1/153 0.09-2.43 <font color=red>aaa</font> 在数学的世界里面有两个超级常数，08-30 20:27:42-08-30 20:27:44<font color=blue>In the world of mathematics, there are two super constants,</font><br>2/153 2.73-3.39 <font color=red>de</font> 家喻户晓，08-30 20:32:44-08-30 20:32:56<font color=blue>Well-known,</font><br>3/153 3.57-4.59 <font color=red>n</font> 一个是圆周率派，08-30 20:27:43-08-30 20:27:44<font color=blue>One is Pi,</font><br>4/153 4.89-5.85 <font color=red>n</font> 这个大家都能理解，08-30 20:27:44-08-30 20:27:45<font color=blue>This is something everyone can understand.</font><br>5/153 6.15-8.91 <font color=red>hp</font> 就是圆的周长除以直径直观又好记。08-30 20:27:44-08-30 20:27:50<font color=blue>The circumference of a circle divided by its diameter is an intuitive and easy-to-remember formula.</font><br>6/153 9.27-11.43 <font color=red>jd</font> 但另外一个常数 e二点七一八，08-30 20:27:45-08-30 20:27:47<font color=blue>However, another constant, e, which is approximately 2.718,</font><br>7/153 11.79-13.41 <font color=red>aaa</font> 很多人就觉得有点莫名其妙了。08-30 20:27:46-08-30 20:27:47<font color=blue>Many people found it quite strange.</font><br>8/153 13.59-16.41 <font color=red>n</font> 为什么这个看起来毫无规律的无限不循环小数，08-30 20:27:46-08-30 20:27:50<font color=blue>Why does this seemingly random, infinite non-repeating decimal look like it?</font><br>9/153 16.71-18.45 <font color=red>n</font> 能跟圆周率平起平坐，08-30 20:27:47-08-30 20:27:49<font color=blue>Can stand shoulder to shoulder with Pi,</font><br>10/153 18.75-21.15 <font color=red>aaa</font> 甚至在微积分和高等数学里无处不在。08-30 20:27:48-08-30 20:27:49<font color=blue>It is ubiquitous, even in calculus and higher mathematics.</font><br>11/153 21.45-24.03 <font color=red>jd</font> 人们甚至还专门把它叫做自然对数的底数，08-30 20:27:48-08-30 20:27:50<font color=blue>People even specifically call it the base of the natural logarithm.</font><br>12/153 24.21-25.59 <font color=red>n</font> 那它到底自然在哪里？08-30 20:27:50-08-30 20:27:51<font color=blue>Where is it exactly located in nature?</font><br>13/153 25.77-26.85 <font color=red>aaa</font> 它又是怎么来的呢？08-30 20:27:51-08-30 20:27:52<font color=blue>How did it come about?</font><br>14/153 27.03-27.81 <font color=red>n</font> 要回答这个问题，08-30 20:27:51-08-30 20:27:52<font color=blue>To answer this question,</font><br>15/153 27.99-30.58 <font color=red>hp</font> 我们得首先回答一个更深层次的终极发问。08-30 20:27:52-08-30 20:27:55<font color=blue>We must first answer a deeper, ultimate question.</font><br>16/153 31.12-32.68 <font color=red>jd</font> 数学里面的这些特殊的书，08-30 20:27:53-08-30 20:27:54<font color=blue>These special books in mathematics,</font><br>17/153 33.04-34.72 <font color=red>n</font> 到底是人类凭空发明的，08-30 20:27:53-08-30 20:27:55<font color=blue>It was invented by humans,</font><br>18/153 34.9-36.88 <font color=red>bg</font> 还是在宇宙中被我们发现的？08-30 20:27:54-08-30 20:28:31<font color=blue>Have we discovered it in the universe?</font><br>19/153 37.24-38.5 <font color=red>aaa</font> 只要想清楚这件事情，08-30 20:27:55-08-30 20:27:56<font color=blue>As long as you think about it carefully,</font><br>20/153 38.62-40.96 <font color=red>n</font> 你才能真正理解意到底有多神奇。