Basics of Mathematics

from Wikipedia, the free encyclopedia

The basics of mathematics are on the one hand part of mathematics , on the other hand they form an important subject of epistemological reflection when it deals with the general principles of human knowledge acquisition. Extent that such mathematics philosophical reflections in the history repeatedly mathematics have influenced the formulation of the fundamentals, they are not only part of mathematics, but lie in an overlap area with the philosophy .

On the history of the basic questions

If one proceeds from a division of mathematics into arithmetic and geometry - as was customary up to modern times - one can ask the "fundamental question" whether the two parts are independent areas of knowledge or whether one of the two is the more fundamental one the other can be led back.

From the ancient Greeks to modern times

In mathematics from ancient times to the Greeks, the greater clarity of geometry meant that many arithmetic problems were solved on a geometric basis. The Pythagoreans found around 500 BC By placing small stones ("Psephoi") in squares and observing the differences between the resulting squares, they found out the laws of square numbers.

Precisely because the Greeks were more familiar with the geometrical, the numbers represented the greater fascination. The Pythagoreans recognized that the arithmetic world of numbers is more comprehensive than the geometrical world of figures, yes they explained the numbers in the sentence "everything is number" to the basis of things in general. In spite of the practical advantage of geometry, numbers were declared to be the real basis of mathematics in philosophical reflection.

While the numbers were hard to grasp, the mathematical systematization of the fundamentals began with the axiomatization of geometry. The around 300 BC The " elements " of Euclid that were created in BC were to remain the paradigm of the foundation of a scientific discipline par excellence until the end of the 19th century. Undoubtedly, this work could only be written under the influence of the rationalist spirit of Greek philosophy; Euclid himself may even have been a student at Plato's Academy .

Descartes ' introduction of the coordinate system , which made it possible to solve geometric problems in the context of algebraic computing, and the invention of differential calculus by Newton and Leibniz brought about great advances in mathematics at the beginning of the modern era and shifted the weights from geometry to arithmetic. The basics of arithmetic remained just as unclear as those of its new sub-disciplines, algebra and analysis .

Arithmetization

Difficulties and uncertainties arose in the 18th and early 19th centuries, particularly in analysis, which arose from calculating with infinitely small quantities. For a while one could not agree on whether every convergent sequence of continuous functions converges to a continuous function or whether the limit function can also be discontinuous. There was evidence for both claims and it proved very difficult to find an error in either evidence. The need to specify the terms and how to deal with them became obvious. In the 19th century, therefore, a conscious “arithmetization” of analysis began, the unclear concept of the infinitely small number was replaced by the “arbitrarily small number greater than zero”, which was often referred to with the letter . This " epsilontics ", promoted primarily by Cauchy and Weierstrass , which turned analysis into a theory of real numbers, meant a breakthrough in terms of its reliability; What remained was the clarification of the concept of the real number or the set of real numbers - apart from the still distant axiomatization of the theory. This term, now considered to be one of the most important foundations of mathematics, was clarified in the 70s and 80s of the 19th century by Dedekind's definition of the real number as an intersection and Cantor's definition as the equivalence class of convergent sequences, which is still in use today. However, these definitions presuppose a general concept of set and thus also infinite sets - avoiding the use of infinitely small quantities was bought at the cost of infinitely large objects: sets with an infinite number of elements. This earned the definitions mentioned an initial philosophical- constructivist criticism: Kronecker was of the opinion that arithmetic should be taken even further in order to avoid talking about infinite sets. In fact, Cantor's transfinite set theory, like Frege's fundamental laws of arithmetic, was affected by Russell's antinomy , which plunged mathematics into a fundamental crisis at the beginning of the 20th century .

crisis

In the course of this crisis, several mathematical-philosophical positions emerged, of which only their view on the question of a uniform basis for mathematics is presented here:

For logicism , the basis of mathematics is simply logic (where it turned out that the logicists used a rather broad logic term, which in today's sense included set theoretic terms). Logicists should be right, insofar as mathematical reasoning can be represented and understood as purely logical reasoning. The rule systems of logical inference provided by formal logic , of which the first-level predicate logic is the most important, form an important basis of mathematics .

The natural numbers form the basis for intuitionism . Brouwer's approach to analysis, the so-called election sequence theory, can be seen as the implementation of Kronecker's demand for complete arithmetic and foregoing the concept of sets.

For formalism , on the other hand, the basis of mathematics is not a subject area consisting of logical objects or numbers, but the axioms of the theory in which one is currently moving plus predicate logic form the basis. This basis is to be secured by proving the consistency of the axioms. This proof should not be carried out within a formal axiomatic theory, since otherwise it would end up being circular, but within the (intuitively given) finite mathematics of natural numbers, whose consistency cannot be doubted. The natural numbers thus form less the basis of mathematics for formalism than for intuitionism, but rather a superstructure, a meta-mathematics , as the formalist Hilbert called it.

Current situation

The formalistic position has largely established itself academically and has led to new sub-disciplines of mathematics, which deal with the fundamentals from a mathematical point of view and are usually summarized under the designation mathematical logic : set theory , proof theory , recursion theory and model theory .

From the formalistic point of view, the search for the basis of mathematics can only mean finding an axiomatic theory in which all other mathematical theories are contained, in which all concepts of mathematics can be defined and all propositions can be proven. According to a widespread opinion among mathematicians, this basis is found with the axiom system of the Zermelo-Fraenkel set theory . However, other set theories are still being investigated as a possible basis. And the question is still being investigated whether Kronecker's requirement cannot be met after all, whether instead of an expansive set-theoretical basis, a much narrower, purely arithmetic basis might not suffice to build the entire mathematics on it. Proof theory conducts such investigations, while recursion theory essentially examines the superstructure of finite mathematics and provides the finest possible methods with which the proof theorists can then conduct their proofs of consistency. Finally, model theory deals with the question of whether a certain axiomatic theory is stronger than another, whether it provides a “model” for it. So has z. B. the impression of the ancient Greeks confirms that arithmetic is much stronger than geometry: The three-dimensional number space, as Descartes introduced it through his coordinate system, is a model of our geometric space, all theorems of geometry can also be in number space, i.e. computationally-algebraically, prove.

literature

  • Oskar Becker: Fundamentals of mathematics in historical development . Suhrkamp, ​​Frankfurt a. M. 1975
  • LEJ Brouwer : Over de grondslagen der wiskunde ("Fundamentals of Mathematics") 1907
  • David Hilbert / Paul Bernays , Fundamentals of Mathematics, I-II, Berlin / Heidelberg / New York 2nd A. 1970
  • P. Mancosu (ed.): From Hilbert to Brouwer. The Debate on the Foundations of Mathematics in the 1920s , Oxford University Press, Oxford, UK 1998.
  • Christian Thiel : Fundamental crisis and fundamental dispute , study of the normative foundation of the sciences using the example of mathematics and social science, Meisenheim am Glan 1972 ISBN 3-445-00883-3

Web links

Wikibooks: Math for Non-Freaks: Fundamentals of Math  - Learning and Teaching Materials