168 Correspondence Principle elements with large contributions to the negative Further Readings elements with large contributions. Cosines and Benze´cri, J. P. (1973). L’analyse des donne´es [Data contributions for the punctuation example are analysis] (2 vol.). Paris: Dunod. given in Tables 3 and 4. Brunet, E. (1989). Faut-il ponderer les donne´es linguistiques [Weighting linguistics data]. CUMFID, 16, 39–50. Clausen, S. E. (1998). Applied correspondence analysis. Multiple Correspondence Analysis Thousand Oaks, CA: Sage. Escofier, B., & Pages, J. (1998). Analyses factorielles CA works with a contingency table that is simples et multiples [Simple and multiple factor equivalent to the analysis of two nominal vari- analysis]. Paris: Dunod. ables (i.e., one for the rows and one for the col- Greenacre, M. J. (1984). Theory and applications of umns). Multiple CA (MCA) is an extension of correspondence analysis. London: Academic Press. CA that analyzes the pattern of relationships Greenacre, M. J. (2007). Correspondence analysis in among several nominal variables. MCA is used practice (2nd ed.). Boca Raton, FL: Chapman & Hall/ to analyze a set of observations described by CRC. Greenacre, M. J., & Blasius, J. (Eds.). (2007). Multiple a set of nominal variables. Each nominal vari- correspondence analysis and related methods. Boca able comprises several levels, and each of these Raton, FL: Chapman & Hall/CRC. levels is coded as a binary variable. For example, Hwang, H., Tomiuk, M. A., & Takane, Y. (2009). gender (F vs. M) is a nominal variable with two Correspondence analysis, multiple correspondence levels. The pattern for a male respondent will be analysis and recent developments. In R. Millsap & A. [0 1], and for a female respondent, [1 0]. The Maydeu-Olivares (Eds.), Handbook of quantitative complete data table is composed of binary col- methods in psychology (pp. 243–263). London: Sage. umns with one and only one column, per nomi- Lebart, L., & Fenelon, J. P. (1971). Statistiques et nal variable, taking the value of 1. informatique appliquees [Applied statistics and MCA can also accommodate quantitative vari- computer science]. Paris: Dunod. Weller, S. C., & Romney, A. K. (1990). Metric scaling: ables by recoding them as ‘‘bins.’’ For example, Correspondence analysis. Thousand Oaks, CA: Sage. ascorewitharangeof − 5to + 5couldbe recoded as a nominal variable with three levels: less than 0, equal to 0, or more than 0. With this schema, a value of 3 will be expressed by the pat- CORRESPONDENCE PRINCIPLE tern 0 0 1. The coding schema of MCA implies that each row has the same total, which for CA The correspondence principle is generally known implies that each row has the same mass. as the Bohr correspondence principle (CP), for Essentially, MCA is computed by using a CA Niels Bohr. It is considered one of Bohr’s greatest program on the data table. It can be shown that contributions to physics, along with his derivation the binary coding scheme used in MCA creates of the Balmer formula. Bohr’s leading idea is that artificial factors and therefore artificially reduces classical physics, though limited in scope, is indis- the inertia explained. A solution for this problem pensable for the understanding of quantum phys- is to correct the eigenvalues obtained from the CA ics. The idea that old science is ‘‘indispensable’’ to program. the understanding of new science is in fact the Herve´ Abdi and Lynne J. Williams main theme in using the concept of correspon- dence; therefore, the CP can be defined as the prin- See also Barycentric Discriminant Analysis; Canonical ciple by which new theories of science (physics in Correlation Analysis; Categorical Variable; Chi-Square particular) can relate to previously accepted theo- Test; Coefficient Alpha; Data Mining; Descriptive ries in the field by means of approximation at a cer- Discriminant Analysis; Discriminant Analysis; tain limit. Historically, Max Planck had Exploratory Data Analysis; Exploratory Factor introduced the concept in 1906. Bohr’s first han- Analysis; Guttman Scaling; Matrix Algebra; Principal dling of the concept was in his first paper after Components Analysis; R World War I, in which he showed that quantum Correspondence Principle 169 formalism would lead to classical physics when of formal correspondence between modern and n → ∞, where n is the quantum number. Although classical physics. there were many previous uses of the concept, the important issue here is not to whom the concept Old Correspondence Principle can be attributed, but an understanding of the var- (Numerical Correspondence) ious ways that it can be used in scientific and phil- Planck stressed the relation between his ‘‘radi- osophic research. cal’’ assumption of discrete energy levels that are The principle is important for the continuity in proportional to frequency, and the classical theory. science. There are two ways of thinking about He insisted that the terms in the new equation such continuity. A theory T covers a set of obser- refer to the very same classical properties. He for- vations S. A new observation s is detected. T can- 1 mulated the CP so that the numerical value of not explain s1. Scientists first try to adapt T to be able to account for s1. But if T is not in principle lim½Quantumphysics = ½Classicalphysics h → 0 able to explain s1, then scientists will start to look for another theory T * that can explain S and s1. He demonstrated that the radiation law for the The scientist will try to derive T * by using CP as energy density at frequency v, a determining factor. In such a case, T * should lead to T at a certain limit. 