08-30 20:27:56-08-30 20:27:58<font color=blue>Only then can you truly understand how amazing the meaning is.</font><br>21/153 41.56-42.16 <font color=red>jd</font> 要搞懂易，08-30 20:27:56-08-30 20:27:57<font color=blue>To understand Yi,</font><br>22/153 42.28-43.36 <font color=red>n</font> 我们得先从一个最俗气，08-30 20:27:57-08-30 20:27:59<font color=blue>We need to start from the most basic,</font><br>23/153 43.48-45.46 <font color=red>hp</font> 但也最直观的事情聊起来就是存钱。08-30 20:27:58-08-30 20:28:02<font color=blue>But the most obvious thing to talk about is saving money.</font><br>24/153 46.18-47.68 <font color=red>aaa</font> 假设有一家慷慨的银行，08-30 20:27:59-08-30 20:28:00<font color=blue>Assume a generous bank,</font><br>25/153 47.8-49.48 <font color=red>de</font> 他给你年利率是百分之百。08-30 20:27:59-08-30 20:28:53<font color=blue>He gave you an annual interest rate of 100%.</font><br>26/153 49.72-51.7 <font color=red>jd</font> 如果你存入一块钱的本金存满一年，08-30 20:28:00-08-30 20:28:02<font color=blue>If you deposit a principal of one dollar, and leave it for one year,</font><br>27/153 51.94-53.08 <font color=red>n</font> 如果你一年结算一次，08-30 20:28:01-08-30 20:28:03<font color=blue>If you settle once a year,</font><br>28/153 53.26-54.7 <font color=red>n</font> 一年后你就能拿到两块钱，08-30 20:28:01-08-30 20:28:04<font color=blue>One year later, you will be able to earn two dollars.</font><br>29/153 55.12-56.56 <font color=red>aaa</font> 这个时候你就开始动脑筋了。08-30 20:28:02-08-30 20:28:04<font color=blue>Now it's time to start thinking.</font><br>30/153 56.74-58.36 <font color=red>hp</font> 如果我让他半年结一次呢，08-30 20:28:03-08-30 20:28:06<font color=blue>If I could make him have a meeting every six months,</font><br>31/153 58.48-60.11 <font color=red>jd</font> 半年的利率就是百分之五十半，08-30 20:28:03-08-30 20:28:05<font color=blue>The interest rate is 50% per half year.</font><br>32/153 60.17-61.61 <font color=red>n</font> 年底连本带利一块五，08-30 20:28:04-08-30 20:28:07<font color=blue>End of the year, with interest, 1.5 million</font><br>33/153 61.91-63.71 <font color=red>aaa</font> 下半年把这一块五继续存进去，08-30 20:28:05-08-30 20:28:08<font color=blue>In the second half of the year, I will continue to store this area.</font><br>34/153 63.95-64.37 <font color=red>n</font> 利滚利，08-30 20:28:06-08-30 20:28:07<font color=blue>Interest accumulates,</font><br>35/153 64.61-66.41 <font color=red>jd</font> 一年后就是二点二五元，08-30 20:28:06-08-30 20:28:09<font color=blue>One year later, it will be 2.25 yuan.</font><br>36/153 66.68-67.34 <font color=red>jd</font> 多赚了两毛五。08-30 20:33:09-08-30 20:33:11<font color=blue>Earned an extra 5 cents.</font><br>37/153 67.75-70.3 <font color=red>hp</font> 那如果是按月结算月月利滚利呢，08-30 20:28:08-08-30 20:28:11<font color=blue>If the interest is calculated monthly and compounded each month,</font><br>38/153 70.42-72.58 <font color=red>n</font> 一年后你就能拿到二点六一三。08-30 20:28:08-08-30 20:28:12<font color=blue>In one year, you will be able to earn $2.613 per day.</font><br>39/153 73.49-74.03 <font color=red>n</font> 看到这里，08-30 20:28:09-08-30 20:28:10<font color=blue>Seeing this,</font><br>40/153 74.15-75.83 <font color=red>de</font> 你的胃口是不是应该被吊起来了？