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DZn_^jZZg]ebckdh]hyaudZ
F_lh^bq_kdb_mdZaZgby
ihZg]ebckdhfmyaudm ^eyklm^_glh\dmjkZfZl_fZlbq_kdh]hnZdmevl_lZ
KhklZ\bl_ebklij_iIjZ\rbgZBF ij_iIjZ\rbg:<
2
F_lh^bq_kdZyaZibkdZ GZklhysb_ f_lh^bq_kdb_ mdZaZgby ij_^klZ\eyxl kh[hc ih^[hj hjb]bgZevguo l_dklh\ ih ki_pbZevghklb ^ey klm^_glh\ fZl_fZlbq_kdh]h nZdmevl_lZ jy^ mijZ`g_gbc k p_evx h[h]Zlblv klm^_glh\ l_jfbgheh]bq_kdhc e_dkbdhc gZmqblv qblZlv fZl_fZlbq_kdb_ kbf\heu kem`blv klbfmehf ^ey fhgheh]bq_kdbob^bZeh]bq_kdbo\ukdZau\Zgbc
3
TEXT 1 The Language of Mathematics Mathematics is not so much a body of Knowledge as a special Kind of language, one so perfect and abstract that – hopefully – it may be understood by intelligent creatures throughout the universerse, however different their organs of sense and perception. The grammar of the language – its proper usage – is determined by the rules of logic. Its vocabulary consists of symbols, such as: numerals for numbers letters for unknown numbers equations for relationships between numbers, π for the ratio of the circumference to the diameter of a circle; sin (for sine), cos (for cosine) and tan (for tangent) for the ratios between sides in a right triangle; √ for a square root ∞ for infinity β, λ, ∑ - or assorted other concepts in higher mathematics. All of these symbols are tremendously helpful to the scientist because they serve to short-cut his thinking. Only part of the vocabulary of mathematics is preempted by science. The rest of it – and all of the grammar – remains in the sphere of general human thought. Indeed, mathematics has as much to do with philosophy, economics, military strategy, musical composition, artistic perspective and parlor games as it has to do with atomic physics. Because of its virtuosity, anyone well taught in it can love it with the same warmth that a devotee feels for the ballet, fine silver, antiques or any other adornment of civilization. A. Pronounce correctly: assort [ ∋‘s :t] circumference [sw‘k∧mf wUwQV@ intelligent [in’telid wnt] OD\PDQ>µOHLPwQ@ parlour [‘pa:l ] precise [pri’sais] ratio [‘rei∫iou] root [ru:t] tangent [‘tænd ∋nt]
-
]jmiibjh\ZlvdeZkkbnbpbjh\Zlv hdjm`ghklv jZamfguc g_ijhn_kkbhgZe kZehg lhqguc hlghr_gb_ dhj_gv dZkZl_evguc lZg]_gk
4
B. Answer the questions: 1. 2. 3. 4. 5.
May mathematics be understood by all intelligent creatures? What does its vocabulary consist of? Why are all of these symbols helpful to the scientist? Does the rest of math vocabulary remain in the sphere of general human thought? What fields of Knowledge does mathematics have much in common?
C. Learn to read math symbols. $.>µælfw ] %>µEL Ww@ => Jæmw@ ∆ /> GHOWw@ ?0> >HSVLOwQ@ =>µ]L Ww@ +>µL Ww@ Θ>µL Ww@ I ι [DL¶RXWw@ K χ [‘kæpw@ Λ λ ['Læmdw@ Fµ [mju:] Ν η [nju:] Ξ ξ [zai] [Ksi:]
-
ZevnZ [_lZ ]ZffZ ^_evlZ wikbehg a_lZ wlZ l_lZ
-
chlZ dZieZ eZf[^Z fx gx dkb
The Greek Alphabet
TEXT 2 Complex Numbers Mathematicians customarily write ¥-1 in such numbers as i, and any complex numbers as a +bi. Ordinary numbers call all be thought of as lying along a single straight line, a continuous stream without gaps in it – what mathematicians call a “continuum”. But a typical complex number a + bi, has no place on the line of ordinary numbers. When two ordinary members are multiplied, the result is a jump along the straight line. When two complex numbers are multiplied, however, the result is a spectacular trapeze like swing within the two-dimensional plane. The excentric behaviour of the complex numbers is important because it matches perfectly – and therefore serves as a literal translation of the behaviour of
5 many quantities in nabure, such as forces, velocities or accelerations, which act in definite directions. Numbers which serve to represent forces, velocities and accelerations acting in more than two dimensions are “hypercomplex numbers” – expressions like a + bi + cj + dk, in which the units i, j and k when multiplied together, produce minus one. The most astonishing thing about these hypercomplex numbers is that they flout a basic role of arithmetic previously thought inviolate. When multiplied together, the same two hypercomplex numbers may produce different results, depending on the order in which they are taken; hypercomplex number “a” times hypercomplex number “b” does not always equal hypercomplex “b” times hypercomplex “a”. A. Pronounce correctly: acceleration [ æk selw‘reiwQ@ DSSO\>wµSODL@ match [mæt@ velocity [vi’l siti ]
