tag:blogger.com,1999:blog-216830402024-03-12T20:00:34.596-07:00數理情色甚麼是『數理情色』?這是個以輕鬆手法討論科學知識的blog,當然也是吸引人來看我的blog的綽頭名稱!韋言仁語http://www.blogger.com/profile/08599435936360249361noreply@blogger.comBlogger564125tag:blogger.com,1999:blog-21683040.post-33372239576654129572019-03-07T07:51:00.001-08:002019-03-07T07:51:01.721-08:00人類乳房演化之謎?Discover Magazine 有篇"科學家仍困惑於人類乳房的演化"(<a href="http://blogs.discovermagazine.com/crux/2019/03/06/why-do-humans-have-breasts/#.XIEQysdIWK9.blogger">Scientists Still Stumped By The Evolution of Human Breasts - The Crux</a>)<br /><br />
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內容其實了無新意, 說的就是為何女人的乳房全天候堅挺, 不似其他靈長類動物. 來來去去的就是說是因為人類能全年發情, 又或是作為備用脂肪能量儲備. 反正沒有啥可能驗證真正答案, 自然永遠停留在困惑層面呀!<br /><br />
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(有學生問我點解冇寫嘢, 咁就寫下囉....)韋言仁語http://www.blogger.com/profile/08599435936360249361noreply@blogger.com0tag:blogger.com,1999:blog-21683040.post-74774585302726328362018-12-21T19:41:00.000-08:002018-12-21T19:41:39.649-08:00數學的迷人曲線唔轉發對唔對我個Blog嘅title.<br />
<br />
<a href="https://www.ettoday.net/dalemon/post/40528#ixzz5aNdkpvLZ" style="border: none; box-sizing: border-box; color: #003399; margin: 0px; padding: 0px; text-decoration-line: none; vertical-align: baseline;">驚覺數學有多美!浪費才能「歐派函數」連日本東大生都著迷 | 鍵盤大檸檬 | ETtoday新聞雲</a><br />
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日本多間大學理工科的數學愛好者,組成「歐派函數熱情注入委員會」,研究函數之美,以下是其中一間大學的作品:<br />
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<img alt="â²â¼ï¼åï¼ç¿»æèªscoopnest.comï¼" src="https://cdn2.ettoday.net/images/3776/3776572.jpg" />韋言仁語http://www.blogger.com/profile/08599435936360249361noreply@blogger.com0tag:blogger.com,1999:blog-21683040.post-6950496786319827412018-05-18T04:40:00.000-07:002019-11-15T21:33:21.243-08:00為何3×2=2×3? (再續)如果有兩種不同定義的運算, 給發現如來得出相同結果, 數學家就會把兩者看作一樣.<br />
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換句話說, 假設一開始時數學家定義了兩種乘法:<br />
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<span style="color: red;">m x n = m+...+m (n 個)</span><br />
<span style="color: red;">m * n = n+...+n (m 個)</span><br />
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接着數學家知道了 mxn=m*n, 他們就會毫不猶疑也接受 m x n = n+...+n (m 個)<br />
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換句說話, 除了開頭不知兩者相等的情況, 其他時候數學家會隨便使用兩個定義.<br />
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唯一例外情況, 是出習作時, 會故意強迫學生使用其中一個定義, 以強化他們的邏輯思維.<br />
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<span style="color: blue;">例如: 從(課本上的)定義出發, 證明<span style="background-color: white; font-family: "helvetica" , "arial" , sans-serif; font-size: 14px;">2 × 3 = </span><span style="background-color: white; font-family: "helvetica" , "arial" , sans-serif; font-size: 14px;"> 3 + 3.</span></span><br />
<span style="background-color: white; font-family: "helvetica" , "arial" , sans-serif; font-size: 14px;"><span style="color: blue;"><br /></span></span>
<span style="color: blue;"><span style="font-family: "helvetica" , "arial" , sans-serif;"><span style="background-color: white; font-size: 14px;">正確答案: 從定義出發, </span></span><span style="background-color: white; font-family: "helvetica" , "arial" , sans-serif; font-size: 14px;">2 × 3 = 2 + 2 + 2=6, 而 3+3=6, 所以</span><span style="background-color: white; font-family: "helvetica" , "arial" , sans-serif; font-size: 14px;">2 × 3 = </span><span style="background-color: white; font-family: "helvetica" , "arial" , sans-serif; font-size: 14px;"> 3 + 3.</span></span><br />
<span style="background-color: white; color: #1d2129; font-family: "helvetica" , "arial" , sans-serif; font-size: 14px;"><br /></span>
<span style="background-color: white; color: #1d2129; font-family: "helvetica" , "arial" , sans-serif; font-size: 14px;">這裡, 我們不能直接寫 </span><span style="background-color: white; color: #1d2129; font-family: "helvetica" , "arial" , sans-serif; font-size: 14px;">2 × 3 = 6, 而一定要寫 </span><span style="background-color: white; color: #1d2129; font-family: "helvetica" , "arial" , sans-serif; font-size: 14px;">2 × 3 = 2 + 2 + 2=6, 皆因我們要從定義出發.</span><br />
<span style="background-color: white; color: #1d2129; font-family: "helvetica" , "arial" , sans-serif; font-size: 14px;"><br /></span>
<span style="background-color: white; color: #1d2129; font-family: "helvetica" , "arial" , sans-serif; font-size: 14px;">能力稍遜的學生, 通常不喜歡這類題目, 因為由定義出發, 通常較煩且少了很多可用的工具.</span><br />
<span style="background-color: white; color: #1d2129; font-family: "helvetica" , "arial" , sans-serif; font-size: 14px;"><br /></span>
<span style="background-color: white; color: #1d2129; font-family: "helvetica" , "arial" , sans-serif; font-size: 14px;">在數學發展史上, 若有兩個看似不同的概念, 給證明了是同一回事, 必定是一件大新聞. 因為本來的兩個概念, 都各自有不同的思考方法及數學工具. 兩者結合, 就多了很多思考方法及工具可用, 會造成數學的一次大躍進.</span>韋言仁語http://www.blogger.com/profile/08599435936360249361noreply@blogger.com0tag:blogger.com,1999:blog-21683040.post-51635208821926543712018-05-17T19:35:00.000-07:002018-05-17T19:35:14.730-07:00為何3×2=2×3? (續)朋友在談論<span style="background-color: white; color: #333333; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 18px;">3×2=2×3時,指出</span>小學老師提及<br />
<span style="background-color: white; color: #1d2129; font-family: Helvetica, Arial, sans-serif; font-size: 14px;">2 × 3 = 2 + 2 + 2</span><br />
<span style="background-color: white; color: #1d2129; font-family: Helvetica, Arial, sans-serif; font-size: 14px;">3 × 2 = 3 + 3</span><br />
<span style="background-color: white; color: #1d2129; font-family: Helvetica, Arial, sans-serif; font-size: 14px;">兩者雖然相等, 但概念不同.</span><br />
<span style="background-color: white; color: #1d2129; font-family: Helvetica, Arial, sans-serif; font-size: 14px;"><br /></span>
這個可以帶出很多話題, 不過我想最多人注意的是:<br />
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<span style="color: blue;">如果有個小學生寫<span style="background-color: white; font-family: Helvetica, Arial, sans-serif; font-size: 14px;">2 × 3 = </span><span style="background-color: white; font-family: Helvetica, Arial, sans-serif; font-size: 14px;"> 3 + 3算唔算錯?</span></span><br />
<span style="background-color: white; color: #1d2129; font-family: Helvetica, Arial, sans-serif; font-size: 14px;"><br /></span>
<span style="background-color: white; color: #1d2129; font-family: Helvetica, Arial, sans-serif; font-size: 14px;">注意, 我強調小學生. 因為過了小學, 再沒有人理會這件事.</span><br />
<span style="background-color: white; color: #1d2129; font-family: Helvetica, Arial, sans-serif; font-size: 14px;"><br /></span>
<span style="background-color: white; color: #1d2129; font-family: Helvetica, Arial, sans-serif; font-size: 14px;">我的答案是: 不清楚. </span><br />
<span style="background-color: white; color: #1d2129; font-family: Helvetica, Arial, sans-serif; font-size: 14px;"><br /></span>
<span style="background-color: white; color: #1d2129; font-family: Helvetica, Arial, sans-serif; font-size: 14px;">原因是"="本身有不同的意思.</span><br />
<span style="background-color: white; color: #1d2129; font-family: Helvetica, Arial, sans-serif; font-size: 14px;"><br /></span>
<span style="background-color: white; color: #1d2129; font-family: Helvetica, Arial, sans-serif; font-size: 14px;">"A=3" 的第一個意思是"把A的數值設為3", 第二個意思是"經過計算, 發現A的數值是3".</span><br />
<span style="background-color: white; color: #1d2129; font-family: Helvetica, Arial, sans-serif; font-size: 14px;"><br /></span>
<span style="background-color: white; color: #1d2129; font-family: Helvetica, Arial, sans-serif; font-size: 14px;">"f(x)=x+4" </span><span style="background-color: white; color: #1d2129; font-family: Helvetica, Arial, sans-serif; font-size: 14px;">的第一個意思是"把f(x)定義為x+4", 第二個意思是"經過計算, 發現f(x)的值是x+4".</span><br />
<span style="background-color: white; color: #1d2129; font-family: Helvetica, Arial, sans-serif; font-size: 14px;"><br /></span>
<span style="color: #1d2129; font-family: Helvetica, Arial, sans-serif;"><span style="background-color: white; font-size: 14px;">有些人為了分開兩種意思, 故意用不同的符號. 例如用":="代表定義, "="是原本第二個的意思.</span></span><br />
<span style="color: #1d2129; font-family: Helvetica, Arial, sans-serif;"><span style="background-color: white; font-size: 14px;"><br /></span></span>
<span style="color: #1d2129; font-family: Helvetica, Arial, sans-serif;"><span style="background-color: white; font-size: 14px;">換句話說, 如果教科書上寫</span></span><span style="color: #1d2129; font-family: Helvetica, Arial, sans-serif;"><span style="background-color: white; font-size: 14px;">"m</span></span><span style="background-color: white; color: #1d2129; font-family: Helvetica, Arial, sans-serif; font-size: 14px;"> </span><span style="background-color: white; color: #1d2129; font-family: Helvetica, Arial, sans-serif; font-size: 14px;">× n:=m+m+...m (n 個 m)", 則</span><span style="background-color: white; color: #1d2129; font-family: Helvetica, Arial, sans-serif; font-size: 14px;">"</span><span style="background-color: white; color: #1d2129; font-family: Helvetica, Arial, sans-serif; font-size: 14px;">2 × 3 := </span><span style="background-color: white; color: #1d2129; font-family: Helvetica, Arial, sans-serif; font-size: 14px;"> 3 + 3"是錯, </span><span style="background-color: white; color: #1d2129; font-family: Helvetica, Arial, sans-serif; font-size: 14px;">而"</span><span style="background-color: white; color: #1d2129; font-family: Helvetica, Arial, sans-serif; font-size: 14px;">2 × 3 = </span><span style="background-color: white; color: #1d2129; font-family: Helvetica, Arial, sans-serif; font-size: 14px;"> 3 + 3"是對的.</span><br />
<span style="background-color: white; color: #1d2129; font-family: Helvetica, Arial, sans-serif; font-size: 14px;"><br /></span>
<span style="background-color: white; color: #1d2129; font-family: Helvetica, Arial, sans-serif; font-size: 14px;">不過, 小學課本基本上沒有區分"="的兩個意思, 所以嘛......</span>韋言仁語http://www.blogger.com/profile/08599435936360249361noreply@blogger.com0tag:blogger.com,1999:blog-21683040.post-9004087432269792582018-05-17T04:02:00.000-07:002019-11-15T21:43:02.998-08:00為何3×2=2×3?近日, 有朋友在網上談及"為何3×2=2×3?" 令我想起一個舊笑話:<br />
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<i>老師: 為何3×2=2×3?</i><br />
<i>學生: 因為都是6.</i><br />
<i>老師: 錯!</i><br />
<i>學生: 因為乘法符合交換律.</i><br />
<i>老師: 對了!</i><br />
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當然, 學生第一個答案沒錯, 第二個答案反而不自然. 就讓我來說一說.<br />
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<u>先來一個日常生活的例子</u><br />
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<span style="color: red;">甲是乙的生母的定義是甲是把乙生下來的女人</span>.<br />
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請看如下兩句:<br />
<ol>
<li><span style="color: purple;">因為沈殿霞生下了鄭欣宜, 所以沈殿霞是鄭欣宜的生母.</span></li>
<li><span style="color: purple;">因為沈殿霞是鄭欣宜的生母, 所以沈殿霞生下了鄭欣宜.</span></li>
</ol>
兩句邏輯上都對, 但第二句卻有些彆扭.<br />
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<u>讓我們入數學正題
</u><br />
<u><br /></u><span style="color: red;">×</span><span style="color: red;">符合交換律的定義是對所有數字a及b, 都有a</span><span style="color: red;">×</span><span style="color: red;">b=b</span><span style="color: red;">×</span><span style="color: red;">a.</span><br />
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請看如下兩句:<br />
<ol>
<li><span style="color: purple;">因為對所有數字a及b, 都有a×b=b×a, 所以乘法符合交換律.</span></li>
<li><span style="color: purple;">因為乘法符合交換律, 所以對所有數字a及b, 都有a×b=b×a.</span></li>
</ol>
同理, 兩句邏輯上都對, 但第二句還是有些不自然.<br />
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好了, 讓我們看看<br />
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<span style="color: blue;">問: 為何2×3=3×2? </span><br />
<span style="color: blue;">答: 因為乘法符合交換律, 所以2×3=3×2.</span><br />
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邏輯正確, 但明明2×3=3×2是交換律的其中一個條件, 這樣寫法有點倒果為因.<br />
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<span style="color: blue;">問: 為何2×3=3×2? </span><br />
<span style="color: blue;">答: 因為2×3=6及3×2=6, 所以2×3=3×2.</span><br />
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清楚, 明白, 簡單, 直接, 自然!<br />
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那麼, 何時才用交換律呢? 就在不要嘗試計算乘積的時候!<br />
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<span style="color: blue;">問: 為何x<sup>2</sup>y=yx<sup>2</sup>? </span><br />
<span style="color: blue;">答: 因為乘法符合交換律, 所以x<sup>2</sup>y=yx<sup>2</sup>.</span><br />
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我們x,y可以是任何數, 我們沒法驗證所有情況, 所以用到交換律.<br />
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<span style="color: blue;">問: 為何1532657883×89234927498274=89234927498274×1532657883? </span><br />
<span style="color: blue;">答: 因為乘法符合交換律, 所以1532657883×89234927498274=89234927498274×1532657883.</span><br />
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我這個覺得可以接受, 因為我不願意花時間把乘積計算出來.韋言仁語http://www.blogger.com/profile/08599435936360249361noreply@blogger.com0tag:blogger.com,1999:blog-21683040.post-46563174869182820732017-09-01T03:55:00.000-07:002017-09-01T04:01:46.608-07:00The Star Thrower and Nature's Way<div class="" data-block="true" data-editor="41oj1" data-offset-key="ep5ou-0-0" style="background-color: white; color: #1d2129; font-family: helvetica, arial, sans-serif; white-space: pre-wrap;">
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<span data-offset-key="ep5ou-0-0" style="font-family: inherit;">A friend has discussed the famous <i><a href="http://www.agiftofinspiration.com.au/stories/attitude/starfish.shtml">The Starfish Thrower</a></i> story on Facebook. This story has moved thousands of people.</span><br />
<span style="font-family: inherit;">I want to show him an interesting sort-of-related story which I read the Chinese translation a </span><span style="font-family: inherit;">long</span><span style="font-family: inherit;"> time ago. As my friend </span><span style="font-family: inherit;">does</span><span style="font-family: inherit;"> not read Chinese, I try to search for the original English version. It turns out that the Chinese translation is very famous, it is used as a Chinese textbook article in mainland China, but the original English is hard to find online. It turns out that many Chinese students have searched for it but failed. As a result, they even question the authenticity of the article.</span><br />
<span style="font-family: inherit;">It seems to be a New York Times article <i>Humans, All Too Human</i> by </span><span style="font-family: inherit;">Michael Blumenthal on 23 July 1989. </span><span style="font-family: inherit;">The only English website I found related to it is a <a href="http://www.nytimes.com/1989/08/27/opinion/l-when-human-and-other-animals-interact-parable-of-the-turtles-418089.html">response posted on New York Times</a> one month later. </span><span style="font-family: inherit;">Then there is a condensed version </span><span style="font-family: inherit;">titled <i>Nature's Way</i> in 1990, which I can found only on Chinese websites.</span></div>
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<span style="font-family: inherit;">If you are interested, you can find the original version and the Chinese version of </span><i style="font-family: inherit;">Nature's Way</i><span style="font-family: inherit;">, and the frustrations of Chinese students on this </span><a class="" href="http://bbs.tianya.cn/post-338-23220-1.shtml" style="font-family: inherit;">website</a><span style="font-family: inherit;">. </span>
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<span style="font-family: inherit;">W</span><span style="font-family: inherit;">hat is </span><i style="font-family: inherit;">Nature's Way</i><span style="font-family: inherit;"> about? It is a story that several men interwind with nature to save a baby turtle, which in turns led to the death of many baby turtles.</span><br />
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<span style="font-family: inherit;">The Starfish Thrower literally tells about saving one starfish to make a difference, Nature's Way reminds us that we have to be careful such that our good deed may not be a wishy-washy thinking.</span></div>
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韋言仁語http://www.blogger.com/profile/08599435936360249361noreply@blogger.com0tag:blogger.com,1999:blog-21683040.post-45170657839145535022017-06-28T19:59:00.000-07:002017-06-28T19:59:02.729-07:00一篇關於June Huh文章的讀後感The quanta Magazine今期有篇<u style="color: blue;">A Path Less Taken to the Peak of the Math World</u>是關於一位名叫June Huh 的數學家。這是一篇高水準的文章,足以讓我這個喜歡過分解讀的人,有一籃子的感想。<br />
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<b>文章大意如下:</b><br />
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「June Huh在美國出生,兩歲時隨父母返回韓國生活。<br />
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在小學時一次數學成績不好,令June Huh自覺不是數學料子。他立志成為詩人,認為詩詞對現實的描繪,勝過無謂的邏輯遊戲。<br />
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他在中學時寫了很多詩,但從未發表過。事實上,他認清了現實,詩人不能糊口,所以把志向改成科普記者。結果,他進入<span style="background-color: white; color: #1a1a1a; font-family: "merriweather" , "georgia" , serif; font-size: 16px;">Seoul National University</span>時選擇了天文物理。<br />
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在大學最後一年,一位Fields Medal得奬人<span style="background-color: white; color: #1a1a1a; font-family: "merriweather" , "georgia" , serif; font-size: 16px;"> </span><a href="http://www.math.harvard.edu/~hironaka/" style="background-color: white; box-shadow: rgba(0, 0, 0, 0) 0px 0px 0px inset, rgb(255, 134, 0) 0px 1px 0px; box-sizing: border-box; font-family: Merriweather, Georgia, serif; font-size: 16px; outline-width: 0px; text-decoration-line: none; transition-duration: 0.2s; transition-property: color, box-shadow; transition-timing-function: ease-in-out;" target="_blank">Heisuke Hironaka</a>到<span style="background-color: white; color: #1a1a1a; font-family: "merriweather" , "georgia" , serif; font-size: 16px;">Seoul National University訪問,教授一年代數幾何的特別課。June Huh上了那一門課,並希望Heisuke Hironaka是他作為科普記者的</span><span style="background-color: white; color: #1a1a1a; font-family: "merriweather" , "georgia" , serif; font-size: 16px;">第一個受訪者。</span><br />
