按：這篇筆記是系列筆記的第六篇，第一部分有4節(jié)，每節(jié)對應1-2篇筆記。
筆記的方式，是引用一段個人覺得比較有亮點的英文原文，再給一段簡化的中文說明，不采用中文版的翻譯，不自行做直接翻譯，只說明要點。因為不可能大段大段地去引用，必然會有語境的丟失，會做一些補充說明，以“按：”開始。對中文版翻譯進行更正或調(diào)整的說明，以“注：”開始。偶爾也會插入自己的議論，以“評：”開始。
前五篇筆記為：
4 Discovering Patterns（發(fā)現(xiàn)模式）
When a problem looks too difficult to solve, one should not give up completely. A much more productive reaction is to formulate related but more approachable questions.
當一個問題看起來太難解決，不應該完全放棄。一種更有成果的應對是提出相關但更易接近的問題。
For some problems, the best approach is to build a highly structured pattern that does what you want, while for others—usually problems for which there is no hope of obtaining an exact answer—it is better to look for less specific arrangements. “Highly structured” in this context often means “possessing a high degree of symmetry.”
對于有的問題，最好的辦法是構建一個高度結構化的模式，使其具備所需要的特性；而對于另一些問題——這些問題通常不太可能獲得一個精確的回答——尋找不那么具體的排布反而更好。在這個語境下，“高度結構化”常常意味著“具有高度對稱性”。
These facts about the Leech lattice illustrate a general principle of mathematical research: often, if a mathematical construction has one remarkable property, it will have others as well. In particular, a high degree of symmetry will often be related to other interesting features.
5 Explaining Apparent Coincidences （解釋表觀巧合）
It is not obvious how seriously we should take this observation, and when it was first made by John McKay opinions differed about it. Some believed that it was probably just a coincidence, since the two areas seemed to be so different and unconnected. Others took the attitude that the function
j(z)and the Monster group are so important in their respective areas, and the number 196 883 so large, that the surprising numerical fact was probably pointing to a deep connection that had not yet been uncovered.
這個觀察結果（按：上文提到橢圓模函數(shù)j(z)的級數(shù)定義中的一個系數(shù)比魔群的最小可能維數(shù)只大了1）應該在多大程度上被嚴肅地對待，并不是那么顯然，而且當John McKay首次觀察到它時，人們的觀點就已經(jīng)有了分歧。有人相信這很可能只是一個巧合，因為這兩個領域看上去很不一樣也沒有關聯(lián)。其他人則采取了這么一種態(tài)度，橢圓模函數(shù)j(z)和魔群在他們各自的領域都那么重要，而196883這個數(shù)字又這么大，這種驚人的數(shù)值上的事實，可能指向某種尚未發(fā)現(xiàn)的深刻聯(lián)系。
Another general principle of mathematical research: if you can obtain the same series of numbers (or the same structure of a more general kind) from two different mathematical sources, then those sources are probably not as different as they seem. Moreover, if you can find one deep connection, you will probably be led to others.
數(shù)學研究的又一個一般性原則：如果你能夠從兩個不同的數(shù)學來源，獲得同樣的數(shù)字序列（或者某種更一般的數(shù)學對象形成的同樣結構），那么這兩個數(shù)學來源很可能沒有它們看上去那么不一樣。甚至，如果你可以找到一個深刻聯(lián)系，還會順藤摸瓜找到更多聯(lián)系。
6 Counting and Measuring （計數(shù)與度量）
This is a simple example of a counting argument, that is, an answer to a question that begins “How many.” However, the word “argument” is at least as important as the word “counting,” since we do not put all the symmetries in a row and say “one, two, three, . . . , sixty,”(resorting to “brute force”) as we
might if we were counting in real life.What we do instead is come up with a reason for the number of rotational symmetries being 5 × 12. At the end of the process, we understand more about those symmetries than merely how many there are.
這（按：“正二十面體有多少個旋轉對稱？”）是“計數(shù)論證”的一個簡單例子，即對“有多少個”這類問題的回答。然而，“論證”這個詞至少和“計數(shù)”一樣重要，因為我們并非把所有的對稱排成一列然后像我們在日常生活中計數(shù)一樣一個一個數(shù)下去（采用蠻力的方式），而是提出了旋轉對稱總數(shù)為 5 × 12的理由。這個過程結束后，我們對這些對稱獲得了更深入的理解，而不只是它們有多少個。
Even if an exact answer does not seem to be forthcoming, it is still very interesting to obtain estimates. In this case, one can try to define an easily calculated function f such that f (n) is always approximately equal to t(n). If even that is too hard, one can try to find two easily calculated functions L and U such that L(n) ≤ t(n) ≤ U(n) for every n. If we succeed, then we call L a lower bound for t and U an upper bound.
