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Born: 1777, Brunswick, Germany

Died: 1855, Gottingen, Germany

Major Works: Disquisitiones Arithmeticae (Arithmetical Investigations) (1801), Theoria Motus Corporum Coelestium in Sectionibus Conicis Solem Ambientium (Theory of the Motion of the Heavenly Bodies Moving about the Sun in Conic Sections) (1809), Disquisitiones Generales Circa Superficies Curvas (General Investigations of Curved Surfaces) (1827)

Major Ideas

Mathematics requires a new rigor in which Greek standards of precision are applied to the subject matter of contemporary mathematics.

The goal of science is the pursuit of truth "for its own sake."

Due to its intrinsic independence from the material and the practical, "mathematics is the queen of the sciences."

Arithmetic, on the other hand, is the "queen of mathematics," since it is the most disinterested and pure.

The logic that holds together mathematics also pervades the universe; we understand the universe by discovering its underlying mathematical theories.

Carl Friedrich Gauss is regarded as the greatest mathematician since Newton. Many consider Gauss to be the greatest mathematician of all time. Like Newton, Gauss was a scientist in a time when there was no clear distinction between the activity of a mathematician and that of a physicist. But, whereas Newton's main interest in mathematics lay in its application to science, Gauss found mathematics to be of intrinsic value. Gauss worked with equal success in pure and applied mathematics, but placed a higher value on the former. The development of a logical and complete theory was the goal of both his mathematical and his scientific work. His experimental activity in science inevitably led to the development of theoretical work.

To Gauss's creativity and depth of insight was added a new sense of rigor that enabled him to reorganize or develop major areas of mathematical and scientific research and that changed the way that mathematics was to be done by his successors.

Mathematics and science were placed in a new relationship in Gauss's scheme. According to eighteenth-century views, mathematics served essentially as a tool for scientific use, and science tended to be valued for its practical advantages. In Gauss's new vision, theoretical considerations held an inherent primacy, resulting in the supreme position of mathematics among the sciences.

Gauss entered elementary school in 1784. His teachers observed and nourished his unusual ability in arithmetic and helped to arrange for his admission to secondary school in 1788, where his studies included Latin and High German.

Gauss entered the Collegium Carolinum in 1792, where he was a student of Latin and Greek and benefited from its excellent library. There he acquired a broad foundation of mathematical knowledge. His early interest in arithmetic led to his investigation of the distribution of prime numbers.

In 1795, Gauss entered Gottingen University, chosen for its extensive mathematical library. There he completed most of his first major work, Disquisitiones Arithmeticae, including its two proofs of the law of quadratic reciprocity (his "golden theorem") in 1796. In that same year he used number theory to prove the constructibility of the regular polygon of seventeen sides, and thereby solved a geometric problem dating back to antiquity

Gauss returned to Brunswick in 1798. He completed his dissertation, in which he proved the fundamental theorem of algebra, and received the doctoral degree from the University of Helmstedt in 1799. Disquisitiones Arithmeticae was published in 1801.

The discovery of the planet Ceres by an Italian astronomer in 1801 led to a shift in Gauss's major interest from pure mathematics to astronomy. Gauss computed the orbit of Ceres and successfully predicted its rediscovery a year later, resulting in his becoming a celebrity in the field of astronomy.

In 1807, Gauss moved to Gottingen to become director of its astronomical observatory.

His second major work, Theoria Motus, was published in 1809. Theoria Motus is a treatise in theoretical astronomy on the determination of the orbits Planets comets. In of and the period from 1807 to 1818 Gauss continued his astronomical work, produced a paper on Gauss sums, second and third proof of the fundamental theorem of algebra, and a paper on the hypergeometric series. The remaining four of his six different proofs of the law of quadratic reciprocity are believed to have been found by him by 1808. In a review written in 1816, Gauss indicated his success in grasping non-Euclidean geometry and he is credited with being the first to do so.

From 1818 to 1832, Gauss participated in a project undertaken to survey the kingdom of Hanover. This work led to two major theoretical works in geodesy and also to an important paper in pure mathematics, Disquisitiones Generales Circa Superficies Curvas, in which he introduced "intrinsic" geometry--concerned with the local properties of a surface such as its curvature--and thus set the foundation for the development of modern differential geometry.

