The science of impossibility
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The science of impossibility

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The science of impossibility The science of impossibility Document Transcript

  • PARADOXES
  • CONTENTS Abstract Introduction-Paradoxes Visual paradox Linguistic Paradox Zeno‘s Paradoxes Physical Paradoxes
  • ABSTRACTIn this paper we have done a brief survey of the fabric and woof of paradoxes. The paperstarts mapping out the various types of paradoxes like visual, linguistic and logical with lucidpictures. Amongst the large varieties of logical paradoxes, physical paradoxes are discussedin detail with certain examples. This paper also illustrates briefly the works of Escher. Thefamous Zeno‘s paradox is also briefly described.“God is not all-powerful as he cannot build a wall he cannot jump.” By RAJESH A View slide
  • Paradox A paradox is truth standing on its head to attract attention.The word ‗paradox‘ is a synthesis two Greek words, Para, beyond, and doxos, belief. It hascome to have a variety of meanings: something which appears contradictory but which is, infact, true; something which appears true but which is, in fact, contradictory; or a harmlesschain of deductions from a self- evident starting point which leads to a contradiction. Aparadox is a situation which gives one answer when analyzed one way, and a different answerwhen analyzed another way, so that we are left in somewhat of a quandary as to actually whatshould happen.Philosophers love paradox. Indeed, Bertrand Russell once remarked that the mark of goodphilosophy is to begin with a statement that is regarded as too obvious to be of interest andfrom it deduce a conclusion that no one will believe.Some philosophers have argued that the existence of certain paradoxes show that theworld in itself is contradictory - and no solution could therefore be given to theseparadoxes. Some paradoxes have, however, been given solutions - for instance, zenosparadoxes (see below), and Galilee‘s paradox about the discovery that there are asmany natural numbers as there are square numbers, which had its solution withCantors development of set theory. To give a solution to a paradox you can either:(1) show that the contradiction was only an apparent one, or(2) show that the paradox rests on invalid or unreasonable grounds.While some paradoxes may be trivial, others reflect profound problems about ourways of thinking and challenge us to re-evaluate them or so seek out unsuspectedinconsistencies in the beliefs that we held to be self-evidently true. Anatol Rapoport,an international authority on strategic analysis—an arena where paradoxical resultsoften result from innocuous beginnings—draws attention to the stimulating role thatthe recognition of paradox has played in many areas of human thinking:Paradoxes have played a dramatic role in intellectual history, often foreshadowingrevolutionary developments in science, mathematics, and logic. Whenever, in any discipline,we discover a problem that cannot be solved within the conceptual framework that supposedlyshould apply, we experience shock. The shock may compel us to discard the old frameworkand adopt a new one. It is to this process of intellectual molting that we owe the birth of manyof the major ideas in mathematics and science. Zeno’s paradox of Achilles and the tortoisegave birth to the idea of convergent infinite series. Antinomies (internal contradictions inmathematical logic) eventually blossomed into Gödel’s theorem. The paradoxical result of theMichelson—Morley experiment on the speed of light set the stage for the theory of relativity.The discovery of wave—particle duality of light forced a reexamination of deterministiccausality, the very foundation of scientific philosophy, and led to quantum mechanics. Theparadox of Maxwell’s demon, which Leo Szilard first found a way to resolve in 1929, gaveimpetus more recently to the profound insight that the seemingly disparate concepts ofinformation and entropy are intimately linked to each other. View slide
  • Visual paradox You arrive at the truth by telling a pack of lies if you are writing fiction, as opposed to trying to arrive at a pack of lies by telling the truth if you are a journalist.The divergence of the artistic and scientific pictures of the world has been made most strikingby the focus of twentieth-century artists upon abstract images and distortions of the everydaypicture of the world. One of the most extraordinary consequences of human consciousness isthe ability it gives us to imagine things which are physically impossible. By this device wecan explore reality in a unique way, placing it in a context defined by impossible events. Inthis way we are able to create resonances of meaning and juxtapositions of ideas which aremind-stretching and stimulating. This we find appealing and novel. Some individuals devotetheir lives to this activity, creating and appreciating these alternative realities in a host ofdifferent media. The affinity that our minds possess for this activity is almost