Mathematics is the study of numbers, shapes, relationships and patterns in the context of axiomatic systems. The most important thing which makes mathematics different from other studies is the rigor with which mathematical systems are proved.
A mathematical proof is an assurance that, given a set of primitives, axioms, criteria and other proofs, a claim is absolutely and always true. A rigorous mathematical proof answers the question, "How do you know that?" in such a way that the answer can only be challenged by challenging the axioms.
According to historians and anthropologists, mathematics was invented to fill the desire to count things and, by extension, measure things. Primitive hunting-gathering cultures only needed a few words for numbers (one, two, more, and many, for example) and measurement (larger or smaller, perhaps). The next level of complexity for human life is the farming community. In farming communities, a person may want to count ears of corn in a grainery. This required larger numbers. In more advanced communities, governments were formed. Government means taxes. Even larger numbers, and ways to measure the size of a farm to calculate taxes were needed.
The first mathematicians were accountants, people who kept track of goods and money for governments. As needs for mathematics increased, government accounts started coming up with ways to do math more quickly and more accurately. A number of algorithms, such as casting out nines were invented.
These early mathematicians also wrote books that today we would call textbooks. These books were in many ways like today's math books. They contained problems to solve. However, since math notation had not been invented yet, the problems were all word problems. An example from ancient Asiatic Indian mathematics is:
One-half, one-sixth, and one-twelfth parts of a pole are immersed respectively under water, clay, and sand. Two hastas are visible. Find the height of the pole? G. R. Kaye. Indian Mathematics (1915). Translated by G. Thibault.
These earlier mathematicians also began struggling with the question, "How do you know that?" Early attempts at answering this question lead to some interesting, if not rigorous, proofs, such as the Chinese proof of the Pythagorean Theorem. The first known published proofs based on axioms is Euclid's Elements. These thirteen books are considered one of the most important landmarks in the advancement of mathematics. For the first time, in 300 B.C.E., mathematics was distinguished from other human endeavors by attempts at a rigorous proof.
Since the time of Euclid, mathematics has grown much larger. Today, most of modern mathematics is based on Set Theory. Different sets are identified and assigned properties using primitives, definitions and axioms. From these properties, other properties are proved as theorems. Some of the branching out happened as follows:
From its early beginnings, mathematics has branched out into many areas of study. It should be noted that these areas of study overlap, and sometimes merge to create new areas of study.
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