In Memory of My Wife, Elise Cawley (1961–2026), with Thanks for 36 Wonderful Years
Something terrible just happened. My wife, Elise Cawley, was recovering from heart surgery and had just attended virtually a celebration for one of our children when she had a freak, vast cardiovascular event—and died instantly. We had been together for 36 years. The picture above was taken just hours before she died.
一件可怕的事刚刚发生。我的妻子伊莉丝·考利正在从心脏手术中康复，刚刚以视频方式参加了我们一个孩子的庆祝活动，随后突发一场罕见而剧烈的心血管事件——当场离世。我们在一起 36 年。上面这张照片拍摄于她去世前几个小时。
In all the writing and public speaking I have done, I have chosen, as a matter of privacy, to say little about my family. But now that Elise is gone I cannot restrain myself from telling the world something about the remarkable person with whom I shared the past 36 years of my life.
在我做过的所有写作和公开演讲里，出于隐私考虑，我一直很少谈及家人。但如今伊莉丝走了，我无法再克制自己，想向世界讲一讲这位与我共度过去 36 年的非凡之人。
It was September 17, 1990, and I was in New York City. The event I had been attending finished a little early, so I decided to drop in on a friend of mine. And there, sitting stylishly on the floor, was Elise, a somewhat bashful but obviously brilliant young pure mathematician. It took me a couple of weeks to call her. But from that moment on we talked essentially every day for 36 years—until the day she died a week ago.
那是 1990 年 9 月 17 日，我在纽约。我参加的活动结束得早了一些，于是我临时决定去看望一位朋友。就在那里，一位年轻的纯数学家伊莉丝正姿态优雅地坐在地板上，有些腼腆，但显然才华出众。我过了两个星期才给她打电话。但从那一刻起，我们几乎每天都交谈，持续了 36 年——直到一周前她去世的那天。
Somehow in the shock of what has happened it feels as if I just met Elise, and now she is gone. But actually there were 36 wonderful years in between, in which my life was lit up by Elise’s intellect, love and energy. She was deeply thematic, able to get, with remarkable ease, to the essence of things. Truth, beauty and our family and four children were her great themes. She always told me that truth and beauty were ultimately the same thing, and that one day I would understand that. What I could see is that she herself loved to create beauty—whether in mathematics, interior design, architecture or personal style. She was a forceful intellect, kind and sweet, but persistent and often feisty in arguing for what she thought was true.
在这场变故的震惊里，我恍惚觉得自己刚刚认识伊莉丝，而她转眼就走了。但事实上，这中间有 36 个美好的年头，我的人生被伊莉丝的才智、爱与活力照亮。她极善于把握主题，能异常轻松地直抵事物的本质。真、美，以及我们的家庭和四个孩子，是她一生的宏大主题。她总对我说，真与美归根结底是同一回事，总有一天我会明白这一点。我所能看到的是，她自己就热爱创造美——无论是在数学、室内设计、建筑还是个人风格上。她的头脑锋利有力，为人善良温柔，但在为她认定的真理辩护时，执着而常常带着一股倔劲。
Despite her deep knowledge and capability in abstract mathematics, she was at some level fundamentally a humanist, thinking about the world in human terms. She decried dry modernism and was drawn to the human connections of classicism. In recent years she thought increasingly about the foundations of mathematics, taking a Platonic view of truth and lamenting the twentieth-century transition to a more formalistic approach.
尽管她对抽象数学有着深厚的造诣和能力，但在某种层面上，她本质上是一个人文主义者，以人的方式来思考世界。她批评枯燥的现代主义，被古典主义中人与人的联结所吸引。近些年，她越来越多地思考数学的基础问题，对真理持一种柏拉图式的看法，并为二十世纪转向更形式化的进路而惋惜。
There was always a certain lively spontaneity to Elise, mixed with a drive towards the highest standards of intellectual and aesthetic rigor. People’s first impressions of Elise tended to be of her personal energy, her lively style and her keen intellect. But as they got to know her, they would also come to see how much she cared about the people around her and how much loving thought and effort she would put into things she did for them. Whether it was a gift or the interior design of a space, Elise would apply her creativity and go back and forth until it was just right.
伊莉丝身上始终有一种鲜活的即兴感，又混合着对智识与审美最高标准的严格追求。人们对伊莉丝的第一印象，往往是她充沛的个人能量、活泼的风格和敏锐的才智。但随着相处渐深，他们也会看到她多么在乎身边的人，为他们所做的事又倾注了多少充满爱意的心思和努力。无论是一份礼物，还是一处空间的室内设计，伊莉丝都会施展她的创造力，反复打磨，直到恰到好处。
Our four children were a defining part of Elise’s life, and she was immensely proud of all of them (as am I!). She had her own unique relationship with each of our children that helped them each develop in their own way. She also built us a wonderful home, full of warmth, architectural beauty and creative design. She took great pleasure in perfecting every detail, maintaining remarkable overall coherence, with dashes of the unexpected.
我们的四个孩子是伊莉丝生命中不可或缺的一部分，她为他们每一个都感到无比骄傲（我也是！）。她与每个孩子都有一种独特的关系，帮助他们各自以自己的方式成长。她还为我们建造了一个美好的家，充满温暖、建筑之美和富有创意的设计。她乐于把每一个细节都做到完美，让整体保持惊人的连贯，又点缀着一些出人意料之处。
Elise was a very fine mathematician, top flight in technical mastery but, more importantly, able to thematically pull together grand conceptual arcs. She was always drawn to elegance, often with an abstract geometrical bent, and by the time I met her she had already established herself with results that continue to be cited today. Later in life Elise would refer to herself as a “retired mathematician”, but if exposed to her kind of math, she would immediately engage with great enthusiasm.
