1、Unit 1 MathematicsPart I EST ReadingReading 1 (/article.cfm?id=what-is-russells-paradoxSection A Pre-reading TaskWarm-up Questions: Work in pairs and discuss the following questions.1. Who is Bertrand Russell?Bertrand Arthur William Russell (b.1872 d.1970) was a British philosopher, logician, essayi

2、st and social critic best known for his work in mathematical logic and analytic philosophy. His most influential contributions include his defense of logicism (the view that mathematics is in some important sense reducible to logic), his refining of the predicate calculus introduced by HYPERLINK /en

3、tries/frege/ Gottlob Frege (which still forms the basis of most contemporary logic), his defense of HYPERLINK /entries/neutral-monism/ neutral monism (the view that the world consists of just one type of substance that is neither exclusively mental nor exclusively physical), and his theories of HYPE

4、RLINK /entries/descriptions/ definite descriptions and HYPERLINK /entries/logical-atomism/ logical atomism. Russell is generally recognized as one of the founders of modern analytic philosophy, and is regularly credited with being one of the most important logicians of the twentieth century.2. What

5、is Russells Paradox?Russell discovered the paradox that bears his name in 1901, while working on his Principles of Mathematics (1903). The paradox arises in connection with the set of all sets that are not members of themselves. Such a set, if it exists, will be a member of itself if and only if it

6、is not a member of itself. The paradox is significant since, using classical logic, all sentences are entailed by a contradiction. Russells discovery thus prompted a large amount of work in logic, set theory, and the philosophy and foundations of mathematics. 3. What effect did Russells Paradox have

7、 on Gottlob Freggs system?At first Frege observed that the consequences of Russells paradox are not immediately clear. For example, “Is it always permissible to speak of the extension of a concept, of a class? And if not, how do we recognize the exceptional cases? Can we always infer from the extens

8、ion of one concepts coinciding with that of a second, that every object which falls under the first concept also falls under the second?Because of these kinds of worries, Frege eventually felt forced to abandon many of his views.4. What is Russells response to the paradox?Russells own response to th

9、e paradox came with the development of his theory of types in 1903. It was clear to Russell that some restrictions needed to be placed upon the original comprehension (or abstraction) axiom of naive set theory, the axiom that formalizes the intuition that any coherent condition may be used to determ

10、ine a set (or class). Russells basic idea was that reference to sets such as the set of all sets that are not members of themselves could be avoided by arranging all sentences into a hierarchy, beginning with sentences about individuals at the lowest level, sentences about sets of individuals at the

11、 next lowest level, sentences about sets of sets of individuals at the next lowest level, and so on Using a vicious circle principle similar to that adopted by the mathematician Henri Poincar, and his own so-called no class theory of classes, Russell was able to explain why the unrestricted comprehe

12、nsion axiom fails: propositional functions, such as the function x is a set, may not be applied to themselves since self-application would involve a vicious circle. On Russells view, all objects for which a given condition (or predicate) holds must be at the same level or of the same type.5. Have yo

13、u ever heard of Zermelo-Fraenkel set theory.? Can you give an account of it?Contradictions like Russells paradox arose from what was later called the unrestricted comprehension principle: the assumption that, for any property p, there is a set that contains all and only those sets that have p. In Ze

14、rmelos system, the comprehension principle is eliminated in favour of several much more restrictive axioms: HYPERLINK /EBchecked/topic/46221/axiom-of-extensionality o Axiom of extensionality Axiom of extensionality. If two sets have the same members, then they are identical. HYPERLINK /EBchecked/top

15、ic/46219/axiom-of-elementary-sets o Axiom of elementary sets Axiom of elementary sets. There exists a set with no members: the null, or empty, set. For any two objects a and b, there exists a set (unit set) having as its only member a, as well as a set having as its only members a and b. HYPERLINK /

16、EBchecked/topic/46240/axiom-of-separation o Axiom of separation Axiom of separation. For any well-formed property p and any set S, there is a set, S1, containing all and only the members of S that have this property. That is, already existing sets can be partitioned or separated into parts by well-f

