(XXIV) Considering the breadth ofundergraduate basic courses, it is understandable that we do not learn somecourses well and shallowly master them, which won’t prevent us from doing thebest research, but if we superficially learn all the undergraduate basiccourses (this makes up a certain proportion in scientific workers), it isdefinitely unacceptable.
Take mathematics as an example, a well knownfact is that for some scholars specializing in analysis, their algebra andtopology are somewhat bad, while for some scholars specializing in algebra andgeometry, their analysis foundation is a little weak, and other scholarsstudying geometry also have certain internal defects in their knowledgestructure; these defects of their knowledge structure will not prevent relevantscholars from making the most outstanding achievements, and mathematicians whomake remarkable contributions in all three major fields-analysis, algebra andgeometry, like Hermann Weyl, are only few. Combining this basic fact, in thispaper, what we emphasize is just: if a scientific worker superficially learnsall the basic courses, then his research and work’s quality cannot be good.
(XXV)Then why do we need to repeat dozens oftimes in the initial stage, to learn certain course well and solve many hardproblems? I think experienced scientific practitioners all know that we need todo a certain amount of hard questions when learning some course; the deeperreason is not hard to understand, and we think there are at least three basicreasons: 1 As we all know that, in the courses we learn, the proof of mosttheorems and approaches are difficult, and it sufficiently proves that, in mostcases, innovations are based on creatively solving hard questions, and due tothis basic feature of innovation, solving hard questions is very important. 2 Onlyby solving hard questions can we understand the spiritual essence of onecourse, because, to solve hard questions, we need to integrate one part ofknowledge and need to creatively use some important concepts, moreover, thecomputations are usually complex, thus, this can train our deeper and overallunderstanding of certain course; therefore, if we just do some simplequestions, we will miss most key points of one course. It is well known thatafter we can solve some hard problems, we will feel especially easy to do easyones. 3 Only by solving relatively hard questions can we feel the pleasure ofstudy and research, and simple ones often just mechanically copy the formulasand their lines of thought are standard, and they do not need flexibletechniques and deep ideas, thus, we cannot feel the charming of particularknowledge. To sum up, solving hard questions is an indispensible part ofscientific learning; though simplicity is also a basic feature in science andengineering fields, solving hard questions is a necessary step which is hard toevade.
The shallow information and deep ideas inscientific fields is an interesting and significant basic problem. We all knowsome science and engineering practitioners just have shallow understandings inevery course and they very shallowly master concrete theories in all courses,meanwhile, they can just solve some easy questions and cannot figure out mosthard ones; therefore, their research can just deal with some shallow, minorproblems, and this superficial research method is wasting everyone’s valuablelife. The reason is that one important feature of scientific fields is everycourse has certain depth, and this requires us to do some hard questions; a lotof knowledge and theories are not that shallow as we mistakenly think, in fact,they all include many deep skills.
Indeed, for scientific fields, in manycases, it is simple ideas that open a new situation (for example, the Taylorexpansion in calculus, congruent standard form in higher algebra, etc), butthese simple ideas are actually based on the deep understanding of the wholefield, based on the researcher’s solid knowledge and thought foundation. To sumup, we need to have sufficient understanding of the depth of scientificcourses.[8]
In a broader sense, as is widely known, inscientific research, like mathematics, physics and chemistry, ‘deep’ is a veryimportant characteristic, then corresponding to the way of repeating for dozensof times in this paper, if we do not repeat reading some courses and papers formany times, how can our research have depth? I think this point is notdifficult to understand. In fact, many good scholars will repeat some coursesand papers for quite a few times.[9]
(XXVI) The large group this paper mainlyaims at is scientific workers whose thinking ability is in the ‘normal’ level,and many of them do not systematically well learn undergraduate courses,instead, they just superficially read recent papers, and the final results ofthis research method are: firstly, their ability is still in a low level;secondly, for the concrete knowledge in undergraduate courses, thesepractitioners also do not proficiently master, and they just superficiallymaster all the undergraduate courses. The superposition of this kind of shallowknowledge structure and low overall ability leads to that they can only makesome peripheral, unimportant innovations, which is surely unacceptable. It iswell known that, in most scientific fields, the most creative period ofrelevant workers is before 45 years old, thus, this process of well learningall the related undergraduate courses needs to start as soon as possible. Thereason why these workers do not start this process is not they are not willingto do so, but they don’t realize that their learning of undergraduate basiccourses is somewhat bad.
