The Mathematics ( federal German High German : [ matematik ], [ matematik ]; Austrian High German : [ matematik ]; ancient Greek μαθηματική τέχνη mathematike téchnē , the art of learning ') is a science which made the study of geometric figures and calculations with numbers arose . For mathematics , there is no universally accepted definition ; today it is usually described as a science that uses logic to examine self-created abstract structures for their properties and patterns by means of logical definitions .
Mathematics is one of the oldest sciences. It experienced its first heyday before antiquity in Mesopotamia , India and China , and later in antiquity in Greece and in Hellenism . From there, the orientation to the task of "purely logical proof" and the first axiomatization , namely Euclidean geometry, date from . In the Middle Ages she survived independently in the early humanism of the universities and in the Arab world.
In the early modern period , François Viète introduced variables, René Descartes opened up a computational approach to geometry through the use of coordinates . The consideration of rates of change ( fluxions ) as well as the description of tangents and the determination of surface areas ("quadrature") led to the infinitesimal calculus by Gottfried Wilhelm Leibniz and Isaac Newton . Newton's mechanics and his law of gravitation continued to be a source of seminal mathematical problems such as the three-body problem in the centuries that followed .
Another key problem in the early modern era was solving increasingly complex algebraic equations. To deal with this, Niels Henrik Abel and Évariste Galois developed the term group , which describes the relationships between symmetries of an object. The more recent algebra and in particular algebraic geometry can be viewed as a further deepening of these investigations .
A new idea at the time in the correspondence between Blaise Pascal and Pierre de Fermat in 1654 led to the solution of an old problem for which there were already other, albeit controversial solutions. The correspondence is seen as the birth of the classical calculus of probability. The new ideas and processes conquered many areas. But over the centuries classical probability theory split up into separate schools. Attempts to define the term “probability” explicitly only succeed in special cases. It was not until Andrei Kolmogorov's textbook Basic Concepts in Probability Theory was published in 1933 that the development of the foundations of modern probability theory was completed, see also History of Probability Theory .
In the course of the 19th century, the calculus found its current strict form through the work of Augustin-Louis Cauchy and Karl Weierstrass . The set theory developed by Georg Cantor towards the end of the 19th century has also become indispensable in today's mathematics, even if it initially made clear through the paradoxes of the naive concept of set the uncertain foundation on which mathematics was previously.
The development of the first half of the 20th century was influenced by David Hilbert's list of 23 math problems . One of the problems has been the attempt to fully axiomatize mathematics; At the same time, there were strong efforts towards abstraction, i.e. the attempt to reduce objects to their essential properties. So developed Emmy Noether the fundamentals of modern algebra, Felix Hausdorff general topology as the study of topological spaces , Stefan Banach probably the most important concept of functional analysis named after him, Banach space . The introduction of category theory by Samuel Eilenberg and Saunders Mac Lane finally created an even higher level of abstraction, a common framework for viewing similar constructions from different areas of mathematics .
Content and methodology
Contents and sub-areas
The following list gives an initial chronological overview of the breadth of mathematical topics:
- calculating with numbers ( arithmetic - antiquity ),
- the investigation of figures ( geometry - antiquity , Euclid ),
- solving equations ( algebra - antiquity , medieval and renaissance , Tartaglia ),
- the investigation of the correct conclusions ( logic - Aristotle ) (partly only included in philosophy, but often also counted as mathematics)
- Investigations on divisibility ( number theory - Euclid, Diophant , Fermat , Euler , Gauß , Riemann ),
- the computational recording of spatial relationships ( analytical geometry - Descartes , 17th century),
- calculating with probabilities ( probability theory - Pascal , Jakob Bernoulli , Laplace , 17th - 19th centuries ),
- the investigation of functions, in particular their growth, curvature, behavior at infinity and the area under the curves ( Analysis - Newton , Leibniz , end of the 17th century),
- the description of physical fields ( differential equations , partial differential equations , vector analysis - Euler , the Bernoullis , Laplace , Gauss, Poisson , Fourier , Green , Stokes , Hilbert , 18th - 19th centuries),
- the perfecting of analysis by including 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 paradoxes of the infinite ( set theory and mathematical logic - Cantor , Frege , Russell , Zermelo , Fraenkel , early 20th century),
- the constant deformation of geometric bodies ( topology - Cantor, Poincaré , Fréchet , Hausdorff , Kuratowski , early 20th century),
- the investigation of structures and theories ( universal algebra , category theory ),
- the collection and analysis of data ( mathematical statistics ).
