The German Culture's influence on Gauss

Gauss was born a few years before Frederick the Great died (while Germany was still known as Prussia). Frederick the Great was a fervent nationalist who led several aggressive expansionist campaigns against neighboring Austria, Poland, and Silesia. The German people revered Frederick the Great (almost idolizing him). In fact, in much of the propaganda that Hitler and the Nazi party distributed during his campaign for power compared him to the next Frederick the Great. Based on this, it is very likely that Gauss held him in high esteem as well. Throughout his life, he was strongly patriotic and loyal to the crown.

Later in life, as the Napoleonic Wars broke out across Europe, Gauss staunchly supported the German monarchy and condemned the revolution and condemned Napoleon as an insurgent. As a result of the Wars, Prussia became a superpower in Europe leading to an even greater upswing in national pride.

This nationalistic attitude likely affected how Gauss approached his career - he was a very independent worker, which may have been sparked as an outgrowth of the national pride. In his life, he was very reluctant to work with others - especially non-Germans. He also took very few students, and all of his most famous students (Bernhard Riemann, Peter Gustav Lejeune Dirichlet, Moritz Cantor, August Ferdinand Möbius) were all German. One of Gauss's only foreign correspondents was Sophie Germain, a French mathematician. Gauss wholeheartedly supported Germain and recommended that she receive an honorary degree for her work in number theory. But, due to prejudice at the time, this request was never honored and she never received her degree and much of her work went unrecognized in her lifetime.

Gauss's Applied Mathematics -- Physics and Statistics

One of Gauss's most important accomplishments is in the development of Gauss's Theorem (also known as the Divergence Theorem), which he used to make several important developments in the physical theory of electromagnetism. Qualitatively, Gauss's Theorem links the flux of a vector field through some closed surface to the divergence of the field inside the surface.Quantitatively, Gauss showed that the following was true:

 

Where V is any volume, F is any field, S is the surface enclosing V, and n is the vector normal to the surface. 

An intuitive way to think about this is that F describes something that is flowing (e.g., a gas or a fluid). The divergence of F would then describe the expansion or compression of the gas or fluid. Gauss's Theorem says that the total expansion (or compression) of the gas with some volume V is equal to how much "stuff" is entering (or leaving) the surface S.

Probably the most important application of Gauss's Theorem is in the physical theory of electromagnetism. Gauss was able to show that the electric flux is equal to the charge enclosed by the volume divided by a constant (ε₀). In other words,

Gauss did the same thing for the magnetic flux and found:

Physically, this shows that the electric field points radially away from positive charges (radially into negative charges) such that the total divergence is Q/ε₀. 

For magnetism, Gauss showed that the magnetic field has no divergence. The magnetic field can circulate, but cannot do anything else. In other words, all field lines that enter the Gaussian surface must also leave the surface. 

 

Gauss's Theorem provided the theoretical foundation for Gauss's laws of electricity and magnetism, which remain the backbone of electromagnetism to this day. 

Of equal importance was Gauss's development in probability theory of a way to handle random variables that each have unknown distributions. Today, this is known as a normal or Gaussian distribution. A fun way to visualize this is with a Galton board.

The Gaussian distribution is incredibly useful because of the central limit theorem, which says that the sum of independent random variables will always tend to a Gaussian distribution. This theorem makes the Gaussian distribution remarkably powerful and applicable, and it can be used to model useful data such as people's heights, blood pressure, IQ scores, and salaries. Because of their incredible applicability, Gaussian distributions are ubiquitous throughout all of branches of Applied Mathematics including physics, economics, and psychology.

Early Mathematical Career

Early Mathematical Career

Carl Gauss was attracted to mathematics from an early age. In 1798 (at the age of 21), Gauss finished what would come to be recognized as a masterpiece of number theory, Disquisitiones Arithmeticae. This work set the foundation of modern number theory and consolidated earlier work done by Fermat, Euler, Lagrange, and Legendre and others. This earlier work was supplemented with original, insightful content provided by Gauss. This was the first book on number theory to bring the entire field together into one cohesive framework, as opposed to a collection of isolated observations and theorems. Disquisitiones also set a organizational convention that would later be mimicked by later books (and one that is still largely followed today). First, a theorem was stated formally, then the theorem was proved, then corollaries of the theorem were explored.

 

Despite the considerable amount of time and effort that Gauss invested in Disquisitiones, he still considered choosing a profession other than mathematics while he was at the University of Gottingen. In particular, he was deeply interested in philology (a field of study similar to linguistics). Another important consequence of Disquisitiones was that it introduced new helpful notation, such as the ≡ symbol for congruence. Finally, Gauss showed in Disquisitiones how a regular heptadecagon (a polygon with 17 equal sides) can be constructed with a compass and a straightedge. This problem had been unsolved since ancient Greece and Gauss's proof was the first progress in the field on constructible polygons (shapes that can be drawn using only a compass and a straightedge) in over 2000 years. Gauss was so proud of this accomplishment that he asked for a heptadecagon to be carved on his tombstone. This last wish was not granted because the engraver said it would be indistinguishable from a circle.