08-30 20:28:10-08-30 20:28:32<font color=blue>Is your appetite getting better?</font><br>41/153 75.95-77.45 <font color=red>jd</font> 如果我让银行按秒计算，08-30 20:28:11-08-30 20:28:13<font color=blue>If I let the bank calculate by the second,</font><br>42/153 77.63-80.919 <font color=red>n</font> 甚至在每个瞬间都在进行无限连续的结算呢，08-30 20:28:12-08-30 20:28:15<font color=blue>Even in each moment, it's performing an infinite series of calculations.</font><br>43/153 81.27-84.24 <font color=red>hp</font> 那我岂不是能利滚利滚出无穷的财富？08-30 20:28:13-08-30 20:28:17<font color=blue>Then wouldn't I be able to generate infinite wealth through a cycle of gains?</font><br>44/153 84.54-84.96 <font color=red>n</font> 很遗憾，08-30 20:28:13-08-30 20:28:14<font color=blue>Unfortunately,</font><br>45/153 85.25-87.41 <font color=red>aaa</font> 宇宙的数学法则在这里就画了一条死线，08-30 20:28:14-08-30 20:28:16<font color=blue>The mathematical laws of the universe have drawn a dead line here.</font><br>46/153 87.72-89.1 <font color=red>jd</font> 不管你结算的有多么的频繁，08-30 20:28:15-08-30 20:28:16<font color=blue>No matter how frequently you settle,</font><br>47/153 89.28-91.5 <font color=red>n</font> 你最终能拿到的钱也无法突破一个极限。08-30 20:28:15-08-30 20:28:19<font color=blue>The amount of money you can ultimately obtain is also limited.</font><br>48/153 91.81-93.84 <font color=red>n</font> 这个极限大约就是二点七八。08-30 20:28:16-08-30 20:28:19<font color=blue>The limit is approximately 2.78.</font><br>49/153 94.14-94.38 <font color=red>jd</font> 没错，08-30 20:28:17-08-30 20:28:18<font color=blue>Right,</font><br>50/153 94.62-95.22 <font color=red>aaa</font> 这就是一、08-30 20:28:18-08-30 20:28:18<font color=blue>This is one,</font><br>51/153 95.89-99.01 <font color=red>hp</font> 它不是哪位数学家在书桌前凭空捏造出来的数字，08-30 20:28:18-08-30 20:28:23<font color=blue>It's not a number conjured up by some mathematician sitting at his desk,</font><br>52/153 99.19-101.23 <font color=red>jd</font> 而是只要你研究连续增长，08-30 20:28:19-08-30 20:28:21<font color=blue>Instead, as long as you study continuous growth,</font><br>53/153 101.41-102.67 <font color=red>xss</font> 你一定会碰到的一堵墙。08-30 20:28:20-08-30 20:29:50<font color=blue>You will encounter a wall.</font><br>54/153 102.85-104.53 <font color=red>aaa</font> 但这里只是 e最基础的能力，08-30 20:28:20-08-30 20:28:22<font color=blue>But here are only the most basic abilities,</font><br>55/153 104.65-109.21 <font color=red>n</font> 它最神奇的地方在于微积分里面微积分的核心是研究变化的速度函数。08-30 20:28:21-08-30 20:28:30<font color=blue>What's most amazing about it is that in calculus, the core of calculus is to study functions whose rate of change is a function.</font><br>56/153 109.33-114.55 <font color=red>n</font> f x 等于 a 的 x 方求导之后会得到 f 撇 x 等于 a 的 x 方乘以某个常数，08-30 20:28:22-08-30 20:28:29<font color=blue>f x equals a times the derivative of x squared, which is f prime x.</font><br>57/153 114.97-116.23 <font color=red>jd</font> 这个常数取决于底数。08-30 20:28:23-08-30 20:28:24<font color=blue>This constant depends on the base.</font><br>58/153 116.29-117.85 <font color=red>aaa</font> a 如果底数等于二常数，08-30 20:28:23-08-30 20:28:25<font color=blue>If the base is equal to a constant of 2,</font><br>59/153 118.03-119.29 <font color=red>hp</font> 约等于零点六九三。