- discrete finite or countably infinite structures ( discrete mathematics , combinatorics , graph theory - Euler, Cayley , Kőnig , Tutte ) with close connections to computer science .
A distinction is also made between pure mathematics, also known as theoretical mathematics , which does not deal with non-mathematical applications, and applied mathematics such as actuarial mathematics and cryptology . The transitions between the areas just mentioned are fluid.
Progress through problem solving
Another characteristic of mathematics is the way in which it progresses through the processing of problems that are “actually too difficult”.
Once a primary school that adding learned natural numbers, he is able to understand the following question and answer by trial and error: "Which number you have to add up to 3 to get to 5" But the systematic solution of these tasks requires the introduction of a new concept: subtraction. The question can then be rephrased to: “What is 5 minus 3?” But as soon as the subtraction is defined, one can also ask the question: “What is 3 minus 5?”, Which refers to a negative number and thus already on elementary school mathematics leads out.
Just as in this elementary example of individual learning, mathematics has also advanced in its history: at every level achieved, it is possible to set well-defined tasks, the solution of which requires much more sophisticated means. Often many centuries have passed between the formulation of a problem and its solution and finally a completely new sub-area has been established with problem solving: In the 17th century, infinitesimal calculus could solve problems that had been open since ancient times.
Even a negative answer, the proof of the unsolvability of a problem, can advance mathematics: for example, group theory emerged from failed attempts to solve algebraic equations.
Axiomatic formulation and language
Since the end of the 19th century, and occasionally since antiquity , mathematics has been presented in the form of theories that begin with statements that are considered to be true; further true statements are then derived from this. This derivation takes place according to precisely defined final rules . The statements with which the theory begins are called axioms , the statements derived from them are called propositions . The derivation itself is a proof of the theorem. In practice, definitions still play a role; they introduce and specify mathematical terms by reducing them to more fundamental ones. Because of this structure of the mathematical theories, they are called axiomatic theories.
Usually one demands from axioms of a theory that they are free of contradictions, i.e. that a proposition and the negation of this proposition are not true at the same time. However, this consistency itself cannot generally be proven within a mathematical theory (this depends on the axioms used). As a result, the consistency of the Zermelo-Fraenkel set theory , which is fundamental to modern mathematics, cannot be proven without the aid of further assumptions.
The subjects covered by these theories are abstract mathematical structures that are also defined by axioms. While in the other sciences the objects treated are given and then the methods for examining these objects are created, in mathematics the other way round, the method is given and the objects that can be examined are only created afterwards. In this way, mathematics always occupies a special position among the sciences.
The further development of mathematics, on the other hand, happened and often happens through collections of propositions, proofs and definitions that are not structured axiomatically but are primarily shaped by the intuition and experience of the mathematicians involved. The conversion into an axiomatic theory only takes place later, when other mathematicians deal with the not so new ideas.
Around 1930 Kurt Gödel showed the incompleteness theorem named after him , which says that in every axiom system of classical logic that allows certain statements about natural numbers to be proven, there are either statements that are just as unprovable as their negation, or the system itself is contradicting itself.
Mathematics uses a very compact language to describe facts, which is based on technical terms and above all formulas. A representation of the characters used in the formulas can be found in the list of mathematical symbols . A specialty of the mathematical terminology consists in the formation of adjectives derived from mathematicians' names like Pythagorean , Euclidean, Eulerian , Abelian , Noetherian and Artinsch .
