「我愛數學！」如果你們想在接下來的幾個小時裡在宴會最不顯眼的角落裡獨啜飲料，那正是該在聚會上要說的話。
And that's because when it comes to this subject - all the numbers, formulas, symbols, and calculations - the vast majority of us are outsiders, and that includes me.
那正是因為一旦涉及到這個主題——所有數字、公式、符號和計算的主題——我們當中的絕大多數人是局外人，包括我在內。
That's why today I want to share with you an outsider's perspective of mathematics - what I understand of it, from someone who's always struggled with the subject. And what I've discovered, as someone who went from being an outsider to making math my career, is that, surprisingly, we are all deep down born to be mathematicians.
這就是為什麼今天我想與大家分享我這個一直掙扎著努力學習數學的局外人，我的觀點和我對數學的理解。身為從局外人轉向以教數學為職志的人，我驚訝地發現我們都天生就是數學家。
But back to me being an outsider. I know what you're thinking: "Wait a second, Eddie. What would you know? You're a math teacher. You went to a selective school. You wear glasses, and you're Asian." Firstly, that's racist. Secondly, that's wrong.
但是先回到我身為局外人。我知道你們在想什麼：「艾迪，且慢，你哪會知道，你是數學老師，你上過好學校。你戴眼鏡，你還是亞洲人。」首先，那是種族歧視。其次，那是錯的。
When I was in school, my favorite subjects were English and history. And this caused a lot of angst for me as a teenager because my high school truly honored mathematics.
上學的時候，我最喜歡的科目是英語和歷史。十幾歲時這就使我非常焦慮，因為我的高中真的很重視數學。
Your status in the school pretty much correlated with which mathematics class you ranked in. There were eight classes. So if you were in math 4, that made you just about average. If you were in math 1, you were like royalty.
學生在學校的地位與所修的數學課密切相關。數學課程有八級。因此，如果修的是數學四，那就是中段的水平。如果修數學一，就像是皇室等級。
Each year, our school entered the prestigious Australian Mathematics Competition and would print out a list of everyone in the school in order of our scores. Students who received prizes and high distinctions were pinned up at the start of a long corridor, far, far away from the dark and shameful place where my name appeared.
我們學校每年都參加著名的澳大利亞數學競賽，還按照分數的高低列出每一個人的排行榜。在長長的走廊的盡頭釘著的是獲得獎項和傑出成就學生的名字，遠在走廊的另一頭是我名字出現的黑暗和羞慚之地。
Math was not really my thing. Stories, characters, narratives - this is where I was at home. And that's why I raised my sails and set course to become an English and history teacher. But a chance encounter at Sydney University altered my life forever.
數學不是我的強項。故事、人物、敘述——我才在行。這就是為什麼我揚帆立志要成為英語和歷史老師的原因。但在雪梨大學的某個偶遇永遠改變了我的生命。
I was in line to enroll at the faculty of education when I started the conversation with one of its professors. He noticed that while my academic life had been dominated by humanities, I had actually attempted some high-level math at school. What he saw was not that I had a problem with math, but that I had persevered with math.
當時我正準備進入教育學院，與一位該學院的教授面談。他注意到，雖然我高中時主要選修的是人文學科，實際上卻在校裡修過一些高等數學。他不是看到我數學學得不怎麼樣，而是看到我堅持到底學了數學。
And he knew something I didn't - that there was a critical shortage of mathematics educators in Australian schools, a shortage that remains to this day. So he encouraged me to change my teaching area to mathematics.
他還知道我所不知——澳大利亞嚴重短缺數學教師，直到今天仍然持續短缺。因此他鼓勵我將教學領域改為數學。
Now, for me, becoming a teacher wasn't about my love for a particular subject. It was about having a personal impact on the lives of young people.
對我來說，當老師的原因並不是我對特定學科的熱愛，而是想影響年輕人的生命。
I'd seen firsthand at school what a lasting and positive difference a great teacher can make. I wanted to do that for someone, and it didn't matter to me what subject I did it in. If there was an acute need in mathematics, then it made sense for me to go there.
