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\noindent{\bf Gustav Theodor FECHNER}\\
b. 19 April 1801 - d. 18 November 1887
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{\bf Summary.} Physicist, psychologist and philosopher, Fechner is noted for the
introduction of quantitative methods into psychology. He also developed a
`theory of collectives' which is built on the frequency interpretation of
probability.
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Gustav Theodor Fechner was born in Gross S\"{a}rchen near Muskau, Lusatia
(Germany), into the family of a protestant 
minister. He studied medicine, but became disenchanted with the subject and 
never wanted to practice it. Instead, he devoted himself to experiments in 
physics in the fashion of the {\it physique exp\'erimentale} of the leading 
French physicists. His translations and revisions of French text books and 
treatises were the chief channel for the reception of French mathematical 
science into Germany at this time, and the reform of German physics that 
resulted from it. His experimental researches, especially in electricity 
theory, eventually earned him a chair in physics at the University of Leipzig 
(in Saxony, Germany) where he remained for the rest of his life. 

Fechner's orientation towards the most advanced physics of his day was 
supplemented, however, by a strong commitment to idealist and romantic 
{\it Naturphilosophie} which was primarily directed against Cartesian dualism 
of mind and body and eighteenth-century French materialism. Like other 
followers of {\it Naturphilosophie}, he argued, that nature is animated and 
that there is an original unity or `identity' of nature and mind which allows 
us to infer nature's laws from the laws of the mind and vice versa.
 
In 1839, Fechner had a nervous breakdown as the result of a depressive 
psychosis. He developed an aversion to food and also to light and experienced 
temporary blindness and complete prostration. Although he kept the title of a 
physics professor, he eventually lost his physics chair to Wilhelm Weber and 
was set on a pension by the university. After his recovery in 1846, he 
continued lecturing on diverse subjects, especially the mind-body problem, 
until 1875.
 
When his crisis was over, Fechner tried to come to terms with the two opposing 
tendencies of his thought, the strict mechanist mathematical physics on the 
one side and the romantic {\it Naturphilosophie} on the other. As a result 
of this, he developed a solution to the mind-body problem called `psychophysical 
parallelism' or `dual aspect theory' which became very popular 
among scientists in the 19th century. This solution is supposed to be 
compatible with science as well as with {\it Naturphilosophie} and it is 
central, both to Fechner's subsequent philosophical as well as scientific and 
mathematical work. According to this view, mind is not to be seen as a 
substance interacting with the body but as a special attribute of matter on 
which it is functionally dependent. In the same way as the appearance of an 
ordinary object, like the back and front of a coin, depends on the perspective 
of the viewer, so a person, as a body with a mental dimension, can be seen 
from the outside as well as from the inside. Mind and body are two different 
aspects of one and the same object. A person appears to outside viewers in 
another way than to herself or himself. As a consequence, it would not make 
any sense to say that the mind acts on the body or vice versa as it would be 
senseless to say that the back of the coin acts on its front when the coin is 
bent. Rather, if there is a change of a person it can be viewed in a mental 
as well as in a physical respect. Similarly, the bending of a coin results in 
a change of the coin's front and of its back at the same time. In thus
rejecting causality as an appropriate category for the mind-body relation, 
Fechner thought he had shown psychophysical parallelism to be compatible 
with the principle of the conservation of energy. On the basis of his 
theory, Fechner founded the science of psychophysics, which became the 
starting point for experimental quantitative psychology.

Before it can be shown what all this has to do with statistics, we have to 
turn to another philosophical development. One of the most influential and 
powerful philosophical systems of the first half of the 19th century was that 
of G. W. F. Hegel. He claimed to have developed a logic which could explain 
history and nature as the necessary conceptual development of the idea on the 
way to self-knowledge. One of the most outspoken critics of this system was 
the Leipzig philosopher Christian Hermann Weisse, Fechner's closest friend. 
Weisse criticised Hegelian ``panlogism" as not giving enough justice to 
the contingent and individual in nature and history. He argued that concrete
reality is not the product of a logically necessary development of ideas as 
Hegel wanted it; there is something in it that transcends all necessity.
 
In two addresses of 1849, Fechner tried to show that taking Weisse's idea of 
indetermination seriously does not preclude the use of mathematics in science. 
He argued that mathematical descriptions only provide a general frame for 
natural phenomena not implying any necessity for the individual case, thus 
being compatible with an indeterminate behaviour on a finer level. He also 
claimed that in order to admit indeterminate events in nature one does not 
have to give up the causal law. This law only says that the same effect will 
recur if the same set of conditions obtains, but it does not preclude the 
emergence of new conditions in the course of time. This discussion might very 
well be the first expression of an indeterministic world-view.
 