Mathematics is applicable in all sciences that are sufficiently formalized . This results in a close interplay with applications in empirical sciences. For many centuries, mathematics has taken inspiration from astronomy , geodesy , physics and economics and, conversely, has provided the basis for the progress of these subjects. For example, Newton developed calculus to mathematically grasp the physical concept “force equals change in momentum”. Solow developed an economic model of the growth of an economy, which forms the basis of the neoclassical growth theory to this day. While studying the wave equation, Fourier laid the foundation for the modern concept of function and Gauss developed the method of least squares and systematized the solving of linear systems of equations as part of his work with astronomy and land surveying . The statistics ubiquitous today emerged from the initial study of gambling.
Conversely, mathematicians have sometimes developed theories that only later found surprising practical applications. For example, the theory of complex numbers for the mathematical representation of electromagnetism, which emerged in the 16th century, has now become indispensable. Another example is the tensor differential forms calculus, the Einstein of the mathematical formulation of general relativity had used. Furthermore, dealing with number theory was for a long time considered an intellectual gimmick with no practical use, without which modern cryptography and its diverse applications on the Internet would be inconceivable today .
Relationship to other sciences
Categorization of mathematics
The question of which category of science mathematics belongs to has been a controversial issue for a long time.
Many mathematical questions and concepts are motivated by questions relating to nature, for example from physics or engineering , and mathematics is used as an auxiliary science in almost all natural sciences. However, it is not itself a natural science in the strict sense, as its statements do not depend on experiments or observations. Nevertheless, in the more recent philosophy of mathematics it is assumed that the methodology of mathematics corresponds more and more to that of natural science. Following Imre Lakatos , a “renaissance of empiricism” is assumed, according to which mathematicians also put forward hypotheses and seek confirmations for them.
Mathematics has methodical and content-related similarities with philosophy ; for example, logic is an area of overlap between the two sciences. Mathematics could thus be counted among the humanities , but the classification of philosophy is also controversial.
At German universities mathematics belongs mostly to the same faculty as the natural sciences, and so mathematicians, after the promotion is usually the academic degree of Dr. rer. nat. (Doctor of Science) awarded. In contrast to this, in the English-speaking world, university graduates achieve the title “Bachelor of Arts” or “Master of Arts”, which are actually awarded to humanities scholars.
Special role among the sciences
Mathematics plays a special role among the sciences with regard to the validity of its findings and the rigor of its methods. For example, while all scientific findings can be falsified by new experiments and are therefore in principle provisional, mathematical statements are produced apart by pure thought operations or are reduced to one another and do not need to be empirically verifiable. For this, however, a strictly logical proof must be found for mathematical knowledge before it is recognized as a mathematical proposition . In this sense, mathematical theorems are in principle final and universal truths, so that mathematics can be regarded as the exact science. For many people it is precisely this exactness that is so fascinating about mathematics. David Hilbert said at the International Congress of Mathematicians in Paris in 1900:
“We shall briefly discuss what justifiable general demands are to be made on the solution of a mathematical problem: I mean above all that the correctness of the answer can be demonstrated by a finite number of inferences on the basis of a finite number of Prerequisites which lie in the problem and which must be precisely formulated each time. This requirement of logical deduction by means of a finite number of inferences is nothing other than the requirement of rigor in the argumentation. Indeed, the requirement of rigor, which, as is well known, has become of proverbial importance in mathematics, corresponds to a general philosophical need of our understanding, and on the other hand, it is only through its fulfillment that the conceptual content and the fruitfulness of the problem come into full effect. A new problem, especially if it comes from the external world, is like a young rice, which only thrives and bears fruit if it is grafted carefully and according to the strict rules of the gardener onto the old trunk, the safe possession of our mathematical knowledge becomes."
"But I maintain that in any particular theory of nature only so much real science can be found as there is mathematics in it."