17、ormed properties. HYPERLINK /EBchecked/topic/46233/axiom-of-power-set o Power-set axiom Power-set axiom. If S is a set, then there exists a set, S1, that contains all and only the subsets of S. HYPERLINK /EBchecked/topic/46245/axiom-of-union o Union axiom Union axiom. If S is a set (of sets), then t

18、here is a set containing all and only the members of the sets contained in S. HYPERLINK /EBchecked/topic/46212/axiom-of-choice o Axiom of choice Axiom of choice. If S is a nonempty set containing sets no two of which have common members, then there exists a set that contains exactly one member from

19、each member of S. HYPERLINK /EBchecked/topic/46225/axiom-of-infinity o Axiom of infinity Axiom of infinity. There exists at least one set that contains an infinite number of members.With the exception of (b), all these axioms allow new sets to be constructed from already-constructed sets by carefull

20、y constrained operations; the method embodies what has come to be known as the “iterative” conception of a set. /entries/russell/Section C Post-reading TaskReading Comprehension1. Directions: Work on your own and fill in the blanks with the main idea.Part 1 (Para. 1): Brief introduction to Russells

21、paradoxPart 2 (Paras. 2-5): The effect of Russells paradox on Gottlob Freges system.Para. 2: Russells paradox dealt a heavy blow to Freges attempts to develop a foundation for all of mathematics using symbolic logic.Para. 3: An illustration of Russells paradox in terms of setsPara. 4: Contradiction

22、found in the set.Para. 5: Frege noticed the devastating effect of Russells paradox on his system and inability to solve it.Part 3 (Paras. 6-8): Solutions offered by mathematicians to Russels paradoxPara. 6: Russells own response to the paradox with his theory of types.Para. 7: Zermelos solution to R

23、ussells paradoxPara. 8: What became of the effort to develop a logical foundation for all of mathematics?Part 4 (Para. 9): Correspondence between Russell and Frege on the paradox2. Directions: Work in pairs and discuss the following questions.1) What is the basic idea of Russells paradox?2) How to e

24、xplain Russells paradox in terms of sets?3) Can you explain the contradiction found in the sets related to Russells paradox 4) Is Russells own response to the paradox workable?5) Do you know Zermelo-Fraenkel set theory? (open)3. Directions: Read the following passage carefully and fill in the blanks

25、 with the words youve learned in the text.Russells own response to the paradox came with the development of his theory of types in 1903. It was clear to Russell that some restrictions needed to be placed upon the original comprehension (or abstraction) axiom of naive set theory, the axiom that forma

26、lizes the intuition that any coherent condition may be used to determine a set (or class). Russells basic idea was that reference to sets such as the set of all sets that are not members of themselves could be avoided by arranging all sentences into a hierarchy, beginning with sentences about indivi

27、duals at the lowest level, sentences about sets of individuals at the next lowest level, sentences about sets of sets of individuals at the next lowest level, and so on. Using a vicious circle principle similar to that adopted by the mathematician Henri Poincar, and his own so-called no class theory

28、 of classes, Russell was able to explain why the unrestricted comprehension axiom fails: propositional functions, such as the function x is a set, may not be applied to themselves since self-application would involve a vicious circle. On Russells view, all objects for which a given condition (or pre

29、dicate) holds must be at the same level or of the same type.Vocabulary and Structure1. Word-buildingDirections: Give the correct form of the word according to the indication in the brackets. Then complete the sentences using the right form for each word. Use each word once.discover (suffix) symbol(s

30、uffix)logic (suffix) form(suffix)correspond(suffix)develop(suffix)describe(suffix)able(prefix)contradict(suffix)equal(suffix)1) The math may not have been new, but Duchin enjoyed the process of_, and she got to work collaboratively with half a dozen other math whizzes.( discovery)2) Packages can be