The first major application of statistical methods by Fechner, and the first 
work, where his ideas on psychophysical parallelism and individual 
indeterminacy come together, are his {\it Elements of Psychophysics}
of 1860. There, Fechner defined psychophysics as the ``exact science of the 
functional or dependency relations between body and mind, or more general: 
between the bodily and the spiritual, physical and psychical, world." 
(Fechner 1860, I, 8) His goal was to measure sensations experimentally and 
thus to arrive at a quantitative science of psychophysics. The major result 
of his research was the so-called Weber-Fechner law which says that the 
intensity of sensation $E$ increases in proportion to the logarithm of a 
stimulus $R$, or: $E = k log R$.  

Fechner conceived of psychophysics as a fundimentally statistical enterprise. 
The rationale behind his reasoning seems to have been the following: If human 
beings are free in their actions and if mind and body are correlated in the 
way as conceived by psychophysical parallelism, then there will be an 
individual variation in the response of a subject to a physical stimulus. This 
response will be physical and manifest itself in a certain bodily reaction, 
but it will also be mental and express itself in a certain judgement. The 
fluctuations are not to be taken as erroneous deviations from the true value 
but as the free mode of reaction of the individual to a stimulus. 

Among the three methods Fechner developed for measuring sensation is the 
``method of right and wrong cases". A subject had to lift a pair of weights, 
$P$ and $P + D$, and to judge which seemed to be the heavier of the two. 
After $n$ trials the ratio $r / n$ of right answers to all trials was 
calculated. Fechner took the ``measure of precision" $h$ that appears in older 
formulations of the Gaussian law as an expression of the differential 
sensitivity of the subject, such that $2r / n - 1 = {\theta}(h D / 2)$, 
where $\theta$ is the Gaussian law. 

In 1878, Fechner published a paper where he developed the notion of the median. 
He later delved into experimental 
aesthetics and thought to determine the shapes and dimensions of aesthetically 
pleasing objects. He mainly used the sizes of paintings as his data base. In 
his {\it Vorschule der Aesthetik} of 1876 he used the method of extreme ranks 
for subjective judgements.

Fechner's most important contribution to statistics is his posthumously 
published book on the measurement of collectives, his {\it Kollektivmasslehre}
of 1897. Fechner defined a `collective' or `collective object' as a 
collection of an indefinite number of individual objects, subject to random 
variation, and embraced under a single specific or generic concept. The main 
examples he treated are to be found in anthropology, zoology, botany, 
meteorology, aesthetics. The object of the enquiry is, as Fechner wrote, 
\begin{quote}
``the 
establishment, by mathematical proof and empirical verification, of a 
generalisation of Gauss's law of accidental variations, whereby the law is 
enabled to transcend the limits of symmetrical probability and comparative 
smallness of the positive and negative deviations from the arithmetical mean, 
and new relations of uniformity are brought to light." (Fechner 1897, vi)
\end{quote}
He developed a set of constants which allowed to characterise different 
distributions and developed a two-sided asymmetric Gaussian law where the two 
branches are treated as if originating from two different distributions.

Fechner took the random variation of the collectives quite literally. He spoke 
of the ``ideal laws of chance" which are realised in true collectives. Chance 
was for him an objective category and not just the expression of ignorance. He 
tried to design tests whereby variation due to factors other than chance could 
be detected by comparing the data under consideration to a random sequence.

Fechner's {\it Kollektivmasslehre} draws on several traditions. There is 
the moral statistics of the Belgian statistician and astronomer Adolphe 
Quetelet (q.v.) who was one of the first to investigate mass phenomena 
and to find numerical regularities in them. (Porter 1986) There is also the 
error theory of the mathematical astronomers Gauss (q.v.), Encke, Bessel and 
Hauber. 
And there is the tradition of the statistical bureaux of state administration. 
Much of Fechner's concept of a collective derives from the 
W\"{u}rttemberg state 
official Gustav R\"umelin (1815-1889) who had distinguished between particulars 
that are typical of their genus and individuals that do not allow a 
straightforward inference as to the nature of the genus as a whole. The latter 
ones form a collective object and are the subject of statistics as the science 
of mass phenomena.

Fechner's {\it Kollektivmasslehre} was of immediate influence on many of his 
colleagues in Leipzig. The psychologists Gottlob Friedrich Lipps, Wilhelm 
Wirth and to some extent also Wilhelm Wundt used the new methods in 
psychophysics. Charles Edward Spearman who obtained his Ph.D. under Wundt 
extended Fechner's ideas and studied the correlation between magnitudes. The 
Leipzig astronomer and mathematician Heinrich Bruns (1848-1919) soon gave a 
general solution to Fechner's problem of a mathematical representation of 
frequency distributions, the so-called Bruns-series (today called 
Gram-Charlier series) and tried to unify Fechner's theory of collectives with 
probability theory. His most illustrious student was Felix 
Hausdorff who carried this tradition further. (Girlich 1996) Bruns and 
Hausdorff, however, dropped Fechner's requirement of chance variation of the 
collective object, thus obscuring any trace of Fechner's indeterminism.