Mathematics is therefore also a cumulative science. Today we know more than 2000 mathematical journals. However, this also harbors a risk: newer mathematical areas make older areas in the background. In addition to very general statements, there are also very special statements for which no real generalization is known. Donald E. Knuth writes in the foreword of his book Concrete Mathematics:
“The course title 'Concrete Mathematics' was originally intended as an antidote to 'Abstract Mathematics', since concrete classical results were 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 that the rest of mathematics was inferior and no longer worthy of attention. The goal of generalization had become so fashionable that 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 was becoming inbred and losing touch with reality; mathematical education needed a concrete counterweight in order to restore a healthy balance. "
“The title of the event 'Concrete Mathematics' was originally intended as a counterpoint to 'Abstract Mathematics', because concrete, classic achievements were quickly removed from the curriculum by a new wave of abstract ideas - commonly called 'New Math' flushed. Abstract math is a wonderful thing that has nothing wrong with: it's beautiful, general, and useful. But their followers mistakenly believed that the rest of the math was inferior and irrelevant. The goal of generalization became so fashionable that an entire generation of mathematicians was no longer able to recognize beauty in particular, to challenge the solution of quantitative problems, or to appreciate the value of mathematical techniques. Abstract mathematics revolved around itself and lost contact with reality; In mathematics training, a concrete counterweight was necessary to restore a stable equilibrium. "
The older mathematical literature is therefore of particular importance.
The mathematician Claus Peter Ortlieb criticizes the - in his opinion - insufficiently reflected application of modern mathematics:
“You have to be aware that there are limits to how mathematics can capture the world. The assumption that it works solely according to mathematical laws leads to the fact that one only looks for these laws. Of course I will also find it in the natural sciences, but I have to be aware that I am looking at the world through glasses that block out large parts from the start. [...] The mathematical method has long been adopted by scientists from almost all disciplines and is used in all possible areas where it actually has no place. [...] Numbers are always questionable when they lead to normalization, although no one can understand how the numbers came about. "
Mathematics in Society
Mathematics as a school subject
Mathematics plays an important role as a compulsory subject in school . Mathematics didactics is the science that deals with teaching and learning mathematics. The lower level is primarily about learning arithmetic skills. In the upper level, differential and integral calculus as well as analytical geometry / linear algebra are introduced and stochastics are continued.
Mathematics as a subject and a profession
People who are professionally involved in the development and application of mathematics are called mathematicians .
In addition to the study of mathematics to graduate can put where you its focus on pure and / or applied mathematics, have recently been more interdisciplinary courses such as industrial mathematics , business mathematics , computer mathematics or biomathematics been established. Furthermore, teaching at secondary schools and universities is an important mathematical profession. At German universities, the diploma is now also being converted to Bachelor / Master degree programs. Budding computer scientists , chemists , biologists , physicists , geologists and engineers must also attend a certain number of hours per week.
The most common employers for graduated mathematicians are insurance companies , banks and management consultancies , especially in the area of mathematical financial models and consulting, but also in the IT area. 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 this kind in Germany is the Mathematisch-Physikalische Salon in Dresden, founded in 1728 . The Arithmeum in Bonn at the Institute for Discrete Mathematics there goes back to the 1970s and is based on the collection of computing devices of the mathematician Bernhard Korte . The Heinz Nixdorf MuseumsForum (abbreviation "HNF") in Paderborn is the largest German museum for the development of computing technology (especially computers), and the Mathematikum in Gießen was founded in 2002 by Albrecht Beutelspacher and is continuously being developed by him. The Math.space , directed by Rudolf Taschner , is located in the Museumsquartier in Vienna and shows mathematics in the context of culture and civilization.
In addition, numerous special collections are housed at universities, but also in more comprehensive collections such as the Deutsches Museum in Munich or the Museum for the History of Technology in Berlin (computer developed and built by Konrad Zuse ).
Aphorisms about math and mathematicians
The following aphorisms by well-known personalities can be found:
- Albert Einstein : Mathematics deals exclusively with the relationships between concepts, regardless of their relation to experience.
- Galileo Galilei : Mathematics is the alphabet that God used to describe the universe.
- Johann Wolfgang von Goethe : Mathematicians are a kind of French: if you speak to them, they translate it into their language, and then it is immediately something completely different.
- Godfrey Harold Hardy : The mathematician is a maker of schemes.
- David Hilbert : Nobody should be able to drive us out of the paradise that Cantor created for us.
