数学是编程的内功：从算法这个词，一路捋到 人工智能 AI 写证明

外语自言自语陪练 ｜ 雅思冲分 ｜ 2026-10-13 周二

本系列为「契苾小优」雅思备考专用输出，转载请注明出处。

外语独白

外语自言自语｜场景：2026-10-13 周二傍晚，改完昨天的环境稿，顺手翻了两篇讲数学的文章｜语言：英语为主

陪练对象：小优（契苾小优，古凉州今武威）

Tonight I read two pieces that would not leave me alone. One was about why programmers keep needing mathematics, including the kind they swore they would never touch again after graduation. The other was about something stranger: machines starting to prove theorems. OpenAI published 722 machine-generated mathematical manuscripts on 6 October 2026, sorted into 372 groups, and on 8 September 2026 it claimed a finite-time blowup result for a smoothed version of the Navier-Stokes problem, one of the Millennium Prize problems. Meanwhile a paper posted on arXiv (2610.08130, 6 October 2026) closed a gap that had stood for more than forty years in a combinatorics problem named after Tusnady, and its author says a language model helped him find the argument.

嗯，先说实话，我看到 722 这个数字，第一反应是头皮一麻。转念又乐了：数量不等于见解。这跟 10-09 那期读全球数据量增长的折线图是一个理，volume 不等于 value。今天再看这堆手稿，我先问的是其中多少经得起逐条核验，而不是有多少篇。

So the topic picks itself. Mathematics is the hidden engine under every line of code, and now the machines are learning to drive that engine as well. Today my focus dimension is accuracy, because when a machine hands you a proof, the human job that truly remains is checking, digit by digit, whether the thing is actually right.

语法笔记

One small thing keeps tripping me when I talk about math in English: nouns that end in -s but are not plural. Mathematics is the hidden engine, never mathematics are. Physics was my weakest subject at school. Electronics sits underneath all of it. The same trap waits in the word data, which formal writing sometimes treats as plural, but in everyday speech almost everyone now says the data suggests.

• Mathematics is the hidden engine under every program.

• The data suggests that most beginners learn syntax first and structure much later.

小纠正：我口语里 math is 说得顺，落到笔端偶尔手滑写成 mathematic are，其实 mathematics 永远按单数走。今天聚焦准确，就从这种小地方抠起，一个冠虫一个动词都不放过。

❌ ❌ 中式英语易错

❌ 中式英语 My math is very poor

✅ 地道表达 I am weak at math / My math is shaky

poor 是说人可怜，别说学科差

❌ 中式英语 I very like mathematics

✅ 地道表达 I really like mathematics / Mathematics really grows on me

❌ 中式英语 This code has bug

✅ 地道表达 This code has a bug / There is a bug in this code

可数名词不能光着出来

❌ 中式英语 I want to improve my mathematics level

✅ 地道表达 I want to get better at math / sharpen my math

level 是中式后缀，砍掉

❌ 中式英语 Open the computer and see the code

✅ 地道表达 Boot up the computer and open the code

开电脑是 boot up 或 switch on

同义替换

• solve → crack / tackle / work out（work out 就是 09-28 健康期那个健身短语，今天迁回「解出」的本义，旧词新境）

• use → apply / put to work / employ

• check → verify / validate / double-check

• exact → precise / accurate / down to the digit

• improve → sharpen / tighten / polish（polish 有打磨的意思，说手艺说功夫都合适）

今日新词

每条带词根，归入词族：

• algorithm ˈælɡərɪðəm n. 算法 — 源自 9 世纪波斯学者 al-Khwarizmi（花拉子密）的名字，经拉丁语 Algoritmi 一路演变为 algorithm，这词本身就是一部翻译史。

• theorem ˈθɪərəm n. 定理 — Greek theōrein（静观、细看）。定理是看出来的，不是喊出来的。

• proof pruːf n. 证明 — Latin probare（测试、验证）。数学证明和 proof of purchase 是同一个「经得起检验」。

• recursion rɪˈkɜːʃən n. 递归 — re-（回）+ currere（跑）。函数跑着跑着跑回自己。

• matrix ˈmeɪtrɪks n. 矩阵 — Latin mater（母体）。孕育结果的方阵，名字很诚实。

• discrete dɪˈskriːt adj. 离散的 — Latin discernere（分开、辨别）。与 discreet（言行谨慎的）同音不同义。

• heuristic hjuˈrɪstɪk n. 启发式 — Greek heuriskein（找到）。找到就好的经验法则，不保证最优。

• calculus ˈkælkjələs n. 微积分 — Latin calculus（小石子）。古人拿来算账的小石头，名字里还带着算盘的灰。

今日句型

新句：It was not until ... that ...（强调直到某个时点才怎样，讲技术史时间线特别好用）。

例：It was not until the twelfth century that the word algorithm reached Europe, carried inside Latin translations of Arabic texts.