alarming. Thesudden appearance of sophisticated computer simulations of alternative realities and the readyavailability of computer games which are indistinguishable from direct human activities haverevealed how seductive such experiences are to young people. They offer a huge range ofvicarious experience without the need to leave the comfort of one‘s chair. Perhaps the appealof these virtual adventures is telling us something about the untapped potential within thehuman mind which is so little used in the cosseted activities of everyday twentieth-centurylife. We have, begun to use the computer interactively in education, but with littleimagination so far. I suspect there is a great opportunity here to teach many subjects—especially science and mathematics—in an adventurous new way. Even a mundanecomputer-based activity like word processing, has done more than make writing and editingmore efficient: it has altered the way in which writers think. Writers used to write becausethey had something to say; now they write in order to discover if they have something to say.The representation of the impossible has become a prominent part of the modern artisticworld. This takes several forms. The graphic style of Maurits Escher employs a form ofprecise drawing which seeks to deceive the viewer into believing that he has entered apossible world which, on closer scrutiny, turns out to be inconsistent with the nature of spacein which we live. Escher likes impossible objects which we could define as two-dimensionalimages of apparent three-dimensional objects which cannot exist as we have interpretedthem: that is, they cannot be constructed in three-dimensional space.The three-dimensional interpretation of these images is a different matter. The eye is led tobuild up different local pictures which, ultimately, cannot be combined into a singleconsistent visual scenario. In modern times impossible objects were drawn first by OscarReutersvard. In 1934 he drew the first known example of an impossible tribar. Escher createdthe first impossible cube in 1958.Escher employed these in his famous drawings Waterfall (1961) and Ascending anddescending (1961). The strange loop is one of the most recurrent themes in Escher‘s work.Comparing its six-step endlessly falling loop with the six-step endlessly rising loop of the―Canon per Tonos‖, we see the remarkable similarity of vision.
  • There are a number of curious older examples of this genre which have been recognizedretrospectively. Hogarth‘s engraving on copper False Perspective (1754) is a beautifulexample. It was drawn by Hogarth to exaggerate the mistakes of inept draughts men. Helabels the picture, ‗whoever makes a Design without the Knowledge of Perspective will beliable to such Absurdities as are seen in this Frontispiece
  • The tribar was rediscovered in 1961 by Lionel and Roger Penrose, who introduced the never-ending staircase.The famous Italian architect and engraver Giovanni Piranesi (1720—78) produced a sinistercollection of designs for a series of labyrinthine dungeons between 1745 and 1760. Thesefantastic creations depicted impossible networks of rooms and stairways. His workingdiagrams reveal that he deliberately set out to create impossible configurations.Breughel‘s The Magpie on the Gallows (1568) deliberately makes use of animpossible four-bar. Unintentional impossible objects can be found at veryearly times. The oldest known example dates from the eleventh century.These impossible figures reveal something more profound than the draughts- man‘s skill.They tell us something about the nature of space and the workings of the brain‘sprogramming for spatial analysis. Our brains have evolved to deal with the geometry of thereal world. They have defense mechanisms to guard against being deceived by false orambiguous perspective. In such a dilemma the brain changes the perspective adopted everyfew seconds as an insurance against having made the wrong choice. A common example isthe Necker cube (fig 1.6), which seems to flit back and forth between two differentorientations.Perhaps we like imaginary worlds that are impossible because their very impossibilityreinforces the appeal of artistic representations of strange environments and circumstanceswhich we can experience safely. They allow us to enter environments which are dangerous,in the sense that they could not possible be part of our (or anyone‘s) experience, but withoutreal risk.Much has been made of the way in which geometrically distorted pictures began to appear ata time when physicists first began to appreciate the physical relevance of geometries other
