CHAPTER 6
THE STRUCTURE OF PROBLEMS
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Each of the philosophers discussed above--Popper, Kuhn, Lakatos and Laudan--proposed a theory of the growth of scientific knowledge in which problems played an important role. Each of them made claims about which type of problem was most characteristic of and/or important for scientific progress.
Popper particularly emphasized the explanatory problems arising out of refutations or violated expectations. Kuhn praised the emphasis on puzzle-solving found in normal science. Lakatos talked about the improvement of models and the importance of novel predictions. Laudan gave a more detailed account of problem weights, which generated the following rough ordering by decreasing importance: external conceptual problems > internal conceptual problems > empirical anomalies > unsolved empirical problems.
This survey of philosophical views is a good reminder of the immense variety of questions and tasks that working scientists confront. There are problems of describing initial conditions and problems of finding general explanatory theories. Sometimes the goal is to fill in the gaps within an already accepted system (such as the search for the "missing elements" in Mendeleev's Periodic Table). Other times we try to design a crucial experiment which will "make or break" a theory (e.g., Eddington's eclipse expedition). The search for a new theory can be driven by the need to resolve an inconsistency (e.g., the clash between quantum mechanics and classical electromagnetism). Or it may be fueled by more aesthetic considerations (Kepler thought it would be nice to find a simple, mathematical representation of planetary distances). Some problems are highly structured (e.g., Plato's problem of describing planetary position in terms of uniform circular motion). Others appear to be almost devoid of constraints (such as the early chemists' problem of understanding combustion).
And this survey of the variety of problems does not even tap non-cognitive factors, such as the moral urgency of understanding violence and the etiology of diseases, or the economic and political dimensions of many scientific research problems. Neither does it include practical factors, such as the availability of funding, or the tastes and technical expertise of the investigator. A comprehensive theory of the evaluation of problems will be very complex indeed.
Let us start modestly by investigating the logical structure of problems. If we can give a formal account first, then it should be easier to fill in the complications which arise from subject matter and pragmatic context. There are three major bodies of literature which give formal analyses of problems: discussions of mathematical heuristics, erotetic logic, and computer simulation studies. Let us look briefly at what each has to say about the structure of problems. Our intention is not to summarize these writings on their own terms, but rather to "raid" them for useful ideas for our program!
6.1 Mathematician's Analyses of Problems
Study of the structure of mathematical problems as a clue to methods of solving them began with Greek geometers who distinguished the task of proving theorems from that of finding or constructing a mathematical object. Only the latter were called problems by the Greeks, but it will be convenient here to use Polya's terminology and talk about problems-to-prove vs. problems-to-find. (Polya, p.119). Familiar examples of problems-to-prove would be that of deriving the Pythagorean theorem from the axioms of geometry or showing there is no greatest prime. Simple problems-to-find would be: "Given two commensurable magnitudes, to find their greatest common measure" (Euclid-Cohen, p.18) and "To construct, in a given rectilinear angle, a parallelogram equal to a given rectilinear figure" (Euclid-Cohen, p.55).
The Greek distinction between theorema and problemata was later extended to mechanics (see Galileo's Two New Sciences) and optics. Newton lists as a theorem the proposition that "the Light of the Sun consists of Rays differently Refrangible" (Opticks, 26) followed by "Proof by Experiments" instead of derivation from axioms. Under problems proper we find a wide diversity of tasks ranging from "To separate from one another the heterogeneous Rays of compound Light" (p.64) to "By the discovered Properties of Light to explain the Colours of the Rainbow" (p.168). We will not pause here to try to analyze exactly how Newton drew the distinction, but the general idea is clear enough. In problems to prove, what is sought is a series of logical steps leading from what is given (the Greeks took this to be the antecedent of the theorem) to what is to be demonstrated (the consequent). In problems-to-find/construct, we are looking for a procedure (a set of permissible operations not limited to deductive logic) which will allow us to move from the starting point or configuration to the desired object or state. The permissible operations in geometry might be limited to what one can do with a straight edge and compass. As we saw, Newton extended the scope of operations to include physical procedures--such as laboratory manipulations of light.
