Mathematics

Mathematics (Greek μαθηματική τέχνη Mathematike techne, the art of learning ', belonging to learning ') is a science, which originated from the study of geometric figures and to calculate with numbers. For mathematics there is no generally accepted definition; today it is usually described as a science, the self-created abstract structures studied by means of logical definitions of logic to their properties and patterns.

• 3.1 Categorization of mathematics
• 3.2 Special role among the Sciences

• 4.1 Mathematics as a school subject
• 4.2 Mathematics as a field of study and career

History

Mathematics is one of the oldest sciences. Your first flower she experienced even before the ancient times in Mesopotamia, India and China. Later, in ancient times in Greece and Hellenism, from there dated the focus on the task of " proving purely logical " and the first axiomatization, namely the Euclidean geometry. In the Middle Ages it survived independently in the early humanism of the universities and in the Arab world.

In the early modern era François Viète introduced variables and René Descartes opened by the use of a mathematical coordinate access to the geometry. The description of tangents and the determination of surface areas ( "quadrature" ) led to the calculus of Gottfried Wilhelm Leibniz and Isaac Newton. Newton's mechanics and his law of gravitation were also in the following centuries a source of trend mathematical problems such as the three- body problem.

Another guiding problem of the early modern period was solving increasingly complicated expectant algebraic equations. To his treatment developed Niels Henrik Abel and Galois Évariste describes the concept of group, the relationship between symmetries of an object. As a further deepening of these studies, the newer algebra and in particular the algebraic geometry can be viewed.

A then-new idea in the correspondence between Blaise Pascal and Pierre de Fermat in 1654 led to the solution of an old problem for which there have been other, however controversial solutions. The exchange of letters is regarded as the birth of classical probability theory. The new ideas and methods to conquer many areas. But over the centuries it comes to splitting of the classical probability theory into separate schools. Attempts to define the term " probability " explicitly, succeed only for special cases. It was not until the release of Andrei Kolmogorov textbook Basic concepts of probability theory in 1933, completes the development of the foundations of modern probability theory.

During the 19th century, calculus was through the work of Augustin- Louis Cauchy and Karl Weierstrass its current strict form. Developed by Georg Cantor in the late 19th century, set theory is from today's mathematics is also here to stay, even if they initially made ​​clear by the paradoxes of naive set term, on what shaky foundation was the math before.

The development of the first half of the 20th century under the influence of David Hilbert's list of 23 mathematical problems. One of the problems was the attempt of a complete axiomatization of mathematics; at the same time, there was a strong effort to abstraction, so the experiment to reduce objects to their essential properties. So Emmy Noether developed the foundations of modern algebra, Felix Hausdorff general topology as the study of topological spaces, Banach probably the most important concept of functional analysis, named after him Banach space. An even higher level of abstraction, a common framework for the consideration of similar structures in various areas of mathematics, eventually created the introduction of category theory by Samuel Eilenberg and Saunders Mac Lane.

Content and methodology

Contents and sub-areas

The following list gives a first chronological overview of the breadth of mathematical topics:

• Calculating with numbers ( arithmetic - Antiquity),
• The study of figures ( Geometry - Antiquity, Euclid )
• The dissolution of equations ( Algebra - Antiquity, Middle Ages and Renaissance, Tartaglia )
• The study of correct conclusions ( Logic - Aristotle ) ( partially counted only to philosophy, but often also to mathematics )
• Studies on the divisibility ( number theory - Euclid, Diophantus, Fermat, Euler, Gauss, Riemann )
• The computational detection of spatial relationships ( Analytical Geometry - Descartes, 17th Century ),
• Calculating with probabilities ( Stochastics -. Pascal, Jakob Bernoulli, Laplace, 17th-19th century),
• The study of functions, especially their growth, curvature of the behavior at infinity of the areas under the curves (Analysis - Newton, Leibniz, late 17th century ),
• The description of physical fields ( differential equations, partial differential equations, vector analysis - Euler, the Bernoulli, Laplace, Gauss, Poisson, Fourier, Green, Stokes, Hilbert, 18-19th century. )
• Perfecting the analysis by the inclusion of complex numbers (Function Theory - Gauss, Cauchy, Weierstrass, 19th century ),
• The geometry of curved surfaces and spaces ( differential geometry - Gauss, Riemann, Levi -Civita, 19th century ),
• The systematic study of symmetries ( group theory - Galois, Abel, Klein, Lie, 19th century),
• The elucidation of the paradoxes of the infinite ( set theory and logic again - Cantor, Frege, Russell, Zermelo, Fraenkel, early 20th century ),
• The study of structures and theories (universal algebra, category theory ),
• The collection and analysis of data ( statistics).