伊莉丝是一位非常出色的数学家，技术功力属于第一流，但更重要的是，她能从主题上把宏大的概念脉络串联起来。她始终被优雅所吸引，常带着一种抽象的几何倾向；在我认识她时，她已经凭借若干成果确立了自己的地位，那些成果至今仍被引用。晚年伊莉丝会称自己是“退休的数学家”，但只要接触到她那一类数学，她会立刻满怀热情地投入进去。
In all my years with Elise we never directly worked together on intellectual projects. Perhaps we were both too strong willed, and with her quick mind, it did not take long for her to get impatient with the differences in our styles of thinking. Still, I would often tell her about work I was doing. And it was remarkable how effortlessly she would see deeper themes, which she would then explain back to me, with varying degrees of patience.
与伊莉丝在一起的这些年里，我们从未在智识项目上直接合作过。也许我们都太固执己见，加上她思维敏捷，没过多久她就会对我们思考风格上的差异失去耐心。不过，我常常会跟她讲我正在做的工作。而她总能毫不费力地看出更深层的主题，然后再讲给我听，只是耐心程度时好时坏。
She liked my concept of computational irreducibility, and I think was pleased that the efforts in science on which I had spent so many of our years together had led to a place that highlighted the limitations of science and of scientism, and aligned well with her humanistic worldview. When I began to form ideas about observers and the ruliad, she was far ahead of me in seeing their philosophical and metaphysical implications.
她喜欢我提出的“计算不可约性”概念。我想，让她感到欣慰的是：我在我们共同度过的那么多年里投入的科学努力，最终走到了一个凸显科学与科学主义之局限的地方，与她人文主义的世界观相当契合。当我开始形成关于观察者和 ruliad 的想法时，她在看清其哲学与形而上学意涵方面，远远走在我前面。
For years, the foundations of mathematics were something of a sore point between us (and our children were highly amused that one of the most vigorous arguments they ever saw between us was about the metaphysical character of the concept of a circle). Elise maintained that raw axiomatic formalization cannot be the true essence of math and that math is really something much more human. And finally, just a few years ago, by a long and circuitous route, I came to realize that—despite my obstinate insistence to the contrary—she’d been right about this all along. But then she made other assertions too, that she thought were quite significant. Like that the fundamental abstractions from which we build math as we know it are the concept of point (which she viewed as equivalent to space, continuity and infinity), and the concept of number. She also asserted that physics differs from math mainly in its introduction of the concept of time.
多年来，数学的基础问题在我们之间一直是个心结（我们的孩子觉得非常好笑：他们见过我们之间最激烈的一次争论，竟是关于“圆”这个概念的形而上学性质）。伊莉丝坚持认为，纯粹的公理化形式化不可能是数学真正的本质，数学其实是某种远更具人性的东西。终于，就在几年前，经过一条漫长曲折的路，我意识到——尽管我曾固执地坚持相反的看法——她在这件事上一直是对的。但她还提出过其他一些论断，她认为相当重要。比如：我们据以构建今日所知数学的那些基本抽象，是“点”的概念（她视之为等同于空间、连续性和无穷）和“数”的概念。她还断言，物理学与数学的主要区别，在于物理学引入了“时间”的概念。
While Elise could be a very original thinker, she also believed in the value of existing wisdom and existing ways of doing things—and was constantly telling me that I shouldn’t try to figure out absolutely everything (including about everyday life) from first principles. Being with Elise was always something of an adventure. There was spontaneity (“no plan; let’s play it by ear”), yet also precision and perfection. And there were the forcefully delivered bursts of insight, not just about math and abstraction, but also about the human condition and the way the world works. Elise would say that there were people of action, and there were philosophers. She saw herself more as the latter. She had strong opinions about how things should be in the world. But she was much more interested in figuring out what should be than in what she saw as the grind of trying to make it actually happen.
伊莉丝可以是一个极富原创性的思考者，但她同样相信既有智慧和既有做法的价值——并且不断告诉我，不该试图凡事（包括日常生活）都从第一性原理去推演。和伊莉丝在一起，总有点像一场冒险。有即兴（“别做计划，走一步看一步”），但也有精确和完美。还有那些强有力迸发出来的洞见，不只关于数学和抽象，也关于人的处境和世界运转的方式。伊莉丝会说，世上有行动者，也有哲学家。她认为自己更属于后者。对于世界应当如何，她有强烈的看法。但比起她眼中“努力让它真正实现”的那种苦役，她远更有兴趣去弄清“应当如何”。
Elise was a person who had both great confidence and great humility. While she was always convinced that she was right in things she thought (and in fact usually was), she was never one to advertise her capabilities. Still, if ever a conversation turned to things she knew or cared about, she wouldn’t restrain herself from making what was often a deeply insightful contribution.
伊莉丝是一个既极自信又极谦逊的人。她总是确信自己所想的是对的（事实上通常也确实如此），却从不张扬自己的能力。不过，只要谈话转到她熟悉或在意的话题，她就不会克制自己，往往会做出极具洞察力的发言。
I always thought Elise and I were an outstanding match—similar in many ways, yet also complementary. We both valued thematic thinking (though in many domains she was better at it than me), and we shared an appreciation for aesthetics and work well done. But for example our ways of interacting with people were somewhat different. My children say I have only one mode of interaction: full on. Elise was at some level more introverted and would imagine herself to be shy—though once engaged she would become highly talkative and animated. And when it was just Elise and me on our own, we always seemed to have things to talk about, in fact an infinite number of things.
