Philosophical Psychology Vol. 23, No. 2, April 2010, 163–172
Functional relations and causality in Fechner and Mach Michael Heidelberger In the foundations of Fechner’s psychophysics, the concept of ‘‘functional relation’’ plays a highly relevant role in three different respects: (1) in respect to the principles of measurement, (2) in respect to the mind-body problem, and (3) in respect to the concept of a law of nature. In all three cases, it is important to explain the difference between a functional dependency of a variable upon another and a causal relationship between two (or more) variables. In all three respects, Ernst Mach developed Fechner’s ideas further and tried to extend the lessons he had learned from using the concept of a functional relation in psychophysics to the whole of science. For all three cases, I explain why they are still relevant for psychophysics and philosophy today. Keywords: Causality; Ernst Mach; Functional Relation; Gustav Theodor Fechner; Measurement
In his Elemente, Fechner (1801–1887) defines psychophysics as an ‘‘exact doctrine on the functional correspondence or interdependence of body and soul’’ (Fechner, 1860, p. 8). ‘‘Functional correspondence’’ [funktionelle Abha¨ngigkeitsbeziehung] is then characterized as a ‘‘constant or lawful relation between both [the material and the mental] such that we can infer from the existence and the changes of one the existence and changes of the other’’ (ibid.). Fechner makes it clear that such a relation is called ‘‘functional’’ because it states the dependency of a psychological variable on a physical one (or the other way around) in the same way as a mathematical function describes a dependency relation between x and y. By using definitions like these, the nature of psychophysics is intrinsically connected with the concept of a mathematical function in a considerable and important way. The question arises why Fechner links up the meaning of his new
Michael Heidelberger is Professor of Philosophy at the University of Tu¨bingen. Correspondence to: Michael Heidelberger, Philosophisches Seminar der Universita¨t Tu¨bingen, 72070 Tu¨bingen, Germany. Email:
[email protected] ISSN 0951-5089 (print)/ISSN 1465-394X (online)/10/020163-10 ß 2010 Taylor & Francis DOI: 10.1080/09515081003727400
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science so closely with the form of a mathematical function. I do not think this is only because Fechner wants psychophysics to be an ‘‘exact science,’’ i.e., a mathematically formulated one. I do not think that anyone in the history of science prior to Fechner seems to have defined a scientific discipline in relation to the functional dependence of one variable upon another. A first glimpse of such a move before Fechner can, however, be made out in the preface of Emil Du Bois-Reymond’s (1848) work on electrophysiology, in which he advocates the ideas of the biophysical movement of his time. He says there that the seeds of a ‘‘physico-mathematical’’ view of physiology lie in the aspiration ‘‘to imagine the causal connection of natural phenomena [in physiology] by the mathematical picture of dependency, by a function’’ (Du Bois-Reymond, 1848/1912, p. 6). Without such a view, Du Bois-Reymond continues, ‘‘even the most exact determinations of measurement would be as fruitless for understanding life-processes as the mere measurement of a machine’s dimension would be for comprehending its workings’’ (ibid.). And later on he adds that the ‘‘true challenge for physiology is not so much to detect the causes’’ of mechanical movement of matter to which the physiological phenomena reduce in the end, ‘‘but to find the laws by which they are governed’’ (Du Bois-Reymond, 1848/1912, p. 15). Fechner seems to have conceived of psychophysics in a related way. He gives two reasons—both of which are somewhat similar to those given by Du BoisReymond—why the psychophysical relation must be treated and conceived as a mathematical function: 1. Doing so makes it possible to avoid any reference to causality. The relation between dependent and independent variables in a mathematical function does not in itself say anything about the causal meaning of this relation, nor about any asymmetry that might be relevant for distinguishing between cause and effect. A mathematical function is itself completely neutral with respect to causality. 2. Doing so makes measurement of the psychological realm possible and thereby provides for a deeper understanding of psychology: ‘‘only the physical can be measured directly whereas the measure of the psychical can be accomplished only in dependence on it [i.e. on the physical]’’ (Fechner, 1860, p. 9). Thus there is an asymmetry in the relation between the psychical and physical (although not a causal one) which can adequately be expressed only by a mathematical function. Hence, outer psychophysics has to concentrate on the functional dependency of the psychological on the physical and not on the reverse dependency (Fechner, 1851/1987, p. 203). But why should we avoid causality and how does a functional relation help us to do so? How does a functional relation make measurement possible? What is the nature of a proposition in which a functional dependency is stated: is it an empirical statement or a conventional definition? In order to find an answer, let us have a closer look at the three respects in which the functional relation plays an important role for Fechner. The first topic concerns the measurement of sensations.
