The Revolutionary Image Of Mathematics

The cognition in the traditional view of mathematics to the cumulativity of mathematics include the cumulativity of the discovery and proof in mathematics, as well as the cognition of the experience-freedom of mathematics, following the attempt to universe the mathematics and illustrate the truth of mathematics via a construction of some foundation. But viewing from the history of mathematics, the above-mentioned efforts all failed, so a correction to the traditional view is needed. Inspired by the Historism in the philosophy of science, someone in the academia realized that "revolution" does not contradict with the nature of mathematics, and in mathematics where processes of revolution which is assimilated with the scientific revolutions is not only logically possible but historically really exist. Accordingly, after the introduction of the Holism with some hierarchy to mathematics, a meaning of revolutions which has a similar characteristic, namely a holistic meaning of revolution, with the scientific revolutions of mathematic s in narrow sense could be established, in addition, owing to B. Cohen’s semantic and historical research to the word "revolution", a kind of mathematical revolutions including some evaluation of value in broad sense could be introduced, so a classification of the revolutions in mathematics could be achieved. To analysis the revolutions in mathematics in narrow sense, the Kuhn’s concepts of "paradigm" and "anomaly" must be introduced to mathematics, then on the basic of predecessors’ research, and combining with the frontier which includes the "Reverse mathematics" and the "Godel Hierarchy" in the contemporary mathematics, we can find that the paradigms in mathematics have their own hierarchy, and due to the existence of this hierarchy, the three types of mathematical anomalies——including new concepts which have contradiction with the view in meta-mathematics and the set of basic concepts&axioms, and also consist some basic and important propositions whose truth-value couldn’t be judged in a given axiomatic system, and in addition to the above, mathematical paradoxes--whose influence have their own impact areas, vitally, with respect to the anomalies in science, the anomalies in mathematics often occur in the kernel parts, and those above could be interpreted via historical materials. The transformations in mathematics replying the three types of mathematical anomalies constructed the three types of revolutions in narrow sense, namely, the revolutions in concepts, a reconstruction of new mathematical theories paralleling to the preceding theories and the adjustments to solve paradoxes; but the revolutions in mathematics which in a broad sense contains some value judgment, and generally speaking, this evaluation is evaluated by the impact of the forces or tendencies which are parts of the "Dynamic factors" impelling the development of mathematics, including vital mathematical methods and mathematical problems leading the mean trend of the research, then the deep roots which caused the revolutions in broad sense lie in the historical limitation of the theories and methods in the preceding mathematics; from the investigation to the revolutions in broad sense also illustrate that the analysis to the paradigms statically is insufficient, but an analysis to the hidden "Dynamic factors" which impelled the evolution of it is needed, and these are the complement to the connotation of earlier concept of "mathematical paradigms". Finally, on the basis of the investigation to the revolutions in mathematics, we could re-examine and rethink the nature of mathematics which containing recognition to the objects researched in mathematics and a cognition to the historic in mathematical proof, and these are what the answer to the question "How the revolutions in mathematics in narrow sense could be possible?"; while for mathematics as a whole, we should regard it as "a quasi-empirical system which means an open deductive system", and the openness as well as the deep connection to experiences are precisely the vita of mathematics.