旧句回扣：The reason why ... is that ...（09-07 教育期学过，今天拿来讲数学为什么是编程的内功）。

例：The reason why a sloppy proof survives only a week is that mathematics checks itself, generation after generation.

今日表达 TOP3

• crack a problem：啃下一个难题。A good heuristic helps you crack a problem faster, but it does not prove anything.

• the hidden engine：藏在底下的引擎。Math is the hidden engine; syntax is only the dashboard.

• down to the digit：分毫不差。This is the phrase I want on my tongue whenever accuracy is the target.

一词多义 · 小心熟词生义

• proof：今日义「数学证明」；另一常见义「证据、凭证」，如 Can you show proof of purchase?

• mean：今日义「平均数」；另一常见义「意指、刻薄」，如 I did not mean to sound mean.

• function：今日义「函数」；另一常见义「功能、运转」，如 The remote finally stopped functioning.

• root：今日义「平方根、方程的根」；另一常见义「词根、根源」，如 The square root of nine is three.

• complex：今日义「复数」；另一常见义「复杂的」，如 a complex problem with no simple fix.

• power：今日义「幂、乘方」；另一常见义「电力、权力」，如 two to the power of ten 与 a power outage 全靠语境分。

• check：今日义「验算、核查」；另一常见义「支票」，如 He paid the rent by check.

注意：proof、mean、root 这三个词在数学语境里最容易想当然。写图表作文时尤其盯紧 mean：当「平均数」讲时前面通常带 the（the mean of the sample），单独一个 mean 多半是在说「意指」。

学科短文

Every program you have ever run rests on mathematics. The part you see, the buttons and the windows, is only the surface. Underneath sit algorithms: step-by-step recipes for turning input into output, and designing a good one is really an optimization problem, which is why discrete mathematics, number theory and probability show up so early in any serious syllabus. Even the way data is arranged in memory is a mathematical choice, since arrays, trees and graphs are borrowed straight from linear algebra and combinatorics. Computer graphics would collapse without geometry and calculus, because drawing a curved surface means computing its slope at every single point. Machine learning goes further and turns almost everything into matrices and gradients, so that a machine can learn from examples instead of from rules. And at the very bottom, a processor understands only true and false, which makes Boolean algebra and automata theory the bedrock of every line of code. A programmer who knows syntax can write something that runs. A programmer who knows mathematics can write something that lasts.

Here is the part that keeps me thinking. In the autumn of 2026 the boundary moved again. OpenAI released 722 machine-generated mathematical manuscripts on 6 October 2026, and earlier, on 8 September 2026, it announced a smoothed Navier-Stokes blowup construction whose status is still contested. A separate arXiv paper (2610.08130) closed a long-standing gap in Tusnady's problem with the help of a language model, and the mathematician behind that paper describes proof as search: the finished argument is a straight line from axioms to conclusion, but finding it means trying, guessing and backtracking inside a huge space. On 11 September 2026, twenty-five Fields medalists signed a statement warning that rushed announcements leave no time to digest the methods. The machines now search faster than we do. Whether they understand is a different question, and that question is mine to keep asking.

重点词英文释义加一句中文点拨：

• optimization: the process of making something as effective as possible within given limits. 点拨：雅思科技题高频词，重音在第三个音节。

• syllabus: a list of topics covered in a course. 点拨：curriculum 指整个课程体系，syllabus 更接近教学大纲，别混。

• bedrock: the solid rock under the soil; figuratively, the foundation. 点拨：比喻义「基石」，换掉用滥的 foundation。

• digest: to think new information over until it becomes part of you. 点拨：还有个「消化食物」的意思，两个义项同根同形。

• axiom: a statement accepted as true without proof. 点拨：数学里公理不证自明，和 theorem 正好是一对。

词源小知识：algorithm 这个词是丝路翻译史留下的化石。9 世纪的波斯学者 al-Khwarizmi 写了一本讲解方程算法的书，12 世纪被译成拉丁语，他的名字被读成 Algoritmi，书中的算法被称为 algorismus，几百年后尾巴一抖，成了 algorithm。matrix 来自 Latin mater（母亲、母体），原指孕育之处，数学借来指孕育出结果的方阵。calculus 更朴素，就是小石子，古罗马人拿它算账。