The Greeks refer to a powerful heuristic for discovering the solutions to both kinds of problems, the famous method of analysis. Pappus describes it as follows:
"In analysis we suppose that which is sought to be already done, and we inquire from what it results, and again what is the antecedent of the latter, until we on our backward way light upon something already known and being first in order ... [A]nalysis is of two kinds. One seeks the truth, being called theoretical. The other serves to carry out what was desired to do, and this is called problematical." (Hintikka, pp.8-9)
There is disagreement amongst modern commentators about exactly what Pappus had in mind. Polya, in How to Solve It, interprets analysis as the process of looking for intermediate steps which are sufficient to produce the goal. Szabo and Lakatos on the other hand, think that analysis consists of deriving consequences from, or operating on, the end state until one produces something which is either impossible or one of the givens. (Hintikka, pp.118-130). Hintikka and Remes produce a more complicated pattern which is based on Pappus' actual practice. (p.36) (See also NK, p.147).
Once again the exegetical question of what exactly the Greeks took the method to be need not concern us; historically a variety of interpretations were proposed. Aristotle compares the mathematical method of analysis with deliberation over practical means to a given end. "They assume the end and consider how and by what means it is to be attained ... as though [they] were analyzing a geometrical construction ..." [Nicomachean Ethics, III, 3, 1112615-20]. In Gassendi's Logic, the analysis is of concepts, thus man is resolved into rational plus animal, animal is resolved into sentient plus living being, etc. (NK, p.141). In his famous 31st Query, Newton extends the notion of analysis to include "... [proceeding] from Compounds to Ingredients, and from Motions to the Forces producing them; and in general, from Effects to their Causes, and from particular Causes to more general ones, till the Argument end in the most general. This is the Method of Analysis." (p.404). So we see there is a long tradition of using mathematical problems and mathematical heuristics as a model for problem-solving in other areas. Thus Isaac Barrow, Newton's mathematical teacher, writes:
"... analysis seems to belong to mathematics no more than to physics, ethics, or any other science. For this is merely a part or species of logic, or a manner of using reason in the solution of questions ... therefore, it is not a part or a species but rather the servant of mathematics ..." (NK, p.148)
Let us now turn to Polya's discussion of the principal components of each of the two types of mathematical problem. We will then see how his formulation might be extended to scientific problems. Polya divides "problems-to-find" into three consistent parts. (pp.119-120) First, a clearly stated problem must specify the general category to which the unknown belongs--are we looking for a geometrical object, a number, or what. Secondly, the problem will lay down one or more specific conditions which the sought for object must satisfy--perhaps the number must be a prime less than 200 which satisfies a certain equation or the figure must be a parallelogram. Thirdly, we may be given some data, such as the values of the constants in the equation or lengths of two line segments which are to be the sides of the parallelogram. Sometimes there are no data above and beyond the general "givens" of the field. So for Polya, problems of this type exhibit the general form; Given [data], find the member[s] of [category], such that [conditions]. The solution to the problem will consist of the subset of objects in the general category which satisfy all the conditions and which can be constructed from the data.
Although Polya does not mention this (he is writing for students and is therefore avoiding complications), there is a strong conventional element in the division of a problem into these three parts. For example, one could either start with the category of number and impose two conditions such as prime and less than 200 or one could begin with the category of prime number and impose only one condition, that of being less than 200. There is a similar flexibility in how we parse conditions and data. Nevertheless mathematical problems-to-find often seem to divide naturally into these three parts. A more important omission from Polya's discussion is the fact (which Pappus stresses) that a problem-to-find can also be "solved" by showing that the sought-for object cannot exist. However, Polya does stress the possibility of negative results in his discussion of proofs.
Since most mathematical propositions have "if-then" as the major logical connective, Polya analyzes problems-to-prove into two principal parts, what he calls the hypothesis and the conclusion. (p.121) So for Goldbach's conjecture the hypothesis is "n is any even integer greater than 4" and the conclusion is "n is the sum of two odd primes".