In 1831, Gauss's major interest shifted to physics. In an 1832 paper he defined an absolute measure of magnetic force, and in 1838 he published a general theory of terrestrial magnetism from which he successfully predicted the location of the magnetic South Pole. He made contributions to potential theory and to the theory of electro-magnetism.

Gauss published a paper on biquadratic residues in 1831 in which he introduced the Gaussian integers and thus began the field of algebraic number theory.

During the final years of his life, from 1838 to 1855, Gauss worked in his astronomical and magnetic observatories in Gottingen, and he continued to pursue his interests in both mathematics and physics. He learned Russian in order to read the work of the Russian geometer Lobachevsky. A fourth, final, and improved version of his doctoral dissertation was presented in 1849 at the time of his golden jubilee.

Gauss made major contributions to every field of mathematics. He reorganized number theory and established its future direction. He founded the subject of modern differential geometry and thus provided the mathematical basis for the later development of the theory of general relativity by Einstein. He set new directions for research in astronomy and geodesy.

Gauss died on February 23, 1855. In the words of mathematician Eric T. Bell, "He lives everywhere in mathematics."

The New "Inner" Rigor

Gauss brought to mathematics a new rigor that was to redefine the standards for the work of his successors. The work of Gauss is characterized not only by the completeness of its overall form but also by a completeness of the details of its inner structure, the latter of which defines him as the first of the modern rigorists.

Gauss had studied the works of Archimedes and Newton, well-known masters of the completed presentation. He held them in high esteem, referring to Newton as "summus." Following their example, Gauss sought to produce work complete in form--logical, unified, and concise. Disquisitiones Arithmeticae, Theoria Motus, and Disquisitiones Generales stand as superb examples of Gauss's ability to produce complete and unified theories.

But Gauss's unique contribution to the standards of modern rigor lay not in his ability to achieve classical completeness of external form hut rather in his ability to further refine the inner workings of his theories. Indeed, he saw the latter as his major occupation. As he simply stated, "One must pursue the tree to all its root fibers. ..." He referred to this as "rigor antiquus," reflecting the fact that the Archimedean standard of precision had been abandoned in the eighteenth century.

Gauss rendered the rigor of the Greeks applicable to the mathematics of his day.

First of all, he insisted that mathematical results, then generally accepted on the basis of intuition or induction, must submit to logical demonstration in order to establish their mathematical validity. In fulfillment of this standard, he provided the first correct proofs of such landmark results as the fundamental theorem of algebra (a theorem in complex analysis that states essentially that every polynomial equation has a complex root), and the law of quadratic reciprocity, which is concerned with the solvability of certain pairs of quadratic congruences and is crucial to the development of number theory.

Second, he tightened the way in which logic is used within a mathematical proof. Rigor is always a matter of degree, and Gauss set a new standard for what may be regarded as evident. His criticism of the work of other mathematicians in his doctoral dissertation and in Disquisitiones Arithmeticae shows his dissatisfaction with the way in which much of recent mathematics was done. In his analysis of Legendre's incorrect proof of the law of quadratic reciprocity, he asserts the unacceptability of using that which is merely plausible or even probable, and he warns of the danger of circular logic.

A third element of Gauss's inner rigor is the setting of new standards of precision in the use of mathematical techniques.

Gauss noted the importance of the use of computational devices in the mathematics of his time, as against that of antiquity. Such devices serve to shorten, simplify, consolidate, and sometimes even to make possible methods of calculation and proof.

But the benefits to mathematics brought by the increased use of such methods were accompanied by serious dangers to logic and beauty. In reaction to the work of his contemporaries and recent predecessors, Gauss warned against the merely mechanical use of techniques that leads to their application in contexts where they have no validity.

Gauss's own work on topics related to the convergence of infinite series provides an illustration of this aspect of his rigor. He saw that methods for computing sums of infinite series must be restricted to those series that actually do converge, so that a precise definition of convergence and convergence tests are required for the development of a sound theory. His paper on the hypergeometric series published in 1813 was the first systematic approach to infinite series. His work set a new standard for the treatment of infinite processes and established for all of mathematics the importance of knowing the exact conditions under which a given method is applicable.