08-30 20:28:24-08-30 20:28:27<font color=blue>Approximately 0.693.</font><br>60/153 119.41-121.99 <font color=red>jd</font> 如果底数等于十常数约等于二点三零二，08-30 20:28:25-08-30 20:28:27<font color=blue>If the base is equal to the constant approximately 2.302,</font><br>61/153 122.26-125.08 <font color=red>aaa</font> 但如果 a 等于二点七一八二八时，08-30 20:28:26-08-30 20:28:28<font color=blue>But if a is equal to 2.7828,</font><br>62/153 125.56-126.76 <font color=red>hp</font> 常数恰好等于一，08-30 20:28:28-08-30 20:28:31<font color=blue>The constant is exactly equal to one.</font><br>63/153 127.0-129.22 <font color=red>jd</font> 它的增长速度会恰好完美的，08-30 20:28:28-08-30 20:28:30<font color=blue>Its growth rate will be just right,</font><br>64/153 129.28-129.82 <font color=red>aaa</font> 等于它自己。08-30 20:28:29-08-30 20:28:30<font color=blue>Equals itself.</font><br>65/153 130.15-131.29 <font color=red>n</font> 这就好比你开着一辆车，08-30 20:28:30-08-30 20:28:32<font color=blue>It's like you are driving a car,</font><br>66/153 131.47-135.19 <font color=red>jd</font> 它的时速表上的数字随时随地都和它已经行驶的距离一模一样。08-30 20:28:30-08-30 20:28:34<font color=blue>The speedometer always shows the same number as the distance it has traveled, no matter where you are.</font><br>67/153 135.53-137.03 <font color=red>aaa</font> 这种极致的干净和优雅，08-30 20:28:31-08-30 20:28:32<font color=blue>This extreme cleanliness and elegance,</font><br>68/153 137.21-138.41 <font color=red>n</font> 在整个数据的宇宙里面，08-30 20:28:32-08-30 20:28:33<font color=blue>In the vast universe of data,</font><br>69/153 138.47-140.09 <font color=red>hp</font> 只有一这一个数字能做到。08-30 20:28:33-08-30 20:28:36<font color=blue>Only one number can do this.</font><br>70/153 140.52-142.38 <font color=red>bg</font> 这到底是人类发明的还是发现的？08-30 20:28:33-08-30 20:29:13<font color=blue>Is this something that humans invented or discovered?</font><br>71/153 142.499-142.86 <font color=red>n</font> 到了这里，08-30 20:28:34-08-30 20:28:37<font color=blue>Here,</font><br>72/153 142.92-144.3 <font color=red>aaa</font> 你应该隐约感受到了答案。08-30 20:28:35-08-30 20:28:36<font color=blue>You should have a vague sense of the answer.</font><br>73/153 144.48-147.48 <font color=red>de</font> 乙确实是在人类的宇宙规律中发现的，08-30 20:28:36-08-30 20:29:10<font color=blue>It was indeed discovered in the human universe.</font><br>74/153 147.66-148.679 <font color=red>n</font> 而不是发明的。08-30 20:28:36-08-30 20:28:38<font color=blue>Instead of inventing.</font><br>75/153 149.14-150.88 <font color=red>jd</font> 人类发明的部分只是一些皮毛。08-30 20:28:37-08-30 20:28:39<font color=blue>Humans have only invented a small part of what they have.</font><br>76/153 151.059-153.82 <font color=red>hp</font> 比如说把它写成十进制的二点七八二八，08-30 20:28:38-08-30 20:28:42<font color=blue>For example, write it as 2.7828</font><br>77/153 154.06-155.74 <font color=red>aaa</font> 或者用字母 e 来代表它，08-30 20:28:38-08-30 20:28:40<font color=blue>Or represent it with the letter e.</font><br>78/153 155.92-157.0 <font color=red>n</font> 这只是我们的命名习惯。08-30 20:28:39-08-30 20:28:41<font color=blue>This is just our naming convention.