Present Age Math
The many parts of research of mathematic in 20th century have been continuing to verify analytic basic and structure of the subjects. It made the research about axiomatics.
Many basic concepts of mathematics developed and the basic fields, the set theory, the abstractive algebra and topology progressed. The general set theory was bumped into the paradox demanding a demonstration. So it became to investigate the logicality using to gain the conclusion from the proposition in mathematics. Finally the mathematical logic was born.The relation logic and philosophy developed as a main sect of the various mathematical philosophy of modern.
And the computer Revolution in 20th century influenced many fields of mathematics. Dedekind built up the basic of mathematics inducing the concept called Schmitt. Not only Klein left many results in analysis but also he announced Erlangen Program, sorted the whole of geometry and served as a stepping stone for the new geometry. The research about geometry axiom became the basic of geometry axiomism. The modern mathematics progressing and advencing over and over again as it promoting his research.
Topology occupies a high position as an integraated mathematics in the 20th century. Topology means position and form. Started from Euler's polyhedron theorem, topology was developed through Poincare's algebraic topology and Brouwer's fixed point theorem. Topology is a feld of mathematics in which topological properties were treated. It is a kind of geometry which includes one-to-one correspondence of space, that is to say, mathematicians study topological properties in topology. For example, an early discovered topological property of a simple closed polyhedral surface is the relation v-e+f=2, where v,e,f denote the number of vertices, edges, and faces, respectively of the polyhedral surface. Many theorems including unicursal problem were known.
As the area of mathematics extended to abstraction, so the object of mathematics expanded from concrete spaces to the abstract ones.
These abstract spaces are called topological spaces. New area, called, mathematical analysic processmg was made by the way of topology and algebra. Thus topological algebra and topological space theorem and topological analysis formed it majorities in mathmatics. As the above, topological geometry is tepresentative mathematics in the 20th century. This topological geometry influenced on the many other flelds of mathematics.
The crisis of mathematics was brought about by the discovery of paradoxes in the fringe of Cantor's general theory of sets. Since so much of mathematics is permeated with set concepts and, for that matter, can actually be made to rest upon set theory as a foundation, the discovery of paradoxes in set theory naturally cast into doubt the validity of the whole foundational structure of mathematics.
Bertrand Russell discovered in 1902 paradox depending on nothing more than just the concept of set itself. Let X denote any set. then, the set of all sets that are not member of themselves by N
Now take X to be N, and we have the contradiction
Known of these was given by Russell himself in 1919 and concerns the plight of the barber of a certain village who has enunciated the principle that he shaves all those persons ans only those persons of the village who do not shave themselves. The paradoxical nature of this situation is realized when we try to answer the question, "Does the barber shave himself?" If he does shave himself, then he shouldn't according to his principle; if he doesn't shave himself, then he should according to his principle.
Other attempts to solve the paradoxes of set theory look for the trouble in logic, and it must be admitted that the discovery of the paradoxes in the general theory of sets has brought about a thorough investigation of the foundations of logic
There have arisen three main philosophies, or schools of thought, concerning the foundations of mathematics the so-called logistic, intuitionist, and formalist schools. Naturally, any modern philosophy of the foundations of mathematics must, somehow or other, cope with the present crisis in the foundations of mathematics
Allfred North Whitehead(1861-1947) and Bertrand Russell(1872-1970) duduced natural number system from hypothesis and set of axiem. They, therefore, identified many parts if mathematics with logic.
To avoid the contradictions of set theory. Principia mathematica employs a "theory of types."
For the intuitionists, a set cannot be thought of as a ready-made collection, but must be considered as a law by means of which the elements of the set can be constructed in a step-by-step fashion. This concept of set rules out the possibilty of such contradictory sets as "the set of all sets."
I summarized the changes of eurogean mathematics' history and I hope that all users of this web page will get a good understanding about the characteristics and essence of mathematics.