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Maths In Nature Essay

How many universes are there? What has made us into who we are? Is there absolute truth?

These are difficult questions, but mathematics has something to say about each of them. It can probe the physical reality that surrounds us, shed light on human interaction and psychology, and it answers, as well as raises, many of the philosophical questions our minds have allowed us to dream up.

On this page we bring together articles and podcasts that examine what mathematics can say about the nature of the reality we live in. They look at physical reality, the mind, consciousness, the emergence of life, philosophy and mathematics itself. The page will be continually updated with new relevant articles, so keep looking and get reading!

We've grouped our articles into three categories:

Mathematics and physical reality

Hooray for Higgs — "It's a great day for particle physics," says Ben Allanach, a theoretical physicist at the University of Cambridge. "It's very exciting, I think we're on the verge of the Higgs discovery." And indeed, it seems like the Large Hadron Collider at CERN has given particle physics an early Christmas present — compelling evidence that the famous Higgs boson exists.
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What is time? — Newton thought that time was absolute. Einstein thought it was relative. Today some people think it doesn't exist at all. Or is it an emergent phenomenon? Paul Davies explores.
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This interview is also available as a podcast.


Is time travel allowed? — We're all on a journey towards the future, but can we travel into the past? Find out with Kip Thorne.


What happened before the Big Bang? — Did the Big Bang mark the beginning of time? Not if we live in a bubble multiverse, says John D. Barrow.


Shattering crystal symmetries —In 1982 Dan Shechtman discovered a crystal that would revolutionise chemistry. He has just been awarded the 2011 Nobel Prize in Chemistry for his discovery — but has the Nobel committee missed out a chance to honour a mathematician for his role in this revolution as well?


Exploding stars clinch Nobel Prize —This year's Nobel Prize in Physics was awarded for a discovery that proved Einstein wrong and right at the same time.


Countdown to the Higgs? — Are we close to finding the Higgs? Ben Allanach explains it is not about catching a glimpse of the beast itself, but instead keeping a careful count of the evidence it leaves behind.


Does quantum physics really describe reality? — Quantum physics is a funny thing. With counterintuitive ideas such as superposition and entanglement, it doesn't seem to resemble reality as we know it, yet it is an incredibly successful theory of how the physical world operates. In this podcast we speak to experts including John Polkinghorne and Roger Penrose about how we can resolve the mysteries of quantum physics with our experience of reality. And we find out why quantum physics is just like riding a bike...


Flying home with quantum physics — Quantum mechanics is usually associated with weird and counterintuitve phenomena we can't observe in real life. But it turns out that quantum processes can occur in living organisms, too, and with very concrete consequences. Some species of birds use quantum mechanics to navigate. Studying these little creatures' quantum compass may help us achieve the holy grail of computer science: building a quantum computer. You can also listen to the interview this article is based on as a podcast.


Biology's next microscope, mathematics' next physics — It is thought that the next great advances in biology and medicine will be discovered with mathematics. As biology stands on the brink of becoming a theoretical science, Thomas Fink asks if there is more to this collaboration than maths acting as biology's newest microscope. Will theoretical biology lead to new and exciting maths, just as theoretical physics did in the last two centuries? And is there a mathematically elegant story behind life?


And the Nobel Prize in Mathematics goes to... — Well, it goes to no one because there isn't a Nobel Prize for maths. Some have speculated that Alfred Nobel neglected maths because his wife ran off with a mathematician, but the rumour seems to be unfounded. But whatever the reason for its non-appearance in the Nobel list, it's maths that makes the science-based Nobel subjects possible and it usually plays a fundamental role in the some of the laureates' work. Here we'll have a look at two of the prizes awarded this year, in physics and economics.


What's happening at the LHC? — This article gives an update on what's happening at the Large Hadron Collider, as of February 2011. It is based on an interview with John Ellis, outgoing leader of the theory division at CERN, who describes some of the mysteries the LHC may solve.


Hidden dimensions — That geometry should be relevant to physics is no surprise. After all, space is the arena in which physics happens. What is surprising, though, is the extent to which the geometry of space actually determines physics and just how exotic the geometric structure of our Universe appears to be. Plus met up with mathematician Shing-Tung Yau to find out more. You can also read our review of a book co-written by Yau, called The shape of inner space.


Exotic spheres, or why 4-dimensional space is a crazy place — The world we live in is strictly 3-dimensional: up/down, left/right, and forwards/backwards, these are the only ways to move. For years, scientists and science fiction writers have contemplated the possibilities of higher dimensional spaces. What would a 4- or 5-dimensional universe look like? Or might it even be true that we already inhabit such a space, that our 3-dimensional home is no more than a slice through a higher dimensional realm, just as a slice through a 3-dimensional cube produces a 2-dimensional square?


The Able Prize 2011 goes to John Milnor — This news story examines the work of John Milnor, who discovered (invented?) the exotic spheres described in the above article.


