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計(jì)算 
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9 7 4 8 6 7 4 1 9 2 9 1 7 讀者對象:廣大IT從業(yè)者、商業(yè)和經(jīng)濟(jì)學(xué)者、普通大眾。 
摩爾定律快要走到盡頭，但計(jì)算革命不會終止。更好的軟件編程、3D芯片和量子計(jì)算等方法應(yīng)運(yùn)而生，其中云計(jì)算將成為業(yè)界應(yīng)對摩爾定律消亡的最佳手段，物聯(lián)網(wǎng)（IoT）的興起將讓我們逼近一個(gè)"消失點(diǎn)”，此前計(jì)算機(jī)的形體從大到小，此后計(jì)算機(jī)將變得"無形”，使計(jì)算無處不在，智能融入日常生活。本書由計(jì)算專業(yè)領(lǐng)域的專家學(xué)者知名吳翰清執(zhí)筆，代表他及背后的阿里公司，對計(jì)算這個(gè)科技終極命題的感悟、展望和深刻洞察。本書為三卷書中的第一卷，著眼于對計(jì)算非常重要的數(shù)學(xué)，覆蓋了量重要的數(shù)學(xué)家、數(shù)學(xué)成就及相關(guān)史實(shí)及其關(guān)聯(lián)。
吳翰清，畢業(yè)于西安交通大學(xué)少年班。清華大學(xué)創(chuàng)新領(lǐng)軍工程博士（人工智能方向，在讀）。2005年加入阿里，創(chuàng)建了阿里巴巴、淘寶、支付寶的安全體系，也是阿里云初創(chuàng)團(tuán)隊(duì)成員，是阿里安全從無到有、從有到強(qiáng)的親歷者。2017年開始致力于城市大腦的研究與建設(shè)，專注于構(gòu)建機(jī)器智能系統(tǒng)。2017年入選 MIT 全球青年科技創(chuàng)新人才榜，是中國互聯(lián)網(wǎng)安全領(lǐng)域入選 TR35 的第一人。2019年入選「大數(shù)據(jù)文摘」評選的「30位新生代數(shù)字經(jīng)濟(jì)人才」。2019年當(dāng)選中國青年科技工作者協(xié)會第六屆理事。公益項(xiàng)目“計(jì)算圖書館”發(fā)起人。目前正致力于人工智能的研究和創(chuàng)業(yè)。
導(dǎo)論 ....................................................................................................................................1
第一部分 計(jì)算的誕生 第 1 章 畢達(dá)哥拉斯的困惑..............................................................................................24 數(shù)的計(jì)算 .......................................................................................................................24 從數(shù)覺到計(jì)數(shù) .......................................................................................................24 文明古國的計(jì)算 ...................................................................................................28 畢達(dá)哥拉斯學(xué)派 ...........................................................................................................30 柏拉圖的理想世界 .......................................................................................................40 第一次數(shù)學(xué)危機(jī) ...........................................................................................................44 無理數(shù)的發(fā)現(xiàn) .......................................................................................................44 芝諾悖論：無窮之辯 ...........................................................................................46 演繹推理：邏輯學(xué)和幾何學(xué) .......................................................................................51 亞里士多德的邏輯學(xué) ...........................................................................................51 歐幾里得的《幾何原本》 ...................................................................................55 悖論：推理的暗面 ...............................................................................................59 第 2 章 計(jì)算之術(shù) ............................................................................................................62 代數(shù)：字符的計(jì)算 .......................................................................................................62 符號與代數(shù) ...................................................................................................................63 零的誕生 ...............................................................................................................63 言辭代數(shù) ...............................................................................................................65 未知量的表示 .......................................................................................................68 還原與對消 ...........................................................................................................70 代數(shù)符號 ...............................................................................................................73 求解多項(xiàng)式方程 ...........................................................................................................77 從數(shù)值解到代數(shù)解 ...............................................................................................77 三次方程的求根公式 ...........................................................................................81 不可約：復(fù)數(shù)的發(fā)現(xiàn) ...........................................................................................84 數(shù)系的擴(kuò)張 ...........................................................................................................89 代數(shù)基本定理 .......................................................................................................92 代數(shù)的結(jié)構(gòu) ...................................................................................................................94 求解一元五次方程 ...............................................................................................94 方程根的結(jié)構(gòu) .......................................................................................................95 伽羅瓦的遺珠 .....................................................................................................101 計(jì)算工具 .....................................................................................................................108 人類計(jì)算員 .........................................................................................................109 面向機(jī)器的計(jì)算思維 .........................................................................................111 第 3 章 萊布尼茨的計(jì)算之夢 ........................................................................................116 數(shù)理邏輯的創(chuàng)立 .........................................................................................................117 人類思想字母表 .........................................................................................................120 思想的大衍術(shù) .....................................................................................................121 計(jì)算之夢 .....................................................................................................................125 思維規(guī)律的研究 .........................................................................................................127 19 世紀(jì)數(shù)理邏輯的復(fù)興 ....................................................................................127 布爾的邏輯代數(shù) .................................................................................................129 第二部分 計(jì)算的數(shù)學(xué)基礎(chǔ) 第 4 章 數(shù)學(xué)的基礎(chǔ).......................................................................................................136 第二次數(shù)學(xué)危機(jī) .........................................................................................................136 微積分的發(fā)明 .....................................................................................................136 消失的鬼魂：貝克萊悖論 .................................................................................139 分析的嚴(yán)格化 .....................................................................................................140 集合論的誕生 .............................................................................................................142 無窮大有多大 .....................................................................................................142 對角線方法 .........................................................................................................146 康托爾的超窮數(shù) .........................................................................................................148 超窮基數(shù)與超窮序數(shù) .........................................................................................148 連續(xù)統(tǒng)假設(shè) .........................................................................................................152 算術(shù)的邏輯化 .............................................................................................................156 弗雷格的“概念文字” .....................................................................................156 自然數(shù)的定義 .....................................................................................................159 