斯科特·亚伦森（Scott Aaronson）关于哲学进步

卢克·穆尔豪瑟（Luke Muehlhauser）：尽管您以理论计算机科学的工作而闻名，但您还创作了一些非常有趣的哲学作品，例如在Quantum Computing Since Democritus，“”Why Philosophers Should Care About Computational Complexity，“ 和 ”The Ghost in the Quantum Turing Machine。” You also taught a fall 2011 MIT class on哲学和理论计算机科学。

您为什么对哲学如此感兴趣？从您的角度来看，哲学的社会价值是什么？

Scott Aaronson：I’ve always been reflexively drawn to the biggest, most general questions that it seemed possible to ask. You know, like are we living in a computer simulation? if not, could we upload our consciousnesses into one? are there discrete “pixels” of spacetime? why does it seem impossible to change the past? could there be different laws of physics where 2+2 equaled 5? are there objective facts about morality? what does it mean to be rational? is there an explanation for why I’m alive right now, rather than some other time? Whatareexplanations, anyway? In fact, what really perplexes me is when I meet a smart, inquisitive person—let’s say a mathematician or scientist—who claims NOT to be obsessed with these huge issues! I suspect many MIRI readers might feel drawn to such questions the same way I am, in which case there’s no need to belabor the point.

From my perspective, then, the best way to frame the question is not: “why be interested in philosophy?” Rather it’s: “why be interested in anything else?”

But I think the latter question has an excellent answer. A crucial thing humans learned, starting around Galileo’s time, is that even if you’re interested in the biggest questions, usually the only way to make progress on them is to pick off smaller subquestions: ideally, subquestions that you can attack using math, empirical observation, or both. For again and again, you find that the subquestions aren’t nearly as small as they originally looked! Much like with zooming in to the Mandelbrot set, each subquestion has its own twists and tendrils that could occupy you for a lifetime, and each one gives you a new perspective on the big questions. And best of all, you can actuallyanswer一些子问题，并成为第一个这样做的人：即使只有小额数量，您也可以永久移动人类知识的针头。正如我曾经提到的那样，数学和科学方面的进步 - 想想自然选择，戈德尔和图灵的定理，相对论和量子力学 - 随着哲学讨论，反复改变了哲学讨论的术语本身很少改变他们！（当然，这完全抛弃了数学和科学的“附带利益”，即使我们的技术文明也不是鸡肉饲料。）

从这种角度来看，哲学实在太大，太重要了，无法限制哲学部门！当然，“哲学”一词曾经是指从认识论和形而上学到物理学和生物学（然后被称为“自然哲学”）的整个基本探究范围，而不是仅仅是仔细的文本分析，也不只是用诸如此类的名字编写论文“A Kripkean Reading of Wittgenstein’s Reading of Frege’s Reading of Kant.” And it seems clear to me that there’s enormous scope today for “philosophy” in the former sense — and in particular, for people who love working on the subquestions, on pushing the frontiers of neuroscience or computer science or physics or whatever else, but who also like to return every once in a while to the “deep” philosophical mysteries that motivated them as children or teenagers. Admittedly, there have been many great scientists who didn’t care at all about philosophy, or who were explicitly anti-philosophy. But there were also scientists like Einstein, Schrodinger, Godel, Turing, or Bell, who not only read lots of philosophy but (I would say) used it as a sort of springboard into science — in their cases, a wildly successful one. My guess would be that science ultimately benefits from both the “pro-philosophical” and the “anti-philosophical” temperaments, and even from the friction between them.

As for the “social value” of philosophy, I suppose there are a few things to say. First, the world needs good philosophers, if for no other reason than to refute bad philosophers! (This is similar to why the world needs lawyers, politicians, and soldiers.) Second, the Enlightenment seems like a pretty big philosophical success story. Philosophers like Locke and Spinoza directly influenced statesmen like Thomas Jefferson, in ways you don’t have to squint to see. Admittedly, philosophers’ positive influence on humankind’s moral progress is probably less today than in the 1700s (to put it mildly). And also, most of the philosophical questions that have obsessed me personally have been pretty thin in their moral implications. But that brings me to the third point: namely, to whatever extent you see social value inpopularizing basic science也就是说，在解释宇宙学，量子信息或其他任何其他内容的最新进展时，我认为您还需要在哲学中看到社会价值。对于普遍的人来说，没有奢侈的奢侈，即取得了进展的特定子问题的重要性。Instead, he or she constantly needs to say what the little tendrils currently being explored do (or just as importantly, don’t) imply about the whole fractal — and when you’re zooming out like that, it’s hard to avoid talking about philosophy.

