SAMPLE 8

　　[數學類]

　　Computer programmers often remark that computing machines, with a perfect lack of discrimination, will do any foolish thing they are told to do. The reason for this lies, of course, in the narrow fixation of the computing machine’s “intelligence” on the details of its own perceptions — its inability to be guided by any large context. In a psychological description of the computer intelligence, three related adjectives come to mind: single-minded, literal-minded, and simple-minded. Recognizing this, we should at the same time recognize that this single-mindedness, literal-mindedness, and simple-mindedness also characterizes theoretical mathematics, though to a lesser extent.

　　Since science tries to deal with reality, even the most precise sciences normally work with more or less imperfectly understood approximations toward which scientists must maintain an appropriate skepticism. Thus, for instance, it may come as a shock to mathematicians to learn that the Schrodinger equation for the hydrogen atom is not a literally correct description of this atom, but only an approximation to a somewhat more correct equation taking account of spin, magnetic dipole, and relativistic effects; and that this corrected equation is itself only an imperfect approximation to an infinite set of quantum field-theoretical equations. Physicists, looking at the original Schrodinger equation, learn to sense in it the presence of many invisible terms in addition to the differential terms visible, and this sense inspires an entirely appropriate disregard for the purely technical features of the equation. This very healthy skepticism is foreign to the mathematical approach.

　　Mathematics must deal with well-defined situations. Thus, mathematicians depend on an intellectual effort outside of mathematics for the crucial specification of the approximation that mathematics is to take literally. Give mathematicians a situation that is the least bit ill-defined, and they will make it well-defined, perhaps appropriately, but perhaps inappropriately. In some cases, the mathematicians literal-mindedness may have unfortunate consequences. The mathematicians turn the scientists’ theoretical assumptions that is, their convenient points of analytical emphasis, into axioms, and then take these axioms literally. This brings the danger that they may also persuade the scientists to take these axioms literally. The question, central to the scientific investigation but intensely disturbing in the mathematical context — what happens if the axioms are relaxed? — is thereby ignored.

　　The physicist rightly dreads precise argument, since an argument that is convincing only if it is precise loses all its force if the assumptions on which it is based are slightly changed, whereas an argument that is convincing though imprecise may well be stable under small perturbations of its underlying assumptions.

　　1. The author discusses computing machines in the first paragraph primarily in order to do which of the following?

　　[A] Indicate the dangers inherent in relying to a great extent on machines.

　　[B] Illustrate his views about the approach of mathematicians to problem solving.

　　[C] Compare the work of mathematicians with that of computer programmers.

　　[D] Provide one definition of intelligence.

　　2. It can be inferred form the text that scientists make which of the following assumptions about scientific arguments?

　　[A] The literal truth of the arguments can be made clear only in a mathematical context.

　　[B] The arguments necessarily ignore the central question of scientific investigation.

　　[C] The arguments probably will be convincing only to other scientists.

　　[D] The premises on which the arguments are based may change.

　　3. According to the text, mathematicians present a risk to scientist for which of the following reasons?

　　[A] Mathematicians may provide theories that are incompatible with those already developed by scientists.

　　[B] Mathematicians may define situations in a way that is incomprehensible to scientists.

　　[C] Mathematicians may convince scientists that theoretical assumptions are facts.

　　[D] Scientists may come to believe that axiomatic statements are untrue.

　　4. The author suggests that the approach of physicists to solving scientific problem is which of the following?

　　[A] Practical for scientific purposes.

　　[B] Detrimental to scientific progress.

　　[C] Unimportant in most situations.

　　[D] Expedient, but of little long-term value.

　　5. The author implies that scientists develop a healthy skepticism because they are aware that

　　[A] mathematicians are better able to solve problems than are scientists.

　　[B] changes in axiomatic propositions will inevitably undermine scientific arguments.

　　[C] well-defined situations are necessary for the design of reliable experiments.

　　[D] some factors in most situations must remain unknown.

