1、 1 Unit 1 Mathematics Part I EST Reading Reading 1 (http:/ Section A Pre-reading Task Warm-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, essayist and social critic best
2、 known for his work in mathematical logic and analytic philosophy. His most influential contributions inc lude his defense of logicism (the view that mathematics is in some important sense reducible to logic), his refining of the predicate calculus introduced by Gottlob Frege (which still forms the
3、basis of most contemporary logic), his defense of 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 definite descriptions and logical atomism. Russell is generally recognized as one of the
4、founders of modern analytic philosophy, and is regularly credited with being one of the most important logicians of the twentieth century. 2. What is Russells Paradox? Russell discovered the paradox that bears his name in 1901, while working on his Principles of Mathematics (1903). The paradox arise
5、s 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 is not a member of itself. The paradox is significant since, using classical logic, all sentences are entailed by a contradiction. Russells discovery th
6、us prompted a large amount of work in logic, set theory, and the philosophy and foundations of mathematics. 3. What effect did Russells Paradox have on Gottlob Freggs system? At first Frege observed that the consequences of Russells paradox are not immediately clear. For example, Is it always permis
7、sible 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 extension 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 thes
8、e 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 the paradox came with the development of his theory of types in 2 1903. It was clear to Russell that some restrictions needed to be placed upon the ori
9、ginal comprehension (or abstraction) axiom of naive set theory, the axiom that formalizes 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 avoide
10、d by arranging all sentences into a hierarchy, beginning with sentences about individuals 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 tha
11、t adopted by the mathematician Henri Poincar, and his own so-called “no class“ theory 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 invo
12、lve 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 you ever heard of Zermelo-Fraenkel set theory.? Can you give an account of it? Contradictions like Russells paradox arose from what was later
13、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 Zermelos system, the comprehension principle is eliminated in favour of several much more restrictive axioms: a. Axiom of extensionality. If
14、two sets have the same members, then they are identical. b. 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. c. Axiom of
15、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-formed properties. d. Power-set axiom. If S is a set, then there exists a
16、set, S1, that contains all and only the subsets of S. e. Union axiom. If S is a set (of sets), then there is a set containing all and only the members of the sets contained in S. f. Axiom of choice. If S is a nonempty set containing sets no two of which have common members, then there exists a set t
17、hat contains exactly one member from each member of S. g. 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 carefully constrained operations; the
18、method embodies what has come to be known as the iterative conception of a set. http:/plato.stanford.edu/entries/russell/ Section C Post-reading Task Reading Comprehension 3 1. Directions: Work on your own and fill in the blanks with the main idea. Part 1 (Para. 1): Brief introduction to Russells pa
19、radox Part 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 sets Para. 4: Contradictio
20、n 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 paradox Para. 6: Russells own response to the paradox with his “theory of types.“ Para. 7: Zermelos solut
21、ion to Russells paradox Para. 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 paradox 2. Directions: Work in pairs and discuss the following questions. 1) What is the basic idea of Russells paradox
22、? 2) How to explain 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 fi
23、ll in the blanks 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
24、 axiom that formalizes 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 sente
25、nces about individuals 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-calle
26、d “no class“ theory 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 gi
27、ven condition (or predicate) holds must be at the same level or of 4 the same “type.“ Vocabulary and Structure 1. Word-building Directions: 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 onc
28、e. discover (suffix) symbol (suffix) 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
29、 math whizzes.( discovery) 2) Packages can be 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
30、_ paradigm was constructed and then compared with 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 org
31、anization has the major advantage that the records 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 opt
32、imal strategies and the value of the game is computationally 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
33、 _ reasoning, have already been gutted and are no 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) 5 2. Directions: Complete the sentences
34、 with the words given in the brackets. Change the form 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.
35、(base) 2) Indeed, this separation of meaning is reflected 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
36、in vehicle emissions by 1980, not only led to him being credited (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 investme
37、nts in education and training has brought a tenfold increase 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 s
38、till new kind of mathematics related to geometry, and to the study 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,
39、 check if that position in the tree is flagged for end-of-word (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, mo
40、re useful materials and medications. (form) 8) A second reason for 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.
41、 “ (serve) 9) Thus the optimum conditions for coastal terrace development 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
42、doors to the booking hall had been replaced by an utterly incongruous picture 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 geome
43、try topologists study those properties of shapes that remain the same when 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,
44、 a higher-processing region in the brain located just above the forehead. 6 3) 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
45、is presented. 4) That has not stopped physicists from devising new algorithms 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 tha
46、t bends light in funny ways. 4. Directions: Change the following sentences into 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,
47、 did not originate until the Hellenistic period. 4) There is a probability that my 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 G
48、od. Discourse Understanding 1. C. A “3 percent margin of error“ means that there is a 95 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 abstraction, 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 (http:/ S