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<span style="color: #1a1a1a; font-family: "merriweather" , "georgia" , serif;"><span style="background-color: white;">特別課很難,很多數學本科生聽不懂,退出了。June Huh也聽不懂,但既然立志做科普記者,只要稍稍明白概念就足夠。他沒退出,甚至主動接觸Heisuke Hironaka。</span></span><br />
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<span style="color: #1a1a1a; font-family: "merriweather" , "georgia" , serif;"><span style="background-color: white;">結果,Heisuke Hironaka看到June Huh的數學天賦,亦成為了他的數學碩士導師。</span></span><br />
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<span style="color: #1a1a1a; font-family: "merriweather" , "georgia" , serif;"><span style="background-color: white;">在</span></span><span style="background-color: white; color: #1a1a1a; font-family: "merriweather" , "georgia" , serif;">Heisuke Hironaka的鼓勵下,June Huh報讀美國的博士課程。但因為不是數學本科生,成績也不算出眾,即使有如此著名的碩士導師撑腰,亦只是</span><span style="background-color: white; color: #1a1a1a; font-family: "merriweather" , "georgia" , serif; font-size: 16px;">the University of Illinois(Urbana-Champaign)接受他的申請。</span><br />
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<span style="background-color: white; color: #1a1a1a; font-family: "merriweather" , "georgia" , serif; font-size: 16px;">在唸博士的第一年,</span>June<span style="background-color: white; color: #1a1a1a; font-family: "merriweather" , "georgia" , serif;"> Huh一鳴驚人,他用代數幾何的方法,解決了一個著名組合數學猜想Rota Conjecture的一個特例!</span><br />
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<span style="background-color: white; color: #1a1a1a; font-family: "merriweather" , "georgia" , serif;">接下來,就是一個數學天才的故事,他與另外兩位天才</span><span style="background-color: white; color: #1a1a1a; font-family: "merriweather" , "georgia" , serif; font-size: 16px;">Eric Katz及Karim Adiprasito合力解決了</span><span style="background-color: white; color: #1a1a1a; font-family: "merriweather" , "georgia" , serif;">Rota Conjecture,現在被視為有力問鼎下屆或再下一屆的</span>Fields Medal。順帶一提,Fields Medal只頒發給四十歲以下的數學家。」<br />
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<b>讀後感</b><br />
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他小時候立志成為詩人,學人寫詩,卻沒有發表,最後放棄。感覺上不是真的因為詩人不能糊口而放棄,而是長大了發覺以前寫的只是少年強說愁的幼稚詩,不能入大雅之堂,明白不是詩人料子!<br />
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他自以為不是數學料子。事實上,他是數學料子,但不是數學考試料子。這種人,給他足夠時間,多難的數學考題都答到滿分,但只給他兩三小時,他便很吃力。他是幸運的,遇到伯樂,大多數同類都給無可奈何的考試抺殺了。<br />
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其實,他的數學成績,一直不算差,否則怎可能入讀天文物理!天文物理的數學,可不是說笑的。<br />
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在撮要中,只有一句"<span style="background-color: white; color: #1a1a1a; font-family: "merriweather" , "georgia" , serif;">他用代數幾何的方法,解決了一個著名組合數學猜想Rota Conjecture的一個特例",但其實在文章中,這段文章佔有三分之一。因為,這反映了他的天才與及影响。</span><br />
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<span style="color: #1a1a1a; font-family: "merriweather" , "georgia" , serif;"><span style="background-color: white;">所謂隔行如隔山,即使是不同的數學分枝,他們的思考方法及通用語言,都有分別。把兩個數學分枝連起上來,難度非常大,而一旦成功,就會給後來的數學家更多有用的功具。</span></span><br />
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<span style="color: #1a1a1a; font-family: "merriweather" , "georgia" , serif;"><span style="background-color: white;">坐標幾何利用算術來研究幾何,萬有引力理論把天上及地上的現象連在一起,電磁學把電與磁合一。博奕論成為生物學與經濟學的基礎工具。可以說,中國文人口中不清不楚的萬法歸一,正在由科學家一點一滴的完成。</span></span><br />
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<span style="color: #1a1a1a; font-family: "merriweather" , "georgia" , serif;"><span style="background-color: white;">作為天才,他是有一點與別不同。其他天才要麼是數學神童,要麼大學時已鋒忘畢露,好像他大學畢業才半途出家做數學,又真是異數。</span></span>
<span style="color: #1a1a1a; font-family: "merriweather" , "georgia" , serif;"><span style="background-color: white;"><br /></span></span>韋言仁語http://www.blogger.com/profile/08599435936360249361noreply@blogger.com0tag:blogger.com,1999:blog-21683040.post-64983472755560940522016-07-07T03:18:00.001-07:002016-08-25T06:41:15.771-07:00試譯Scott Aaronson的《數造傳奇》影評<h2 style="color: #333333; font-family: "Trebuchet MS", "Lucida Grande", Verdana, Arial, sans-serif; font-size: 1.6em; line-height: 24.576px; margin: 30px 0px 0px; text-align: justify;">
<a href="http://www.scottaaronson.com/blog/?p=2707">“Largely just men doing sums”: My review of the excellent Ramanujan film</a></h2>
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<strong style="background-color: white;">[警告<i>:</i>內有劇透,以及連分數算式.]</strong></div>
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<span style="background-color: white; color: #333333; font-family: "lucida grande" , "verdana" , "arial" , sans-serif; font-size: 1.05em; line-height: 1.4em;">這陣子,我和太太忙於照顧小孩,很難得有機會去看《<a href="http://www.imdb.com/title/tt0787524/">數造傳奇</a></span><em style="color: #333333; font-family: "lucida grande", verdana, arial, sans-serif; font-size: 16.128px; line-height: 21.504px;"><span style="color: #b85b5a;"><a href="http://www.imdb.com/title/tt0787524/">The Man Who Knew Infinity</a></span></em><span style="background-color: white; color: #333333; font-family: "lucida grande" , "verdana" , "arial" , sans-serif; font-size: 1.05em; line-height: 1.4em;">》首映</span><span style="background-color: white; font-family: "lucida grande" , "verdana" , "arial" , sans-serif; line-height: 1.4em;"><span style="color: magenta; font-size: x-small;">(鳴!他好幸褔呀!我太太不肯陪我看首映!)</span></span><em style="color: #333333; font-family: "lucida grande", verdana, arial, sans-serif; font-size: 1.05em; line-height: 1.4em;">—</em><span style="color: #333333; font-family: "lucida grande" , "verdana" , "arial" , sans-serif; font-size: 1.05em; line-height: 1.4em;">此片是關於印度數學家<a href="https://en.wikipedia.org/wiki/Srinivasa_Ramanujan">拉馬努金</a></span><a href="https://en.wikipedia.org/wiki/Srinivasa_Ramanujan" style="color: #333333; font-family: "lucida grande", verdana, arial, sans-serif; line-height: 1.4em;"><span style="background-color: white; font-size: 1.05em; line-height: 1.4em;"> </span><span style="color: #b85b5a;"><span style="font-size: 1.05em; line-height: 1.4em;">Srinivasa Ramanujan</span></span></a><span style="color: #333333; font-family: "lucida grande" , "verdana" , "arial" , sans-serif;"><span style="line-height: 1.4em;">與英國數學大師<span style="background-color: white;"><a href="https://zh.wikipedia.org/zh-hk/%E6%88%88%E5%BC%97%E9%9B%B7%C2%B7%E5%93%88%E7%BD%97%E5%BE%B7%C2%B7%E5%93%88%E4%BB%A3">哈代</a></span></span></span><span style="color: #b85b5a; font-family: "lucida grande" , "verdana" , "arial" , sans-serif;"><span style="font-size: 1.05em; line-height: 1.4em;"><a href="https://zh.wikipedia.org/zh-hk/%E6%88%88%E5%BC%97%E9%9B%B7%C2%B7%E5%93%88%E7%BD%97%E5%BE%B7%C2%B7%E5%93%88%E4%BB%A3">G. H. Hardy</a></span></span><span style="color: #333333; font-family: "lucida grande" , "verdana" , "arial" , sans-serif;"><span style="line-height: 1.4em;">的亦師亦友關係。此放映會之後,還可以與該片的導演</span></span><span style="background-color: white; color: #333333; font-family: "lucida grande" , "verdana" , "arial" , sans-serif; font-size: 1.05em; line-height: 1.4em;">Matthew Brown, 該片的藍本傳記作家Robert Kanigel ,及該片的顧</span><span style="background-color: white; color: #333333; font-family: "lucida grande" , "verdana" , "arial" , sans-serif; font-size: 16.8px; line-height: 23.52px;">問<a href="https://zh.wikipedia.org/zh-hk/%E8%8F%B2%E5%B0%94%E5%85%B9%E5%A5%96" style="color: #252525;">費爾茲獎</a><span style="color: #333333;">得主</span></span><a href="https://en.wikipedia.org/wiki/Manjul_Bhargava" style="color: #b85b5a; font-family: "lucida grande", verdana, arial, sans-serif; font-size: 1.05em; line-height: 1.4em; text-decoration: none;">Manjul Bhargava</a><span style="background-color: white; color: #333333; font-family: "lucida grande" , "verdana" , "arial" , sans-serif; font-size: 1.05em; line-height: 1.4em;">面談。</span></div>
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<span style="background-color: white;">我是在90年代初期拜讀Kanigel所著<em style="color: #b85b5a; text-decoration: none;"><a href="http://www.amazon.com/Man-Who-Knew-Infinity-Ramanujan/dp/0671750615" style="color: #b85b5a; text-decoration: none;">The Man Who Knew Infinity</a></em>,對我影响甚深。過千頁的傳記談及拉馬努金謎樣的思考方法,他與</span><span style="background-color: white; font-size: 16.8px; line-height: 23.52px;">哈代及Littlewood的關係,他對數學分析的影响。當然書中也有對當年13歳的書蟲極為震撼的算式:例如</span></div>
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width: 0px;"></span></span><span style="border: 0px; clip: rect(3.003em 1003.5em 5.266em -999.998em); left: 36.5156px; margin: 0px 0px 0px -1.719em; padding: 0px; position: absolute; top: -3.244em; transition: none; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-107" style="border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mn" id="MathJax-Span-108" style="border: 0px; display: inline; font-family: "mathjax_main"; font-size: 14.3671px; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;">1</span><span class="mo" id="MathJax-Span-109" style="border: 0px; display: inline; font-family: "mathjax_main"; font-size: 14.3671px; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;">+</span><span class="mfrac" id="MathJax-Span-110" style="border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="border: 0px; display: inline-block; height: 0px; margin: 0px 0.101em; padding: 0px; position: relative; transition: none; vertical-align: 0px; width: 2.364em;"><span style="border: 0px; clip: rect(3.348em 1001.18em 4.135em -999.998em); left: 24.0156px; margin: 0px 0px 0px -0.588em; padding: 0px; position: absolute; top: -4.327em; transition: none; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-111" style="border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="border: 0px; display: inline-block; height: 0px; margin: 0px; padding: 0px; position: relative; transition: none; vertical-align: 0px; width: 1.183em;"><span style="border: 0px; clip: rect(3.643em 1000.2em 4.135em -999.998em); left: 0.002em; margin: 0px; padding: 0px; position: absolute; top: -3.982em; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-112" style="border: 0px; display: inline; font-family: MathJax_Math-italic; font-size: 10.1606px; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;">e</span><span style="border: 0px; display: inline-block; height: 3.987em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; left: 0.248em; margin: 0px; padding: 0px; position: absolute; top: -4.13em; transition: none; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-113" style="border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-114" style="border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mo" id="MathJax-Span-115" style="border: 0px; display: inline; font-family: "mathjax_main"; font-size: 10.1606px; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;">−</span><span class="mn" id="MathJax-Span-116" style="border: 0px; display: inline; font-family: "mathjax_main"; font-size: 10.1606px; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;">4</span><span class="mi" id="MathJax-Span-117" style="border: 0px; display: inline; font-family: MathJax_Math-italic; font-size: 10.1606px; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;">π<span style="border: 0px; display: inline-block; height: 1px; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0.002em;"></span></span></span></span><span style="border: 0px; display: inline-block; height: 3.987em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span></span><span style="border: 0px; display: inline-block; height: 3.987em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; clip: rect(3.102em 1002.22em 4.528em -999.998em); left: 24.0156px; margin: 0px 0px 0px -1.129em; padding: 0px; position: absolute; top: -3.244em; transition: none; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-118" style="border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mn" id="MathJax-Span-119" style="border: 0px; display: inline; font-family: "mathjax_main"; font-size: 10.1606px; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;">1</span><span class="mo" id="MathJax-Span-120" style="border: 0px; display: inline; font-family: "mathjax_main"; font-size: 10.1606px; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;">+</span><span class="mfrac" id="MathJax-Span-121" style="border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="border: 0px; display: inline-block; height: 0px; margin: 0px 0.101em; padding: 0px; position: relative; transition: none; vertical-align: 0px; width: 1.331em;"><span style="border: 0px; clip: rect(3.348em 1001.18em 4.135em -999.998em); left: 13.5156px; margin: 0px 0px 0px -0.588em; padding: 0px; position: absolute; top: -4.277em; transition: none; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-122" style="border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="border: 0px; display: inline-block; height: 0px; margin: 0px; padding: 0px; position: relative; transition: none; vertical-align: 0px; width: 1.183em;"><span style="border: 0px; clip: rect(3.643em 1000.2em 4.135em -999.998em); left: 0.002em; margin: 0px; padding: 0px; position: absolute; top: -3.982em; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-123" style="border: 0px; display: inline; font-family: MathJax_Math-italic; font-size: 10.1606px; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;">e</span><span style="border: 0px; display: inline-block; height: 3.987em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; left: 0.248em; margin: 0px; padding: 0px; position: absolute; top: -4.13em; transition: none; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-124" style="border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-125" style="border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mo" id="MathJax-Span-126" style="border: 0px; display: inline; font-family: "mathjax_main"; font-size: 10.1606px; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;">−</span><span class="mn" id="MathJax-Span-127" style="border: 0px; display: inline; font-family: "mathjax_main"; font-size: 10.1606px; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;">6</span><span class="mi" id="MathJax-Span-128" style="border: 0px; display: inline; font-family: MathJax_Math-italic; font-size: 10.1606px; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;">π<span style="border: 0px; display: inline-block; height: 1px; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0.002em;"></span></span></span></span><span style="border: 0px; display: inline-block; height: 3.987em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span></span><span style="border: 0px; display: inline-block; height: 3.987em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; clip: rect(3.495em 1001.18em 4.184em -999.998em); left: 13.5156px; margin: 0px 0px 0px -0.588em; padding: 0px; position: absolute; top: -3.638em; transition: none; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-129" style="border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mn" id="MathJax-Span-130" style="border: 0px; display: inline; font-family: "mathjax_main"; font-size: 10.1606px; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;">1</span><span class="mo" id="MathJax-Span-131" style="border: 0px; display: inline; font-family: "mathjax_main"; font-size: 10.1606px; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;">+</span><span class="mo" id="MathJax-Span-132" style="border: 0px; display: inline; font-family: "mathjax_main"; font-size: 10.1606px; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;">⋯</span></span><span style="border: 0px; display: inline-block; height: 3.987em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; clip: rect(0.888em 1001.33em 1.232em -999.998em); left: 0.002em; margin: 0px; padding: 0px; position: absolute; top: -1.178em; transition: none; vertical-align: 0px;"><span style="border-color: initial; border-top-style: solid; border-width: 1.3px 0px 0px; display: inline-block; height: 0px; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none; vertical-align: 0.002em; width: 1.331em;"></span><span style="border: 0px; display: inline-block; height: 1.085em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span></span></span><span style="border: 0px; display: inline-block; height: 3.987em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; clip: rect(0.888em 1002.36em 1.232em -999.998em); left: 0.002em; margin: 0px; padding: 0px; position: absolute; top: -1.227em; transition: none; vertical-align: 0px;"><span style="border-color: initial; border-top-style: solid; border-width: 1.3px 0px 0px; display: inline-block; height: 0px; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none; vertical-align: 0.002em; width: 2.364em;"></span><span style="border: 0px; display: inline-block; height: 1.085em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span></span></span><span style="border: 0px; display: inline-block; height: 3.987em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; clip: rect(0.888em 1003.59em 1.232em -999.998em); left: 0.002em; margin: 0px; padding: 0px; position: absolute; top: -1.277em; transition: none; vertical-align: 0px;"><span style="border-color: initial; border-top-style: solid; border-width: 1.3px 0px 0px; display: inline-block; height: 0px; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none; vertical-align: 