Here again there are some conventional elements, the most obvious of which is the option of forming the contrapositive of the original assertion. However, once again there are generally "natural" formulations of problems-to-prove. And on Hintikka's account of the Greek method of analysis, the two fragments played different roles when people were looking for proofs. (See the diagram in NK, p.147). More troubling perhaps is Polya's neglect of the "background givens" in problems-to-prove, such as axioms, previously proved theorems, and rules of inference. These will be taken for granted by beginning mathematics students, but as Lakatos reminds us, historically the "givens" can also be questioned. In any case, the typical form of these problems is: [Given the background], to prove that if [hypothesis] then [conclusion].
Let us now see the extent to which scientific problems fit either of these paradigms. We'll begin with problems-to-prove. When scientists are working with a rather complex formal theory, there are often problems of ascertaining exactly what can be derived within that theory. For example quantum chemists had to figure out exactly what quantum mechanics predicted about the spectra of molecules, and the calculation for even the simplest case of the hydrogen molecule ion (H2+) consisting of only two protons and one electron was rather difficult. We could look at this as a problem-to-prove in which the antecedent is a description of the H2+ molecule, the consequent is a description of the observed spectrum, and the background theory is quantum mechanics.
The most difficult aspect of such problems in the scientific case, however, may not lie in discovering the steps in the mathematical proof. Often the activity which requires the most creativity is figuring out how to give a simplified model of the system, perhaps using symmetry considerations or hunches about which effects are second-order and may be safely ignored. In this case the difficult problem is to find a model of the initial conditions such that one can then prove that the theory does or does not account for the experimental results. Still and all, since the overall goal is to show logical connections between statements, we might well consider these cases to be problems-to-prove. Another classic case of a scientific problem-to-prove was the debate about whether altruism could be an adaptive trait according to Darwinian theory.
As we saw in Newton's Opticks, one might also extend this category to include proofs of theory from experiments. Normally, such inferences will not be deductive. (See Dorling's "Deductions from Phenomena" for exceptions.) However, as in the mathematics case, the problem explicitly states two propositions--here we can put the evidence as antecedent and the theory-under-test as consequent--and the task at hand is to ascertain the strength of the connection between them, given the usual background information. This classification works most smoothly if one believes in inductive logic, for then what I've called the "strength of connection" really is a generalization of logical entailment. However, it seems to me that subjective Bayesians, Popperian corroborationists, or those who adopt Lakatosian measures of support might also wish to classify the general problem of appraising theories by means of evidence as a sort of problem-to-prove: the relata are statements; what we seek is a relation between them. We could also include here various sophisticated statistical techniques such as path analysis.
What are the scientific analogues of the mathematician's problems-to-find, i.e., problems where we're trying to locate items within a prescribed category which satisfy certain conditions? Here I would include all of the problems which eventuate in what are usually called scientific discoveries. First, there are what we might call theory-driven discoveries: Mendeleev's early Periodic Table had three gaps--places for additional elements. From the Table itself one could infer their valences, and approximate values for their atomic weights, and other physical and chemical properties. Thus one knew in which types of minerals they were likely to be found. There are many other examples: the existence of the outer planets, Uranus and Neptune, various fundamental particles, e.g. the neutrino and the omega-minus particle, and the gravitational redshift were all directly predicted from physical or astronomical theories. We could also include cases such as Kekule's discovery of the benzene ring. Here the theory did not give a complete specification of the structure, but it did place severe constraints on the permissible solutions. (Remember that mathematical problems do not all have unique solutions, although in math any object which satisfies the conditions is as good as any other; in science there is also an extra "reality check".)
Another common problem-to-find is the quest for explanations. Here the sought-for object is often a generalization about causal processes. It is constrained by the explananda (although as Popper emphasizes, it may correct them, thus Newton's theory corrects Kepler's Laws while explaining them). The explanations may also have to satisfy other theoretical conditions. According to an anecdote about Bohr, at a conference he once told the reader of a speculative paper, "You can say anything you like as long as you conserve energy!". (However, the Bohr-Kramer-Slater theory violated energy conservation!) And the prevailing research tradition/metaphysics may also place constraints on the candidates. However, unlike the case of mathematical conditions, these metaphysical constraints are "soft" and may be challenged.