8phv3 uðvÞ = ð1Þ Nonetheless, sometimes there may be a set of c3ðehv=kT 1Þ, new observations, S1, for which it turns out that a direct derivation of T * from T that might in prin- corresponds numerically in the limit h → 0tothe ciple account for S1 is not possible or at least does classical Rayleigh–Jeans law: not seem to be possible. Then the scientist will try 8pkTv2 to suggest T * separately from the accepted set of uðvÞ = , ð2Þ boundary conditions and the observed set of S and c3 S . But because T was able to explain the set of 1 where k is Boltzmann’s constant, T is the tempera- observations S, then it is highly probable that T has ture, and c is the speed of light. This kind of corre- a certain limit of correct assumptions that led to its spondence entails that the new theory should ability to explain S. Therefore, any new theory T * resemble the old one not just at the mathematical that would account for S and S should resemble T 1 level but also at the conceptual level. at a certain limit. This can be obtained by specifying a certain correspondence limit at which the new for- malism of T * will lead to the old formalism of T. Configuration Correspondence These two ways of obtaining T * are the general Principle (Law Correspondence) forms of applying the correspondence principle. The configuration correspondence principle Nevertheless, the practice of science presents us claims that the laws of new theories should corre- with many ways of connecting T * to T or parts of spond to the laws of the old theory. In the case of it. Hence it is important to discuss the physicists’ quantum and classical physics, quantum laws corre- different treatments of the CP. Moreover, the inter- spond to the classical laws when the probability den- pretation of CP and the implications of using CP sity of the quantum state coincides with the classical will determine our picture of science and the future probability density. Take, for example, a harmonic development of science; hence, it is important to oscillator that has a classical probability density discuss the philosophical implications of CP and pffiffiffiffiffiffiffiffiffiffi 1 the different philosophical understandings of the PCðxÞ = =ðp x2x2Þ, ð3Þ 0 concept. where x is the displacement. Now if we superim- pose the plot of this probability onto that of the Formal Correspondence 2 quantum probability density jjcn of the eigen- In the current state of the relation between modern states of the system and take (the quantum num- physics and classical physics, there are four kinds ber) n → ∞, we will obtain Figure 1 below. As 170 Correspondence Principle

the high quantum number domain turns out to C be displaced as

vn + 1 = vn + h=2md, Q q q where m is the particle’s mass and d is the length of the box. Such a spectrum does not collapse toward the classical frequency in the limit of large

Prob. Density quantum numbers, while the spectrum of the parti- cle does degenerate to the classical continuum in the limit h → 0 It can be argued that such corre- + −x 0 x spondence would face another obvious problem C = classical prob. Q = quantum prob. relating to the assumption that Planck’s constant goes to zero. What is the meaning of saying that ‘‘a constant goes to zero’’? A constant is a number Figure 1 Classical Versus Quantum Probability that has the same value at all times, and having it Density as zero is contradictory, unless it is zero. A reply to this problem might be that in correspondence, we Richard Liboff, a leading expert in the field, has ought to take the real limiting value and not the noted, the classical probability density PC does not abstract one. In the case of relativity, the limit, ‘‘c 2 follow the quantum probability density jjcn . goes to infinity’’ is an abstract one, and the real Instead, it follows the local average in the limit of limit should be ‘‘v/c goes to zero.’’ Now, when large quantum numbers n: dealing with corresponding quantum mechanics to classical mechanics, one might say that we ought Zx + e → ∞ → ED to take the limit n as a better one than h 0 = = 2 = 1 2 The point here is that values like c and h are con- PCðxÞ PQðxÞ jjcn 2e jjcnðyÞ dy: stantsandwouldnottendtogotozeroortoinfin- xe ity, but n and n/c are variables—n = (0,1,2,3,.. ð4Þ . ) and n/c varies between 0 (when n = 0) and 1 (when nc). (Of course, this point can also count Frequency Correspondence Principle against the old CP of Planck, the first correspon- The third type of correspondence is the offi- dence principle in our list, because it is built on the cially accepted form of correspondence that is assumption that the limit is of Planck’s constant known in quantum mechanics books as the Bohr going to zero.) Correspondence Principle. This claims that the classical results should emerge as a limiting case of Form Correspondence Principle the quantum results in the limits n (the quantum number) and h → 0 (Planck’s constant). Then in The last type of correspondence is form CP, the case of frequency, the quantum value should which claims that we can obtain correspondence if be equal to the classical value, i.e., Q = C.In the functional (mathematical) form of the new the- most cases in quantum mechanics, the quantum ory is the same as that of the old theory. This kind frequency coalesces with the classical frequency in of correspondence is especially fruitful in particu- the limit n → ∞ and h → 0. lar cases in which other kinds of correspondence Nevertheless, n → ∞ and h → 0. are not uni- do not apply. Let us take the example used in fre- versally equivalent, because in some cases of the quency correspondence (quantum frequency). As quantum systems, the limit n → ∞ does not seen in the case of the particle in a cubical box, the yield the classical one, while the limit h → 0 outcome of n → ∞ does not coincide with the does; the two results are not universally equiva- outcome of h → 0. Hence the two limits fail to lent. The case of a particle trapped in a cubical achieve the same result. In cases such as this, form box would be a good example: the frequency in correspondence might overcome the difficulties Correspondence Principle 171 facing frequency correspondence. The aim of form Brian David Josephson proved that the relation correspondence is to prove that classical frequency between the phase difference and the voltage is and quantum frequency have the same form. So, if {d = 2e = h {d given by {t h V,thatis,thevoltageV 2e {t. Q denotes quantum frequency, C classical fre- Now, by the assertion that the Josephson junction quency, and E energy, then form correspondence is would behave as a classical circuit, the total cur- satisfied if CðEÞ has the same functional form as rent would be QðEÞ. Then, by using a dipole approximation, Liboff showed that the quantum transition 2 I = I sin d + Z dd + ZC d d: ð7Þ between state s + n and state s where s >> n gives c 2Re dt 2e dt2 the relation This equation relates the current with the phase n 2 1=2 difference but without any direct reference to the ðEÞ ≈ nðEs=2ma Þ : ð5Þ Q voltage. Furthermore, if we apply form correspon- He also noticed that if we treat the same system dence, Equation 7 is analogous to the equation of classically (particles of energy E in a cubical box), a pendulum in classical mechanics. The total tor- the calculation of the radiated power in the nth que τ on the pendulum would be vibrational mode is given by the expression τ = Md2y + Ddy + τ sin y, 8 dt2 dt 0 ð Þ n ≈ 2 1=2 CðEÞ nðE=2ma Þ : ð6Þ where M is the moment of inertia, D is the viscous Both frequencies have the same form, even if damping, and τ is the applied torque. one is characterizing quantum frequency and the Both these equations have the general mathe- other classical, and even if their experimental matical form treatment differs. Hence, form CP is satisfied. But such correspondence is not problem free; in dx d2x Y = Y sin x þ B þ A : ð9Þ the classical case, E denotes the average energy 0 dt dt2 value of an ensemble of nth harmonic frequency, but in the quantum case, it denotes the eigenenergy This kind of correspondence can be widely used of that level. Also, in the quantum case, the energy to help in the solution of many problems in phys- is discrete, and the only way to assert that the ics. Therefore, to find new horizons in physics, quantum frequency yields the classical one is by some might even think of relating some of the new saying that when the quantum number is very big, theories that have not yet applied CP. Such is the the number of points that coincide with the classi- case with form corresponding quantum chaos to cal frequency will increase, using the dipole classical chaos. The argument runs as follows: approximation, which asserts that the distance Classical chaos exists. If quantum mechanics is to between the points in the quantum case is assumed be counted as a complete theory in describing small. Hence the quantum case does not resemble nature, then it ought to have a notion that corre- the classical case as such, but it coincides with the sponds to classical chaos. That notion can be average of an ensemble of classical cases. called quantum chaos. But what are the possible