-
mkdhj_gb_ ijbdeZ^u\Zlv ih^[bjZlv kdhjhklv
B. Answer the questions: 1. 2. 3. 4. 5. 6.
What do mathematicians call a “continuum”? What is the result when two ordinary numbers are multiplied? Why is the excentric behaviour of the complex numbers important? How does it serve? What numbers are called “hypercomplex numbers”? What is the most astonishing thing about hypercomplex numbers?
C. Learn to read math symbols.
O o [ mikrwn ] I>SDL@ J!>URX@ />VLTPw ] L2>W@ ¥ >MXSVLOwQ ] N3>IDL@ O$>KDL@ Ψ%>VDL@
&>PL Tw ]
-
hfbdjhg ib jh kb]fZ lZm xikbihg nb ob ikb hf_]Z
The Geek alphabet
6
TEXT 3 The Main Principles of Axiomatic Methods The axiomatic method consists simply in making a complete collection of the basic concepts as well as the basic facts from which all concepts and theorems of a science can be derived by definition and deduction respectively. If this is possible, then the scientific theory in question is said to be definite according to Husseri. Such is the case for the theory of space. Of course, from the axioms of geometry we cannot possible deduce the law of gravitation. Similarly the axioms of geometry fail to disclose whether Zurich is farther from Hamburg than Paris. Though the question deals with a geometrical relation, the relation is one between individually exhibited locations. Thus precisely speaking, what is supposed to be deducible from the axioms are the pertinent general true prepositions. An axiom system must under all circumstances be free from contradictions, in which case it is called consistent; that is to say, it must be certain that logical inference will never lead from the axioms to a proposition “a” while some other proof will yield the opposite proposition “a”. If the axioms reflect the truth regarding some field of objects, then, indeed, there can be no doubt as to their consistency. But the facts do not always answer our questions as unmistakably as might be desirable, a scientific theory rarely provides a faithful redition of the data but is almost invariably a bold construction. Therefore the testing for consistency is an important check; this task is laid into the mathematician’s hands. Not indispensable but desirable is the independence of the individual axioms of an axioms system. It should contain no superfluous components, no statements which are already demonstrable on the basis of the other axioms. The question of independence is closely connected with that of consistence, for the proposition “a” is independent of a given set of axioms if and only if the proposition “a” is consistent with them. The dependence of a proposition “a” on other propositions A (an axiom system) is established as soon as a concrete proof of “a” on the basis of A is given. In order to establish the independence on the other hand, it is required to make sure that no combination of inferences, however intricate, is capable of yielding the proposition “a”. There are some methods at one’s disposal of reaching this goal; by what has been said above, each of them qualifies also for proving the consistency of an axiom system. (1) The first method is based on the following principle: if “a” contains a new original concept, not defined in terms of those occurring in A, then “a” cannot be a consequence of A. For example: a ship is 250 feet long and 60 feet wide; how old is its captain? Only in the most trivial cases does this simple idea accomplish our objective.
7 (2) The construction of a model. Objects and relations are exhibited which, upon suitable naming, satisfy all of the propositions A, and yet fail to satisfy “a”. This method has been the most successful so far invented. A. Pronounce correctly: pertinent proposition inference yield consistency rendition bold construction indispensable intricate trivial
[pw:tinwnt] [pr pwziwn] [LQIwrwns] [ji:ld] [kwn′sistwnsi ] [rendi∫wn] [bould] [indis′penswbl] [ ′intrikit] [ ′triviwl]
-
mf_klguc ml\_j`^_gb_ \u\h^ ^Z\ZlvmklmiZlv ihke_^h\Zl_evghklv bgl_jij_lZpby kf_euc g_h[oh^bfh keh`guc h[uqguc
B. Answer the questions: 1. What are the main aspects of the axiomatic method? 2. What must any system of axioms be free from? 3. Why is the question of independence of axioms closely connected with that of consistency? 4. How do we establish the independence of axioms? 5. Why is it necessary to test the consistency of any set of axioms? 6. What is the meaning of the term “completeness” of a set of axioms? C. Learn to read math symbols. a=b
a equals b a is equal to b
a≠b
a doesn’t equal b a isn’t equal to b
a ∈A
a belongs to A capital
a ∉A
a doesn’t belong to A capital
A⊂B
A is contained in B
B⊃C A⊄B
B contains C A isn’t contained in B
8