Even more ill-defined is the case in which two comprehensive scientific theories are found to be inconsistent and one is trying to remove the contradiction by finding a new, better explanatory theory (an example of one of Laudan's external conceptual problems). Here we are not even sure of the explananda. If Aristotelian physics is inconsistent with Copernican astronomy, should we seek a new physics which is compatible with Copernicus, a new astronomy compatible with Aristotle, or a new astronomical-physics which supercedes them both? Of course the context may give us a preferred direction of inquiry, but problems arising out of inconsistencies are unusually messy. (The same is true in mathematics when paradoxes are uncovered. In such cases we know that something is wrong amongst the set of propositions which are normally taken for granted and which are thus normally constraints on all problem solutions.)
We could also include under scientific problems-to-find the process of designing experimental apparatus to carry out particular measurements to a specified degree of accuracy. Once again we know (more or less) what we're looking for and can recognize it when we find it.
There seems to be a rough parallel between the mathematician's distinction between problems-to-find and problems-to-prove and the traditional dichotomy within philosophy of science between the contexts of discovery and justification. Thus we might expect problems-to-find to demand more creativity (or luck!) than problems to prove. But there are also exceptions. The problem of designing a severe test, which as Popper reminds us may be a very difficult task, is really a problem-to-find although it is central to the context of justification. (cf. Lenat on this point)
Those interested in mathematical heuristics have also written on other topics relevant to our inquiry, such as the notion of subproblem or the importance of how a problem is represented but we will deal with these issues later in the form which they have taken within computer science.
6.2 Erotetic Logic and the Structure of Questions
The primary problem for writers on the logic of interrogatives has been that of the semantics of questions. Often the forthcoming solutions have somehow analyzed the meanings of questions in terms of their answers. Thus on the propositional approach, the yes-no question "Will John stay?" is identified with a set consisting of two propositions, "John will stay" and "John will not stay". (Kiefer, p.2) On the categorial approach questions are functions from categorial answers to propositions. So the question, "Who stayed?" will be analyzed as Gxc (xc stayed) where xc ranges over the category of people. (Kiefer, p.2 and ). On the imperative approach, questions are understood as commands, e.g. "Bring it about that I know that John will stay or that I know that John will not stay." (Kiefer, p.2) Other operators have been proposed, such as Aqvist's "Tell me truly" approach (Kiefer, p.9).
It will not be useful for us to pursue the debates between the various schools, but we can profit from their classifications of questions and their analyses of the presuppositions of questions. There is general agreement in this field that questions fall into two basic groups, those which can be formalized in an erotetic variant of propositional logic and those which require quantifiers. We have already seen an example of each, "Will John stay?" vs. "Who stayed?" The two types have received various designations. Belnap and Steel speak of "whether-questions vs. which questions" (B, p.19). Others use the labels "yes-no questions vs. fill-in-the-blank questions" or "Entsheidungsfragen vs. Erga"nzungsfragen" (A, p.47)
Logicians realize that certain questions in ordinary language could plausibly be placed in either grouping, e.g. "Which of the Sipes sisters, Molly and Polly, stayed?" could be rendered as "Is it the case that Molly stayed or Polly stayed?" (A, p.47). And in their classic essay the Priors noted that these two elementary groupings do not include hypothetical questions such as "If you go to College, which one will it be?" However, one can easily extend the classification by adding conditional-whether and conditional-which questions. (A, p.49)
Let us now apply the whether/which dichotomy to scientific problems, and compare the cut which it makes with that effected by the mathematician's prove/find distinction. Under whether-questions it is very natural to put queries about directly observable states of affairs, e.g. questions about the truth of what Popper called `basic' statements, e.g., "Is it the case that this swan is white?" Under which-questions one can obviously put simple problems such as, "Which potassium salts are not soluble in water?" "What is the velocity of light in a vacuum?" "When did dinosaurs become extinct?" "Who is primarily responsible for giving American children moral instruction?" "Where was Atlantis?"
The structure of these who-what-when-where-questions so beloved by journalists is basically the same as that of the problems-to-find: The question either explicitly or implicitly indicates a category (e.g., potassium salts, velocities, times, places, people) and then lays down one or more conditions to be satisfied (e.g., insoluble in water).
At first glance one might also be tempted to draw a parallel between whether-questions and problems-to-prove since both are centered around propositions. However, although a proof does indeed allow us to answer the question of whether the core proposition is true (at least in the sense of following from the axioms), a proof does more because it also gives reasons for the conclusion by connecting it to the premises. So in a sense to-prove problems are a special sort of which-question: Which string of statements connected by inference rules leads from the approved premises to the specified proposition?