  • than Euclid‘s. Pioneering cubists like Picasso always denied that scientific developmentsmotivated them in any direct way. Escher, on the other hand, seemed to appreciate the studiesthat mathematicians made of other geometries. Indeed, his work may even have stimulatedsome explorations of new tessellations of space.There is also a complementary literary style which trades on impossibility and paradox. Thegreatest early exponent of this was probably the Victorian surrealist Lewis Carroll. We seeit‘s more eclectic and fantastic manifestations in the short stories of Jorge Luis Borges, andothers. The conjuring up of worlds that don‘t quite fit remains a strangely attractive creativeactivity: the only way to be truly original.The interesting feature of all these examples is the way in which they show our recognition ofthe impossible. The impossible is not necessarily something that lies outside our mentalexperience even if it falls outside our physical experience. We can create mental worldswhich are quite different from the one we experience. Indeed, some people clearly relishthese images of impossible worlds as much as any that could be made of this one
  • Linguistic paradox The supreme triumph of reason is to cast doubt upon its own validity.Impossible figures are examples of visual paradoxes, or perhaps we should say invertedparadoxes. A paradox is usually something which, although seeming to be false, is in facttrue. Impossible figures are things which, despite seeming true, are in reality false. We mighthave expected that our reaction to paradox would be one of confusion or aversion.Paradoxically, it is apparently quite the opposite. We enjoy paradox: it lies at the heart ofmany forms of humour, stories, pictures, and a host of well-appreciated quirks of humancharacter.Paradoxes spun for amusement have a habit of subsequently proving deeply profound.History is strewn with examples. Zeno‘s paradoxes have stimulated our understanding of theinfinite. Zeno was Greek philosopher of the fifth century BC who is best known for theseparadoxes, which appear to show that motion is impossible. His most famous example is thatof the race between Achilles and the tortoise. Suppose that the tortoise is given a 100-metrestart but Achilles runs a hundred times faster than the tortoise. While Achilles runs 100metres, the tortoise covers 1 metre; while Achilles runs 1 metre, the tortoise covers 1centimetre; and so on, for an infinite number of steps. As a result:Achilles will never catch the tortoise! The problem can be resolved if we recognize thatalthough an infinite number of instants of time will have elapsed before Achilles catches thetortoise, it is not necessarily true that an infinite number of instants of time must add up tomake an infinitely long time.In modern science the term ‗paradox‘ is usually reserved for a counter- intuitive finding thatis believed to shed light upon something fundamental. Thus we have the ‗twin paradox‘ ofrelativity, Schrödinger‘s ‗cat paradox‘, the ‗Einstein—Podolsky—Rosen (EPR) paradox‘, the‗Klein paradox; of quantum field theory, and the paradox of ‗Wigner‘s Friend‘ in quantummeasurement. These ‗paradoxes‘ may be created by some incompleteness of our knowledgeof what is going on, either at the level of the theory supposed to describe it, or in thespecification of the state of affairs that is observed. Alternatively, they may appearparadoxical only because our expectations are simply wrong and derive from very limitedexperience of reality (as in the case of the ‗twin paradox‘). We can expect that further d ofour understanding will either resolve the apparent paradox or reveal that there is in fact noparadox.Linguistic and logical paradoxes are not like this at all. They are simple enough for everyoneto appreciate. They affect the very tools that we use to think about everything and aretherefore more deeply disturbing. Logic seems to be the final stop for human thinking. Wecan reduce science to mathematics and mathematics to logic, but there seems to be nothing towhich we might reduce logic. The buck stops there.Logical paradoxes have a long history. The most famous is repeated by St Paul in his Epistleto Titus when he remarks that ‗all Cretans are liars, one of their own poets has said so.‘ Thisis the Epimenides (or ‗Liar‘) Paradox. For centuries such paradoxes appeared to be little morethan isolated curiosities that could safely be ignored because they never seemed to arise insituations of practical importance. But during the twentieth century their importance hasgrown into something fundamental. They are consequences of logical structures which arecomplex enough to permit self-reference but arise when we are insufficiently careful todistinguish statements made in a particular language from those made in another language.Far from confining the linguistic paradoxes to the world of triviality, this distinction ends up