For those individuals whose thinking abilityis in the ‘normal’ level, perhaps some of them once scatteredly repeated someundergraduate courses, but they did not embark on the huge process of relearningall the knowledge related to their research in undergraduate, and there are 3reasons: firstly, they did not realize the hardness of this process, even formathematical analysis, I was finally able to solve most of its problems afterrepeating it for 3 years and 7 months, and obviously, this is a long anddifficult process. Secondly, they did not realize the breadth of this process,and some students probably repeated some parts of the undergraduate courses,but they did not realize that they needed to repeat all the basic coursesrelated to their own research. Thirdly, they did not realize the highimportance of this process, because these workers didn’t have an overall anddeep understanding of the situation that their foundation is not solid enough,they maybe thought that it was just a minor problem, but the fact is not likethis, the fundamental knowledge in undergraduate courses has a significantmeaning to every individual. Among scientific workers, there is a universalmisunderstanding: perhaps our learning of undergraduate basic courses is notvery good, but this does not affect too much, and moreover, if we want to learnthem well, it is not so hard; thus, this is not a very serious problem, however,in our opinion, this is a very wrong view and there is an enormous differencein the mastering of basic courses (even for mathematical analysis in thefreshman year) between good students and normal ones. In actual life, thesethree reasons are often intertwined, and therefore, they lead to the fact thatnot too many workers embarked on the highly important process of relearningundergraduate courses.
We also need to point out that, amongscientific workers, a small part of them do possess some fundamental knowledgeand they can solve some of the after-class problems (for instance, somestudents in this level can solve the majority of problems in abstract algebraand algebraic topology, but they cannot figure out basic problems in functionalanalysis and PDE); for these students, their thinking ability also needs toimprove and they need to be able to solve almost all the hard problems, andonly by doing this can they achieve the improvement of their depth of thought;otherwise, what we do is just a simple accumulation of knowledge width in thesame thinking depth.
(XXVII)In the meanwhile, we need to pointout 8 basic facts: firstly, many scientific workers actually realize the basicfact that the thinking ability of different people has big differences, thus,this is not a new insight; experienced scientific teachers, students andworkers often know that the talents between different people has largedifferences, however, they do not know that these differences in thinkingability can be overcome, and, to some extent, must be overcome by us. Secondly,in the process of relearning undergraduate courses, due to the difference offoundation among people, the repeating times needed perhaps vary widely, butthey must meet the standard of solving most after-class problems. Thirdly, thelearning and work methods of different people often vary, and this paper valuesand emphasizes the holistic property of knowledge, but many people’s work styleis not like this (as revealed by Dyson in the essay ,some mathematicians like general intellectual framework, while othermathematicians pay more attention to isolated and concrete problems), thus, weneed to use our own creativity and seek work method which fits our ownpersonalities. Fourthly, about the process of relearning undergraduate courses,different people may work in different ways, here what I adapt is the way ofrepeating textbooks for dozens of times (with watching videos and discussingwith classmates), for different people, we can choose the ways which best suitourselves, but the final criterion is unified, namely, we need to solve almostall the problems. Fifthly, this paper emphasizes the importance and benchmarkof solving problems, but the intension of learning is much richer than simplysolving problems; take mathematics as an example, understanding the latentthought essence in knowledge is probably a more important basic step behindsolving problems, and meanwhile, the importance of problem solving should alsonot be overestimated, as the great mathematician Atiyah once said: “I don’t payvery much attention to the importance of proofs. I think it is more importanceto understand something.” “A proof is important as a check on yourunderstanding. I may think I understand, but the proof is the check that I haveunderstood, that’s all. It is the last stage in the operation-an ultimatecheck-but it isn’t the primary thing at all.”[10]This view of Atiyah isvery reasonable. Sixthly, the problem solving we deal with in this paper isactive problem solving, namely, we can actively solve these problems, notunderstanding the line of thought after reading the solutions, and such kind offiguring out problems by seeing the solutions is not very meaningful, since itdoes not include the process of creative thought and positively active thought,and it won’t cost us so much energy; namely, we must actively solve relatedproblems, otherwise, it doesn’t have essential meaning. Seventhly, we may notstrictly follow the standard of solving all the after-class problems, but wemust be able to solve most hard questions. Eighthly, for the students in ‘normal’level, they actually do not master almost all the contents in undergraduatecourses, namely, many science and engineering students actually know verylittle about their own major at graduation, which is a somewhat shocking basicfact.