我一直觉得伊莉丝和我是极好的一对——在很多方面相似，又互为补充。我们都看重主题式的思考（尽管在许多领域她比我更擅长），也都欣赏美感和做得漂亮的工作。但比如说，我们与人相处的方式有些不同。我的孩子们说我只有一种交往模式：火力全开。伊莉丝在某种程度上更内向，会想象自己是害羞的——尽管一旦投入交谈，她会变得非常健谈、神采飞扬。而当只有伊莉丝和我两个人时，我们似乎总有话可说，事实上是说不完的话。
Elise was born in 1961 in Urbana, IL—ironically not far from the headquarters of our company—when her father was a physics graduate student at the University of Illinois. The second of four children, she grew up mostly in Columbia, MD, as a top student, particularly recognized for her math and her thematic essays. As she often mentioned to our children, in 6th grade her teacher characterized her writing as “the cat’s pajamas”—and for a while she imagined growing up to be a writer. In high school she had the formative experience of going to the Hampshire summer math program, where for the first time she met other people who were intensely interested in math.
伊莉丝 1961 年出生于伊利诺伊州厄巴纳——说来讽刺，离我们公司总部不远——当时她父亲是伊利诺伊大学的物理学研究生。她是四个孩子中的老二，大部分时间在马里兰州哥伦比亚长大，是个尖子生，尤其以数学和主题作文见长。正如她常向我们的孩子提起的，六年级时老师把她的文章形容为“the cat’s pajamas”（1920 年代俚语，意为“顶呱呱”）——有一阵子她想象自己长大会成为作家。高中时，她有过一次影响深远的经历：参加汉普夏暑期数学项目，第一次遇到其他同样对数学抱有强烈兴趣的人。
She went to MIT for college, falling in with a group that included several later prominent mathematicians. She always told me that what really ignited her interest in conceptual pure mathematics was an epiphany she had in a class she took about topology. At MIT, she supported herself by working in the thesis bindery, using her calligraphic skills to label theses. She also made up language puzzles for a language acquisition study and had a summer job working on algorithms at an early speech recognition company. A family crisis took her back to Maryland, where she finished degrees in both math and physics at the University of Maryland. She also had an internship working on a PET scanner at NIH, volunteered at some hospitals and considered going to medical school. But in the end she decided on graduate school.
她进入麻省理工学院读本科，融入了一个后来出了好几位知名数学家的圈子。她总告诉我，真正点燃她对概念性纯数学兴趣的，是她在一门拓扑学课上的一次顿悟。在麻省理工，她靠在学位论文装订室打工养活自己，用书法功底为论文题写标签。她还为一项语言习得研究编写语言谜题，并在一家早期语音识别公司做过研究算法的暑期工。一场家庭危机把她带回马里兰，她在马里兰大学同时完成了数学和物理两个学位。她还在美国国立卫生研究院（NIH）实习，参与一台 PET 扫描仪的工作，在几家医院做过志愿者，并考虑过读医学院。但最终她决定去读研究生。
She couldn’t make her mind up between physics and math but ended up applying in physics. One of her physics professors had said she was the best student he’d ever seen but would be most suited to very theoretical physics. She had her choice between Harvard and Berkeley. And in her ever-typical fashion she waited until the last minute, picking Berkeley because she was told she’d be able to switch to math there.
她在物理和数学之间拿不定主意，最后按物理专业申请。她的一位物理教授说，她是他见过最优秀的学生，但最适合非常理论化的物理。她在哈佛和伯克利之间做选择。以她一贯的作风，她拖到最后一刻，选了伯克利——因为有人告诉她，在那里可以转去数学。
After an intense “think on your feet” interview process, she received a Hertz Foundation fellowship, which generously supported her graduate studies. Before even arriving at Berkeley she’d settled on math, and having impressed people with her elegant solution to a small open problem, she joined a group working on geometrical approaches to dynamical systems. After four years in graduate school she completed her PhD thesis on “Smooth Markov Partitions and Toral Automorphisms”. It was quintessential Elise. An elegant thematic idea implemented with clean technical proficiency.
经过一场紧张的“临场应变”式面试，她获得了赫兹基金会的奖学金，慷慨地资助了她的研究生学业。还没到伯克利，她就已经定下了数学；凭着对一个小型未解问题的优雅解答让人印象深刻，她加入了一个用几何方法研究动力系统的团队。读研四年后，她完成了博士论文《光滑马尔可夫分划与环面自同构》。这是最典型的伊莉丝：一个优雅的主题性构想，以干净利落的技术功力实现出来。
It made her a hotshot young mathematician, sought after by many universities. She ended up with two jobs: a tenure-track professorship at Stony Brook and an NSF postdoc sponsored by the prominent mathematician Dennis Sullivan. She chose to defer the professorship and take the postdoc for a year, partly at CUNY in New York City and partly at IHES near Paris.
这让她成为一位炙手可热的年轻数学家，许多大学都想要她。她最后同时拿到两份工作：石溪大学的终身教职轨道教授职位，以及由著名数学家丹尼斯·沙利文资助的美国国家科学基金会（NSF）博士后。她选择推迟教授职位，先做一年博士后，一部分在纽约市立大学（CUNY），一部分在巴黎附近的法国高等科学研究所（IHES）。
While at IHES, Elise lived in the very center of Paris, only steps from the Jardin du Luxembourg, in an unused studio owned by an artist who was the sister of another mathematician. It was a period of intense mathematical work for Elise about which she was very enthusiastic. And her thematic and aesthetic sensibilities drew her to the study of Teichmüller spaces, which are spaces of possible geometrical objects, or in a sense spaces of possible spaces. It was also at this time that she embarked on what would be a long-term interest in the so-called thermodynamic formalism, or Gibbs theory.