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1. Functional Relations and Principles of Measurement Fechner first argued for a general principle of measurement in which the concept of a measuring instrument plays a central role. Fechner’s reasoning can be reconstructed as follows: An instrument A measures a dependent variable Q by another one RA (that is realized in A) only if: (a) every value of Q serving as input to A is associated with a unique value of RA, and (b) the order of values of Q in A is such that it uniquely correlates to an order of values of RA in A. If a measurement apparatus A fulfills these properties, then RA functions as a representative of Q, and A realizes a constant correlation between Q and RA. Take for example a mercury thermometer that measures the temperature Q of a liquid (independent variable) in virtue of the length RA of the mercury column (dependent variable). We can infer from the length of the column the temperature of the measured liquid because the behavior of mercury serves as a representative of the state of temperature. The thermometer realizes a constant correlation between the temperature and the independent variable RA such that the height of the mercury column correlates with the size of the temperature. If we calibrate the thermometer in the right way, it can express the values of the temperature quantitatively. Can the conditions stated above be taken as sufficient for measurement? Not yet, because, as Fechner argues, we have to be sure that the equality of values of Q can be determined empirically, and we have to know when Q is zero (Fechner, 1851/1987, p. 203; 1887/1987, p. 213; 1860, pp. 58f, 63f ). This means that we have to add a further necessary condition to the above definition for it to be sufficient: (c) it must be empirically possible to determine differences between values of Q as being equal to each other, and this equality must be preserved by the measurement formula. If (c) were not added, the choice of the measurement formula would be completely arbitrary. Note that the measurement formula does not state that a certain natural law holds between Q and RA. Rather, it describes how Q and R are related to each other by virtue of the measuring instrument that was chosen or constructed for the measurement of Q. The measurement formula here works as a conventionally and freely chosen functional relation and is as such to be distinguished from a law of nature. It is true that the choice among possible conventions is guided by empirical facts, but this does not make it a natural law. In order to have a simple and easily manageable measuring instrument, we choose or construct a representative and calibrate it in such a way that equal differences between values of R correspond to equal differences between values of Q. If we call the equation between the values of Q and those of RA a ‘‘measurement
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formula,’’ then the most convenient and useful measurement formula for the thermometer would be Q ¼ n f(RA), with n ¼ 1 and f as a linear function. In Fechner’s time, the objection was raised that RA must be caused by Q in order to be measurable (e.g., by Elsas, 1886). In the example with the measurement of heat this would mean that the thermometer measures temperature because the temperature causes the mercury column to expand. There must be a causal relationship between the temperature of the mercury and the length of the column. Fechner responded to this criticism that this requirement is too strong: it is enough for a measuring instrument that the relation in question is functional in order to fulfill its purpose. The controversy boils down to the question whether condition (c) is really adequate. Let us consider how this reasoning can be transferred from the physics of heat and temperature to the problem that is at the center of Fechner’s interest, the problem of measuring sensations. This measurement requires, as every measurement does, a suitable measuring instrument for which we know the measurement formula involved. The only variable of which we know by experience that it varies with sensation S seems to be the physical intensity I of the stimulus. This intensity seems to be the only feasible candidate to serve as a representative of S in a measuring instrument. (There might be other, neurophysiological variables, but they are—at least in Fechner’s time—empirically out of reach.) Consequently, the only device which could possibly serve as a measurement apparatus relating S to I is the living human body. So, instead of being able to construct a measuring instrument with a most convenient measurement formula, as it is the case in physics, we can, in the case of psychophysics, choose only a single one with an unknown measurement formula. We find ourselves, as it were, in the position of someone who finds an unknown physical measuring instrument fabricated by an alien culture, for which the measured variable and its representative are known, but not the relevant measurement formula. Consequently, the first task of outer psychophysics must be precisely to find the measurement formula for measuring sensation. It follows from Weber’s law that the measurement formula for mental measurement cannot be as simple as in ‘normal’ physical measurement. As is well known, Fechner eventually arrived at his logarithmic solution S ¼ k log I/ I0, where I0 is the absolute threshold of a stimulus intensity below which a stimulus is not sensed. To find the measurement formula means to determine I0, and also the stimulus increment necessary to produce a just noticeable difference when added to the starting stimulus level I0. It also means that its non-linear character makes the measurement formula more difficult to handle than in the case of physical measures. For Fechner, the logarithmic formula also plays a role in inner psychophysics, indeed a central one, yet this time not as a measurement formula, but as a fundamental empirical law of nature that governs the relationship between the subjective dimension and its neural correlate or ‘‘psychophysical activity’’ as Fechner calls it. According to Fechner, precisely inner psychophysics has to deal with this relation, the relation of psychophysical activity or ‘‘excitation,’’ i.e., of the nervous activity of the brain parts immediately connected with the mind to subjective phenomena.
