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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 \& E. G.
Boring,
Holt, Rinehart \& 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 \& 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}
\end{thebibliography}
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