Now let us turn to one of the most characteristic kinds of scientific problems--the search for explanations. What does erotetic logic tell us about "How" and "Why" questions? Not much, and it is difficult to see how they fit in. (See Gary's paper). How and why questions typically apply to propositions, but they are definitely not whether-questions. Sometimes how-questions can smoothly be reformulated as which-questions, but not always. Thus "How can I get from here to the Mall?" can be construed as "Which route accessible to me leads from here to the Mall?" "How is Aids transmitted?" also can be seen as asking which "route" in some extended sense that includes a wide variety of causal processes. On the other hand, it is difficult to parse Semmelweis' problem into a which-question with a natural category. Semmelweis asked how the women in Ward I (which had a high incidence of child-bed-fever) were different from those in Ward II of the Vienna General Hospital. We can, of course, write down a translation, "Which property is such that Ward I and Ward II differ in possessing it?" However, what Semmelweis was looking for was a property which might be causally relevant to the occurrence of childbed fever and this set is not a category in any ordinary sense and neither is the set of differences between the Wards.
Why-questions are even more recalcitrant, "Why do you want to go to the Mall?" might be recast as "Which motive/goal is served by your going to the Mall?" "Why does the heart pump blood?" would presumably be construed as asking for a function, but is the set of functions a category? Well, maybe it is no more loosely defined than the set of human goals or motives, especially to the experienced evolutionary biologist. But when pre-Galilean natural philosophers asked why Nature abhorred a vacuum, they had no idea where the explanation would be found. Would it be in an abstract principle of plenitude, or mysterious attractive force (cf. Galileo's forza del vacuo), or, as it turned out, a combination of cohesive forces and atmospheric pressure?
There are two immediate morals to be drawn from these attempted analyses. Why-and-how questions are very context dependent. (As Gary and van Fraassen pointed out, we may well be asking for a modern variant of Aristotle's four causes.) Secondly, it may well be that scientists are reluctant to pursue such questions until there is a background theory or research program which makes it possible to construe them as which-questions with a more-or-less well-defined category. To naively ask for the cause or explanation of a phenomenon without having any idea of whether the expected answer will be in terms of God's will, symmetry properties of molecules, or the social class of the investigator, is probably not a high priority research question, simply because of its amorphous form--it lacks a definite category in which to search for the answer. (We are reminded here of Laudan's observation that problems which cannot be located within a discipline are given low weights.)
Let us now turn briefly to what the erotetic logicians have said about the presuppositions of questions. Presuppositions are of interest to us for two major reasons. First of all whatever the details of our theory of problem evaluation turn out to be, it will have to include the possibility of negative appraisals and one important criticism of a problem would be to show that it presupposes a falsehood (even more devastating would be to show its presupposition is not even approximately true). Secondly, we might wish to show how certain problems cluster together because of shared presuppositions.
However, the task of analyzing questions for their presuppositions turns out not to be a purely formal one. Contrast the following dialogues (based on K, p.87).
1) Q. Who has studied water pollution?
A. There is no water pollution.
2) Q. Who has studied water pollution?
A. Nobody.
Some claim that the "jolt" we experience in the first case is so much greater than in the second case that we should somehow mark the distinction. Thus Kiefer would say that in case #1, the answer violates a presupposition of the question, namely that there is water pollution, while in case #2 only a background assumption is corrected.
There are even subtler cases:
3) Q. When did John take Intro. Logic?
A. Never.
4) Q. Which full professors teach Intro. Logic?
A. None.
Here Joshi claims that the answer in #3 violates a presupposition of the question, while in case #4 only a presumption is violated. (K., p.87). Since #2, #3, #4 are formally the same (all give negative pronouns as answers) logicians have had to search for other clues in order to ascertain when a which-question should be construed as presupposing a positive answer and when it should be read as a which-if-any question. Their answers for ordinary language refer to intonation patterns or the "topic/focus articulation" of questions. (See H & S in K).
We will not pursue this excursion into pragmatics. But it is worth noting that the same sorts of gradations in "jolt" occur in science.
