Page 25 - Science Focus (Issue 19)

Q&A 1 Do you have a favorite piece of scientific work, or perhaps a

particular scientist whose work you love?
આఃᛇࡳɓ΅߅ኪഹЪdɦא݊ϞӚϞɓЗઆఃᛇՉ޼Ӻٙ߅ኪ࢕k

Prof. Ivan Ip : My favorite piece of work is Euclid’s Elements.
葉智皓教授：我最喜歡的作品是歐幾里得的《幾何原
Euclid was a mathematician who lived around 300 BC, and
本》。歐幾里得是公元300年左右的一位數學家，現時在中
many of the geometry results you learn in high school are based
學數學課學到的不少幾何定理也是基於這本書裡的結果。
upon results in this book. It is probably the first ever mathematical
數學作品，書裡的所有結果和定理都是從最初提出的五條
in the books were derived logically from the five axioms at the
公理 推論出來的。（那五條公理是一些「符合常理」的假
beginning that were assumed to be true. (The five axioms are
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things that just “make sense” as assumptions.) Euclid derived lots
設。）歐幾里得從公理推導出很多幾何定理，譬如說，畢氏
with HKuST Scientists work that uses strict, logical proofs; all the results and theorems 《幾何原本》大概是第一本用嚴密數學邏輯和證明寫成的
of results from them – for example, Pythagoras’ theorem is #48
定理就是書中第48條定理，他只用了五條公理和前面47條
in the book, and was derived solely from the axioms and the 定理就成功把它推導出來。
߅ɽ߅ኪ࢕ਪၾഈ previous 47 theorems.
It had a huge impact on me as a teenager, since in secondary 《幾何原本》對中學時期的我有很大的影響，畢竟在中學數
學課中涉獵到的大多都是計算，或是短短兩行的幾何證明，
By Sonia Choy 蔡蒨珩 school math, you only learn how to calculate, or just write two- 幾乎沒有提及《幾何原本》裡面那種正式的數學證明。
line proofs in geometry – the sort of formal proofs seen in Euclid is
almost never introduced in schools.
In October 2017, Stephen Hawking allowed his PhD thesis Properties of Expanding Universes to be available online
to the public, hoping to inspire others to think, learn and “look up at the stars and not down on their feet”. It was
accessed more than two million times within just a few days. We chatted with two scientists from different fields of Prof. Julie Semmelhack : There are a couple of papers that Julie Semmelhack教授：其實有幾份科學論文
science at HKUST to learn more about their favorite scientific work, inspiration and advice for students. influenced me. When I was applying for graduate school in 2001, I 都對我帶來了影響。當我在2001年報讀研究院的時候，我
was a molecular biologist interested in neuroscience, but I wasn’t 從事的是分子生物學的研究，可是我亦對神經科學很有興
在2017 年 10 月，Stephen Hawking（斯蒂芬霍金）在網上公開了他的博士論文 ─〈宇宙膨脹的屬性〉，希望能夠啟發他 sure if I could do both of these disciplines at the same time, since 趣。我不肯定自己可否同時從事這兩方面的研究，因為在當
at the time it wasn’t common. Then I read a paper about the
人思考和學習。在短短幾天之內，論文已被閲讀超過 200 萬次。我們與科大兩位來自不同科學領域的科學家作專訪，了解 study of memory using fruit flies, and the idea that something as 時這並不普遍。然後我看了一份在果蠅上研究記憶的論文，
覺得即使是記憶那樣抽象的東西也可以用基因工具研究這
他們最喜愛的科學研究、獲得的啟發及予學生的建議。 abstract as memory could be studied using genetic tools seemed 一點很吸引和有趣。
really cool and interesting.
Some mutant flies were lacking a certain gene, causing some
parts of their mushroom body, the part of the brain involved in 有些突變果蠅缺乏了某一個特定基因，令參與腦部產生記
making memories, to be undeveloped. The idea was to do some 憶的蕈形體（mushroom body）的某些部分發育不全。基
genetic surgery to show that different parts of the mushroom 本上就是要透過在這些果蠅上做一些基因手術，來證明蕈
body were involved in making short and long-term memories. 形體的不同部分分別參與產生短期和長期記憶。

2A How have you been inspired or intrigued by Euclid’s Elements?
How does it change your journey of science education and research?
఻Оࡡ͉‘ܣᅵ઼೯אіˏഹઆkί߅ኪɪ̴࿁આٙӋኪձ޼Ӻ͛ૹϞഹʡჿᅂᚤk