TEXT 4 Boole I “Oh, we never read anything the English mathematicians do.” This characteristically Continental remark was the reply of a distinguished European mathematician when he was asked whether he had seen some recent work of one of the leading English mathematicians. The “we” of his superiority included Continental mathematicians in general. This is not the sort of story that mathematicians like to tell on themselves, but as it illustrates that characteristic of British mathematicians – their originality – which has been the chief claim to distinction of the British school, it is an ideal introduction to the life and work of one of the most original mathematicians England has produced, George Boole. The fact is that British mathematicians have often gone their own way, doing the things that interested them personally as if they were playing cricket for their own amusement only, with a self-satisfied disregard for what others have assured the world is of supreme importance. Sometimes indifference to the leading fashions of the moment has cost the British school dearly, but in the long run the take-it-or-leave-it attitude of this school has added more new fields to mathematics than an exact imitation of the Continental masters could ever have done. The theory of invariance is a case in point; Maxwell’s electrodynamic field theory is another. Although the British school has had its share of powerful developers of work started elsewhere, its greater contribution to the progress of mathematics has been in the direction of originality. Boole’s work is a striking illustration of this. When first put out it was ignored as mathematics, except by a few, chiefly Boole’s own countrymen, who recognized that here was something of great interest fofr all mathematics. Today the natural development of what Boole started is rapidly becoming one of the major divisions of pure mathematics, with a number of workers in practically all countries extending it to all fields of mathematics. As Bertrand Russel remarked some years ago, pure mathematics was discovered by George Boole in his work The Laws of Thought published in 1854. This may be an exaggeration, but it gives a measure of the importance in which mathematical logic and its branches are held today. Others before Boole, notably Leibniz and De Morgan, had dreamed of adding logic itself to the domain of algebra; Boole did it. II George Boole was born in 1815 at Lincoln, England, and was the son of a small trader. If we can credit the picture drawn by English writers themselves of those old days – 1815 was the year of Waterloo – the whole class to which Boole’s father belonged was treated with a contempt. The “lower classes”, into whose ranks Boole
9 had been born, simply did not exist in the eyes of the “upper classes”. It was taken for granted that a child in Boole’s station should dutifully and gratefully master the shorter catechism and so live as never to go beyond the strict limits of obedience. Boole was permitted to attend the school that was designed with the end in view of keeping the poor in their proper place. Of course no Latin was taught there. Boole decided that he must learn Latin and Greek. By the age of twelve he had mastered enough Latin to translate an ode of Horace into English verse. Boole got his early mathematical instruction from his father, who had gone considerably beyond his own meagre schooling by private study. The father had also tried to interest his son in another hobby, that of making optical instruments. After finishing his common schooling Boole took a commercial course. By the age of sixteen he saw that he must contribute at once to the support of his parents. School teaching offered the most immediate opportunity of earning steady wages – in Boole’s day assistant teachers were not paid salaries but wages. There is more than a monetary difference between the two. He taught at two schools. Boole spent four more or less happy years teaching in these elementary schools. The nights, at least, long after the pupils were safely asleep, were his own. At last he found himself. His father’s early instruction now bore fruit. In his twentieth year Boole opened up a civilized school of his own. To prepare his pupils properly he had to teach them some mathematics as it should be taught. His interest was aroused. Soon the ordinary textbooks of the day awoke his wonder, then his contempt. Was this nonsense mathematics? Incredible. What did the great masters of mathematics say? Like Abel and Galois, Boole went directly to the masters. It must be remembered that he had had no mathematical training beyond the rudiments. To get some idea of his mental capacity we can imagine the lonely student of twenty mastering, by his own unaided efforts, the Mécanique céleste of Laplace, one of the toughest masterpieces ever written for a student to assimilate. A. Pronounce correctly: I distinguished superiority originality claim supreme indifference invarience share pure exaggeration domain
>GLV¶WLTZLW@ [sju,piwri’ riti] [w,rid i’næliti] [kleim] [sju’pri:m] [in’difrwns] >LQ¶Y0wriwns] [ 0w] [pjuw] [iq’zæd wreiLwn] [dw’mein]
-
\u^Zxsbcky ij_\hkoh^kl\h hjb]bgZevghklv ij_l_g^h\Zlv ij_\hkoh^guc jZ\gh^mrguc bg\ZjbZglghklv ^heyqZklv qbkluc ij_m\_ebq_gb_ h[eZklv
10
II credit treat take for granted catechism obedience meagre arouse incredible able aid tough assimilate
[‘kredit] [tri:t] [qra: ntid] [kætikizwm] [w’bi:djens] [‘mi:qw] [w’rauz] [in’kredwbl] [eibl] [eid] [t∧f] [w’simileit]
^h\_jb_ ljZdlh\Zlv kqblZlvkZfhkh[hcjZamf_xsbfky k\h^ijZ\be ih^qbg_gb_ g_agZqbl_evguc \hagbdZlv g_\_jhylguc kihkh[guc ihfhsv `_kldbc mk\Zb\Zlv