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Consequently, Fechner claims that taking the living human body as a measuring instrument of sensations the values of S resulting from measurements in outer psychophysics are, as a matter of fact, logarithmically related to the fundamental neural activity. It is thus only a coincidence that the same formula figures in outer as well as in inner psychophysics. If in a different world Weber’s law looked different, the fundamental law could very well differ from the measurement formula.
2. Functional Relations and the Mind-Body Problem The second aspect of Fechner’s psychophysics for which the difference between ‘‘law of nature’’ and ‘‘functional relation’’ is important is mind-body theory, the view of how the two are related to each other. We have seen that Fechner defines psychophysics as the doctrine of the functional dependence of body and soul. This is now meant in a different sense than just discussed. It means that psychophysics should investigate only the ‘‘phenomenal side of the material and the mental world’’ and refrain from saying anything about the essences behind these phenomena and their relation (Fechner, 1860, p. 8). Any reference to such essences, i.e., to matter and soul, would belong to metaphysics. Any hypothesis which would regard the connection between mental and material phenomena as stronger than a functional relation, for example as a causal one or as a relation of identity of mental states and brain states, would beg the question and be metaphysical. Fechner is not against metaphysics as a matter of principle, but against premature metaphysics. Only a very mature science can suggest a plausible ‘‘metaphysical complement’’ for its starting principles, but a young and not fully formed science like psychophysics ought to refrain from such an attempt for a long time yet, just as in physics one had to wait quite long before atomism became really plausible. Psychophysics turns into metaphysics as soon as one asserts something ‘‘about the reason [for a functional dependency], about its interpretation and its scope’’ (ibid.) that goes beyond the actually given phenomena and their relations. At the beginning, any realistic assumption about the nature of mind and body and the relation between them has to be avoided. It seems as if Fechner distinguished two components of a realistic statement about the mind-body relation: a core element stating a functional relation between a material and a mental variable, and an additional component that interprets this relation in a realist way, as is done, for example, by Cartesian mindbody dualism or by a materialist identity theory. Fechner maintains as a basic principle of psychophysics that every mental event has a ‘‘psychophysical substrate’’—a ‘‘support’’ or ‘‘basis’’ in the brain. This conception has survived until today in the idea of the ‘‘neural correlate’’ of the mental. Such a neural correlate is the minimal set of neurons, whose firing directly correlates with the relevant mental activity of the subject at a given time. Conversely, stimulating these neurons in the right manner with some suitable, yet possibly still unheard-of, technology should give rise to the same mental activity as
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before (see, e.g., Koch, 2004, p. 16). Fechner and today’s neurophysiology agree with each other that the investigation of the underlying substrate of a subjective behavior—whether called ‘‘neural correlate’’ or ‘‘psychophysical activity’’—does not by itself commit one to a particular metaphysical view about the nature of matter and mind or about the relation between them. This is precisely the reason why Fechner calls the relation between a mental activity and its neural correlate a ‘‘functional relation’’ and avoids for now any causal terminology in characterizing it further. We can observe this Fechnerian heritage even today: psychophysics is still a science concentrating on functional dependencies, avoiding talk of causal relations, and postponing the question of how precisely mind and body are related to each other. ‘‘By keeping to it [i.e. to the mere admission of an empirical parallelism given by the functional dependency of mind and body]’’ William James wrote, ‘‘our psychology will remain positivistic and non-metaphysical,’’ although he thought like Fechner that ‘‘this is certainly only a provisional halting-place, and things must some day be more thoroughly thought out’’ (James, 1890, p. 182).