在法国高等科学研究所期间，伊莉丝住在巴黎正中心，离卢森堡公园只有几步之遥，租住在一位艺术家闲置的画室里——那位艺术家是另一位数学家的姐妹。那是伊莉丝一段紧张、而她本人非常投入的数学工作时期。她对主题和美感的敏锐，把她引向对泰希米勒空间的研究——那是由可能的几何对象构成的空间，某种意义上说，是“可能的空间”构成的空间。也正是在这一时期，她开始了对所谓“热力学形式化”（即吉布斯理论）的长期兴趣。
With connections to mathematical physics, Gibbs theory provides a way to talk about infinite collections of infinite structures, and their development, for example through time. (Ironically enough—as Elise would repeatedly mention—concepts such as transversals that appear in Gibbs theory are now directly relevant to my own work on infrageometry and the foundations of physics.)
吉布斯理论与数学物理相关联，它提供了一种方式，用来谈论无穷多个无穷结构的集合，以及它们的演化（比如随时间的演化）。（说来颇为讽刺——伊莉丝会反复提起——吉布斯理论中出现的诸如“横截线”之类的概念，如今与我自己关于亚几何和物理学基础的工作直接相关。）
Rather untypically for a mathematician, Elise was always notable for her distinctive personal style—often an interpretation of current fashion, but with her own spunky and colorful touches. As one of my daughters tells it, Elise’s interest in style began to expand with the funding she received from her Hertz fellowship and developed further during her time in New York City and Paris.
与多数数学家颇不一样，伊莉丝一向以她鲜明的个人风格引人注目——常常是对当下时尚的一种诠释，但带着她自己那份泼辣而缤纷的处理。据我一个女儿说，伊莉丝对风格的兴趣，是从她拿到赫兹奖学金的资助开始扩展的，又在她待在纽约和巴黎的日子里进一步发展。
Upon returning from Paris, Elise was about to start at Stony Brook, but while waiting for her apartment to be ready, was staying with her long-time friend Lisa Goldberg in New York City. And that’s how, on September 17, 1990, Elise and I first met.
从巴黎回来后，伊莉丝即将到石溪大学就任，但在等公寓准备就绪的这段时间，她住在纽约的老朋友丽莎·戈德堡那里。就这样，1990 年 9 月 17 日，伊莉丝和我第一次见面。
Lisa’s apartment was an elegant one, but I was not quite so elegant that day. I had been at a tech industry event where I had received a rather bulky award, which was in my jacket pocket weighing it down. Elise was sitting on the floor, with her bangs flopped across her face, in a pose that I thought might be yoga. We talked just a bit, but it was enough.
丽莎的公寓很雅致，但我那天不太雅致。我刚参加完一个科技行业活动，领了一座颇为笨重的奖，就揣在夹克口袋里，把衣服都坠得往下沉。伊莉丝坐在地板上，刘海垂落在脸上，那姿势我以为可能是瑜伽。我们只聊了一会儿，但这就够了。
The next day I went back to Champaign, IL, where I was based at the time. It had been two and a half years since we’d released Mathematica 1.0 and our company was growing rapidly. I had started my career young, so even though I was only two years older than Elise (she was close to 29 and I had just turned 31), I had already done quite a bit, both in academia and the tech industry.
第二天我回到伊利诺伊州香槟——我当时的据点。距我们发布 Mathematica 1.0 已经过去两年半，公司正在迅速壮大。我出道早，所以尽管我只比伊莉丝大两岁（她将近 29 岁，我刚满 31 岁），我在学术界和科技行业都已经做了不少事。
Early versions of Mathematica came with a book, that I had to write. And in September 1990 I was rushing to finish the second edition of that book, to accompany Mathematica 2.0. Two weeks after I returned to Champaign, I did finish, though promptly got sick with a high fever. And that was when I finally called Elise.
Mathematica 的早期版本会附带一本书，得由我来写。1990 年 9 月，我正赶着完成那本书的第二版，好配合 Mathematica 2.0 一起发布。回到香槟两周后，我确实写完了，但随即发起高烧病倒。也正是在那时，我终于给伊莉丝打了电话。
Soon I was taking trips to Stony Brook, and she to Champaign. And when apart we were talking by phone for hours each day. Lisa Goldberg made a point of telling me, “She’s a very fine mathematician, you know”. Elise had met another person a little earlier, and Lisa had dismissed them as unsuitable. But—as I just now learned from one of my children—when Elise asked Lisa about me she immediately responded with enthusiasm.
很快，我开始往石溪跑，她往香槟跑。分开时，我们每天通电话好几个小时。丽莎·戈德堡特意对我说：“你知道，她是位非常出色的数学家。”稍早之前伊莉丝遇到过另一个人，丽莎认为那人不合适。但——我刚刚从一个孩子那里得知——当伊莉丝向丽莎问起我时，丽莎立刻热情地回应。
One notable trip was the result of a call I got from Elise where she explained that she’d mixed up contact lens solution and hydrogen peroxide and now had to patch both her eyes for a few days. I immediately got on the next flight. And while the only food I could cook at that time (and still today) was an omelette, she viewed this “rescue” as an act of chivalry that—as she told our children—was what definitively sold her on me.
有一次难忘的行程，起因是伊莉丝打来电话，说她把隐形眼镜护理液和双氧水弄混了，接下来几天两只眼睛都得贴上眼罩。我立刻搭上了下一班飞机。虽然那时（直到今天也是）我唯一会做的菜就是煎蛋卷，但她把这次“营救”看作一种骑士精神——正如她对我们的孩子所说，正是这件事让她彻底认定了我。
On Elise’s 29th birthday in December 1990 I sent some flowers. She responded by email, now bittersweet:
1990 年 12 月伊莉丝 29 岁生日时，我送了些花。她发邮件回复——如今读来百感交集：
Hi there. Some amazing flowers just arrived. I think I can make to 92. I’m heading off to aerobics — tonight is the amazing Margaret (who plays great music). I has some good math conversation with caroline s. today. And got a little bit of real work done.