Is space like a chessboard? — Physicists at the University of California, Los Angeles set out to design a better transistor and ended up with a discovery that may lead to a new explanation of electron spin and possibly even the nature of space.


Benoît Mandelbrot has died — Benoît Mandelbrot, the father of fractal geometry, died in October 2010 at the age of 85. His work shed important light on how complexity can arise from simple rules.


Classroom activity: The game of life — One of the amazing things about life is its sheer complexity. How can a bunch of mindless cells combine to form something as complex as the human brain, or as delicate, beautiful and highly organised as the patterns on a butterfly's wing? Maths has some surprising answers you can explore yourself with this interactive activity.


The illusory Universe — With online socialising and alternative realities like Second Life it may seem as if reality has become a whole lot bigger over the last few years. In one branch of theoretical physics, though, things seem to be going the other way. String theorists have been developing the idea that the space and time we inhabit, including ourselves, might be nothing more than an illusion, a hologram conjured up by a reality which lacks a crucial feature of the world as we perceive it: the third dimension. Plus talks to Juan Maldacena to find out more.


Particle hunting at the LHC — It's hard to avoid CERN these days. Last year's successful switch-on of CERN's Large Hadron Collider, followed by a blow-out which is currently being fixed, sparked wide-spread media coverage, and currently CERN stars in the Tom Hanks movie Angels and Demons. So what goes on at CERN and why the hubbub about the Large Hadron Collider, known as the LHC?


Lambda marks the spot — the biggest problem in theoretical physics — The mathematical maps in theoretical physics have been highly successful in guiding our understanding of the universe at the largest and smallest scales. Linking these two scales together is one of the golden goals of theoretical physics. But, at the very edges of our understanding of these fields, one of the most controversial areas of physics lies where these maps merge: the cosmological constant problem.


Outer space: Are the constants of nature really constant? — Are the unchanging features of the Universe really unchanging?


Mathematics and the mind

Gut logic — Human reasoning is biased and illogical. At least that's what a huge body of psychological research seems to show. But now a psychological scientist from the University of Toulouse in France has come up with a new theory: that logical and probabilistic thinking is an intuitive part of decision making, only its conclusions often lose out to heuristic reasoning.


Convex is complex — If computers are ever going to be as clever as humans they'll have to be able to solve really hard problems. But what do we mean by "hard" and how hard are the hardest problems? This news story explores the concept of NP completeness using a result on the convexity of mathematical functions as an example.


Born to count — People as well as animals are born with a sense for numbers. But is this inborn number sense related to mathematical ability? A new study suggests that it is.


Guilt counts — Guilt, so some people have suggested, is what makes us nice. When we do someone a favour or choose not to exploit someone vulnerable, we do it because we fear the guilt we'd feel otherwise. A team of neuroscientists, psychologists and economists has produced some new results in this area, using a model from psychological game theory.


Wiring up brains — The human brain faces a difficult trade-off. On the one hand it needs to be complex to ensure high performance, and on the other it needs to minimise "wiring cost" — the sum of the length of all the connections — because communication over distance takes a lot of energy. It's a problem well-known to computer scientists. And it seems that market driven human invention and natural selection have come up with similar solutions.


Neurotweets: #hashtagging the brain — We like to think of the human brain as special, but as we reported in the above article, it has quite a lot in common with worm brains and even with high-performance information processing systems. But how does it compare to online social networks? In a recent lecture the psychiatrist Ed Bullmore put this question to the test.


Struggling with your maths? — If you are, then you may be one of the 5 to 7% of the population suffering from dyscalculia, the mathematical equivalent of dyslexia. But unlike many dyslexia sufferers, you probably haven't received the help you need to cope with your condition. As a recent article published in the journal Science points out, dyscalculia is the "poor relation" of dyslexia.


Baby robots feel the love — Researchers have unveiled the first prototypes of robots that can develop emotions and express them too. If you treat these robots well, they'll form an attachment to you, looking for hugs when they feel sad and responding to reassuring strokes when they are distressed. But how do you get emotions into machines that only understand the language of maths?


Spaceships are doing it for themselves — It requires only a little processing power, but it's a giant leap for robotkind: engineers at the University of Southampton have developed a way of equipping spacecraft and satellites with human-like reasoning capabilities, which will enable them to make important decisions for themselves.


Finding your way home without knowing where you are — Foraging ants have a hard life, embarking on long and arduous trips several times a day, until they drop dead from exhaustion. The trips are not just long, they also follow complex zig-zag paths. So how do ants manage to find their way back home? And how do they manage to do so along a straight line? Their secret lies in a little geometry.


Uncoiling the spiral: Maths and visual hallucinations — Think drug-induced hallucinations, and the whirly, spirally, tunnel-vision-like patterns of psychedelic imagery immediately spring to mind. But it's not just hallucinogenic drugs that conjure up these geometric structures. People have reported seeing them in near-death experiences, following sensory deprivation, or even just after applying pressure to the eyeballs. So what can these patterns tell us about the structure of our brains?