0.002em; width: 3.594em;"></span><span style="border: 0px; display: inline-block; height: 1.085em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span></span></span><span style="border: 0px; display: inline-block; height: 3.987em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; clip: rect(0.888em 1005.76em 1.232em -999.998em); left: 0.002em; margin: 0px; padding: 0px; position: absolute; top: -1.277em; transition: none; vertical-align: 0px;"><span style="border-color: initial; border-top-style: solid; border-width: 1.3px 0px 0px; display: inline-block; height: 0px; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none; vertical-align: 0.002em; width: 5.758em;"></span><span style="border: 0px; display: inline-block; height: 1.085em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span></span><span class="mo" id="MathJax-Span-133" style="border: 0px; display: inline; font-family: "mathjax_main"; margin: 0px; padding: 0px 0px 0px 0.298em; position: static; transition: none; vertical-align: 0px;">=</span><span class="mrow" id="MathJax-Span-134" style="border: 0px; display: inline; margin: 0px; padding: 0px 0px 0px 0.298em; position: static; transition: none; vertical-align: 0px;"><span class="mo" id="MathJax-Span-135" style="border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0.002em;"><span style="border: 0px; font-family: "mathjax_size4"; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;">(</span></span><span class="msqrt" id="MathJax-Span-136" style="border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="border: 0px; display: inline-block; height: 0px; margin: 0px; padding: 0px; position: relative; transition: none; vertical-align: 0px; width: 4.479em;"><span style="border: 0px; clip: rect(2.315em 1003.45em 4.823em -999.998em); left: 0.986em; margin: 0px; padding: 0px; position: absolute; top: -3.982em; transition: none; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-137" style="border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mfrac" id="MathJax-Span-138" style="border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="border: 0px; display: inline-block; height: 0px; margin: 0px 0.101em; padding: 0px; position: relative; transition: none; vertical-align: 0px; width: 3.249em;"><span style="border: 0px; clip: rect(3.003em 1003.1em 4.282em -999.998em); left: 33px; margin: 0px 0px 0px -1.572em; padding: 0px; position: absolute; top: -4.671em; transition: none; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-139" style="border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mn" id="MathJax-Span-140" style="border: 0px; display: inline; font-family: "mathjax_main"; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;">5</span><span class="mo" id="MathJax-Span-141" style="border: 0px; display: inline; font-family: "mathjax_main"; margin: 0px; padding: 0px 0px 0px 0.248em; position: static; transition: none; vertical-align: 0px;">+</span><span class="msqrt" id="MathJax-Span-142" style="border: 0px; display: inline; margin: 0px; padding: 0px 0px 0px 0.248em; position: static; transition: none; vertical-align: 0px;"><span style="border: 0px; display: inline-block; height: 0px; margin: 0px; padding: 0px; position: relative; transition: none; vertical-align: 0px; width: 1.331em;"><span style="border: 0px; clip: rect(3.151em 1000.45em 4.135em -999.998em); left: 0.839em; margin: 0px; padding: 0px; position: absolute; top: -3.982em; transition: none; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-143" style="border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mn" id="MathJax-Span-144" style="border: 0px; display: inline; font-family: "mathjax_main"; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;">5</span></span><span style="border: 0px; display: inline-block; height: 3.987em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; clip: rect(0.937em 1000.49em 1.281em -999.998em); left: 0.839em; margin: 0px; padding: 0px; position: absolute; top: -1.916em; transition: none; vertical-align: 0px;"><span style="border-color: initial; border-top-style: solid; border-width: 1.3px 0px 0px; display: inline-block; height: 0px; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none; vertical-align: -0.047em; width: 0.494em;"></span><span style="border: 0px; display: inline-block; height: 1.085em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; clip: rect(3.052em 1000.84em 4.331em -999.998em); left: 0.002em; margin: 0px; padding: 0px; position: absolute; top: -4.031em; transition: none; vertical-align: 0px;"><span style="border: 0px; font-family: "mathjax_main"; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;">√</span><span style="border: 0px; display: inline-block; height: 3.987em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span></span></span><span style="border: 0px; display: inline-block; height: 3.987em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; clip: rect(3.151em 1000.45em 4.135em -999.998em); left: 33px; margin: 0px 0px 0px -0.244em; padding: 0px; position: absolute; top: -3.293em; transition: none; vertical-align: 0px;"><span class="mn" id="MathJax-Span-145" style="border: 0px; display: inline; font-family: "mathjax_main"; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;">2</span><span style="border: 0px; display: inline-block; height: 3.987em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; clip: rect(0.888em 1003.25em 1.232em -999.998em); left: 0.002em; margin: 0px; padding: 0px; position: absolute; top: -1.277em; transition: none; vertical-align: 0px;"><span style="border-color: initial; border-top-style: solid; border-width: 1.3px 0px 0px; display: inline-block; height: 0px; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none; vertical-align: 0.002em; width: 3.249em;"></span><span style="border: 0px; display: inline-block; height: 1.085em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span></span></span><span style="border: 0px; display: inline-block; height: 3.987em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; clip: rect(3.594em 1003.5em 3.889em -999.998em); left: 0.986em; margin: 0px; padding: 0px; position: absolute; top: -5.556em; transition: none; vertical-align: 0px;"><span style="border: 0px; display: inline-block; height: 0px; margin: 0px; padding: 0px; position: relative; transition: none; vertical-align: 0px; width: 3.495em;"><span style="border: 0px; font-family: "mathjax_main"; left: -0.096em; margin: 0px; padding: 0px; position: absolute; top: -3.982em; transition: none; vertical-align: 0px;">−<span style="border: 0px; display: inline-block; height: 3.987em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; font-family: "mathjax_main"; left: 2.806em; margin: 0px; padding: 0px; position: absolute; top: -3.982em; transition: none; vertical-align: 0px;">−<span style="border: 0px; display: inline-block; height: 3.987em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; font-family: "mathjax_main"; left: 0.396em; margin: 0px; padding: 0px; position: absolute; top: -3.982em; transition: none; vertical-align: 0px;">−<span style="border: 0px; display: inline-block; height: 3.987em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; font-family: "mathjax_main"; left: 0.839em; margin: 0px; padding: 0px; position: absolute; top: -3.982em; transition: none; vertical-align: 0px;">−<span style="border: 0px; display: inline-block; height: 3.987em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; font-family: "mathjax_main"; left: 1.331em; margin: 0px; padding: 0px; position: absolute; top: -3.982em; transition: none; vertical-align: 0px;">−<span style="border: 0px; display: inline-block; height: 3.987em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; font-family: "mathjax_main"; left: 1.823em; margin: 0px; padding: 0px; position: absolute; top: -3.982em; transition: none; vertical-align: 0px;">−<span style="border: 0px; display: inline-block; height: 3.987em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; font-family: "mathjax_main"; left: 2.315em; margin: 0px; padding: 0px; position: absolute; top: -3.982em; transition: none; vertical-align: 0px;">−<span style="border: 0px; display: inline-block; height: 3.987em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span><span style="border: 0px; display: inline-block; height: 3.987em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; clip: rect(2.069em 1001.04em 5.365em -999.998em); left: 0.002em; margin: 0px; padding: 0px; position: absolute; top: -4.13em; transition: none; vertical-align: 0px;"><span style="border: 0px; font-family: "mathjax_size4"; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;">√</span><span style="border: 0px; display: inline-block; height: 3.987em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span></span><span class="mo" id="MathJax-Span-146" style="border: 0px; display: inline; font-family: "mathjax_main"; margin: 0px; padding: 0px 0px 0px 0.248em; position: static; transition: none; vertical-align: 0px;">−</span><span class="mfrac" id="MathJax-Span-147" style="border: 0px; display: inline; margin: 0px; padding: 0px 0px 0px 0.248em; position: static; transition: none; vertical-align: 0px;"><span style="border: 0px; display: inline-block; height: 0px; margin: 0px 0.101em; padding: 0px; position: relative; transition: none; vertical-align: 0px; width: 3.249em;"><span style="border: 0px; clip: rect(3.003em 1003.05em 4.282em -999.998em); left: 33px; margin: 0px 0px 0px -1.572em; padding: 0px; position: absolute; top: -4.671em; transition: none; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-148" style="border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="msqrt" id="MathJax-Span-149" style="border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="border: 0px; display: inline-block; height: 0px; margin: 0px; padding: 0px; position: relative; transition: none; vertical-align: 0px; width: 1.331em;"><span style="border: 0px; clip: rect(3.151em 1000.45em 4.135em -999.998em); left: 0.839em; margin: 0px; padding: 0px; position: absolute; top: -3.982em; transition: none; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-150" style="border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mn" id="MathJax-Span-151" style="border: 0px; display: inline; font-family: "mathjax_main"; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;">5</span></span><span style="border: 0px; display: inline-block; height: 3.987em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; clip: rect(0.937em 1000.49em 1.281em -999.998em); left: 0.839em; margin: 0px; padding: 0px; position: absolute; top: -1.916em; transition: none; vertical-align: 0px;"><span style="border-color: initial; border-top-style: solid; border-width: 1.3px 0px 0px; display: inline-block; height: 0px; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none; vertical-align: -0.047em; width: 0.494em;"></span><span style="border: 0px; display: inline-block; height: 1.085em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; clip: rect(3.052em 1000.84em 4.331em -999.998em); left: 0.002em; margin: 0px; padding: 0px; position: absolute; top: -4.031em; transition: none; vertical-align: 0px;"><span style="border: 0px; font-family: "mathjax_main"; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;">√</span><span style="border: 0px; display: inline-block; height: 3.987em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span></span><span class="mo" id="MathJax-Span-152" style="border: 0px; display: inline; font-family: "mathjax_main"; margin: 0px; padding: 0px 0px 0px 0.248em; position: static; transition: none; vertical-align: 0px;">+</span><span class="mn" id="MathJax-Span-153" style="border: 0px; display: inline; font-family: "mathjax_main"; margin: 0px; padding: 0px 0px 0px 0.248em; position: static; transition: none; vertical-align: 0px;">1</span></span><span style="border: 0px; display: inline-block; height: 3.987em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; clip: rect(3.151em 1000.45em 4.135em -999.998em); left: 33px; margin: 0px 0px 0px -0.244em; padding: 0px; position: absolute; top: -3.293em; transition: none; vertical-align: 0px;"><span class="mn" id="MathJax-Span-154" style="border: 0px; display: inline; font-family: "mathjax_main"; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;">2</span><span style="border: 0px; display: inline-block; height: 3.987em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; clip: rect(0.888em 1003.25em 1.232em -999.998em); left: 0.002em; margin: 0px; padding: 0px; position: absolute; top: -1.277em; transition: none; vertical-align: 0px;"><span style="border-color: initial; border-top-style: solid; border-width: 1.3px 0px 0px; display: inline-block; height: 0px; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none; vertical-align: 0.002em; width: 3.249em;"></span><span style="border: 0px; display: inline-block; height: 1.085em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span></span><span class="mo" id="MathJax-Span-155" style="border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0.002em;"><span style="border: 0px; font-family: "mathjax_size4"; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;">)</span></span></span><span class="mroot" id="MathJax-Span-156" style="border: 0px; display: inline; margin: 0px; padding: 0px 0px 0px 0.15em; position: static; transition: none; vertical-align: 0px;"><span style="border: 0px; display: inline-block; height: 0px; margin: 0px; padding: 0px; position: relative; transition: none; vertical-align: 0px; width: 2.315em;"><span style="border: 0px; clip: rect(2.954em 1001.28em 4.135em -999.998em); left: 0.986em; margin: 0px; padding: 0px; position: absolute; top: -3.982em; transition: none; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-157" style="border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="border: 0px; display: inline-block; height: 0px; margin: 0px; padding: 0px; position: relative; transition: none; vertical-align: 0px; width: 1.281em;"><span style="border: 0px; clip: rect(3.397em 1000.45em 4.135em -999.998em); left: 0.002em; margin: 0px; padding: 0px; position: absolute; top: -3.982em; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-158" style="border: 0px; display: inline; font-family: MathJax_Math-italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;">e</span><span style="border: 0px; display: inline-block; height: 3.987em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; left: 0.445em; margin: 0px; padding: 0px; position: absolute; top: -4.376em; transition: none; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-159" style="border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-160" style="border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mn" id="MathJax-Span-161" style="border: 0px; display: inline; font-family: "mathjax_main"; font-size: 14.3671px; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;">2</span><span class="mi" id="MathJax-Span-162" style="border: 0px; display: inline; font-family: MathJax_Math-italic; font-size: 14.3671px; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;">π<span style="border: 0px; display: inline-block; height: 1px; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0.002em;"></span></span></span></span><span style="border: 0px; display: inline-block; height: 3.987em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span></span><span style="border: 0px; display: inline-block; height: 3.987em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; clip: rect(3.594em 1001.33em 3.889em -999.998em); left: 0.986em; margin: 0px; padding: 0px; position: absolute; top: -4.72em; transition: none; vertical-align: 0px;"><span style="border: 0px; display: inline-block; height: 0px; margin: 0px; padding: 0px; position: relative; transition: none; vertical-align: 0px; width: 1.331em;"><span style="border: 0px; font-family: "mathjax_main"; left: -0.096em; margin: 0px; padding: 0px; position: absolute; top: -3.982em; transition: none; vertical-align: 0px;">−<span style="border: 0px; display: inline-block; height: 3.987em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; font-family: "mathjax_main"; left: 0.642em; margin: 0px; padding: 0px; position: absolute; top: -3.982em; transition: none; vertical-align: 0px;">−<span style="border: 0px; display: inline-block; height: 3.987em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; font-family: "mathjax_main"; left: 0.248em; margin: 0px; padding: 0px; position: absolute; top: -3.982em; transition: none; vertical-align: 0px;">−<span style="border: 0px; display: inline-block; height: 3.987em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span><span style="border: 0px; display: inline-block; height: 3.987em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; clip: rect(3.003em 1001.04em 4.479em -999.998em); left: 0.002em; margin: 0px; padding: 0px; position: absolute; top: -4.13em; transition: none; vertical-align: 0px;"><span style="border: 0px; font-family: "mathjax_size1"; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;">√</span><span style="border: 0px; display: inline-block; height: 3.987em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; clip: rect(3.495em 1000.25em 4.135em -999.998em); left: 0.445em; margin: 0px; padding: 0px; position: absolute; top: -4.523em; transition: none; vertical-align: 0px;"><span class="mn" id="MathJax-Span-163" style="border: 0px; display: inline; font-family: "mathjax_main"; font-size: 10.1606px; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;">5</span><span style="border: 0px; display: inline-block; height: 3.987em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span></span></span><span style="border: 0px; display: inline-block; height: 3.987em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span><span style="border-color: initial; border-left-style: solid; border-width: 0px; display: inline-block; height: 6.015em; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none; vertical-align: -3.53em; width: 0px;"></span></span></nobr></span></div>