-
B. Answer the questions: I 1. 2. 3. 4. 5. 6. 7. 8.
What was the Continental remark of a distinguished European mathematician? What does this story illustrate? What way have British mathematicians often gone? Has this indifference cost the British school dearly? What new fields of research has the British school added to mathematics? Is Boole’s work a striking illustration of originality? What did Boole’s own contrymen recognize? What was discovered by George Boole? II
1. When was George Boole born? 2. How was the whole class to which Boole’s father belonged treated with a contempt? 3. Was Boole permitted to attend the school? 4. Had Boole’s father gone considerably beyond his own meagre schooling by private study? 5. What did he see by the age of 16? 6. Where did Boole spend four happy years? 7. When did Boole open up a school of his own? 8. What did he have to do to prepare his pupils properly?
11 C. Learn to read math symbols: a (B)
a of B
K ∉φK
K doesn’t belong to the empty subset of K
A′ ⊂ A
the set A prime is contained the set A
A ⊃ A′
the set A contains the set A prime
A ∪ A′
the union of A and A prime
A ∩ A′
the intersection of A and A prime
TEXT 5 Sets and Propositions. Propositional Algebra I Let us come back to the Boolean algebra of sets which plays the most important role in the present book. Let us discuss the methods for specifying the sets which are the elements of this algebra. It is obvious that the simplest method is to specify a given set by tabulation, that is by enumerating all the elements of the set. For instance, we can consider the “set of the pupils: Peter, John, Tom and Mary” or the “set of the numbers: 1, 2, 3, 4” or the “set of the four operations of arithmetic: addition, subtraction, multiplication and division”. In mathematics the elements of a set which is defined by tabulation are usually written in curly brackets; for instance, the sets we have mentioned can be written as A = {Peter, John, Tom, Mary} B = {1, 2, 3, 4, 5} and C = {+, ×, : } (in the last expression the signs of the operations symbolize the operations themselves). However, this method of representing a set is highly inconvenient in case there are very many elements in the set; it becomes completely inapplicable when the set in
12 question is infinite (we cannot enumerate an infinite number of the elements of the set!). Besides, even in those cases when a set can be defined by tabulation and the tabulation is quite simple it may nevertheless happen that the enumeration itself does not indicate why these elements are collected to form the set. II Therefore another method which specifies the sets implicitly by description is more widely used. When a set is defined by description we indicate a property characterizing all the elements of the set. For instance, we can consider the “set of all excellent pupils in your class” (it may turn out that the set A mentioned above coincides with this set of excellent pupils) or the “set of all integers X such that O ≤ X ≤ 5” (this set exactly coincides with the set B mentioned above) or the “set of all animals in a zoo”. The descriptive method for specifying sets is quite applicable for the definition of infinite sets such as the “set of all integers” or the “set of all triangles with area equal to 1”; moreover, as has been mentioned, infinite sets can be defined by description only. The descriptive method of representing sets connects the sets with propositions which are studied in mathematical logic. Namely, the essence of the method is that we fix a collection of the objects we are interested in (for instance, the collection of the pupils in your class or the collection of the integral numbers) and then state a proposition which is true for all the elements of a set under consideration and only for these elements. For instance, if we are interested in the sets whose elements are some (or all) pupils in your class then such propositions can be “he is an excellent pupil”, “he is a chess-player”, “his name is George” and the like. The set A of all those elements of the universal set I in question (for instance, the set of the pupils, the set of the numbers, etc.) which satisfy the condition mentioned as the characteristic property in a given proposition a is called the truth set of this proposition). Thus, there is a “two-way connection” between sets and propositions: every set is described by a proposition (in particular, such a proposition may simply reduce to the enumeration of the elements of a given set, for instance, “the name of the pupil is Peter or John, or Tom, or Mary”) and to every proposition there corresponds a definite set. A. Pronounce correctly: I preposition specify enumerate curly bracket inconvenient inapplicable
[ prwpw’ziwn] [‘spesifZi] [i’nju:mwreit] [kwili brækit] [inkwn’vi:njwnt] [in′wpi:kwbl]
\ukdZau\Zgb_ \u^_eblv \uqbkeylv nb]mjgZykdh[dZ g_m^h[guc g_ijb_fe_fuc
13 infinite define
[in’finit] di’fain]
- [_kdhg_qguc - hij_^_eylv
II implicitly coincide applicable under property two-way connection
[im’plisitli] [ koun’said] [‘ æplikwbl] [∧ndw] [‘pr pwti] [tu wei kw′nek∫wn]
- [_ah]h\hjhqgh - kh\iZ^Zlv - ijb_fe_fuc - ih^ - k\hckl\h - ^\mklhjhggyy k\yav
B. Answer the questions: I 1. 2. 3. 4. 5.
What methods do we discuss? What is the simplest method? How are the elements of set usually written? When is this method of representing a set highly inconvenient? Can the enumeration indicate why these elements are collected to form the set? II
1. 2. 3. 4. 5. 6.
