SBP Sociedade Brasileira de Psicologia

The Creation of Scientific Psychology

By David J. Murray / Paper presented virtually at the 2021 Brazilian Celebration of Fechner Day, October 22.


Gustav Theodore Fechner (1801 – 1887) is unquestionably the hero of THE CREATION OF SCIENTIFIC PSYCHOLOGY(CSP), a history of 19th century psychophysics written by Murray and Link (2021). This paper has four sections one concerning Fechner, one concerning Johann Friedrich Herbart (1776 – 1841), one concerning Ernest Mach (1838-1916) and a final conclusion.


CSP on Fechner’s Contributions

At the core of CSP is our account of how Fechner, a professor specializing in the physics of electricity, was afflicted in his early 40s by an illness that probably arose from overwork.  We must remember that he had to read and write by candlelight. The dimness of candlelight has been illustrated by Maurer & Maurer (2019, pp. 88-89). During the illness he had the following insight.

Let the physical intensity of a stimulus be strong enough to arouse a consciously felt sensation. Fechner (1860/1966, vol. 2) gave reasons for asserting that the magnitude of that sensation was a monotonic function of the physical intensity of that stimulus. Moreover, he claimed that the function could be specified as being a “natural logarithmic” function. 

“Natural logarithms” are logarithms to the base e. The irrational number e, whose numerical value is 2.71828…, appears in functions that represent how the value of any variable, say y, grows or decays as a function of the value of another variable, say t (time). Mathematically, e is the limit, as n tends to infinity, of the expansion of [1+(1/n)]n. Many interesting properties of e are listed in an easy-to-read volume by Maor (1994).

Fechner then carried out a very large experiment that investigated the affects of five variables on the accuracy with which a participant could report that a stimulus, A, was greater or smaller in intensity than a stimulus B. In the experiment, A and B were liftable weights that were presented close together in time, and the difference in perceived heaviness between A and B was only barely detectable by the “just noticeable difference”.  Fechner, in his large experiment, used (r/n) as a measure of performance, where r= the number of correct “different” responses, and ntotals responses in the experiment.

Pioneering investigations of just noticeable differences were published earlier by a professor of anatomy and physiology, Ernst Heinrich Weber (1795-1878).  Both Fechner and Weber worked at the University of Leipzig in the country then known as Saxony. Weber’s Law (so named by Fechner 1860/1966, p. 54) himself, formed the foundation-stone from which Fechner’s Law was derived. 

In CSP Weber’s Law was introduced on p. 63 as: 

k = ΔI / I                                                       (1)


In Equation 1, k is a constant, and Δ I is the just noticeable difference when a stimulus of physical intensity I was increased by appropriate intensity. k is often called the “Weber fraction” or “Weber’s constant” with respect to any given stimulus material (e.g. a light, a tone, lifted weight).

S = K loge(I /I0)                                                 (2)


In Equation 2, K is a constant and I0 is the smallest stimulus intensity that can be consciously detected against the background. Fechner denoted this as the “absolute threshold” value of stimulus intensity.

A second version of Fechner’s Law was given on page 86 as:

S = K loge[(1 + k)i]                           (3)

Please note that Weber’s constant, k appears in the second version. The valueirepresents the number of small increments in stimulus intensity that need to be added to the absolute threshold in order for the sensation-magnitude S to be obtained.

The large experiment was counter balanced, an important step in the history of the design of psychological experiments (CSP, figure 4.1, p. 93). Fechner hoped to demonstrate the validity of Weber’s Law by showing, for various values are starting physical intensity I, whenever Δ I / I was constant, so too would be the proportion of correct “different” responses in the experiment. To Fechner’s surprise, even though Δ I / I was chosen to be identical as I increased, the proportion of correct “different” responses, instead of remaining identical, also increased. Fechner explained this finding as an artefact of failing to include arm-weight in the equations concerning Weber’s Law. More surprising, though, is the fact that Link (1992) has arrived at his own independent explanation of Fechner’s findings. This explanation is based on his “wave theory of difference and similarity” (CSP, p.192.

Fechner’s magnum opus was a book in two volumes titled Elements of Psychophysics (Elemente der Psychophysik)(Fechner, 1860/1964). Volume 1 was translated into English (Fechner 1860/1966), but Volume 2 remains untranslated. The contents of Volume 1 include Fechner’s specification of Weber’s Law and Fechner’s Law, a discussion of three “psychophysical methods”, and the account of his large experiment.  Volume 2 starts by deriving Fechner’s Law (Equation 2) from Weber’s Law (Equation 1), using calculus. It is important here to know that, if y = logex, then dy / dx = (1/x).