Mathematics as the Queen of the Sciences

For Gauss, "mathematics is the queen of the sciences, and arithmetic is the queen of mathematics."

The eighteenth-century belief in the mathematical nature of the universe was raised to a new level in the work of Gauss. Mathematics was indeed a tool for scientific use. But for Gauss, the relationship of mathematics to science was not that of mere servant. In his view, mathematics governed the behavior of the universe; to understand the universe, therefore, we must discover and develop its underlying mathematical theories.

Mathematics, moreover, provided a model for the way in which scientific theories were to be developed. Gauss's first major work, Disquisitiones Arithmeticae, provided such a model for his later work in both mathematics and science. His classic treatise in astronomy, Theoria Motus, like its mathematical predecessor, is described as rigorous, concise, and complete. Theoria Motus contains Gauss's first publication of the method of least squares, a mathematical method whose essential role in his analysis of data in both astronomy and geodesy gave further evidence of the fundamental congruity of mathematical thought and the behavior of the universe.

Gauss's elevation of mathematics from servant to queen of the sciences arose also from his belief, expressed in the inaugural Lecture on Astronomy (undated), that the primary goal of science is the pursuit of truth "for its own sake." In this work, he praises Archimedes for ranking pure mathematics as first among the sciences, a position that it warrants by virtue of its intrinsic independence from the material and the practical.

As a consequence.: of Gauss's vision, the very nature of mathematics was redefined for his successors. Mathematics regained its identity as a distinct science and the near identification of mathematics and physics that had prevailed since the time of Newton ended. Disquisitiones Arithmeticae stands as a monument to Gauss's new vision.

Gauss not only redefined mathematics, he also redefined arithmetic as a subject within it. In his preface to Disquisitiones Arithmeticae, Gauss distinguishes higher arithmetic, which is concerned with general properties of numbers, from elementary arithmetic, which is concerned with counting and calculation. Higher arithmetic, now called number theory, forms. the subject matter of Disquisitiones Arithmeticae, in which he refers to it as "this divine science." Gauss values higher arithmetic as the most disinterested and pure of all of mathematics. He values its supreme elegance of theory and notes the special joy and passion that accompany its study. Qualities such as rigor and conciseness which are perhaps only useful to the other sciences are absolutely necessary for arithmetic.

Gauss's own work was a major contribution to the new position of arithmetic within the mathematical sciences. For, in Disquisitiones Arithmeticae, he placed the rather isolated results of his predecessors on a sound basis within a newly organized theory, and by adding his own work defined the subject of modern number theory.

Further Reading

Buhler, Walter K. Gauss: A Biographical Study. Berlin, Heidelberg, and New York: Springer-Verlag, 1981. Designed for the contemporary mathematician and scientist. Provides a summary of the contents of Gauss's work and includes many passages from his writings.

Dunnington, G. Waldo Carl Friedrich Gauss: Titan of Science New York: Exposition Press, 1955. A comprehensive and authoritative biography that describes Gauss's life against its historical setting Contains a chronological listing of Gauss's 155 published works and a bibliography of; secondary literature.

Gauss, Carl F Disquisitiones Arithmeticae. Translated by Arthur A. Clarke. New Haven: Yale University Press, 1966.

____. General investigations of Curved Surfaces. Translated by Adam Hiltebeitel and James Morehead. New York: Raven Press, 1965.

____. Theory of the Motion of Heavenly Bodies. New York: Dover, 1963.

____. Inaugural Lecture on Astronomy and Papers on the Foundations of Mathematics. Translated by G. Waldo Dunnington. Baton Rouge: Louisiana State University, 1937.

Hall, Tord. Carl Friedrich Gauss: A Biography. Cambridge and London: MIT Press, 1970. A readable explanation of the problems Gauss sought to solve, together with his methods of solution. Designed for the general reader.
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Publication:Great Thinkers of the Western World
Article Type:Biography
Date:Jan 1, 1999

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