I’ll be back around 9pm.
elise
嗨。刚收到一些美得惊人的花。我想我能撑到 92。我要去做有氧了——今晚是那位了不起的玛格丽特（她放的音乐特别棒）。今天我和 caroline s. 聊了些不错的数学。也做成了一点点真正的工作。
我大概晚上 9 点回来。
伊莉丝
The very next email I received from her was about a bug in the Mathematica Residue function.
我从她那里收到的下一封邮件，讲的是 Mathematica 的 Residue 函数里的一个 bug。
My day job was (and still is) tech CEO. But what made me originally develop our technology was that I wanted to use it myself—to do science. By the spring of 1991 I was getting frustrated that I could inject new ideas into our company faster than it could absorb them. So I decided I should take something of a sabbatical and work on science for a while. Meanwhile, Elise still had time left on her NSF postdoc, that she could take anywhere. And so it was that we decided to get a house together in the hills above Oakland, CA, and she returned to Berkeley.
我的本职工作当时是（现在仍是）科技公司 CEO。但我最初开发我们这套技术，是因为我想自己用它——用来做科学。到 1991 年春天，我越来越沮丧：我往公司里注入新想法的速度，比公司能消化的速度还快。于是我决定，该给自己放一段类似学术休假的时间，专心做一阵科学。与此同时，伊莉丝的 NSF 博士后还有剩余期限，可以在任何地方用。就这样，我们决定在加州奥克兰上方的山里合租一处房子，她也回到了伯克利。
Before moving into the house that she found for us, she came with me on an 18-city tour of Europe+Russia to promote Mathematica 2.0. After that, we settled into a good routine. She would go off each day to do math. I would stay home to be a remote CEO, and work on science. And from time to time she’d jet off to Texas, Montana, France or wherever to give talks.
在搬进她为我们找到的房子之前，她陪我一起做了一次横跨欧洲和俄罗斯、走访 18 座城市的巡回，为 Mathematica 2.0 做推广。之后，我们过上了一种不错的日常。她每天出门去做数学。我留在家里当远程 CEO，做科学。她还时不时飞去得克萨斯、蒙大拿、法国或别的什么地方做讲座。
By late 1992 we were thinking about next steps. Elise was adamant that she couldn’t live in Champaign-Urbana (despite having been born there). But she thought the idea of living in the Chicago area was interesting. Meanwhile, she was highly in demand as a mathematician, and there were multiple opportunities for her there.
到 1992 年底，我们开始考虑下一步。伊莉丝坚决表示自己没法住在香槟—厄巴纳（尽管她就出生在那里）。但她觉得住在芝加哥一带这个想法挺有意思。与此同时，她作为数学家非常抢手，那里有好几个机会等着她。
Elise took charge, and explored many different locations and houses around Chicago, eventually picking a house in Hinsdale, IL. We moved there in the middle of 1993, and Elise took up a position at the University of Chicago. It was a nice house, but Elise thought it could be nicer, and set about renovating it to have top-of-the-line interior design. She started working with a designer who she discovered by reading magazines, and learned that her aesthetic and geometric sensibilities translated well into creative interior design. Elise was particularly keen on integrating lively colors and unexpected elements into traditional design themes.
伊莉丝主导了这件事，考察了芝加哥周边许多不同的地段和房子，最终选定了伊利诺伊州欣斯代尔的一栋。我们 1993 年年中搬了过去，伊莉丝到芝加哥大学任职。那是一栋不错的房子，但伊莉丝觉得它可以更好，于是着手翻修，做出顶级的室内设计。她开始与一位设计师合作——那是她翻杂志时发现的——并意识到自己在美感和几何上的敏锐，能很好地转化为富有创意的室内设计。伊莉丝尤其热衷于把鲜活的色彩和出人意料的元素，融进传统的设计主题里。
Meanwhile, at the University of Chicago, Elise was teaching a course about Gibbs theory. She wrote extensive notes which she planned eventually to turn into a book. Over the years, people asked for the book many times, and Elise had recently started talking about finally finishing it. Ironically enough, our daughter Catherine, now herself a mathematician, has ended up working in somewhat related areas.
与此同时，在芝加哥大学，伊莉丝开了一门关于吉布斯理论的课。她写了大量讲义，打算最终整理成一本书。这些年来，不断有人问起这本书，伊莉丝最近也开始说要把它最终写完。说来讽刺，我们的女儿凯瑟琳如今自己也是数学家，最后从事的正是与之多少相关的领域。
Elise interacted a lot with Bob Zimmer, who was then chairman of the math department (and would later become president of the University of Chicago). Together they devised the concept of a new activity for the math department: a professional master’s degree in financial mathematics, basically aimed at quants, and taught by math professors. Though Elise was never too interested in the messiness of actual market data, she liked the pure mathematics of market models, and helped develop a curriculum around it. The master’s degree ended up being highly successful and continues to this day.
伊莉丝与鲍勃·齐默有很多往来——他当时是数学系主任（后来成为芝加哥大学校长）。两人一起构想了数学系的一项新活动：一个金融数学的专业硕士学位，基本面向量化从业者，由数学教授授课。尽管伊莉丝对真实市场数据的杂乱从没太大兴趣，但她喜欢市场模型的纯数学，并帮忙围绕它开发了一套课程。这个硕士学位最终非常成功，一直延续至今。
In the fall of 1994 Elise and I married. And then, in December 1995, as Elise would later describe it, “the world overnight had a new center”: our first child was born. The next few years were dominated by the arrival of children, with Catherine being born in 1997 and Christopher in 1998—leaving Elise, as she put it, with “three under three”.