<div style="font-family: "lucida grande", verdana, arial, sans-serif; line-height: 1.4em;">
<span style="color: #333333; font-size: 15.36px;">拜讀此書後數年,我在加拿大遇到過真正認識哈代的數學大師</span><a href="https://en.wikipedia.org/wiki/Richard_K._Guy" style="color: #b85b5a; font-size: 1.05em; line-height: 1.4em; text-decoration: none;">Richard K. Guy</a><span style="background-color: white; color: #333333; font-size: 1.05em; line-height: 1.4em;">, 我的中學同學沒有一個知道哈代嘛!</span><span style="background-color: white; line-height: 1.4em;"><span style="color: magenta; font-size: x-small;">(我在卡加里大學時也和Guy說過話呢!)</span><span style="color: #333333;">除為甚麼我這麼興奮,因為我對所有數學家的電影及傳記都很感興趣。</span></span></div>
<ul style="color: #333333; font-family: "lucida grande", verdana, arial, sans-serif; font-size: 15.36px; line-height: 1.4em; list-style: none; margin-left: 0px; padding: 0px 0px 0px 10px; text-indent: -10px;">
</ul>
<div style="color: #333333; font-family: "lucida grande", verdana, arial, sans-serif; font-size: 1.05em; line-height: 1.4em;">
<span style="background-color: white;">閒話說完,《數造傳奇》究竟如何?當然不可能滿分喲!但這肯定是到目前為止,最好最接近史實的數學家電影。趁未落畫,好去睇啦!</span><br />
<span style="background-color: white;"><br /></span>
<span style="background-color: white;">讓我先從數學講起。</span><br />
<span style="background-color: white;"><br /></span>
<span style="background-color: white;">在電影開場時,便很貼題的以<a href="https://www.google.com.hk/url?sa=t&rct=j&q=&esrc=s&source=web&cd=8&cad=rja&uact=8&ved=0ahUKEwi8m47J9-DNAhUMG5QKHU2cDdMQFgg2MAc&url=https%3A%2F%2Fen.wikipedia.org%2Fwiki%2FRamanujan%27s_lost_notebook&usg=AFQjCNFGslCa-b4G-Wh8nKPFVNhv6TGK_g&sig2=1ij4J3lpShMFrIhdfaAFug">拉馬努金的筆記</a>作為背景。在片中的後半段,很正確地解釋了拉馬努金與</span><span style="background-color: white; font-size: 16.8px; line-height: 23.52px;">Hardy最令人稱為的成果,就是發現了</span></div>
<div class="MathJax_Display" style="color: #333333; font-family: "lucida grande", verdana, arial, sans-serif; font-size: 15.36px; line-height: 1.4em; margin: 1em 0em; max-height: none; max-width: none; min-height: 0px; min-width: 0px; position: relative; text-align: center; width: 450px;">
<span class="MathJax" id="MathJax-Element-3-Frame" style="border: 0px; direction: ltr; display: inline; float: none; font-size: 16.128px; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; padding: 0px; text-align: left; white-space: nowrap; word-spacing: normal; word-wrap: normal;" tabindex="0"><nobr style="border: 0px; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; padding: 0px; transition: none; vertical-align: 0px;"><span class="math" id="MathJax-Span-164" role="math" style="background-color: white; border: 0px; display: inline-block; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 9.152em;"><span style="border: 0px; display: inline-block; font-size: 20.3213px; height: 0px; margin: 0px; padding: 0px; position: relative; transition: none; vertical-align: 0px; width: 7.234em;"><span style="border: 0px; clip: rect(0.494em 1007.18em 3.397em -999.998em); left: 0.002em; margin: 0px; padding: 0px; position: absolute; top: -2.26em; transition: none; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-165" style="border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><a href="http://mathworld.wolfram.com/PartitionFunctionP.html"><span class="mi" id="MathJax-Span-166" style="border: 0px; display: inline; font-family: MathJax_Math-italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;">P<span style="border: 0px; display: inline-block; height: 1px; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0.101em;"></span></span><span class="mo" id="MathJax-Span-167" style="border: 0px; display: inline; font-family: "mathjax_main"; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;">(</span><span class="mi" id="MathJax-Span-168" style="border: 0px; display: inline; font-family: MathJax_Math-italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;">n</span><span class="mo" id="MathJax-Span-169" style="border: 0px; display: inline; font-family: "mathjax_main"; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;">)</span></a><span class="mo" id="MathJax-Span-170" style="border: 0px; display: inline; font-family: "mathjax_main"; margin: 0px; padding: 0px 0px 0px 0.298em; position: static; transition: none; vertical-align: 0px;">≈</span><span class="mfrac" id="MathJax-Span-171" style="border: 0px; display: inline; margin: 0px; padding: 0px 0px 0px 0.298em; position: static; transition: none; vertical-align: 0px;"><span style="border: 0px; display: inline-block; height: 0px; margin: 0px 0.101em; padding: 0px; position: relative; transition: none; vertical-align: 0px; width: 3.249em;"><span style="border: 0px; clip: rect(2.856em 1003.15em 4.135em -999.998em); left: 33px; margin: 0px 0px 0px -1.572em; padding: 0px; position: absolute; top: -4.671em; transition: none; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-172" style="border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="border: 0px; display: inline-block; height: 0px; margin: 0px; padding: 0px; position: relative; transition: none; vertical-align: 0px; width: 3.151em;"><span style="border: 0px; clip: rect(3.397em 1000.45em 4.135em -999.998em); left: 0.002em; margin: 0px; padding: 0px; position: absolute; top: -3.982em; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-173" style="border: 0px; display: inline; font-family: MathJax_Math-italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;">e</span><span style="border: 0px; display: inline-block; height: 3.987em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; left: 0.445em; margin: 0px; padding: 0px; position: absolute; top: -4.327em; transition: none; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-174" style="border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-175" style="border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-176" style="border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-177" style="border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mo" id="MathJax-Span-178" style="border: 0px; display: inline; font-family: MathJax_Math-italic; font-size: 14.3671px; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;">π</span></span></span><span class="msqrt" id="MathJax-Span-179" style="border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="border: 0px; display: inline-block; height: 0px; margin: 0px; padding: 0px; position: relative; transition: none; vertical-align: 0px; width: 2.216em;"><span style="border: 0px; clip: rect(3.298em 1001.48em 4.331em -999.998em); left: 0.691em; margin: 0px; padding: 0px; position: absolute; top: -3.982em; transition: none; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-180" style="border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mn" id="MathJax-Span-181" style="border: 0px; display: inline; font-family: "mathjax_main"; font-size: 14.3671px; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;">2</span><span class="mi" id="MathJax-Span-182" style="border: 0px; display: inline; font-family: MathJax_Math-italic; font-size: 14.3671px; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;">n</span><span class="texatom" id="MathJax-Span-183" style="border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-184" style="border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mo" id="MathJax-Span-185" style="border: 0px; display: inline; font-family: "mathjax_main"; font-size: 14.3671px; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;">/</span></span></span><span class="mn" id="MathJax-Span-186" style="border: 0px; display: inline; font-family: "mathjax_main"; font-size: 14.3671px; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;">3</span></span><span style="border: 0px; display: inline-block; height: 3.987em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; clip: rect(0.937em 1001.48em 1.281em -999.998em); left: 0.691em; margin: 0px; padding: 0px; position: absolute; top: -1.67em; transition: none; vertical-align: 0px;"><span style="border-color: initial; border-top-style: solid; border-width: 1.3px 0px 0px; display: inline-block; height: 0px; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none; vertical-align: -0.047em; width: 1.478em;"></span><span style="border: 0px; display: inline-block; height: 1.085em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; clip: rect(3.249em 1000.74em 4.381em -999.998em); left: 0.002em; margin: 0px; padding: 0px; position: absolute; top: -3.982em; transition: none; vertical-align: 0px;"><span style="border: 0px; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="border: 0px; font-family: "mathjax_size1"; font-size: 14.3671px; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;">√</span></span><span style="border: 0px; display: inline-block; height: 3.987em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span style="border: 0px; display: inline-block; height: 3.987em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span></span><span style="border: 0px; display: inline-block; height: 3.987em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; clip: rect(3.003em 1002.41em 4.282em -999.998em); left: 33px; margin: 0px 0px 0px -1.227em; padding: 0px; position: absolute; top: -3.195em; transition: none; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-187" style="border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mn" id="MathJax-Span-188" style="border: 0px; display: inline; font-family: "mathjax_main"; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;">4</span><span class="msqrt" id="MathJax-Span-189" style="border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="border: 0px; display: inline-block; height: 0px; margin: 0px; padding: 0px; position: relative; transition: none; vertical-align: 0px; width: 1.331em;"><span style="border: 0px; clip: rect(3.151em 1000.45em 4.135em -999.998em); left: 0.839em; margin: 0px; padding: 0px; position: absolute; top: -3.982em; transition: none; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-190" style="border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mn" id="MathJax-Span-191" style="border: 0px; display: inline; font-family: "mathjax_main"; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;">3</span></span><span style="border: 0px; display: inline-block; height: 3.987em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; clip: rect(0.937em 1000.49em 1.281em -999.998em); left: 0.839em; margin: 0px; padding: 0px; position: absolute; top: -1.916em; transition: none; vertical-align: 0px;"><span style="border-color: initial; border-top-style: solid; border-width: 1.3px 0px 0px; display: inline-block; height: 0px; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none; vertical-align: -0.047em; width: 0.494em;"></span><span style="border: 0px; display: inline-block; height: 1.085em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; clip: rect(3.052em 1000.84em 4.331em -999.998em); left: 0.002em; margin: 0px; padding: 0px; position: absolute; top: -4.031em; transition: none; vertical-align: 0px;"><span style="border: 0px; font-family: "mathjax_main"; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;">√</span><span style="border: 0px; display: inline-block; height: 3.987em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span></span><span class="mi" id="MathJax-Span-192" style="border: 0px; display: inline; font-family: MathJax_Math-italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;">n</span></span><span style="border: 0px; display: inline-block; height: 3.987em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; clip: rect(0.888em 1003.25em 1.232em -999.998em); left: 0.002em; margin: 0px; padding: 0px; position: absolute; top: -1.277em; transition: none; vertical-align: 0px;"><span style="border-color: initial; border-top-style: solid; border-width: 1.3px 0px 0px; display: inline-block; height: 0px; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none; vertical-align: 0.002em; width: 3.249em;"></span><span style="border: 0px; display: inline-block; height: 1.085em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span></span><span class="mo" id="MathJax-Span-193" style="border: 0px; display: inline; font-family: "mathjax_main"; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;">,</span></span><span style="border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span><span style="border-color: initial; border-left-style: solid; border-width: 0px; display: inline-block; height: 3.412em; margin: 0px; overflow: hidden; 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<span style="background-color: white;">及給出正確證明。</span></div>
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<span style="background-color: white; line-height: 1.4em;">電影很清楚的說明,數學家醉心的並不是數學的應用,而是數學的美。虔試的拉馬努金認為他是在寫下女神的旨意,而無神論者哈代亦以數學作為他的心靈歸宿。</span></div>
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<span style="background-color: white; line-height: 1.4em;">關於數學的正確性,未受過正規訓練拉馬努金很信任直觀,而哈代則強調證明,因而產生了張力。片中對兩者的描述都頗中肯。不似荷理活電影老是貶低嚴謹思維的重要性。</span></div>
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<span style="background-color: white; line-height: 1.4em;">不過,對於數學愛好者來說,片中對數學的描述還未足夠。</span></div>
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<span style="background-color: white; line-height: 22.5792px;">接着,讓我談談電影的其他支節。</span></div>
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<span style="background-color: white; color: #333333; line-height: 22.5792px;">就我所知,電影中的南印度服飾,音樂,藝術,宗教,都符合現實。至於印度人交談用英語,而非泰米爾語,據導演</span><span style="background-color: white; color: #333333; line-height: 23.52px;">Brown</span><span style="background-color: white; line-height: 22.5792px;"><span style="color: #333333;">所講,是一種妥協:既要照顧觀眾,主角及一些印度演員也不懂泰米爾語!</span><span style="color: magenta; font-size: x-small;">(作為國家,印度有最多不同的語言。)</span></span></div>
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<span style="background-color: white; color: #333333;">有人不滿演員太老。當年哈地不過四十,</span><span style="background-color: white; color: #333333; line-height: 23.52px;">Jeremy Irons來演老了三十年。拉馬努金的妻子</span><span style="background-color: white; color: #333333; line-height: 23.52px;">Janaki</span><span style="background-color: white; color: #333333; line-height: 23.52px;">結婚時只有九歳(!),丈夫去英國時只是十四,卻找個成熟的</span><span style="background-color: white; line-height: 1.4em;"><span style="color: #333333;">Devika Bhise來演。</span><span style="color: magenta; font-size: x-small;">(如果要找個年紀相乎的演員,我突然想起黑白片時代的馮寶寶!)</span></span><span style="background-color: white; color: #333333; line-height: 1.4em;">此外 </span><a href="https://en.wikipedia.org/wiki/John_Edensor_Littlewood" style="color: #b85b5a; line-height: 1.4em; text-decoration: none;">J. E. Littlewood</a><span style="background-color: white; color: #333333; line-height: 1.4em;"> 由一個感覺似諧星的演員來做,很難令人感到他也是一代的數學大師。</span></div>
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<span style="background-color: white; line-height: 24.696px;">在片中,對數學邏輯有傑出貢獻的哲學家</span><span style="background-color: white; line-height: 23.52px;"><a href="https://en.wikipedia.org/wiki/Russell%27s_paradox">Bertrand Russell</a>,只是個過場角色。雖則現實裡他與哈代是好朋友兼同時是第一次大戰的和平主義者,但哈代向他請教如何幫助拉馬努金,應是子虛烏有。</span></div>
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<span style="background-color: white;">其實這些都不重要。最令我介懷的,是片中誇大了的種族岐視。</span><span style="background-color: white; line-height: 23.52px;">有個數學教授感到拉馬努金侮辱而趕了他出課室?</span><span style="background-color: white; line-height: 1.4em;">拉馬努金被英國士兵拳打腳踢?哈代見到他的傷勢不聞不問?是的,拉馬努金會受岐視及感到寂寞,但劍橋社會不見得會如此赤裸裸。</span><span style="background-color: white; line-height: 24.696px;">電影選擇直接見得到的傷害</span><span style="background-color: white; line-height: 1.4em;">,避開難拍的隱藏岐視,反而減弱它的吸引力。</span></div>
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<span style="background-color: white;">還有,電影如實的拍出劍橋三一書院變成戰地醫院及學生操兵出陣,但要拉馬努金親眼見到屍體?不用罷,現實經已很殘酷。</span><br />
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<span style="background-color: white;">電影似乎引導觀眾接馬努金是死於肺癆。但近年有研究顯示,他是死於由印度帶過去的<a href="https://en.wikipedia.org/wiki/Amoebiasis">阿米巴病</a>—這是一種當年很難確診但醫很好的病。片中沒有講到拉馬努金在回去印度的那一年,其實不斷與哈地書信往來,直到病殁。很多人認同這抺去這一年是好的選擇,但若然是我,我會拍出來!</span></div>
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夠了!以荷里活電影的角度,以上的批評已是吹毛求疵。要記住,<span style="background-color: white; line-height: 23.52px;">Brown用了近十年來尋找資金,來拍這套電影;他本來可以早些籌到金錢,但他拒住加插拉馬努金與由英國女星主演的白人護士熱戀的吸金情節!可以想像,加插P(n)解釋時的一番鬥爭。</span></div>
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<span style="background-color: white;">最後,讓我們一看</span><a href="http://www.nme.com/reviews/various-artists/16439" style="color: #b85b5a; line-height: 1.4em; text-decoration: none;">Olly Richards</a>為<span style="background-color: white; line-height: 1.4em;">此劇寫出的負面評價</span><span style="background-color: white; line-height: 1.4em;">:</span></div>
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<span style="background-color: white;"><i>看着一大堆公式,不會讓你興奮。以感情為主題,你很難說不喜歡,但它就是枯燥。拍出水準,但它其實只是一班男人在計數。這些算式很重要,但直至最後一幕談及黑洞,觀眾是不能領略它對科學發展有多重要。</i></span></div>
