Mathematical object
As mathematical objects, the abstract objects are known, which are described in the various areas of mathematics and studied. Basic examples are numbers, quantities and geometrical body, outgoing, for example, graphs, integrals and Kohomologien. The questions about the existence and nature of mathematical objects are central in the philosophy of mathematics. The contemporary mathematics, however, excluded from these issues and deal with them within structurally. This includes areas such as set theory, predicate logic, model theory and category theory with one in which the (otherwise superior ) mathematical structures such as axioms, inference rules and proofs are investigated, which are thus themselves to mathematical objects. The views on what mathematical objects are, have changed greatly over the history of mathematics.
History
The first objects of mathematical considerations were numbers and geometrical figures. Even the Mathematics in Ancient Egypt and the Babylonian mathematics reckoned with numbers and was able to solve simple equations. Even the Pythagoreans presented, however, that there are both fractions also incommensurable ratios, they could not yet be quantified but these figures. Until the 19th century there was great uncertainty in mathematics in calculations with infinitesimals, which only changed after the mid -19th century by Karl Weierstrass. Today, several construction possibilities of the real numbers from the rational numbers are known. Over the real numbers are also complex numbers and quaternions of practical importance.
Euclid (about 300 BC ) the first time put some properties of geometric objects such as point, line and triangle in Euclidean geometry by its postulates, axioms similar to today's fixed. However, a complete and consistent axiomatization of geometry was not until David Hilbert 1899. Developed in the second half of the 19th century, Georg Cantor 's set theory, with the mathematical objects can be described as sets. A little further on he took the concept of class, real class as the universal class no longer represent quantities. However, the naive set theory was not consistent, probably the most famous paradox is the Russell's antinomy. The axiomatization of set theory was only completed by Ernst Zermelo and Abraham Fraenkel Adolf in the 1920s with the Zermelo -Fraenkel set theory.
In constructive mathematics of the 20th century demanded that mathematical objects must be constructible. In the foundational crisis of mathematics in the 1920s and 1930s, however, the formalism compared to the intuitionism prevailed. More important than the mathematical objects themselves are therefore their inter-relationships, which are defined by axioms. These axioms, not the objects themselves, form the basis of modern mathematical theories dar.
Respect to formal systems for foundations of mathematics
The formalist point of view, according to the math always works in formal systems. Influenced by it has become a claim of modern mathematics that records that are placed in mathematics, can be at least prinizipiell construed as a set of a formal system must. Thus they are considered valid, it must be recognized in this formal system as provable, regardless of how the system from a philosophical point of view is to be considered as basal. The most widely used such systems for foundations of mathematics are based on the classical first-order logic ( based on compared to other logics ). Such work with variables, which are arbitrary symbols (in the sense of a token, not in the sense of a symbol for something ) that can be used in the formal system in a special way. This way, it is similar to intuitive ideas about that they denote objects. For example, a formal expression of the form is as there is exactly one, so ... read. If you have demonstrated an expression of this form, it can also be combined in certain ways with other expressions in which that can be used, and one speaks of a definition of the object. The acceptance of a mathematical statement that makes use of such a variable, so is not a reference to any objects, whatever they may be, but only the correct use within the formal system.
To get to the predicate logic a rich system in which most of the known mathematics can be operated, you can SHapINg the system predicates and axioms. The most common are different approaches that are referred to as set-theoretic Foundations. Lead you into the formal system, the element relation. Instead of objects in the above sense we speak of quantities and reads as the set is an element of the set. Certain axioms guarantee a varied approach, meaning a variety of possible evidence and thus, inter alia, also multiple possible definitions in the above sense. The most common choice of such axiom system is the Zermelo -Fraenkel set theory with the axiom of choice ( ZFC ). In mathematical parlance, it happens that in spite of a foundation by ZFC of "objects " talks, which behave similarly in non-formal descriptions, such as the so-called volumes, of which, however, it turns out that it is impossible to formalize in the same way as known quantities with variables can be set in conjunction, because when such an attempt formalization taking into account the desired properties contradictions to the axioms arise. We then speak of a real class. This can also be called mathematical object, but not a lot, this word is reserved for above closer view. There are also systems of axioms, such as the Neumann - Bernays - Gödel set theory and the Ackermann set theory, which allow a formalization of the concept of a real class with real classes then also to mathematical objects in the above strict sense.