哪怕準確的答案看上去不易找到，獲得其估計也是非常有趣的。這種情況下，我們可以嘗試定義一個容易計算的函數(shù)f，使得f(n)總是近似地等于t(n)。即使這太難，我們也可以嘗試找兩個容易計算的函數(shù)L和U，使得對于所有n，L(n) ≤ t(n) ≤ U(n)均成立。如果成功找到了，我們就稱L為下界，而稱U為上界。
Given a set of objects, one may wish to know, besides its size, roughly what a typical one of those objects looks like. Many questions of this kind take the form of asking what the average value is of some numerical parameter that is associated with each object.
給定數(shù)學對象的一個集合，除了它的大小，我們可能希望知道，這個集合的一個典型對象大概是什么樣子的。這類問題很多具有同樣的形式，即求與每個對象關聯(lián)的某種數(shù)值參數(shù)的平均值。
There are many problems in mathematics where one wishes to maximize or minimize some quantity in the presence of various constraints. These are called extremal problems. As with counting questions, there are some extremal problems for which one can realistically hope to work out the answer exactly, and many more for which, even though an exact answer is out of the question, one can still aim to find interesting estimates.
數(shù)學中有很多問題，都是希望在各種約束下，最大化或最小化某些量。這些問題被稱為極值問題。就像計數(shù)問題一樣，有的極值問題可以獲得精確答案，而更多的問題，雖然精確答案不可能獲得，但依然可以找到有趣的估計。
7 Determining Whether Different Mathematical Properties Are Compatible （判定不同的數(shù)學性質是否相容）
Suppose that we wish to determine whether G has some property P that some groups have and others do not. Since we cannot prove that the property P follows from the group axioms, it might seem that we are forced to abandon the general theory of groups and look at the specific group G. However, in many situations there is an intermediate possibility: to identify some fairly general property Q that the group G has, and show that Q implies the more particular property P that interests us.
假定我們希望判定群G是否具備某種性質P，該特性有的群具備而其他群不具備。因為我們不能證明性質P服從群公理，我們可能被迫放棄群的一般理論而考察具體的群G。然而，很多情況下，存在一種間接的可能：識別出群G具備某種比較一般的性質Q，而證明性質Q可以導出我們感興趣的更為具體的性質P。
8 Working with Arguments that Are Not Fully Rigorous （利用不完全嚴格的論證）
A mathematical statement is considered to be established when it has a proof that meets the high standards of rigor that are characteristic of the subject. However, nonrigorous arguments have an important place in mathematics as well. For example, if one wishes to apply a mathematical statement to another field, such as physics or engineering, then the truth of the statement is often more important than whether one has proved it. However, this raises an obvious question: if one has not proved a statement, then what grounds could there be for believing it? There are in fact several different kinds of nonrigorous justification.
當一個數(shù)學命題有了一份滿足該門學科特有的嚴格性的高標準的證明，它才被認為成立。然而，不完全嚴格的論證在數(shù)學里也有非常重要的地位。例如，如果我們希望應用一個數(shù)學命題到另外一個領域，比如物理或者工程，命題是否為真要比命題是否被證明更為重要。然而，這就提出了一個顯然的問題：如果一個命題尚未被證明，那相信其為真有什么基礎呢？實際上，存在著幾種非嚴格的“正當化”論證。
There are large numbers of papers with theorems that are proved only under the assumption of some version of the Riemann hypothesis. Therefore, anybody who proves the Riemann hypothesis will change the status of all these theorems from conditional to fully proved. How should one regard a proof if it relies on the Riemann hypothesis? One could simply say that the proof establishes that such and such a result is implied by the Riemann hypothesis and leave it at that. But most mathematicians take a different attitude. They believe the Riemann hypothesis, and believe that it will one day be proved. So they believe all its consequences as well, even if they feel more secure about results that can be proved unconditionally.