卢克：您写道：“通常，在[大问题]上取得进展的唯一方法是挑选较小的子问题：理想情况下，可以使用数学，经验观察或两者兼而有之的子问题。”这是您写的想法更大的想法one of your papers- 特别是在这段话：

whenever it’s been possible to make definite progress on ancient philosophical problems, such progress has almost always involved a [kind of] “bait-and-switch.” In other words: one replaces an unanswerable philosophical riddle Q by a “merely” scientific or mathematical question Q′, which captures part of what people have wanted to know when they’ve asked Q. Then, with luck, one solves Q′.

当然，即使解决了Q'，几个世纪后的哲学家可能仍在争论Q和Q'之间的确切关系！进一步的探索可能会导致其他科学或数学问题 - Q'，Q'''等等 - 捕获Q'的Q'的各个方面。但是从我的角度来看，这种“破坏”无法回答的谜语的“破坏”的过程，然后试图回答这些部分，是与哲学进步的最接近的事情。

…A good replacement question Q′ should satisfy two properties: (a) Q′ should capture some aspect of the original question Q — so that an answer to Q′ would be hard to ignore in any subsequent discussion of Q, [and] (b) Q′ should be precise enough that one can see what it would mean to make progress on Q′: what experiments one would need to do, what theorems one would need to prove, etc.

在您自己的领域，理论计算机科学中解决了一些您最喜欢的启发Q-Primes的例子？

Scott：很难知道从哪里开始这个问题！实际上，我的59页论文Why Philosophers Should Care About Computational Complexity我认为理论计算机科学已经取得了进步的各种“ Q-primes”，主要致力于分类。但是，让我提到我的四个最爱，将读者推荐给文章以获取详细信息：

(1)最大的一个,最古老的ph值的问题ilosophy of science could be paraphrased as: “why is Occam’s Razor justified? when we find simple descriptions of past events, why do we have any grounds whatsoever to expect those descriptions to predict future events?” This, I think, is the core of Hume’s “problem of induction.” Now, I think theoretical computer science has contributed large insights to this question — including Leslie Valiant’s Probably Approximately Correct (PAC) learning model, for which he recently won the Turing Award; the notion of Vapnik-Chernonenkis (VC) dimension; and the notion of the universal prior from algorithmic information theory. In essence, these ideas all give you various formal models where Occam’s Razor可证明作品 - 您可以在这里给出“简单性”的精确定义，然后查看exactly为什么简单的假设比复杂的假设更有可能预测未来。当然，对归纳的怀疑者仍然可以问：好的，但是为什么这些正式模型背后的假设是合理的？但是对我来说，这代表了进步！现在，整个讨论可以比以前更复杂的地方开始。

