Wigner Quotes

Physics is becoming so unbelievably complex that it is taking longer and longer to train a physicist. It is taking so long, in fact, to train a physicist to the place where he understands the nature of physical problems that he is already too old to solve them.

What, to use Noble Laureate Wigner's classic phrase, accounts for the 'unreasonable effectiveness of mathematics'?

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.

The enormous usefulness of mathematics in the natural sciences is something bordering on the mysterious and … there is no rational explanation for it. —Eugene Wigner, 1960

In 1967, the second resolution to the cat problem was formulated by Nobel laureate Eugene Wigner, whose work was pivotal in laying the foundation of quantum mechanics and also building the atomic bomb. He said that only a conscious person can make an observation that collapses the wave function. But who is to say that this person exists? You cannot separate the observer from the observed, so maybe this person is also dead and alive. In other words, there has to be a new wave function that includes both the cat and the observer. To make sure that the observer is alive, you need a second observer to watch the first observer. This second observer is called “Wigner’s friend,” and is necessary to watch the first observer so that all waves collapse. But how do we know that the second observer is alive? The second observer has to be included in a still-larger wave function to make sure he is alive, but this can be continued indefinitely. Since you need an infinite number of “friends” to collapse the previous wave function to make sure they are alive, you need some form of “cosmic consciousness,” or God. Wigner concluded: “It was not possible to formulate the laws (of quantum theory) in a fully consistent way without reference to consciousness.” Toward the end of his life, he even became interested in the Vedanta philosophy of Hinduism. In this approach, God or some eternal consciousness watches over all of us, collapsing our wave functions so that we can say we are alive. This interpretation yields the same physical results as the Copenhagen interpretation, so this theory cannot be disproven. But the implication is that consciousness is the fundamental entity in the universe, more fundamental than atoms. The material world may come and go, but consciousness remains as the defining element, which means that consciousness, in some sense, creates reality. The very existence of the atoms we see around us is based on our ability to see and touch them.

But why has our physical world revealed such extreme mathematical regularity that astronomy superhero Galileo Galilei proclaimed nature to be “a book written in the language of mathematics,” and Nobel Laureate Eugene Wigner stressed the “unreasonable effectiveness of mathematics in the physical sciences” as a mystery demanding an explanation?

Wigner maintains that human consciousness collapses one lucky universe into being from all of the possible ones. Had my mother’s universe now uncollapsed? Were cards flying back across the baize back into the dealer’s pack?

I have known a great many intelligent people in my life. I knew Max Planck, Max von Laue, and Wemer Heisenberg. Paul Dirac was my brother-in-Iaw; Leo Szilard and Edward Teller have been among my closest friends; and Albert Einstein was a good friend, too. And I have known many of the brightest younger scientists. But none of them had a mind as quick and acute as Jancsi von Neumann. I have often remarked this in the presence of those men, and no one ever disputed me. [...] But Einstein's understanding was deeper than even Jancsi von Neumann's. His mind was both more penetrating and more original than von Neumann's. And that is a very remarkable statement. Einstein took an extraordinary pleasure in invention. Two of his greatest inventions are the Special and General Theories of Relativity; and for all of Jancsi's brilliance, he never produced anything so original.

Out of the prospering but vulnerable Hungarian Jewish middle class came no fewer than seven of the twentieth century’s most exceptional scientists: in order of birth, Theodor von Kármán, George de Hevesy, Michael Polanyi, Leo Szilard, Eugene Wigner, John von Neumann and Edward Teller.

Early in April 1933, the German government passed a law declaring that Jews (defined as anyone with a Jewish grandparent) could not hold an official position, including at the Academy or at the universities. Among those forced to flee were fourteen Nobel laureates and twenty-six of the sixty professors of theoretical physics in the country. Fittingly, such refugees from fascism who left Germany or the other countries it came to dominate—Einstein, Edward Teller, Victor Weisskopf, Hans Bethe, Lise Meitner, Niels Bohr, Enrico Fermi, Otto Stern, Eugene Wigner, Leó Szilárd, and others—helped to assure that the Allies rather than the Nazis first developed the atom bomb. Planck

In science, it is not speed that is the most important. It is the dedication, the commitment, the interest and the will to know something and to understand it — these are the things that come first.