我在學生時代親身體驗過出色的老師能夠帶來多麼持久而積極的改變。我也想那樣。對我來說，這與教哪個學科無關。如果迫切需要數學教師，那我去當數學老師就合情合理。
As I studied my degree, though, I discovered that mathematics was a very different subject to what I'd originally thought. I'd made the same mistake about mathematics that I'd made earlier in my life about music.
然而在修學位時，我發現數學與我最初想的完全不同。我在數學上犯的錯誤與我早年在音樂上犯的錯誤相同。
Like a good migrant child, I dutifully learned to play the piano when I was young. My weekends were filled with endlessly repeating scales and memorizing every note in the piece, spring and winter.
像每個聽話的移民小孩一樣，我從小就乖乖地學鋼琴。我的周末充滿了無止境的重複音階練習，強記春季和冬季樂章中的每個音符。
I lasted two years before my career was abruptly ended when my teacher told my parents, "His fingers are too short. I will not teach him anymore." At seven years old, I thought of music like torture. It was a dry, solitary, joyless exercise that I only engaged with because someone else forced me to. It took me 11 years to emerge from that sad place.
我兩年的音樂生涯在老師對我的父母說了這話後嘎然而止：「他的手指太短。我不再教他了。」七歲的我一想到音樂就像受酷刑一樣。那是一種枯燥、孤獨、無聊的鍛鍊，我只因被逼而學。我花 11 年的時間才從那悲慘之地走了出來。
In year 12, I picked up a steel string acoustic guitar for the first time. I wanted to play it for church, and there was also a girl I was fairly keen on impressing. So I convinced my brother to teach me a few chords. And slowly, but surely, my mind changed. I was engaged in a creative process.
在第 12 年，我第一次拿起鋼弦木吉他。我想為教堂演奏，還很想給某個女孩留下深刻的印象。所以我說服哥哥教我一些和弦。可以肯定的是我的想法慢慢變了。我進入創作的過程。
I was making music, and I was hooked. I started playing in a band, and I felt the delight of rhythm pulsing through my body as we brought our sounds together. I'd been surrounded by a musical ocean my entire life, and for the first time, I realized I could swim in it. I went through an almost identical experience when it came to mathematics.
我迷上了音樂，開始在樂隊裡演奏。每當我們將樂聲匯聚在一起時，我感到節奏在我體內跳動。我一生都被音樂之洋包圍著，那卻是我首次意識到自己能夠悠遊其中。我在數學方面也經歷了幾乎相同的經歷。
I used to believe that math was about rote learning inscrutable formulas to solve abstract problems that didn't mean anything to me. But at university, I began to see that mathematics is immensely practical and even beautiful,
我曾經認為數學是死記硬背深奧難懂的公式來解決抽象的問題，對我來說沒有任何意義。但在大學裡，我初見數學非常實用，甚至很美，
that it's not just about finding answers but also about learning to ask the right questions, and that mathematics isn't about mindlessly crunching numbers but rather about forming new ways to see problems so we can solve them by combining insight with imagination.
數學不僅在於尋找答案，還在於學會正確提問。數學不是無意識地處理數字，而是以新的方式看問題，結合洞察力和想像力來解決問題。
It gradually dawned on me that mathematics is a sense. Mathematics is a sense just like sight and touch; it's a sense that allows us to perceive realities which would be otherwise intangible to us. You know, we talk about a sense of humor and a sense of rhythm. Mathematics is our sense for patterns, relationships, and logical connections. It's a whole new way to see the world.
我漸漸意識到數學是一種知覺。數學就像視覺和觸覺一樣，是種使我們能夠感知現實的知覺，要不然我們就看不見這些現實了。我們談幽默感和節奏感。數學是我們對模式、關係和邏輯連結的感知。這是一種看待世界的全新方式。
Now, I want to show you a mathematical reality that I guarantee you've seen before but perhaps never really perceived. It's been hidden in plain sight your entire life.