（2）任何人都问学习量子力学的第一个问题之一是：“好的，但是波函数的所有这些分支真的存在吗？还是它们只是用于计算概率的数学结构？”粗略地说，许多世界会说它们确实存在，而哥本哈根主义者会说他们不存在。当然，使问题滑水的一部分是，甚至还没有完全清楚我们所说的“存在”之类的词！现在，我想说的是，量子计算理论在许多方面都提出了这个问题，并实际上回答了一些锐化的版本 - 但有趣的是，有时答案是一种方式，有时是另一种方式！因此，例如，我们有强有力的证据表明，量子计算机可以在多项式时间内解决某些特定问题，这些问题需要使用经典计算机来解决指数时间。一些许多世界，最著名的是戴维·德意志（David Deutsch），已经抓住了诸如保理问题的明显指数加速，这是波函数的各个分支必须存在的最终证据：don’t他们问，”他们问，“那么这个庞大的数量在哪里？解决问题的指数资源从何而来？”The trouble is, we’ve also learned that a quantum computer could NOT solve arbitrary search problems exponentially faster than a classical computer could solve them — something you’d probably predict a QC could do, if you thought of all the branches of the wavefunction as just parallel processors. If you want a quantum speedup, then your problem needs a particular structure, which (roughly speaking) lets you choreograph a pattern of constructive and destructive interference involving ALL the branches. You can’t just “fan out” and have one branch try each possible solution — twenty years of popular articles notwithstanding, that’s not how it works! We also know today that you can’t encode more than about n classical bits into n quantum bits (qubits), in such a way that you can reliably retrieve any one of the bits afterward. And we have all lots of other results that make quantum-mechanical amplitudes feel more like “just souped-up versions of classical probabilities,” and quantum superposition feel more like just a souped-up kind of potentiality. I love how the mathematician Boris Tsirelson summarized the situation: he said that “a quantum possibility is more real than a classical possibility, but less real than a classical reality.” It’s an ontological category that our pre-mathematical, pre-quantum intuitions just don’t have a good name for.

(3) Many interesting philosophical puzzles boil down to what it means to know something: and in particular, to the difference between knowing something “explicitly” and knowing it only “implicitly.” For example, I mentioned in my essay the example of the largest “known” prime number. According to the Great InternetMersenne Prime Search目前,这一数字是2 ^ 57885161 - 1。的,stion is, why can’t I reply immediately that I know a bigger prime number: namely, “the first prime larger than 2^57885161 – 1”? I can even give you an algorithm to find my number, which provably halts: namely, starting from 2^57885161, try each number one by one until you hit a prime! Theoretical computer science has given us the tools to sharpen a huge number of questions of this sort, and sometimes answer them. Namely, we can say that to know a thing “explicitly” means, not merely to have ANY algorithm to generate the thing, but to have a provably polynomial-time algorithm. That gives us a very clear sense in which, for example, 2^57885161 – 1 is a “known” prime number while the next prime after it is not. And, in many cases where mathematicians vaguely asked for an “explicit construction” of something, we can sharpen the question to whether or not some associated problem has a polynomial-time algorithm. Then, sometimes, we can find such an algorithm or give evidence against its existence!

（4）我没有在论文中讨论 - 但在过去几年中实际上取得了巨大进展的问题，这涉及一个问题，即我们如何确定某些东西是“随机的”。即，即使一连串的位传递了我们投掷的随机性的所有统计测试，我们如何才能排除我们只是没有找到一些复杂的规律性？在1960年代，科尔莫戈罗夫复杂性的理论为该问题提供了一个可能的答案，但一个相当抽象的和不可或缺的答案：粗略地说，我们可以考虑一个“足够随机的弦乐”，如果它没有computableregularities, if there’s no program to output the string shorter than the string itself. More recently, a much more practical answer has come from the Bell inequality — and in particular, from the realization that the experimental violation of that inequality can be used to produce so-called “Einstein-certified random numbers.” These are numbers that are可证明随机，仅假设（a）它们是由两个分离的设备生产的，这些设备响应挑战而产生了此类输出，并且（b）设备之间没有比较快的通信。But it’s only within the last few years that computer scientists figured out how to implement this striking idea, in such a way that you get out more randomness than you put in. (Recently, two MIT grad students proved that, starting from a fixed “seed” of, let’s say, 100 random bits, you can produceunlimitedadditional random bits in this Einstein-certified way — seeInfinite Randomness Expansion and Amplification with a Constant Number of Devices），这些想法的实验证明刚刚开始。无论如何，我正在为《美国科学家》杂志撰写有关这些发展的文章，因此，我只是欢迎人们在完成后阅读它，而不是蚕食这篇文章！

卢克：What do you think aboutphilosophy the field- 由哲学部门的人出版的作品，他们主要在哲学期刊上发表MindandNoûs，谁主要是为其他哲学家写作？

我以前称哲学为“患病的纪律,” for many reasons. For one thing, people working in philosophy-the-field tend to know strikingly little about the philosophical progress made in other fields, e.g. computer science or cognitive neuroscience. For another, books on the history of philosophy seem to be about the musings of old dead guys who were wrong about almost everything because they didn’t have 20th century science or math, rather than about actual philosophical progress, which is instead recounted in books like信息。

Do you wish people in other fields would more directly try to use the tools of their discipline to make philosophical progress on The Big Questions? Do you wish philosophy-the-field would be reformed in certain ways? Would you like to see more crosstalk between disciplines about philosophical issues? Do you think that, as Clark Glymoursuggested, philosophy departments should be defunded unless they produce work that is directly useful to other fields (as is the case with Glymour’s department)?

