Can I do actuarial science without math?
The mathematics (Greek μαθηματική τέχνη mathēmatikē téchnē: "The art of learning, part of learning"; colloquialmaths) is the science that emerged from studying figures and calculating with numbers. For mathematics there is no generally accepted definition; today it is usually described as a science that examines self-created abstract structures for their properties and patterns.
Mathematics is one of the oldest sciences. It experienced its first heyday before antiquity in Mesopotamia, India and China. 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, dates. 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 and René Descartes opened up a computational approach to geometry through the use of coordinates. The description of tangents and the determination of 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. For its treatment, 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.
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, through the paradoxes of the naive concept of set, it initially made clear the uncertain foundation on which mathematics stood before.
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. Emmy Noether developed the foundations of modern algebra, Felix Hausdorff the general topology as the investigation of topological spaces, Stefan Banach the most important concept of functional analysis, the Banach space named after him. An even higher level of abstraction, a common framework for viewing similar constructions from different areas of mathematics, was ultimately created by the introduction of category theory by Samuel Eilenberg and Saunders Mac Lane.
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),
- the investigation of figures (geometry - pre-classical civilizations, Euclid),
- the investigation of the correct conclusions (logic - Aristotle) (partly only part of philosophy, but often also part of mathematics)
- solving equations (Algebra - Tartaglia, Middle Ages and Renaissance),
- Investigations on divisibility (number theory - Euclid, Diophant, Fermat, Leonhard Euler, Gauß, Riemann),
- the computational recording of spatial relationships (analytical geometry - Descartes, 17th century),
- calculating with probabilities (stochastics - Pascal, Jakob Bernoulli, Laplace, 17th - 19th centuries),
- the investigation of functions, especially 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 - Leonhard Euler, the Bernoullis, Laplace, Gauss, Poisson, Fourier, Green, Stokes, Hilbert, 18th - 19th centuries),
- perfecting 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 again logic - Cantor, Frege, Russell, Zermelo, Fraenkel, early 20th century),
- the investigation of structures and theories (category theory).
A little off the beaten track in this list is numerical mathematics, which provides algorithms for solving concrete continuous problems from many of the above-mentioned areas and examines them.
A distinction is also made between pure mathematics, too theoretical math which does not deal with extra-mathematical applications, such as those used by the Briton Andrew Wiles and the German Gerd Faltings, 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 an elementary school student has learned to add natural numbers, they will be able to understand the following question and answer it through trial and error: "What number do you have to add to 3 to get 5?" The systematic solution of such tasks, however, 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 leads to a negative number and thus already beyond elementary school mathematics.
Just as in this elementary example of individual learning, mathematics has also advanced in its history: at every level reached, it is possible to set well-defined tasks, the solution of which requires much more sophisticated means. Many centuries have passed between the formulation of a problem and its solution, and finally a completely new sub-area has been established with the problem-solving process: In the 17th century, for example, infinitesimal calculus was able to 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 Axiomsthat are derived from it are called sentences. The derivation itself is a proof of the sentence. In practice they still play Definitions a role, but they are part of the logic tools that are assumed. Because of this structure of the mathematical theories, they are known as axiomatic theories.
Usually one demands that axioms of a theory are free of contradictions, that is, 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, for example, 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 dealt with 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 with it 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.
However, there are also limits to the axiomatization of mathematics. Kurt Gödel showed around 1930 in the incompleteness theorem named after him that either true but not provable statements exist in every mathematical system of axioms, or that the system is contradictory.
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 table of mathematical symbols. A specialty of the mathematical terminology consists in the formation of adjectives derived from mathematicians' names such as 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. Over 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 infinitesimal calculus in order to mathematically grasp the physical concept “force equals change in momentum”. 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.
