Foundations of mathematics

The foundations of mathematics on the one hand part of mathematics, on the other hand, they are an important object of epistemological reflection, if this deals with the general principles of human intelligence. In this respect have taken multiple influence on the formulation of the foundations of mathematics mathematics such philosophical reflections in history, they are not only part of mathematics, but lie in a region overlapping with philosophy.

History of the fundamental questions

If we - as far as the modern era usual - a subdivision of mathematics in arithmetic and geometry, so you can make the "fundamental question " whether the two parts are independent of each other knowledge areas or if one of the two is more fundamental to the can be attributed to the other.

From the ancient Greeks to modern times

In mathematics, the ancient Greeks up to the higher clarity of geometry meant that many arithmetic problems were solved on a geometric basis. So found about the Pythagoreans around 500 BC laws of square numbers out by laying small stones ( " Psephoi " ) to squares and considered the differences of the resulting squares.

Just because the Greeks the Geometric was familiar, presented the numbers the greater fascination represents the Pythagoreans realized that the arithmetic world of figures relative to the geometric world of figures is the more comprehensive, so they declared the numbers in the phrase " all is number" the basis of things in general. Despite the practical preference of the geometry so the numbers were declared the very foundation of mathematics in philosophical reflection.

While the numbers were not quite grasp the mathematical systematization of the foundations began with the axiomatization of geometry. The resulting 300 BC "Elements" of Euclid should the paradigm of the foundation of a scientific discipline to stay until the end of the 19th century par excellence. Undoubtedly, this work could only be written under the influence of the rationalistic spirit of Greek philosophy, perhaps even self Euclid was student at Plato's Academy.

Descartes ' introduction of the coordinate system, which enabled the solution of geometric problems within the framework of algebraic computing, as well as the invention of the differential calculus by Newton and Leibniz effected at the beginning of the modern era although much progress in mathematics and moved it towards the weights of the geometry to arithmetic. However, the foundations of arithmetic remained as unclear as that of their new sub-disciplines, algebra and calculus.

Arithmetization

In particular, in the analysis occurred in the 18th and early 19th centuries, difficulties and uncertainties that stemmed from calculations with infinitely small quantities. So you could for a while unable to agree on whether any convergent sequence of continuous functions, in turn, converges to a continuous function, or whether the limit function can also be discontinuous. There were proofs of both assertions and it proved very difficult to find a fault in one of the proofs. Highly visible herein was to clarify the necessity of the concepts and how to deal with them. In the 19th century therefore established a conscious " arithmetization " of analysis, the vague concept of infinitely small number has been replaced by " arbitrarily small number greater than zero ", which was often referred to with the letter. This advanced especially by Cauchy and Weierstrass ' epsilontics "which had become a theory of the real numbers, the calculus, marked a breakthrough for their reliability; what remained was the clarification of the concept of real numbers or the set of real numbers - apart from the still lying in the distance axiomatization of the theory. This term is now regarded as one of the main foundations of mathematics learned his clarification in the 70s and 80s of the 19th century by Dedekind's definition of real number as a cut and Cantor's definition as an equivalence class of convergent sequences, which is still in use today. However, these definitions presuppose a general concept of set and therefore also infinite sets - avoiding the talk of infinitely small quantities was therefore redeemed by means of infinite objects: flat amounts with infinitely many elements. This contributed to the definitions mentioned a first -constructivist philosophical criticism: Kronecker was of the opinion that one must drive the arithmetization further in order to avoid even talking about infinite sets. In fact, Cantor's transfinite set theory was, like Frege's Basic Laws of Arithmetic attacked by Russell's antinomy, which overthrew the math at the beginning of the 20th century in a foundational crisis.

Crisis

In the course of this crisis, several mathematics philosophical positions formed out of which only their view is shown to the question of a uniform basis of mathematics:

For the logicism the basis of mathematics is simply the logic (where it turned out that the logicists used a rather broad term logic, the set-theoretical in the present sense with terms included ). Law should retain the logicists insofar as the mathematical Close can be represented as a pure logical reasoning and understanding. Thus form an important basis of the mathematics provided by the formal logic control systems of logical inference, of which the first-order predicate logic is the most important.

For intuitionism the natural numbers form the basis. Brouwers access to analysis, the so-called selection order theory, can be used as implementation of Kronecker's demand for full arithmetization and abandonment of the concept of a set to see.

For the formalism of the foundation of mathematics, however, is not a subject area, which consists of logical objects or numbers, but the basis form the axioms of the theory in which one moves, plus predicate logic. Hedge this basis by the proof of the consistency of the axioms. This proof should now itself can not be performed within a formal axiomatic theory, since it would otherwise circular at the end, but is within the ( intuitively given ) finite mathematics of natural numbers, not to doubt their consistency. The natural numbers thus form less for the formalism of mathematics as the basis for intuitionism, but rather a superstructure, a meta- mathematics, as the formalist Hilbert called them.

Today's situation

The formalist position has academically largely prevailed and led to new disciplines of mathematics, which treat of mathematical side the basics and are usually grouped together under the name of mathematical logic: set theory, proof theory, recursion theory and model theory.

From the formalist point of view can only mean the search for the basis of mathematics, to find an axiomatic theory in which all other mathematical theories are included in the therefore define all mathematical concepts and can prove all sentences. According to a widespread opinion among mathematicians that foundation is with the axiom system of Zermelo -Fraenkel set theory have been found. There but different amount teachings continue to be investigated as a possible basis. And it continues to be followed up with the question of whether Kronecker demand can not yet meet, if instead of a sweeping set theory not only a much narrower basis arithmetic could be enough to build all of mathematics on it. Such studies lead to proof theory, recursion theory during much studied the superstructure of finite mathematics and fine as possible provides methods by which the proof theorist can then lead their consistency proofs. The model theory finally deals with the question whether a particular axiomatic theory is stronger than another, whether it provides a "model" for this. So, for example, has the impression of the ancient Greeks confirmed that the arithmetic is much stronger than the geometry: The three-dimensional number space, as it Descartes introduced by its coordinate system is a model of our geometrical space, all the propositions of geometry can be also in the number space, so mathematically - algebraically prove.