（XXVIII）Forrepeating over 55 times (it will cost over 3 years’ time) and the difficulttask to solve most hard problems, it is somewhat unexpected, but it also fitssome basic rules of human society, such as: science and engineering work shouldalways strive for excellence, and any valuable thing is not easy to get and requiresarduous effort;[11]hardworking is the foundation of modern society; anything should begin bylaying a solid foundation and should start from the root and proceedstep-by-step (from this principle, we should repeat undergraduate courses byfirstly repeating freshman and sophomore years’ courses), and we cannot makeimportant innovations without a good foundation ( minor innovations are stillpossible). Broadly speaking, the process of relearning all the courses relatedto our own research in the undergraduate stage is a hard process with constantstruggles, meanwhile (somewhat complementarily), it is also a pleasant journey.
(IXXX) We think the problems we discuss inthis paper are meaningful to the following several kinds of people: 1 somemathematical workers still cannot proficiently write down the second mean valuetheorem for integrations at 37 or 38, such a condition of weak foundation is prevalentin many majors’ PhDs, faculties and workers, like electronic engineering (somebasic courses like digital analog circuit, circuit principle, engineeringmathematics), chemistry (courses like organic chemistry), mechanicalengineering (courses like material mechanics, mechanical principle), physics,statistics, aerospace engineering, computer science, chemical engineering, etc;we should pay sufficient attention to this phenomenon. 2 To some workers whoneed mathematical and physical knowledge, like architecture (structuralmechanics, elastic mechanics, etc), they should also realize the long processof learning these courses. For the above several kinds of people, the problemsdiscussed in this paper may help them to open a magnificent intellectualhorizon.
Meanwhile, some science and engineeringworkers may do not need original activities, while laying a solid foundation inbasic courses are still of rich value for them. Because mastering basicknowledge in undergraduate and improving their thinking ability can make thembetter face their work, for instance, learning the pointer, array and othercontents in C language, mastering some central contents in higher algebra, likethe solution of linear equations, the method of figuring out Jordan standardform, well learning important knowledge in calculus, like definite integral,improper integral, multivariate differential calculus, function series,multiple integral, Fourier series, etc, all of these can enable them to bettersolve various problems in their work since the affected area of these knowledgeis so broad. Many scientific workers do not really master these knowledge, thus,this has a negative impact on their practical work, while some ideas putforward above can change some workers’ situation. For me, before repeatingundergraduate courses, I am always not confident about C language, but inAugust, 2016, I feel that my mastery of C language is already in place, at thesame time, my thinking ability has enhanced, and due to the basic changes ofthese two aspects, my confidence about C language has greatly improved (this isalready 9 years after I entered college); considering the broad application ofC language, we think the ideas raised in this paper is valuable. Namely, if wejust superficially master 10 courses, when facing concrete problems in realwork, we often cannot solve them, thus, we don’t have executive ability, and itis less valuable than thoroughly learning 3 to 4 courses. (However, forinnovation, only well learning 3 to 4 courses is not sufficient.)
To sum up, the ideas we articulate in thispaper have double meanings: research and learning.