Book review: Natural computing — Computing is at the heart of our modern world, but what are its frontiers? This book presents new trends in this fast growing field. Although the topics covered range from spacecraft control to embedding intelligence in bacteria, they all coincide in one fundamental point: the future of computing is a synthesis with nature.


Mathematics, philosophy and truth

Freedom and physics — Most of us think that we have the capacity to act freely. Our sense of morality, our legal system, our whole culture is based on the idea that there is such a thing as free will. It's embarrassing then that classical physics seems to tell a different story. And what does quantum theory have to say about free will?


Free, from top to bottom? — A traditional view of science holds that every system — including ourselves — is no more than the sum of its parts. To understand it, all you have to do is take it apart and see what's happening to the smallest constituents. But the mathematician and cosmologist George Ellis disagrees. He believes that complexity can arise from simple components and physical effects can have non-physical causes, opening a door for our free will to make a difference in a physical world.


John Conway: discovering free will (part I) — On August 19, 2004, John Conway was standing with his friend Simon Kochen at the blackboard in Kochen’s office in Princeton. They had been trying to understand a thought experiment involving quantum physics and relativity. What they discovered, and how they described it, created one of the most controversial theorems of their careers: The Free Will Theorem.


John Conway: discovering free will (part II) — In the second part of our interview, John Conway explains how the Kochen-Specker Theorem from 1965 not only seemed to explain the EPR Paradox, it also provided the first hint of Conway and Kochen's Free Will Theorem.


John Conway: discovering free will (part III) — Inthe second part of our interview, John Conway explains how the Kochen-Specker Theorem from 1965 not only seemed to explain the EPR Paradox, it also provided the first hint of Conway and Kochen's Free Will Theorem.


This is not a carrot: Paraconsistent mathematics — Paraconsistent mathematics is a type of mathematics in which contradictions may be true. In such a system it is perfectly possible for a statement A and its negation not A to both be true. How can this be, and be coherent? What does it all mean?


The philosophy of applied mathematics — We all take for granted that mathematics can be used to describe the world, but when you think about it this fact is rather stunning. This article explores what the applicability of maths says about the various branches of mathematical philosophy.


Searching for the missing truth — Many people like mathematics because it gives definite answers. Things are either true or false, and true things seem true in a very fundamental way. But it's not quite like that. You can actually build different versions of maths in which statements are true or false depending on your preference. So is maths just a game in which we choose the rules to suit our purpose? Or is there a "correct" set of rules to use? We find out with the mathematician Hugh Woodin.


Picking holes in mathematics — In the 1930s the logician Kurt Gödel showed that if you set out proper rules for mathematics, you lose the ability to decide whether certain statements are true or false. This is rather shocking and you may wonder why Gödel's result hasn't wiped out mathematics once and for all. The answer is that, initially at least, the unprovable statements logicians came up with were quite contrived. But are they about to enter mainstream mathematics?


The revelation game — Is it rational to believe in a god? The most famous rational argument in favour of belief was made by Blaise Pascal, but what happens if we apply modern game theory to the question? This article has been adapted from the book Game theory and the humanities, which has been reviewed in Plus.


Constructive mathematics — If you like mathematics because things are either true or false, then you'll be worried to hear that in some quarters this basic concept is hotly disputed. In this article Phil Wilson looks at constructivist mathematics, which holds that some things are neither true, nor false, nor anything in between.


Unreasonable effectiveness — When it comes to describing natural phenomena, mathematics is amazingly — even unreasonably — effective. In this article Mario Livio looks at an example of strings and knots, taking us from the mysteries of physical matter to the most esoteric outpost of pure mathematics, and back again.


Infinite investigators: Part I — What's the nature of infinity? Are all infinities the same? And what happens if you've got infinitely many infinities? This article explores how these questions brought triumph to one man and ruin to another, ventures to the limits of mathematics and finds that, with infinity, you're spoilt for choice.


Infinite investigators: Part II — We continue the investigation into Cantor and Cohen's work, looking at the continuum hypothesis, the question that caused Cantor so much grief.


Teacher package: Logic — In some sense, all of maths should come under the label "logic", but mathematical logic has shown that mathematics isn't entirely logical. Makes sense? If not, then this teacher package may help.


Book review: The big questions, mathematics — With twenty skillfully written essays Tony Crilly paints a broad-stroke picture of modern mathematics, focusing on some of the most exciting topics. This book is intended for people whose acquaintance with mathematics is limited to their high school years, but who want to know "what all this fuss is about". It is ideal for those who have heard that mathematicians talk about imaginary numbers and unbreakable codes, and want to know how much of it, if any, is true.


The Unreasonable Effectiveness of Mathematics in the Natural Sciences

by Eugene Wigner

"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

[1 The remark to be quoted was made by F. Werner when he was a student in Princeton.] [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.] [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).]

[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.]

[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.] [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.] [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.]

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 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).]

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 examples�many 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 concepts�the 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.]

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

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