有大量的論文中存在許多定理，它們都是在黎曼假說的某種形式成立的前提下進行證明的。因此，任何人只要證明了黎曼假說，就將這些定理的狀態(tài)從有條件的變成了完全被證明的。我們應該如何對待一個依賴黎曼假說的證明？我們可以簡單地說該證明證明了黎曼假說蘊含了該結論，然后不再深究。但大多數(shù)數(shù)學家采取不同的態(tài)度。他們相信黎曼假說，而且相信它有一天會被證明。所以他們相信黎曼假說的所有推論，雖然他們覺得具備無條件證明的結論更為可靠。
There is far more to a conjecture than simply a wild guess: for it to be accepted as important, it should have been subjected to tests of many kinds. For example, does it have consequences that are already known to be true? Are there special cases that one can prove? If it were true, would it help one solve other problems? Is it supported by numerical evidence? Does it make a bold, precise statement that would probably be easy to refute if it were false?
It requires great insight and hard work to produce a conjecture that passes all these tests, but if one succeeds, one has not just an isolated statement, but a statement with numerous connections to other statements. This increases the chances that it will be proved, and greatly increases the chances that the proof of one statement will lead to proofs of others as well. Even a counterexample to a good conjecture can be extraordinarily revealing: if the conjecture is related to many other statements, then the effects of the counterexample will permeate the whole area.
猜想比瞎猜具備更豐富的內(nèi)涵：要被接受為重要的猜想，它需要接受多種多樣的測試。例如，它是否存在已知為真的推論？有沒有可以被證明的特例？如果它成立，是否可以幫助解決其他問題？它是否為數(shù)值上的證據(jù)所支持？它是否作出大膽而精確的預言，其如果非真，很容易被證偽？
需要非常強的洞察力與大量的工作才能提出一個能夠通過如上測試的猜想，不過如果成功了，得到的就不僅僅是一個孤立的命題，卻是一個與其他命題有著無數(shù)聯(lián)系的命題。這增加了它被證明的可能性，也極大地增加了一個命題的證明會引出其他命題的證明的可能性。即使一個好的猜想的反例也能揭示許多東西：如果該猜想與許多其他的命題相關，該反例的影響會滲透到整個領域。
The more precise the predictions that follow from a conjecture, the more impressive it is when they are confirmed by later numerical evidence. Of course, this is true not just of mathematics but of science more generally.
一個猜想的預言越精確，當它后來被數(shù)值證據(jù)證明時，它就更令人印象深刻。當然這不僅僅對于數(shù)學成立，對于更一般的科學也是如此。
Although almost nothing has been rigorously proved, physicists have a collection of nonrigorous methods that, if used carefully, seem to give correct results. With their methods, they have in some areas managed to establish statements that go well beyond what mathematicians can prove. Such results are fascinating to mathematicians, partly because if one regards the results of physicists as mathematical conjectures then many of them are excellent conjectures, by the standards explained earlier: they are deep, completely unguessable in advance, widely believed to be true, backed up by numerical evidence, and so on. Another reason for their fascination is that the effort to provide them with a rigorous underpinning often leads to significant advances in pure mathematics.
物理學家有一系列不嚴格的方法，雖然幾乎都沒有被嚴格證明過，但如果小心使用，似乎能給出正確的結果。采用他們的方法，能夠在一些領域建立一些命題，而這些命題遠非數(shù)學家們所能夠證明。這些結論對于數(shù)學家是非常有吸引力的，一部分是因為如果把這些物理學家的結果當作數(shù)學猜想，按照之前解釋過的標準，它們中的許多都是優(yōu)秀的猜想：它們是深刻的，不可能事先猜測出來，被廣泛認為是真實的，具備數(shù)值證據(jù)的支持，等等。其吸引力的另外一個理由是，用嚴格的基礎證明它們，往往帶來純數(shù)學的重大進展。
One might wonder whether rigor is important: if the results established by nonrigorous arguments are clearly true, then is that not good enough? As it happens, there are examples of statements that were “established” by nonrigorous methods and later shown to be false, but the most important reason for caring about rigor is that the understanding one gains from a rigorous proof is frequently deeper than the understanding provided by a nonrigorous one. The best way to describe the situation is perhaps to say that the two styles of argument have profoundly benefited each other and will undoubtedly continue to do so.