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<span style="background-color: white;">作為一個從小就喜愛數學的人,我向Brown表達由衷的敬意。拍一套數學家電影,</span><span style="background-color: white; line-height: 23.52px;">Brown是在打一場幾乎不可能勝的仗。趁未落畫,不管你走路,坐船,或乘搭車牌是</span><span style="background-color: white; line-height: 23.52px;"><a href="https://zh.wikipedia.org/zh-hk/%E7%9A%84%E5%A3%AB%E6%95%B8">#1729</a>的士,去看這套戲罷!</span></div>
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韋言仁語http://www.blogger.com/profile/08599435936360249361noreply@blogger.com0tag:blogger.com,1999:blog-21683040.post-12970488259045977912016-07-06T07:53:00.000-07:002016-07-06T07:54:26.830-07:00短文兩則<div>
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一味雪榚濃湯,放在口中可以同時感受到冰涼及溫暖。想像不難,但煮不出來的,不是食物。</div>
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一隻大理石鳥,沒有任何由地面來的支撐。想像不難,但彫刻不出來的,不是彫塑。</div>
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一條符號算式,例如3+7=9。想像不難,但證明不到的,不是數學。</div>
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一切學問,都由想像開始。然後,以想像為指引,嘗試落手去幹。有時成功,有時失敗。</div>
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要澄清一點,落手去幹,可能是假手於人去幹。<a href="https://zh.wikipedia.org/zh-hk/%E5%88%97%E5%A5%A5%E7%BA%B3%E5%A4%9A%C2%B7%E8%BE%BE%C2%B7%E8%8A%AC%E5%A5%87">逹文西</a>不是每幅畫所有部分都自己畫,<a href="https://zh.wikipedia.org/zh-hk/%E5%8F%B2%E8%92%82%E5%A4%AB%C2%B7%E4%B9%94%E5%B8%83%E6%96%AF">喬布斯</a>不是電腦每一部分都自己造,<a href="https://zh.wikipedia.org/zh-hk/%E6%96%AF%E9%87%8C%E5%B0%BC%E7%93%A6%E7%91%9F%C2%B7%E6%8B%89%E9%A9%AC%E5%8A%AA%E9%87%91">拉馬努金</a>不是每條數學定理都自己證明。</div>
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若有很多成功的往績,其他人漸漸相信你的直覺。這個,對於倚靠外來資源的創作,尤其重要。因為其他人要相信你的直覺,才肯給與幫助,完成作品。(當然啦,很多時給與幫助,就是給錢!)</div>
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英文把一切學問的最前沿,稱為"state of art"(藝術境界)。只有懂得這門學問的,才可真正感受到學問之美。</div>
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不懂品酒的人,怎知紅酒是否美酒?不懂繪畫的人,怎分得出行貨定佳品?不懂數學的人,也不可能真正了解數學之美。</div>
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不過,紅酒與圖畫,還可以用價錢粗略分辨。對於不懂數學的人,實用範圍便成了一個很方便的指標。</div>
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朋友,如果你在笑一個老粗不識藝術,只懂買貴貨;想想你談數學時,是否只懂談及數學多有用?</div>
韋言仁語http://www.blogger.com/profile/08599435936360249361noreply@blogger.com0tag:blogger.com,1999:blog-21683040.post-32710996938141271022016-06-16T21:18:00.000-07:002016-06-16T21:18:18.788-07:00由《數造傳奇》談天才之路我在之前的一個POST講過,有天份也要教育,始可成才。<br />
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《數造傳奇(<a data-href="https://www.pinterest.com/pin/538250592946801101/" href="https://www.google.com.hk/url?sa=t&rct=j&q=&esrc=s&source=web&cd=19&cad=rja&uact=8&ved=0ahUKEwi79P2oma7NAhXKchQKHf7LBoA4ChAWCFIwCA&url=https%3A%2F%2Fwww.pinterest.com%2Fpin%2F538250592946801101%2F&usg=AFQjCNEvr-4crncyAmjyivUQ_3O9y-Y7cw&sig2=mmgLeG7EvfLBWez6IUUoiA" style="color: #660099; cursor: pointer; font-family: arial, sans-serif; font-size: 18px; white-space: nowrap;"> The Man Who Knew Infinity</a>)》中的<a href="https://en.wikipedia.org/wiki/Srinivasa_Ramanujan" style="background: none rgb(255, 255, 255); color: #0b0080; font-family: sans-serif; font-size: 14px; line-height: 22.4px;" title="Srinivasa Ramanujan">Ramanujan</a>,沒有受過正統數學訓練。但他在成長的關鍵時期,卻遇到了一位好導師—<a href="https://en.wikipedia.org/wiki/G._S._Carr" style="background: none rgb(255, 255, 255); color: #0b0080; font-family: sans-serif; font-size: 14px; line-height: 22.4px; text-decoration: none;" title="G. S. Carr">G. S. Carr</a>的署作《<i style="background-color: white; color: #252525; font-family: sans-serif; font-size: 14px; line-height: 22.4px;"><a href="https://en.wikipedia.org/wiki/Synopsis_of_Pure_Mathematics" style="background: none; color: #0b0080;" title="Synopsis of Pure Mathematics">A Synopsis of Elementary Results in Pure and Applied Mathematics</a></i>》,激發了他的數學潛能。若以學術成就論英雄,他接受了非常適合他的教育。<br />
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另一位香港人較熟悉的數學家,六歳學習微積分的<span style="background: none rgb(255, 255, 255); font-family: inherit; line-height: 22.4px;"><a href="https://zh.wikipedia.org/wiki/%E9%99%B6%E5%93%B2%E8%BD%A9" style="background-attachment: initial; background-clip: initial; background-image: none; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; color: #0b0080; line-height: 22.4px;" title="陶哲軒">陶哲軒</a>。其父母用盡方法去平衡他的過人天賦及正常的校園生活,結果人所共知:註定永遠沒有電影商拍攝陶哲軒傳,皆因這個天才的生活太平凡!</span><br />
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<span style="background-color: white; line-height: 22.4px;">Ramanujan有過人的直觀,陶哲軒有超強邏輯思維。對數學而言,好的直觀用以找出正確方向,好的邏輯思維用以找出正確證明。直觀正確而證明力到位,就會有好定理;直觀有徧差而證明力到位,也可以適時調整目標;直觀正確而證明力不足,可以找人幫忙,很多數學家會提出問題與人討論,不時會出現數百年也未有人能證明的猜想;若直觀與證明力都不足,放棄是個好選擇。最槽榚的直觀與邏輯思維都不足,但自以為是的傢伙—每年都有人宣稱可以<a href="https://zh.wikipedia.org/wiki/%E4%B8%89%E7%AD%89%E5%88%86%E8%A7%92">三等分任意角</a>或證明了</span><a href="https://zh.wikipedia.org/wiki/%E5%93%A5%E5%BE%B7%E5%B7%B4%E8%B5%AB%E7%8C%9C%E6%83%B3" style="background: none rgb(255, 255, 255); color: #0b0080; font-family: sans-serif; line-height: 22.4px;" title="哥德巴赫猜想">哥德巴赫猜想</a>—他們老喜歡寫信給數學家要求認同,就是這種人差點把Ramanujan埋沒了。<br />
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<span style="color: #252525; font-family: sans-serif;"><span style="background-color: white; line-height: 22.4px;">至於天才何以成為天才,答案其實很簡單。除非相信印度教或佛教中的業報,否則只有一個答案:好運氣。大多數人的能力都是中度,即所謂正常。徧離正軌的,要麼是弱能,要麼是超能,統計學告訴我們這兩者必然出現。</span></span><span style="background-color: white; color: #252525; font-family: sans-serif; line-height: 22.4px;">李麗珊,李慧思,曹星如,亞里,保特,岩士唐,貝多分,莫扎特,梵高,莫奈,… 很多人一樣努力,但就是沒有他們的天才。</span><br />
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<span style="color: #252525; font-family: sans-serif;"><span style="background-color: white; line-height: 22.4px;">不過呢,即使是天份,若沒有受到合適的教育,也不能成才。一個英國小鎮的孩子,可能很有圍棋天份,但誰會知呢?說起來,真有點悲哀,原來運氣真的很重要!</span></span>韋言仁語http://www.blogger.com/profile/08599435936360249361noreply@blogger.com0tag:blogger.com,1999:blog-21683040.post-9722539369138998682016-06-04T05:57:00.000-07:002016-06-04T05:57:53.708-07:00論天才由一個單細胞開始,按基因的指令成長。在這個成長過程,其實是不停受外界因素影响:就如鱷魚蛋孵出是雄是雌,由環境溫度決定。<br />
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即使極簡單的封閉系統,都會出現不可預測的混沌,更何況生命這個開放的複雜系統!<br />
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有人提出所謂「基因決定論」,不過這個"決定",是概率性的。即是主軸決定了,但大誤差也有可能。這種誤差,甚至是長尾的,即大幅徧離主軸也是尋常。<br />
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對人類社會來說:如果徧離主軸的方向是"壞",這個人就是有問題;方向是"好",這個人就是有天份。<br />
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我們通常以出生的剎那,作為生命的原點。在原點,身體的各部分,包括思想、神經、肌肉,也對世界有個糢糊的感覺,然後隨着生活的經驗累積,這個感覺遂步清晰。可以說,整個身體,是由不同的反饋迴路組成。<br />
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有些人,在出生時的感覺,就徧離了主軸。有些人,卻是在吸收經驗時走徧了。<br />
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天生有問題的人,最終未必現問題—天生社交功能障礙,未必不能融入社會。有天份的人,最終未必成大器—容易與人交朋友,未必成為公關能手。<br />
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所以天生有問題,需要教育令其不致太差。天生有才華的人,需要教育令他成大器。<br />
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若果是天生沒問題沒才華的人,好的教育會讓他成為一個較好的人,卻不可能成大器。有說成功靠1%天才99%努力,其實只是用來強調努力的重要。努力而能成功,原因是本身的天才讓人能夠在努力中吸收!<br />
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有天份,有合適的教育,還要有運氣。要成為一代大家,在吸收經驗時要嚴重走徧,跳離前人知識的框架!<br />
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年初的李世乭與AlphaGo之戰。兩者其實都是天生棋手(後者的圍棋程式),從老師處學懂圍棋(後者的老師是程式員),然後不斷透過博奕過善自己的棋藝。兩者都曾在對戰中使出「神的一手」取勝,而所謂「神的一手」其實是徧離正路無人預見的怪棋—若然輸了,就是糟榚的壞棋。當出現「神的一手」,肯定革新了人類對圍棋的認知。「昭和棋聖」吳清源就是因發現了新圍棋佈局而震驚棋壇。<br />
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「神的一手」,亦顯示了所謂圍棋高手也是對圍棋"知其然而不知其所以然"。這並不是貶低他們,而是圍棋本身的特性。數學家Ernst Zermelo很早就證明,所有不含運氣的二人博奕,就會有個不敗玩法。撲克不知對手的牌,飛行棋要撙骰,都是含有運氣。圍棋一切都清清楚楚,應該不含運氣;但現實是變化太多,不能把對手的所有下子法都分析清楚,結果要靠概率估算對方的步法,亦即是含有運氣。<br />
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真正清清楚楚的二人博奕,根本沒有人喜歡玩。大人通常不玩打井游戲嘛!<br />
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事實上,世間大部分學問,都非常複雜,所以我們都是用概率的方法的思考。經濟學說人是理性的,意思是人都會把自身利益極大化。但自身利益的計算異常複雜,如果真的要統統計算,肯定會「機關算盡太聽明」,浪費過多的精力而徒勞!所以自身利益極大化,要用概率的想法去看,所以見到有人徧離了,也不要大驚小怪。<br />
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我們的思想都是概率性,都是以徧概全。科學非黑即白,可人類慣常思考卻複雜十倍。因此,現代數學及科學所要求的嚴謹性,與人類本來的思想模式不太想同。可能,這就是人們感到學習理科困難的原因。怪不得身兼數學家、電腦先驅、博奕論之父John von Neumann說過:"若人們不相信數學簡單,只因他們未意識到生命之複雜!"<br />
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順帶一提,一門學問出現突破,未必會成為該門學問的大家。John Conway研究圍棋,發明出一門新的數學。雖然他本身不是頂級棋手,但他卻帶給我們,頂級棋手也不知道的圍棋知識。若有人喜歡一門學問,卻自知沒可能成為專家,或者可以考慮從另一個方向思考,為喜歡的學問添加新的認知。<br />
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<span style="color: #252525; font-family: sans-serif;"><span style="font-size: 15.104px; line-height: 24.1664px;"><br /></span></span>韋言仁語http://www.blogger.com/profile/08599435936360249361noreply@blogger.com0tag:blogger.com,1999:blog-21683040.post-27866279519564050192016-04-11T10:27:00.000-07:002016-04-11T10:27:55.311-07:00活在當下的女子原文: <a href="http://www.wired.com/2016/04/susie-mckinnon/">http://www.wired.com/2016/04/susie-mckinnon/</a> <br />
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就如很多美國的中產夫婦,Susie Mckinnon與Eric Green不時坐遊輪去度假。他們在華盛頓州效區的家,放滿了記念品。一對普普通通,將快退休的夫婦,和我談及他們的少年時代以及各種旅遊經歷。不過,Susie很明確的告訴我,她其實一點兒也不記得。她不記那些遊輪假期,不記得與丈夫一起的任何一天,不記得自己的婚前往事。可是,她並非失憶,她從未記得過。<br />
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多年來,科學家都猜測有這種人存在。他們與普通人沒有分別,生活如常,可就是沒有正常的長期記憶。終於,在2006,科學家找到了Susie—或者應該說,是Susie找上了科學家。Susie是史上第一個被診斷為SDAM(severely deficient autobiographical memory)的人,她知道自己生命中發生的事,卻不可以在腦海中重構出來。<br />
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Susie解釋:「我知道甚麼發生了,只是沒能夠回想起來。例如我孩童時候應該很矮小,但我卻完全記不起相關經驗。我很多時需要猜度,Cayman Islands應該很熱,我和丈夫應該在那裡行了許多轉,時間應該是介乎2000到2010。」
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在賣座電影〈玩轉腦朋友〉,主角失去核心記憶後,連個性也失去了。但Susie似乎沒有核心記憶,個性卻很鮮明。她是思想開通的白人女子,再保守父親的抗議下嫁了個黑人;她曾經是天主教徒,卻決定要離開宗教;她有工作(從事退休規劃),有興趣,有大堆朋友。問題來了:我們常以為重要的記憶,究竟真的有多重要?<br />
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曾幾何時,科學家以為長期記憶只有一種。但在1972, 加拿大學者Endel Tulving提出長期記憶有多種。一種是關於技能(semantic memory),例如記住一個字怎麼唸,一個菜怎麼做。但你未必記得幾時學唸哪個字,幾時學會煮哪個菜。另一重是關於情景(episodic memory),把時間及感覺重構成一個畫面,例如記住幾時學會唸一個字,記住學會煮一個菜的喜悅。<br />
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<span style="color: blue;">(以下的兩段,細節由</span><a href="http://www.thestar.com/life/2015/04/28/susie-mckinnon-cant-remember-events-in-her-life-just-facts.html">http://www.thestar.com/life/2015/04/28/susie-mckinnon-cant-remember-events-in-her-life-just-facts.html</a><span style="color: blue;">補充,原文寫得不大清楚。)</span><br />
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Susie是在1977年發現自己與眾不同。她的一位中學時的朋友在唸護理,問她是否願意幫忙完成一份關於記憶的功課。當朋友問到一些她小時候的基本問題,她答:「你問這些作甚?誰會記起?」她以前聽過有人自稱記得往事,她一向以為是老作,就如她自己一般。如果有些舊事她會清楚"記得",只因為她不斷重覆向其他人說那些故事。朋友驚訝於她的答案,她則驚訝於她朋友的驚訝。她去問其他人,始發現只有自己的記憶是老作,而她一向以為老作記憶是正當的行為。<br />
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如是者,三十年斷斷續續的追尋答案,終於在2004,她讀到了Endel Tulving的文章,找到了答案的鎖匙。Tulving 的文章是關於失憶症者K. C.,30 歳時單車意外腦部受損,只能記着最近一兩分鐘發生的事,但以往所學技能(例如數學及歷史)並未失去,且仍可以透過練習取得新技能。雖然所有已知失憶症者原因都是後天腦部受傷,亦因而難以再過正常生活,但Tulving猜測,現實中應該有生活正常的失憶症者。<br />
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為了不想驚動太出名的Tulving,改為接觸他的合作伙伴Brian Levine。她在006年八月廿五日的電郵中寫到:「我應該是你們要找的失憶症者。」在Susie之後,Brian又發現兩位SDAM患者,兩位都有穩定工作,其中一人更是博士。Brian發表這三人的研究文章,引起轟動,很多人宣稱自己也是這種病患者。即使當中只有小部分證實有病,亦顯示出這並非罕見。Brian想問:若沒有這種長期記憶也可以生活得好,為何我們會演化出這種能力?<br />
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沒有長期記憶不盡是壞的。Susie的丈夫曾被一班白人毒打,負傷回家,嚇得Susie哭了出來,但Susie之後忘記了情景,沒有留下陰影。她也很少吵架,吵了也不記得。她只記得自己是60歳的女人,沒有不復年青的感慨。她見自己小時候的相片,只覺得是看着個不認識的女孩子。Susie喜歡看故事,她每次看完都會忘記,再看就有新鮮感。<br />
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順帶一提,與Susie剛好相反的,是一個叫Price的女人,她可以記得生活一切鎖碎事。超强記憶令她一時成為傳媒追訪對象 。不過,太强的記憶似乎佔據她的生活,以至年過四十依然要倚靠她的父母。<br />
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你可能認為,現代電腦記憶科技,可以彌補Susie不足。不過Susie跟本沒與趣。她很少影相,不享受錄影,也很少用社交媒體。她不特別用電郵作記錄,也不常用面書。不過,相片不能代替記憶:她一向以來不知一位好友有出席她的婚體,結婚相冊沒有好友的照片,但Eric告訴她,好友就是攝影師!話說回來,我們其實變得更似Susie,因為科技令到我們不用太着意記憶!<br />
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如大多數的失憶症者一般,Susie沒有想象力。她不能想象沙灘的情景。她沒有白日夢。她下棋時只能想到眼前的一步。她沒有能力考慮將來!<br />
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她從未回憶過去,從不放眼將來,她是真真正正的活在當下!
韋言仁語http://www.blogger.com/profile/08599435936360249361noreply@blogger.com0tag:blogger.com,1999:blog-21683040.post-6883026143336217702015-12-22T04:14:00.000-08:002015-12-22T04:14:08.417-08:00解迷的解之迷,再現曙光....<span style="background-color: white; font-family: Georgia, Times, 'Times New Roman', serif; font-size: 30px; font-weight: inherit; line-height: 36px;"><a href="https://www.quantamagazine.org/20151221-hope-rekindled-for-abc-proof/">Hope Rekindled for Perplexing Proof</a></span><br />
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<span style="background-color: white; color: #3b3b3b; font-family: Georgia, Times, 'Times New Roman', serif; font-size: 19px; line-height: 24.89px;">(三年前,一位出色數學家為abc猜想提出了難以明白的解。近期的一個會議,為</span><span style="color: #3b3b3b; font-family: Georgia, Times, Times New Roman, serif;"><span style="font-size: 19px; line-height: 24.89px;">這個謎樣的解,帶來不大清晰的曙光。)</span></span><br />
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本月初,數學界的注意力轉向牛津大學,為一個困惑了大家三年的謎,尋找曙光。</div>
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京都大學天才數學家<a href="http://www.kurims.kyoto-u.ac.jp/~motizuki/top-english.html" style="background: none; border: 0px none !important; color: #384da0; font-family: inherit; font-style: inherit; font-weight: inherit; margin: 0px; min-height: 0px !important; outline: 0px; padding: 0px !important; text-decoration: none; vertical-align: baseline;">Shinichi Mochizuki</a>在2012年8月發表了四篇解以理解卻不可勿略的論文(<a href="http://www.kurims.kyoto-u.ac.jp/~motizuki/papers-english.html" style="background: none; border: 0px none !important; color: #384da0; font-family: inherit; font-style: inherit; font-weight: inherit; margin: 0px; min-height: 0px !important; outline: 0px; padding: 0px !important; text-decoration: none; vertical-align: baseline;">see here)</a>,創始了一套理論並命名為 “inter-universal Teichmüller theory” (IUT theory) ,當中包含<a href="https://www.quantamagazine.org/tag/number-theory/" style="background: none; border: 0px none !important; color: #384da0; font-family: inherit; font-style: inherit; font-weight: inherit; margin: 0px; min-height: 0px !important; outline: 0px; padding: 0px !important; text-decoration: none; vertical-align: baseline;">數論</a>中著名<em style="background-color: transparent; border: 0px; font-family: inherit; font-weight: inherit; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;">abc</em>猜想的解。</div>
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不過,很快數學界就發現,這套Mochizuki獨力用了二十年來發展的理論,幾乎無人可以理解。公認的天才數學家,以解難及注意細節聞名,沒有人可以忽視他的論文。而然,共超過500頁的論文,太多新的慨念及定義,根本難以閱讀。再加上,這位日本數學家拒絕外訪,以致曾經拜讀該論文的數學家無法解開疑問而放棄。</div>
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過了三年,這套理論幾乎被廢棄了。終於,在今年12月7日,世界有名的數學家在牛津大學Clay Mathematics Institue舉行<a href="http://www.claymath.org/events/iut-theory-shinichi-mochizuki" style="background: none; border: 0px none !important; color: #384da0; font-family: inherit; font-style: inherit; font-weight: inherit; margin: 0px; min-height: 0px !important; outline: 0px; padding: 0px !important; text-decoration: none; vertical-align: baseline;">會議</a>,目的是嘗試了解這套理論. 牛津大學數學家<a href="https://www.quantamagazine.org/20151221-hope-rekindled-for-abc-proof/Minhyong%20Kim" style="background: none; border: 0px none !important; color: #384da0; font-family: inherit; font-style: inherit; font-weight: inherit; margin: 0px; min-height: 0px !important; outline: 0px; padding: 0px !important; text-decoration: none; vertical-align: baseline;">Minhyong Kim</a>是會議三個主辦人之一,認為這會議其實已經遲了。</div>
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Kim說:「大家,包括我與Mochizuki,開始不耐煩。應該有人做些事情,這是對數學界,對朋友,對自己,及對Mochizuki,負上一點責任。」</div>
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會議包括三天基礎講座,及兩天 IUT theory深入討論,包括關於第四篇論文的講解。第四篇論文內有<em style="background-color: transparent; border: 0px; font-family: inherit; font-weight: inherit; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;">abc</em>猜想的解。無人認為只用一星期的會議就能明白這套理論,大家都只是希望能得到個大概,認清這套理論是否正確,是否有用,是否值得繼續研究。</div>
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<strong style="border: 0px; font-family: inherit; font-style: inherit; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;"><span style="font-weight: normal;">三天基礎講座,似乎令人感覺有希望。</span></strong></div>
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<span style="border: 0px; font-family: inherit; font-style: inherit; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;"><b>解題新方向</b></span></div>
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<a href="https://en.wikipedia.org/wiki/Abc_conjecture"><em style="background-color: transparent; border: 0px; font-family: inherit; font-weight: inherit; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;">abc</em>猜想</a>的解介:除了有限個例外,若然正整數 <em style="border: 0px; font-family: inherit; font-weight: inherit; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;">a </em>+ <em style="border: 0px; font-family: inherit; font-weight: inherit; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;">b</em> = <em style="border: 0px; font-family: inherit; font-weight: inherit; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;">c</em>,則 <em style="border: 0px; font-family: inherit; font-weight: inherit; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;">c</em> 的大小會被 <em style="border: 0px; font-family: inherit; font-weight: inherit; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;">a</em>與 <em style="border: 0px; font-family: inherit; font-weight: inherit; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;">b</em>的質因數所規限。</div>
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質數作為整數乘法之本,與整數加法,從來沒有明顯的關係。所以任何把兩者連在一起的命題,都是非常重要。</div>