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.

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.

As chairman of the physics department, Professor Max Born nurtured the work of Heisenberg, Eugene Wigner, Wolfgang Pauli and Enrico Fermi. It was Born who in 1924 coined the term “quantum mechanics,” and it was Born who suggested that the outcome of any interaction in the quantum world is determined by chance. In 1954 he would be awarded the Nobel Prize for physics. A pacifist and a Jew, Born was regarded by his students as an unusually warm and patient teacher. He was the ideal mentor for a young student with Robert’s delicate temperament.

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.

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. If there were no phenomena which are independent of all but a manageably small set of conditions, physics would be impossible.

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 if for worse, to our pleasure, even though perhaps also to our bafflement, to wide branches of learning.

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.

Wheeler wasn’t the first to point out that quantum mechanics slips into paradox the minute you introduce a second observer. The Nobel Prize–winning physicist Eugene Wigner, for one, had emphasized it with a Schrödinger’s-cat-style thought experiment that became known as “Wigner’s friend.” It went something like this: Inside a lab, Wigner’s friend sets up an experiment in which an atom will randomly emit a photon, producing a flash of light that leaves a spot on a photographic plate. Before Wigner’s friend checks the plate for signs of a flash, quantum mechanics shows that the atom is in a superposition of having emitted a photon and not having emitted a photon. Once the friend looks at the plate, however, he sees a single outcome—the atom flashed or it didn’t. Somehow his looking collapses the atom’s wavefunction, transforming two possibilities into a single reality. Wigner, meanwhile, is standing outside the lab. From his point of view, quantum mechanics shows that until his friend tells him the outcome of the experiment, the atom remains in a superposition of having emitted a photon and not having emitted a photon. What’s more, his friend is now in a superposition of having seen a spot of light on the plate and not having seen a spot of light on the plate. Only Wigner, quantum theory says, can collapse the wavefunction by asking his friend what happened in there. The two stories are contradictory. According to Wigner’s friend, the atom’s wavefunction collapsed when he looked at the plate. According to Wigner, it didn’t. Instead, his friend entered a superposition correlated with the superposition of the atom, and it wasn’t until Wigner spoke to his friend that both superpositions collapsed. Which story is right? Who is the true creator of reality, Wigner or his friend?

In this book, you will encounter various interesting geometries that have been thought to hold the keys to the universe. Galileo Galilei (1564-1642) suggested that "Nature's great book is written in mathematical symbols." Johannes Kepler (1571-1630) modeled the solar system with Platonic solids such as the dodecahedron. In the 1960s, physicist Eugene Wigner (1902-1995) was impressed with the "unreasonable effectiveness of mathematics in the natural sciences." Large Lie groups, like E8-which is discussed in the entry "The Quest for Lie Group E8 (2007)"- may someday help us create a unified theory of physics. in 2007, Swedish American cosmologist Max Tegmark published both scientific and popular articles on the mathematical universe hypothesis, which states that our physical reality is a mathematical structure-in other words, our universe in not just described by mathematics-it is mathematics.

The German mathematician Emmy Noether proved in 1915 that each continuous symmetry of our mathematical structure leads to a so-called conservation law of physics, whereby some quantity is guaranteed to stay constant-and thereby has the sort of permanence that might make self-aware observers take note of it and give it a "baggage" name. All the conserved quantities that we discussed in Chapter 7 correspond to such symmetries: for example, energy corresponds to time-translation symmetry (that our laws of physics stay the same for all time), momentum corresponds to space-translation symmetry (that the laws are the same everywhere), angular momentum corresponds to rotation symmetry (that empty space has no special "up" direction) and electric charge corresponds to a certain symmetry of quantum mechanics. The Hungarian physicist Eugene Wigner went on to show that these symmetries also dictated all the quantum properties that particles can have, including mass and spin. In other words, between the two of them, Noether and Wigner showed that, at least in our own mathematical structure, studying the symmetries reveals what sort of "stuff" can exist in it.

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.