Obviously, our discussion is very applicableto scientific fields, like mathematics and physics; while for engineeringfields, if we relate the actual industrial world, we think this paper also hasrich values; firstly, it is of direct meaning to the practitioners in a numberof companies in electronic engineering, like Sumsung, Cisco, IBM, etc, and alsoincluding many small and medium-sized enterprises in this field. For manycompanies whose major business is big data and artificial intelligence, ourdiscussion is very applicable to their workers too, since we think there is adeep difference between a worker who has solid foundations in mathematicalanalysis and higher algebra and another one who just know R software; forinstance, in the summer of 2017, I worked in a big data company for a shorttime, at that time I fully felt the great benefit brought by my deep foundationin calculus and linear algebra (stemming from many repetitions which cost muchtime), because I felt especially easy when learning some new knowledge andskills and it was easy for me to well master them. In the meantime, it alsoapplies to workers in mechanical engineering, like medical instruments (likePhilips, Siemens, Johnson and Baxter) and large instruments. The latent impactof our discussion to other fields, like civil engineering, is also similar. Tosum up, the basic issues we deal with in this paper have high potential valuesto both the science and engineering fields.
(XXX)In the following part, we want tofurther analyze the possible actual impacts of this paper, and the complexreactions possibly triggered by it:
For many students in mathematics andphysics, if they have normal talent, then it is very difficult for them to makefirst-class contributions by systematically repeating undergraduate courses andlaying a good foundation, because the golden age in scientific fields is before45 years old. The above example, Carleson, is a successful case, and Hardy isanother example, he said that (also, it is an objective fact) his professionalcareer truly began at 34 years old when he met Littlewood, but, even with somepast cases, for these students, it is still considerably difficult to makeoutstanding contributions. Therefore, our paper is just partially applicable tothis type of students. However, our paper is of certain value to another typeof workers in mathematics and physics, namely, for a few excellent professorsin some top universities, like Harvard and Yale, their knowledge structure hascertain internal defect-their analysis side is probably very fantastic, butthey are afraid to do algebra and topology problems and are afraid to touch them,or they do very well in algebra, but they are afraid to touch any analyticalquestions (Mr. Shing-Tung Yau once mentioned this phenomenon); this way ofdoing research may be not bad, but there is certainly a limitation in it, andfor this type of workers, the analytical framework and deep experience weprovide in this paper perhaps have some value.
For the workers in engineering fields, likeelectronic engineering, aerospace engineering and statistics, their researchand work may not require exceptional genius, and their courses are not too deepand broad, since part of their job is routine, and they can better qualify for theirwork by laying down a solid foundation. Meanwhile, the number of workers inthese engineering fields is much larger than mathematics and physics,therefore, many of the practitioners in these fields have a weak foundation,and for these people, they need to have a broad plan and should not onlyconsider short-term (2 or 3 months) goals. Considering the enormous number ofengineering workers, the impact to engineering majors is probably the majorimpact of this paper.
The reaction to this paper may be: (1) Aboutthose students whose foundation is weak, their learning about all the basiccourses is superficial, and they did not realize this point before (alsoprobably realized it), but they quickly realize this basic fact with ourextensive analysis, namely, they have an objective assessment about their weakfundamental; considering that most scientific workers have a clear understandingof their own fundamental, thus, they have a sober judgment about whether theycan solve problems; since they realize this point, they will probablysystematically relearn some courses and build certain foundations for their ownwork, meanwhile, they may also choose the way of superficially learning andcontinue to publish some ordinary papers, which is also understandable.(However, a small group of students will perhaps have a disordered and falsejudgment about their knowledge foundation and real ability, meanwhile, when oneperson is 35 years old, the crowd will form a somewhat objective judgment aboutthe ability of a particular worker, since the judgment of many experts togetherwill be objective.) (2) About those good students whose foundation is solid,they do not have any experience of the phenomenon described in this paper, andthey just need to learn graduate’s courses and directly embark on independentresearch and work. However, another basic issue analyzed in this paper, namely,the thought in speculative level and artistic level is extremely important fortheir long-term and deep development (we have analyzed it in another paper). Weshould note that there are so many science and engineering majors, and thestudents’ foundation and professional pursuits are also very complex; thus, ourdiscussion in this paper will create different reactions in differentindividuals.[12]
What we really want to stress is thatscientific workers should have long-term views about their own work. For instance,for practitioners in electronic engineering, if they keep superficiallylearning (hastily read recent books, papers and related literature withoutthinking, roughly learn new techniques and softwares), they will still lack amastery of some necessary basic courses, and this kind of foundation isproblematic in actual work. If one worker spends 3 years in doing so-calledfrontier research or work, after 3 years, his foundation won’t change too muchand he also cannot achieve essential overall progress; not as good as spending3 years’ time in learning several basic courses and truly improve one’sability; after all, three years’ time is not so short and also not so long, andspending 3 years in well learning several courses will give us enough confidencein our own work.