Something stands apart in this enumeration Numerical Analysis, which examines provides for continuous concrete problems of many of the above areas and algorithms to solve them.

Differences are further pure mathematics, also referred to theoretical mathematics that does not deal with non- mathematical applications, and mathematics such as actuarial science and cryptology applied. The transitions of the aforementioned areas are fluid.

Progression through problem solving

Characteristic of mathematics continues to be the way they "really serious " progresses through the editing problems.

Once an elementary school student has learned the addition of natural numbers, he is able to understand the following question and answer by trial and error, "What number must you add to 3 to get 5" The systematic solution of such tasks but requires the introduction a new concept: the subtraction. The question can then be rephrased: "What is 5 minus 3" But when the subtraction is defined, you can also ask the question: " What is 3 minus 5? " Which to a negative number and thus already have the basic school mathematics addition leads.

Just as in this elementary example, when individual learning mathematics is also advanced in its history: on every standard achieved, it is possible to provide well-defined tasks which can be solved much more sophisticated means are necessary. Often many centuries between the formulation of a problem and its solution is passed and the problem solution eventually an entirely new branch has been established: so could the calculus in the 17th century, problems are solved, which were open since antiquity.

Even a negative answer, the proof of the unsolvability of a problem, can promote mathematics: it is from failed attempts to resolve algebraic equations to group theory emerged.

Axiomatic formulation and language

Since the end of the 19th century, isolated since antiquity, mathematics is presented in the form of theories that begin with statements that are considered true; from other true statements are then derived. This derivation takes place by the well-defined rules of inference. The statements with which begins the theory, called axioms, which derived from it are called sentences. The derivation itself is a proof of the theorem. In practice, definitions play a role, through them mathematical concepts are introduced and clarified by returning to more basic. Due to this construction of mathematical theories they are called axiomatic theories.

Usually, this requires axioms of a theory that these are consistent, so that not the same one set and the negation of this proposition to be true. This consistency itself can be but generally not within a mathematical theory to prove ( this is dependent on the used axioms ). This has the consequence that about the consistency of Zermelo -Fraenkel set theory, which is fundamental to modern mathematics, is not provable without the aid of additional assumptions.

The treated of these theories are abstract mathematical objects structures, which are also defined by axioms. While in the other studies, the treated articles are set, and thereafter the methods for the study of these objects to be created, the method is defined in mathematics reversed and the inspectable objects to be created afterwards. In this manner, takes and always took the math a special position among the sciences.

The development of mathematics was done and on the other hand is often done through collections of sets, proofs and definitions that are not structured axiomatic, but mainly by the intuition and experience of the mathematicians involved are marked. The conversion to an axiomatic theory is made later when other mathematicians deal with the then not-so- new ideas.

However, the axiomatization of mathematics has its limits. Kurt Gödel showed in 1930 in the incompleteness theorem named after him, that in any mathematical axiom system either true but unprovable statements exist, or the system is contradictory.

Mathematics used to describe situations in a very compact language that is based on technical terms and, in particular formulas. A representation of the symbols used in the formulas can be found in the list of mathematical symbols. A special feature of the mathematical jargon is the formation of derived from mathematician name adjectives like Pythagorean, Euclidean, Euler tour, abelian, noetherian and artinian.

Areas of application

Mathematics is applicable in all the sciences that are sufficiently formalized. This results in a close interplay with applications results in empirical sciences. For many centuries, the mathematics ideas from astronomy, geodesy, physics and economics has taken and vice versa provided the basis for the progress of these subjects. For example, Newton developed the calculus to grasp the physical concept of " force equal to change in momentum " mathematically. Fourier laid the foundations for the modern concept of function in the study of the wave equation and Gauss developed the method of least squares as part of his employment with astronomy and surveying and systematized solving systems of linear equations.