Is another method widely used? What do we indicate when a set is defined by description? What is the descriptive method for specifying sets applicable for? Can infinite sets be defined by description only? What does the descriptive method of representing sets connect? Is there a “two-way connection” between sets and prepositions?
C. Learn to read math symbols: A−B
the difference of A and B
CA B
the complement of B with respect to A
A×B
the product of A and B
F (B) = X
the image of the set B by the mapping F coincides with X
a ∈ F-1(b)
the element a belongs to the inverse image of the element b
14 A → (a1 F(a))
the mapping which establishes a correspondence between the pair (a1 F(a)) and A
(Aβ)β ∈ B
the family of the elements A sub β denoted by the elements of the set β
Text 6 Non-Metric Versus Metric I We must now see very sketchily how topology as a species of geometry differs from both the common Euclidean geometry of the elementary high-school course and the classical projective geometry of a college course. It is also far different from the extension of Descartes’ geometry to that of a space of more than three dimensions. A root of the fundamental distinction between topology and other kinds of geometry is in the kinds of permissible transformations. To see the difference, we recall what we were allowed to do in school geometry. It was assumed there that triangles and other figures could be slid about in a plane without alteration of measures of distance – lengths of sides – and sizes of angles. It was also sometimes tacitly assumed that one triangle could be lifted out of a plane and then be superimposed on another without having undergone distortions of measurements in the process. Neither assumption is justified without explicit postulation, as there are “spaces” and “geometries” in which at least one is not permissible. Euclid, evidently relying on what to him seemed obvious, omitted both assumptions, among several others, equally “obvious”, from his list of postulates. But in projective geometry the lengths of the sides of the figures projected into others had no relevance for the theorems sought and exhibited. As a simple but sufficient example, if two lines in the original figure intersect they continue to intersect in the projection. (The sophisticated reader will make allowances for the region at infinity and the full projective plane. The unsophisticated reader will get the essential sense of what has been said from his visual experience. This is enough for an account like the present.) In central projection, for instance, the projected figures are the shadows of others thrown on a screen by a pointsource of light. Among the permissible transformations in Euclidean geometry are those allowing rigid bodies to be moved about freely in space without distortion. In projective geometry when phrased algebraically the permissible transformations are linear (of the first degree) in the coordinates. In topology the transformations are not restricted to be even algebraic, and it is required to find what remains invariant under certain very general types of transformations. These will be accurately described later.
15 Topology originated in what now would be called mathematical recreations. I shall recall only two of several. It seems that in the eighteenth century two islands in the Pregel were connected by a bridge and with Königsberg by six bridges, and some disturbing soul asked whether it was possible to start from the mainland and traverse each of the seven bridges, each one only once, and return to the mainland. As a matter of fact it was impossible, but this required proof. Clearly it would not affect the problem if the island and the mainland were swelled or shrunk without touching, and if the seven bridges were twisted in any way so long only as no two of them were permitted to intersect. Euler (1736) disposed of this puzzle. The second recreation, so-called, goes very much deeper. A solution was still lacking in 1950. Practical cartographers had noticed that a map of any plane area, say that of a continent cut up into countries, could be coloured in at most four coloures. In such a map the same colour may be used for each of several different countries provided no two touching along a line boundary are coloured alike. The ocean, of course, is counted as just another country, surrounding all the others if the land mapped is an island. The reader may easily construct a map which requires exactly four colours. The problem is to prove that, no matter how complicated the map, four colours suffice. The problem was noticed as early as 1840 by Moebius, and again about 1850 by A. De Morgan, a founder with Boole of modern logic. Cayley (1878) advertised the problem to professional mathematicians and commended it to their attention. He evidently has spent some time on it. Several attempts to prove the sufficiency of only four colours for a plane map followed Cayley’s appeal for a rigorous solution. Some seemed promising, but all were incomplete or fallacious. It makes no difference to the problem, as in that of the Königsberg bridges, whether the map to be coloured is deformed in any way that introduces no new boundaries. Nor does it matter whether the map is drawn on a plane or on a sphere. But as the reader will see, the map problem on a doughnut differs from that on a plane – the connectivities are not the same for a plane and a doughnut. The problem is thus one of topology, that division of modern geometry which deals with the properties of figures unchanged by continuos deformations. The map problem belongs to what is called combinatorial topology, in which continuity, as for functions of a real variable is of comparatively minor importance. II Definite progress in the four-colour problems is disappointing. It has been limited to proving that solutions exist for a map with a given rather small number of regions. The labored record up to 1940 does not seem to have been beaten; for any map of thirty-five or fewer regions four colours suffice. The problem, like Fermat‘s, is one of those easily stated and deceptively simple things that amateurs had better leave alone. The professionals seem to have given it up. G. D. Birkhoff said shortly before his death that in spite of all his efforts, to crack the four-colour problem wide open he
