Mathematics in the Age of Enlightenment

Leonhard Euler: Mathematics Pioneer

April 15, 1707 – September 18, 1783

Leonhard Euler was born on April 15, 1707, in Basel, Switzerland. He died on September 18, 1783, in St. Petersburg, Russia. He lived through a period of Europe known as the Age of Enlightenment—a time when Europeans were turning away from the Age of Faith that had characterized the Middle Ages and embracing an Age of Reason. During this time, it was popular to look at the world in a naturalistic, rationalistic way. Nonetheless, Euler's early life was steeped in religious influences. His father was a Calvinist preacher, and his mother herself descended from a line of pastors. In fact, the family initially envisioned a religious path for young Euler. However, his innate mathematical abilities soon became apparent, and his course in life took a turn the more resembled the scientific bent of the times rather than the Puritanical religious bent that persisted (Calinger, 2016).

Euler's academic studies began at the University of Basel where he demonstrated a penchant for mathematics and caught the attention of famous mathematician Johann Bernoulli. Under Bernoulli's mentorship, Euler's love for mathematics only deepened.

At the heart of the Enlightenment was the belief that human beings could apply reason and logic to understand the world around them and improve society (Calinger, 2016). This was a significant departure from the previous centuries, where religious doctrine guided the way most people looked at life and the world. The Enlightenment philosophers argued against Original Sin and the Fall (concepts that Euler’s own parents would have promoted heavily in the house). Enlightenment philosophy emphasized empirical evidence (Calinger, 2016).

This shift in thinking had a big impact on Euler. Indeed, his work epitomized the Enlightenment's values. He approached problems with an emphasis on empirical evidence and systematic investigation and thus he was able to make foundational contributions to calculus, as his textbook "Introductio in analysin infinitorum" which became a seminal work in the field (Dunham, 1999).

Euler also dealt with graph theory, presenting the first theorem through the Seven Bridges of Königsberg problem. Before Euler's work on the Königsberg bridge problem, there was no formalized study of what is now recognized as graph theory. His approach was in abstracting the problem from a real-world scenario into a mathematical one. He represented the land masses as nodes (or vertices) and the bridges as connections (or edges) between these nodes. This abstraction allowed for a more generalized approach to problem-solving, a method that has since become fundamental in mathematical modeling. One of the remarkable aspects of Euler's solution was that he did not provide a method to traverse the bridges without crossing the same one twice (because it was impossible). Instead, he demonstrated why such a path could not exist. This non-constructive approach to problem-solving became a valuable tool in mathematical proofs (Dunham, 1999).

Number theory also benefitted from Euler's influence. He introduced the totient function and his groundbreaking proof regarding the infinitude of prime numbers are celebrated milestones. Among his many contributions, Euler's Formula stands out. The formula, e^ {ix} = cos(x)+isin(x), is a testament to Euler's brilliance, weaving together exponential functions, trigonometry, and complex numbers. Euler's formula is foundational to the Fourier transform, a mathematical tool that decomposes a function into its constituent frequencies and which has had applications in signal processing, image analysis, and quantum physics (Calinger, 2016).