3. Functional Relations and Laws of Nature Fechner’s peculiar distinction in psychophysics between a functional relation and a natural law implies a third aspect, namely, a special view of the notion of a natural law that is at variance with a widely popular view today. In the greater part of today’s philosophical literature, the paradigm of a law of nature is a conditional proposition like: ‘‘if mercury is heated, then it expands’’ or ‘‘if you are pricked with a needle, then you feel pain’’; i.e., it is taken as a relation between events or conditions. Consequently, one has to make an interpretative effort in order to show that a law like the 2nd law of Newtonian mechanics, F ¼ ma, is only a different formulation of such a conditional proposition. For Fechner, however, the paradigm of a natural law is precisely such a functional equation and not an ‘‘if . . . . , then . . . ’’-clause. Fechner was convinced that the more equations we can find for a certain object realm, the closer we will come to a causal understanding of it. There is a relatively recent exception to mainstream philosophy of science that comes close to Fechner’s idea. More than forty years ago, Herbert A. Simon and Nicholas Rescher rejected the idea that a causal relation can be reduced to the asymmetry of the conditional and tried to relate it instead with the concept of the functional relation (Simon & Rescher, 1966). They defined a ‘‘self-contained structure’’ as a set of functions involving as many functions as variables. A certain asymmetry can obtain in the functional relations appearing in these equations such that subsets of equations could be solved for certain variables without solving for others, but not vice versa. The asymmetric order of the variables thereby induced can be interpreted as a causal ordering. The inverse can be shown as well that if a set of functions is not (yet) self-contained, it cannot determine causal
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relations among the variables. Fechner would certainly have liked this result because it shows that: 1. a ‘thick’ structure of purely functional dependencies can eventually reveal a causal order among the variables, and 2. a functional dependency is a necessary, but not sufficient ingredient of a causal ordering.
4. Functional Relations in the Hands of Ernst Mach Whereas for Fechner the functional relation with its acausal character served as a methodological precaution for psychophysics to avoid any premature smuggling-in of metaphysics, Ernst Mach (1838–1916) took it as the cornerstone for a whole new conception of science in general. His basic idea concerns the order between physics and psychophysics and turns the tables: it is wrong to take physics as the fundamental base and erect a science of sensation upon it, as many exponents of the mechanical worldview propagated at Fechner’s and Mach’s time. Mach, however, thought that it is not psychophysics that needs reinterpretation in the light of physics in order to harmonize with ordinary scientific criteria, but science itself (and its philosophy) must change so that it suits psychophysics! According to Mach then, the task of physical measurement is in the first place to find a suitable representative for sensations that can be used in a functional relation. We know through experience that the expansion of volume can serve, for example, as an indicator or representative of heat sensations. However, since physics is primarily a science of external phenomena, it has to relate volume not only to sensations of heat but also to other phenomena in the outer world. In the course of time, the relation of the representative to other physical phenomena will get the upper hand and the relation to heat sensation will lose its importance and will get sacrificed eventually. We try to find a representative of the temperature scale that allows us to make simple and productive claims about the relations existing between volume and other physical phenomena and we do not care anymore about the dependence it originally had with our sensations. For example, if the sensation of heat in my hand varies with a certain former condition of the hand (whether it was immersed before into cold or hot water), we do not say anymore that the temperature itself has changed, because this would be impractical for our physical knowledge. The necessary deviation from the original functional relation is, however, minimized as much as possible. If we are clever enough in selecting the right indicator for heat, it will enable us to better understand the relations of things in the external world and to make predictions about heat phenomena that are superior to those we would get sticking with exactly representing pure heat sensation. Other reasons for changing the initial choice of a representative of sensation can lie in the limited reliability of our bodily constitution. Instead of using our individual hands and feet to determine spatial extension without perceiving spatial changes in them, we select a publicly accessible standard that is more rigid (Mach, 1886, p. 157). For Mach, measuring
