Mathematical rigor#Mathematical rigour

In mathematical rigor ( in a slightly different context often mathematical precision ) is understood to her based sciences a clear logical approach within mathematics and others. On the one hand it includes the axiomatic approach based on sharp definitions and other compelling evidence. Further, the method of systematic deduction is sought. As a consequence, mathematical theorems are basically final and universal truths, so that mathematics can be regarded as an exact science. Mathematical rigor is to enable an end in itself, but a necessary means to lasting progress in mathematics. She is also in the Greek sense a good school of thought. In the after-effect is due to the mathematical rigor and simplification of mathematical considerations.

History

Already found in Greek mathematics, in particular in Euclid in his Elements (late fourth century BC), the first attempts by axiomatization mathematical rigor and systematic mathematical deduction. However, it was in ancient times often less rigorous treatment of mathematics preferable than the Euclidean. It was also clear that the principle of mathematical rigor can not be applied to all sciences. Aristotle writes, " Mathematical rigor is not to require in all things, but rather in the non-material. " After a long period of stagnation began only in the 17th century, a revival of the mathematical sciences with analytic geometry and calculus. However, the Greek ideal of axiomatic and systematic deduction was productive mathematicians of this time a hindrance. The results played a greater role than the way there. A strong intuitive feeling and an almost blind belief in the power of the newly invented methods initially justified this approach. The age of onset of the industrialization supported this form even further. With this confidence still said 1810 Sylvestre Lacroix: " Such sophistry with which abquälten the Greeks, we do not need today."

Only at the beginning of the 19th century, widespread belief in progress was replaced by a new awakening of self-criticism. There was the need to ensure the results and clarity. This process was supported by the French Revolution by a strong popularization of science.

The Disquisitiones Arithmeticae of Carl Friedrich Gauss regarded as one of the first exemplary works of mathematical rigor. It has all the style of Theorem - Proof - Corollary written, contains no motivation of the chosen proof directions carefully and hides the way, as Gauss came to his discoveries. The last aspect, however, is based in part on the requirement of mathematical rigor and not to a specific character of Gauss. It depends with the requirement discussed below for absolute " no redundancy " together.

Through the work of Augustin Louis Cauchy and Karl Weierstrass in particular the calculus was put on a safe and rigorous basis. The 19th century was thus characterized by a successful reflection on the classical ideal of precision and rigor of argument, where the model of the Greek science was exceeded. Even before Cauchy Bernhard Bolzano in 1817 with the work of analytical pure proof of the theorem that delivered between Zwey Werthen that grant an opposite result, lies at least one real root of the equation is an important contribution to the rigorous treatment of analysis.

Quotes

One of the chief proponents of mathematical rigor associated with enormous versatility, David Hilbert was. He formulated at the International Congress of Mathematicians in Paris in 1900:

"We discuss briefly what general requirements must be met by the solution of a mathematical problem: I mean especially that it is possible to demonstrate the correctness of the solution by a finite number of conclusions, on the basis of a finite number of conditions, which are the problem and must be formulated precisely every time. This requirement of logical deduction by means of a finite number of circuits is none other than the requirement of rigor in reasoning. In fact, the requirement of rigor, which is known to become in mathematics proverbial meaning, corresponds to a general philosophical necessity of our understanding, and on the other hand comes through their fulfillment alone just the thought content and the suggestiveness of the problem attain their full effect. A new problem, especially if it comes from the outer world of appearance, is like a young twig, which thrives and bears fruit when grafted onto the old trunk, the established achievements of our mathematical knowledge, carefully and according to the strict horticultural rules will. Moreover, it is a mistake to believe that the rigor in the proof is the enemy of simplicity. Numerous examples we find confirmation on the contrary, that the rigorous method is the simpler and the more easily comprehended at the same time. The very effort for rigor forces us to find out simpler methods of proof it also paves the way for us often methods that are viable than the old methods of lesser severity. "

Alexander Danilovich Alexandrov said:

" Morally teaches mathematics to adhere strictly to act against what is claimed to be the truth, what is brought forth as an argument or what is cited as evidence. The mathematics calls for clarity of concepts and assertions, and tolerates no fog and no unprovable statements. "

Redundancy freedom

The above- indicated personal characteristics of Carl Friedrich Gauss were virtually " internalized " by mathematicians, by implicitly or explicitly demanded principle of redundancy: none of the superfluous statements are to be eliminated and the understanding of what is said is up to the reader ( veracity and importance provided ). In a typical mathematical work therefore still giving rise to statements of the following type are out of set statements, conditions and performing proof steps best if desired: " This result is important because ...", so that the statements be made at least in the right contexts. This principle of " no redundancy " is useful or necessary for the realization of mathematical rigor and personal forbids colored accessories as " unnecessary and possibly even harmful for the cause ", but is also one of the biggest obstacles for the intelligibility of many mathematical statements, or generally a major reason for the often lamented incomprehensibility of " mathematical style" with its lemmas, theorems and corollaries, including the opacity of many Related evidence.

The Bourbakisten

Particularly pronounced, and more abstract, the " mathematical style " was in the works, comprehensive manuals, a group of eminent French mathematician who sought in 1934 from an overall view of mathematics published under the pseudonym Nicolas Bourbaki. After decades of dominating influence of this authors' collective but the trend of increasing rigor and abstraction is currently seem rather slight decline.