人們可能懷疑嚴格性是否還重要：如果非嚴格的論證建立的結論明確是真的，這還不夠好嗎？有例子表明，確實有不嚴格的方法“建立”起來的命題后來被證明是非真的，但在乎嚴格性的更重要的理由是人們從一份嚴格的證明中獲得的理解往往遠比一份不嚴格的證明中獲得的理解要深刻得多。對這種局面最好的描述是，這兩種論證的風格都深深受益于彼此，而以后無疑還會繼續(xù)如此。
9 Finding Explicit Proofs and Algorithms （尋求顯式的證明和算法）
A fundamental dichotomy in mathematics: If you are proving that a mathematical object exists, then sometimes you can do so explicitly, by actually describing that object, and sometimes you can do so only indirectly, by showing that its nonexistence would lead to a contradiction.
數(shù)學中基礎的兩分法：如果你在證明一個數(shù)學對象存在，那么有時你可以通過實際描述這個對象來顯式地證明，而有時你只能通過證明其不存在會帶來矛盾來間接證明其存在性。
Just as, all else being equal, a rigorous argument is preferable to a nonrigorous one, so an explicit or algorithmic argument is worth looking for even if an indirect one is already established, and for similar reasons: the effort to find an explicit argument very often leads to new mathematical insights. (Less obviously, as we shall soon see, finding indirect arguments can also lead to new insights.)
就像，一切其他條件相同時，一個嚴格的論證優(yōu)于一個不嚴格的論證，一個顯式的或有算法的論證是值得尋找的，即使一個間接的論證已經(jīng)建立?；谙嗨频睦碛桑簩ふ乙粋€顯式論證的努力常常帶來全新的數(shù)學洞察。（不那么顯然地，正如我們很快會看到的，尋找間接論證也能帶來新的洞察。）
10 What Do You Find in a Mathematical Paper? （在數(shù)學論文中可以找到什么？）
A typical paper is usually a mixture of formal and informal writing. Ideally (but by no means always), the author writes a readable introduction that tells the reader what to expect from the rest of the paper. And if the paper is divided into sections, as most papers are unless they are quite short, then it is also very helpful to the reader if each section can begin with an informal outline of the arguments to follow. But the main substance of the paper has to be more formal and detailed, so that readers who are prepared to make a sufficient effort can convince themselves that it is correct.
一篇典型的數(shù)學論文通常是形式的和非形式的寫作風格的混合物。理想情況下（但并非總是如此），作者會寫一段可讀的介紹來告訴讀者能從論文中的其他部分讀到什么。而且如果論文被分成幾個部分（大部分論文都如此，除非它們非常短），如果每一部分都以一段后繼論證的非形式的大綱開始，對讀者會有很大幫助。但論文的主要實質部分應該是比較形式、比較詳細的，使得做好付出充分努力準備的讀者，可以說服他們自己該論文是正確的。
The most important of these statements are usually called theorems, but one also finds statements called propositions, lemmas, and corollaries. One cannot always draw sharp distinctions between these kinds of statements, but in broad terms, this is what the different words mean. A theorem is a statement that you regard as intrinsically interesting, a statement that you might think of isolating from the paper and telling other mathematicians about in a seminar, for instance. The statements that are the main goals of a paper are usually called theorems. A proposition is a bit like a theorem, but it tends to be slightly “boring.” It may seem odd to want to prove boring results, but they can be important and useful. What makes them boring is that they do not surprise us in any way. They are statements that we need, that we expect to be true, and that we do not have much difficulty proving.
數(shù)學命題中最重要的通常被稱之為定理，不過論文中還有一些被稱之為命題、引理、推論（又稱系）。這些類型的命題很難做清楚的區(qū)分，但它們的字面意思大概說明了這種區(qū)分。一個定理，是一個本質上有趣的命題，比如它可以從論文中單獨抽出來在研討會上告訴其他數(shù)學家。一個命題有點像一個定理，但它們傾向于有點乏味。似乎證明乏味的結論很奇怪，但它們可能很重要而且有用。使得它們乏味的只是它們并不在任何角度上帶來驚喜。它們是我們需要的命題，我們期待它們?yōu)檎?，而且它們證明起來并不難。
Often, if you are trying to prove a theorem, the proof becomes long and complicated, in which case if you want anybody to read it you need to make the structure of the argument as clear as possible. One of the best ways of doing this is to identify subgoals, which take the form of statements intermediate between your initial assumptions and the conclusion you wish to draw from them. These statements are usually called lemmas.