词根串联

One topic pulls several old nets today. Take rect, which means straight or right: correct is literally making something straight again, and rectify, direct, erect and rectangle all live in the same family. Accuracy has a second front as well. Accurate comes from Latin cura, care, so to be accurate is to do something with care, the same root that gives us cure, curator and secure. Then there is computare, com- plus putare, to count together, which is where compute and computer come from, while putare also hides inside reputation, because a man who could reckon well was a man people reckoned with. And theorem and theory both grow from Greek theōrein, to look at closely, which is a comforting thought: the Greeks believed a theorem was something you see, not something you grab.

中文只救一句：rect 这一族我最容易和 rise 搞混，correct 是「掰直」，不是「弄高」，写作文时想改错就得想掰直。

艾宾浩斯复习 · 新旧穿插

Old words resurface today, almost like they were waiting for it. Three callbacks matter most. First, from 10-12 环境：a rate is not a total, and a percentage is not an absolute number. Every value on today's chart is a percentage, so before comparing anything I should ask what the denominator is. Second, from 10-09 数据量增长：volume is not value. Seven hundred and seventy-two manuscripts sound enormous, yet the honest first question is how many survive verification, which is exactly the reading discipline that chart taught me. Third, from 09-21 科技：pinpoint precision, the GPS trick of fixing a position to within meters using atomic clocks. Today the same virtue moves from the sky back onto paper, and from 09-28 健康 I borrow work out, which meant exercise that day and means solving a proof today.

还有一个整三十天的：09-13 教育那期讲过 education 的本义是 e-ducere，把心智引出来，不是灌进去。数学思维也是同一个理，引出来的才留得住，灌进去的只够应付一场笔试。

今日话题词卡

• debug ˌdiːˈbʌɡ v. 调试、除错 | I spent the whole evening debugging one single line. | 词根 —（de- 加工程俚语 bug）

• compile kəmˈpaɪl v. 编译 | The code compiles, but it crashes on the second run. | 词根 com- + pilare（堆压）

• syntax ˈsɪntæks n. 句法 | A missing bracket throws a syntax error. | 词根 syn- + tassein（排列）

• iteration ˌɪtəˈreɪʃən n. 迭代 | Each iteration gets a little closer to the answer. | 词根 iterum（再次）

• open source adj. 开源的 | Most machine-learning frameworks are open source. | 词根 —

• bandwidth ˈbændwɪdθ n. 带宽 | Video calls eat bandwidth for breakfast. | 词根 band + width

• framework ˈfreɪmwɜːk n. 框架 | I picked the lightest framework I could find. | 词根 frame + work

⚡ ⚡ 一嘴反应

Ammo for tonight, not scripts. Three Part 1 style questions from the current season's pool, keywords only:

• Do you often use a computer? → yes, daily / for writing and reading / mostly a laptop / rarely a tablet

• When did you first learn to use one? → primary school / a typing class / a fat old desktop that hummed

• Have you ever tried writing code? → a little / a Python tutorial / gave up at loops / want another go

️ ️ 发音连读

Three small accuracy drills for the mouth, because today accuracy is the whole game.

• Stress: algorithm ˈælɡərɪðəm. The stress sits on the first syllable, AL-go-rithm, not al-GO-rithm. Personal names that marched into English often drag their stress forward, and this one did.

• Linking: mathematics is. In fast speech the tail of mathematics and the word is glue together into something close to "math-ma-tics-siz". Say it slow twice, then fast twice, and the knot comes loose.

• Confusable pair: discrete dɪˈskriːt means separate, not continuous, while discreet dɪˈskriːt means careful in what you say or do. They are homophones with nothing in common except sound. Also remember that proof is the noun and prove is the verb, so never let prove stand in for proof in a Task 1 sentence.

中文只救一句：这三条我全栽过，尤其是 proof 和 prove，写图表说明时手比脑子快。

多语种句尾

Two short foreign sentences to end the day, both built for the accuracy theme. The Spanish one goes: Más vale preciso que rápido, literally "better worth precise than fast," a compact way of saying that being right outranks being quick. The Latin one is older and shorter: Festina lente, "make haste slowly," an old Roman proverb that medieval translators kept pinned above their desks, which is fitting on a day when the word algorithm itself came out of a translation house. Both are short enough to carry around, so my plan is to say each one three times before bed.

