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 simpleminded. Recognizing this, we should at the same time recognize that this single-mindedness, literal-mindedness, and simplemindedness 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.
21. The author discusses computing machines in the first paragraph primarily in order to do which 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
E) Emphasize the importance of computers in modern technological society
22. According to the passage, scientists are skeptical toward their equations because scientists
(A) work to explain real, rather than theoretical or simplified, situations
(B) know that well-defined problems are often the most difficult to solve
(C) are unable to express their data in terms of multiple variables
(D) are unwilling to relax the axioms they have developed
(E) are unable to accept mathematical explanations of natural phenomena
23. It can be inferred from the passage 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 conclusions of the arguments do not necessarily follow from their premises.
(E) The premises on which the arguments are based may change.
24. According to the passage, mathematicians present a danger to scientists 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.
(E) Scientists may begin to provide arguments that are convincing but imprecise.
25. The author suggests that the approach of physicists to solving scientific problems 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
(E) Effective, but rarely recognized as such
26. The author suggests that a mathematician asked to solve a problem in an ill-defined situation would first attempt to do which of the following?
(A) Identify an analogous situation
(B) Simplify and define the situation
(C) Vary the underlying assumptions of a description of the situation
(D) Determine what use would be made of the solution provided
(E) Evaluate the theoretical assumptions that might explain the situation
27. 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) mathematical solutions can rarely be applied to real problems
(E) some factors in most situations must remain unknown