# Mathematics

Mathematics (from Greek mathema "a lesson"; literally "that which is learned") is the study of quantity, structure, space, and motion.

# 1   History

Nick Szabo_ claims that the origins of mathematics can be traced to commercial accounting techniques. Geometry was invented from a need to define property rights for farmland; algebra was invented from a need to create balance sheets; calculus was invented in order to generate balance sheets from income statements.

## 1.1   al-Khwarazmi

Muhammad ibn Musa al-Khwarizmi (780-850) was a Persian mathematician who, among others, introduced algebra to the West.

Al-Khwarizmi's popularizing treatise on algebra ("The Compendious Book on Calculation by Completion and Balancing", ca. 813-833 CE) presented the first systematic solution of linear and quadratic equations in Arabic, for which he is credited as the father of algebra. The term algebra itself comes from the title of his book (specifically the word al-jabr meaning "completion" or "rejoining").

In the 12th century, Latin translations of his textbook on arithmetic (Algorithmo de Numero Indorum) which codified the various Indian numerals, introduced the decimal positional number system to the Western world.

## 1.2   Newton

The equals sign was first invented in 1557 by Robert Recorde, who was tired of having to write 'is equal to' over and over again and settled on parallel lines as a perfect symbol for equality, just five years before Galileo was born.

A scant century later, in 1667, Newton discovered gravity, the binomial theorem, optics and calculus, and the rest is history.

## 1.3   Leibniz

Gottfried Wilhelm Leibniz (1646) was a German mathematician. He completed his bachelors degree of philosophy in one year at age 15, and went on to become a prominent figure in the history of philosophy.

Leibniz contributed to calculus, and also refined the binary numerical system.

He also invented mechanical calculators. In 1685, he designed the pinwheel calculator, which was popular in the 19th and 20th centuries. The Leibniz wheel, invented in 1673, powered the first mass-produced calculating machine and was used for three centuries until the invention of the electronic calculator in the 1970s.

## 1.4   Modern mathematics

An unexpected of the development of predicate logic in the late 19th century was the discovery of weaknesses in the foundation of mathematics. For example, Euclid's Elements turned out to be full of logical mistakes. This realization created a crisis in the foundation of mathematics. Mathematicians started rebuilding the foundations of mathematics from the bottom up. In 1889, Giuseppe Peano developed axioms for arithmetic, and in 1899, David Hilbert did the same for geometry. [1]

Hilbert also outlined a program to formalize the remainder of mathematics, with specific requirements that any such attempt should satisfy including:

Completeness
There should be a proof that all true mathematical statements can be proved in the formal system.
Decidability
There should be an algorithm for deciding the truth or falsity of any mathematical statement.

Rebuilding mathematics in this way became known as "Hilbert's program". Up through the 1930s, this was the focus of a core group of logicians including Hilbert, Russel, Kurt Godel, John Von Neumann, Alonzo Church, and Alan Turing. [1]

Hilbert's program proceeded on at least two fronts. On the first front, logicians created logical systems that tried to prove Hilbert's requirements either satisfiable. On the second front, mathematicians used logical concepts to rebuild classical mathematics. For example, Peano arithmetic_. [1]

Russel made the first notable use of the liar's paradox_ in mathematical logic. He showed that Frege's system allowed self-contradicting sets to be derived via Russel's paradox_. [1]

Russel and his colleague Alfred North Whitehead put forth the most ambitious attempt to complete Hilbert's program with the Principia Mathematica, published in three volumes between 1910 and 1913. The Principia's method was so detailed that it took over 300 pages to get to that proof that 1 + 1 = 2. [1] Russel and Whitehead tried to resolve Frege's paradox by introducing they they called type theory_. The idea was to partition formal languages into multiple levels or types. Each level could make reference to levels below, not to their own or higher levels. This resolved self-referential paradoxes by banning self-reference. [1]

Self-referential paradoxes ultimately showed that Hilbert's program could never be successful. The first blow came in 1931, when Godel published his now famous incompleteness theorem, which proved that any consistent logical system powerful enough to encompass arithmetic must also contain statements that are true but cannot be proven to be true. [1] The final blow came with Turing and Church independently proved that no algorithm could exist that determined whether an arbitrary mathematical statement was true or false. [1]

## 1.5   Notation

Diophantus was one of the first mathematicians to give letters to unknown quantities. They would have previously been referred to simply as 'first quantity', 'second quantity', etc.

Descartes created the more modern notation for algebra, and used it to show that everything in geometry can be represented algebraically. He used the x- and y- axes to plot things out, added the z-axis for 3-d, and transformed Euclidean Geometry into Cartesian Geometry.

Descartes is credited with inventing the coordinate plane, even though he never graphed an equation.

Euler introduced the following symbols and notations: $$e$$, $$\pi$$, $$i$$, $$f(x)$$, and $$\sum$$.