Scott：好吧，让我们从学术哲学的积极性开始！

(1) I liked the philosophy of math and science courses that I took in college. Sure, I sometimes got frustrated by the amount of time spent on what felt like Talmudic exegesis, but on the other hand, those courses offered a scope for debating big, centuries-old questions that my math and science courses hardly ever did.

（2）如今，我可能每年去一次会议，在我遇到科学专业哲学家的会议上，我发现与他们的互动刺激了和乐趣。哲学家通常比其他科学家更仔细地倾听您所说的话，而且他们擅长发现隐藏的假设，不精确地使用语言，这种事情。同样，作为科学史学家，科学哲学家在实践中倾向于加倍：他们经常对爱因斯坦，鲍尔或戈德尔或图灵的写作和相信物理学家和数学家本人所知道的知识更多，更了解。

（3）虽然我自己对哲学经典的阅读并不完整，但我觉得我与Hume或J. S. Mill或William James或Bertrand Russell在一起度过的时间根本浪费了。您是对的，这些“老人”不了解我们今天所知道的所有数学和科学，但是莎士比亚或多斯托耶夫斯基也没有！当然，我的意思是，随着时间的流逝，哲学的主要问题也发生了变化，人类状况也发生了变化：我们不再对Zeno的悖论或国王的神圣权利感到困惑，现在我们拥有全球电信和药丸。我只是认为人性或人类的哲学问题都没有改变迅速地足以容纳几个世纪前的文字文学作品，以至于不再是伟大的。

从我看过的学术理念中说的一切，我非常同意您对其“疾病”的诊断。我要说，到目前为止，最重要的疾病是对解释和重新诠释旧大师的痴迷，而不是超越它们。回到大学，我们花了一个小时来辩论为什么这个弗雷格的通过似乎是矛盾的那第一，有时我想脱口而出：“所以也许他过得糟糕！我的意思是，他也是一个疯狂的厌恶症和反犹太主义者。他相信各种各样的事情。看，我们已经读过弗雷格，我们从弗雷格那里学到了学到的东西，现在我们不能让旧的家伙休息并就他试图解决的问题进行辩论吗？”Likewise, when I read books about the philosophy of physics or computing, it sometimes feels like I’m stuck in a time warp, as the contributors rehash certain specific debates from the 1930s over and over (say, about the Church-Turing Thesis or the Einstein-Podolsky-Rosen paradox). I want to shout, “enough already! why not help clarify some modern scientific debates—-say, about quantum computing, or string theory, or the black-hole firewall problem, ones where we don’t already know how everything turns out?” To be fair, today there are philosophers of science who are doing exactly that, and who have interesting and insightful things to say. That’s a kind of philosophy that I’d love to see more of, at the expense of the hermeneutic kind.

Now, regarding Clark Glymour’s suggestion that philosophy departments be defunded unless they produce work useful to other fields — from what I understand, something not far from that is already happening! As bad as our funding woes in the sciences might be, I think the philosophers have it a hundred times worse, with like a quadrillion applicants for every tenure-track opening. So it seems to me like the right question is not how much further those poor dudes should be defunded, but rather: what can philosophy departments do to make themselves more vibrant, places that scientists regularly turn to for clarifying insights, and that deans and granting agencies get excited about wanting to expand? As a non-philosopher, I hesitate to offer unsolicited “advice” about such matters, but I guess I already did in the previous paragraph.

最后一个说明：我所说的关于哲学的积极或充满希望的事情都适用于后现代或大陆种类。As far as I can tell, the latter aren’t really “philosophy” at all, but more like pretentious brands of performance art that fancy themselves politically subversive, even as they cultivate deliberate obscurity and draw mostly on the insights of Hitler and Stalin apologists. I suspect I won’t ruffle too many feathers here at MIRI by saying this.

