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Mathematics is the science that deals with the logic of shape, quantity and arrangement. Math is all around us, in everything we do. It is the building block for everything in our daily lives, including mobile devices, architecture (ancient and modern), art, money, engineering, and even sports. Since the beginning of recorded history, mathematic discovery has been at the forefront of every civilized society, and in use in even the most primitive of cultures. The needs of math arose based on the wants of society. The more complex a society, the more complex the mathematical needs. Primitive tribes needed little more than the ability to count, but also relied on math to calculate the position of the sun and the physics of hunting. Mathematics is not about answers, it's about processes. Let me give a series of parables to try to get to the root of the misconceptions and to try to illuminate what mathematics IS all about. None of these analogies is perfect, but all provide insight. Other mathematicians hate to admit it, but when you get right down to it, math is about numbers. Of course it involves logic, spatial relations, proofs, all sorts of things. But that’s not the heart of it. Math is the study of numbers in all their fascinating aspects and manifestations. But since that’s so mundane - people can actually understand the definition - it just doesn’t sound mysterious enough for many mathematicians. "Mathematics is the art of giving the same name to different things." -Henri Poincaré. I would define mathematics as the study of structure divorced from context.
In mathematics, we study various structures: numbers, groups, geometric objects, etc. We study their patterns and figure out how they work and interconnect. I would make the argument that anything existing in the universe and anything that can be cooked up by the human mind that has some sort of structure to it can be studied mathematically. Of course, what one might argue is that disciplines like physics, chemistry, and biology do the same thing: they search for the physical patterns and structures that exist out in the world. What is the key difference between these pursuits and mathematics? The key is that in all of the above-mentioned physical sciences, any problem that you consider has a certain context, a certain specific interpretation. An oscillating pendulum is not the same thing as a vibrating string which is not the same thing as a spring with a mass attached to it. However, from a mathematical point of view, all of these systems are essentially the same thing. Once you strip away all of the physical details and particular context of a problem, what remains is its mathematical content. The beauty of this is that by considering things so abstractly, you begin to see connections that you would not otherwise recognize. The motion of a pendulum, a string, and a spring are all described using sinusoidal functions. That same periodic behavior might be observed just as well for sound waves, and light waves, and ocean waves. When you have a perspective this broad, you can begin by looking at a problem one way, flip it around, realize that you can contextualize it in a completely different way, and then start using the tools of a seemingly unconnected theory to solve your problem.
History of Mathematics
In Babylonia mathematics developed from 2000 BC. Earlier a place value notation number system had evolved over a lengthy period with a number base of 60. It allowed arbitrarily large numbers and fractions to be represented and so proved to be the foundation of more high powered mathematical development. Number problems such as that of the Pythagorean triples (a,b,c) with a2+b2 = c2 were studied from at least 1700 BC. Systems of linear equations were studied in the context of solving number problems. Quadratic equations were also studied and these examples led to a type of numerical algebra. Geometric problems relating to similar figures, area and volume were also studied and values obtained for p. The Babylonian basis of mathematics was inherited by the Greeks and independent development by the Greeks began from around 450 BC. Zeno of Elea's paradoxes led to the atomic theory of Democritus. A more precise formulation of concepts led to the realization that the rational numbers did not suffice to measure all lengths. A geometric formulation of irrational numbers arose. Studies of area led to a form of integration. The theory of conic sections shows a high point in pure mathematical study by Apollonius. Further mathematical discoveries were driven by the astronomy, for example the study of trigonometry. Several civilizations — in China, India, Egypt, Central America and Mesopotamia — contributed to mathematics as we know it today. The Sumerians were the first people to develop a counting system. Mathematicians developed arithmetic, which includes basic operations, multiplication, fractions and square roots. The Sumerians’ system passed through the Akkadian Empire to the Babylonians around 300 B.C. Six hundred years later, in America, the Mayans developed elaborate calendar systems and were skilled astronomers. About this time, the concept of zero was developed.
