CAJ | 학술논문

在前期工作的基础上，提出强而有力的证据，进一步论述了在现有微积分基础理论中，贝克莱悖论没有、也根本不可能如很多人所认为并声称的那样被消除，而只是在若干繁复、生僻的概念、定义、步骤之下被掩盖。基于此认识，把先前提出的唯一可以彻底解决此类矛盾、并且根本无涉极限、潜无穷、无穷小概念的导数的全新定义，推广到微分、积分领域，以彻底解决、澄清数学分析基础理论中的一些疑难问题。可以看出，如想真正消除微积分基础理论、导数求导、微分、积分理论中的固有矛盾，笔者前期及本文提出的理论解释———特别是关于导数的新定义———就不能不是必需的。显然，分析理论中的矛盾、悖论即消，可使整个理论非常明确并大为简化。同时，正是由于现有的、仅仅致力于“掩盖”分析理论中固有的矛盾、悖论的做法本身的窘境，而使中外学者甚感困扰的微积分基础理论的教学困难，也必将由于理论中矛盾的彻底澄清而变得极其简单、干净。
재전기공작적기출상，제출강이유력적증거，진일보논술료재현유미적분기출이론중，패극래패론몰유、야근본불가능여흔다인소인위병성칭적나양피소제，이지시재약간번복、생벽적개념、정의、보취지하피엄개。기우차인식，파선전제출적유일가이철저해결차류모순、병차근본무섭겁한、잠무궁、무궁소개념적도수적전신정의，추엄도미분、적분영역，이철저해결、징청수학분석기출이론중적일사의난문제。가이간출，여상진정소제미적분기출이론、도수구도、미분、적분이론중적고유모순，필자전기급본문제출적이론해석———특별시관우도수적신정의———취불능불시필수적。현연，분석이론중적모순、패론즉소，가사정개이론비상명학병대위간화。동시，정시유우현유적、부부치력우“엄개”분석이론중고유적모순、패론적주법본신적군경，이사중외학자심감곤우적미적분기출이론적교학곤난，야필장유우이론중모순적철저징청이변득겁기간단、간정。
Based on the preliminary work ,this paper provides strong evidence for the opinion that in the current calculus theory ,Berkeley’s paradox is not eliminated and it is also impossible to be elimi‐nated as many people think and claim .It is just covered by several complicated and uncommon concepts , definition and procedures .On this basis ,the new definition ,which is proposed to solve such kind of contradiction and does not involve the concepts of limit ,potential infinity ,and infinitesimal ,can be ex‐tended to the fields of differential and integral to completely solve and classify some complicated prob‐lems in the basic theory of mathematic analysis .It can be seen that the theoretical explanations proposed by the author previously and in this paper ,especially the new definition of the derivative ,are necessary for the elimination of the contradictions in the basic theory of calculus and theories of derivation ,differ‐ential and integral .Obviously ,the elimination of contradictions and paradox in the theory of analysis would make the whole theory very clear and simplified .At the same time ,the teaching difficulties of the basic theory of calculus ,which confuse the scholars at home and abroad because of the practice of trying to cover the inherent contradictions and paradox in the theory of analysis ,would surely be solved with the complete classification of the contradictions in the theory .