  • by giving them a central role in formal proofs of the logical incompleteness of logicalsystems.One of the most notable modern thinkers to be troubled by paradoxes was the philosopherBertrand Russell, who wrote about his discovery, in June 1901, that logic contains afundamental inconsistency. Subsequently, it became known as the ‗Russell Paradox‘It seems to me that a class sometimes is, and sometimes is not, a member of itself.The class of teaspoons, for example, is not another teaspoon, but the class of things that arenot teaspoons, is one of the things that are not teaspoons. . . [Let me to consider the classesthat are not members of themselves; and these, it seemed, must form a class. I asked myselfwhether this class is a member of itself or not. If it is a member of itself, it must possess thedefining properties of the class, which is to be not a member of it. If it is not a member ofitself, it must not possess the defining property of the class, and therefore must be a memberof itself. Thus each alternative leads to its opposite and there is a contradiction.The most memorable formulation that Russell gave to this difficulty of the set of all sets thatare not members of themselves was to tell us of a town in which there is a barber who shavesall those who do not shave themselves. Who shaves the barber? What worried Russell somuch about this paradox was its infiltration of logic itself. If any logical contradiction existsit can be employed to deduce that anything is true. The entire edifice of human reasoningwould fall. Russell was deeply pessimistic of the outcome:Every morning I would sit down before a blank sheet of paper. Throughout the day, with abrief interval for lunch, I would stare at the blank sheet. Often whenEvening came it was still empty . . . it seemed quite likely that the whole of the rest of my lifemight be consumed in looking at that blank sheet of paper. What m it more annoying was thatthe contradictions were trivial, and that my time was spent in considering matters thatseemed unworthy of serious attention.Later, we shall discover that these seemingly innocuous linguistic paradoxes revealed thepresence of profound problems for the whole of logic and mathematics, showing there to be atrade-off between our ability to determine whether statements are true or false and our abilityto show that the system of reasoning we are employing is self-consistent. We can have one orthe other, but not both. We shall find that there are limits to what mathematics can do for us:“Limits that are not merely consequences of human fallibility.”
  • Zeno’s ParadoxesZeno of Elea was an ancient Greek (born around 490 B.C.) who lived in what is nowsouthern Italy. He became a disciple of the philosopher Parmenides, a philosopher whowent around telling people that reality was an absolute, unchanging whole, and thattherefore many things we take for granted, such as motion and plurality, were simplyillusions. This kind of thing must no doubt have brought on ridicule from the morecommon-sensical Eleatics, and so Zeno set out to defend his master‘s position byinventing ingenious problems for the common-sense view. Ever since then, Zeno‘sparadoxes have been debated by philosophers and mathematicians.Space Paradox―If there is such a thing as space, it will be in something, for all being is in something,and that which is in something is in some space. So this space will be in a space, and soon ad infinitum. Accordingly, there is no such thing as space.”Aristotle analyzed four paradoxes of motion: the Racetrack (or Dichotomy), Achilles and theTortoise, the Arrow, and the Stadium (or Moving Rows).The Racetrack (or Dichotomy)One can never reach the end of a racecourse, for in order to do so one would first have toreach the halfway mark, then the halfway mark of the remaining half, then the halfway markof the final fourth, then of the final eighth, and so on ad infinitum. Since this series offractions is infinite, one can never hope to get through the entire length of the track (at leastnot in a finite time). Start ____________________1/2__________3/4_____7/8__15/16... FinnishBut things get even worse than this. Just as one cannot reach the end of the racecourse, onecannot even begin to run. For before one could reach the halfway point, one would have toreach the 1/4 mark, and before that the 1/8 mark, etc. As there is no first point in this series,one can never really get started (this is known as the Reverse Dichotomy).Achilles and the TortoiseAchilles and the tortoise is similar. Suppose that the swift Achilles is having a race with atortoise. Since the tortoise is much slower, she gets a head start. When the tortoise hasreached a given point a, Achilles starts. But by the time Achilles reaches a, the tortoise hasalready moved beyond point a, to point b. And by the time Achilles reaches b the tortoise hasalready moved a little bit farther along, to point c. Since this process goes on indefinitely,Achilles can never catch up with the tortoise.The ArrowAn arrow in flight is really at rest. For at every point in its flight, the arrow must occupy alength of space exactly equal to its own length. After all, it cannot occupy a greater length,nor a lesser one. But the arrow cannot move within this length it occupies. It would needextra space in which to move, and it of course has none. So at every point in its flight, thearrow is at rest. And if it is at rest at every moment in its flight, then it follows that it is at restduring the entire flight.The StadiumWith reference to equal bodies moving in opposite directions past equal bodies inthe stadium with equal speed, some form the end of the stadium, others from themiddle, Zeno thinks half the time equal to twice the time.