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instruments in physics are not devices to make measures objective in some sense, but artifacts that originally served as substitutes for our sensations and were later adjusted to the purpose of predicting the relations between external phenomena. This view of things allowed Mach to solve some of the problems he had encountered in physics. First of all, measuring is not a matter of discovering an intrinsic property of an object, but of determining a relation between the thing measured and the freely chosen standard of measurement. If one keeps in mind its ‘old’ relation to sensation, one can understand, why some physicists and philosophers think that it makes sense to look for the ‘true’ measure of a magnitude. The fact that we often seek an ‘objective’ length, an ‘objective’ time-period or a ‘true’ temperature must be explained by a psychological mechanism: since physics, as we have seen, very often replaces the indicator of a sensation by a representative that favors the relations of external objects among each other in such a way that the sensation and representative does not anymore run (entirely) in parallel, ‘‘we secretly and unconsciously harbor a notion of the original sensation as the core of our ideas’’ (Mach, 1896, p. 51). This ‘‘shadowy core’’ acquires then the imagined role of being the ‘real’ magnitude which is at most approximated by the indications of a measuring instrument, but never exactly portrayed. The notions of absolute space and time are such relics of the pre-physical perception of space and time prior to the institution of clocks and rulers. ‘‘Newton’s notion of ‘absolute time,’ ‘absolute space,’ and so forth . . . originated analogously. In our ideas of time, sensations of duration play the same part for variously measured portions of time as the sensation of heat does [for measured temperatures in our idea of heat]. The situation is similar for space’’ (Mach, 1896, p. 52). As is well known, Fechner favored a dual-aspect (or better: a double-perspective) theory as solution to the mind-body problem that was later, somewhat misleadingly, called ‘‘psychophysical parallelism.’’ According to this view, the living human body can be perceived from two perspectives: from the outside, if seen by another person (i.e., from an external point of view), or from the inside, if perceived by the person herself (i.e., from the point of view of the perceiver, the inner point of view). In order to clarify now the relation of Mach’s mind-body theory to Fechner’s solution, one has to distinguish between the psychophysicist’s renunciation of a causal interpretation of the psychophysical law and the rejection of causality by the mind-body philosopher trying to spell out the most plausible solution to the mind-body problem in view of the progress of psychophysics. In other words, one should tease apart the methodological postulate to treat the mind-body relation as a functional relation from the noncausal interpretation of the psychophysical law in mind-body theory. Fechner’s philosophical (or if you wish: inductively metaphysical) doctrine is the result of just such an interpretation. For him, mind and body are related to each other like the two sides of a coin. They are aspects of one and the same object and cannot be said to have a causal relation between them. In order to avoid any speculative reference to an unknown reality underlying the phenomena that is neither material nor mental, Fechner rejected the concept of substance from very early on and conceived of the living human body as a bundle of lawfully connected
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phenomena or, as he called it, ‘‘appearances.’’ When, in 1886, Mach rejected Fechner’s view, he was criticizing, as he admitted in a later note, Fechner’s pre-1855 viewpoint, when Fechner had not yet argued for the bundle theory of substance (for more details on this see Heidelberger, 2000). By 1886, Mach came to reject Fechner’s notion of perspective and wrote that ‘‘the elements given in experience, whose connection we are investigating, are always the same, and are of only one nature, though they appear, according to the nature of the connection, at one moment as physical and at another as psychical elements’’ (Mach, 1886, p. 51). This means that Mach replaces the two different perspectives by two kinds of functional dependency that can obtain between the elements. This is the basic idea of his ‘neutral monism’, as it came to be called later. Mach’s reason for this move might be the urge to avoid any remnants of an irreducible ego that might be implied by the notion of a perspective of a person toward herself as in Fechner’s mind-body theory. Surprisingly enough, however, Mach fell back upon Fechner’s mature theory later in 1905 (see again Heidelberger, 2000). We have seen that for Fechner, psychophysics turns into metaphysics as soon as one surmises about objects (including the soul) or relations that are behind the appearances. He argued, however, for the possibility of a good and tamed inductive metaphysics that is based on science as its basis against a speculative one that has no roots in science. Instead of working out criteria telling us under which circumstances an inductive metaphysics is scientifically acceptable and plausible, Mach vehemently rejected Fechner’s outlook—not only for psychophysics, but for science in general. This means not only that for Mach psychophysics will never suggest a solution to the mind-body problem or that physics has to forego atoms forever and the like, but that the concept of causality has to go as well and that functional relation must take its place for good and everywhere: ‘‘the old-fashioned idea of causality,’’ Mach wrote, ‘‘is a little clumsy: A dose of cause is followed by a dose of effect. This represents a kind of primitive, pharmaceutical Weltanschauung, like in the doctrine of the four elements . . . . The connections in nature are rarely ever so simple that one can identify one cause and one effect. Consequently, I have tried for a long time to replace the concept of cause by the mathematical concept of the functional relation: dependency of the appearances from each other’’ (Mach, 1886, p. 74). Fechner’s distinction between a functional relation and a (causal) law has gone a long way to Ernst Mach.