1994 年秋天，伊莉丝和我结婚。接着，1995 年 12 月，用伊莉丝后来的话说，“世界一夜之间有了一个新的中心”：我们的第一个孩子出生了。接下来几年被孩子们的到来主导——凯瑟琳 1997 年出生，克里斯托弗 1998 年出生——用她的话说，让伊莉丝有了“三个不到三岁的娃”。
With our family growing—and our book collection reaching over 5000 volumes—we were outgrowing our house in Hinsdale, and started looking for a new home. At first, Elise found real estate in Lake Forest, IL. But when the deal for it fell through we started thinking more broadly and decided that since the Boston area was the place where together we knew the most people, that was where we should go. But which town in particular? Elise’s #1 initial criterion was that it should already have a Starbucks. From the list of such towns, she then started looking at USGS aerial photographs (or what would now be satellite images), and soon settled on Concord, MA. But finding no suitable existing house to buy there, she set about designing a house to build.
随着家庭壮大——藏书也超过了 5000 册——我们在欣斯代尔的房子渐渐不够住了，于是开始找新家。起初，伊莉丝在伊利诺伊州森林湖看中一处房产。但那笔交易告吹后，我们开始更宽地考虑，并认定：既然波士顿一带是我们俩加起来认识人最多的地方，那我们就该去那儿。可具体是哪个镇？伊莉丝最初的第一条标准是：那里得已经有一家星巴克。从符合条件的镇子清单里，她接着开始查美国地质调查局（USGS）的航拍照片（放到今天就是卫星图像），很快选定了马萨诸塞州康科德。但在那里没找到合适的现成房子可买，她便着手设计一栋房子来建。
It was a complex process. The house had to accommodate what was by then a library of nearly 10,000 books, as well as space for my remote-CEO office and for our three children. Elise had been scouring architecture books, and interviewed several top-of-the-line architects before settling on one to work with. With her mathematical and aesthetic sensibilities she easily absorbed principles of architecture and soon developed her own intuition for it. Sometimes she’d bring in serious math—like number theory to design a staircase with aligned squiggly banisters, or circle packing to lay out an Italian-inspired tiling based on circles and Sierpinski patterns. She started off making sketches, but soon learned the latest CAD tools. And in the end her ideas defined much of the design of the house. (At the beginning, I had tried making some “suggestions” about the plans, but quickly realized that this was an Elise project, and I should stay out of it.)
这是一个复杂的过程。房子得容得下当时已近一万册的藏书，还要有我远程 CEO 办公室的空间，以及我们三个孩子的房间。伊莉丝把建筑类书籍翻了个遍，面试了几位顶级建筑师，才定下合作人选。凭着她在数学和美感上的敏锐，她轻松吸收了建筑学的原理，很快形成了自己的直觉。有时她会动用正经数学——比如用数论来设计一段栏杆蜿蜒且彼此对齐的楼梯，或用圆填充来铺排一种受意大利风格启发、基于圆和谢尔宾斯基图案的拼砖。她一开始画草图，但很快学会了最新的 CAD 工具。最后，房子的设计在很大程度上是由她的想法定义的。（一开始，我曾试着对方案提一些“建议”，但很快意识到这是伊莉丝的项目，我该置身事外。）
It took several years to build the house (with Elise regularly traveling to Boston to oversee the project) but in the spring of 2002 the project was finished—coincidentally at basically the same time as my 10-year project of writing A New Kind of Science. Other than in CAD drawings I’d only seen the house once, early in its construction, so when we drove from Hinsdale and arrived at the house I was in for a surprise. It was much bigger than I expected! And when we went inside I could see that it was a masterpiece, filled with personality: beautiful, elegant, creative—quintessential Elise.
房子建了好几年（伊莉丝定期往波士顿跑，监督工程），到 2002 年春天完工——巧的是，基本和我耗时十年的写作项目《一种新科学》同时收尾。除了 CAD 图纸，我只在施工早期看过房子一次，所以当我们从欣斯代尔一路开车抵达时，我大吃一惊。它比我预想的大得多！进到里面，我看得出这是一件杰作，充满个性：美丽、优雅、富有创意——最典型的伊莉丝。
And then there was the interior design. Bold and colorful, yet classic. Each room with a distinct character, gracefully flowing into the next. Over the years, Elise revised and updated the interior design of the house many times. She was often to be seen sketching on graph paper new possible designs based on new ideas. She kept the style fresh, constantly breathing new life into the house—even if it meant that our children would comment that every time they showed up the furniture seemed to have been rearranged. And 24 years after I first saw the house, I still find myself wowed by it.
然后是室内设计。大胆、缤纷，却又古典。每个房间都有鲜明的性格，优雅地过渡到下一个。这些年来，伊莉丝多次修订和更新房子的室内设计。人们常能看到她在方格纸上画草图，根据新想法勾勒新的可能方案。她让风格保持新鲜，不断为房子注入新的生命——哪怕这意味着我们的孩子会吐槽：每次回来，家具好像都被重新摆过了。在我第一次看到这栋房子 24 年之后，我仍然会被它惊艳。
By 2003, our oldest was encouraging us to have another child, noting, among other things, that the symmetry of the house had given it four children’s bedrooms. And so it was that in the fall of 2004 our fourth child, Elizabeth, was born.
到 2003 年，我们的老大在鼓动我们再要一个孩子，理由之一是：房子的对称性让它有了四间儿童卧室。于是，2004 年秋天，我们的第四个孩子伊丽莎白出生了。
In her earlier years, Elise had always said that she “couldn’t imagine her life not doing math”. But partly through me, and, more importantly through our children, her world had been broadened, and her perspectives had shifted. She would later describe her life as having several chapters, and this was a chapter about children.