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1985年提出的<em style="background-color: transparent; border: 0px; font-family: inherit; font-weight: inherit; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;">abc</em>猜想,在Mochizuki的四篇論文之前,進展不大。但數家界早知些猜想與其他著名數學問題有好大關連。例如在 1986年德國Max Planck Institute for Mathematics的 <a href="https://www.hcm.uni-bonn.de/people/profile/gerd-faltings/" style="background: none; border: 0px none !important; color: #384da0; font-family: inherit; font-style: inherit; font-weight: inherit; margin: 0px; min-height: 0px !important; outline: 0px; padding: 0px !important; text-decoration: none; vertical-align: baseline;">Gerd Faltings</a> 憑着證明Mordell猜想取得費爾茲獎, 以後美國哈佛大學的 <a href="http://www.math.harvard.edu/~elkies/" style="background: none; border: 0px none !important; color: #384da0; font-family: inherit; font-style: inherit; font-weight: inherit; margin: 0px; min-height: 0px !important; outline: 0px; padding: 0px !important; text-decoration: none; vertical-align: baseline;">Noam Elkies</a> 發現<em style="border: 0px; font-family: inherit; font-weight: inherit; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;">abc</em>猜想可以改良Faltings的定理。</div>
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<em style="background-color: transparent; border: 0px; font-family: inherit; font-weight: inherit; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;">abc</em>猜想亦等價於由法國數學家<a href="https://www.gc.cuny.edu/Page-Elements/Academics-Research-Centers-Initiatives/Doctoral-Programs/Mathematics/Faculty/Lucien-Szpiro/" style="background: none; border: 0px none !important; color: #384da0; font-family: inherit; font-style: inherit; font-weight: inherit; margin: 0px; min-height: 0px !important; outline: 0px; padding: 0px !important; text-decoration: none; vertical-align: baseline;">Lucien Szpiro</a>在1980代提出的Szpiro猜想。<em style="background-color: transparent; border: 0px; font-family: inherit; font-weight: inherit; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;">abc</em>猜想是關於整數的,而Szpiro猜想則是關於橢圓曲線。將整數與橢圓曲線拉上關係,是一個常用策略。一方面它使得數論問題變得抽象及難於表達,但另一方面它引入更多解題技巧。在1994年<a href="https://www.maths.ox.ac.uk/people/andrew.wiles" style="background: none; border: 0px none !important; color: #384da0; font-family: inherit; font-style: inherit; font-weight: inherit; margin: 0px; min-height: 0px !important; outline: 0px; padding: 0px !important; text-decoration: none; vertical-align: baseline;">Andrew Wiles</a>利用這策略解決了費馬大定理(若 <em style="background-color: transparent; border: 0px; font-family: inherit; font-weight: inherit; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;">n></em>2,則 <em style="border: 0px; font-family: inherit; font-weight: inherit; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;">a<span style="border: 0px; bottom: 1ex; font-family: inherit; font-size: 12px; font-style: inherit; font-weight: inherit; height: 0px; line-height: 0; margin: 0px; outline: 0px; padding: 0px; position: relative; vertical-align: baseline;">n</span></em> +<em style="border: 0px; font-family: inherit; font-weight: inherit; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;">b<span style="border: 0px; bottom: 1ex; font-family: inherit; font-size: 12px; font-style: inherit; font-weight: inherit; height: 0px; line-height: 0; margin: 0px; outline: 0px; padding: 0px; position: relative; vertical-align: baseline;">n</span></em> = <em style="border: 0px; font-family: inherit; font-weight: inherit; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;">c<span style="border: 0px; bottom: 1ex; font-family: inherit; font-size: 12px; font-style: inherit; font-weight: inherit; height: 0px; line-height: 0; margin: 0px; outline: 0px; padding: 0px; position: relative; vertical-align: baseline;">n</span></em> 沒有整數解),事實上他更進一步把橢圓曲線連繫上 “Galois representation” ,並利用Galois representation找到問題的解。</div>
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Mochizuki把相同的策略用於<em style="border: 0px; font-family: inherit; font-weight: inherit; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;">abc</em>猜想,他把Szpiro猜想連繫上他的新發明:Frobenioid。本來,他花了很多時間去發展一套“Hodge-Arakelov 橢圓曲線理論”,卻發現並不足夠,但在過程中,他得到了Frobenioid,一種新的幾何物件的代數特徵。</div>
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簡單幾何物件的對稱,連繫着 “群”;利用抽象的群,可以研究更上一層的對稱。橢圓曲線的對稱,便連繫着Frobenioid;利用Frobenioid的等性,可以研究整數深層的對稱性。</div>
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Kim說:「Mochizuki認為,解題的關鍵,在於由一個更基本的角度,去審視整數的特性。」</div>
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在第三天結尾及第四天開首,聖地牙哥加州大學的<a href="http://math.ucsd.edu/~kedlaya/" style="background: none; border: 0px none !important; color: #384da0; font-family: inherit; font-style: inherit; font-weight: inherit; margin: 0px; min-height: 0px !important; outline: 0px; padding: 0px !important; text-decoration: none; vertical-align: baseline;">Kiran Kedlaya</a>,解釋Mochizuki希望如何用Frobenioid去對代<em style="border: 0px; font-family: inherit; font-weight: inherit; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;">abc</em>猜想。他在演講澄清了Mochizuki方法的中心思想,是這個會議最大的進展。Faltings,即Mochizuki的博士導師,在電郵中稱讚Kedlaya的演講在啟發性。</div>
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史丹褔大學的 <a href="http://math.stanford.edu/~conrad/" style="background-attachment: initial; background-clip: initial; background-color: transparent; background-image: none; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; border: 0px none !important; color: #384da0; font-family: inherit; font-style: inherit; font-weight: inherit; margin: 0px; min-height: 0px !important; outline: 0px; padding: 0px !important; text-decoration: none; vertical-align: baseline;">Brian Conrad</a>亦認同Kedlaya的演講是會議的高潮。聽過Kedlaya啟發性的演講,他期望接着的演講會更有趣味。</div>
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可惜事有願違。</div>
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<strong style="border: 0px; font-family: inherit; font-style: inherit; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;">‘好的混亂’</strong></div>
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明白Mochizuki如何把<em style="border: 0px; font-family: inherit; font-weight: inherit; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;">abc</em>猜想聯繫上Frobenioid是令人振奮的發展。但明白Mochizuki如何利用Frobenioid去解決問題卻是另一回事。</div>
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Purdue University的<a href="http://www.mcm.ac.cn/faculty/cpmok/201409/t20140916_255899.html" style="background-attachment: initial; background-clip: initial; background-color: transparent; background-image: none; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; border: 0px none !important; color: #384da0; font-family: inherit; font-style: inherit; font-weight: inherit; margin: 0px; min-height: 0px !important; outline: 0px; padding: 0px !important; text-decoration: none; vertical-align: baseline;">Chung Pang Mok</a>和京都大學的<a href="http://www.kurims.kyoto-u.ac.jp/~yuichiro/index_e.html" style="background-attachment: initial; background-clip: initial; background-color: transparent; background-image: none; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; border: 0px none !important; color: #384da0; font-family: inherit; font-style: inherit; font-weight: inherit; margin: 0px; min-height: 0px !important; outline: 0px; padding: 0px !important; text-decoration: none; vertical-align: baseline;">Yuichiro Hoshi</a> 及 <a href="http://www.kurims.kyoto-u.ac.jp/~gokun/" style="background-attachment: initial; background-clip: initial; background-color: transparent; background-image: none; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; border: 0px none !important; color: #384da0; font-family: inherit; font-style: inherit; font-weight: inherit; margin: 0px; min-height: 0px !important; outline: 0px; padding: 0px !important; text-decoration: none; vertical-align: baseline;">Go Yamashita</a>,負責在最後兩天解釋Mochizuki該四篇關於IUT理論的論文。 可以說,沒有人真的明白他們的解說。</div>
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University of Texas的<a href="http://www.ma.utexas.edu/users/voloch/" style="background: none; border: 0px none !important; color: #384da0; font-family: inherit; font-style: inherit; font-weight: inherit; margin: 0px; min-height: 0px !important; outline: 0px; padding: 0px !important; text-decoration: none; vertical-align: baseline;">Felipe Voloch</a>連續<a href="https://plus.google.com/106680226131440966362/posts/LLHPN3QLoqX" style="background-attachment: initial; background-clip: initial; background-color: transparent; background-image: none; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; border: 0px none !important; color: #384da0; font-family: inherit; font-style: inherit; font-weight: inherit; margin: 0px; min-height: 0px !important; outline: 0px; padding: 0px !important; text-decoration: none; vertical-align: baseline;">五天</a>在社交網站交代<a href="https://plus.google.com/106680226131440966362/posts/49TR2Qeb9WR" style="background: none; border: 0px none !important; color: #384da0; font-family: inherit; font-style: inherit; font-weight: inherit; margin: 0px; min-height: 0px !important; outline: 0px; padding: 0px !important; text-decoration: none; vertical-align: baseline;">會議情況</a>。如同Conrad,他的希望變為失望。所有與會者都迷失在奇怪的術語內。</div>
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迷失的原因有多方面。首先是在最後兩天的技術難度大幅提昇,這也可算在預期之內。其次是兩位日本講者慣了單向講解,不同與西方講者與聽眾互動的習慣。</div>
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雖然會議並未能帶出多少實質的發展,但的確燃起一些與會者的興趣。Kedlaya打算在明年七月在京都大學舉辦多一次會議。</div>
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有些人認為Mochizuki應該主動解釋他的工作。Faltings在電郵中提到: 「除非Mochizuki願意寫出一篇讓人明白的論文去解釋,否則問題無法解決。」</div>
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Kim卻認為未必需要,因為 “好的混亂”可以刺激人思考學習。Kim說:「在會議前,無人明白IUT文章,現在起碼有了一個概念。雖然仍然有問題,但卻是更有深度的問題。」</div>
韋言仁語http://www.blogger.com/profile/08599435936360249361noreply@blogger.com0tag:blogger.com,1999:blog-21683040.post-84433387657223557172015-07-25T23:16:00.000-07:002015-07-25T23:16:18.397-07:00陶哲軒的不凡思路<a href="http://www.nytimes.com/2015/07/26/magazine/the-singular-mind-of-terry-tao.html?_r=0">The Singular Mind of Terry Tao</a> <span style="color: red;">(受朋友所托, 試譯)</span><br />
<br />
<h2>
陶哲軒的不凡思路</h2>
<span style="font-family: nyt-mag-sans, arial, helvetica, sans-serif;"><span style="background-color: white; font-size: 18px; line-height: 22px;">—神童如何成為大數學家 <span style="color: red;">(怪獸家長注意,文章可能沒啥看頭!) </span></span></span><br />
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<span style="background-color: white; color: #333333; font-family: georgia, 'times new roman', times, serif; font-size: 16px; line-height: 23px;">今年四月,<a href="http://www.ucla.edu/">加州大學</a>辦公室內,陶哲軒思索水會否爆炸。抽述水般流體活動的方程式,始乎沒有抺殺爆炸的可能。簡單來說,當足夠的能量聚集,拋一個錢幣入校園的水池,就會摧毁整個南加州。</span><br />
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<span style="color: #333333; font-family: georgia, times new roman, times, serif;"><span style="background-color: white; line-height: 23px;">數十年來,科學家都嘗試了解,</span></span><span style="background-color: white; color: #333333; font-family: georgia, 'times new roman', times, serif; font-size: 16px; line-height: 23px;">這種爆炸會不會發生。</span><span style="background-color: white; color: #333333; font-family: georgia, 'times new roman', times, serif; font-size: 16px; line-height: 23px;">陶氏希望用一個新方法來解決這個老問題</span><span style="background-color: white; color: #333333; font-family: georgia, 'times new roman', times, serif; font-size: 16px; line-height: 23px;">。這個方法,據陶氏講,古怪猶如愛麗思夢遊仙境。</span><span style="background-color: white; font-family: georgia, 'times new roman', times, serif; font-size: 16px; line-height: 23px;"><span style="color: red;">(愛麗思夢遊仙境的作家<a href="https://en.wikipedia.org/wiki/Lewis_Carroll">Lewis Carroll</a>,也是數學家。)</span></span><br />
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<span style="color: #333333; font-family: georgia, times new roman, times, serif;"><span style="background-color: white; line-height: 23px;">他說:「如果有個水做的機器,能自我繁殖,生出更小更快的下一代,最終有一代會去到極限,引發爆炸。」他一邊用手比劃,一邊笑着解釋:「我不是說有能力製造這種機器,只是作為思想實驗,用另一個角度考慮水爆炸的問題。」若然他能推導出這並非不可能,便順道解決了</span></span><span style="background-color: white; color: #333333; font-family: georgia, 'times new roman', times, serif; font-size: 16px; line-height: 1.4375rem;"><a href="https://en.wikipedia.org/wiki/Navier%E2%80%93Stokes_existence_and_smoothness">Navier-Stokes global regularity problem</a>,一個上百年的數學問題。</span><span style="background-color: white; font-family: georgia, 'times new roman', times, serif; font-size: 16px; line-height: 1.4375rem;"><span style="color: red;">(這條數學問題,是<a href="https://en.wikipedia.org/wiki/Millennium_Prize_Problems">百萬美金千禧大奬</a>的題目。)</span></span><br />
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<span style="color: #333333; font-family: georgia, times new roman, times, serif;"><span style="background-color: white; line-height: 23px;">陶氏身穿polo裇牛仔褲涼鞋,今年40。辦公室內:書桌在窗旁;黑板滿佈公式;梳化與牆邊有距離—好攞放上班用的單車;當然有個大書櫃,有很多書,包括</span></span><span style="background-color: white; color: #333333; font-family: georgia, 'times new roman', times, serif; font-size: 16px; line-height: 1.4375rem;"> ‘‘Compactness and Contradiction’’ 及 ‘‘Poincaré’s Legacies, Part I’’—這是陶氏自少年時代出版的16冊書其中兩本。</span><br />
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<span style="background-color: white; color: #333333; font-family: georgia, 'times new roman', times, serif; font-size: 16px; line-height: 1.4375rem;">生於南澳洲的陶氏,很早就成為新閒人物。家鄉的頭條:</span><span style="background-color: white; color: #333333; font-family: georgia, 'times new roman', times, serif; font-size: 16px; line-height: 1.4375rem;"> ‘‘TINY TERENCE, 7, IS HIGH-SCHOOL WHIZ.’’ (七歲小童唸中學),內文見陶氏蹲在椅上,才夠高用書桌,和他共用書桌的女同童年紀大他一倍。老師告訴記者,陶氏正在自學,比全班快兩堂。據說陶氏2歲時已懂閱讀。</span><br />
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<span style="color: #333333; font-family: georgia, times new roman, times, serif;"><span style="background-color: white; line-height: 23px;">報導之後數個月,還在學期中,陶氏升讀12班數學。三年後,陶氏成為史上最年青國際數學奧林匹克得奬者。之後,陶氏獲奬無數,包括</span></span><span style="background-color: white;"><span style="color: #333333; font-family: georgia, times new roman, times, serif;"><span style="line-height: 23px;"><a href="https://en.wikipedia.org/wiki/MacArthur_Fellows_Program">MacArthur ‘‘genius’’ grant</a> 及<a href="https://zh.wikipedia.org/zh-hk/%E8%8F%B2%E5%B0%94%E5%85%B9%E5%A5%96">菲爾茲獎</a>,後者更有數學諾貝爾奬之稱。今天,陶氏並認為是同期最出色的數學家。</span></span></span><br />
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<span style="background-color: white; color: #333333; font-family: georgia, 'times new roman', times, serif; font-size: 16px; line-height: 23px;">陶氏告訴筆者,他對數學的認知,與兒時有很大的不同。</span><span style="background-color: white; color: #333333; font-family: georgia, 'times new roman', times, serif; line-height: 23px;">「我成長時,早知我會做數學家,但卻不知甚麼是數學家。我以為是或者有個委員會,給我一些難題去解決。」真實是,做數學家與數學系學生有很大的分別。有些成績很好的學生,成不了好的數學家。由古至今,數學的真蒂並不需要做得快,而需要耐性及創意。有時要與人合作,有時又要即興。聽來,和音樂家差不多。陶氏認為,神童時代的他根本不是在做數學。只懂得彈奏技巧及樂理,並不表示懂得音樂。</span><span style="background-color: white; font-family: georgia, 'times new roman', times, serif; line-height: 23px;"><span style="color: red;">(我曾帶過三名中學生做暑期研究。沒有固定題目的思考,嚇怕了他們。)</span></span><br />
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<span style="color: #333333; font-family: georgia, times new roman, times, serif;"><span style="background-color: white;"><span style="line-height: 23px;">十八世紀未的</span><a href="https://en.wikipedia.org/wiki/Carl_Friedrich_Gauss" style="line-height: 23px;">數學王子高斯</a><span style="line-height: 23px;">,可能是有史以來最偉大的數學家,把很多研究成果收埋,認為世人並未能接受他的發現。他死後,後人才從他的遺物整理出一個個結果,以至他死後很多年還有論文發表。其他的著名數學家,如暴燥的<a href="https://en.wikipedia.org/wiki/Isaac_Newton">牛頓</a>,有幻覺的<a href="https://en.wikipedia.org/wiki/John_Forbes_Nash,_Jr.">John Nash</a>,及最近期謝絕一切奬項的隱居者</span></span></span><span style="background-color: white; color: #333333; font-family: georgia, 'times new roman', times, serif; font-size: 16px; line-height: 23px;"><a href="https://en.wikipedia.org/wiki/Grigori_Perelman">Perelman</a>,似乎都有點怪。</span><br />
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陶哲軒看來就太正常了!他寧願打份舒服的工,拒絕一些著名學府的招聘。他教書時,喜歡強調數學有趣的一面。他的學生捉狹說:<span style="background-color: white; color: #333333; font-family: georgia, 'times new roman', times, serif; line-height: 23px;">「陶哲軒打破了荷里活天才學者的模型,他性格樂觀,生活正常,家庭和睦。肯定沒有人會拍電影講他!」</span><span style="background-color: white; font-family: georgia, 'times new roman', times, serif; line-height: 23px;"><span style="color: red;">(加州大學夠著名了罷?另外,其實九成天才,都是普通人一個)</span></span><br />
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<span style="color: #333333; font-family: georgia, times new roman, times, serif;"><span style="background-color: white; line-height: 23px;">天才的正常生活,源於正常的童年。儘管在外人看來,仍是不大正常。陶哲軒的父母來自香港,1971年移民澳洲,三年後誕下陶哲軒,之後還有兩個孩子。三兄妹經常一起玩耍,例如用不同棋類的配件,設計新的棋類遊戲。陶氏喜歡看幻想小說,上堂無聊時會畫幻想世界的地園。</span></span><br />
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<span style="color: #333333; font-family: georgia, times new roman, times, serif;"><span style="background-color: white; line-height: 23px;">陶氏8歲時,在SAT數學部分考獲760分,可以入大學了。但應該嗎?父母帶他到美國尋求協助。心理學家<a href="https://en.wikipedia.org/wiki/Julian_Stanley">Julian Stanley</a>建議,不要太急進,要有時間給陶氏發展正常的社交生活。在1985年,9歲的陶氏同時就讀中學及附近的</span></span><span style="background-color: white; color: #333333; font-family: georgia, 'times new roman', times, serif; font-size: 16px; line-height: 23px;"><a href="http://www.flinders.edu.au/">Flinders University</a>。</span><br />