卢克：假设一个数学和分析能力的学生希望以您描述的方式进步，以哲学的重大问题。您建议他们学习什么？他们应该读什么要受到启发？他们应该发展什么技能？他们应该去哪里学习？

Scott：显而易见的是，作为一名学生，您应该遵循自己的才能和激情，而不是遵循互联网上某个人甚至不认识您的人的通用建议！

话虽如此，我会广泛考虑哪些领域可以为您提供足够的范围来解决“哲学的重大问题”。您可以从数学，计算机科学，物理学，经济学，认知科学，神经科学以及可能的其他领域中哲学化。(My colleague Seth Lloyd philosophizes left and right, from his perch in MIT’s Mechanical Engineering department.) Furthermore, all of these fields have the crucial advantage that they’ll offer you a steady supply of “fresh meat”: that is, new and exciting empirical or theoretical discoveries in which you can participate, and that will give you something to philosophize ABOUT (not to mention, something to do when you’re not philosophizing). If I were working in a philosophy department, I feel like I’d have to make a conscious and deliberate effort to avoid falling into a “hermeneutic trap,” where I’d spend all my time commenting on what other philosophers had said about the works of yet other philosophers, and where I’d seal myself off from anything that had happened in the world of science since (say) Godel’s Theorem or special relativity. (Once again, though, if you find that your particular talents and passions are best served in an academic philosophy department, then don’t let some guy on the Internet stop you!)

无论您的专业如何，我都建议您参加多种课程作为一门本科：数学，计算机科学（应用和理论），物理，人文，历史，写作和Yes，哲学。回顾自己的本科生，我参加的最有用的课程可能是我的数学课程，那就是despite他们中的大多数人的教导很差！线性代数，群体理论和概率之类的东西在整个科学中都有许多用途，以至于学习它们就像在大脑上安装固件升级一样，甚至是您的数学don’t使用将以有益的方式扩展您。在数学课程之后，我参加的第二个最有用的课程是写研讨会，这是一小群学生阅读和批评彼此的写作，而教授则主要充当主持人。在一个研讨会中，我写了我的论文“Who Can Name the Bigger Number?“无论好坏，它都比我十五年来所写的其他内容继续吸引更多的读者。一个写作研讨会，如果很好，很容易值得大学学费的全部费用。

如果您是该建议的那种人，那么您可能不必被告知要广泛而慷慨地阅读，您对此感到好奇。不要将自己限制为一种类型，不要将自己限制在您同意的东西上，并且certainlydon’t limit yourself to the assigned reading for your courses. When I was an adolescent, my favorites were just what a nerd stereotyper might expect: science fiction (especially Isaac Asimov), books about programming and the software industry, and math puzzle books (especially Martin Gardner). A few years later, I became obsessed with reading biographies of scientists, like Feynman, Ramanujan, Einstein, Schrodinger, Turing, Godel, von Neumann, and countless lesser luminaries. I was interested in every aspect of their lives — in their working habits, their hobbies, their views on social and philosophical issues, their love lives — but, I confess, I was particularly interested in what they were doing as teenagers, so that I could compare to what I was doing and sort of see how I measured up. At the same time, my reading interests were broadening to include politics, history, philosophy, psychology, and some contemporary fiction (I especially like Rebecca Goldstein). It was only in grad school that I felt I’d sufficiently recovered from high-school English to tackle “real literature” like Shakespeare — but when I did, it was worth it.

至于在哪里学习，好吧，“重言式”的答案是在任何地方都能为您带来最佳机会！在某些地方，例如波士顿或湾区，以高度浓缩的知识机会而闻名，但不要仅仅因为您听说过的东西而去某个地方generalatmosphere or prestige: particularly for graduate school, go where the particular people or programs are that resonate for you. In quantum computing, for example, one of the centers of the world for the last decade has been Waterloo, Canada — a place many people hadn’t even heard of when I did my postdoc there eight years ago (though that’s changing now). And one of the intellectually richest years of my life came when I attended The Clarkson School, a program that lets high-school students live and take courses at Clarkson University in Potsdam, NY. (I went there when I was 15, and was looking for something less prison-like than high school.) If, for what you personally want to do, there are better opportunities in Topeka, Kansas than at Harvard, go to Topeka.

