# Niels Henrik Abel

## Advocate of Rigorous Mathematical Proofs

Abel was born on August 5, 1802, the son of a poor Lutheran minister in Finnoy, Norway. Poverty and famine were common in Norway and times were difficult for the clergyman and his seven children. At age 13 Abel entered the cathedral school in Christiana (now called Oslo). With his math instructor Bernt Michael Holomboe, Abel studied the works of ISAAC NEWTON, LEONHARD EULER, Joseph Louis Lagrange and CARL FRIEDRICH GAUSS. He read critically and found gaps in their arguments, quickly becoming a first-rate mathematician.

In 1820 Abel’s father died. Although Abel was the second son, at age 18 he took on the responsibility of supporting his mother and his six siblings with money that he earned by taking on private students. Holomboe also helped to find and donate funds so that Abel was able to continue his education at the University of Christiana.

About a year later Abel produced a stunning result on quintic (fifth degree) equations of the form ax^{5} + bx^{4} + cx^{3} + dx^{2} + ex + f = 0. Solutions of second degree equations were given by quadratic equations (in the form ax^{2} + bx + c = 0), and solutions to the third and fourth degree equations were also known. Mathematicians had searched for more than 200 years for an algebraic solution to quintic equations. Abel proved that such a solution did not exist. He sent the result to Gauss, who dismissed it without even reading it.

In 1825 the government of Norway sponsored Abel for travel and study in France and Germany. In Berlin he began an association with August Leopold Crelle with whom he founded the first journal of mathematical research. The first three issues of Crelle’s Journal of Pure and Applied Mathematics contained 22 of Abel’s papers. In 1826 Abel submitted to the French Academy of Sciences his paper on transcendental functions (all nonalgebraic functions such as exponential, logarithmic, trigonometric functions, etc.). The well established mathematicians AUGUSTIN LOUIS CAUCHY and Andrien Marie Legendre were to review the paper but both forgot about it. Abel realized he was wasting his time and returned to Norway.

The rigorous Abel was unhappy that the proofs of correct theorems rested on shaky reasoning. Among other things, Abel felt that “divergent series are the invention of the devil, and it is a shame to base on them any demonstration whatsoever.” He created proofs of his own – most notably he provided the first proof for the binomial theorem.

Abel taught part time at the University of Christiana and continued to publish papers on the theory of equations and elliptic functions (those that are doubly periodic). Abel died of tuberculosis on April 6, 1829, in Froland, Norway, at the age of 26. In response to a diplomatic inquiry, Cauchy found Abel’s paper on transcendental functions in 1830 and it was published in 1841, 12 years after Abel’s death.

Abel introduced rigor into mathematical proofs, suggested creative methodology for investigating mathematical problems, and left a host of new theorems, proofs, and equations for future mathematicians to learn from.

A pioneer of a wide range of mathematical ideas, Abel helped change the mathematical standards of what constitutes a rigorous proof. He put mathematics on a firmer foundation, one that supported the creative mathematical work of future generations. Abel also put the discipline of mathematics into a professional stance supported by its tint journal.

One of his intellectual heirs was GEORGE BOOLE, an elementary school teacher who bought math books to read because they were less expensive than any others available to him. When he studied Abel, he discovered that algebra could be a logical delight, not just a symbolic system for real numbers.

Two of Abel’s famous discoveries came from inverting a problem that others had proposed. For example, instead of continuing to search for the formula that would give a solution to the quintic, consider the possibility that none exists, and prove that! Now it is standard practice to consider the inverse or the “upside down” model to get a new perspective for creative work.

Modern math texts reflect the range of Abel’s original work by calling his result on transcendental functions “Abel’s theorem” and attaching his name to Abelian functions, Abelian equations, and Abelian groups. As HENRI POINCARÉ’s teacher, Charles Hermite, claimed, Abel laid out 500 years of work for future mathematicians.

Niels Henrik Abel – 1802-1829