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Tuesday, May 30, 2017

Mathematical Advancements During the First Memorial Day

The idea of honoring our fallen veterans began in the late 1800's after losing nearly 750,000 soldiers in the Civil War. In 1868, General John A. Logan, one of the leaders of a Northern Civil War veteran's organization, declared that there would be a nationally recognized day for the fallen soldiers. “The 30th of May, 1868, is designated for the purpose of strewing with flowers, or otherwise decorating the graves of comrades who died in defense of their country during the late rebellion, and whose bodies now lie in almost every city, village and hamlet churchyard in the land,” he proclaimed. He decided to call this special day Decoration Day and chose the date because there were no battle anniversaries on that day. Decoration Day would eventually be known as Memorial Day in 1971, which fully encompassed American military personnel who died in all wars. The memory of American troops stands for the protection of freedom within the world, a freedom to advance together as a collected human race.

Around the time of the Civil War and then the creation of Decoration day in the 19th century, there were numerous mathematicians who were able able to make vast advancements in the field.

George Boole: This British mathematician and philosopher was one of the few who focused on logic and reasoning as well as their usefulness in mathematics. A few years prior to the Civil War, Boole introduced a new form of algebra (now called Boolean Algebra or Boolean Logic) which only consisted of operators 'AND', 'OR', and 'NOT'. These algebraic operators could be used to solve logic problems as well as some mathematical functions. Boole also composed an approach to logical systems with a form of binary, where he would process two objects (yes-no, true-false, 1-0, etc.). Under Boolean Logic, 1 + 1 = 1 (True ∧ True = True). This was a major advancement in modern mathematical logic, but people at the time didn't recognize it as one. It wasn't until American logician Charles Sanders Pierce recognized Boole's work, revised some of it, and then elaborated on his ideas in 1864. Nearly seventy years later, Boolean logic would go on to be used for electrical switches to process logic and become the basis for computer science.

Bernhard Riemann: A very well-known mathematician, from Germany, who made great contributions to differential geometry, analysis, and number theory in the 1850s (a few years prior to the Civil War). In college, Riemann began to take stride under the wing of his professor, Carl Friedrich Gauss (ranked as one of History's most influential mathematicians). One of Riemann's contributions was that of elliptic geometry and also Riemannian Geometry, which essentially generalized the ideas of surfaces and curves. Riemann's new contributions changed how we view the higher dimensional world we live in. He would go on to break away from 2 and 3 dimensions, and look at n dimensional surfaces which would contribute to further conceptualize relativity on curved surfaces. Another big breakthrough for Riemann came from working with the zeta function, which Euler had first experimented with in the 18th century. Using the zeta function to build a 3-dimensional landscape, he noticed that "the zeroes" (where the graph dipped to zero) of his landscape had a connection to the way prime numbers are distributed. This relationship between his zeta function and prime numbers brought him instant fame in 1859, when his findings were published. Unfortunately, Riemann passed away at the age of 39 in 1866 and his incomplete findings on this relationship, known as the Riemann Hypothesis, remain unsolved 160 years later. A prize of $1 million has been offered as a prize for a final solution by the Clay Mathematics Institute.

George Cantor: Also a German mathematician, became a full Professor at the University of Halle at the age of 34, which was essentially unheard of. One of his major contributions to mathematics is the first foundation of set theory, which helped explain the notion of infinity and became very common in all of mathematics. He also delved into the concept of the infinities of infinity, where he showed there may be infinitely many sets of infinite numbers. His work undoubtedly changed how mathematicians now view sets and the concept of infinity. This all started in the early 1870s when Cantor considered an infinite series of natural numbers (1, 2, 3, ...) and an infinite series of multiples of 10 (10, 20, 30, ...). He could clearly state that the the series of multiples of 10 was a subset of the series of natural numbers, but he could also recognize the sets could be matched one-to-one (1 with 10, 2 with 20, 3 with 30, etc). He used this process, which is known as bijection, to show that the sets were the same size. Cantor realized he could do the same sort of thing comparing rational numbers and natural numbers, concluding that they are of the same infinity even though fractions would seem to outnumber natural numbers. Cantor also looked at irrational numbers, and argued that there exists an infinite amount of irrational numbers between each and every rational number. In later eras, Cantor would go on and refine his set theory and introduce new ideas such as ordinality and cardinality.

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