現在，我想向你們展示一個數學上的現實，我保證你們以前見過，但也許未曾真正察覺，一直以來視而不見。
This is a river delta. It's a beautiful piece of geometry. Now, when we hear the word geometry, most of us think of triangles and circles. But geometry is the mathematics of all shapes, and this meeting of land and sea has created shapes with an undeniable pattern.
這是河流的三角洲。它是塊美麗的幾何圖形。在聽到「幾何」一詞時，我們大多數人會想到三角形和圓形。但是幾何是所有形狀的數學，這陸地和海洋相會之處創造出具有不可否認模式的形狀。
It has a mathematically recursive structure. Every part of the river delta, with its twists and turns, is a micro version of the greater whole. So I want you to see the mathematics in this.
它具有數學遞歸結構。三角洲的每個部分有其曲折，是整體的微觀版本。我希望你們能看到其中的數學。
But that's not all. I want you to compare this river delta with this amazing tree. It's a wonder in itself. But focus with me on the similarities between this and the river. What I want to know is why on earth should these shapes look so remarkably alike? Why should they have anything in common?
但這還不是全部。我希望你們將此河的三角洲與這棵令人驚奇的樹相比較。它本身就是一個奇蹟。但是，請把重點放在樹與河的相似之處。我想知道為什麼這些形狀看起來如此相似？它們為什麼有共同點？
Things get even more perplexing when you realize it's not just water systems and plants that do this. If you keep your eyes open, you'll see these same shapes are everywhere. Lightning bolts disappear so quickly that we seldom have the opportunity to ponder their geometry. But their shape is so unmistakable and so similar to what we've just seen that one can't help but be suspicious.
意識到不僅水流系統和植物這樣更加令人困惑。睜開眼睛，到處都看得到這些相同的形狀。閃電消失得如此之快，以至於我們很少有機會思考它們的幾何形狀。但是它們的形狀如此明顯，與我們剛剛看到的如此相似，讓人不禁疑惑。
And then there's the fact that every single person in this room is filled with these shapes too. Every cubic centimeter of your body is packed with blood vessels that trace out this same pattern. There's a mathematical reality woven into the fabric of the universe that you share with winding rivers, towering trees, and raging storms.
還有另一事實就是這裡的每個人也都充滿了這些形狀。身體的每立方釐米都充滿了血管，這些血管可以勾畫出相同的模式。在與蜿蜒的河流、參天大樹和洶湧的暴風雨共享的宇宙中，編織著一種數學現實。
These shapes are examples of what we call "fractals," as mathematicians. Fractals get their name from the same place as fractions and fractures - it's a reference to the broken and shattered shapes we find around us in nature.
這些形狀是數學家稱為「碎形」的例子。碎形與分數和斷裂的得名相同——是我們給予自然界中所發現的破碎形狀的名字。
Now, once you have a sense for fractals, you really do start to see them everywhere: a head of broccoli, the leaves of a fern, even clouds in the sky. Like the other senses, our mathematical sense can be refined with practice. It's just like developing perfect pitch or a taste for wines. You can learn to perceive the mathematics around you with time and the right guidance.
一旦對碎形有感，就處處看得到它們：青花菜、蕨類植物的葉子，甚至天上的雲彩。與其他知覺一樣，我們的數學感可以經由練習來改善。就像發展完美的音感或品味葡萄酒一樣。可以通過時間和正確的指導學習感知周圍的數學。
Naturally, some people are born with sharper senses than the rest of us, others are born with impairment. As you can see, I drew a short straw in the genetic lottery when it came to my eyesight. Without my glasses, everything is a blur.
當然，有些人天生比我們其他人敏銳，其他人天生就有些障礙。如大家所見，我的視力屬於基因樂透中的劣勢。不戴眼鏡就一片模糊。
I've wrestled with this sense my entire life, but I would never dream of saying, "Well, seeing has always been a struggle for me. I guess I'm just not a seeing kind of person."
我一生都為視力掙扎，然而我永遠不會說：「好吧，既然一直以來我的視力差，我大概不是個【善視者】吧。」
Yet I meet people every day who feel it quite natural to say exactly that about mathematics. Now, I'm convinced we close ourselves off from a huge part of the human experience if we do this. Because all human beings are wired to see patterns.
