Math object

As mathematical objects are abstract objects referred to in the various areas of mathematics are described and analyzed. Basic examples are numbers , sets and geometrical bodies , followed by graphs , integrals and cohomologies . Questions about the existence and nature of mathematical objects are central to the philosophy of mathematics . Contemporary mathematics, on the other hand, leaves out these questions and deals with them internally . This includes areas such as set theory , predicate logic , model theory and category theory , in which the (otherwise superordinate) mathematical structures such as axioms , inference rules and proofs are explored, which thus themselves become mathematical objects. Views of what mathematical objects are have changed dramatically over the course of the history of mathematics .

history

The first objects of mathematical considerations were numbers and geometrical figures . Even mathematics in ancient Egypt and Babylonian mathematics calculated with natural numbers and positive fractions and could thus solve simpler equations . However, the Pythagoreans already established that there are also incommensurable numerical relationships , but they have not yet been able to quantify them. Until the 19th century there was great uncertainty in mathematics when calculating with infinitesimal quantities , which only changed from the middle of the 19th century through Karl Weierstrass . Today, several ways of constructing real numbers from rational numbers are known. In addition to real numbers, complex numbers and quaternions are of practical importance.

Euclid (approx. 300 BC) first established some properties of geometric objects, such as points , straight lines and triangles , in his Euclidean geometry through postulates, comparable to today's axioms . However, a complete and consistent axiomatization of the geometry was only achieved by David Hilbert in 1899. In the second half of the 19th century Georg Cantor developed his set theory , with which mathematical objects can be described as elements of sets , whereby these elements can also be sets:

Elements are certain well-differentiated objects of our intuition or our thinking.

He defined the term class a little further , whereby real classes like the all-class no longer represent sets. The naive set theory , however, was not free of contradictions; the best-known paradox is Russell's antinomy . The axiomatization of set theory was only completed by Ernst Zermelo and Abraham Adolf Fraenkel in the 1920s with the Zermelo-Fraenkel set theory .

In the constructive mathematics of the 20th century, it was required that mathematical objects must be constructible. In the fundamental crisis of mathematics in the 1920s and 1930s, however, formalism prevailed over intuitionism . More important than the mathematical objects themselves are their relationships to one another, which are determined by axioms. These axioms, not the objects themselves, are the basis of modern mathematical theories, Hilbert is said to have once said:

"Instead of 'points, straight lines, levels ', you have to be able to say ' tables, chairs, beer mugs '."

Relation to formal systems for the foundation of mathematics

From the formalistic point of view , mathematics always works in formal systems . Influenced by this, it has become a requirement of modern mathematics that sentences that are set up in mathematics must be able to be understood, at least in principle, as sentences of a formal system. In order for them to be considered valid, they must be recognized as provable in this formal system, regardless of the extent to which the system is to be regarded as fundamental from a philosophical point of view. The most widespread such systems for laying the foundations of mathematics are those based on classical first-order predicate logic (in comparison to other logics based). Such work with variables , which are freely selectable symbols (in the sense of a token , not in the sense of a carrier of meaning ) that can be used in the formal system in a special way. This is similar to intuitive notions of denoting objects . For example, a formal expression of the form is read as exactly one such as ... Once an expression of this form has been proven, it can also be combined in certain ways with other expressions in which this can be used, and one speaks of a definition of the object . The decisive factor for the acceptance of a mathematical statement that makes use of such variables is not a reference to any objects, whatever they may be, but only their correct use within the formal system. ${\ displaystyle \ exists! x \ ldots}$${\ displaystyle x}$${\ displaystyle x}$${\ displaystyle x}$

In order to get a rich system with predicate logic in which most of the known mathematics can be carried out, one can design the system with predicates and axioms . The most widespread are different approaches, which are referred to as set theoretical foundations . They introduce the element relation into the formal system . Instead of objects in the above sense one speaks of sets and reads as the set is an element of the set . Certain axioms guarantee a diverse handling, that is, diverse possible proofs and thus u. a. also various possible definitions in the above sense. The most common choice of such an axiom system is the Zermelo-Fraenkel set theory with axiom of choice (ZFC). In mathematical parlance it happens that, despite a foundation by ZFC, one speaks of "objects" which behave similarly to the so-called sets in non-formal paraphrases, but of which it turns out that they cannot be formalized in the same way as So-called sets can be associated with variables, since when attempting such a formalization, taking into account the desired properties, contradictions with the axioms arise. One then speaks of a real class . This can also be called a mathematical object , but not a quantity , this word is reserved for the above narrower understanding. Axiom systems also exist, such as Neumann-Bernays-Gödel set theory and Ackermann set theory , which allow the concept of a real class to be formalized, with real classes then also becoming mathematical objects in the narrower sense described above. ${\ displaystyle \ in}$${\ displaystyle x \ in y}$${\ displaystyle x}$${\ displaystyle y}$