APPEARED IN BULLETIN OF THE
논문은 "controversy"에 대한 이야기로 시작합니다. 이 논문은 1992년 1월에 출판되었으며, 두 가지 의미의 논쟁을 다룹니다.
첫 번째는 "complexity theory"라는 개념에 대한 서로 다른 관점으로, 두 명의 저자가 강력한 언급을 사용하여 이에 대해 다른 의견을 내놓습니다. 두 번째는 이 논문이 연구-설명 논문 섹션에서 출판되는지 여부에 관한 것입니다.
편집자들은 논쟁의 본질에 대해 설명합니다. 수학자는 다른 분야에서는 종종 격렬한 분쟁으로 이어지는 것과 같은 의견 차이점을 objective하게 해결할 수 있습니다. 그러나, 다른 수학적 논쟁은 less objective nature가 있고, 쉽게 해결되지 않습니다.
수학의 역사에서 예를 들어 Cantor의 proof가 transcendental numbers의 존재를 보여주기 위해 algebraic numbers의 countability와 real numbers의 non-countability을 증명했지만, 실제로 constructible한 transcendental 수를 보여주지 않았다는 주장과 같은 논쟁이 있습니다. Hilbert의 Invariant Theory에서 abstract methods와 nonconstructive 방법을 사용한 그의 접근도 controversy를 일으켰습니다.
현재, Catastrophe Theory의 abuse에 대한 sharp controversy가 있었고, fractals과 chaos에 대한 overpopularization이 논쟁의 대상입니다. 수학과 이론물리학의 관계에 대한 discussion도 있습니다.
편집자들은 논문이 mathematically rigorous한 방법으로 proof되지 않았지만, physical world의 new insights를 제시한다면, 어떤 기준을 적용해야 하는지에 대해 고민해야 한다고 말합니다. rational discussion와 debate가 대화의 의미를 밝히는 데 도움이 될 수 있습니다.
결론적으로, 논문은 controversial한 주제에 대한 debate가 학술 저널에 포함될 필요성이 있음을 제안합니다. 논문이 intrinsic interest와 merits로 evaluated되고, 논쟁스럽지만 serious concern과 long-term significance가 있다면, 그 논쟁이 이 저널에서 다루어질 수 있습니다.
영어 요약 시작:
APPEARED IN BULLETIN OF THE
arXiv:math/9201262v1 [math.HO] 1 Jan 1992APPEARED IN BULLETIN OF THEAMERICAN MATHEMATICAL SOCIETYVolume 26, Number 1, Jan 1992, Pages 1-2EDITORS’ REMARKSMorris W. Hirsch and Richard S. PalaisThe two papers that follow are controversial—in two senses. First, the authorsexpress opposing and strongly worded views on what “complexity theory” shouldbe.
Second, the decision to open the Research-Expository Papers section of theBulletin as a forum for such a debate may also be considered controversial; and, infact, the wisdom of this decision was the subject of some dispute within the editorialboard. It was made by us jointly as present and past chairs of the editorial board.As to the controversy in the first sense, we let the papers speak for themselves.However, since their publication is a precedent of sorts, we feel it is important toclarify our general attitudes toward articles of a controversial nature.As mathematicians we have the good fortune to be able to settle in a straight-forward and objective way one sort of controversy which, in other disciplines, oftenleads to quite rancorous disputes.
While there are occasional disagreements overthe correctness of a paper, the strictly logical nature of mathematical proof usuallypermits a quick resolution of such issues that is agreed to by all sides. But thisshould not blind us to other mathematical controversies that are less objective intheir nature, and not so easily settled.For example, we have probably all heard the story that some mathematiciansfelt it was scandalous for Cantor to claim that, in demonstrating that the algebraicnumbers were countable while the real numbers were not, he had given a new proofof the existence of transcendental numbers.
After all, his proof gave no way to con-struct even a single transcendental number. Echoes of this controversy are hearddown to the present day in the now somewhat muffled debate over “Constructivismversus Classical Mathematics.” Similarly, we read that Hilbert’s approach to Invari-ant Theory, using his Basis Theorem and other nonconstructive, abstract methods,provoked controversy in a mathematical world still steeped in the concrete methodsof the classical tradition, where solving a particular problem in Invariant Theoryhad always meant exhibiting a specific basis for the invariants.
Other controversiesinclude debates over the status of infinitesimals, irrationals, imaginary numbers,large cardinals; the proper treatment of geometry, logic, set theory, foundations ofmathematics; the role of computer science; the use of mathematics in the socialsciences; and perennial issues in mathematical education.And not all such controversies are ancient history! Fifteen years ago there was asharp controversy over purported excesses in the applications of Catastrophe The-ory, and currently there is a similar controversy concerning what some see as anoverselling and overpopularization of “fractals” and “chaos.” Another simmeringdebate has grown out of the current renewal of the on-again, off-again love affairbetween Mathematics and Theoretical Physics.
We have learned to accept thatdifferent standards of mathematical rigor may be appropriate when mathematicsc⃝1992 American Mathematical Society0273-0979/92 $1.00 + $.25 per page1
2M. W. HIRSCH AND R. S. PALAISis being used as a tool to gain new insights about the physical world.
But whatstandards should we apply to judge a paper that uses nonrigorous or semirigorousmethods from physics to suggest important new insights into our own mathemat-ical world, particularly if those insights seem beyond the reach of current rigorousmathematics?Especially because such questions cannot always be answered by logical principlesalone, we believe that it is important for mathematicians to confront them. Evenwhen rational discussion and debate does not completely resolve differences, at leastit may clarify the issues.Traditionally debate about issues of this sort has been carried on in nonscholarlyjournals, and for questions that are less weighty or more transitory in significancethis is appropriate.
We certainly have no intention to open these pages to emotionaldebate over whether the C programming language is better or worse than Pascal!But when a controversial matter comes up that is of serious concern and long-termsignificance to the mathematical community, and so deserving of careful debate,then such a debate belongs in an archival journal. This does not mean we areinviting authors to submit some new category of “controversial issue” paper to theResearch-Expository Papers section.
On the contrary, as always, any paper willjudged on its intrinsic interest and merits, and controversial papers will no doubthave to jump through a few extra hoops.What we are saying is that we willnot reject a paper solely because the ideas presented in it may not be universallyaccepted or subject to mathematical proof or disproof.Department of Mathematics, University of California at Berkeley, Berkeley,California 94720Department of Mathematics, Brandeis University, Waltham, Massachusetts 02154-9110
출처: arXiv:9201.262 • 원문 보기