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<span style="color: #333333; font-family: georgia, times new roman, times, serif;"><span style="background-color: white; line-height: 23px;">即使以不要太急進的速度,陶氏還是在17歲完成碩士論文</span></span><span style="background-color: white; color: #333333; font-family: georgia, 'times new roman', times, serif; font-size: 16px; line-height: 1.4375rem;">(‘‘Convolution Operators Generated by Right-Monogenic and Harmonic Kernels’’) 並赴<a href="http://www.princeton.edu/main/">普林斯頓</a>唸博士。陶氏的入學申請,包括來自</span><span style="background-color: white;"><span style="color: #333333; font-family: georgia, times new roman, times, serif;"><span style="line-height: 1.4375rem;"><a href="https://en.wikipedia.org/wiki/Paul_Erd%C5%91s">Paul Erdos</a>的</span><span style="line-height: 23px;">推薦信—</span><span style="line-height: 1.4375rem;"> 這位匈牙利數學家是近代最多產的數學大師。天才陶氏在普林斯頓的第一年,剛巧是三百年的<a href="https://en.wikipedia.org/wiki/Fermat%27s_Last_Theorem">費馬猜想</a>被證實的一年,陶氏震驚地發現自已原來對很多數學領域一無所知!</span></span></span><br />
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<span style="background-color: white;"><span style="color: #333333; font-family: georgia, times new roman, times, serif;"><span style="line-height: 1.4375rem;">尤如很多年青學生一樣,他晚上沉迷電腦遊戲</span></span></span><span style="background-color: white; color: #333333; font-family: georgia, 'times new roman', times, serif; font-size: 16px; line-height: 1.4375rem;"> Civilization。(他現在避開電腦遊戲,因為太易沉迷。)他又與朋友去漫畫書店玩遊戲。這是他首次正式與同齡朋友交往,亦是他逃避</span><span style="background-color: white; color: #333333; font-family: georgia, 'times new roman', times, serif; font-size: 16px; line-height: 23px;">普林斯頓讀書</span><span style="background-color: white; color: #333333; font-family: georgia, 'times new roman', times, serif; font-size: 16px; line-height: 1.4375rem;">壓力的方法。事實上,陶氏與多數天才兒童一樣,害怕難以駕御的挑戰。他在</span><span style="background-color: white; color: #333333; font-family: georgia, 'times new roman', times, serif; font-size: 16px; line-height: 23px;">Flinders University時,在量子物理一科,沒有準備,萬料不到要考量子物理的歷史,當場大哭被送離考場,結果當然不合格!</span><br />
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<span style="background-color: white; color: #333333; font-family: georgia, 'times new roman', times, serif; font-size: 16px; line-height: 23px;">不過,陶氏還未有學乖。在</span><span style="background-color: white; color: #333333; font-family: georgia, 'times new roman', times, serif; font-size: 16px; line-height: 23px;">普林斯頓,要通過三位教授的口試,才能繼續學業。口試的範圍很廣,通常要很多時間準備。陶氏又來個臨急抱佛腳,最終招架不來。據陶氏講,他當時覺得很對不起他的博士導師</span><span style="background-color: white; color: #333333; font-family: georgia, 'times new roman', times, serif; font-size: 16px; line-height: 23px;"><a href="https://en.wikipedia.org/wiki/Elias_M._Stein">Elias Stein</a>。事實上,過了口試,才是考驗的開始。</span><br />
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唸碩士有時還可以抄抄寫寫做論文,但唸博士卻要為數學增添新知識。新知識要靠嚴謹證明來支持。在尋找證明的過程中,你可能寫了很多很多紙,到頭來卻發現出了個邏輯問題,又要從頭試過。普林斯頓的<span style="background-color: white; color: #333333; font-family: georgia, 'times new roman', times, serif; font-size: 16px; line-height: 1.4375rem;"><a href="https://en.wikipedia.org/wiki/Charles_Fefferman">Charles Fefferman</a>,本身也是神童變成</span><span style="color: #333333; font-family: georgia, times new roman, times, serif;"><span style="line-height: 23px;">菲爾茲獎</span></span><span style="background-color: white; color: #333333; font-family: georgia, 'times new roman', times, serif; font-size: 16px; line-height: 1.4375rem;">得主,把尋找證明比喻為與魔鬼博奕,不過遊戲方式很奇特:魔鬼技法超高,但你可以無限次數回棋;每次被魔鬼逼回,你就會試新的步法;若你足夠聰明,你終於會找到一絲擊敗魔鬼的曙光,但你仍然要面對很多次失敗。你可能需要一週,一月,一年去尋找證明。若你能夠成功寫完博士論文,表示你有聰明和毅力,去迎接研究的工作。</span><br />
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<span style="background-color: white; color: #333333; font-family: georgia, 'times new roman', times, serif; font-size: 16px; line-height: 1.4375rem;">陶氏的研究,最著名的是關於質數,即只有兩個因數(1及自身)的正整數,如2,3,5,7和11。1不是質數,4也不是,6也不是。質數被視作數學的元素,任何一個正整數都可是分解為質數的積,例如</span><span style="background-color: white; color: #333333; font-family: georgia, 'times new roman', times, serif; font-size: 16px; line-height: 23px;">12 = 2 x 2 x 3。</span><br />
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<span style="background-color: white; color: #333333; font-family: georgia, 'times new roman', times, serif; font-size: 16px; line-height: 23px;">質數雖基本,但神秘。你永遠估計不到下一個質數出現在何處。有人甚至用音樂或詩篇去談論質數。質數連結很多不同的數學領域,假如量子力學。想象一個外太空的高等文明,他們不會懂說地球語言,但他們肯定知道質數及為之着迷。</span><br />
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<span style="background-color: white; color: #333333; font-family: georgia, 'times new roman', times, serif; font-size: 16px; line-height: 23px;">陶氏的工作,源自法國數學家</span><span style="background-color: white; color: #333333; font-family: georgia, 'times new roman', times, serif; font-size: 16px; line-height: 23px;"> </span><span style="background-color: white; color: #333333; font-family: georgia, 'times new roman', times, serif; font-size: 16px; line-height: 23px;"><a href="https://en.wikipedia.org/wiki/Alphonse_de_Polignac">Alphonse de Polignac</a>於1849提出的<a href="https://en.wikipedia.org/wiki/Twin_prime">雙生質數猜想</a>:有無限對雙生質數。雙生質數即是相隔</span><span style="background-color: white; color: #333333; font-family: georgia, 'times new roman', times, serif; font-size: 16px; line-height: 23px;">2的質數:5及7, 11及13,17及19,等等。在數線上,質數出現的頻率逾來逾小,雙生質數更是罕有,在2,237 及 2,239 之後,是2,267及2,269;在31,391之後 31,393,是31,511及31,513。順帶一提,古希臘的<a href="https://en.wikipedia.org/wiki/Euclid">歐基里德</a>早證明了存在無限的質數。</span><span style="background-color: white; color: #333333; font-family: georgia, 'times new roman', times, serif; font-size: 16px; line-height: 23px;"> </span><br />
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<span style="background-color: white; color: #333333; font-family: georgia, 'times new roman', times, serif; font-size: 16px; line-height: 23px;">當數學家面對不能應付的問題,他們會考慮相關但似乎較易的題目,期望得到靈感。陶氏於2004年與<a href="http://www.ox.ac.uk/">牛津</a>的</span><span style="background-color: white; color: #333333; font-family: georgia, 'times new roman', times, serif; font-size: 16px; line-height: 1.4375rem;"><a href="https://en.wikipedia.org/wiki/Ben_Green_(mathematician)">Ben Green</a>合作,解決了一個雙生質數的相關問題。他們證明了Green-Tao定理:存在任意長質數算術數列。例如,3, 7 及11是長度為3的質數算術數例; 11, 17</span><span style="background-color: white; color: #333333; font-family: georgia, 'times new roman', times, serif; font-size: 16px; line-height: 23px;">, 23 及29是長度為4</span><span style="background-color: white; color: #333333; font-family: georgia, 'times new roman', times, serif; font-size: 16px; line-height: 23px;">的質數算術數例。於當年二月,Green來加州大學探訪陶氏,討論兩個月,證明了這個定理。該證明使用了很多不同領域的數學,創造了新的證明技巧,給其他數學家很多新的思考方向。</span><br />
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<span style="background-color: white; color: #333333; font-family: georgia, 'times new roman', times, serif; font-size: 16px; line-height: 23px;">與人合作是陶氏研涯生涯的特點。大部分數學家都非常專門,但陶氏卻很廣泛,喜歡從合作中學習並很到新發現。他的一個老友及多年研究夥伴</span><span style="background-color: white; color: #333333; font-family: georgia, 'times new roman', times, serif; font-size: 16px; line-height: 23px;"><a href="http://www.math.umn.edu/~keel/">Markus Keel</a>,形容陶氏為</span><span style="background-color: white; color: #333333; font-family: georgia, 'times new roman', times, serif; font-size: 16px; line-height: 23px;"> </span><span style="background-color: white; color: #333333; font-family: georgia, 'times new roman', times, serif; font-size: 16px; line-height: 23px;">‘‘The Matrix’’ 的Neo,可以瞬間下載新知識及運用。</span><span style="color: #333333; font-family: georgia, times new roman, times, serif;"><span style="line-height: 23px;">菲爾茲獎簡</span></span><span style="background-color: white; color: #333333; font-family: georgia, 'times new roman', times, serif; font-size: 16px; line-height: 23px;">介中,提到陶氏對</span><span style="background-color: white; color: #333333; font-family: georgia, 'times new roman', times, serif; font-size: 16px; line-height: 23px;">Horn’s conjecture的頁獻。據稱他在唸博士時,與朋友在打玩中途,完成證明。這是一個與他以往研究大為不同的領域。在</span><span style="color: #333333; font-family: georgia, times new roman, times, serif;"><span style="line-height: 23px;">菲爾茲獎簡</span></span><span style="background-color: white; color: #333333; font-family: georgia, 'times new roman', times, serif; font-size: 16px; line-height: 23px;">介中,形容這是英語小說家,突然改行出版優等俄文小說!</span><br />
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<span style="background-color: white; color: #333333; font-family: georgia, 'times new roman', times, serif; font-size: 16px; line-height: 23px;">Green-Tao定理亦是如此。Green是<a href="https://en.wikipedia.org/wiki/Number_theory">數論</a>專家,陶氏原本是做<a href="https://en.wikipedia.org/wiki/Harmonic_analysis">調和分析</a>。與魔鬼博奕,需要各種不同的手段,惟有從觀摩其他高手的證明才能學習。 合作是減少學習時間,亦行少了寃枉路。</span><br />
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<span style="background-color: white; color: #333333; font-family: georgia, 'times new roman', times, serif; font-size: 16px; line-height: 23px;">陶氏與</span><span style="background-color: white; color: #333333; font-family: georgia, 'times new roman', times, serif; font-size: 16px; line-height: 1.4375rem;">Green</span><span style="background-color: white; color: #333333; font-family: georgia, 'times new roman', times, serif; font-size: 16px; line-height: 1.4375rem;">參考前人的證明,改正錯的,拋棄方向不同的,最終修成正果。 若要探討他倆的結果,不可以忽略其他人的貢獻,例如英國的</span><span style="background-color: white; color: #333333; font-family: georgia, 'times new roman', times, serif; font-size: 16px; line-height: 23px;"><a href="https://en.wikipedia.org/wiki/Timothy_Gowers">Timothy Gowers</a>與匈牙利的</span><span style="background-color: white; color: #333333; font-family: georgia, 'times new roman', times, serif; font-size: 16px; line-height: 1.4375rem;"><a href="https://en.wikipedia.org/wiki/Endre_Szemer%C3%A9di">Endre Szemeredi</a>;但這些人的工作又建基於更最期的Erdos, <a href="https://en.wikipedia.org/wiki/Klaus_Roth">Klaus Roth</a>及<a href="https://en.wikipedia.org/wiki/Frank_P._Ramsey">Frank Ramsey</a>等等。數學就是這樣一代代承傳下來。</span><br />
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<span style="background-color: white; color: #333333; font-family: georgia, 'times new roman', times, serif; font-size: 16px; line-height: 23px;">陶氏與</span><span style="background-color: white; color: #333333; font-family: georgia, 'times new roman', times, serif; font-size: 16px; line-height: 1.4375rem;">Green研究的問題,已經流傳多年,且當年很多數學家認為短期內無法解決。是以</span><span style="background-color: white; color: #333333; font-family: georgia, 'times new roman', times, serif; font-size: 16px; line-height: 23px;">Green-Tao定理的證實,出乎數學界意料。這天筆者與陶氏午饍,談及近來有關雙生質數猜想的突破。陶氏認為,這個百年猜想,可能在十多年內得到解決。</span><span style="background-color: white; font-family: georgia, 'times new roman', times, serif; font-size: 16px; line-height: 23px;"><span style="color: red;">(其中一個突破,來自可以做勵志片主角,年青時籍籍無名的<a href="https://en.wikipedia.org/wiki/Yitang_Zhang">張益唐</a>。)</span></span><br />
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<span style="background-color: white; color: #333333; font-family: georgia, 'times new roman', times, serif; font-size: 16px; line-height: 23px;">晚飯時筆者去了陶氏在校園西邊的五睡房屋子。陶氏本來要帶12歲的兒子去學琴,但結果兒子去了拍廣告片。他的兒子早已在汽車廣告中出現。當陶氏的妻子帶兒子回來時,四歲的女兒剛吃完晚飯,正在食甜品。女兒見到媽媽,從椅子走下來,高興地跑來跑去。</span><br />
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<span style="background-color: white; color: #333333; font-family: georgia, 'times new roman', times, serif; font-size: 16px; line-height: 23px;">陶氏是數學界的建橋者,把不同領域的數學家連繫起來。他現在亦是一位勤力的博客,在網上討論不同的數學,亦不嗇談及他的研突進展。他又發起網上數學合作運動,希望集眾人之力,解決數學問題。陶氏,被譽為廿一世紀數學家的典範。</span><span style="background-color: white; font-family: georgia, 'times new roman', times, serif; font-size: 16px; line-height: 23px;"><span style="color: red;">(可惜,新人不好學陶氏,否則閣下的貢獻將被埋在眾多的合作者之間,不利求職!)</span></span><br />
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訪問陶氏其間,筆者注意到他唯一的數學家特徵:專注思考,以致善忘。他時常不見書,着錯衫,忘了穿襪子—所以他着涼鞋算了。近來,他在坐飛機時最能做數,因為其他時間很多人找他給意見。<br />
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陶氏的兒子唸六年級,開始從網上學習高等數學,但他最愛最作,尤其是神怪作品。他亦喜愛<span style="background-color: white; color: #333333; font-family: georgia, 'times new roman', times, serif; font-size: 16px; line-height: 23px;">Minecraft</span>,但近來發現頗難過關。在晚飯時,提及與朋友嘗試證明1=0,但發現原來0不能作分母。陶氏似乎反一反白眼。<br />
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證明1=0不會有甚麼成果。但敢於嘗試一些匪夷所思的想法,對數學很有用處。很久以前,數學家發明了-1的開方,並把這個子虛烏有的數稱為<a href="https://en.wikipedia.org/wiki/Imaginary_number">虛數</a>。虛數是現代工程及物理的重要概念。<br />
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中小學的數學有誤導性。學生以為知道應用前人的方法很到答案,就是得到答案。但數學,是需要去實驗,去撞版。數學是一本創意藝術。據說,上世紀初最具影响力的數學家<span style="background-color: white; color: #333333; font-family: georgia, 'times new roman', times, serif; font-size: 16px; line-height: 23px;"><a href="https://en.wikipedia.org/wiki/David_Hilbert">David Hilbert</a>,知道有同事改行寫小說,回應:</span><span style="background-color: white; color: #333333; font-family: georgia, 'times new roman', times, serif; line-height: 23px;">「確實,他不夠想象力做數學。」</span><br />
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<span style="background-color: white; color: #333333; font-family: georgia, 'times new roman', times, serif; line-height: 23px;">數學是抽象。兩個蘋果及兩個橙,抽出共同之象就是2。但陶氏似乎抽得過了籠。據說,有次他在考慮一個新的變動坐標時,躺在地上轉來轉去觀看。陶氏笑說</span><span style="background-color: white; color: #333333; font-family: georgia, 'times new roman', times, serif; font-size: 16px; line-height: 23px;">:</span><span style="background-color: white; color: #333333; font-family: georgia, 'times new roman', times, serif; line-height: 23px;">「我的姨姨剛好看見,但我解釋不了我做甚麼。」</span><br />
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<span style="background-color: white; color: #333333; font-family: georgia, 'times new roman', times, serif; line-height: 23px;">陶氏最近關於水爆炸的工作,源於一位哈薩克的教授宣稱解決了</span><span style="background-color: white; color: #333333; font-family: georgia, 'times new roman', times, serif; font-size: 16px; line-height: 23px;">Navier-Stokes問題。陶氏看過後,肯定證明是錯了。他想進一步證明該教授的研究方向不可能解決談問題,所以深入去看,結果激發了他的想象力,一部會繁殖的水機器。數學未解決的猜想,同時是心理及數學的挑戰。陶氏一年前還認為</span><span style="background-color: white; color: #333333; font-family: georgia, 'times new roman', times, serif; font-size: 16px; line-height: 23px;">Navier-Stokes未能解,但今日卻覺得看到一線希望。當然,這個看似魔鬼的弱點,可能是另一個陷阱。</span><span style="background-color: white; color: #333333; font-family: georgia, 'times new roman', times, serif; line-height: 23px;">陶氏說</span><span style="background-color: white; color: #333333; font-family: georgia, 'times new roman', times, serif; font-size: 16px; line-height: 23px;">:</span><span style="background-color: white; color: #333333; font-family: georgia, 'times new roman', times, serif; line-height: 23px;">「做研究要學會懷疑,要小心翼翼。」</span><br />
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<span style="color: #333333; font-family: georgia, times new roman, times, serif;"><span style="background-color: white; line-height: 23px;">這就是數學的吸引力,亦是數學的可怕。局勢在暗地裡變動。遊戲在繼續.</span></span>韋言仁語http://www.blogger.com/profile/08599435936360249361noreply@blogger.com0tag:blogger.com,1999:blog-21683040.post-88966394479156413892015-07-14T06:04:00.001-07:002015-07-14T06:12:23.095-07:00不再捉蟲, 改做蝠居, 食屎要大氮!標題解釋: 放棄捉蟲習性, 葉桶變大吸引蝙蝠入住, 靠其糞便供給氮元素<br />
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出租葉桶都要賣包裝, 反射聲納的設計最能吸引想找地方落腳的蝙蝠.<br />
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<a href="http://phenomena.nationalgeographic.com/2015/07/09/with-sonar-reflecting-leaves-plant-lures-bats-to-poo-in-it/#.VaUIuew1YxA.blogger">With Sonar-Reflecting Leaves, Plant Lures Bats to Poo in it</a>韋言仁語http://www.blogger.com/profile/08599435936360249361noreply@blogger.com0tag:blogger.com,1999:blog-21683040.post-63373122133509765662014-12-04T06:25:00.001-08:002014-12-04T19:05:36.535-08:00量子攝影, 無形有相<a href="http://discovermagazine.com/2015/jan-feb/10-quantum-photo-finish#.VIBmquXCYok.blogger">Quantum Photo Finish | DiscoverMagazine.com</a><br />
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科學家製造了兩組糾纏的光束, 一弱一強.<br />
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糾纏, 量子力學的慨念, 指不受距離阻隔的連繫.<br />
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兩組光線方向不同: 弱光束穿過了貓形的孔, 強光速直接打在攝影器上.<br />
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結果, 攝影器捕捉了一個貓的影象!<br />
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原來, 貓形孔改變穿越它的弱光束, 亦同時改變未有穿越它的強光束.<br />
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這就是糾纏的力量.<br />
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在光源與攝影器之間, 沒有貓形, 卻攝得貓相.<br />
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無形有相.<br />
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將來, 對生物學或其它科學攝影大有益處.<br />
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因為用弱光束射向目標物, 就不會造成太大影響; 同時糾纏的強光束就可以製造目標的相片.<br />
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<img alt="entanglement" src="http://discovermagazine.com/~/media/Images/Issues/2015/jan-feb/entanglement.jpg?mw=900" height="243" width="320" /><br />
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<br />韋言仁語http://www.blogger.com/profile/08599435936360249361noreply@blogger.com0tag:blogger.com,1999:blog-21683040.post-55855973870772058902014-07-06T08:27:00.002-07:002014-07-06T08:28:29.464-07:00 一個物種?今天看了National Geographic的<a href="http://phenomena.nationalgeographic.com/blog/not-exactly-rocket-science/">NOT EXACTLY ROCKET SCIENCE</a>. 其中有兩篇文章可以一拼來看: <a href="https://www.blogger.com/one-lichen-species-is-actually-126-and-probably-more/">One lichen species is actually 126 and probably more</a> 及 <a href="http://phenomena.nationalgeographic.com/2014/07/02/sex-with-extinct-humans-passed-high-altitude-gene-to-tibetans/">Extinct humans passed high altitude gene to tibetans</a>.<br />
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第一篇談及, "一個品種"的青苔經過基因分析後, 發現原來是百多個品種. 首先, 何謂不同品種? 有個定義說, 若兩個體不能交配生育, 即成兩個品種--這定義當然不適合單性繁殖的生物. 但即使是兩性生物, 何謂不能交配生育? 可能在某個環境不會, 在另一個環境卻會!長頸鹿就是好例子, 在嘗試分類不同的長頸鹿時, 就出現這個情況. 有個定義說, 若兩個體的基因有足夠差異, 即成兩個品種--但多少差異才算足夠?