Because the Nazis considered theoretical physics and quantum mechanics too abstruse and “Jewish,” they had replaced them years before with a more homegrown and homespun curriculum—the rudimentary Deutsche Physik—and as a result of the switch, half of the country’s nuclear scientists had been relieved of, or driven from, their posts. A plethora of the continent’s brightest lights had also taken flight. Not just Einstein, but Hans Bethe, Max Born, Erwin Schrödinger, Eugene Wigner, Otto Stern, Lise Meitner, Robert Frisch, Enrico Fermi, Edward Teller, Maria Goeppert-Mayer—the list went on and on.

In a famous article,8 the physicist Eugene Wigner has written of “the unreasonable effectiveness of mathematics.

The distinction between mathematics and science is pretty well settled. It remains mysterious to us why mathematics that is invented for reasons having nothing to do with nature often turns out to be useful in physical theories. In a famous article,8 the physicist Eugene Wigner has written of “the unreasonable effectiveness of mathematics.

Popular-science news about quantum mechanics is to me as baffling as it is frustrating. Hand me an equation, and I can deal with it. But if you tell me that quantum mechanics allows one to separate a cat from its grin or that an experiment shows "an irreconcilable mismatch between the friends and the Wigners," I'll back out of the room quietly before anyone demands I explain this mess. I have suffered through countless well-intended introductions to quantum mechanics featuring quantum shoes, quantum coins, quantum boxes, and entire zoos of quantum animals that went in and out of those boxes. If you actually understand those explanations, I salute you, because if I hadn't known already how quantum mechanics works, I still wouldn't know.

Szilard’s good friend and fellow Hungarian, the theoretical physicist Eugene Wigner, who was studying chemical engineering at the Technische Hochschule at the time of Szilard’s conversion, watched him take the University of Berlin by storm.

From the horrible weapon which they were about to urge the United States to develop, Szilard, Teller and Wigner—“the Hungarian conspiracy,” Merle Tuve was amused to call them—hoped for more than deterrence against German aggression.1194 They also hoped for world government and world peace, conditions they imagined bombs made of uranium might enforce.

Wigner told me of Hahn’s discovery.1030 Hahn found that uranium breaks into two parts when it absorbs a neutron. . . . When I heard this I immediately saw that these fragments, being heavier than corresponds to their charge, must emit neutrons, and if enough neutrons are emitted . . . then it should be, of course, possible to sustain a chain reaction. All the things which H. G. Wells predicted appeared suddenly real to me.

Hungary was drawn along in the same vortex of intellectual excitement and scientific progress that enveloped the rest of the empire. An extraordinary constellation of the twentieth century’s leading physicists and mathematicians were the product of its equally exceptional educational system at the turn of the century—John von Neumann, Edward Teller, Leo Szilard, Eugene Wigner, Theodor von Kármán, Paul Erdös, and George Pólya, among many others. All came from Hungary’s Jewish middle class, all would flee Hitler’s Europe, and many would end up working during the Second World War for the Manhattan Project, helping to ensure that America, and not Germany, would be the first to build the atomic bomb. The educational reforms instituted in the era of ascendant liberal values in the last decades of the Austro-Hungarian Empire emphasized creative thinking and experimental curiosity over rote learning.

The slow, careful checking continued through the morning. A crowd began to gather on the balcony. Szilard arrived, Wigner, Allison, Spedding whose metal eggs had flattened the pile. Twenty-five or thirty people accumulated on the balcony watching, most of them the young physicists who had done the work.

The physicist Eugene Wigner talked of ”the unreasonable effectiveness of mathematics“—in delineating the present, disinterring the past, and baking a trustier fortune cookie.

A plethora of the continent’s brightest lights had also taken flight. Not just Einstein, but Hans Bethe, Max Born, Erwin Schrödinger, Eugene Wigner, Otto Stern, Lise Meitner, Robert Frisch, Enrico Fermi, Edward Teller, Maria Goeppert-Mayer—the list went on and on.

The scientists and technicians working on America’s secret atom bomb project at Los Alamos during the 1940s called them the ‘Martians’. The joke was that with their strange accents and exceptional intellects, the Hungarians among them were from some other planet. The Martians themselves differed on why one small country should churn out so many brilliant mathematicians and scientists. But there was one fact upon which they were all agreed. If they came from Mars, then one of their number came from another galaxy altogether. When the Nobel Prize-winning physicist and Martian Eugene Wigner was asked to give his thoughts on the ‘Hungarian phenomenon’, he replied there was no such thing. There was only one phenomenon that required any explanation. There was only one Johnny von Neumann.