In Richard von Mises' (q.v.) theory of probability of 1919, however, 
this condition becomes one of two central requirements. As an axiomatic basis 
of his theory, von Mises postulated 1. the existence of the limiting values of 
the relative frequencies and 2. the randomness of the way how the attributes 
are mapped unto the elements of the collectives. Randomness is thereby defined 
as the invariance of the frequencies under any place selection - 
a criterion which clearly shows traces of Fechner's above mentioned test of 
homogeneity. Von Mises' theory gives a precise formulation of Fechner's 
basic intuitions. It marks the final defeat of the subjective Laplacean 
interpretation of probability, consolidates the frequency interpretation and 
conceives of probability as an empirical science of chance phenomena. One can 
only speculate what the course of statistics would have been if Fechner's 
work had been published before  K. Pearson (q.v.) developed his biometrics in the 
early 1890s. 
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\begin{thebibliography}{3}
\bibitem{1}  Fechner, Gustav Theodor 1860. {\it Elemente der Psychophysik}, 2 
vols.,  
Breitkopf und H\"{a}rtel, Leipzig; 2nd. ed. ibid. 1889. (English transl. of vol 1 by 
Helmut 
E. Adler: {\it Elements of Psychophysics}, ed. by D. H. Howes \&amp; E. G. 
Boring,  
Holt, Rinehart \&amp; Winston, New York, 1966).

\bibitem{2}  Fechner, Gustav Theodor 1878. Ueber den Ausgangswerth der kleinsten 
Abweichungssumme. In: {\it Abhandlungen der K\"{o}niglich S\"{a}chsischen 
Gesellschaft 
der Wissenschaften}, math.-phys. Klasse 11, 1-76 (Sometimes cited with the 
year 1874).

\bibitem{3}  Fechner, Gustav Theodor 1897. {\it Kollektivmasslehre}, ed. by 
Gottlob Friedrich Lipps, Engelmann, Leipzig.

\bibitem{4}  Girlich, Hans-Joachim 1996. Hausdorffs Beitr\"{a}ge zur 
Wahrscheinlichkeitstheorie. In: E. Brieskorn (ed.), {\it Felix Hausdorff zum 
Ged\"{a}chtnis}. Vol. 1: {\it Aspekte seines Werkes},  Vieweg, Braunshweig, 
31-69.

\bibitem{5}  Heidelberger, Michael 1987. Fechner's Indeterminism: From Freedom to 
Laws of Chance. In: {\it The Probabilistic Revolution}, vol 1: 
{\it Ideas in History}, 
ed. by Lorenz Kr\"uger, Lorraine J. Daston and Michael Heidelberger,  
MIT Press/ Bradford Books, Cambridge, Mass., 117-156.
 
 Heidelberger, Michael 1993. {\it Die innere Seite der Natur: Gustav 
Theodor Fechners wissenschaftlich-philosophische Weltauffassung} 
(Philosophische 
Abhandlungen, Band 60), Klostermann, Frankfurt am Main. (Comprehensive study 
of Fechner's life and work; Ch. 5 on Fechner's psychophysics, Ch. 7 on 
Fechner's indeterminism, his theory of collectives and its reception)

\bibitem{6}  Mises, Richard von 1928. {\it Wahrscheinlichkeit, Statistik und 
Wahrheit}, 
Springer, Wein (2nd revised English ed. prepared by Hilda Geiringer: 
{\it Probability, Statistics and Truth}, Allen \&amp; Unwin, London
1957. 1st English 
edition 1939.)

\bibitem{7}  Plato, Jan von 1994. {\it Creating Modern Probability: its 
Mathematics, Physics and 
Philosophy in Historical Perspective}, Cambridge University Press, Cambridge. 
(Ch. 6 on von Mises)

\bibitem{8}  Stigler, Stephen M. 1986. {\it The History of Statistics: The 
Measurement of 
Uncertainty before 1900}, Harvard University Press, Cambridge, MA. (pp. 
242-254 on Fechner's use of statistics in psychophysics)

\bibitem{9}  Witting, Hermann 1990. Mathematische Statistik. In: Gerd Fischer 
et al. (eds.), {\it Ein Jahrhundert Mathematik 1890-1990: Festschrift zum 
Jubil\"{a}um 
der DMV}, Vieweg, Braunschweig, 781-815. (pp. 786-787 on Fechner's 
{\it Kollektivmasslehre}, p. 788 on Lipps and Spearman)
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\hfill{Michael Heidelberger}


 
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