Consider this case:
5) Q. What is lost when a metal is calcined?
A. Nothing.
A phlogistonist would be more surprised by the answer than would a student of Lavoisier, yet their question and answer format are the same. And the following would be even more shocking to the phlogistonist:
6) Q. What are the properties of phlogiston?
A. It doesn't exist.
Here, of course, there is a linguistic difference--the answer to #5 is a negative pronoun; in #6 we have a negative existential claim. Yet a beginning student of Lavoisier (who might mistakenly believe that if `phlogiston' is just an old-fashioned name for hydrogen) would be much less shocked by #6 than s/he would by:
7) Q. What is gained when a metal is calcined?
A. Nothing.
Maybe the strength of a presupposition/background assumption/presumption/allegation of a question has more to do with the location of the assumption in our Quinean net than it does with either the linguistic structure or the articulation of the question.
Erotetic logicians have also tried to explicate various logical relationships between questions--when are two questions inconsistent and when is one question a consequence of another--but the accounts given are sensitive to the details of the approach adopted and we will not pursue them here.
6.3 Computer Science Definitions of Problems
There is a rapidly expanding literature on problem solving by computers. Some of it is motivated by attempts to understand human cognition through simulation studies; in other cases one actually hopes to surpass or supplement human problem solving by means of expert systems. One nice feature of computer simulation studies is that one is forced to make everything explicit and so, not surprisingly, in this area we find very clear conceptions of problems and problem solving. Programmers are also keenly aware of the practical importance of finding a felicitous representation of a problem and we will find similar concerns within the context of scientific problem solving.
What is a problem according to computer scientists? Newell and Simon in their classic Human Problem Solving introduce two general types of representation. I will present both of them although the second kind involving search procedures has come to dominate the field.
Newell and Simon define what they call the set-theoretic representation of a problem as follows:
"Given a set U, to find a member of a subset of U having specified properties (called the goal-set G)." (N & S, p.74) The problem is "well defined if a test exists, performable by the system, that will determine whether an object proposed as a solution is in fact a solution. By performable we mean, more specifically, performable with a relatively small amount of processing effort." (N & S, p.73)
This characterization resembles the logician's analysis of which-questions in terms of a category and conditions but adds the practical requirement (at least for well-defined problems) that the problem-solving system have an operational method of deciding whether the conditions are met. Some of the problems-to-find encountered by mathematicians have this practical decidability feature; others do not. When U is large and G is dispersed through it, it will typically be very difficult to find the solutions. All one can do is generate members of U and then test to see if they fall into the G subset. If G is defined in terms of several properties we may be able to be selective in our generative trials. For example, if we are looking for a crossword puzzle and need a nine letter word, beginning with h, meaning "guide to a solution", we could begin by limiting our generation procedure to words beginning with h or by eliminating non-h words straightaway without bothering to check on length or meaning.
The more common computer formulation of problems, what N & S call the search representation is in terms of states and operations. Here we are given a desired goal state; at the outset we (or the machine) are in some initial state; there are a certain set of allowable operations which transform the present state; when we reach the goal state, the problem is solved. Proving trigonometric identities and solving the Rubick's cube are naturally thought of in this way. One begins with the complicated equation (or starting configuration of the cube) and manipulates the formula or cube in an acceptable manner (no fair dividing by zero or taking the cube apart) until one ends up in a "winning" state. In many cases the search path we take is not randomly chosen (we don't substitute into trigonometric identities willy-nilly or close our eyes when we turn the cube); instead we constantly make comparisons between the present state and the goal state and select our next move accordingly.
Most mathematician's problems-to-prove are a special case of a search problem. Their problems-to-find on the other hand would most naturally be represented in terms of goal sets. In addition to the contrast between the two general types of representations of problems described above, computer scientists have also been interested in the practical features of more specific representations, e.g., the computational advantages of list structures over sentential representations. (Langley, p.320). The issue is not one of logical strength but of operational economy. Scientists are also very sensitive to modes of notation and representation and their effect on our ability to grasp or compute new results.