The main thrust of form correspondence is that things that resemble chaotic behavior in quantum it can relate a branch of physics to a different systems? The reply gave rise to quantum chaos. branch on the basis of form resemblance, such as However, it turns out that a direct correspondence in the case of superconductivity. Here, a quantum between the notion of chaos in quantum mechan- formula corresponds to classical equations if we ics and that in classical mechanics does not exist. can change the quantum formula in the limit into Therefore, form correspondence would be fruit- a form where it looks similar to a classical form. ful here. Instead of corresponding quantum chaos The case of Josephson junctions in superconductiv- to classical chaos, we can correspond both of them ity, which are an important factor in building to a third entity. Classical chaos goes in a certain superconducting quantum interference devices, limit to the formj, and quantum chaos goes to the presents a perfect demonstration of such concept. same form at the same limit: 172 Correspondence Principle

Lim classicalchoas = j subject. But in general, realists are the defenders of n → ∞ the concept, whereas positivists, instrumentalists, = Limn → ∞ quantumchoas j, empiricists, and antirealists are, if not opposed to the principle, then indifferent about it. Some might but because we have only classical and quantum accept it as a useful heuristic device, but that does theories, then the correspondence is from one to not give it any authoritarian power in science. the other, as suggested by Gordon Belot and John Even among realists there is more than one Earman. position. Some such as Elie Zahar claim that the In addition to these four formal forms of corre- CP influence ought to stem from old theory and spondence, many other notions of correspondence arrive at the new through derivative power. might apply, such as conceptual correspondence, Heinz Post is more flexible; he accepts both ways whereby new concepts ought to resemble old con- as legitimate and suggests a generalized corre- cepts at the limited range of applicability of such spondence principle that ought to be applied to concepts. In addition, there is observational corre- all the developments in science. In doing so, he is spondence, which is a weak case of correspon- replying to Thomas Kuhn’s Structure of Scientific dence whereby the quantum will correspond to Revolutions, rejecting Kuhn’s claim of paradigm what is expected to be observed classically at a cer- shift and insisting on scientific continuity. In tain limit. Structural correspondence combines ele- doing so, Post is also rejecting Paul Feyerabend’s ments from both form correspondence and law concept of incommensurability. correspondence. Hence, scientific practice might So is CP really needed? Does correspondence need different kinds of correspondence to achieve relate new theories to old ones, or are the new the- new relations and to relate certain domains of ories deduced from old theories using CP? Can old applicability to other domains. theories really be preserved? Or what, if anything, can be preserved from the old theories? What about incommensurability between the new and the old? Philosophical Implications How can we look at correspondence in light of Usually, principles in science, such as the Archi- Kuhn’s concept of scientific revolution? What hap- medes principle, are universally accepted. This is pens when there is a paradigm shift? All these ques- not the case with CP. Although CP is considered tions are in fact related to our interpretation of CP. by most physicists to be a good heuristic device, it CP, realists say, would help us in understanding is not accepted across the board. There are two developments in science as miracle free (the no- positions: The first thinks of development in sci- miracle argument). Nevertheless, by accepting CP ence as a mere trial-and-error process; the second as a principle that new theories should uphold, we thinks that science is progressive and mirrors real- in effect are trapped within the scope of old theo- ity, and therefore new theories cannot cast away ries. This means that if our original line of reason- old, successful theories but merely limit the old ing was wrong and still explains the set of ones to certain boundaries. observations that we obtained, then the latter the- Max Born, for example, believed that scientists ory that obeys CP will resolve the problems of the depend mainly on trial and error in a shattered old theory within a certain limit, will no doubt jungle, where they do not have any signposts in continue to hold the posits of the wrong theory, science, and it is all up to them to discover new and will continue to abide by its accepted bound- roads in science. He advised scientists to rely not ary conditions. This means that we will not be able on ‘‘abstract reason’’ but on experience. However, to see