contrast to the article of faith of the theoretical physicist mentioned before, this is the nightmare of the theorist.

Let us consider a few examples of "false" theories which give, in view of their falseness, alarmingly accurate descriptions of groups of phenomena. With some goodwill, one can dismiss some of the evidence which these examples provide. The success of Bohr's early and pioneering ideas on the atom was always a rather narrow one and the same applies to Ptolemy's epicycles. Our present vantage point gives an accurate description of all phenomena which these more primitive theories can describe. The same is not true any longer of the so-called free-electron theory, which gives a marvelously accurate picture of many, if not most, properties of metals, semiconductors, and insulators. In particular, it explains the fact, never properly understood on the basis of the "real theory," that insulators show a specific resistance to electricity which may be 1026 times greater than that of metals. In fact, there is no experimental evidence to show that the resistance is not infinite under the conditions under which the free-electron theory would lead us to expect an infinite resistance. Nevertheless, we are convinced that the free-electron theory is a crude approximation which should be replaced, in the description of all phenomena concerning solids, by a more accurate picture.

If viewed from our real vantage point, the situation presented by the free-electron theory is irritating but is not likely to forebode any inconsistencies which are unsurmountable for us. The free-electron theory raises doubts as to how much we should trust numerical agreement between theory and experiment as evidence for the correctness of the theory. We are used to such doubts.

A much more difficult and confusing situation would arise if we could, some day, establish a theory of the phenomena of consciousness, or of biology, which would be as coherent and convincing as our present theories of the inanimate world. Mendel's laws of inheritance and the subsequent work on genes may well form the beginning of such a theory as far as biology is concerned. Furthermore,, it is quite possible that an abstract argument can be found which shows that there is a conflict between such a theory and the accepted principles of physics. The argument could be of such abstract nature that it might not be possible to resolve the conflict, in favor of one or of the other theory, by an experiment. Such a situation would put a heavy strain on our faith in our theories and on our belief in the reality of the concepts which we form. It would give us a deep sense of frustration in our search for what I called "the ultimate truth." The reason that such a situation is conceivable is that, fundamentally, we do not know why our theories work so well. Hence, their accuracy may not prove their truth and consistency. Indeed, it is this writer's belief that something rather akin to the situation which was described above exists if the present laws of heredity and of physics are confronted.

Let me end on a more cheerful note. The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve. We should be grateful for it and hope that it will remain valid in future research and that it will extend, for better or for worse, to our pleasure, even though perhaps also to our bafflement, to wide branches of learning.

Merci W. Cooper

Histoire et philosophie des mathmatiques Le quasi-empirisme en philosophie des mathmatiques. Une presentation Liens mathmatiques en relation indirecte avec le quasi-empirisme Retour la page d'accueil

Further reading: 
Max Tegmark, "The Mathematical Universe." October 2007