Consider, for example, the way natural philosophers in the Middle Ages pictured uniformly accelerated motion:
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Insert Fig. 6.1 about here
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On such a diagram, the distance traversed is an area, which at first hardly seems like a natural method of representing it. Yet from the above diagram (using the dotted lines and some trivial geometrical facts) one can directly read off the so-called "odd-number rule" (which applies to objects falling down inclined planes) and the mean-speed theorem. The Medievals had neither algebra nor the integral calculus but by means of a clever geometrical representation it was easy to derive these results. (See Galileo and Oresme.) It is not clear the extent to which the Medieval scientists thought of the motion of actual falling bodies as conforming to such rules. And certainly their concept of motion was substantially different from post-Galilean definitions. (For Aristotle and his medieval followers, being in motion was a quality which could be intensified or remitted.) But because of a serendipitous representation of the problem of motion, they arrived at interesting results.
Computer scientists (especially Simon) have developed another concept which will be useful in our discussion of scientific problems, that of a cue-action pair, or production. (Langley, p.9). As problem-solvers (humans or sophisticated machines) go through a series of similar searches for problem solutions, they may learn to index features of complicated situations and couple them with felicitous responses. A clear but atypically explicit case would be a handbook of chess openings. At the other extreme would be the famous Japanese chicken-sexers, or nursemaids who can tell from the sound of a baby's cry what's disturbing it. Such cue-action pairs may underlie many of the rather ephemeral references to the expert scientist's Fingerspitzegefu"hl (Holton) or tacit knowledge (Polanyi). And probably one major purpose of the set textbook problems emphasized by Kuhn is the fairly systematic inculcation of thousands of these "productions". (Simon believes it takes about ten years to become a world-class expert [Pittsburgh paper] and that ordinary professionals have 50,000 productions at their disposal!) (Langley, p.9).
Productions are sensitive to very specific attributes of the state of search. What people in artificial intelligence call heuristics are also guide-lines for action, but these are cued to much more general features of the problem situation. We will postpone for now discussion of so-called scientific problem-solving programs such as BACON.
6.4 Preliminary Conclusion
What can we now say at this point about the formal structure of scientific problems? To me, the most natural way of representing them is to use an eclectic mixture of concepts drawn from the three formal disciplines surveyed above. In rough order of increasing "depth", I propose the following tentative list:
1. The problem of verifying an "observation" claim (or "basic" statement) (e.g., does the height of the barometer fall as we ascend the mountain?): A whether-question. Is it the case that p?
2. The problem of deriving a prediction or explanandum from a theory (e.g., will liquid flow through a leaky siphon when it is submerged?): A to-prove problem--Given the theory and initial conditions, does the effect follow?
3. The problem of extracting the best statistical summary of the data: A computer-search problem--Given the data, allowable operations, and productions/heuristics, to find a parsimonious general description.
4. The problems of modelling complex initial conditions, of applying theories in new domains, of finding predicted new phenomena, of designing new instruments for a specific purpose: Which-questions or to find problems--Against a background theory, to find that member of a certain general category which satisfies certain special conditions.
5. The problem of producing fundamental explanations or resolving inter-theoretic inconsistencies: Why-questions -- the answer is constrained by the phenomena to be explained, but we really don't know in which general category the answer will be found. (We may need new concepts and a new metaphysics and we may even end up redescribing or correcting the data.)
What computer scientists try to do is to "reduce" `Type 4 questions to Type 3 if possible. The role of a metaphysical research program is to "reduce" Type 5 questions to Type 4.
Chapter VI: Bibliography
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Belnap, Nuel D., Jr. and Thomas B. Steel, Jr. (1976). The Logic of Questions and Answers. New Haven: Yale University Press.
Cohen, Morris R., Ph.D. and I.E. Drabkin, Ph.D. (1958). A Source Book in Greek Science. Cambridge: Harvard University Press.
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Hintikka, J. and U. Remes. (1974). The Method of Analysis: Its Geometrical Origin and Its General Significance. Dordrecht: D. Reidel Publishing Company.
Kiefer, Ferenc. (1983). Questions and Answers. Dordrecht: D. Reidel Publishing Company.
Koertge, Noretta. (1980). "Analysis as a Method of Discovery During the Scientific Revolution," in T. Nickles, Scientific Discovery, Logic, and Rationality, 139-157.
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