The mathematical technique of modelling consists of ignoring this trouble and speaking about your deductive model in such a way as if it coincided with reality. The fact that this path, which is obviously incorrect from the point of view of natural science, often leads to useful results in physics is called "the inconceivable effectiveness of mathematics in natural sciences" (or "the Wigner principle").

Adamson scoffed. It generally takes two wars to develop a new weapon, he said; besides, it was “morale,” not research, that led to victory. Shifting in his chair, the formal and ever-polite Wigner could not contain his impatience. “Perhaps,” he told Adamson in a high-pitched but steady voice, enunciating every syllable, “it would be better if we did away with the War Department and spread the military funds among the civilian population. That would raise a lot of morale.”49

[The double-slit experiment] has in it the heart of quantum mechanics. In reality, it contains the only mystery. —Richard Feynman296 The mystery Feynman was referring to in the preceding quote is the curious fact that a quantum object behaves like a particle when it is observed, but it behaves like a wave when it’s not observed. This can be easily demonstrated in a double-slit interferometer, which is a simple device in which one sends particles of light (or electrons, or any elementary particle) through two tiny slits and then records the pattern of light that emerges onto a screen, or a camera. One might expect that if particles of light (called photons) behaved like separate hunks of stuff, like tiny marbles, then the pattern of light emerging from two slits would always be two bright bands of light. And indeed, if you track each photon as it passes through the slits, then that is what you will see on the screen. However, if you do not trace the photons’ paths, then you will see an alternating sequence of light and dark bands, called an “interference pattern.” This then is the mystery of the dual nature of light—whether you see a wavelike or particle pattern on the screen depends on how you’re looking at it. It’s as though all matter—photons, electrons, molecules, and so on297—“knows” that it is being watched. This exquisitely sensitive bashfulness, known in physics jargon as wave-particle complementarity, lies at the heart of quantum mechanics. It is also known as the quantum measurement problem, or QMP. It’s a problem because it violates the commonsense assumption that we live in an objective reality that is completely independent of observers. The founders of quantum theory, including Neils Bohr, Max Planck, Louis de Broglie, Werner Heisenberg, Erwin Schrödinger, and Albert Einstein, knew that introducing the notion of the observer into quantum theory was a radical change in how physics had been practiced, and they all wrote about the consequences of this change. A few physicists, like Wolfgang Pauli, Pascual Jordan, and Eugene Wigner, believed that consciousness was not merely important but was fundamentally responsible for the formation of reality. Jordan wrote, “Observations not only disturb what has to be measured, they produce it.… We compel [the electron] to assume a definite position.… We ourselves produce the results of measurement

In a 1978 memoir von Weizsäcker remembers discussing the possibility of a bomb with Otto Hahn in the spring of 1939. Hahn opposed secrecy then partly on the grounds of scientific ethics but also partly because he “felt that if it were to be made, it would be worst for the entire world, even for Germany, if Hitler were to be the only one to have it.” Like Szilard, Teller and Wigner, von Weizsäcker remembers realizing in discussions with a friend “that this discovery could not fail to radically change the political structure of the world”:1207

We are all guests in this world.

Various factors, it seems, contribute to what we call genius and the forms it may take: speed of thought (at which von Neumann, by all accounts, was exceptional), depth of understanding (at which, according to Wigner, Einstein excelled), originality, creativity, and so forth. Sometimes, too, genius may be narrow in its focus – as in the case of Einstein or Ramanujan – while at other times, as illustrated by von Neumann, and to an even greater extent by some Renaissance figures such as Leonardo da Vinci, it can range over many subjects.

I had the mistaken idea, based on what happened in World War I, that we would stay out of the war, and it is very unfortunate that I felt like that. If I had been more convinced, as Wigner and Szilard were, that we were going to get into the war, I would have pushed harder to begin making the bomb. I figured out that roughly half a million to a million people were being killed a month in the later stages of the war. Every month by which we could have shortened the war would have made a difference of a half million to a million lives, including the life of my own brother.