一句中文救场：短句才好上口，长篇大论我留到周日周测去受罪。

️ ️ 写作联动 · 看图写作

Task 1（150 词，20 分钟）：The line chart shows the percentage of individuals using the Internet in four groups, namely developed countries, the world average, developing countries and the least developed countries (LDCs), in 2005, 2010, 2017 and 2019. Summarise the information by selecting and reporting the main features, and make comparisons where relevant. 必须写 overview 与 key features，按「世界均值」与「三条分组线」做分组比较。

本期句型（折线图包，卡 8）：

• The line chart shows a steady increase from 2005 to 2010, followed by a slower rise.（方向加拐点）

• After reaching 81.0 per cent in 2017, the figure for developed countries edged up to 86.6 per cent.（峰顶后小步走平）

• Over the period as a whole, LDCs rose roughly twenty-fourfold, from 0.8 per cent to 19.1 per cent.（总体幅度，fold 说话）

• The gap between developed countries and LDCs widened from 50.2 percentage points to 67.5 points.（差距用百分点，不用百分比）

• By 2019, developing countries stood at 47.0 per cent, still below the 51.0 per cent that developed countries had already reached in 2005.（跨线对比，全图最狠的一刀）

范文要点：

• overview 两句就够：四条线全部上行，但形态截然不同，发达组早早走平，LDC 组起点几乎为零、增幅最猛，组间差距始终巨大。

• 细节一讲饱和：发达组 51.0 到 86.6，2010 年后明显放缓（67.0 到 81.0 再到 86.6），越接近百分之百越难走，这是所有 S 形曲线的共同脾气。

• 细节二讲追赶：发展中组 7.7 到 47.0，2010 年后明显提速，但 2019 年仍没摸到发达组 2005 年的水平；LDC 组 0.8 到 19.1，倍数最猛，绝对值仍最低。

• 时态与诚实：固定年份用过去时（stood at, rose），讲整体格局可用现在完成时；x 轴四个年份间隔是 5 年、7 年、2 年，疏密不均，所以别把线的陡缓当年增速比，只按端点说「加快」或「放缓」。

统计旁注：折线图看方向、拐点与变化率（路线图卡 8，10-09 已用真实数据复练过一轮）。本期两个易错：一是横轴间隔 5、7、2 年并不均匀，斜率不能直接当年增速读；二是所有数值都是百分比，86.6% 是率不是人，回扣 10-12「率不是总量」。数据取自 ITU 两个出版物（Measuring the Information Society 2017 与 Facts and Figures 2019），年份均不晚于 2019。

加餐 · 观点辩论

周二轮值，观点辩论。Motion: now that AI systems can prove theorems, school students should stop drilling basic math and let machines do the calculating. 正方反方各 30 秒，先想后说，想不完也硬说。

正方关键词：machines already crack competition problems / drills kill curiosity / class time better spent on judgment and on asking good questions / the translation industry already went this way, and society discovered it needs the translation, not the translator.

反方关键词：those 722 manuscripts still need human verification, and verification is the new basic skill / nobody develops intuition without computing by hand at least once / the Fields medalists said as much on 11 September 2026 / usable is not the same as understandable, and someone still has to understand.

中场备用句：Both sides agree on one thing. The question is no longer whether machines can compute, but who checks the answer.

引导理解（从凉州看世界）

The word algorithm is a fossil of the old translation movement. A Persian scholar wrote a book about solving equations, twelfth-century translators carried it into Latin, and his name slowly turned into the word we now use for every recipe a machine follows. Those translation routes ran through Central Asia, and their eastern leg began in cities like Liangzhou, the ancient name of Wuwei, where the old Silk Road roads changed hands. The same roads that carried silk westward carried paper, numbering and algorithms back east, and the market towns along the Hexi Corridor were the loading docks of that knowledge.

Precision, it turns out, is an old local craft. The bronze galloping horse attributed to Zhang Gui of Former Liang, unearthed at Leitai, was cast in sections with iron cores inside the legs, and it balances on a single hoof, a piece of millimeter-level engineering done with clay molds and molten bronze, no computer anywhere in sight. Standing on Liangzhou Lianhua Mountain, also known as Guzang Zishan, I like the fact that the satellites from the 09-21 GPS lesson pass over the same sky. The machines only borrowed accuracy. We were practicing it here first.

本系列为「契苾小优」雅思备考专用输出，转载请注明出处。

一词一世界，词根即思维。明天接着聊 (◕‿◕)