  • Physical ParadoxesIn physics there are never any real paradoxes because there is only one correct answer; atleast we believe that the nature will act in only one way (that is the right way, naturally). Soin physics a paradox is only confusion in our own understanding.Here is our paradox-1:Imagine that we construct a device like that shown in the figure. There is a thin, circularplastic disc supported on a concentric shaft with excellent bearings, so that it‘s quite free torotate. On the disc is a coil of wire in the form of a short solenoid concentric with the axis ofrotation. This solenoid carries a steady current I provided by small battery, also mounted onthe disc. Near the edge of the disc and spaced uniformly around its circumference are anumber of small metal spheres insulated from each other and from the solenoid by the plasticmaterial of the disc. Each of these small conducting spheres is charged with the sameelectrostatic charge q. Everything is quite stationary, and the disc is at rest. Suppose now thatby some accident –or by prearrangement- the current in the solenoid is interrupted, without,however, any intervention from the outside. So long as the current continued, there was amagnetic flux through the solenoid more or less parallel to the axis of the disc. When thecurrent is interrupted, this flux must go to zero. There will, therefore, be an electric fieldinduced which will circulate around in circles centered at the axis. The charged spheres onthe perimeter of the disc will all experience an electric field tangential to the perimeter of thedisc. This electric force is in the same sense for all the charges and so will result in net torqueon the disc. From these arguments we would expect that as the current in the solenoiddisappears, the disc would begin to rotate. If we knew the moment of inertia of the disc, thecurrent in the solenoid, and charges on the small spheres, we could compute the resultingangular velocity.But we could also make a different argument. Using the principle of the conservation ofangular momentum, we could say that the angular momentum of the disc with all itsequipment is initially zero. There should be no rotation when the current is stopped. Whichargument is correct? Will the disc rotate or will it not? We will leave this question for you tothink about.We should warn you that the correct answer does not depend on any non-essential feature,such as asymmetric position of a battery, for example. In fact, you can imagine an idealsituation such as the following: The solenoid is made of superconducting wire through whichthere is a current. After the disc has been carefully placed at rest, the temperature of thesolenoid is allowed to rise slowly. When the temperature of the wire reaches the transitiontemperature between superconductivity and normal conductivity, the current in the solenoidwill be brought to zero by the resistance of the wire. The flux will, as before, fall to zero, andthere will be an electric field around the axis. When you figure it out, you will havediscovered an important principle of electromagnetism.
  • Paradox-2:Attempts to construct perpetual motion machines still continue in our days, too. It is knownthat in the absence of a dielectric field electrons moving in any conductor are in a state ofperpetual motion. The total randomness of the motion might lead to a situation in which theupper part of a conductor contains amore electrons than the lower one. Here the question iselectron density fluctuations. These fluctuations will result in a potential difference across theconductor ends which can be used to charge a capacitor. A detector will prevent the capacitorfrom discharging when the potential difference across the conductor ends changes sign. Thecharged capacitor can then be used as a source of ―gratuitous‖ energy. The power will ofcoarse be low, but it is the principle that is important.Paradox-3:A device consuming 50w is connected through another resistor of 40 ohm to a power supplyof 120v.Let us calculate the current through the device from these data.To solve the problem, note that the voltage across the device and the voltage across the otherresistor must be equal to the sum of the network voltage, i.e.Udev + Ures =UnetBy expressing the first term on the left hand side in terms of the power consumed by thedevice divided by the current running through it and second term, as the product of the otherresistance and the same current , we obtain the following equation:W/I + IR = UnetAll the quantities here except for the current are known. By substituting in the numericalvalues we get50/I + 40I = 120If we solve this quadratic equation we obtain two values for the current, Viz. I1=0.5A andI2=2.5 A
  • Bibliography Gödel, Escher, Bach: an Eternal Golden Braid by Douglas R. Hofstadter. Physical Paradoxes and Sophisms by V.N Lange. Lectures in Physics by Richard Feynman. Impossibility-Limits of Science And Science of Limits by John D Barrow