早些年，伊莉丝总说她“无法想象自己的人生不做数学”。但一部分是通过我，更重要的是通过我们的孩子，她的世界被拓宽了，视角也发生了转变。她后来会把自己的人生描述为有好几个篇章，而这一章是关于孩子的。
Through her children showing interest in math, she ended up coaching some middle-school MathCounts math competition teams. She wasn’t a fan of the way math was typically taught in schools, complaining that the beauty of math was being lost in the attempt to make math seem “applicable”, and noting that “most of math in modern textbooks is not really math at all”. In an email to me in 2010 Elise spiritedly went further:
因为孩子们对数学表现出兴趣，她最后当上了几支中学 MathCounts 数学竞赛队的教练。她并不欣赏学校里通常的数学教法，抱怨说，在把数学弄得“看起来有用”的努力中，数学的美被弄丢了，并指出“现代教科书里的大部分数学，其实根本算不上数学”。在 2010 年给我的一封邮件里，伊莉丝更是激动地进一步写道：
What has happened to beauty in math? The beauty derives from the simplicity and elegance. That has been lost in the proliferation of extraneous terminology, meant to make things “more conceptual”, from early endless excursions with “place value” to absurd variations on algorithms, to gummy descriptions of everything using contortions of language, rather than the language of mathematics itself.
数学里的美怎么了？那种美源于简洁与优雅。而它已经在多余术语的泛滥中丢失了——这些术语号称要让东西“更具概念性”：从早年围绕“位值”没完没了的兜圈子，到对算法荒唐的花样翻新，再到用扭曲的语言（而不是数学本身的语言）去黏黏糊糊地描述一切。
Elise did nevertheless do some applications of math herself. During the pandemic, for example, she developed the idea that infection was not a binary phenomenon, but that instead each person had a constantly changing viral load and immunity, creating a “viral field” at the population level. Elise was always up to date on world affairs, and had a deep interest in understanding the thematic arcs at work.
尽管如此，伊莉丝自己也做过一些数学的应用。比如在疫情期间，她提出一个想法：感染不是一种非此即彼的二元现象，相反，每个人都有不断变化的病毒载量和免疫力，从而在人群层面形成一个“病毒场”。伊莉丝一向紧跟世界大事，对理解其中运行的主题脉络抱有浓厚兴趣。
Elise loved the small-town atmosphere of Concord, MA, and enjoyed the feeling of being in a place “where everyone knows your name”. She was a regular at local cafes and exercise classes, bringing her liveliness wherever she went. She had a circle of friends with many different backgrounds, often met through their children or simply through repeatedly running into them at Starbucks. She took great pleasure in applying her knowledge and thinking to help her friends in all sorts of ways. Whether it was doing research on medical issues, solving interior design problems, or simply using her keen insight and judgment to provide life advice, she would reliably go above and beyond.
伊莉丝热爱马萨诸塞州康科德那种小镇氛围，享受身处一个“人人都叫得出你名字”的地方的感觉。她是本地咖啡馆和健身课的常客，走到哪儿都把她的活力带到哪儿。她有一圈背景各异的朋友，往往是通过各自的孩子认识的，或者只是因为在星巴克反复碰面。她极其乐于运用自己的知识和思考，以各种方式帮助朋友。无论是查医学问题的资料、解决室内设计上的难题，还是仅仅用她敏锐的洞察和判断力提供人生建议，她总是可靠地做得比分内更多。
Our family had a definite routine, with dinner each day at about 6pm. Every Friday night—following a tradition from Elise’s own family growing up—we would go out to a movie (and, yes, that meant we saw many questionable movies). Before our children were old enough, it was just Elise and me; later it included various contingents of children too. Elise and I would talk after each movie, with her often being quite appalled by my lack of understanding of what seemed to her like the movie’s obvious thematic elements. (Elise also gave me a hard time about my failure to read fiction, claiming that many ideas, particularly about human nature, are best communicated in fiction.)
我们家有固定的作息，每天大约傍晚 6 点吃晚饭。每个星期五晚上——沿袭伊莉丝自己成长时的家庭传统——我们会出门看电影（是的，这意味着我们看了不少可疑的片子）。在孩子们够大之前，只有伊莉丝和我；后来也会带上不同阵容的孩子。每场电影之后，伊莉丝和我都会聊一聊，她常常被我的理解迟钝惊到——那些在她看来分明是电影显而易见的主题元素，我却看不出来。（伊莉丝还老拿我不读小说这件事说事，说很多想法，尤其是关于人性的，最好是通过小说来传达。）
A dozen years ago two of our children were at the Stanford Online High School, and Elise was recruited as an inaugural member of the school’s Community Advisory Group. Originally this was just supposed to be for a year. But a decade later she was still there, a highly valued member of the group. She spent considerable effort thinking about the overall strategy and issues of the organization, and developed many ideas and opinions. Despite this, she’d often tell us that she didn’t think she’d end up saying anything at a given meeting. But with great consistency it would later transpire that actually she had been the most talkative and engaged person there.
十几年前，我们有两个孩子在斯坦福在线高中就读，伊莉丝被招募为该校社区顾问小组的创始成员。本来这只该做一年。但十年过去，她还在那里，是小组里极受重视的一员。她花了相当多的精力思考这个机构整体的战略和问题，形成了许多想法和意见。尽管如此，她常对我们说，她觉得自己在某次会议上大概不会发言。但几乎每次事后都表明，她其实是全场最健谈、最投入的人。
In the early 2000s, the Hertz Foundation, from which Elise had received her graduate school fellowship, started reaching out to her, and she enjoyed engaging with the emerging Hertz community, and meeting fellows from a wide range of fields. In 2018 she was also recruited to the board of the Hertz Foundation. And here again she became a key contributor, often expecting not to say anything, but then ending up as one of the most vocal people there.