Other topics treated in detail in Volume 2 include several special phenomena in the visual sense modality, and Fechner’s distinction between “inner” and “outer” psychophysics.  His large experiment was an example of “outer psychophysics”. His theorizing about the locus in the nervous system of the logarithm transformation of stimulus intensity into sensation-magnitude are examples of his “inner psychophysics”.


CSP on Herbart’s Contributions


Currently we are enduring a pandemic caused by the COVID-19 virus and its variants. Just over 350 years ago, in 1665, England suffered from a pandemic caused by the bacillus Pasteurella pestis. This pandemic was named the “great plague of London”. The disease was known as the “bubonic plague” had two serious variants, known as “pneumonic plague” and “septicemic plague”. The skin of patients who died from septicemic plague was deep purple, and it was because of this that a wide-spread pandemic of bubonic plague that had devastated Europe in the Middle Ages was called the “Black Death”.

In 1665, Isaac Newton (1642 – 1727) was 23 years old studying and teaching at Trinity College in the well-populated University of Cambridge.  In order to isolate himself from the plague, he returned to his mother’s country home in Lincolnshire, an agrarian region with a relatively low population.  We learned from a friend, writing much later, that Newton “observed” an apple falling from a tree and wondered whether the gravitational force that brought the apple down to the ground might also serve to keep the moon in its orbit round the earth and the earth in its orbit round the sun.  The geometry used by Newton (1726/1999) was supplemented by calculus, a method invented independently by Newton and by G. W. Leibnitz (1646 – 1716).

The philosopher Herbart, known at the time for his criticisms of the usefulness of the words “ego” and “self” in metaphysics, took it upon himself to provide, for mental events, a mathematical model that would parallel, in reliability and completeness, Newton’s model of physical events.  At the age of 46, he published an article claiming that it was both possible and necessary to replace metaphysics by mathematics when writing about psychology (Herbart 1822/1877). Herbart’s magnum opus on the topic was a book titled Psychology as Science (PsychologiealsWissenschaft) (Herbart, 1824/1890).  If presented a closely reasoned mathematical model of how mental events that were consciously experienced (by only one person) could influence each other. A mental representation of a physical object was known to Herbart as a Vorstellung, but so was a sensation (e.g. of a touch) or a body-feeling (e.g. a feeling of thirst).

Properties of Herbartian Vorstellungen are discussed in CSP (chapter 1, pp. 4-5).  Herbart’s model cannot be conveniently summarized in a few words. But, because mental events have no special dimensions, his model does not include variables analogous to Newton’s use of special coordinates, momentum, impetus, velocity, or acceleration.  His model does however include time-duration as a variable. Herbart’s task can be summarized briefly as followed. At time t0, let VorstellungA be in consciousness.  At t1, let a weaker Vorstellung B enter consciousness. At t2, let A and B compete to survive in consciousness. Herbart ascribed a temporary “strength” in consciousness to each Vorstellungin consciousness. He proved that two Vorstellungen competing to stay in consciousness would inhibit each other and thereby lose strength, but neither would leave consciousness. A mathematical equation (CSP, chapter 1, p. 14 determined the strength that a third Vorstellung, C would need in order not to leave consciousness because of inhibition from A and B.

Two Vorstellungenconcurrently in consciousness can combine to form a new Vorstellung (CSP, chapter 1, pp. 14-15) in a “complication” the two Vorstellugen “remain independent that travel in pairs, so to speak” (CSP, p. 14) in a “fusion” the two Vorstellung do not remain independent of each other. In a partial fusion, a proportion of Vorstellung A fuses with a proportion of Vorstellung B. If A enters consciousness, the partly fused proportion of B will also be brought, by A, into consciousness. On the other hand, if A and B are identical in content, all of A can fuse of all of B. The two Vorstellungen A and B will be fused into one Vorstellung whose content can be labelled A or B as preferred.

The following thought-experiment illustrates an argument originally presented by Murray (2002). Imagine that there are six different stimuli (e.g. single digits such as 5 4 8 7 6 1) presented one after the other so quickly that all six stimuli give rise to six corresponding mental Vorstellungen that are concurrently in consciousness.  Let an interval of well under 2 seconds precede the presentation of a seventh single digit. If that seventh digit is identical to one of the six digits, then their corresponding Vorstellung should be identical also.  The two should therefore fuse into one single Vorstellung. To give an example:  if the seventh digit be 4, its Vorstellung could fuse with that of the second digit in the series 5 4 8 7 6 1.