第 5 章 第三次數(shù)學(xué)危機(jī) ...............................................................................................163 危機(jī)：羅素悖論 .........................................................................................................163 集合論悖論 .........................................................................................................163 自我指涉 .............................................................................................................165 悖論的解決方法 .................................................................................................168 邏輯主義進(jìn)路 .............................................................................................................169 直覺主義進(jìn)路 .............................................................................................................173 公理集合論進(jìn)路 .........................................................................................................176 ZFC 公理集合論.................................................................................................177 選擇公理 .............................................................................................................180 NBG 公理集合論................................................................................................182 第三部分 計(jì)算理論的形成 第 6 章 計(jì)算理論的奠基：希爾伯特進(jìn)路......................................................................186 數(shù)學(xué)的無冕之王 .........................................................................................................186 希爾伯特問題 .............................................................................................................188 數(shù)學(xué)的世紀(jì)之問 .................................................................................................188 希爾伯特的第 10 個(gè)問題 ...................................................................................189 幾何的算術(shù)基礎(chǔ) .........................................................................................................192 歐幾里得的第五公設(shè) .........................................................................................192 模型化方法 .........................................................................................................194 桌子、椅子和啤酒杯：形式系統(tǒng)思想..............................................................195 “形式主義”之父 .....................................................................................................196 有窮主義證明論 .........................................................................................................198 希爾伯特綱領(lǐng) .............................................................................................................201 可判定性問題 .....................................................................................................201 王者的落幕 .........................................................................................................202 第 7 章 計(jì)算不能做什么：終結(jié)者哥德爾......................................................................204 昨日的世界 .................................................................................................................204 我們必須知道，我們必將知道 .........................................................................204 偉大的友誼 .........................................................................................................205 哥德爾的發(fā)現(xiàn) .....................................................................................................207 編碼思想：哥德爾數(shù) .................................................................................................209 哥德爾證明 .................................................................................................................213 不完備性定理 .....................................................................................................213 塔斯基定理 .........................................................................................................215 希爾伯特計(jì)劃的破滅 .........................................................................................216 哥德爾綱領(lǐng) .................................................................................................................217 自亞里士多德以來 .....................................................................................................218 第 8 章 計(jì)算理論的誕生：圖靈的可計(jì)算數(shù) ..................................................................221 圖靈的學(xué)業(yè) .................................................................................................................221 圖靈機(jī).........................................................................................................................223 模擬人類計(jì)算員 .................................................................................................223 圖靈機(jī)模型 .........................................................................................................224 可計(jì)算數(shù) .............................................................................................................226 丘奇-圖靈論題............................................................................................................229 判定性問題的證明 .....................................................................................................231 圖靈的證明 .........................................................................................................231 停機(jī)問題 .............................................................................................................234 忙碌的海貍 .................................................................................................................235 快速增長函數(shù) .....................................................................................................235 不可計(jì)算的函數(shù) .................................................................................................238 圖靈的命運(yùn) .................................................................................................................239 第四部分 計(jì)算的極限 第 9 章 計(jì)算復(fù)雜性.......................................................................................................242 難解的計(jì)算問題 .........................................................................................................243 旅行商問題 .........................................................................................................243 多項(xiàng)式時(shí)間與指數(shù)時(shí)間 .....................................................................................245 P/NP 問題....................................................................................................................249 NP 問題...............................................................................................................249 NP 完全問題.......................................................................................................251 柯爾莫哥洛夫復(fù)雜度 .........................................................................................254 庫克-萊文定理....................................................................................................255 計(jì)算的局部性原理 .............................................................................................257 P=NP 嗎 ......................................................................................................................257 P=NP 的世界.......................................................................................................258 認(rèn)知的邊界 .........................................................................................................259 P≠NP 的若干推論.............................................................................................260 站在兩個(gè)世界之間 .....................................................................................................266 未分類的問題 .....................................................................................................266 因數(shù)分解問題 .....................................................................................................267 圖同構(gòu)問題 .........................................................................................................268 近似計(jì)算 .....................................................................................................................268 丹齊格的線性規(guī)劃 .............................................................................................270 挑戰(zhàn)旅行商問題 .................................................................................................272 PCP 定理與不可近似性 .....................................................................................281 并行計(jì)算 .....................................................................................................................284 計(jì)算的時(shí)空平衡性 .............................................................................................284 并行計(jì)算的極限 .................................................................................................285 挑戰(zhàn)極限 .............................................................................................................287 第 10 章 量子計(jì)算 ........................................................................................................293 計(jì)算是數(shù)學(xué)的，更是物理的 .....................................................................................293 量子計(jì)算的啟蒙 .................................................................................................293 量子的特性 .........................................................................................................295 計(jì)算的最小能量 .................................................................................................296 量子比特 .....................................................................................................................298 從經(jīng)典比特到量子比特 .....................................................................................298 量子優(yōu)勢 .............................................................................................................300 量子門與量子線路 .............................................................................................300 量子算法 .....................................................................................................................303 從 BPP 到 BQP ...................................................................................................303 Shor 算法.............................................................................................................305 量子霸權(quán) .....................................................................................................................307 量子計(jì)算機(jī)的實(shí)現(xiàn) .............................................................................................307 展望量子霸權(quán) .....................................................................................................308 第 11 章 復(fù)雜性計(jì)算.....................................................................................................310 什么是復(fù)雜 .................................................................................................................310 反饋與控制 .................................................................................................................312 現(xiàn)代復(fù)雜性研究思潮 .................................................................................................318 復(fù)雜性的簡單算法 .............................................................................................318 生命游戲 .............................................................................................................321 涌現(xiàn) .....................................................................................................................323 耗散結(jié)構(gòu) .............................................................................................................324 網(wǎng)絡(luò)科學(xué) .............................................................................................................326 進(jìn)化計(jì)算 .....................................................................................................................330 生物系統(tǒng)的信息處理 .........................................................................................330 邏輯深度 .............................................................................................................334 企業(yè)的進(jìn)化計(jì)算 .................................................................................................335 第 12 章 機(jī)器能思考嗎.................................................................................................338 模擬大腦的結(jié)構(gòu) .........................................................................................................338 機(jī)器智能大論戰(zhàn) .........................................................................................................340 模仿游戲與中文屋 .............................................................................................340 符號主義與連接主義 .........................................................................................344 AlphaGo 與李世石..............................................................................................350 ChatGPT 與烏鴉.........................................................................................................355 人工智能的圣杯 .................................................................................................355 ChatGPT 的原理 .................................................................................................356 350 多年的等待 ..................................................................................................362 聰明的烏鴉 .........................................................................................................365 未來的方向 .........................................................................................................366 機(jī)器的意識 .................................................................................................................374 第 13 章 自然哲學(xué)的計(jì)算原理......................................................................................379 計(jì)算的邊界 .................................................................................................................379 時(shí)空的桎梏 .........................................................................................................379 宇宙是一臺計(jì)算機(jī)嗎 .........................................................................................380 圖靈極限 .............................................................................................................384 邊界之外 .....................................................................................................................386 無窮時(shí)間的計(jì)算 .................................................................................................386 無窮空間的計(jì)算 .................................................................................................387 一種計(jì)算主義的世界觀 .............................................................................................392 后記 ................................................................................................................................397 附錄 A 科研范式進(jìn)化史綱要.........................................................................................399 附錄 B 提問與求解的藝術(shù) ............................................................................................404 附錄 C 世界需要什么樣的智能系統(tǒng)..............................................................................416 附錄 D 機(jī)器智能宣言 ...................................................................................................423 參考文獻(xiàn) .........................................................................................................................425
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