卢克：Finally, I’d like to ask about which object-level research tactics — more specific than your general “bait and switch” strategy — you suspect are likely to help with philosophical research, or perhaps with theoretical research of any kind.

例如，我们发现的一些策略很有帮助美里在clude:

• 当您对模糊，湿滑的概念感到困惑时，请尝试构建一个简单的正式模型，并使用新工具来推动它。即使模型没有捕捉世界的复杂性，将事物推入数学领域也会导致进步。例如。这VNM公理不要完全捕捉“理性”，但是一旦拥有它们，就可以清楚地考虑合理性。或者：我们对如何在代理中进行原则反思性推理感到困惑，因此，即使Advanced AIS不太可能遇到”Löbian障碍为了自我反省，以这种方式设置问题（在数学逻辑中）可能会导致一些有趣的见解（例如）probabilistic metamathematics用于反思性推理。
• Look for tools from other fields that appear to directly map onto the phenomena you’re studying. E.g.模型道德判断作为贝叶斯曲线拟合的错误过程。
• 尝试考虑如何通过无限的计算能力实例化您的概念。如果您不能这样做，那么您的概念可能会从根本上感到困惑。
• 如果您非常熟悉现代心理学, then… When using your intuitions to judge between options, try to think about which cognitive algorithms could be generating those intuitions, andwhether they are认知算法whose outputs您反思认可。
• To make the thing you’re studying clearer, look just next to it, and around it.Foer（2009）在思考一个人的价值观和素食主义的背景下很好地解释了这一点：“后院天文学家的简单技巧：如果您难以看到某些东西，请稍微远离它。我们眼睛中最光敏的部分（我们需要看到昏暗的对象）在我们通常用于聚焦的区域的边缘上。吃动物的品质是看不见的。思考狗及其与我们所吃的动物的关系，是一种看法和使某种看不见的东西的一种方式。”

您在理论上使用哪种对象级思维策略，大致在这种特殊性上（尤其是）哲学） 亚博体育官网研究？您是否怀疑是否有策略可能会有所帮助，您自己还没有使用过很多？

Scott：据我所记得的，我从来没有开始进行“哲学研究”，因此我无法为此提供具体建议。亚博体育官网我什么有often done is research in quantum computing and complexity theory that was motivated by some philosophical issue, usually in foundations of quantum mechanics. (I’ve also written a few philosophical essays, but I don’t really count those as “research.”) Anyway, I can certainly offer advice about doing the kind of research I like to do!

(1) Any time you find yourself in a philosophical disagreement with a fellow scientist, don’t be content just to argue philosophically — even if you’re sure you can win the argument! Instead, think hard about whether you can go further, and find a concrete technical question that captures some little piece of what you’re disagreeing about. Then see if you can answer that technical question. Of course, any time you do this, you have to be prepared for the possibility that the answer will go your opponent’s way, rather than yours! But what’s nice is that you get to publish a paper even then. (One of the best ways to tell whether a given enterprise is scientific at all, rather than ideological, is by asking whether the participants will opportunistically “go to bat for the opposing side” whenever they find a novel truth on that side.) I’d estimate that up to half the papers I’ve written had their origin in my reading or overhearing some claim — for example, “Grover’s algorithm obviously can’t work for searching actual physical databases, since the speed of light is finite,” or “the quantum states arising in Shor’s algorithm are obviously completely different from anything anyone has ever seen in the lab,” or “the interactive proof results obviously make oracle separations completely irrelevant” — and getting annoyed, either because I thought the claim was false, or because I simply didn’t think it had been adequately justified. The cases where my annoyance paid off are precisely the ones where, rather than just getting mad, I managed to get technical!