31、sealed and can contain personal _if it relates to the contents of the package.( correspondence)3) New research indicates that the brain region may prefer_ notation to other numeric representations .( symbolic)4) To do this, an ideal model based on the _ paradigm was constructed and then compared wit

32、h a neutral model reflecting the further education system as it existed before the Act took effect.( equality)5) Is this not in flagrant _to Einsteins rule that signals do not travel faster than the velocity of light?( contradiction)6) Sequential organization has the major advantage that the records

33、 are stored in a _ order, presumably that sequence to which the records are normally required for printing and for soft copy reports.( logical)7) The mathematical _ of a zero-sum two-person game is not difficult to construct, and determining the optimal strategies and the value of the game is comput

34、ationally straightforward.( description)8) The proof we now know required the _ of an entire field of mathematics that was unknown in Fermats time.( development)9) Williams adds that many courses in geometry, “the one high school class that demands _ reasoning,” have already been “gutted” and are no

35、 longer proof-based.( formal)10) The concept of total aircraft ownership will become increasingly important should the traditional trade structure be _to cover the expanse of technologies economically.( unable)2. Directions: Complete the sentences with the words given in the brackets. Change the for

36、m if necessary.1) The key to unraveling such apparent paradoxes is to characterize the initial set of possibilities (initial meaning before you receive any extra information) and then to eliminate possibilities based on that extra information. (base)2) Indeed, this separation of meaning is reflected

37、 by the definition of weak in the OALD, with a distinct sense reserved for its use when pertaining to that of solutions (definition)3) The resulting radical pollution control programme outlined by Nixon, calling for a 90 per cent reduction in vehicle emissions by 1980, not only led to him being cred

38、ited (albeit briefly) as policy initiator of an environmental clean-up but also provided him with the chance to deal a blow to one of his most important opponents in the 1972 elections, Edmund Muskie (blow)4) Singapores continuing investments in education and training has brought a tenfold increase

39、in our pool of Information Technology professionals and the Singapore worker has been consistently rated by BERI as the worlds best in terms of technical skills, attitude and productivity. (term)5) In this work he was led to topology, a still new kind of mathematics related to geometry, and to the s

40、tudy of shapes (compact manifolds) of all dimensions. (lead)6) If there is no allowable string which spans the whole graph, then we can search in the same way as described above, but wherever the required path does not exist in the tree, check if that position in the tree is flagged for end-of-word

41、(way)7) During the past century, steps forward in physics have often come in the form of newly found particles; in engineering, more complex devices; in astronomy, farther planets and stars; in biology, rarer genes; and in chemistry, more useful materials and medications. (form)8) A second reason fo

42、r measurements is the more theoretical, put by Love as the discovery of numerical relations between the quantities that can be measured to serve as a basis for the inductive determination of the form of the intrinsic energy function. (serve)9) Thus the optimum conditions for coastal terrace developm

43、ent would seem to be areas with small tidal ranges. Finally, tidal range is an important factor in the generation of tidal currents which may locally become of geomorphological importance (become)10) The original double entrance doors to the booking hall had been replaced by an utterly incongruous p

44、icture window as had adjacent booking hall and waiting room windows. (replace)3. Directions: Reorder the disordered parts of a sentence to make a complete sentence.1) A simple way to describe topology is as a rubber sheet geometry topologists study those properties of shapes that remain the same whe

45、n the shapes are stretched or compressed.2) Since the mid-1990s scientists have floated the idea that representations of numeric quantities, whether expressed as digits or as written words, are codified by the parietal cortex, a higher-processing region in the brain located just above the forehead.3

46、) As activity was monitored, located just above the forehead ,researchers noted changes under the assumption that the brain reduces activity as it becomes accustomed to a stimulus and then reactivates when a novel stimulus is presented.4) That has not stopped physicists from devising new algorithms

47、for the devices, which can calculate a lot faster than ordinary computersin fact, exponentially faster, in quite a literal sense.5) Such a device would be made of metamaterial, a thicket of metal rings or other shapes that bends light in funny ways.4. Directions: Change the following sentences into