</font><br>79/153 157.33-160.93 <font color=red>n</font> 但这个数字本身所代表的规律却是绝对客观存在的。08-30 20:28:40-08-30 20:28:42<font color=blue>However, the pattern represented by this number is absolutely objective.</font><br>80/153 161.17-163.09 <font color=red>jd</font> 即便在几亿光年外的外星文明里面，08-30 20:28:41-08-30 20:28:43<font color=blue>Even among extraterrestrial civilizations located several billion light-years away,</font><br>81/153 163.21-164.35 <font color=red>aaa</font> 如果他研究复利、08-30 20:28:41-08-30 20:28:42<font color=blue>If he studies compounding,</font><br>82/153 164.59-165.25 <font color=red>n</font> 研究微积分，08-30 20:28:42-08-30 20:28:43<font color=blue>Study calculus,</font><br>83/153 165.43-167.53 <font color=red>hp</font> 不管他们用什么样的静制什么样的符号，08-30 20:28:43-08-30 20:28:47<font color=blue>No matter what symbols they use, whether custom-made or not.</font><br>84/153 167.71-169.39 <font color=red>n</font> 他们一定会算出同样一个常数，08-30 20:28:43-08-30 20:28:45<font color=blue>They will definitely arrive at the same constant.</font><br>85/153 169.63-170.59 <font color=red>aaa</font> 不可能有第二种结果。08-30 20:28:44-08-30 20:28:45<font color=blue>There is no other outcome.</font><br>86/153 170.95-171.79 <font color=red>jd</font> 在客观规律面前，08-30 20:28:45-08-30 20:28:46<font color=blue>In the face of objective laws,</font><br>87/153 172.03-174.61 <font color=red>n</font> 全宇宙的智慧生命都会收敛到同一个智。08-30 20:28:45-08-30 20:28:48<font color=blue>All intelligent life in the universe will converge to a single intelligence.</font><br>88/153 174.85-175.93 <font color=red>de</font> 虽然意识被发现的，08-30 20:33:47-08-30 20:34:06<font color=blue>Although consciousness has been discovered,</font><br>89/153 176.05-178.51 <font color=red>n</font> 但它发现的大前提却是由人类发明的。08-30 20:28:47-08-30 20:28:49<font color=blue>However, the fundamental premise it discovered was invented by humans.</font><br>90/153 178.63-179.29 <font color=red>aaa</font> 为什么这么说呢？08-30 20:28:48-08-30 20:28:49<font color=blue>Why is that so?</font><br>91/153 179.41-181.93 <font color=red>jd</font> 因为数字并不是像物理那样研究眼前的实体，08-30 20:28:48-08-30 20:28:51<font color=blue>Because numbers are not like physical objects, which are studied in terms of their immediate existence.</font><br>92/153 182.31-184.95 <font color=red>hp</font> 它是建立在一套由人类选定的公里之上的。08-30 20:28:49-08-30 20:28:53<font color=blue>It is built on a set of human-selected kilometers.</font><br>93/153 185.28-186.6 <font color=red>n</font> 最经典的例子就是几何学。08-30 20:28:50-08-30 20:28:51<font color=blue>The most classic example is geometry.</font><br>94/153 186.84-187.62 <font color=red>aaa</font> 两千多年以来，08-30 20:28:50-08-30 20:28:52<font color=blue>For over two thousand years,</font><br>95/153 187.74-190.88 <font color=red>n</font> 大家都觉得欧基里德的平行公理过直线外一点。08-30 20:28:51-08-30 20:28:54<font color=blue>Everyone thinks that Euclid's parallel postulate goes beyond a line.</font><br>96/153 191.18-192.44 <font color=red>jd</font> 有且只有一条平行线，08-30 20:28:52-08-30 20:28:53<font color=blue>There is only one parallel line.</font><br>97/153 192.56-193.22 <font color=red>n</font> 是绝对的正理。