We now use this innocuous-sounding string of consecutive experiences as the basis for a psychological task. The task is to judge whether the probe digit is “old” or “new” with respect to the six different target digits. If a Herbartian fusion occurs because the probe is old then the number of Vorstellung in consciousness, using L as the length of the target list, will be [(L + 1) – 1] = (L + 0).  If there are no Herbartian fusions because the probe is new, then the number of Vorstellung in contiousness immediately after the probe will be (L + 1).

Let E denote the effortlessness or ease of deciding whether a probe is old or new. The larger E is the easier the task:

E = [(L+1)/L] – [(L + 0) / L ] = 1/L                     (4)


Finally, we need to predict the response-time, R, associated with a correct judgement that the probe was old. Let R vary inversely with E =

R = a / E   (5)

where a is a constant of proportionality. It follows that:

R = a / E = a / (1 / L ) = La(6)

That is to say, the response time increases as a linear function of L and the slope of that function is a.

The notion of a “Herbartian fusion” gave rise to Equation 5 that there was no need in the derivation of Equation 5 to incorporate the notion of any kind of mental “scanning” procedure. A Herbartian perspective can conceptualize this task as a psychophysical task (in valuing the mental comparison of numerosities of Vorstellung) rather than as a recognition task (Monsell, 1978).

CSP on Mach’s Contributions

In CSP the discussion of Ernst Mach’s contributions to psychophysics was restricted to chapter 7 (pp. 149-152).  This chapter was devoted entirely to the sensation objective the argument that a mental entity (e.g., a sensation) cannot be measured in the same way as can an entity composed of matter (e.g. a chair).  Kant’s distinction between “expensive” and “intensive” measurement-units was discussed in chapters 1 and 2.  The role played by concatenation as a criterion of the validity of a measurement unit is relevant to any discussion of Fechner’s law.  Fechner considered that a “sensation-magnitude” could be constructed by concatenating of “jnds” each of which was objectively equal in magnitude to each of the other jnds. 

Hering (1875) however, protested that jnds were not subjectively equal in magnitude. Indeed, the most famous thought-experiment in the history of psychophysics was surely that of Hering (1875). Hering

“asserted that it would be obvious to anybody that, IF you have a light weight W, and double it, and that, IF you have a heavy weight W+ and double it, then the experience increase in heaviness will not be identical for the two weights.” (CSP, chapter 8, p. 159)

Fechner’s psychophysics were also criticized for his claim that a “sensation” could be assigned a numerical magnitude at all. Mathematicians in particular insisted that it was scientifically unacceptable to extrapolate, from the established numerical magnitudes used to measure e.g. the length of a table, to the artificially constructed numerical magnitudes used to “measure” the strength of the sensation. Mach’s rescued Fechner from this uncomfortable situation by re-examining what was meant by the word “measurement” itself.  This argument can be encapsulated in three logical steps as follows.

1.       Measurement-events apply only to phenomena that can be observed by scientists.

2.       There are cases in which aspects of non-physical entities can be assigned measurements, provided that these properties of those metaphysical entities can be derived from sensations made by the scientist.

3.       Because “observations” necessarily involve the participation of the scientist’s sensory apparatus, all scientists should base their inferences on sensations themselves, rather than on events surmised to have caused those sensations.

Mach’s argument was also congenial to Albert Einstein (1879 – 1938). Using a thought-experiment in which an individual is imagined to be travelling in an elevator, Einstein pointed out that

IF the sensations experienced by person A standing inside an elevator travelling at such-and-such a velocity will be different from those experienced by a person B outside that same elevator, THEN the coordinates system employed by A to describe his or her movement inside the elevator may differ from the coordinates system employed by B to describe how he or she perceives, from outside the elevator, the movements of the elevator itself. (CSP, chapter 7, p. 151)

More surprising perhaps, is the fact that Mach’s views were also incorporated into the beginnings of quantum physics. Rovelli (2021, chapter 1) considers that modern-day quantum physics originated in a puzzle concerning the pathway taken by electrons in action.  An electron orbiting the nucleus of an atom was capable of ‘jumping’, apparently instantaneously, from one orbit to another. No successful model existed of why this ‘quantum leap’ (as it was called even then) took place.

