Critical Essay by
Ahmed Sebbar
Professor
D.C. Struppa Chair in Mathematics, Chapman University
Schmid College of Science and Technology
View Bio
Carl Friedrich Gauss (1777-1855)
The early life. Carl Friedrich Gauss, born on April 30th, 1777, in Braunschweig, Germany, emerged from a very humble family. His exceptional brilliance became evident at a young age. One day, Gauss’s teacher assigned the class an exercise: to add the numbers from 1 to 100. As students completed their calculations on tablets, they placed them in the center of a table. Before the teacher could finish articulating the question, ten-year-old Gauss confidently placed his answer -5050- on the table, without showing any other work, and declared, "Ligget se!"(There it is).
Gauss went on to earn his Doctor of Philosophy degree in 1799. He is often hailed as the "Prince of Mathematicians" and considered as the "greatest mathematician since antiquity" due to his significant contributions to mathematics and physics. Alongside Leonard Euler and Henri Poincaré, Gauss stands as one of the rare universal minds in science. The legacy of Gauss extends to numerous concepts named after him, including Gauss’s law, Gaussian distribution, Gaussian integrals, Gaussian elimination, Gaussian integers, and the Gaussian plane. One of the most widely used theorems in mathematics is Gauss’s theorem, also known as the divergence theorem. It establishes a relationship between the volume integral of the divergence of a vector-valued function over a region and the flux or surface integral over the region’s surface.
In a letter dated September 2, 1808, the Hungarian mathematician János Bolyai, a pioneer in non-Euclidean geometry, received these words from C. F. Gauss: "It is not knowledge, but the act of learning; not possession, but the act of getting there, which grants the greatest enjoyment." This insight sheds light on the mindset of these intellectual giants and their approach to knowledge dissemination. It comes as no surprise that some of Gauss’s students, including Riemann, Dedekind, and Eisenstein, went on to become highly influential mathematicians. Gauss was always open to working privately with any interested student who sought his guidance.
Gauss’s Eureka Theorem.
EY PHKA : num = Δ + Δ + Δ.
In 1796, Carl Friedrich Gauss began keeping a mathematical diary known as the Notizenjournal. Most of the entries are succinct and occasionally cryptic statements of results in Latin. One notable entry, dated July 10, 1796, echoes Archimedes’ triumphant cry of discovery. In it, Gauss provides a proof for a conjecture by Fermat: that any positive integer can be expressed as the sum of three or fewer triangular numbers (i.e., numbers of the form n (n + 1)/2, for an integer n). Interestingly, Gauss’s theorem implies Lagrange’s theorem (1772), which asserts that every natural number can be represented as the sum of four or fewer square numbers. Legendre further demonstrated in 1798 that the set of positive integers not expressible as the sum of three or fewer squares corresponds to the integers of the form n = 4s(8m+7), where m and s are integers. Gauss going way beyond Legendre, actually obtained a formula for the number of primitive representations of an integer as a sum of three squares. According to many historians of mathematics, no simple proof of this theorem has been found up to date.
Gauss’s diary was rediscovered in 1897 and published by Klein(1903) [1], [3].The journal is a small book of nineteen pages consisting of a total of 146 discoveries.
Heritage and "Germinality". C.F. Gauss’s ideas resemble acorns: small seeds that eventually grow into mighty oaks, gaining strength and beauty over time. As the writer of these lines, I hold dear three mathematical gems from Gauss’s legacy, each leaving an indelible mark on my academic journey
The Gem of Arithmetic, "Theorema Aureum".
Tsze-kung asked, saying, "Is there one word which may serve as a rule of practice for all one’s life?"
The Master said: Is not RECIPROCITY such a word?
[4, Confucius, XXIII, p.226]
Carl Friedrich Gauss bestowed upon the law of Quadratic Reciprocity two illustrious titles: "Theorema Aureum"(the golden theorem) and the "Gem of Higher Arithmetic." His mathematical diary reveals that he discovered the complete proof for Quadratic Reciprocity on April 18th, 1796. However, it wasn’t until 1801 that his seminal work, Disquisitiones Arithmeticae [2], was published. Remarkably, Gauss went on to find three additional proofs for this fundamental law before its formal publication. Now there are more than 50 proofs of the Quadratic Reciprocity formula [5]. Its history intertwines with the development of algebraic number theory. Notable extensions include the Quartic Reciprocity formula, the Eisenstein, Kummer, Hilbert, Hasse, Artin, and Tate Reciprocity Laws. As our pursuit of absolute knowledge continues, new reciprocity formulas will undoubtedly emerge, perpetuating this fascinating journey through mathematical discovery.
The Gauss-Bonnet theorem. This fundamental theorem can be interpreted as a connection between the Gaussian curvature of a compact Riemann surface X and its Euler characteristic. This theorem is a true result of reciprocity. It is the seed from which profound theorems of modern times emerge. Notable among these are the Riemann-Roch theorem, the Atiyah-Singer theorem or the Selberg trace formula.
The hypergeometric function. A beautiful function that transcends multiple domains: analysis, topology, and differential equations. It stands as a beacon of reciprocity, weaving intricate connections, culminating in the so-called Riemann-Hilbert problem [6].
References
[1] F. Klein, Gauß’ wissenschafftliches Tagebuch 1796-1814, mit Anmerkungen herausgegeben von Felix Klein. Math. Annalen 57 (1903), pp. 1-34. Entry 146, July 9, 1814 is on page 33. http://link.springer.com/journal/208/57/1/page/1.
[2] Gauss, C. F., Disquisitiones Arithmeticae, Fleischer, Leipzig, 1801; reprinted in vol.I of Gauss’sWerke: Disq. Arith.
[3] Gray, J. J. (1984), A commentary on Gauss’s mathematical diary, 1796-1814, with an English translation, Expositiones Mathematicae, 2 (2): 97-130.
[4] Legge, J. The life and teachings of Confucius, with explanatory notes. Sixth edition, Trübner, 1887.
[5] Lemmermeyer, F., Reciprocity Laws: From Euler to Eisenstein, Springer, NewYork (2000).
[6] Yoshida, M., Hypergeometric Functions, My Love: Modular Interpretations of Configuration Spaces (Aspects of Mathematics) 1997th Edition.