In fact, it took the resources of three countries to produce the bomb: the United States, Great Britain, and Canada. But there was more to it than that. In some sense it took some of the most valuable scientific talent of all Europe to do it. Consider this partial list: the Hungarians John von Neumann, Eugene Wigner, and Edward Teller; the Germans Hans Bethe and Rudolf Peierls; the Poles Stanislaw Ulam and Joseph Rotblat; the Austrians Victor Weisskopf and Otto Frisch; the Italians Enrico Fermi and Emilio Segrè; Felix Bloch from Switzerland; and, from Denmark, the Bohrs, Niels and his son Aage. This talent, the B-29 heavy bomber program that could deliver the bombs, plus Manhattan Project efforts—all together cost more than fifty billion in today’s dollars. Wilhelm

When the rods were pushed back in and the clicking had died down, we suddenly experiences a let-down feeling, for all of us understood the language of the counter. Even though we had anticipated the success of the experiment, its accomplishment had a deep impact on us. For some time we had known that we were about to unlock a giant; still we could not escape an eerie feeling when we had actually done it. We felt as, I presume, everyone feels who has done something that he knowns will have very far-reaching consequences which he cannot foresee.

In contrast to classical physics, with its exclusive focus on material causation, quantum physics offers a mechanism that validates the intuitive sense that our conscious thoughts have the power to affect our actions. Quantum theory, in the von Neumann-Wigner formulation as developed by Henry Stapp, offers a mathematically rigorous alternative to the impotence of conscious states: it allows conscious experience to act back on the physical brain by influencing its activities. It describes a way in which our conscious thoughts and volitions enter into the causal structure of nature and focus our thoughts, choose from among competing possible courses of action, and even override the mechanical aspects of cerebral processes. The quantum laws allow mental effort to influence the course of cerebral processes in just the way our subjective feeling tells us it does. How? By keeping in focus a stream of consciousness that would otherwise diffuse like mist at daybreak. Quantum theory demonstrates how mental effort can have, through the process of willfully focusing attention, dynamical consequences that cannot be deduced or predicted from, and that are not the automatic results of, cerebral mechanisms acting alone. In a world described by quantum physics, an insistence on causal closure of the physical world amounts to a quasi-religious faith in the absolute powers of matter, a belief that is no more than a commitment to brute, and outmoded, materialism.

According to the Copenhagen Interpretation, it is the observer who both decides which aspect of nature is to be probed and reads the answer nature gives. The mind of the observer helps choose which of an uncountable number of possible realities comes into being in the form of observations. A specific question (Is the electron here or there?) has been asked, and an observation has been performed (Aha! the electron is there!), corralling an unruly wave of probability into a well-behaved quantum of certainty. Bohr was silent on how observation performs this magic. It seems, though, as if registering the observation in the mind of the observer somehow turns the trick: the mental event collapses the wave function. Bohr, squirming under the implications of his own work, resisted the idea that an observer, through observation, is actually influencing the course of physical events outside his body. Others had no such qualms. As the late physicist Heinz Pagels wrote in his wonderful 1982 book The Cosmic Code, “There is no meaning to the objective existence of an electron at some point in space… independent of any actual observation. The electron seems to spring into existence as a real object only when we observe it!” Physical theory thus underwent a tectonic shift, from a theory about physical reality to a theory about our knowledge. Science is what we know, and what we know is only what our observations tell us. It is unscientific to ask what is “really” out there, what lies behind the observations. Physical laws as embodied in the equations of quantum physics, then, ceased describing the physical world itself. They described, instead, our knowledge of that world. Physics shifted from an ontological goal—learning what is—to an epistemological one: determining what is known, or knowable. As John Archibald Wheeler cracked, “No phenomenon is a phenomenon until it is an observed phenomenon.” The notion that the wave function collapses when the mind of an observer registers a new bit of knowledge was developed by the physicist Eugene Wigner, who proposed a model of how consciousness might collapse the wave function—something we will return to. But why human consciousness should be thus privileged has remained an enigma and a source of deep division in physics right down to today.

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.

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.

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.

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.

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.

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.