　　[答案與考點解析]

　　1. 【答案】B

　　【考點解析】這是一道寫作手法題。旨在考察考生的語言基本功和對文章段落結構的認識。這是一道比較難的題目。通過仔細閱讀第一段并且把第一段和本文其它段落相聯系，可推導出本題的正確選項是B。原文作者借用大家所熟悉的“computing machines”作為例子來幫助我們更好的了解“theoretical mathematics”的特點�？忌�在解題時一定要識別出作者的寫作手法和意圖。

　　2. 【答案】D

　　【考點解析】這是一道審題定位與引申推導題。通過本題題干中的“arguments”一詞可將本題的答案信息來源迅速確定在尾段。通過仔細閱讀理解尾段的內容并且根據尾段的內容進行推導，可得知本題的正確選項應該是突出“the premises…may change”的選項D。考生在解題時即要具備迅速審題定位的能力，又要具備理解原文深層含義的能力。

　　3. 【答案】C

　　【考點解析】這是一道審題定位與細節(jié)推導題。根據本題題干中的“risk”一詞可將本題的答案信息來源確定在原文倒數第二段的倒數第二句，通過仔細閱讀倒數第二段的倒數第二句和第三句，尤其是倒數第二句中“danger”一詞后面的同謂語從句，我們可以推斷出本題的正確選項是突出“theoretical assumptions are facts”的選項C�？忌�在解題時一定要善于深入理解原文的含義，千萬不能只停留在文字的表面。

　　4. 【答案】A

　　【考點解析】本題是一道審題定位與關鍵詞語題。通過本題題干中的“physicists”可將本題的答案信息來源迅速確定在第二段的倒數第二句。通過閱讀本句以及它的前后句，尤其是它后面的一句話(第二段的尾句)，我們可以推斷出本題的正確選項應該是表達“healthy”(有益的)含義的選項A，因為該選項中的“practical”是一個表示肯定的詞語。考生在解題時一定要善于審題定位，更要善于理解句間關系所表達的內容。

　　5. 【答案】D

　　【考點解析】這是一道指代詞題型。根據本題題干中的“healthy skepticism”可將本題的答案信息來源迅速確定在第二段的尾句，該句中的指代詞“this”暗示我們本題的真正答案信息來源在第二段的倒數第二句，通過仔細閱讀和理解倒數第二段的第二句話，我們可以得知強調“invisible”的選項D是本題的正確選項，因為該選項中含有“unknown”一詞�？忌�在解題時應注意指代詞的作用和功能，更要注意對原文細節(jié)的理解和把握。

　　[參考譯文]

　　計算機程序員經常說計算機器會執(zhí)行任何愚蠢的命令，因為它們完全缺乏辨別力。當然，其原因在于計算機的智力在其感知細節(jié)上的狹窄固定性——它不能被冗長的上下文所引導。三個相關的形容詞可以用來對計算機進行心理上的描述：單一的思維，刻板的思維，簡單的思維。在認識到這一點的同時，我們也應該認識到這種單一的思維，刻板的思維，簡單的思維同樣可以用來(盡管只是在較低的程度上)描述理論數學。

　　由于科學總是處理現實的問題，即使比較精確的科學也通常在或多或少的不完全理解的近似的基礎上探討現實，對此，科學家必須保持適當的懷疑。例如，當數學家得知關于氫原子的薛定諤方程并不是對這種原子的精確的描述，而只是在考慮了旋轉、磁極以及相對論作用的基礎上一個稍微正確的近似方程，并且這個所謂的正確方程自身也只是對一個無窮量子理論場的不完美的近似時，他們一定深感震驚。當物理學家看到比較初的薛定諤方程時，他們從中感知到在可見的各種關系之外，仍然存在著許多不可見的關聯，而這種感知就會激勵物理學家合理地忽略方程中純技術的特色。這種非常有益的懷疑態(tài)度對于數學領域而言則是較為陌生的。

　　數學家必須研究精確界定的情況。因此，數學家依賴數學以外的努力來對數學照字面意義理解的近似性加以詳細的說明。當給予數學家一個不確定程度較小的情形時，他們會把它轉化成一個完全確定的狀態(tài)。這種轉化可能是合適的，也可能是不合適的。在某些情形下，數學家的這種刻板思維可能會產生不幸的后果。數學家把科學家的理論假設，也就是科學家分析重點的權宜之點，轉化成公理，然后依據精確字義理解這些公理。他們可能還會說服科學家依據字義理解公理，這就會帶來危險。科學家調查的中心問題，在數學領域則成為極其擾亂人心的問題，因而被忽略——如果公理不嚴謹會發(fā)生什么情況?

　　物理學家懼怕精確的論斷是對的，因為一個僅僅因其精確性而使人信服的論斷，當它所基于的假設略微改變時，其說服力就會喪失殆盡，而一個盡管不精確卻具有說服力的論斷，卻會在它的基礎假設受到小干擾時，依舊巋然不動。