16 had not even scratched it. Still, some uninhibited explorer breaking a new trail may reach the end tomorrow. There is another very famous topological problem that at a first look seems as hard as the map problem. In fact Poincaré abandoned it, and shortly before his death (1912) proposed it to all mathematicians as a question worth their consideration. For about a year it was called “Poincaré’s Last Theorem”, in analogy with Fermat’s because it was anticipated that whatever Poincaré could not settle must be really likely to remain a challenge for many years. It is given that a continuous one-one transformation takes the ring bounded by two concentric circles into itself in such a way as to advance the points of the outer circle positively and those of the inner circle negatively, and at the same time to preserve areas. It is to be proved that there are at least two points invariant (left lixed) under this transformation. This may seem like a useless puzzle. Actually Poincaré has reduced a difficult problem in dynamical astronomy – the restricted problem of three bodies – to this one in topology. A. Pronounce correctly: I [slid , slaid] slid or slide superimpose >VMXSw′impouz] omit [w′mit] relevance [′relivwns] infinity [in′finiti] infinite [′infinit] [dis′pouz] dispose [sw′fis] suffice commend [kw′mend] doughnut [dou′n∧t] connectivity [kw′nektiviti] continuity [k nti′nju:iti] II deceptively [di′septivli] amateur [ æmwtw] in spite of crack uninhibited explorer trail abandon anticipate challenge outer circle
[spait] [kræk] [ ∧ninhibitid] [ikspl ′rw] [treil] [w′bændwn] [ æn′ticipeit] [′t∫ælind ] [′autwsw:kl]
-
kdhevablv \deZ^u\Zlvqlh-lhgZqlh-lh himkdZlv hlghr_gb_ [_kdhg_qghklv [_kdhg_qguc
- jZkiheZ]Z_l - o\ZlZlv - hlebqZlv - ihgqbd - k\yav -^ebl_evghklv - h[fZggh - ex[bl_ev g_ijhn_kkbhgZe - g_kfhljygZ - s_edgmlva^jZa]jualv - jZkdh\Zggucg_aZfhj_gguc - bkke_^h\Zl_ev - lZsblvb^lbihke_^m - ihdbgmlv - ihgbfZlv - [jhkZlv\uah\ -\g_rgbcdjm]
17 B. Answer the questions: I 1. 2. 3. 4. 5. 6. 7. 8. 9.
What is the fundamental difference between topology and other geomentries? What transformations are permissible in the common Euclidean geometry? What requirements are imposed on the transformations in topology? Are the transformations in topology restricted? What did topology originate? What is the puzzle of two islands and bridges? What is the second recreation? Who was the first to deal with map colouring problem? Is the four-colour problem solved now? II
1. What did Birkhoff say? 2. Is there another very famous topological problem that seems as hard as the map problem? 3. What problem was called “Poincaré’s Last Problem”? 4. Did Poincaré abandon it? 5. What is given in the problem? 6. What must be proved?
C. Learn to read math symbols: a En
- the number a is less than b sub n or equal to b sub n
|x| < n
- the module (the absolute value) of x is less than n
Y = {y}
- the set Y contains the unique element y
y ⇒
- y tends to infinity or y is tending to infinity
sin y = h
- the sine of y is equal to h
b (¥ - b of the square root of two equals two
18
TEXT 7 Common Sense and the Universe I Speaking last December (1941) at the annual convention of the American Association for the Advancement of Science, Professor Edwin Hubble, of the Mount Wilson Observatory, made the glad announcement that the universe is not expanding. This was good news indeed, if not to the general public who had no reason to suspect that it was expanding, at least to those of us who humbly attempt to “follow science”. For some twenty-five years past, indeed ever since the promulgation of this terrific idea in a paper published by Professor W. de Sitter in 1917, we had lived as best we could in an expanding universe, one in which everything else. It suggested to us the disappointed lover in the romance who leaped on his horse and rode madly off in all directions. The idea was majestic in its sheer size, but it somethow gave an uncomfortable sensation. Yet we had to believe it. Thus, for example, we had it on the authority of Dr. Spencer Jones, the British Astronomer Royal, as recently as in his new and fascinating book of 1940, “Life on Other Worlds”, that “a distant universe in the constellation of Boots has been found to be receding with a velocity of 24,300 miles a second. We can infer that this nebula is at a distance of 230,000,000 light-years”. I may perhaps remind my fellow followers of science that a light-year means the distance traveled in one year by light, moving at 186,000 miles a second. In other words, this “distant universe” is now 1,049,970,980,000,000, 000,000 miles away. Some distance! as Mr. Churchill would say. But now it appears that that distant universe has not been receding at all; in fact, it isn’t away out there. Heaven knows where it is. Bring it back. Yet not only did the astronomers assert the expansion but they proved it, from the behavior of the red band in the spectrum, which blushed a deeper red at the revelation of it, “The Expanding Universe”, to bring it down to our level. Astronomers at large accepted this universal explosion in all directions as calmly as they once accepted the universal fall of gravitation, or the universal death in the cold under Carnot’s Second Law of Thermodynamics. But the relief brought by Professor Hubble is tempered by certain doubts and afterthoughts. It is not that I have any disbelief or disrespect toward science. But we begin to doubt whether science can quite keep on believing in and respecting itself. If we expand today and contract tomorrow; if we get reconciled to dying a martyr’s death at one general temperature of 459 degrees below zero, the same for all, only to find that the world is perhaps unexpectedly warming up again – then we ask, where are we? To which, of course, Einstein answers “Nowhere”, since there is no place to be. So we must pick up our little book again, follow science and wait for the next astronomical convention.