Conversely, mathematicians have sometimes developed theories that have later found surprising practical applications. For example, has become indispensable in the 16th century, the resulting theory of complex numbers for the mathematical representation of electromagnetism. Another example is the tensor differential form calculus, which Einstein had used for the mathematical formulation of general relativity. Furthermore, the study of number theory has long been understood as an intellectual game of no practical use without them today, however, the modern cryptography and its diverse applications in the Internet is not feasible.

Relationship to other sciences

Categorization of Mathematics

The question of which category of Sciences mathematics heard has been controversial for a long time.

Many mathematical problems and concepts are motivated by the nature of questions concerning, for example, physics or engineering, and mathematics is used as an auxiliary science in almost all natural sciences. However, it is itself not a natural science in the strict sense, since their statements do not depend on experiments or observations. Nevertheless, it is assumed in the recent philosophy of mathematics believe that the methodology of mathematics corresponds more to that of natural science. Following Imre Lakatos a "renaissance of empiricism " is suspected, which also set up mathematician hypotheses and look for these confirmations.

Mathematics has methodological and content in common with the philosophy; For example, the logic is an area of ​​overlap of the two sciences. Thus, one could expect the math to the humanities, but also the incorporation of philosophy is controversial.

For these reasons categorize some mathematics - in addition to other disciplines such as computer science - as a structural science or formal science.

At German universities mathematics mostly belongs to the same faculty as the natural sciences, and so will mathematicians after graduation usually the academic degree Dr. rer. nat. Awarded ( Doctorate in Science ). In contrast, in the English speaking university graduates reached the title "Bachelor of Arts" or " Master of Arts" which are actually awarded to scholars.

Special role among the Sciences

A special role among the Sciences takes the math regarding the validity of their findings and the severity of its methods. For example, while all scientific knowledge can be falsified by new experiments and are therefore in principle temporary, mathematical statements are produced by pure thoughts operations apart or back out to each other and do not need to be empirically verifiable. But a strictly logical proof must be found for mathematical knowledge before they are recognized as a mathematical theorem. In this sense, mathematical theorems are basically final and universal truths, so that mathematics can be regarded as an exact science. It is this accuracy is the fascinating thing about mathematics for many people. So said David Hilbert at the International Congress of Mathematicians in Paris in 1900:

"We discuss briefly what general requirements must be met by the solution of a mathematical problem: I mean especially that it is possible to demonstrate the correctness of the solution by a finite number of conclusions, on the basis of a finite number of conditions, which are the problem and must be formulated precisely every time. This requirement of logical deduction by means of a finite number of circuits is none other than the requirement of rigor in reasoning. In fact, the requirement of rigor, which is known to become in mathematics proverbial meaning, corresponds to a general philosophical necessity of our understanding, and on the other hand comes through their fulfillment alone just the thought content and the suggestiveness of the problem attain their full effect. A new problem, especially if it comes from the outer world of appearance, is like a young twig, which thrives and bears fruit when grafted onto the old trunk, the established achievements of our mathematical knowledge, carefully and according to the strict horticultural rules will. "

Joseph Weizenbaum of the Massachusetts Institute of Technology described the mathematics as the mother of all sciences.

" But I maintain that in every special doctrine of nature only so much real science can be found, as there is mathematics to be found. "

Mathematics is therefore a cumulative science. Are now known over 2000 mathematical journals. However, this also poses a danger: by newer mathematical areas get older areas in the background. Apart from very general statements, there are also very specific statements for which no real generalization is known. Donald E. Knuth writes in the preface of his book Concrete Mathematics:

