If math is more than proof, we need to better celebrate the rest of it
If math is more than proof, we need to better celebrate the rest of it
如果数学不仅仅是证明，我们需要更好地赞美它的其他部分
A sentiment echoing throughout the mathematics community right now is that solving problems and generating proofs have always served as proxies for the true goal of mathematicians, which is to further human understanding. When proofs can be generated without that understanding, it undermines their value as a proxy. This immediately raises a question: What other proxies should we use instead?
目前数学界普遍存在一种共识：解决问题和生成证明一直以来都是数学家真正目标——即增进人类理解——的替代指标。当证明可以在缺乏这种理解的情况下被生成时，它们作为替代指标的价值就会被削弱。这立即引发了一个问题：我们应该使用什么其他的替代指标呢？
I want to propose that we more firmly define a notion of a “motivated explanation” and that we give novel and compelling motivated explanations academic credit similar to what generating new proofs of open problems has had historically. Further, I believe this is an important step to help those outside of math better understand what it is that mathematicians contribute.
我想提议，我们应该更明确地定义“动机性解释”（motivated explanation）的概念，并给予新颖且引人入胜的动机性解释以学术认可，其地位应类似于历史上解决开放性问题所获得的认可。此外，我认为这是帮助数学界以外的人更好地理解数学家贡献的重要一步。
If outsiders believe that proof-generating machines render mathematicians obsolete, while insiders see that as a misconception of what researchers add, it’s incumbent on this community to better project its true values through the kind of work that it rewards. Outsiders can be forgiven for this misunderstanding if the work most celebrated skews heavily toward generating proofs, while clarification and exposition are treated as second-class.
如果外行认为证明生成机器使数学家变得过时，而内行认为这是对研究者价值的误解，那么数学界就有责任通过奖励那些体现其真正价值的工作来更好地展示自己。如果最受推崇的工作严重偏向于生成证明，而澄清和阐述被视为二等工作，那么外行产生这种误解是可以被原谅的。
I should acknowledge up front an obvious personal bias. I have a non-traditional career in math, focused on producing videos about the topic. This shares the goal of “furthering human understanding”, but my focus has been on explanations and intuitions that resonate with the public, not on solving outstanding problems. A cynic could easily read this proposal as shamelessly self-elevating. As a practical matter, though, my own career and funding exist outside academia, and I have no skin in the game for what this community assigns credit to.
我首先要承认一个明显的个人偏见。我的数学职业生涯是非传统的，专注于制作关于数学的视频。这与“增进人类理解”的目标是一致的，但我的重点在于与公众产生共鸣的解释和直觉，而不是解决悬而未决的问题。愤世嫉俗者很容易将此提议解读为无耻的自我吹捧。然而，从实际情况来看，我的职业生涯和资金来源都在学术界之外，对于这个社区如何分配学术认可，我并没有切身利益。
Moreover, in proposing that we elevate the status of motivated explanations, I don’t mean popularization. I mean any work which primarily aims to answer the question “how would you think of that?”, even if the subject matter requires deep expertise to appreciate. The examples I highlight below show this is nothing new. Practicing mathematicians already devote a meaningful amount of mindshare to work like this. The proposal here is mainly to 1) more clearly define this work, and 2) elevate its status.
此外，我提议提升动机性解释的地位，并不意味着科普。我指的是任何主要旨在回答“你是怎么想到这一点的？”这一问题的工作，即使该主题需要深厚的专业知识才能欣赏。我在下面强调的例子表明，这并不是什么新鲜事。从业数学家已经投入了相当多的精力在这样的工作上。这里的提议主要是：1）更清晰地定义这类工作，2）提升其地位。
What defines a motivated explanation? Although it might be clear what this phrase “motivated explanation” is intended to mean, it’s worth briefly contrasting it with proof. In a proof, definitions sit at the start. It is common and expected to begin with a new construction and proceed by analyzing its properties. In a motivated explanation, definitions sit in the middle. New constructions are only allowed to enter the vocabulary if the problem they are addressing has been clearly established.