where old science went wrong. In reply, con- to accept CP means that we accept abstract reason; ventional realists and structural realists would it also means that we do not depend on trial and argue that the well-confirmed old theories are error but reason from whatever accepted knowl- good representations of nature, and hence any new edge we have to arrive at new knowledge. theories should resemble them at certain limits. The philosophical front is much more complex. Well, this is the heart of the matter. Even if old the- There are as many positions regarding correspon- ories are confirmed by experimental evidence, this dence as there are philosophers writing on the is not enough to claim that the abstract theory is Correspondence Principle 173 correct. Why? Mathematically speaking, if we the old theory as a whole; we can save only the have any finite set of observations, then there are representative part. Structural realists, such as many possible mathematical models that can John Worrell and Elie Zahar, claim that only the describe this set. Hence, how can we determine mathematical structure need be saved and that CP that the model that was picked by the old science is capable of assisting us in saving it. Philip Kitcher was the right one? asserts that only presupposition posits can survive. But even if we accept CP as a heuristic device, Towfic Shomar claims that the dichotomy should there are many ways that the concept can be be horizontal rather than vertical and that the only applied. Each of these ways has a different set of parts that would survive are the phenomenological problems for realists, and it is not possible to models (phenomenological realism). Stathis Psillos accept any generalized form of correspondence. claims that scientific theories can be divided into The realist position was challenged by many phi- two parts, one consisting of the claims that con- losophers. Kuhn proved that during scientific revo- tributed to successes in science (working postu- lutions the new science adopts a new paradigm in lates) and the other consisting of idle components. which the wordings of the old science might con- Hans Radder, following Roy Bhaskar, thinks tinue, but with different meanings. He demon- that progress in science is like a production line: strated such a change with mass: The concept of There are inputs and outputs; hence our old mass in relativity is not the same as Newtonian knowledge of theories and observations is the mass. Feyerabend asserted that the changes between input that dictates the output (our new theories). new science and old science make them incommen- CP is important in the process; it is a good heuris- surablewitheachother.Hence, the realist notion of tic device, but it is not essential, and in many cases approximating new theories to old ones is going it does not work. beyond the accepted limits of approximation. But is CP a necessary claim for all kinds of real- The other major recent attacks on realism come ism to account for developments in science? Some, from pessimistic metainduction (Larry Laudan) on including Shomar, do not think so. Nancy Cart- one hand and new versions of empiricist arguments wright accepts that theories are mere tools; she (Bas van Fraassen) on the other. Van Fraassen thinks that scientific theories are patchwork that defines his position as constructive empiricism. Lau- helps in constructing models that represent different dan relies on the history of science to claim that the parts of nature. Some of these models depend on realists’ explanation of the successes of science does tools borrowed from quantum mechanics and not hold. He argues that the success of theories can- account for phenomena related to the microscopic not offer grounds for accepting that these theories world; others use tools from classical mechanics are true (or even approximately true). He presents and account for phenomena in the macroscopic a list of theories that have been successful and yet world. There is no need to account for any connec- are now acknowledged to be false. Hence, he con- tion between these models. Phenomenological real- cludes, depending on our previous experience with ism, too, takes theories as merely tools to construct scientific revolutions, the only reasonable induction phenomenological models that are capable of repre- would be that it is highly probable that our current senting nature. In that case, whether the fundamen- successful theories will turn out to be false. Van tal theories correspond to each other to some extent Fraassen claims that despite the success of theories or not is irrelevant. The correspondence of theories in accounting for phenomena (their empirical ade- concerns realists who think that fundamental theo- quacy), there can never be any grounds for believ- ries represent nature and approximate its blueprint. ing any claims beyond those about what is Currently, theoretical physics is