08-30 20:28:52-08-30 20:28:54<font color=blue>It is absolutely right.</font><br>98/153 193.4-194.6 <font color=red>aaa</font> 但是后来数学家们想，08-30 20:28:53-08-30 20:28:54<font color=blue>However, mathematicians later thought that...</font><br>99/153 194.87-195.83 <font color=red>hp</font> 如果换了这个规则，08-30 20:28:54-08-30 20:28:57<font color=blue>If we change this rule,</font><br>100/153 196.13-197.03 <font color=red>de</font> 规定过一点，08-30 20:28:55-08-30 20:29:10<font color=blue>Regulations have been made.</font><br>101/153 197.15-199.55 <font color=red>n</font> 有无数条平行线或者一条平行线都没有，08-30 20:28:56-08-30 20:28:59<font color=blue>There are countless parallel lines or no parallel lines.</font><br>102/153 199.67-199.97 <font color=red>n</font> 行不行？08-30 20:28:57-08-30 20:28:59<font color=blue>Is it okay? / Is this alright?</font><br>103/153 200.21-203.51 <font color=red>aaa</font> 结果他们真的创造出了截然不同的非 o几何。08-30 20:28:57-08-30 20:29:00<font color=blue>As a result, they truly created a completely different non-Euclidean geometry.</font><br>104/153 204.02-205.46 <font color=red>jd</font> 这不仅逻辑完全自洽，08-30 20:28:58-08-30 20:29:00<font color=blue>This is not only logically consistent,</font><br>105/153 205.64-209.24 <font color=red>hp</font> 而且爱因斯坦后来也发现我们真实的宇宙时空在大尺度上，08-30 20:28:59-08-30 20:29:04<font color=blue>Furthermore, Einstein later discovered that our real universe is on a large scale,</font><br>106/153 209.42-211.94 <font color=red>n</font> 其实就是被引力弯曲的非欧几何。08-30 20:29:00-08-30 20:29:03<font color=blue>It's actually a non-Euclidean geometry that is curved by gravity.</font><br>107/153 212.059-214.34 <font color=red>n</font> 这就说明数学不是在描述唯一的真实，08-30 20:29:00-08-30 20:29:03<font color=blue>This shows that mathematics is not describing a unique truth,</font><br>108/153 214.52-216.5 <font color=red>jd</font> 而是在玩一个如果那么的游戏，08-30 20:29:01-08-30 20:29:03<font color=blue>And then, in a game of "if so,"</font><br>109/153 216.98-217.76 <font color=red>aaa</font> 这就像在下棋。08-30 20:29:02-08-30 20:29:03<font color=blue>It's like playing chess.</font><br>110/153 217.94-219.56 <font color=red>jd</font> 围棋象棋的规则是人类发明的，08-30 20:29:04-08-30 20:29:05<font color=blue>The rules of chess and Go are invented by humans.</font><br>111/153 219.74-221.24 <font color=red>aaa</font> 但是一旦规则定下来之后，08-30 20:29:04-08-30 20:29:06<font color=blue>However, once the rules are established,</font><br>112/153 221.48-223.88 <font color=red>n</font> 在这个残局的红方有没有必胜策略，08-30 20:29:05-08-30 20:29:08<font color=blue>Does the red side have a winning strategy in this endgame?</font><br>113/153 224.22-227.19 <font color=red>n</font> 那就是一个绝对客观等着被发现的事实了。08-30 20:29:06-08-30 20:29:09<font color=blue>That's an absolutely objective fact waiting to be discovered.</font><br>114/153 227.84-228.2 <font color=red>hp</font> 同样的，08-30 20:29:06-08-30 20:29:08<font color=blue>Similarly,</font><br>115/153 228.32-229.64 <font color=red>jd</font> 我们人类选择了经典的逻辑，08-30 20:29:07-08-30 20:29:09<font color=blue>We humans have chosen the classic logic.</font><br>116/153 229.88-230.78 <font color=red>aaa</font> 选择了集何卵。