" The course title ' Concrete Mathematics ' which originally Intended to as antidote to 'Abstract Mathematics ', since concrete classical results were proceed rapidly being swept out of the modern mathematical curriculum by a new wave of abstract ideas popularly called the ' New Math '. Abstract mathematics is a wonderful subject, and there 's nothing wrong with it: It's beautiful, general and useful. But its adherents had become deluded did the rest of mathematics which inferior and no longer worthy of attention. The goal of generalization had become so fashionable did a generation of mathematicians had become unable to relish beauty in the Particular, to enjoy the challenge of solving quantitative problems, or to appreciate the value of technique. Abstract mathematics what becoming inbred and losing touch with reality; mathematical education needed a concrete counterweight in order to restore a healthy balance. "

" The event title, Concrete Mathematics' was originally as a counterpoint to, abstract mathematics ' thought, because concrete, classical achievements were of a new wave of abstract ideas - commonly, New Math '(' new math ') called - at a rapid pace from the curriculum rinsed. Abstract mathematics is a wonderful thing, where nothing is to be exposed: She is beautiful, universally valid and useful. But their supporters came to the erroneous view that the rest of mathematics was inferior and not worthy of attention. The goal of generalization was so fashionable that a whole generation of mathematicians was no longer able to recognize beauty in particular, to understand the solution of quantitative problems as a challenge or to estimate the value of mathematical techniques. The abstract mathematics only turned himself and lost touch with reality; in mathematical education a concrete counterweight was necessary to re-establish a stable equilibrium. "

It thus comes the older mathematical literature on a special significance.

The mathematician Claus Peter Ortlieb criticized - in his opinion - too little reflected application of modern mathematics:

"One must be aware that the acquisition of the world through mathematics has its limits. The assumption that she was working alone according to mathematical laws, means that you hold out only in accordance with these laws. Of course I will find them in the natural sciences, but I have to be aware that I look at the world through a pair of glasses through which hides from the outset large parts. [ ... ] The mathematical method has long since been taken over by scientists almost all disciplines and is applied in all kinds of areas where it has nothing to look for. [ ... ] Are alarming numbers whenever they lead to normalizations, although no one can understand how the numbers were arrived at. "

Mathematics in Society

The German Federal Ministry of Education and Research ( BMBF) oriented annually since 2000, Science, 2008 was the Year of Mathematics.

Mathematics as a school subject

Mathematics plays an important role in the school as a compulsory subject. Mathematics education is the science that deals with the teaching and learning of mathematics. In the lower and middle level is not just about learning numeracy. In the upper level then the differential and integral calculus and analytic geometry / linear algebra are introduced and continued to stochastics.

Mathematics as a field of study and career

People who are professionally engaged in the development and application of mathematics, called mathematicians.

In addition to the study of mathematics at diploma, in which one can put his focus on pure and / or applied mathematics, more interdisciplinary courses such as industrial mathematics, business mathematics, computer mathematics and biomathematics have been set up in recent times. In addition, teachers at secondary schools and universities is an important mathematical profession. At German universities, the Diploma to Bachelor / Master programs will be changed now. Must occupy a certain number of hours per week also budding computer scientists, chemists, biologists, physicists, geologists and engineers.

The most common employers for graduate mathematicians are insurance companies, banks and consulting companies, particularly in the area of mathematical models and financial consulting, but also in the IT field. In addition, mathematicians are used in almost all industries.

Mathematical museums and collections

Mathematics is one of the oldest sciences and also an experimental science. These two aspects can be illustrated very well by museums and historical collections.

The oldest institution of its kind in Germany, founded in 1728 Mathematical- Physical Salon in Dresden. The Arithmeum in Bonn at the Institute for Discrete Mathematics goes back to the 1970s and is based on the collection of computing devices of the mathematician Bernhard Korte. The Heinz Nixdorf Museum Forum ( abbreviation " HNF " ) in Paderborn is the largest German Museum for the development of computer technology (especially the computer), and the Mathematikum in casting was founded in 2002 by Albrecht Beutelspacher and is continuously being developed by him. In the Museum Quarter in Vienna, the led by Rudolf Taschner math.space showing the mathematics in the context of culture and civilization.

In addition, numerous special collections are housed at universities, but ( designed and built by Konrad Zuse computer ) also in broader collections such as the Deutsches Museum in Munich or at the Museum of History of Technology in Berlin.