Prehistoric people must have used simple arithmetic. However when people became civilized mathematics became far more important. Proper record keeping was essential. In Iraq a people called the Sumerians counted in sets of 60. We still divide hours into 60 minutes and minutes into 60 seconds. We also divide circles into 360 degrees. There were a number of great Indian mathematicians during this era. Among them were Aryabhata (c. 476-550) and Brahmagupta (c 598-670). A Persian named Al-Khwarizmi was also a famous mathematician. He lived in the early 9th century. He wrote about Indian numerals and algebra. In the 19th century Carl Friedrich Gauss (1777-1855) made contributions to algebra, geometry and probability. Charles Babbage (1791-1871) is called the father of the computer because he designed a mechanical calculating machine he called an analytical engine (although it wasn't actually built in his lifetime). Babbage was assisted by another great mathematician called Ada Lovelace (1815-1852). George Boole (1815-1864) created Boolean algebra. Meanwhile in 1801 William Playfair (1759-1823) invented the pie chart. (Florence Nightingale did not invent the pie chart although she did use them). John Venn (1834-1923) invented the Venn diagram.
Mathematics as Science
Many others will answer this question by referring to the various definitions and associations that come with the word 'science'. However, these are not even consistent between languages as English, German, Swedish, Japanese, Mandarin, all have words for Science that are not 1:1 matches.
If Math would be seen as something that assists Science, then mathematicians would be degraded to assistants. No good. So, while the test/verify/measure/model paradigms in Science quite don't apply to Math, I prefer thinking of Math’s as a part of Science for mentioned reasons. Traditionally, the development of physics, which is a science, was paralleled with the refinement of the mathematical techniques employed to handle the mathematizing way of the (theoretical) method in physics. This gives rise to two different approaches to mathematics (since up to the mid-1950s the most influential science to mathematics apart from mathematics itself was physics) a pure and an applied one. Content wisely, both approaches can encompass the same branches of mathematics, e.g., Lie Group Theory, which can be further developed from a pure approach, can also be instrumental zed and further developed in order to meet the needs of physicists who investigate symmetries in their theories or that study symmetries of solution spaces to e.g. Partial differential equations.
If we look on mathematics as an imaginary world with no connection to reality, we can still do science. Let's call that "Platonic mathematics." If instead we claim that mathematics is nature's language, then we can explore nature through mathematics, and again, its legitimate science. Let's call that "natural mathematics."
The kind of mathematics having no essential connection with the physical world. I say "essential connection" because links between nature and esoteric mathematical notions spring up in unexpected places, like the surprising connection between locusts and prime numbers.
The kind of mathematics arising from the uncanny efficiency with which mathematics describes nature.
In this article we will explore science derived from both Platonic and natural mathematics.
Foundation of Mathematics
The content of a theory, describing its model(s), is made of components which are pieces of description (concepts and information, described in 1.3). A theory starts with a choice of foundation made of a logical framework and an initial version of its content (hopefully rather small, or at least simply describable). The components of this initial version are qualified as primitive.
The study of the theory progresses by choosing some of its possible developments: new components resulting from its current content and that can be added to it to form its next content. Any other possible development (not yet chosen) can still be added later, as the part of the foundation that could generate it remains. Thus, the totality of possible developments of a theory, independent of the order chosen to process them, already forms a kind of «reality» that these developments explore. it is a misconception to think that natural structures in mathematics have simplest or best definitions and the same applies to the axioms for mathematics itself. The definition requires you to specify some sort of theoretical computer such as a Turing machine, but this is completely arbitrary. There are lots of alternatives that are equally good and all you can do is show that they are all equivalent. What makes concepts interesting and useful in mathematics is not how good their base definition is, but rather their universality. You may think the best definition of PI is in terms of a circle, but it could also be defined in terms of infinite series or the period of functions obeying simple differential equations. What makes PI so interesting is that it comes up all over mathematics, not just in geometry. This is what we mean by universality. So mathematics itself also has many different starting points but they all lead to the same thing. Some starting points or axioms may be more convenient given our current preferred mathematical interests, but they are not really more fundamental than any other starting point that leads to the same body of mathematical ideas.