48、nominalized ones. 1) The passage of night could be marked by the appearance of 18 of these stars.2) The full proof of Fermats Last Theorem is contained in these two papers.3) The concept of fixed-length hours, however, did not originate until the Hellenistic period.4) There is a probability that my

49、first sock is red because only one of the remaining three socks is red.5) The importance of accurate data in quantitative modeling is central to using Bayess theorem to calculate the probability of the existence of God.Discourse Understanding1. C. A 3 percent margin of error means that there is a 95

50、 percent chance that the survey result will be within 3 percent of the population value.2. E. How is it that a survey of only 1,000 people can reach this level of accuracy?3. G. The margin of error depends inversely on the square root of the sample size.4. A. The margin of error is a mathematical ab

51、straction, and there are a number of reasons why actual errors in surveys are larger.5. F. Finally, the 3 percent margin of error is an understatement because opinions change.Reading 2 (/article.cfm?id=mandelbrot-set-1990-horgan)Section A Pre-reading TaskWarm-up Questions HYPERLINK /article.cfm?id=m

52、andelbrot-set-1990 /article.cfm?id=mandelbrot-set-19901. Who discovered the Mandelbrot set? This is not a trick question, not easy to answer. Many people including Mandelbrot have laid claim to the discovery.2. Why was the set named after Benoit B. Mandelbrot?The set is named after Benoit B. Mandelb

53、rot, a mathematician at the IBM Thomas J. Watson Research Center because he coined the term fractal to describe phenomena (such as coastlines, snowflakes, mountains and trees) whose patterns repeat themselves at smaller and smaller scales.3. Why has there been so much controversy about who discovere

54、d the Mandelbrot set?Mandelbrot claims that he and he alone discovered the Mandelbrot set, but there are other mathematicians who have challenged his claim.4. What did the challengers say about Mandelbrots claim of discovery?Two mathematicians said that they independently discovered and described th

55、e set at about the same time as Mandelbrot did. And another mathematician also asserted that his work on the set not only predated Mandelbrots efforts but also helped to guide them5. Why did some professors support Mandelbrots claim?Mandelbrot deserves to have the set named after him, because his ef

56、forts brought the set to the attention of both the public and of the pure-mathematics community. Section C Post-reading TaskReading and Understanding1. Choose the best summary of the passage.C(/issue40/features/devaney/ ) A( HYPERLINK /r/programming/comments/cmpvt/beno%EEt_mandelbrot_on_fractals_ted

57、/ /r/programming/comments/cmpvt/beno%EEt_mandelbrot_on_fractals_ted/) B( HYPERLINK /DYSYS/FRACGEOM/FRACGEOM.html /DYSYS/FRACGEOM/FRACGEOM.html) D2. Complete the sentences based on the text.1) The term Mandelbrot set is used to refer both to a general class of _ and to a particular instance of such a

58、 set. （ HYPERLINK /Fractal.html fractal sets）2) 2.In general, a Mandelbrot set marks the set of points in the _ such that the corresponding HYPERLINK /JuliaSet.html Julia set is HYPERLINK /ConnectedSet.html connected and not HYPERLINK /ComputableNumber.html computable. （ HYPERLINK /ComplexPlane.html

59、 complex plane）3) The Mandelbrot set is the set obtained from the _ z_(n+1)=z_n2+c with z_0=c, where points in the HYPERLINK /ComplexPlane.html complex plane for which the orbit of z_n does not tend to infinity are in the HYPERLINK /Set.html set. （ HYPERLINK /QuadraticRecurrenceEquation.html quadrat

60、ic recurrence equation）4) Setting z_0equal to any point in the set that is not a _ gives the same result. （periodic point）5) The Mandelbrot set was originally called a _ by Mandelbrot. J. Hubbard and A. Douady proved that the Mandelbrot set is HYPERLINK /ConnectedSet.html connected（molecule） HYPERLI