什么是动机性解释？虽然“动机性解释”这个短语的含义可能很明确，但有必要将其与证明进行简要对比。在证明中，定义位于开头。通常的做法是先引入一个新的构造，然后分析其性质。而在动机性解释中，定义位于中间。只有当它们所解决的问题被明确建立后，新的构造才被允许进入词汇表。
In a proof, all statements must be correct, each claim following as a necessary implication from what comes before. In a motivated explanation, it is okay and often desirable to start with an idea that is not quite right and requires correction, but whose origins are relatable. A genre of motivated explanation I’m fond of is “discovery fiction”, a term coined by Michael Nielsen. You develop an idea with a narrative that starts with a simple-but-wrong solution to a problem, see where it breaks down, fix that problem, discover a new problem, and so on.
在证明中，所有陈述必须正确，每个主张都必须是前文的必然推论。而在动机性解释中，从一个不太正确、需要修正但其起源易于理解的想法开始，是可以的，甚至往往是可取的。我喜欢的一种动机性解释类型是迈克尔·尼尔森（Michael Nielsen）创造的术语——“发现小说”（discovery fiction）。你通过一个叙事来发展一个想法，从一个简单但错误的问题解决方案开始，观察它在何处失效，修复该问题，发现一个新问题，依此类推。
The scope of a proof is to explain why a particular theorem is true. The scope of a motivated explanation is not only to clarify why a theorem is true, but why the theorem is the right one to pose in the first place, and how it is used in the surrounding context.
证明的范围是解释为什么某个特定定理是正确的。动机性解释的范围不仅是阐明为什么定理是正确的，还要阐明为什么这个定理最初就是值得提出的，以及它在周围语境中是如何被使用的。
One clear shortcoming of a motivated explanation is that its validity is not binary the way a proof’s is. This is a big reason proof is so useful a way to measure progress: You can clearly define what does and does not have a proof yet. There will never be Lean for motivated explanations. If we’re serious about the goal of advancing human understanding, there’s no way around the fact that this aim is intrinsically squishier than that of finding proofs, because defining human understanding itself is squishier.
动机性解释的一个明显缺点是，其有效性不像证明那样是二元的。这是证明作为衡量进展方式如此有用的一个重要原因：你可以清楚地定义什么是已经有证明的，什么还没有。动机性解释永远不会有像 Lean（自动定理证明器）那样的工具。如果我们认真对待增进人类理解这一目标，就无法回避这样一个事实：这个目标本质上比寻找证明要模糊得多，因为定义“人类理解”本身就是模糊的。
The reason I’m leaning on the word “motivated”, as opposed to other potential choices like “lucid” or “demystifying”, is that this is a more verifiable property. It’s not quite as rigidly verifiable as a proof; almost nothing is. But it’s enough to be a practical measure. In my own work, I often repeat the phrase “I want this to feel like you could have discovered it yourself”. I say this not just to placate a viewer, but because it’s an actionable guideline for myself to assess whether an explanation feels complete or not. For each new idea introduced, you can ask whether it’s clear where that idea comes from. The answer is not quite a binary yes or no, but it’s close enough for practical purposes.
我倾向于使用“动机性”（motivated）这个词，而不是“清晰”（lucid）或“去神秘化”（demystifying）等其他潜在选择，是因为这是一个更具可验证性的属性。它不像证明那样具有严格的可验证性（几乎没有什么能做到这一点），但它足以成为一种实用的衡量标准。在我自己的工作中，我经常重复一句话：“我希望这让人感觉像是你自己也能发现的。”我这么说不仅仅是为了安抚观众，更是因为这是一个可操作的准则，用来评估一个解释是否完整。对于引入的每一个新想法，你都可以问：这个想法的来源是否清晰？答案虽然不是简单的“是”或“否”，但对于实际目的来说已经足够接近了。
Exemplars of motivated explanations: One of the best repositories I can think of for motivated explanations is Part IV of the Princeton Companion to Mathematics. It covers over two dozen active fields of research, each one introduced by an expert with a talent for clear communication. Whether it’s Andrew Granville explaining analytic number theory, or David Ben-Zvi introducing moduli spaces, these articles offer a level of intuition and motivation more typically found in one-on-one conversation at a blackboard. The background on this book is noteworthy.
动机性解释的典范：我能想到的最好的动机性解释库之一是《普林斯顿数学指南》（Princeton Companion to Mathematics）的第四部分。它涵盖了二十多个活跃的研究领域，每个领域都由一位擅长清晰表达的专家介绍。无论是安德鲁·格兰维尔（Andrew Granville）解释解析数论，还是大卫·本-兹维（David Ben-Zvi）介绍模空间，这些文章提供的直觉和动机水平，通常只出现在黑板前的一对一交谈中。这本书的背景值得注意。