19 Let us take this case of the famous Second Law of Thermodynamics, which condemned the universe – or at least all life in it – to die of cold. I look back now with regret to the needless tears I have wasted over that, the generous sympathy for the last little band of survivors, dying at 459 degrees below our zero (-273 0 centigrade), the absolute zero of cold when the molecules cease to move and heat ends. This famous law was first clearly enunciated in 1824 by the great French physicist Nicolas Carnot. It showed that all bodies in the universe kept changing their temperature, hot things heated cold, and cold things chilled hot. Thus they pooled their temperature. Like the division of a rich estate among a flock of poor relations, it meant poverty for all. We must all share ultimately the cold of absolute space. It is true that a gleam of hope came when Ernest Rutherford and others, working on radioactivity, discovered that there might be a contrary process of “stoking up”. Atoms exploding into radioactivity would keep the fires burning in the sun for a long time. This glad news meant that the sun was both much older and much younger than Lord Kelvin had ever thought it was. But even at that it was only a respite. The best they could offer was 1,500,000,000 years. After that we freeze. II One must not confuse Rutherford’s work on atoms with Einstein’s theories of space and time. Even in his later days his life without reference to Einstein Even in his later days at the Cavendish Laboratory at Cambridge when he began, ungratefully, to smash up the atom that had made him, he needed nothing from Einstein. I once asked Rutherford – it was at the height of the populer interest in Einstein, in 1923 – what he thought of Einstein’s relativity. “Oh, that stuff!” he said. “We never bother with that in our work!” His admirable biographer, Professor A.S. Eve, tells us that when the German physicist Wien told Rutherford answered, “No, they have too much sense”. But it was Einstein who made the real trouble. He announced in 1905 that there was no such thing as absolute rest. After that there never was. But it was not till just after the Great War that the reading public caught on to Einstein and little books on “Relativity” covered the bookstalls. Einstein knocked out space and time as Rutherford knocked out matter. The general viewpoint of relativity towards space is very simple. Einstein explains that there is no such place as here. “But”, you answer, “I’m here; here is where I am right now”. But you’re moving, you’re spinning round as the earth spins; and you and the earth are both spinning round the sun, and the sun is rushing through space towards a distant galaxy, and the galaxy itself is beating it away at 26,000 miles a second. Now where is that spot that is here! How did you mark it? You remember the story of the two idiots who were out fishing, and one said, “We should have marked that place where we got all the fish”, and the other said, “I did, I marked it on the boat”. Well, that’s it. That’s here. You can see it better still if you imagine the universe swept absolutely empty: nothing in it, not even you. Now put a point in it, just one point. Where is it? Why,
20 obviously it’s nowhere. If you say it’s right there, where do you mean by there? In which direction is there? In that direction? Oh! hold on, you’re sticking yourself in to make a direction. It’s in no direction; there aren’t any directions. Now put in another point. Which is which? You can’t tell. They both are. One is no the right, you say and one on the left. You keep out of that space! There’s no right and no left. The discovery by Einstein of the curvature of space was greeted by the physicists with the burst of applause. That brilliant writer just mentioned Sir Arthur Eddington, who can handle space and time with the imagery of a poet, and even infiltrate humor into gravitation, as when he says that a man in an elevator falling twenty stories had an ideal opportunity to study gravitation – Sir Arthur Eddington is loud in his acclaim. Without this curve, it appears, things won’t fit into their place. The fly on the globe, as long as he thinks it flat (like Mercator’s map), finds things shifted as by some unaccountable demon to all sorts of wrong distances. Once he gets the idea of a sphere everything comes straight. So with our space. The mystery of gravitation puzzles us, except those who have the luck to fall in an elevator, and even for them knowledge comes too late. They weren’t falling at all: just curving. “Admit a curvature of the world”, wrote Eddington in his Gifford Lectures of 1927, “and the mysterious agency disappears. Einstein has exercised this demon”. But it appears now, fourteen years later, that Einstein doesn’t care if space is curved or not. He can take it either way. A prominent physicists of today, head of the department in one of the greatest universities of the world, wrote me on this point: “Einstein had stronger hopes that a general theory which involved the assumption of a property of space, akin to what is