Professional ability and technicalfoundation is obviously a central problem in scientific practitioner’s work,and considering the complex analysis of this paper, we can get one exhilaratingbasic conclusion: the professional foundation of scientific workers can bechanged, and we can also change our professional capability, but we must startfrom basic courses and make long-term plans. Admission to top universities andentering good companies cannot change our professional strength, since strengthcan only stem from long-term (over 3 years) systematic accumulation and it willnot change with the change of external environment; the professional knowledgeof science and engineering majors is too broad and delicate, and it requireslong-term accumulation, how can it be suddenly mastered by us just because weenter top companies? We think that for various scientific majors,systematically learning basic courses is probably the only way to change ourprofessional strength and superficially doing research and work will make usstamp on the same ground, and it is virtually impossible to reinforce ourtechnical foundation by piecemeal learning, and seeking short-term success andplaying petty tricks can hardly bring us a solid, fundamental progress; whiletime (for instance, 2 years, 3 years or 5 years) will quickly pass in the busylife and work. Meanwhile, considering the heavy work of job assignments, thebustle of life and it will cost over 3 years to repeat undergraduate courses,this is indeed a knotty problem and requires some wisdom to deal with it.
In a word, we think this paper will have acertain degree of impact for scientific workers, but is also limited.Meanwhile, with the passage of time, the impact of this paper will graduallydeepen.
(XXXI) We need to point out that the twocategories of scientific students’ talents in this paper-‘normal’ and ‘good’are just a coarse approximate analysis; the actual situation is much morecomplicated and there are many different levels of talent. The analysis oftalent differences among students is just a basic ingredient of this paper andit is naturally not the major purpose of it, our main purpose is to display thelong and complex process of learning some scientific courses.
(XXXII) Finally, a basic awareness we needto establish is: laying a solid foundation in undergraduate courses iscertainly not the eventual goal, and our ultimate aim is to make essentialinnovations in our own area; thus, for the knowledge we have thoroughly mastered,we should not waste much time in learning them, and we need to think thepossible directions of creative developments. In academic fields, we will findsome students with good foundation cannot get original results, which isbecause that they lack a systematic understanding about research and they alsolack independent thinking of particular knowledge, thus, they don’t know how tofind new research areas and problems, and we need to avoid this kind ofbehavior. Thus, in the process of relearning undergraduate courses, we need tothink about many important issues in other aspects, including how to find novelresearch direction, the interconnection of many courses, understanding thoughtsystems, fostering deep intuitions, creating unknown tools, developing existingtechniques, refining latent concepts, capturing flashed inspirations, etc. Weneed to creatively understand undergraduate knowledge, and also need to keepindependent research views and cutting-edge consciousness; without the richunderstanding and broad reserve of academic original awareness, it is hard forus to make important contributions in our research. The nature of research isto find new knowledge, not merely learn existing knowledge, and therefore, weneed to look ahead and think about new directions and not stay too long inknown knowledge. To sum up, the return to foundation is for better facing thefuture, if we cannot face the future, return to foundation will lose most ofits meaning.