 

In 1799, Gauss finished his dissertation by discovering a new proof of the fundamental theorem of algebra (every polynomial of degree n, has n roots, counting multiplicity). Prior proofs of the theorem (done by Lagrange, Euler, Laplace, and others) were mostly correct but were technically not acceptable by today's standards. Their proofs assumed that every polynomial has n roots and showed that the form of these roots was a + bi. Gauss improved upon their work by not assuming the existence of roots. Gauss's proof was mostly geometric, and -- ironically enough -- is also technically incorrect due to topological reasons.

Another important discovery of Gauss's was that every number can be written as a sum of at most three triangular numbers. The proof of this result is beautiful and makes use of the result of a different proof. The proof of the triangular number theorem starts with the fact that every positive integer congruent to 3 mod 8 can be expressed as the sum of at most three squares.

 

This is a neat result, particularly since it is not immediately obvious.

Collectively, these examples really hammer home the magnitude of Gauss's genius. He accomplished all of this by the time he was 22!

Background

Carl Friedrich Gauss was born in Germany in April 1777 and would go on to make significant contributions in mathematics (especially in the areas of algebra, number theory, and statistics) and physics (particularly in astronomy, optics, and electromagnetism). Gauss devoted his life to mathematics even incorporating the symbols for pi and integration into his signature.

Born to a poor family, Gauss learned to solve "real-world" problems using math from an early age. As an example, Gauss's mother was was unable to read or write and as a result could not record when he had been born -- she could only remember that he had been born on a Wednesday, 8 days before the Feast of Ascension. (The Feast of Ascension is a mostly-abandoned Protestant holiday that always takes place on a Thursday, 40 days after Easter). Gauss was able to figure out when he was born by devising a method to calculate the date for Easter in any given year. Not bad for a young boy from a poor, uneducated family!

A more famous example of Gauss’s problem-solving abilities took place when he was an 8-year-old student, and his teacher instructed him to sum up all of the integers between 1 and 100 as a punishment for misbehaving. To their astonishment, Gauss solved the problem within seconds [1]. His reasoning is sketched out below.

Later in life, he was able to make several important contributions, particularly in number theory. Gauss was the first to prove the Triangular Number Theorem (any positive integer can be written as the sum of at most three triangular numbers), was the first to conjecture the Prime Number Theorem, was the first to calculate the orbit of a dwarf planet named Ceres, and provided the foundation for the modern study of statistics, optics, and electromagnetism. Of these, probably the most important mathematical accomplishment was his conjecture of the Prime Number Theorem, which hypothesizes that the prime numbers are distributed roughly as N/log(N). In other words, the probability that a randomly chosen integer greater than N is prime is given approximately by 1/log(N) [2]. This approximation becomes more accurate as N approaches infinity.

Gauss's personal life was tumultuous. His father had a dominating personality leading to a distant relationship between the two. In 1805, he married a woman named Johanna Osthoff who tragically died just a few years later in 1809. They had two children together - one son and one daughter. The son died a year after Johanna. In 1810, Gauss remarried but was a markedly different man as the result of the grief from his first wife's and son's deaths. He began do treat his children aggressively in the same way that his own father had treated him, leading to tension within the family [1].

Gauss was a zealous perfectionist (his personal motto was pauca sed matura: "few, but ripe"), diligent worker, and devout Lutheran. These traits may have been influenced in part by the nationalistic attitudes in Germany at the time (Gauss rarely collaborated with other mathematicians, especially non-Germans) together with the nihilistic philosophies espoused by the German philosophers, Immanuel Kant and Friedrich Nietzsche, who may have influenced some of Gauss’s more unorthodox religious beliefs [3].

In 1855, Gauss passed away from a heart attack and was buried in Gottingen, Germany [1]. After his death, Gauss’s brain was studied by medical experts who found it to be slightly bigger than average. They also found his brain had more pronounced convolutions (the folds on the surface of the brain), which have been speculated by some as a physiological explanation for his genius [4].

References

[1] Bruno, Leonard C. (2003). Math and mathematicians: the history of math discoveries around the world U·X·L, 1999

[2] Hoffman, Paul (1998). The Man Who Loved Only Numbers. New York: Hyperion Books

[3] Bühler, Walter Kaufmann (1987). Gauss: a biographical study. Springer-Verlag

[4] Bardi, Jason (2008). The Fifth Postulate: How Unraveling A Two Thousand Year Old Mystery Unraveled the Universe. John Wiley & Sons