通常，如果你想要證明一個定理，證明會變得很長和很復雜，這種情況下，如果你想讓任何人來閱讀它，你需要使得論證的結構越清晰越好。其中最好的辦法就是識別出子目標，其形式就是位于最初前提與你想要證明的最終結論中間的中介命題。這些命題通常被稱為引理。
A corollary of a mathematical statement is another statement that follows easily from it. Sometimes the main theorem of a paper is followed by several corollaries, which advertise the strength of the theorem. Sometimes the main theorem itself is labeled a corollary, because all the work of the proof goes into proving a different, less punchy statement from which the theorem follows very easily. If this happens, the author may wish to make clear that the corollary is the main result of the paper, and other authors would refer to it as a theorem.
一個數(shù)學命題的推論是另外一個很容易從其推出的命題。有時一篇論文的主定理后面會跟著一系列的推論，它們會展現(xiàn)出定理的威力。有時主定理本身被標為一個推論，因為所有證明的工作都是為了證明一個不同的、不那么簡練有力的命題，而主定理能從中很容易地推導出來。如果這種情況發(fā)生，作者可能希望說明這個推論是論文的主要結果，而其他作者會將其當作定理來引用。
A purely formal proof would be very long and almost impossible to read. And yet, the fact that arguments can in principle be formalized provides a very valuable underpinning for the edifice of mathematics, because it gives a way of resolving disputes. If a mathematician produces an argument that is strangely unconvincing, then the best way to see whether it is correct is to ask him or her to explain it more formally and in greater detail. This will usually either expose a mistake or make it clearer why the argument works.
一份純粹形式的證明會非常的長，而且?guī)缀鯖]法閱讀。然而，論證原則上可以被形式化，為數(shù)學大廈提供了非常有價值的基礎，因為它提供了一種解決爭端的途徑。如果一個數(shù)學家給出了一份奇怪的沒有說服力的論證，那檢驗它是否正確的最佳辦法就是要求他或她采用更為形式化、具備更豐富細節(jié)的方式解釋該論證。這通常能揭示出一個錯誤或者使得該論證為什么有效變得更清晰。
Some mathematicians will tell you that the main aim of their research is to find the right definition, after which their whole area will be illuminated. Yes, they will have to write proofs, but if the definition is the one they are looking for, then these proofs will be fairly straightforward.
有的數(shù)學家會告訴你，他們研究的主要目標是找到正確的定義，有了這些定義，他們的整個領域都被照亮了。是的，他們必須要寫證明，但如果定義正是他們所尋找的那個，這些證明會是相當直接了當?shù)摹?/p>
The main aim of an article in mathematics is usually to prove theorems, but one of the reasons for reading an article is to advance one’s own research. It is therefore very welcome if a theorem is proved by a technique that can be used in other contexts. It is also very welcome if an article contains some good unsolved problems.
一篇數(shù)學文章的主要目的常常是證明定理，但閱讀一篇文章的其中一個理由是推進自己的研究。因此，如果一個定理是被一種可以應用在其他場景的技巧所證明的，這篇文章會非常受歡迎。如果一篇文章包含一些好的尚未解決的問題，它也會很受歡迎。
Perhaps the most important feature of a good problem is generality: the solution to a good problem should usually have ramifications beyond the problem itself. A more accurate word for this desirable quality is “generalizability,” since some excellent problems may look rather specific.
一個好問題的最重要的特性，可能是其一般性：一個好問題的解，通常應該具備超越其自身的衍生影響。對于這個理想性質的一個更為精確的詞是“可泛化性，因為某些優(yōu)秀的問題可能看上去非常具體。
It is quite common for a good problem to look uninteresting until you start to think about it. Then you realize that it has been asked for a reason: it might be the “first difficult case” of a more general problem, or it might be just one well-chosen example of a cluster of problems, all of which appear to run up against the same difficulty.
很常見的一種情況是：在開始真正思考它之前，一個好問題看上去挺無趣的。然后你會意識到問出這樣的問題背后的原因： 它可能是一個更一般問題的首個困難特例，或者它可能是一簇問題的精心挑選的特例，它們?nèi)紩龅酵瑯拥睦щy。
Sometimes a problem is just a question, but frequently the person who asks a mathematical question has a good idea of what the answer is.
有時一個問題只是一個提問，但通常提出一個數(shù)學問題的人，對于答案應該是怎么樣的，也已經(jīng)有了很好的概念。