Here, one part of argument given by Beveridgeis very helpful to our discussion, he wrote: “Charles Nicolle distinguished (a)the inventive genius who cannot be a storehouse for knowledge and who is notnecessarily intelligent in the usual sense, and (b) the scientists with a fineintelligence who classifies, reasons and deduces but is, according to Nicolle,incapable of creative originality or making original discoveries. The formeruses intuition and only calls on logic and reason to confirm the finding. Thelatter advances knowledge by gradual steps like a mason putting brick on brickuntil finally a structure is formed. Nicolle says that intuitions were sostrong with Pasteur and Metchnikoff that sometimes they almost published beforethe experiment results were obtained. Their experiments were done mainly toreply to their critics.”[13]This passage is verythought-provoking, some researchers with good foundation cannot make originaldiscoveries, and this fact is widely known to us (one of the reasons is perhapsthe lack of accumulation in thought level), thus, we need to pay high attentionto original spirit; to keep a blooming condition in all the human scientificfields, innovative spirit and original ability have an overriding importance.Of course, innovation needs solid knowledge and thought foundation, Beveridge’sexposition just reminds us: originality, making new discoveries, finding newphenomenon and creating new technology are the most important and most decisivecriteria in all science and engineering fields. One sentence of Whitehead probablygives the best generalization of the fundamental value of innovation: “Passivelyunderstanding the past will lose the entire value contained in the past. Aliving civilization needs learning, but not only learning.”[14]
(XXXIII)Correspondingly, the basic ideasraised here do not aim at the few students whose talent is ‘good’, because theyhave well mastered the undergraduate basic knowledge in the four undergraduateyears, thus, they do not need to repeat them, and for these students, they don’tneed to relearn undergraduate courses, they can just master graduate coursesand begin independent research. Even for the scientific students whosefoundation is not good, in the actual process of work and research, they areunlikely to spend a whole period of 3 years and 7 months like me in justrepeating freshman and sophomore years’ courses; thus, we need to mainly focuson our work and frontier research, however, we also should pay attention tobasic courses.
In summary, the 4 central thought themes ofthis paper are: systematically repeating undergraduate courses, problem solving(as pointed out above, the problem solving we emphasize in this paper must beactive problem solving, not solving problems after reading the keys), theholistic improvement of thinking ability and independent thinking; though wemainly discuss the higher education of mathematics, I think the same learningrules also apply to a number of science and engineering disciplines, includingphysics, chemistry, mechanical engineering, electronic engineering, computerscience, statistics, aerospace engineering, petroleum engineering, civilengineering, chemical engineering, communication engineering, etc.
Our overall belief is: nearly all thescience and engineering workers who make real contributions have a solidfoundation (for instance, when we explore the field of differential geometry,if our foundation is not strong enough, then even we can vaguely guess theexistence of Gauss-Bonnet formula, we will necessarily be unable to completethe complex proof, or when we explore the field of quantum mechanics, even we canguess the general direction of Dirac equation, we cannot detailedly get theoverall properties of this equation, or perhaps a more intuitive argument isthat for those scientific workers who get good results in independent researchstage, most of them are also good students in undergraduate; to sum up, solidfoundation is fundamentally important for scientific practitioners, which is asomewhat obvious fact), all the science and engineering workers who make reallysignificant contributions have extremely strong independent thinking ability(in the process of learning, independent thinking can doubly enrich theconcrete information, thus, the process of scientific learning is a positiveand active procedure). All the conclusions in this paper are based on these twoobvious basic facts.
Inconclusion, one secret lurking in the undergraduate science and engineeringeducation for over 100 years is uncovered. (This secret is not discovered for along time because it looks easy, but in fact it is not, and it is the mergenceof five fundamental insights: 1 The difference of thinking ability, most peoplejust realize the basic fact that there are differences of thinking abilityamong scientific students, but they don’t realize that we need to make enormouseffort to improve form the ‘normal’ level to the ‘good’ level. 2 Relearningundergraduate courses, some people may realize this point, but they did not repeatfor sufficient times and their repetition was less than 55 times, and they didnot insist on the basic principle that most after-class problems need to besolved, thus, they falsely believed that they had mastered these courses afterrepeating for some times, in fact, there was still a long distance ahead;meanwhile, some students did relearn some undergraduate courses, but they didn’trealize that we need to repeat all the undergraduate courses related to our ownresearch. 3 One purpose of relearning undergraduate courses is to improve ourability, and the process of our learning is one process in which our thinkingability keeps improving, and we can evidently feel this by comparing ourintellectual condition between senior year and freshman year, but, almostnobody realized: to thoroughly solve the issue of improvement of our overallability, we must return to undergraduate courses. After repeating theundergraduate courses for over 55 times, our professional thinking ability willgreatly enhance, and we will feel especially easy when facing concreteknowledge and problems, namely, in the process of relearning undergraduatecourses, the mastery of knowledge and the improvement of ability are deeplyintertwined, which is certainly a pleasant feeling. 4 In learning, the accumulationof knowledge is one aspect, and the understanding in thoughtful and artisticlevels of particular knowledge is another indispensible aspect, namely, we needto actively master relevant knowledge in our own way, independent thinking candoubly enrich our scientific information; in a word, when facing basicknowledge, we not only need long-term accumulation, we also need creativeaccumulation. 5 As to the time dimension, relearning undergraduate courses willrequire a long time, and even we don’t do research and read any recentliterature, and just take on the assignment of relearning undergraduatecourses, relearning undergraduate courses will cost us at least 4 to 5 yearsafter graduation. The above 5 points perhaps can be discovered isolatedly by somepeople, but only by merging them together can we form an overall insight ofundergraduate science and engineering education.)
