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Papers and Books on History of Mathematics

Dirichlet's Contributions to Mathematical Probability Theory, Historia Mathematica 21 (1994), 39--63.

Summary: The article discusses central topics in Dirichlet's printed papers and unpublished lecture notes on probability, error theory, and, more generally, on analysis:

- Dirichlet's deduction of the asymptotic normal distributions of medians, connected with a criticism of least squares
- improvement of Laplace's method of approximation, as applied to Stirling's formula, and to
- the Central Limit Theorem
- Dirichlet's analytic style between fintitistic reasoning and calculations with infinitesimals.


 

 

 

Die verschiedenen Formen und Funktionen des zentralen Grenzwertsatzes in der Entwicklung von der klassischen zur modernen Wahrscheinlichkeitsrechnung
(The different forms and applications of the central limit theorem in its development from classical to modern probability theory.) 
Aachen: Shaker, 2000.

 

Summary:
This study discusses the historical development of the probabilistic central limit theorem from about 1810 through 1940. The central limit theorem was originally deduced by Laplace as a statement about approximations for sums of independent random variables in the framework of classical probability which focused upon specific problems and applications. Making this theorem an autonomous mathematical object was very important for the development of  modern probability theory. The mathematical perspectives and the analytic methods connected with the central limit theorem, and its applications to mathematical statistics are discussed in this book thoroughly from a historical point of view.
Single topics:
Laplace's way towards the central limit theorem; Laplace's method of approximations; applications of the central limit theorem by Laplace and by his successors; Poisson's modifications; reconstruction of Dirichlet's proof; the central limit theorem in Cauchy's and Bienaymé's controversy over the method of least squares; the development of the hypothesis of elementary errors from Hagen through Edgeworth, and its meaning for the ``statistics of variations''; roots of the theory of moments; theory of moments and central limit theorem in Chebyshev's and Markov's works; first steps into modern probability by Lyapunov; contributions by Pólya, v. Mises, Lindeberg, Lévy, Bernshtein, and Cramér in the tension field between formalism and practical applications; generalizations towards dependent random variables and nonnormal limit laws; Lévy's and Feller's necessary conditions, reconstruction of Lévy's original proof; general limit problems.


Jakob Friedrich Fries und die Grenzen der Wahrscheinlichkeitsrechnung

in: Form, Zahl, Ordnung, Seising, R. & Folkers, M. & Hashagen, U., Hsg, Franz Steiner Verlag Stuttgart, 2004, S. 277-299.

In this article, the influence of Kantianism as represented by the philosopher Fries (1773-1843) on the decline of the classical notion of probability is discussed. Based on a strictly objectivistic interpretation of probabilities, Fries set very narrow limits to the application of probability theory, excluding the entire field of "moral" issues. It is shown, that  this limitation was based on Kant's restriction of philosophy to a "secure range of science"  which can be confirmed by experience.


Laplace's Approximation of the Gamma Function, a Direct Approach, Preprint KU 2006-01

In this note Laplace's method for the deduction of an asymptotic expansion for $Gamma(s+1)$ ($s$ a complex number with a positive real part) is described from the historical point of view. It is shown that this method can be used in its original fashion for a direct and rigorous treatment of the expansion.


Die Geschichte des Integrals  , eine Geschichte der Analysis in der Nußschale, Mathematische Semesterberichte 54 (2007), 13-30.

This is a history of the famous integral from Euler to Hardy reflecting the history of real and complex analysis from the beginning of 19th up to the beginning of 20th century.


 

 

 

This is an entirely revised English version of the 2000 book above.

See Springer 2011 for more details 

 

 

 

 

 


For more books and papers since 2011 see the survey of Hans Fischer in KU.edoc