Scientific structure

Author: Ernest Nagel (USA), translated by Xu Xiangdong
Translator: Xu Xiangdong
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Publication Date: 2002-04-01
Features: Regarding the example of ice, the second point mentioned above—where the explained item is logically not entailed by the premises—does not raise significant controversy as a general requirement for explanation. If this condition is not met, the conjunction of the premises would be logically equivalent to the explained item; in such a case, the premises would merely restate the law to be explained. For example, consider the law: The time it takes for a freely falling body to travel a certain distance is proportional to the square root of the distance. This law is logically derived from the following law: The distance traveled by a freely falling body is proportional to the square of the time interval of fall. However, since this premise is exactly an equivalent numerical transformation of the explained item, no one would regard it as an explanation of this item. (This example violates the requirement that an explanation must have more than one premise. Due to its complexity, it is not possible to elaborate on an example that does not violate this condition but where the explained item is still logically equivalent to the premises. For instance, Newton's formulation in mechanics and the formulation given by the 18th-century theoretical physicist Joseph-Louis Lagrange are equivalent, although the former is familiar to beginners in physics, while the latter is less well-known due to its use of advanced mathematics.) If someone were to do so, they might be treating the explained item as an explanation of itself. Therefore, it is clear that we expect, in a satisfactory explanation, that the premises used to assert something should be more numerous than what the explained item asserts. More specifically, in the explanation of a particular law, at least one premise should meet the following requirement: when combined with other suitable assumptions, it should be able to explain other laws, not just this one; on the other hand, even when those other assumptions are combined with this law, it should not be possible to explain this premise using the specific law. If no premise in an explanation meets this requirement, two undesirable results will follow: first, it would be impossible to obtain evidence for the premises without the evidence provided by the explained item itself; second, except in isolated cases, since known facts and yet-to-be-discovered facts are not yet related, the explanation would not meaningfully organize the subject matter into a system. The premise must not be equivalent to the explained item, and this requirement is sufficient to exclude many pseudo-explanations, in which the premises merely rename the facts to be explained. A classic example of a pseudo-explanation is the target of Molière's satirical works, where he mocks those who use the claim that opium has a sedative effect to explain the fact that opium induces sleep. Sometimes, subtle examples of explanation can be found in popular science explanations, such as the statement that if an object is not acted upon by unbalanced external forces, its velocity will remain constant because all objects possess an intrinsic inertial force. This is a pseudo-explanation because the term "inertia" is merely another name for the fact explained by the law. It has been argued that the idea used to clarify the concept of causality is vague, and strong objections have been raised—particularly against the common-sense concepts of spatial continuity and temporal continuity, as these concepts are fraught with confusion. Moreover, in advanced sciences like mathematical physics, this idea is indeed redundant, which is undoubtedly true; when the explanation of this concept of causality (as in the above example) is analyzed in terms of modern physical theory, it even raises controversy whether the four conditions mentioned earlier are actually met. However, regardless of how unsuitable this concept may be for the purposes of theoretical physics, it continues to play a role in many other research fields. Even in practical affairs, in laboratories, when abstract physical theories are used to process suitable means in order to obtain various results, this concept is firmly embedded in the language we adopt. In fact, the causal language becomes a legitimate and convenient way to describe the relationships between many events because certain things can be handled, not the other way around. On the other hand, not all natural laws are causal laws in the sense already indicated. A brief examination of the various laws used as explanatory premises in different sciences will make this clear. Let us make the implications of this clearer. When an expression is said to be "explicitly" defined, it can always be removed from any context in which it appears, because it can be replaced by the expression used to define it without altering the meaning of the context. Thus, the expression "x is a triangle" is defined by the expression "x is a closed plane figure bounded by three line segments." This allows the former (the defined expression) to be removed from any context favorable to the latter (the defining expression); for example, the statement "the area of a triangle is half the product of its base and height" can be replaced by the logically equivalent statement "the area of a closed plane figure bounded by three line segments is half the product of its base and height." On the other hand, for this theoretical expression in Bohr's theory, "x is the radiation wavelength emitted when an electron transitions from an approximately small allowed orbit of a hydrogen atom to a small allowed orbit," it is not explicitly defined when equated to an expression of the form "y is a spectral line appearing at a certain position in the hydrogen spectrum." In fact, these two expressions have quite different implications, which is evident. Therefore, although a definite connection is established between the two expressions through the correspondence rule, in statements like "in about ten percent of hydrogen atoms, the transition of an electron from its approximately small orbit to a small allowed orbit occurs," the former cannot replace the latter. If one attempts such a substitution, the result would in fact be meaningless. There is no conclusive proof, and perhaps no such proof is possible, that theoretical concepts adopted in current science cannot be explicitly defined in terms of experimental concepts. The issue raised here will be discussed more fully in the next chapter. However, it should be noted that no one has yet successfully constructed such a definition. Moreover, there are sufficient