然而我每天都會遇到一些人坦然宣稱自己沒有數學細胞。我深信這樣子會讓我們錯失許多人類的經驗，因為全人類與生俱來會看到模式。
We live in a patterned universe, a cosmos. That's what cosmos means - orderly and patterned - as opposed to chaos, which means disorderly and random. It isn't just seeing patterns that humans are so good at. We love making patterns too. And the people who do this well have a special name.
我們生活在有序的宇宙體系裡。宇宙的含義就是——有秩序、有模式——與混亂恰恰相反，混亂意味著雜亂和隨機。人類不僅善於看到模式，也喜歡製作模式。做得很好的人有特殊的名字。
We call them artists, musicians, sculptors, painters, cinematographers - they're all pattern creators. Music was once described as the joy that people feel when they are counting but don't know it.
我們稱他們為藝術家、音樂家、雕塑家、畫家、攝影師——他們都是模式創作者。音樂曾經被描述為人們在不自覺地計數時所感受到的快樂。
Some of the most striking examples of mathematical patterns are in Islamic art and design. An aversion to depicting humans and animals led to a rich history of intricate tile arrangements and geometric forms.
數學模式最引人注目的一些例子在伊斯蘭藝術和設計裡見得到，其對描繪人類和動物的禁忌引領出錯綜複雜的瓷磚排列和幾何形狀的豐富歷史。
The aesthetic side of mathematical patterns like these brings us back to nature itself. For instance, flowers are a universal symbol of beauty. Every culture around the planet and throughout history has regarded them as objects of wonder. And one aspect of their beauty is that they exhibit a special kind of symmetry.
這些數學模式的美學將我們帶回了自然界。例如，花朵是普遍的美麗象徵。地球上、整個歷史上的每種文化都將花朵視為奇觀。花朵美麗的特點之一是它們表現出特殊的對稱性。
Flowers grow organically from a center that expands outwards in the shape of a spiral, and this creates what we call "rotational symmetry." You can spin a flower around and around, and it still looks basically the same. But not all spirals are created equal. It all depends on the angle of rotation that goes into creating the spiral.
花以螺旋狀由內向外有機地擴展生長，這產生了我們所謂的「旋轉對稱」。任意旋轉一朵花，它看起來基本上還是一樣。但並非所有螺旋都相同。這完全取決於產生螺旋的旋轉角度。
For instance, if we build a spiral from an angle of 90 degrees, we get a cross that is neither beautiful nor efficient. Huge parts of the flowers area are wasted and don't produce seeds. Using an angle of 62 degrees is better and produces a nice circular shape, like what we usually associate with flowers. But it's still not great. There's still large parts of the area that are a poor use of resources for the flower.
例如，如果以 90 度角構建螺旋，就會得到既不美觀也無效率的十字。大部分花的面積被浪費掉，不產生種子。使用 62 度角好一些，會產生一個很好的圓形，就像通常與花朵關聯的形狀那樣。但這仍然不夠好，仍有很大一部分未充分利用花卉的面積資源。
However, if we use 137.5 degrees, we get this beautiful pattern. It's astonishing, and it is exactly the kind of pattern used by that most majestic of flowers - the sunflower. Now, 137.5 degrees might seem pretty random, but it actually emerges out of a special number that we call the "golden ratio."
然而，如果用 137.5 度，就得到這個美麗的圖案。令人驚訝，這正是最壯觀的花朵——向日葵所用的那種圖案。137.5 度看似隨機，但它實際上出現在一個特殊的數字中，我們稱之為「黃金比例」。
The golden ratio is a mathematical reality that, like fractals, you can find everywhere - from the phalanges of your fingers to the pillars of the Parthenon. That's why even at a party of 5000 people, I'm proud to declare, "I love mathematics!"
黃金比例是一個數學上的現實，就像碎形一樣，處處找得到它——從手指的指骨到帕臺農神廟的柱子。這就是為什麼即使身處五千人的聚會之中，我仍然自豪地宣稱：「我愛數學！」