ordinarily called curvature, would be more useful than he now believes to be the case”. Plain talk for a professor. Most people just say Einstein has given up curved space. It’s as if Sir Isaac Newton years after had said, with a yawn, “Oh, about that apple – perhaps it wasn’t falling”. III But we cannot understand the full impact of the Quantum Theory, in shattering the world we lived in, without turning back again to discuss time in a new relation, namely, the forward – and – backwardness of it, and to connect it up again with the Second Law of Thermodynamics – the law, it will be recalled, that condemns us to die of cold. Only we will now call it by its true name, which we had avoided before, as the Law of Entropy. All physicists sooner or later say. “Let us call it Entropy”, just as a man says, when you get to know him, “Call me Charlie”. And now let us try to explain. Entropy means the introduction into things that happen of a random elements, as opposed to things that happen and ‘unhappen’, like a turning wheel, good either way, or a ball falling and bouncing as high as it falls, or the earth going around the sun. These primary motions are “reversible”. As far as they are concerned, time could just as well go back as forward. But now consider a pack of cards fresh from the maker, all in suits, all in order. Shuffle them. Will they ever come all in order again? They might, but they won’t. Entropy.
21 Here then is Entropy, the smashing down of our world by random forces that don’t reverse. The heat and cold of Carnot’s Second Law are just one case of it. This is the only way by which we can distinguish which of two events came first. It’s only clue as to which way time is going. If procrastination is the thief of time, Entropy is the detective. The Quantum Theory beings with the idea that the quantities of “disturbance” in the atom are done up, at least they act that way, in little fixed quantities (each a Quantum – no more, no less), as if sugar only existed by the pound. The smallness of the Quantum is beyond comprehension. A Quantum is also peculiar. A Quantum in an atom flies round in an orbit. This orbit may be a smaller ring or a bigger ring. But when the Quantum shifts from orbit to orbit it does not pass or drift or move from one to the other. No, sir. First, it’s here and then it’s there. Believe it or not, it has just shifted. Its change of place is random, and not because of anything. Now the things that we think of as matter and movements and events (things happening) are all based, infinitely far down, on this randow dance of Quantums. Hence, since you can’t ever tell what a Quantum will do, you can’t ever say what will happen next. Cause and effect are all gone. But as usual in this bright, new world of the new physics, the statement is no sooner made than it is taken back again. There are such a lot of Quantums that we can feel sure that one at least will turn up in the right place – by chance, not by cause.
A. Pronounce correctly:
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22 II reference spin galaxy curvature acclaim
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akin
[ri′fw:rwns] [spin] [gælwksi] [kw:vwt∫w] [wk′leim] [w′kin]
III random procrastinate quantity
[′rændwm] [prou′kræstineit] [kw ntiti]
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B. Answer the questions: I 1. What announcement did Professor Edwin Hubble make? 2. What idea was majestic? 3. What does a light year mean? 4. What does it appear now? 5. How did astronomers accept this universal explosion in all directions? 6. What did the famous Second Law of Thermodynamics condemn the universe? 7. Who was the famous law enunciated by? 8. What did it show? 9. What did Rutherford discover? 10.What did each new advance of science unveil? II 1. 2. 3. 4.
What did Einstein announce in 1905? What example did Einstein use explaining the theory of relativity? What Einstein is theory was greeted by the physicists? What did Eddington write about gravitation? III
1. 2. 3. 4.
Why cannot we understand the full impact of the Quantum Theory? What does Entropy mean? What does the Quantum Theory begin with? What is a Quantum?
23 C. Learn to read math symbols: lim cos (2 πn) = 1
the limit of the cosine of 2 n is equal to one
x 2 (y, x) = 0
the function x sub two of y and x is equal to 0
ƒx (y) dy
the integral of the function ƒ of y with respect to y from a to b
bn< ε
the absolute value of b to the n is less than epsilon
[m, n]
the segment whose origin and terminus are the points m and n
B – { M0««Mn-1}
the difference of the set B and the finite set consisting of the elements M sub zero and so on, M sub n minus 1
24 Ebl_jZlmjZ 1. Stefan Kulczycki. Non-Euclidean Geometry. – New York. Oxford, 1989. 2. R. Couranf, H. Robbins. Plateu’s Problem. - New York. Oxford, 1993. 3. James T. Culbertson. Mathematics and Logic for Digital Devices. - London. Toronto, 1996. 4. L. Fox. The Numerical Solution of Two-Point Boundary Problem in Ordinary Differential Equations. - Oxford, 1997.
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