facing a dead- observable. That is, we cannot say that such theo- lock; as Lee Smolin and Peter Woit have argued, ries are real or that they represent nature; we can the majority of theoretical physicists are running only claim that they can account for the observed after the unification of all forces and laws of phys- phenomena. ics. They are after the theory of everything. They Recent trends in realism tried to salvage realism are convinced that science is converging toward from these attacks, but most of these trends a final theory that represents the truth about depend on claiming that we do not need to save nature. They are in a way in agreement with the 174 Covariate realists, who hold that successive theories of Krajewski, W. (1977). Correspondence principle and ‘‘mature science’’ approximate the truth more and growth in science. Dordrecht, the Netherlands: Reidel. more, so science should be in quest of the final the- Liboff, R. (1975). Bohr’s correspondence principle for ory of the final truth. large quantum numbers. Foundations of Physics, 5(2), Theoretical representation might represent the 271–293. Liboff, R. (1984). The correspondence principle revisited. truth about nature, but we can easily imagine that Physics Today, February, 50–55. we have more than one theory to depend on. Makowski, A. (2006). A brief survey of various Nature is complex, and in light of the richness of formulations of the correspondence principle. nature, which is reflected in scientific practice, one European Journal of Physics, 27(5), 1133–1139. may be unable to accept that Albert Einstein’s Radder, H. (1991). Heuristics and the generalized request for simplicity and beauty can give the cor- correspondence principle. British Journal for the rect picture of current science when complexity and Philosophy of Science, 42, 195–226. diversity appear to overshadow it. The complexity Shomar, T. (2001). Structural realism and the of iphysics forces some toward a total disagreement correspondence principle. Proceedings of the conference on Mulla Sadra and the world’s with Einstein’s dream of finding a unified theory for contemporary philosophy, Kish, Iran: Mulla Sudra everything. To some, such a dream directly contra- Institute?. dicts the accepted theoretical representations of Zahar, E. (1988). Einstein’s revolution: A study in physics. Diversity and complexity are the main heuristics. LaSalle, IL: Open Court. characteristics of such representations. Nonetheless, CP is an important heuristic device that can help scientists arrive at new knowledge, but scientists and philosophers should be careful as to how much of CP they want to accept. As COVARIATE long as they understand and accept that there is more than one version of CP and as long as they Similar to an independent variable, a covariate is accept that not all new theories can, even in princi- complementary to the dependent, or response, var- ple, revert to old theories at a certain point, then iable. A variable is a covariate if it is related to the they might benefit from applying CP. One other dependent variable. According to this definition, remark of caution: Scientists and philosophers also any variable that is measurable and considered to need to accept that old theories might be wrong; have a statistical relationship with the dependent the wrong mathematical form may have been variable would qualify as a potential covariate. A picked, and if they continue to accept such a form, covariate is thus a possible predictive or explana- they will continue to uphold a false science. tory variable of the dependent variable. This may be the reason that in regression analyses, indepen- Towfic Shomar dent variables (i.e., the regressors) are sometimes called covariates. Used in this context, covariates See also Frequency Distribution; Models; Paradigm; are of primary interest. In most other circum- Positivism; Theory stances, however, covariates are of no primary interest compared with the independent variables. Further Readings They arise because the experimental or observa- tional units are heterogeneous. When this occurs, Fadner, W. L. (1985). Theoretical support for the their existence is mostly a nuisance because they generalized correspondence principle, American may interact with the independent variables to Journal of Physics, 53, 829–838. obscure the true relationship between the depen- French, S., & Kamminga, H. (Eds.). (1993). dent and the independent variables. It is in this cir- Correspondence, invariance and heuristics: Essays in honour of Heinz Post. Dordrecht, the Netherlands: cumstance that one needs to be aware of and Kluwer Academic. make efforts to control the effect of covariates. Hartmann, S. (2002). On correspondence. Studies in Viewed in this context, covariates may be called History & Philosophy of Modern Physics, 33B, by other names, such as concomitant variables, 79–94. auxiliary variables, or secondary variables. This