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**“…das Wesen der reinen Mathematik verherrlichen”.**
*(German)*
Zbl 0836.01009

Against the trends of his time, C. G. J. Jacobi (1804-1851) advocated an autonomous pure mathematics being independent from experiences and applications. Starting from this position he tried to explain progress in mathematics and its applicability for describing real world phenomena.

The authors approach Jacobi’s notion of mathematics and its tension with mathematical natural philosophy in three sections. In the first section Jacobi is presented as a typical representative of the neo-humanistic ideal of scientific knowledge in early 19th century Germany as it was already claimed by Alexander von Humboldt in 1797 who said that mathematics as a pure object of speculation not being applicable to the solution of practical problems will not loose anything from its dignity: everything is important which extends the limits of our knowledge. The differences between Jacobi’s idealistic pure mathematics and French mathematics is illustrated by a comparison with Fourier’s conception of mathematics as a pre-existing element of the universe. The authors maintain that both conceptions were opposed diametrically in theory, but were not consequently realized in practice.

The authors explain the emergence and development of pure mathematics in Germany as a result of the new interest in foundations of mathematics which was a reaction to the 18th century emphasis of applied mathematics. They stress the influence of Kant’s distinction between pure and empirical intuition which was used by G. S. Klügel for the distinction between pure and applied mathematics in his ‘Mathematisches Wörterbuch’ (1808).

The second section gives for the first time a German translation of Jacobi’s inaugural lecture for entering the philosophical faculty of the University of Königsberg in Prussia on July 7, 1832. The lecture was published in its original Latin by W. von Dyck [Sitzungsber., Bayer. Akad. Wiss., Math.-Naturwiss. Kl. 31, 203-208 (1901); reprinted in Math. Ann. 56, 252-256 (1903)]. In his lecture Jacobi pleads for an inner progress in mathematics being independent from mathematical solutions of real world problems. The mathematical ideas inherent in nature could not have been recognized if the human mind had not already erected mathematics according to these ideas. The necessary development of mathematics is, according to Jacobi, the true cause of mathematical progress. The translation is commented comprehensively with some excellent recursions, e.g., into the history of the term “analysis” (pp. 122sq.).

In the third section Jacobi’s applicability problem is treated. Although Jacobi never abandoned his high esteem for speculative mathematics he worked since the 1830s on questions of mathematical physics which had their origin in 18th century analytical mechanics and the calculus of variations, i.e. directions with connections to empirical sciences. The conflict is illustrated by Jacobi’s hitherto unpublished lectures on analytical mechanics held at Berlin 1847/48, in which he explicitly warns against regarding analytical mechanics as a discipline of pure mathematics. The mathematical symbolism is applied to something apart from it, a convention is needed, i.e. a stipulation which cannot be made by mathematics itself. Jacobi seems to be the first (half a century before Poincaré) who had characterized mechanical principles as conventions.

The authors have presented an excellent article full of stimulations for the history and philosophy of mathematics.

The authors approach Jacobi’s notion of mathematics and its tension with mathematical natural philosophy in three sections. In the first section Jacobi is presented as a typical representative of the neo-humanistic ideal of scientific knowledge in early 19th century Germany as it was already claimed by Alexander von Humboldt in 1797 who said that mathematics as a pure object of speculation not being applicable to the solution of practical problems will not loose anything from its dignity: everything is important which extends the limits of our knowledge. The differences between Jacobi’s idealistic pure mathematics and French mathematics is illustrated by a comparison with Fourier’s conception of mathematics as a pre-existing element of the universe. The authors maintain that both conceptions were opposed diametrically in theory, but were not consequently realized in practice.

The authors explain the emergence and development of pure mathematics in Germany as a result of the new interest in foundations of mathematics which was a reaction to the 18th century emphasis of applied mathematics. They stress the influence of Kant’s distinction between pure and empirical intuition which was used by G. S. Klügel for the distinction between pure and applied mathematics in his ‘Mathematisches Wörterbuch’ (1808).

The second section gives for the first time a German translation of Jacobi’s inaugural lecture for entering the philosophical faculty of the University of Königsberg in Prussia on July 7, 1832. The lecture was published in its original Latin by W. von Dyck [Sitzungsber., Bayer. Akad. Wiss., Math.-Naturwiss. Kl. 31, 203-208 (1901); reprinted in Math. Ann. 56, 252-256 (1903)]. In his lecture Jacobi pleads for an inner progress in mathematics being independent from mathematical solutions of real world problems. The mathematical ideas inherent in nature could not have been recognized if the human mind had not already erected mathematics according to these ideas. The necessary development of mathematics is, according to Jacobi, the true cause of mathematical progress. The translation is commented comprehensively with some excellent recursions, e.g., into the history of the term “analysis” (pp. 122sq.).

In the third section Jacobi’s applicability problem is treated. Although Jacobi never abandoned his high esteem for speculative mathematics he worked since the 1830s on questions of mathematical physics which had their origin in 18th century analytical mechanics and the calculus of variations, i.e. directions with connections to empirical sciences. The conflict is illustrated by Jacobi’s hitherto unpublished lectures on analytical mechanics held at Berlin 1847/48, in which he explicitly warns against regarding analytical mechanics as a discipline of pure mathematics. The mathematical symbolism is applied to something apart from it, a convention is needed, i.e. a stipulation which cannot be made by mathematics itself. Jacobi seems to be the first (half a century before Poincaré) who had characterized mechanical principles as conventions.

The authors have presented an excellent article full of stimulations for the history and philosophy of mathematics.

Reviewer: V.Peckhaus (Erlangen)

### MSC:

01A55 | History of mathematics in the 19th century |