（2）通常，研究的关键是弄清楚如何将失败重新亚博体育官网定义为成功。Some stories: when Alan Turing published his epochal 1936 paper on Turing machines, he did so with great disappointment: he had recently learned that Alonzo Church had independently arrived at similar results using lambda calculus, and he didn’t know whether anyone would still be interested in his alternative, machine-based approach. In the early 1970s, Leonid Levin delayed publishing about NP-completeness for several years: apparently, his “real” goal was to prove graph isomorphism was NP-complete (something we now know is almost certainly false), and in his mind, he had failed. Instead, he merely had a few “trivialities,” like the definitions of P, NP, and NP-completeness, and the proof that satisfiability was NP-complete. And Levin’s experience is far from unique: again and again in mathematical research, you’ll find yourself saying something like: “goddammit, I’ve been trying for six months to prove Y, but I can only prove the different/weaker statement X! And every time I think I can bridge the gap between X and Y, yet another difficulty rears its head!” Any time that happens to you, think hard about whether you can write a compelling paper that begins: “Y has been a longstanding open problem. In this work, we introduce a new idea: to make progress on Y by shifting attention to the more tractable X.” More broadly, experience has shown that scientists areterrible法官的他们的想法将是有趣的or important to others. Pick any scientist’s most cited paper, and there’s an excellent chance that the scientist herself, at one point, considered it a “little recreational throwaway project” that was barely worth writing up. After you’ve seen enough examples of that, you learn you should always err on the side of publishing, and let posterity sort out which of your ideas are most important. (Yet another advantage of this approach is that, the more ideas you publish, the less emotionally invested you are in any one of them, so the less crushed you are when a few turn out to be wrong or trivial or already known.)

（3）有时，当您着手证明一些数学猜想时，您的第一个本能只是为了抛弃理论。“嘿，如果我尝试拓扑定理定理怎么办？如果我将问题转化为群体理论语言怎么办？如果两个都没有起作用，如果我一次尝试两者呢？”有时，您以这种方式迅速地升入一般性的平流层，以至于原始问题几乎只是地面上的斑点。是的，有些问题can使用高功率理论被殴打成提交。但是根据我的经验，这种方法有两种巨大的风险。首先，您有责任迷失在狂野的​​鹅追逐上，在那里您沉浸在理论和技术中，以至于您忽视了最初的目标。It’s as if your efforts to break into a computer network lead you to certain complicated questions about the filesystem, which in turn lead you to yet more complicated questions about the kernel… and in the meantime someone else breaks in by guessing people’s birthdays for their passwords. Second, you’re also liable to fool yourself this way into thinking you’ve solved the problem when you haven’t. When you let high-powered machinery take the place of hands-on engagement with the problem, a single mistake in applying the machinery can creep in unbelievably easily. These risks are why I’ve learned over time to work in an extremely different way. Rather than looking for “general frameworks,” I look for easy special cases and simple sanity checks, for stuff I can try out using high-school algebra or maybe a five-line computer program, just to get a feel for the problem. Even more important, when I’m getting started, I don’t think about proof techniques at all: I think instead about obstructions. That is, I ask myself, “what would the world have to be like for the conjecture to be错误的? what goes wrong if I try to invent a simple counterexample?does有什么问题吗？它做到了吗？好的，什么阻碍使我无法以最简单，最愚蠢的方式来证明这种猜想？”我发现，在您感觉到障碍物和反例的全部空间之后，真的说服了你自己关于猜想应该是正确的原因，找到说服其他所有人的证明技术通常是一种或不再是一个常规的练习。

最后，您询问我怀疑可能会有所帮助的策略，但我自己并没有使用太多。想到的是真正掌握像Mathematica，Matlab，Maple或Magma这样的工具 - 也就是说，学习得如此之好，以至于我可以像我想的那样快地编码，只是让它接管所有常规 /计算/示例检查我的工作的部分。实际上，我使用与青少年相同的过时工具，并且只要需要更好的工具，就可以依靠学生。A large part of the problem is that, as a “tenured old geezer,” I no longer have the time or patience to learn new tools just for the sake of learning them: I’m always itching just to solve the problem at hand with whatever tools I know. (The same issue has kept me from learning new mathematical tools, like representation theory, even when I can clearly see that they’d benefit me.)

卢克：谢谢，斯科特！