5. Conclusion In using Fechner’s ideas of the ‘‘measurement formula’’ in physics Mach does not get in conflict with Fechner the psychophysicist (although perhaps with Fechner the physicist). Mach’s idea of measurement as relating phenomena with each other in functional relationships has found a subtle ‘Fechnerian’ counterpart in modern psychophysics where scales and numbers are more and more attached to the inner perceptual worlds of different species and thus not only the human inner world related to the outer one but the inner worlds of different species to each other.
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‘‘Mapping these worlds quantitatively will allow us to compare them between related species operating in different environments, to see how the architecture of perceptual spaces is adapted to mirror biologically useful information from the real world in each species’’ (Chittka & Brockmann, 2005, p. 137). Whether we will finally be able to tighten functional relations in psychophysics so as to make out a causal ordering, not to mention whether it will lead to new insight into the mind-body problem, are still open questions. Causality shows its intricate nature especially clearly in the case of the mind-body relation. There is hope, however, that the new and exciting imaging methods of investigating the brain will bring us closer to a causal understanding. Mach’s severe criticism on causality has not cut back on the use of causal terminology in science at all, although, in mature physics, symmetry principles (which are functional relations after all) have largely taken over the role of natural laws. The causal reticence of psychophysics and mathematical psychology has not changed much since Fechner’s and Mach’s time. Perhaps Mach is right that to look for proximate causes of the mental is futile, but this does not exclude that an evolutionary point of view invoking ultimate causes can lead us in the end closer to the truth. References Chittka, L., & Brockmann, A. (2005). Perception space—the final frontier. PLoS Biology, 3, 564–568. Du Bois-Reymond, E. (1912). Ueber die Lebenskraft [On vital force]. In E. Du Bois-Reymond (Ed.), Reden [Popular lectures] (2nd ed., Vol. 1, pp. 1–26). Leipzig: Veit, (Original work published 1848). Elsas, A. (1886). Ueber die Psychophysik. Physikalische und erkenntnisstheoretische Betrachtungen [On psychophysics: Physical and epistemological observations]. Marburg: Elwert. Fechner, G. Th. (1860). Elemente der Psychophysik, 2 Vols [Elements of psychophysics] (Vol. 1). Leipzig: Breitkopf & Ha¨rtel. Fechner, G. Th. (1987). Outline of a new principle of mathematical psychology. Psychological Research, 49, 203–207 (Original work published 1851). Fechner, G. Th. (1987). My own viewpoint on mental measurement. Psychological Research, 49, 203–207 (Original work published 1887). Heidelberger, M. (2000). Fechner und Mach zum Leib-Seele-Problem [Fechner and Mach on the mind-body problem]. In A. Arndt & W. Jaeschke (Eds.), Materialismus und Spiritualismus: Philosophie und Wissenschaften nach 1848 [Materialism and spiritualism: Philosophy and the sciences after 1848] (pp. 53–67). Hamburg: Meiner. Heidelberger, M. (2004). Nature from within: Gustav Theodor Fechner and his psychophysical worldview. Pittsburgh: University of Pittsburgh Press. James, W. (1890). The principles of psychology. London: Macmillan. Koch, C. (2004). The quest for consciousness: A neurobiological approach. Englewood: Roberts. Mach, E. (1886). Analyse der Empfindungen [The analysis of sensations] (5th ed.). Jena: Fischer. Mach, E. (1896). Die Principien der Wa¨rmelehre [Principles of the theory of heat] (3rd ed.). Leipzig: Barth. Mach, E. (1905). Erkenntnis und Irrtum [Knowledge and error]. Darmstadt: Wissenschaftliche Buchgesellschaft. Simon, H. A., & Rescher, N. (1966). Cause and counterfactual. Philosophy of Science, 33, 323–340.