08-30 20:29:08-08-30 20:29:09<font color=blue>What egg was chosen.</font><br>117/153 230.9-231.62 <font color=red>hp</font> 在这个规则之下，08-30 20:29:09-08-30 20:29:11<font color=blue>Under this rule,</font><br>118/153 231.94-233.26 <font color=red>n</font> 我们盖起了现代数学的大厦，08-30 20:29:10-08-30 20:29:12<font color=blue>We have built the modern edifice of mathematics.</font><br>119/153 233.5-235.18 <font color=red>n</font> 而 e 等于二点七八，08-30 20:29:10-08-30 20:29:13<font color=blue>And e is equal to 2.78,</font><br>120/153 235.36-238.78 <font color=red>aaa</font> 就是在这个地基上被规则锁死的不可改动的客观真理。08-30 20:29:11-08-30 20:29:14<font color=blue>It is on this foundation that objective truths, immutable and locked by rules, are established.</font><br>121/153 239.13-239.85 <font color=red>jd</font> 更震撼的是，08-30 20:29:12-08-30 20:29:13<font color=blue>More shocking is,</font><br>122/153 240.03-241.17 <font color=red>de</font> 即便是这栋大厦，08-30 20:29:12-08-30 20:29:32<font color=blue>Even this building,</font><br>123/153 241.35-242.25 <font color=red>de</font> 也不是一成不变的。08-30 20:29:13-08-30 20:29:35<font color=blue>It's not static.</font><br>124/153 242.54-243.14 <font color=red>hp</font> 一九三一年，08-30 20:29:14-08-30 20:29:16<font color=blue>1931,</font><br>125/153 243.2-245.54 <font color=red>n</font> 伟大的数学家歌德尔证明了不完备定律，08-30 20:29:14-08-30 20:29:17<font color=blue>The great mathematician Gödel proved the incompleteness theorems.</font><br>126/153 245.81-246.59 <font color=red>jd</font> 任何足够强、08-30 20:29:15-08-30 20:29:16<font color=blue>Any sufficiently strong,</font><br>127/153 246.77-251.87 <font color=red>n</font> 无矛盾的数学规则系统里面都一定存在一些既无法证明也无法证伪的命题。08-30 20:29:16-08-30 20:29:20<font color=blue>In a consistent mathematical system, there must be some statements that are neither provable nor refutable.</font><br>128/153 252.47-252.95 <font color=red>bg</font> 也就是说，08-30 20:29:16-08-30 20:29:37<font color=blue>That is to say,</font><br>129/153 253.25-254.45 <font color=red>aaa</font> 哪怕游戏规则定好了，08-30 20:29:17-08-30 20:29:19<font color=blue>Even if the rules of the game are set,</font><br>130/153 254.51-256.07 <font color=red>jd</font> 有些问题也永远没有标准答案。08-30 20:29:18-08-30 20:29:19<font color=blue>Some questions also have no standard answers.</font><br>131/153 256.19-258.84 <font color=red>hp</font> 这时候数学家可以选择加一条规则规定，08-30 20:29:18-08-30 20:29:23<font color=blue>At this time, mathematicians can choose to add a rule that:</font><br>132/153 259.02-259.44 <font color=red>n</font> 它是对的，08-30 20:29:19-08-30 20:29:20<font color=blue>It's right,</font><br>133/153 259.56-261.12 <font color=red>aaa</font> 也可以加一条规则规定。08-30 20:34:21-08-30 20:34:22<font color=blue>You can also add a rule to specify.</font><br>134/153 261.3-264.78 <font color=red>aaa</font> 它是错的这就意味着数学不是一栋单一的大楼，08-30 20:29:20-08-30 20:29:23<font color=blue>It is wrong to say that, because it means mathematics is not a single building.</font><br>135/153 264.9-266.88 <font color=red>jd</font> 而是一棵可以无限分叉的树。08-30 20:29:21-08-30 20:29:23<font color=blue>It is a tree that can branch infinitely.