2000 年代初，曾资助伊莉丝读研的赫兹基金会开始与她联系，她很享受与正在成形的赫兹社群互动，结识来自各个领域的学者。2018 年，她还被招入赫兹基金会董事会。在这里，她再一次成为关键贡献者——常常预期自己不会发言，最后却成了场上发声最多的人之一。
By 2023, our three older children had all graduated from college, but our youngest, Elizabeth, was just starting at Yale. Elizabeth has always been involved in a great many activities, and Elise took great pleasure in attending every possible concert, sporting or other event—and also for example hosting Elizabeth’s entire a cappella group for a retreat at our house.
到 2023 年，我们三个大孩子都从大学毕业了，最小的伊丽莎白则刚进耶鲁。伊丽莎白一向参加很多活动，伊莉丝极其乐于去看每一场能去的音乐会、体育比赛或其他活动——比如还把伊丽莎白整个阿卡贝拉团请到我们家做集训。
A bit more than a decade ago Elise also discovered a love for the Southwest and for hiking. And with her brother Jason—with whom she was always very close—living in the Phoenix, AZ, area, we started spending time there each winter.
十多年前，伊莉丝还发现自己爱上了美国西南部和徒步。她的弟弟贾森——她一向和他很亲近——住在亚利桑那州凤凰城一带，于是我们开始每年冬天去那里待一阵。
With Elizabeth off in college we were finally empty nesters, and Elise was ready for a new chapter in her life. She considered returning to where she’d left off in Gibbs theory 30 years ago, being told by an old colleague that her work there was still very current. She also considered getting more deeply involved in the philosophy of mathematics and was on the brink of writing a piece about it. She was also quite engaged in thinking about the purpose and direction of higher education, and the importance of free expression in the search for truth.
伊丽莎白去上大学后，我们终于成了空巢父母，伊莉丝也准备好开启人生的新一章。她考虑重新拾起 30 年前在吉布斯理论上中断的工作——一位老同事告诉她，她当年的成果如今依然很前沿。她也考虑更深地投入数学哲学，并且即将动笔写一篇这方面的文章。她还相当投入地思考高等教育的目的与方向，以及在追寻真理的过程中自由表达的重要性。
In some ways Elise and I were becoming “the old married couple”, with her admonishing me about my messiness in eating ice cream cones or her desire that I switch on self-driving to avoid her being subjected to my own human driving.
某种程度上，伊莉丝和我正在变成“结婚多年的老夫老妻”——她会数落我吃甜筒吃得一团糟，或者希望我打开自动驾驶，免得她还要忍受我这个人类亲自开的车。
Elise had always been a fit and healthy person, and was also very medically knowledgeable. So when she started noticing jaw pain when hiking in Colorado and Arizona, she was concerned. Tests at first showed nothing alarming, though, as she already knew, she had a small congenital heart issue that was gradually worsening. But just a few weeks ago her symptoms worsened and more invasive tests revealed that she needed urgent open heart surgery. She had an outstanding team of top doctors, the complex surgery went very well, and she was soon continuing her recovery from home.
伊莉丝一直身体健康、体格良好，医学知识也很丰富。所以当她开始注意到在科罗拉多和亚利桑那徒步时会下颌疼痛，她很警惕。检查起初没查出什么令人担忧的东西，不过正如她早已知道的，她有一个轻微的先天性心脏问题，正在逐渐加重。但就在几周前，她的症状恶化，更具侵入性的检查显示她需要紧急做心脏直视手术。她有一支出色的顶尖医生团队，复杂的手术非常顺利，她很快就回到家中继续康复。
Meanwhile there was a long-planned celebration for one of our children that Elise hoped to attend. But though it was not possible for her to be there in person, we arranged for her to attend virtually. It was a wonderful event, and Elise could be seen via video call beaming through all of it. But then, soon after, disaster struck, and she collapsed, the victim of a massive cardiovascular event. My only solace in this tragedy is that the end came instantly, after a day filled with nothing but joy.
与此同时，我们一个孩子有一场筹划已久的庆祝活动，伊莉丝很希望能参加。虽然她没法亲自到场，我们安排了她以视频方式参加。那是一场美好的活动，视频里能看到伊莉丝自始至终笑意盈盈。但随后不久，灾难降临，她倒下了，成为一场剧烈心血管事件的受害者。在这场悲剧里，我唯一的慰藉是：结局来得瞬间，而在那之前是充满喜悦、别无其他的一天。
There is so much that Elise was looking forward to. The imminent arrival of our first grandchild. Elizabeth’s graduation from college. The next chapter of her life. But Elise will not be here for any of these.
伊莉丝有那么多期待着的事。我们第一个孙辈即将出生。伊丽莎白大学毕业。她人生的下一章。但这些，伊莉丝一样都赶不上了。
Elise had no idea the end was near. And even though she expected more, she had already had many chapters in her life, and many achievements of which she was proud. Elise had a beautiful life, full of things she valued greatly, especially her children. That she is gone is a tragedy for us all; she will be deeply missed.
伊莉丝完全不知道终点将近。尽管她期待更多，但她的人生已经有过许多篇章，许多让她引以为傲的成就。伊莉丝拥有美丽的一生，充满了她极为珍视的东西，尤其是她的孩子。她的离去对我们所有人都是悲剧；我们会深深地想念她。
Thank you, Elise, for all those wonderful years.
谢谢你，伊莉丝，谢谢那些美好的岁月。
(Elizabeth Wolfram helped me a lot in the writing of this piece.)
（伊丽莎白·沃尔弗拉姆在这篇文章的写作上帮了我很多。）