Sept 29, 2018
[8]Wecan refer to Whitehead’s exposition: “Whenever a textbook is written of realeducational worth, you may be quite certain that some reviewer will say that itwill be difficult to teach from it. Of course it will be difficult to teachfrom it. If it were easy, the book ought to be burned; for it cannot beeducational.” See the above quoted paper “The Aims of Education”.
[9]Asthe great mathematician Chern says: “Like reading books or seeing paintings,for some great works, they are still interesting even if we read them for onehundred times…mathematical works are also like this”, seeCollected Essays of S.S. Chern, Part I, p. 57, East China NormalUniversity Press, 2002. In fact, in scientific research, to make theirunderstanding about certain problems ‘deep’ enough and delicate enough, manygood scholars will repeat reading some classical books and papers in theirfields for many times, and meanwhile, considering the complexity of reality,some scholars will not do like this (as an example, famous mathematicianGrothendieck only read few existing books and papers), but reading some booksand papers for many times is perhaps a somewhat common and universal researchmethod. As is well known, when many good scientists learn some new courses andnew knowledge, they are somewhat slow, like Einstein, Hilbert, Perelman, Bott,etc, because they want to deeply understand these things, and do not just wantto learn many new things as quickly as possible (for example, Hilbert thinksthat, if one person wants to truly understand some mathematical knowledge, heneeds to repeat at least 5 times). In a word, reading scientific books isdifferent from reading literary books, and because scientific knowledge issomewhat complex and difficult, to learn them well and detailedly, we need torepeat many times, or else our learning is easy to be crude and shallow, whichI think is not difficult to understand.
[10]Seethis view of Atiyah in “An Interview with Michael Atiyah”,The Mathematical Intelligencer, 6 (1984), p. 17. Atiyah hasrepeatedly emphasized this notion，and in another article “Advice to a YoungMathematician”, he writes: “I believe the search for an explanation, forunderstanding, is what we should really be aiming for. Proof is simply part ofthat process, and sometimes its consequence.”
[11]Famousphysicist Feynman once wrote: “You see, I have the advantage of having foundout how hard it is to get to really know something, how careful you have to beabout checking the experiments, how easy it is to make mistakes and foolyourself. I know what it means to know something,…they haven’t done the carenecessary.” One people as brilliant as Feynman also has such a feeling, andthus, my tortuous experience is probably a normal basic phenomenon. See thepaper “The Pleasure of Finding Things Out” in the bookThe Pleasure of Finding Things Out, p. 22, Perseus Books, 1999.
[12]Ourpaper can explain one universal phenomenon in science and engineering fields,namely, the enormous difference between a hazy understanding and a true one; asdescribed by Professor Whitehead: “ In the past half century, in the east andwest coast of Atlantic, I have hired teachers for many times. How todistinguish between loud and vigorous, how to distinguish between making aracket and originality, how to distinguish between mental instability andhighly talented, how to distinguish between rigid knowledge and reallearning-nothing is more difficult than this.” See the paper “Harvard: TheFuture”, section V, an address at the Tercentenary of Harvard University, 1936.From our complex analysis, we can feel two basic facts: firstly, for theunderstanding of one or several courses, there is indeed an enormous differencebetween hazy understanding and real mastery; secondly, for most scientificstudents, since their talent is ‘normal’, it will be a long and hard process totruly master several basic courses.
[13]SeeThe Art of Scientific Investigation,Chapter 11, pp. 148, 149, Norton & Company Inc, 1957.
[14]SeeAdventures of Ideas, Chapter IXX,section III, The Free Press, 1967.