In 1924, Werner Karl Heisenberg (1901-1976 ) claimed that no such model would ever come into being if the suppositions of that model were based on speculations about unobserved physical events. He proposed doing exactly what Mach had described, namely, basing a model of quantum leaps entirely on what was seen/observed by the scientists. A quantum leap is accompanied by a stream of light-photons that can be detected using special apparatus. Imagine that an electron jumped from orbit 3 (near the nucleus) to orbit 5 (further away from the nucleus).  Heisenberg proposed describing this event as an entry in a table (a “matrix”) where the rows represented the departure orbit (here, orbit 3) and the columns represented the arrival orbits, (here, orbit 5) with m rows and n columns, this leap would be recorded as “m3n5”into that cell into that matrix corresponding to row 3, column 5.

“Quantum mechanics” then burgeoned, with tables as numbers (matrices) dominating the relevant calculations. The replacement of numbers by matrices led to at least one unexpected consequence. The fact is that matrix multiplication is non-commutative. If I multiply matrix A by matrix B, I arrive at a new matrix C. But if I multiply matrix B by matrix A, I arrive at a new matrix that is not identical to C. In ordinary arithmetic, 3 x 5 =15 = 5 x 3. This is not true if numbers are replaced by matrices.

There is a particular reason why many educated laymen confess to having avoided both Einstein’s relativity theory and quantum mechanics. The reason is that both kinds of mathematics introduced more than three “dimensions”, that is, coordinates system. Few people are capable of “visualizing events taking place in more than three dimensions”. Even their brains appear to have evolved in such a way as to suggest that three dimensions – horizontal, vertical, and in-depth – suffice for everyday usage.

The anatomical evidence is startling in our face. The sensory (“proprioceptive”) vestibular system in the inner ear is designed to tell us where we are located in space, and to warn us if we are losing our balance.  This system consists in part of three “semicircular canals”. The adoption of three coordinates (typically labelled x y and z) dominated classical mechanics, not merely since Newton’s time, but from ancient times. So-called “solid geometry” was pioneered alongside two-dimensional geometry by ancient Greek and Chinese mathematicians (CSP, Prologue, pp. xxiv – xxvi). Normal aging has the effect of thickening the gel in each semicircular canal, with the result that old people can more easily “lose their balance” than can young people (Bryson, 2019, p.87).


It would be wrong to end this talk without mention of the work of Ehtibar N. Dzhafarov and his colleagues. They are incorporating quantum mechanics into an analysis of how a human participant decides that two sensory stimuli are identical intensity. Quantum mechanics was traditionally a vehicle for describing physical events taking place at a sub-microscopic level.  But the neural activity itself lies on the borderline of being “sub-microscopic”. Quantum mechanics has by no means been excluded from the psychophysics of the future.



Bryson, B. (2019). The Body: The Guide for Occupants.  Doubleday Canada

Fechner, G. T. (1964). Elemente der Psychophysik[Elements of Psychophysics] (2 vols.). Amsterdam, Netherlands. (E. J Bonset, Ed.) (Original work published 1860)

Fechner, G. T. (1966). Elements of Psychophysics, volume 1. (H. E. Adler, Trans. D.H. Howes and E. G. Boring, Eds.).  New York, NY: Holt, Reinhart and Winston. (Original work published 1860)

Herbart, J. F. (1877). Possibility and necessity of applying mathematics in psychology (Hennel, trans.). Journal of Speculative Philosophy, 11, 251-264. (Original work published 1822)

Herbart, J. F. (1890). PsychologiealsWissenschaft[Psychology as Science] in K. Kehrbach& O. Flügel (Eds.), Joh. Fr. Herbart’s sämtliche Werke InchronologischeReihenfolge (Part 1, Vol 5, 177 – 434). Langensalza, Germany: Hermann Buier und Söhne. (Original work published 1824).

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Maurer, C. and Maurer, D. (2019). Pretty Ugly: Why We Like Some Songs, Faces, Foods, Plays, Pictures, Poems, etc. and Dislike Others. New Castle upon Tyne, U.K.: Cambridge Scholars Publishing.

Monsell, S. (1978). Recency, immediate recognition memory, and reaction time. Cognitive Psychology, 10, 405-501.

Murray, D. J. (2002). The SOC framework and short-term memory. Behavioral and Brain Sciences, 25, 347-348.

Murray, D. J. & Link, S. W. (2021). The Creation of Scientific Psychology. London, U.K. and New York, NY: Routledge (Taylor & Francis Group).

Newton, I. (1999). The Principia: Mathematical Principles of Natural Philosophy (I.B. Cohen & A. Whitman, trans.) (pages 371 – 938). Cambridge, MA: Harvard University Press (Original work published 1687; and ed. 3rd edition 1726).

Rovelli, C. (2021). Helgoland: Making Sense of the Quantum Revolution. New York, NY: Riverhead Books.








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