Prof. Ivan Ip : This book had changed my impression of math 葉智皓教授：《幾何原本》改變了我對數學的看法 — 對
as a subject – math became something extremely logical. There 我而言，數學從那時起變了一門極之講究邏輯的科目。數學
are no grey areas in math – if A implies B and B is false, then A 裡沒有灰色地帶的存在；如果A意味著B，而B是不正確的
must also be false – which is something unique to the subject. 話，那麼A一定也不正確。這一點跟其他學科不一樣，譬如
Literature is obviously extremely subjective, different economists
might interpret the same set of data differently, and there are 說文學明顯是十分主觀的，不同經濟學家對同樣的數據可
Professor Ivan Chi Ho IP Professor Julie SEMMELHACK always errors in the sciences whenever you perform experiments. 能會有不同的理解，就算是科學實驗也少不免會有誤差，但
葉智皓教授 Julie SEMMELHACK 教授 However, math has no room for any of that – if it is logically 是數學就沒有這種不確定性：如果邏輯正確的話，結論也
correct, then the result must be correct – it doesn’t depend on
就必然正確 — 並不會因為你說話大聲就代表你是正確。這
Assistant Professor, Department of Mathematics Assistant Professor, Division of Life Science whether you’re the loudest person in the room. The book really 本書真的影響了我，令我在大學選擇修讀數學。
數學系助理教授 生命科學部助理教授 influenced me in terms of choosing math as the subject I wanted
to pursue in university.
《幾何原本》也告訴我現代數學遠遠並不只是計算，而是講
Prof. Ivan Ip obtained his bachelor’s degree Prof. Julie Semmelhack earned her bachelor’s This book also revealed to me that modern mathematics is far 求證明和邏輯導向的。讓我們回到《幾何原本》，歐氏幾何
from HKUST and PhD degree from Yale degree from Princeton University and PhD more than arithmetic, but proof and logic oriented. When we 其實是一門可以以畫圖表達的學問，但是如果我們把線和
look back at Elements, Euclidean geometry is a topic we can
University. Before returning to HKUST as an degree from University of California, San Diego. draw by hand using diagrams, but when you turn the points and 點換成變數和符號的話，那其實就是抽象代數（abstract
Assistant Professor in 2018, he first served as a Before joining HKUST in 2017, she worked as lines into variables and symbols, it is actually the precursor of algebra）的前身。任何實在的物件都可以化為抽象，
這就是我直至現在也在做的，關於抽象代數和表現理論
abstract algebra. Everything concrete can be turned abstract,
Project Researcher at Kavli IPMU, University of a postdoctoral researcher at University of and this is what I am doing to this day, in my research on abstract （representation theory）的研究，基本上是把幾何意義
Tokyo, and then an Assistant Professor at Kyoto California, San Francisco and then Max Planck algebra and representation theory – interpreting the geometric 理解為背後的代數結構。
meaning as algebraic structures behind.
University. Institute for Neurobiology.
Also, an interesting little story here – I was quite naughty in 這裡說一個有趣的小故事吧：我在中學的時候也挺頑皮的，
葉智皓教授於科大取得其學士學位，並於耶魯大學 Julie SEMMELHACK 教授於普林斯頓大學取得其 secondary school, so on a school math exam, there were a few 在一次學校數學考試中有幾條跟三角形相關的幾何問題，
questions on the geometry of triangles – just very short proofs
取得博士學位。在 2018 年以助理教授身份回到科 學士學位，並於加州大學聖地牙哥分校取得博士學 to prove that, for example, the triangles were congruent. After 當中只要求一些簡單的證明，例如證明它們是全等之類；在
看完《幾何原本》後，我嘗試從歐幾里得的五條公理開始，
大之前，他先後於東京大學科維理宇宙物理學與數 位。她在 2017 年加入科大，此前她曾先後於加州大 reading Elements, I tried writing the proof from scratch, starting 一步一步地從頭推論，於是原本只是三兩行的功夫，變成了
from Euclid’s axioms and deriving the results step by step. What
學研究所擔任項目研究員，以及在京都大學擔任助 學舊金山分校和馬克斯•普朗克神經生物學研究所 required three lines ended up taking two pages! After that, my 兩大版的數學證明！事後數學老師當然有責備我，但是最後
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理教授。 擔任博士後研究員。 math teacher did tell me off, but he ended up giving me full 在那條題目也給了我滿分 。這就是數學只講邏輯的最好證
marks on that question . That’s an example of math being right
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明 — 只要邏輯正確，條條大路通羅馬。
when it’s logically sound – just that there are many methods to
get there.