</font><br>136/153 267.22-269.77 <font color=red>n</font> 你往左边选汇入一个新的数学宇宙，08-30 20:29:22-08-30 20:29:25<font color=blue>You choose to enter a new mathematical universe on the left,</font><br>137/153 269.89-272.66 <font color=red>n</font> 你往右边选又会进入另外一个自洽的世界。08-30 20:29:22-08-30 20:29:26<font color=blue>If you choose to go right, you will enter another self-consistent world.</font><br>138/153 273.05-273.65 <font color=red>hp</font> 只要你愿意，08-30 20:29:23-08-30 20:29:26<font color=blue>As long as you want,</font><br>139/153 273.89-276.9 <font color=red>jd</font> 理论上还有无数多个数学宇宙在等着我们去构建。08-30 20:29:24-08-30 20:29:26<font color=blue>In theory, there are countless mathematical universes waiting for us to construct.</font><br>140/153 277.52-278.18 <font color=red>aaa</font> 绕了这一大圈，08-30 20:29:25-08-30 20:29:26<font color=blue>After going around in circles,</font><br>141/153 278.36-280.7 <font color=red>n</font> 我们再来看 e为什么我们今天如此的重视义，08-30 20:29:26-08-30 20:29:28<font color=blue>Let's take a look at why we are so important today.</font><br>142/153 280.82-282.14 <font color=red>aaa</font> 甚至还专门给他起了一个名字？08-30 20:29:26-08-30 20:29:28<font color=blue>Did they even give him a special name?</font><br>143/153 282.32-286.09 <font color=red>hp</font> 是因为在我们人类目前已经居住的这颗数学之树的主干上，08-30 20:29:27-08-30 20:29:31<font color=blue>Because on the mathematical tree that we currently live in,</font><br>144/153 286.49-289.85 <font color=red>n</font> 在我们研究的连续增长微积分和现实物理的这个分支里面，08-30 20:29:28-08-30 20:29:31<font color=blue>In the continuous growth calculus and this branch of real physics that we are studying,</font><br>145/153 290.29-291.43 <font color=red>jd</font> e 是一颗最璀璨、08-30 20:29:28-08-30 20:29:30<font color=blue>e is the most brilliant,</font><br>146/153 291.61-292.81 <font color=red>aaa</font> 最无法绕过去的明珠。08-30 20:29:29-08-30 20:29:30<font color=blue>The most unmissable Pearl.</font><br>147/153 293.05-294.07 <font color=red>n</font> 我们选择了经典逻辑，08-30 20:29:30-08-30 20:29:31<font color=blue>We chose classical logic.</font><br>148/153 294.19-294.85 <font color=red>jd</font> 选择了微积分，08-30 20:29:30-08-30 20:29:32<font color=blue>I chose calculus.</font><br>149/153 295.11-296.85 <font color=red>aaa</font> 选择了研究连续变化的这个方向。08-30 20:29:31-08-30 20:29:33<font color=blue>I chose this direction of research, which involves studying continuous change.</font><br>150/153 297.23-298.25 <font color=red>hp</font> 一旦做出这些选择，08-30 20:29:32-08-30 20:29:34<font color=blue>Once these choices are made,</font><br>151/153 298.61-301.37 <font color=red>n</font> e 那就在路口静静的等着我们不可能是别的值。08-30 20:29:32-08-30 20:29:36<font color=blue>Then we just wait at the intersection, there's no other option.</font><br>152/153 301.55-302.99 <font color=red>n</font> 它不是我们凭空发明的工具，08-30 20:29:33-08-30 20:29:36<font color=blue>It is not a tool we invented out of thin air,</font><br>153/153 303.17-306.83 <font color=red>jd</font> 它是我们在这个奇妙的数学宇宙里面收到的最美妙的礼物。08-30 20:29:34-08-30 20:29:36<font color=blue>It is the most wonderful gift we have received in this wonderful mathematical universe.</font><br>