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第二篇談及, 西藏人的祖先在多年前曾與一種現今已絶種的古人類交合, 遺傳了該古人類的一個突變基因, 所以能夠適應高原生活. 該古人類名為<a href="https://en.wikipedia.org/wiki/Denisovan">Denisovan</a>, 只有小量骸骨被發現, 但已足夠作基因分析--這個古人類的研究發展已是很好的故事. Denisovan並不是唯一與現代人祖先交配的古人種.<br />
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現在生物學書中的進化樹, 分支再分支, 分支之間不會相交. 但現實不是這樣. 只要相距不太大, 仍然有可能交合. 在一個品種及兩個品種之間, 可能有很多不同的狀態. 是時候考慮兩個體的物種距離了: 不是單純的基因距離, 為要考慮每個基因的用途作出調整. (有人在做這類研究, 不過我唔熟行, 唔識!)
韋言仁語http://www.blogger.com/profile/08599435936360249361noreply@blogger.com0tag:blogger.com,1999:blog-21683040.post-86939852077233693922013-12-26T23:45:00.001-08:002013-12-26T23:45:59.305-08:00激走記憶National Geographic: <a href="http://phenomena.nationalgeographic.com/2013/12/22/shocking-memories-away/#.Ur0keEZCMDY.blogger">Shocking Memories Away</a><br />
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在(以科學發展來說)古代(1968年,近50年前),心理學家<a href="http://news.usc.edu/#!/article/30224/In-Memoriam-Donald-J-Lewis-88" style="font-family: minion-pro, Georgia, serif; line-height: 25.989582061767578px;" target="_blank">Donald Lewis</a>發現,剛學會害怕某聲響的老鼠被電擊頭被後,會忘卻了該恐懼。這個發現推翻了「新鮮記憶才會被忘卻」的假設,原來只要記憶重新被呼喚,就有可能將之刪除。<br />
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這個大發現並沒有產生震撼的回響,原因是:其他科學家的實驗,並不都能重覆Lewis的結果。記憶實驗真的是很難很難。<br />
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在2000年,神經學家<span style="color: black; font-family: minion-pro, Georgia, serif; line-height: 26px;"><a href="http://www.psych.mcgill.ca/faculty/nader.html" style="font-family: minion-pro, Georgia, serif; line-height: 26px;" target="_blank">Karim Nader</a>把Lewis的學說重新提出來。他發現,老鼠呼喚舊記憶是需要一種特殊的蛋白。只要在重構記憶的過程用藥物防止蛋白製造,就可以把記憶抹掉。</span><br />
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</span> <span style="color: black; font-family: minion-pro, Georgia, serif; line-height: 26px;">今次的研究,終於引人注目。皆因有可能用藥物對心靈創傷者,忘卻不快的記憶。</span><br />
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</span> <span style="color: black; font-family: minion-pro, Georgia, serif; line-height: 26px;">在2013年,另一位神經學家</span><a href="http://www.ru.nl/people/donders/kroes-m/">Marijn Kroes</a>發表了如下研究:<br />
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一群志願者觀看了兩套幻燈片,簡稱A及B。兩星期後,研究員讓志願者看A的其中一張,以激活記憶;然後,其中部分志願者被電擊。最後,研究員問關於A及B的問題。相關A的題目,受過電擊的一批答得較差;相關B的題目,兩批相若。<br />
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Lewis的學說,得到期待已久的驗證。<br />
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不過,我們真的能選擇性的刪除記憶嗎?道德上可以嗎?韋言仁語http://www.blogger.com/profile/08599435936360249361noreply@blogger.com0tag:blogger.com,1999:blog-21683040.post-37099781345890310892013-10-10T06:21:00.001-07:002013-10-10T06:26:54.619-07:00天才與瘋子一線之隔?<h1 class="postTitle" id="postTitle2" style="background-color: white; border: 0px; font-family: Brunel-for-Titles, georgia, times, serif; font-size: 18px; line-height: 26px; margin: 0px 0px 5px; outline: 0px; padding: 0px; vertical-align: baseline;">
<span style="color: red;"><a href="http://blogs.scientificamerican.com/beautiful-minds/2013/10/03/the-real-link-between-creativity-and-mental-illness/"><span style="font-size: 18px; line-height: 26px;">Scientific </span>American : <span style="font-size: medium;">The Real Link Between Creativity and Mental Illness</span></a></span></h1>
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<span style="font-family: Times, Times New Roman, serif;">俗語有云: 天才與瘋子一線之隔.</span><br />
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<span style="font-family: Times, Times New Roman, serif;">研究人員着眼於此說的真偽. 全文可點擊以上連結, 在此我嘗試攝要我的理解.</span><br />
<span style="font-family: Times, Times New Roman, serif;">研究人員發現: 精神分裂傾向與創意澎拜有一個共同點, 就是腦部不同思維工作分工不清--例如, </span><br />
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<span style="font-family: Times, Times New Roman, serif;">在進行某些思考活動時, 不太相干的記憶功能照樣啟動. 分別在於, 前者是負面可導致思相混亂; </span><span style="font-family: Times, 'Times New Roman', serif;">後者是正面可引發跳出框框的念頭, 即潮語fusion.</span><br />
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<span style="font-family: Times, Times New Roman, serif;">研究人員進一步指出: 這些瘋子不一定有創意. 但他們的心理正常近親, 卻因為擁有這樣的"腦部缺撼", 反而最有可能是創意天才.</span><br />
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<span style="font-family: Times, Times New Roman, serif;">原來, 天才與瘋子, 可能是一代之隔.</span>韋言仁語http://www.blogger.com/profile/08599435936360249361noreply@blogger.com0tag:blogger.com,1999:blog-21683040.post-27591778800518737372013-08-23T19:07:00.001-07:002013-08-23T19:07:54.798-07:00孔雀蛛之舞<a href="http://blogs.discovermagazine.com/d-brief/2013/08/23/video-peacock-spider-does-a-crazy-booty-shaking-dance/#.UhgUidm86Mc.blogger">Video: Peacock Spider Does a Crazy Booty-Shaking Dance - D-brief | DiscoverMagazine.com</a><br />
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韋言仁語http://www.blogger.com/profile/08599435936360249361noreply@blogger.com0tag:blogger.com,1999:blog-21683040.post-41026133099389526532013-08-21T22:53:00.001-07:002013-08-23T20:42:39.100-07:00海中的圓形 <a href="http://newswatch.nationalgeographic.com/2013/08/15/whats-this-mysterious-circle-on-the-seafloor/#.UhWj4QmYgGM.blogger">What’s This Mysterious Circle on the Seafloor?</a> (National Geographic)<br />
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20年前, 日本海底發現神秘圓形, 闊達兩米...</div>
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<a href="http://newswatch.nationalgeographic.com/files/2013/08/photo2.jpg" style="background-color: transparent; background-position: initial initial; background-repeat: initial initial; border: 0px; color: #044e8e; font-size: 14px; margin: 0px; outline: 0px; padding: 0px; text-decoration: none; vertical-align: baseline;"><img alt="seafloor circle picture" class="size-medium wp-image-104588" height="450" src="http://newswatch.nationalgeographic.com/files/2013/08/photo2-600x450.jpg" style="background-color: transparent; background-position: initial initial; background-repeat: initial initial; border: 0px; display: block; font-size: 14px; margin: 5px 5px 0px; outline: 0px; padding: 0px; vertical-align: baseline;" width="600" /></a><br />
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A male pufferfish (center) made this nest to lure females in Japan in 2012.<br />
Photograph courtesy Kimiaki Ito</div>
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現在, 科學家證實是一個新品種的<a href="http://en.wikipedia.org/wiki/Tetraodontidae">魨</a>, 所築的愛巢. 雄性(約13厘米長)利用身體在沙上造成美麗的圖案, 然後雌性會來撿視是否合乎心意. 合乎心意的, 雌性會在巢中心產卵.</div>
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<a href="http://newswatch.nationalgeographic.com/files/2013/08/photo1.jpg" style="background-color: transparent; background-position: initial initial; background-repeat: initial initial; border: 0px; color: #044e8e; font-size: 14px; margin: 0px; outline: 0px; padding: 0px; text-decoration: none; vertical-align: baseline;"><img alt="pufferfish picture" class="size-medium wp-image-104589" height="400" src="http://newswatch.nationalgeographic.com/files/2013/08/photo1-600x400.jpg" style="background-color: transparent; background-position: initial initial; background-repeat: initial initial; border: 0px; display: block; font-size: 14px; margin: 5px 5px 0px; outline: 0px; padding: 0px; vertical-align: baseline;" width="600" /></a><br />
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A male pufferfish uses his body to create peaks and valleys in the sand. Photograph courtesy Kimiaki Ito</div>
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雄性會在以後六天守謢愛巢, 直至魚卵孵化為止.</div>
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之後, 雄性會另覓地點, 重新築巢, 吸引另一位異性.</div>
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<span style="color: #333333; font-family: Arial, Helvetica, sans-serif;"><span style="font-size: 14.166666030883789px; line-height: 21.9921875px;"><u><b>Update</b>: 有片段</u></span></span><br />
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韋言仁語http://www.blogger.com/profile/08599435936360249361noreply@blogger.com0tag:blogger.com,1999:blog-21683040.post-22651289970404596352013-08-21T19:59:00.001-07:002013-08-21T19:59:45.743-07:00看看你的腦電波, 費用全免!<a href="http://blogs.discovermagazine.com/neuroskeptic/2013/08/21/can-you-see-your-own-brain-waves/#.UhV91l5IrDh.blogger">Can You See Your Own Brain Waves? - Neuroskeptic | DiscoverMagazine.com</a><br />
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有科學家提出, 你在下圖見度的"不存在動運", 其實就是你的腦電波.<br />
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咁, 就要看多幾次:<br />
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<img src="http://blogs.discovermagazine.com/neuroskeptic/files/2013/08/flicker.png" />韋言仁語http://www.blogger.com/profile/08599435936360249361noreply@blogger.com0tag:blogger.com,1999:blog-21683040.post-78415319769832893122013-07-30T05:29:00.000-07:002013-07-30T05:29:09.231-07:00大數據的聯想....第一步: 收集大量數據;<br />
第二步: 整合資訊, 篩走多餘.<br />
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話說, 人腦的發展正是這樣. 孩童時代, 腦神經元過量生產; 然後用一生時間去蕪存菁, 成為智慧.<br />
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經濟泡沬也是這樣. 新產業誔生, 各種瘋狂意念出現, 其實是收集市場資訊; 到整合時期, 泡沬爆破, 留下的就是王者.<br />
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生物演化也是這樣. 環境劇變, 希奇古怪的異種得以繁衍, 乜乜記大爆發正是如此; 到環境穩定下來, 留下的只有少數.<br />
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科學研究也是這樣. 各式各樣的實驗就是要收集數據; 因為數據太多, 會有預設機制篩選, 然後就要看看有沒有理論與之配合.<br />
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市場研究也是這樣. 可能是單純市場調查, 可能是不停開設新分店, 可能是不斷推出各種服務; 到整合時期, 只有最合適的經營項目可以留低. 例如<a href="https://zh.wikipedia.org/wiki/%E6%96%BD%E6%B0%B8%E9%9D%92">施永青</a>的所謂無為而治, 又如<a href="http://www.google.com/">Google</a>最近<a href="http://www.androidauthority.com/google-services-shut-down-238066/">整合旗下服務</a>.<br />
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當然, 最新的<a href="http://www.nokia.com/">Nokia</a>手機<a href="http://www.nokia.com/global/products/phone/lumia1020/">Lumix1020</a>照相. 先拍攝41MP影像; 處理過後只剩下5MP相片.<br />
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<a href="http://www.facebook.com/">Facebook</a>的<a href="http://en.wikipedia.org/wiki/Social_graph">Social Graph</a>, 開始利用手頭的大量資料.<br />
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<a href="http://www.euronews.com/2013/07/29/us-and-french-ad-agencies-merge-to-form-world-s-largest-agency/">兩間全球最大的廣告公司合併</a>, 其中一個原因是要聯合龐大的數據庫, 抗衡Google及Facebook的威脅.<br />
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<a href="http://www.wired.com/">Wired Magazine</a> 今期有篇<a href="http://www.wired.com/opinion/2013/07/facebooks-at-its-breakpoint-and-heres-how-to-save-it-from-growth-hacking/">文章</a>, 認為互聯網的連結已到臨界點, 需要整合才更能發揮作用.<br />
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亦即是說, 大數據時代(<a href="http://en.wikipedia.org/wiki/Big_data">BIG DATA</a>), 已經降臨.<br />
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<br />韋言仁語http://www.blogger.com/profile/08599435936360249361noreply@blogger.com0tag:blogger.com,1999:blog-21683040.post-64276214457618483682013-07-24T03:29:00.002-07:002013-07-24T03:29:55.667-07:00海豚朋友, 你的名字是?<a href="http://blogs.discovermagazine.com/d-brief/2013/07/22/dolphins-call-each-other-by-names/#.Ue-pNxIycoE.blogger">Dolphins Respond to Hearing Their Own Name - D-brief | DiscoverMagazine.com</a><br />
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科學家發現, 每隻海豚都有一組特殊的聲調組合. 當海豚聽到自己的聲調組合, 就會作出反應; 若聽到的熟豚的聲調組合, 就偶然有反應; 若聽到陌生豚的聲調合, 就全無反應.<br />
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因此推測, 這組特殊的聲調組合, 便是海豚的"自我身份", 即"名字".韋言仁語http://www.blogger.com/profile/08599435936360249361noreply@blogger.com0tag:blogger.com,1999:blog-21683040.post-26162035089617461462013-06-21T16:51:00.002-07:002013-06-21T16:51:44.093-07:00羊食草, 草食羊....<a href="http://www.isciencetimes.com/articles/5472/20130621/sheep-eating-plant-puya-chilensis-animals-england.htm">Sheep-Eating Plant: How Does Puya Chilensis Trap And Eat Animals?</a>
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有草名普亞, 罕開花, 一開七日.<br />
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身有長刺, 羊觸刺, 困而餓死, 化為草之糧食. <br />
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南美印第安人見普亞, 放火燒殺, 使其不能傷其羊. <br />
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<img alt="sheep eating plant" height="320" src="http://cdn.isciencetimes.com/data/images/full/2013/06/21/4385-sheep-eating-plant.jpg" width="216" />韋言仁語http://www.blogger.com/profile/08599435936360249361noreply@blogger.com0