Pure and Applied Mathematics
Applied mathematics seeks to produce tools to help us understand the physical world. Pure mathematics does not. There are some really great answers here, so I'm not sure if I can really add to this, but I am going to try. Hopefully I don't sound like too much of a broken record. Applied mathematicians are typically motivated by problems arising from the physical world. They use mathematics to model and solve these problems. These models are really theories and, as with any science, they are subject to testify ability and falsifiability. As the amount of information regarding the problem increases, these models will possibly change. Pure and applied are not necessarily mutually exclusive. There are many great mathematicians who tread both grounds. German Mathematics soon outstripped the French, with Gottingen becoming the center of a golden age of Mathematicians. The terms pure mathematics and applied mathematics seem to have been invented in the 19th century, but they really aren't two different kinds of mathematics. The distinction is more of intent than of content. Any applied mathematics contains pure mathematics, and any pure mathematics may have been or may be applied sometime in the future. The applications are to some kind of science. Mathematics that is studied with the intent to be applied to natural science, biological science, or social science can be called applied mathematics. Scientists use a lot of mathematics in their work. If a scientist develops some new mathematics in the course of studying science that could be called applied mathematics. If that same scientist goes a little further studying that mathematics without the intent of applying it, you could call that pure mathematics. There are many problems pursued by pure mathematicians that have their roots in concrete physical problems – particularly those that arise from relativity or quantum mechanics. Typically, in a deeper understanding of such phenomena, various “technicalities” arise (believe me when I tell you these technicalities are very difficult to explain). These become abstracted away into purely mathematical statements that pure mathematicians can attack. Solving these mathematical problems then can have important applications.
Pure mathematics is simply mathematics considered without regard to applications outside of mathematics. It may or may not have been applied. It may or may not be applied sometime in the future. The French solved a lot of the problems of the time, mostly in probability, but the Germans solved problems that didn't exist, but were going to be vital as the 20th Century started. Notably the Mathematics behind relativity had been worked out, as had general solutions involving abstract algebra. New technologies such as radio and electricity required Mathematics that wasn't based on things you could see. Numbers themselves turned out to be a whole lot more complicated and Math’s moved into non-numerical fields (Boolean algebra and Graph theory) Pure Math’s at its heart is a quest for truth, it has an almost mystical element. Pure Math’s is the equivalent of a mystery tour, you could end up anywhere, but the point of a journey is not the destination. That is pure math. Applied mathematics occurs when we search for a collection of axioms and conclusions which help us describe the physical world. Much of mathematics is based on such quests. Indeed, due to the historical nature of mathematics being applied to these quests, a large portion of our body of mathematical theory is related to the physical world, right down to the axioms that we use. This is one reason why so much that mathematicians produce ends up being useful.
The Fields Medal
The Fields Medal is one of the most recognized prizes given to mathematicians who have achieved something great during their career. The official title for this prize is the International Medal for Outstanding Discoveries in Mathematics and it is given out once every four years to up to four mathematicians under the age of 40.
This prestigious prize is actually presented by a king – the King of Norway – to a mathematician who is outstanding in their field of study.
Wolf Prize in Mathematics
The Wolf Foundation of Israel awards six different prizes each year and one of them is the Wolf Prize in Mathematics. This prize has been awarded since 1978 and it is also a very prestigious honor to receive one in the fields in which they are given.
One of the newer prizes in mathematics is the Chern Meda, which began recognizing lifetime achievements for mathematicians in 2010, is awarded every four years. It is given out at the International Congress of Mathematicians and it includes a prize of $250,000. The first recipient in 2010 was Louis Nirenberg and the 2014 winner was Philllip Griffiths.