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.

¡ßTopology

    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,fdenote 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

    (X¡ôN)¡ê(XX)
Now take X to be N, and we have the contradiction

(N¡ôN)¡ê(NN)

    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

¡ÝRussell and Whitegead's LOGICISM:The logistic thesis is that mathematics is a branch of logic.  Rather than being just a tool of mathematics, logic becomes the progenitor of mathematics.  All mathematical concepts are to be formulated in terms of logical concepts, and all theorems of mathematics are to be developed as theorems of logic; the distinction between mathematics and logic become merely one of practical convenience.
    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."

¡ÝBrower's INTUITIONISM:The intuitionist thesis is that mathematics is to be built solely by finite constructive methods in the intuitively given sequence of natural numbers,  According to this view, then, at the very base of mathematics lies a primitive intuition, allied, no doubt, to our temporal sense of before and after, that allows us to conceive a single object, then one more, then one more, and so on endlessly.
    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."

¡ÝHilbert's FORMALISM:The formalist thesis is that mathematics is concerned with formal symbolic systems.  In fact, mathematics is regarded as a collection of such abstract developments, in which the terms are mere symbols and the atatements are formulas involving these symbols; the ultimate base of mathematics does not lie in logic but only in a collectin of prelogical marks or symbols and in a set of operations with these marks.  Since, from this point of view, mathematics is devoid of concrete content and contains only ideal symbolic elements,the establishment of the consistency of the various bravches of mathematics becomes an important and necessary part of the formalist program.   Without such an accompanying consistency proof, the whole study is essentially senseless.  In the formalist thesis, we have the axiomatic development of mathematics pushed to its extreme.
  In his Grundlagen der Geometrie.(1899).
  Hilbert had sharpened the mathematical method from the material axiomatics of Euclid to the formal axiomatics of the present day.  The formalist point of view was developed later by Hilbert to meet the crisis caused by the paradexes of set theory and the challenge to classical mathematics caused by intuitionisric criticism.

¡ßPostscript

    Because the area of 'history of mathematics ' is so wide that I referred to written by Lee Woo-young and Shin Hang-gyun which is the translation of ¡¸An Introduction to the History of Mathematics¡¹by Howard Eves.
    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.