- Novalis : All of mathematics is actually an equation for the rest of the sciences.
- Friedrich Nietzsche : We want to drive the subtlety and rigor of mathematics into all sciences as far as this is at all possible; not in the belief that we will know things in this way, but in order to determine our human relation to things. Mathematics is only the means of general and ultimate knowledge of human nature.
- Bertrand Russell : Mathematics is the science of which you do not know what you are talking about or whether what you are saying is true.
- Friedrich Schlegel : Mathematics is, as it were, a sensual logic; it relates to philosophy like the material arts, music and sculpture, to poetry.
- James Joseph Sylvester : Mathematics is the music of reason.
- Ludwig Wittgenstein : Mathematics is a method of logic.
- Richard Courant , Herbert Robbins: What is Mathematics? Springer-Verlag, Berlin / Heidelberg 2000, ISBN 3-540-63777-X .
- Georg Glaeser: The mathematical toolbox. Elsevier - Spektrum Akademischer Verlag, Munich, Heidelberg 2004, ISBN 3-8274-1485-7 .
- Timothy Gowers : Mathematics. German first edition, translated from English by Jürgen Schröder, Reclam-Verlag, Stuttgart 2011, ISBN 978-3-15-018706-7 .
- Hans Kaiser, Wilfried Nöbauer: History of Mathematics. 2nd Edition. Oldenbourg, Munich 1999, ISBN 3-486-11595-2 .
- Mario Livio : Is God a Mathematician? Why the book of nature is written in the language of mathematics. CH Beck Verlag, Munich 2010, ISBN 978-3-406-60595-6 .
- Timothy Gowers (Ed.), June Barrow-Green (Ed.), Imre Leader (Ed.): The Princeton Companion to Mathematics . Princeton University Press 2008 (Introductory Encyclopedia)
- Portals and knowledge databases
- Link catalog on mathematics at curlie.org (formerly DMOZ )
- MadiPedia (Society for Mathematics Didactics)
- Mathe-Online.at - mathematical backgrounds and lexicon
- Mathematik.de - DMV portal for mathematics with a wide range of content
- math.space - founded by Rudolf Taschner in the Museumsquartier in Vienna
- Solve Wolframalpha , formulas and tasks online
- Mathworld.Wolfram.com - extensive math resource , engl.
- Zentralblatt für Mathematik: MATH database
- Specialized information service mathematics
- School math
- Collection of professional learning videos for use in mathematics lessons at high school using new media and technologies (e.g. GeoGebra )
- Mathe1.de - school knowledge of grades 1–11
- Mathematics summary (of school mathematics) - overview of many topics of mathematics in the form of a 1 m × 1 m coordinate system (PDF)
- Mathematics in ZUM-Wiki.de - mathematics for teachers
- thema-mathematik.at - Mathematics knowledge of the AHS upper level (grades 9-12)
- Ethnomathematics ( Spectrum of Science - Special Issue 2/2006)
- "Women in the History of Mathematics" (Lecture slides Prof. Blunck, University of Hamburg)
- Images of Some Famous Mathematical Works (images of famous mathematical works)
- Sumerian based math
- Testimonials from mathematicians
- Austrian pronunciation database , http://www.aussprache.at
- Helmut Hasse : Mathematics as humanities and means of thinking in the exact natural sciences . In: Studium generale . tape 6 , 1953, pp. 392-398 (on- line ). online ( Memento from April 25, 2013 in the Internet Archive )
- David Hilbert. Mathematical Problems ( January 19, 2012 memento in the Internet Archive ). Lecture given at the international mathematicians' congress in Paris 1900.
- Oliver Link: The world cannot be calculated. Interview with Claus Peter Ortlieb, brand eins 11/2011, accessed on January 1, 2012.
- Lothar Schmidt : Aphorisms from A – Z. The great handbook of winged definitions . Drei Lilien Verlag, Wiesbaden 1980, p. 288-289 . (Lothar Schmidt holds a degree in economics and taught political science at the Johann Wolfgang Goethe University in Frankfurt am Main .)