reasons to believe that in practical applications, the correspondence rule is not used to explicitly define theoretical concepts in terms of experimental concepts. One of these reasons has already been noted. When a theory is expressed through a model, the language used to clarify the model often has implications that experimental procedures do not have. Therefore, as noted above, in Bohr's theory, the expression referring to electron transitions is not equivalent to the expression referring to spectral lines. Thus, in this case, since the defining expression and the defined expression are equivalent in meaning in an explicit definition, it is impossible for the correspondence rule to provide an explicit definition. Perhaps another more compelling reason is that the correspondence rule often coordinates a theoretical concept with more than one experimental concept. As has been argued, a theoretical concept is implicitly defined by the axioms of a theory (even when the theory is proposed through a model). Therefore, as a logical necessity, an infinite number of experimental concepts correspond to a single theoretical concept. For example, in Bohr's theory, the theoretical concept of electron transitions corresponds to the experimental concept of a spectral line; but (through Planck's radiation law, which can be derived from Bohr's theory) this theoretical concept can also be equated to the temperature change in blackbody radiation that can be experimentally determined. Therefore, in cases where a specific theoretical concept corresponds to two or more experimental ideas (although likely in different contexts or problem situations), it would be absurd to think that the theoretical concept is defined by each of these experimental ideas in turn. The lack of a unique correspondence between theoretical and experimental concepts warrants further discussion and examples. A familiar fact is that in science (especially, though not entirely, in mathematical physics), theories are generally formulated with great care, and the relationships between theoretical concepts (whether they are primitives of the theory system or defined in terms of these primitives) are precisely stated. This precision is essential for the deductive reasoning of theoretical hypotheses. On the other hand, the correspondence rule that connects theoretical and experimental ideas is generally not explicitly stated; in practice, the equivalence between the two is rather loose and rough. For instrumentalist views of theory, these kinds of difficulties do not arise, because according to this view, the appropriate question about theories is not whether they are true, but whether they are effective methods for expressing and inferring experimental phenomena. The fact that theories contain expressions that refer to or describe things that do not actually exist, or that contain expressions not related to experimental concepts, is actually taken to confirm the claim that theories must be analyzed in terms of their instrumental function as media in research, rather than in terms of their appropriateness as objective descriptions of a subject matter. From this perspective, for example, the use of restrictive concepts like point particles, instantaneous velocity, and perfectly elastic collisions in the kinetic theory of gases is not a flaw of the theory. Because the task of the theory is not to faithfully depict the evaporation of gases, but to provide a way to analyze and symbolically process certain properties of gases, so that in specific experimental situations, when information about such properties is available, the theory can derive information about other properties with the required precision. Similarly, when studying the thermal properties of gases, we use a theory that analyzes gases as aggregates of discrete particles, while when studying the acoustic phenomena associated with gases, we adopt a theory that represents gases as continuous media. This does not cause trouble for instrumentalist views. The two theories appear incompatible when interpreted as statements that are either true or false. But when interpreted as techniques or principles of reasoning, the two theories are different but complementary, each being an effective tool for dealing with a range of specialized problems. In any case, when physicists use one theory to address one class of problems and another apparently inconsistent theory to address another class of problems, they do not feel any obvious guilt. When dealing with problems of light diffraction and polarization, they adopt the more general wave theory of light, according to which light phenomena are represented as periodic waves, but when dealing with reflection and refraction, they continue to use the simpler geometric optics theory, according to which light is analyzed as a straight-line propagation. When using quantum mechanics to analyze the fine structure of spectral lines, they introduce relativistic considerations; when using quantum theory to analyze the nature of chemical bonds, they ignore such considerations. Examples of this kind can be multiplied; if they do not prove anything else, they at least show that when theories are used in experimental research, the literal truth of the theory is not the primary concern. However, suppose that mechanical techniques gradually advance, and we learn how to polish or cut objects so that one object's surface can fit precisely against another's. Eventually, we might think of taking three objects, polishing their surfaces until any two of them can fit together smoothly. This method seems to provide an excellent objective standard for surfaces with maximum smoothness, regardless of whether we are prepared to call such surfaces "planes." It is clearly meaningless to ask whether such surfaces are "really" planes, because they are considered planes through definition, by assuming that there is no other standard for "being a plane" except the one just described. Also note that when judging whether two surfaces fit together precisely, we can use some optical test, such as one that would show that light does not pass through when the two surfaces are in contact. However, although we can use such an optical test, we do not assume (explicitly or implicitly) that light travels in "straight lines," so our method is not actually circular. We are merely using an observational fact as a condition for saying that two surfaces are in contact. Note that this point has essential significance: when we say that a surface is a plane, the key question is whether the surface satisfies the specified conditions for contact between the surfaces. In particular, it should be noted that when we assign the name "plane" to such surfaces, it does not involve assumptions associated with Euclidean geometry.