Conversely, mathematicians have sometimes developed theories that only later found surprising practical applications. For example, the theory of complex numbers, which emerged as early as the 16th century, has now become indispensable for the mathematical representation of electromagnetism, or Boolean algebra is widely used in digital technology and electrical control technology for machines and systems. Another example is the tensor calculus of differential forms, without which the general theory of relativity could not be mathematically formulated. Furthermore, dealing with number theory was long considered an intellectual gimmick with no practical use, without which modern cryptography and its diverse applications on the Internet would be inconceivable today.
See also: Applied Mathematics
Relationship to other sciences
Categorization of mathematics
The question of which category of science mathematics belongs to has been the subject of controversy 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 in the broader sense, but the classification of philosophy is also controversial.
For these reasons, too, some categorize mathematics - alongside other disciplines such as computer science - as structural science or formal science.
At German universities, mathematics usually belongs to the same faculty as the natural sciences, and so mathematicians are usually awarded the academic degree of Dr. rer. nat. (Doctor of Science) awarded. In contrast, 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 from one another through pure thought operations or reduced to one another and need not be empirically verifiable. For this, however, a strictly logical proof must be found for mathematical knowledge before it can be recognized as a mathematical proposition. In this sense, mathematical propositions are in principle final and generally valid truths, so that mathematics as the exact science can be considered. It is precisely this exactness that is so fascinating about mathematics for many people. This is how David Hilbert said at the International Congress of Mathematicians in Paris in 1900:
“We shall briefly discuss which justified general demands are to be made on the solution of a mathematical problem: I mean above all that it is possible to demonstrate the correctness of the answer by a finite number of inferences, namely 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 is known to have 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 intellectual 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."
Joseph Weizenbaum of the Massachusetts Institute of Technology called mathematics the mother of all sciences.
"I maintain, however, that in any particular theory of nature there can only be found as much actual science as there is mathematics to be found in it."
- Immanuel Kant: Metaphysical Beginnings of Natural Science, A VIII - (1786)
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 into the background. In addition to very general statements, there are also very special statements for which no real generalization is known. Donald Ervin 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 known as “New Math” Abstract math is a wonderful thing that has nothing wrong with: it's beautiful, general, and useful. But its adherents have misconstrued that the rest of the math is inferior and irrelevant. The goal of generalization got so in It was fashionable that an entire generation of mathematicians was no longer able to recognize beauty in particular, to see the solution of quantitative problems as a challenge, or to appreciate the value of mathematical techniques. Abstract mathematics only revolved around itself and lost it Contact with reality; there was a specific counterpart in mathematics training weight necessary to restore a stable equilibrium.)
The older mathematical literature is therefore of particular importance.
Mathematics in Society
Mathematical skills are indeed a characteristic feature of humans, but bonobos and some other animal species are also capable to a limited extent of simple mathematical achievements (see also: Differentiation of quantities in animals).
Mathematics as a school subject
Mathematics plays an important role as a compulsory subject in school. Mathematics didactics is the science that is concerned with teaching mathematics. In the lower and middle grades, however, the subject of mathematics is mostly limited to learning arithmetic skills. In the upper level, differential and integral calculus are introduced.
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 studying mathematics for a diploma, in which one can focus on pure and / or applied mathematics, more and more interdisciplinary courses such as industrial mathematics, business mathematics, computer mathematics or biomathematics have recently been set up. 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 courses. Budding computer scientists, chemists, biologists, physicists, geologists and engineers also have to take 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 Museum Quarter 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).
- Richard Courant, Herbert Robbins: What is math? Springer-Verlag, Berlin / Heidelberg 2000, ISBN 3-540-63777-X.
- Georg Glaeser: The math toolbox. Elsevier - Spektrum Akademischer Verlag, 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. C. H. Beck Verlag, Munich 2010, ISBN 978-3-406-60595-6.
Portal: mathematics - Overview of Wikipedia content on mathematics
- Portals and knowledge databases
- School math
- ↑ David Hilbert. Math problems. Lecture given at the International Congress of Mathematicians in Paris in 1900
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