这是这次访谈中我最喜欢的一个部分之一：Vlad（@vladtenev）终于公开了他自己的 Putnam 成绩， 以及他对数学的热爱。 “就像孩子在数学能力上超过了父母，然后事情就开始变得奇怪了。” P.S. 你知道什么是 Erdős–Bacon 数吗？ “@HarmonicMath 的北极星目标一直是：我们是否真的能解决那些极其重要的数学问题，比如黎曼猜想，或者霍奇猜想？” “有一类数学问题已经悬而未决几百年了，它们被称为‘千禧年大奖难题（Millennium Prize Problems）’，被认为极其重要、极其困难，而且具有真正的价值。所以这就是我们的长期指引方向。” “我们之所以想做这件事，是因为如果你能解决这些问题，那么数学之后的一切——比如理论物理——都有可能被彻底解锁。接着你就可以想象，去解决那些极其困难的物理问题。” “我不知道你对这个有没有兴趣，但我个人非常感兴趣。我本科是学物理的。我们能不能把强相互作用、弱相互作用、电磁力和引力统一起来？” “现在的情况是：宇宙中有这四种基本力，其中三种我们已经有理论，认为它们在宇宙早期是从同一种力演化而来的，并且我们有假说可以把这三种力统一起来。但引力如何融入其中，我们完全不知道。” “我认为这就是理论物理的圣杯：如何把引力和其他三种力统一进一个‘万有理论（Theory of Everything）’。” “而且如果你真的能解决这个问题，那么在工程层面将会出现各种令人兴奋的可能性。根据这个理论长什么样，你甚至可以想象像超光速旅行这样的事情，听起来就非常疯狂。” “不过，我们也必须设定一些短期里程碑，因为我们不可能从第一天起就直接冲着黎曼猜想或大统一理论去。” “所以我们从竞赛数学问题开始。结果我们成为最早一批真正拿到国际数学奥林匹克竞赛（IMO）金牌的模型之一——这是世界上最难的数学竞赛。” “在那之后，我当时的心理模型是：好，我们会发布这个产品、开始商业化，而真正解决尚未被证明的数学难题，大概还要一年时间。” “因为在我看来，竞赛数学问题和真正未解决的数学问题之间，差距是非常、非常大的。” “但就在去年 11 月，上个月，Aristotle 做了一次分析。Aristotle 协助解决或完全解决了 11 个 Erdős 问题。” “我给你解释一下什么是 Erdős 问题。Paul Erdős 被认为是史上最多产的数学家。他会像巡回推销员一样，到各地朋友家里住上两周，而这些朋友全是数学家。他会帮助他们一起解决数学问题，因此他发表了成千上万篇论文。” “后来就有了所谓的 Erdős 数，类似于 Bacon 数。你听说过 Bacon 数吗？它是指你和演员 Kevin Bacon 在电影中的‘距离’。” “如果你和 Kevin Bacon 一起拍过电影，你的 Bacon 数就是 0；如果你和一个和 Kevin Bacon 拍过电影的人拍过电影，你的 Bacon 数就是 1。” “Erdős 数也是类似：如果你和 Paul Erdős 合著过论文，你的 Erdős 数是 0；如果你和一个与 Erdős 合著过论文的人合著过论文，你的 Erdős 数是 1。” “甚至还有 Erdős–Bacon 数，把两者加在一起。比如有人 Erdős–Bacon 数是 3，因为他在《心灵捕手》里和一位数学家一起出演，而这位数学家曾与 Paul Erdős 合著论文。” “总之，有大约 1100 个 Erdős 问题，其中差不多一半至今仍未解决，是开放问题。” “而现在，借助 Aristotle，人们开始系统性地推进这些问题。有一个问题是 Aristotle 完全自主解决的，还有很多问题是 Aristotle 协助解决，或者将它们转换成了形式化数学语言。” “这比我原本预期的，人们使用这个产品加速数学研究的速度，要快得多。” “然后就在上周，我们公布了一个相当酷的成果。有一个叫 William Lowell 的数学竞赛。如果你看过《美丽心灵》，里面提到过这个竞赛。因为诺贝尔奖得主 John Nash 从来没在 Putnam 竞赛中取得好成绩，所以他一直有点耿耿于怀。” “而 Aristotle——这是我们公司构建的数学模型的名字，并且已经以消费者产品形式免费提供——在 Putnam 竞赛中答对了 12 题中的 10 题，这比我当年做得好多了。” “我当时的成绩是（消音——请看视频）。” “其实我甚至都没去查。我可能得了一点分，但我知道自己做得太差了，干脆没看结果。” “所以是的，就像孩子在数学能力上超过了父母，然后事情就开始变得奇怪了。” MO：“所以 Harmonic 是不是要去解决时间旅行了？” Vlad：“我希望如此。如果真的有解的话。” “你可以想象，向未来旅行在某种意义上其实更容易——你只需要把自己冷冻起来。向过去旅行则可能违反物理学的基本定律。” “这个就更难了。超光速旅行我觉得反而稍微更有可能一点，尽管它同样非常困难，甚至可能根本不可能。” 挺有意思的。 One of my favorite parts of the interview: Vlad ( @vladtenev) finally reveals his own Putnam score.. and how much he loves math. "it’s like the child has surpassed the parent in mathematical ability, & then things get weird." P.S. Do you know what an Erdős-Bacon number is? "The [@HarmonicMath] North Star was: can we actually solve really, really important math problems like the Riemann hypothesis or, you know, the Hodge conjecture? There’s this group of math problems that have been open for hundreds of years that are called the Millennium Prize Problems, and they’re considered very big, difficult, and actually valuable. So that was kind of the North Star. And the reason we wanted to do that was if we could solve those problems, well, maybe everything downstream of math—like theoretical physics—becomes unlocked. So then you can imagine solving really hard physics problems. I don’t know if you’re interested in this, but I was very interested. I was a physics major. How can we unify the strong, the weak, and the electromagnetic force with gravity? So it’s sort of like there are these four forces. Three of them we kind of have a theory for how they originated from one single force in the early part of the universe, and then we have a hypothesis that we can unify these three. Gravity somehow fits into there too, but we just have no idea how it fits in. And I think this is the holy grail of theoretical physics: how to unify gravity with the other three forces into a theory of everything. And actually, if you can solve that, then there’s all kinds of exciting engineering developments. Depending on how that theory looks, you can imagine things like faster-than-light travel, and it gets really crazy. But anyway, we had to think of some near-term milestones, because we can’t just go after the Riemann hypothesis or a grand unification on day one. We started solving competition math problems, and we became one of the first models to actually get a gold medal on the International Math Olympiad, which is the hardest math competition in the world. And then after that, my mental model was, okay, we’re going to release this product and we’re going to commercialize, and we’re probably a year away from actually solving unsolved math problems. Because it seemed like the gap between hard competition problems and unsolved math problems was very, very large. But in November, just this last month, Aristotle did this analysis. Aristotle assisted in or solved 11 Erdős problems. I’ll tell you what an Erdős problem is. So Paul Erdős was—he’s considered the most prolific mathematician. He would go around and basically stay with his friends for two weeks at a time. His friends were all mathematicians, and he would kind of help them work on their math problems. He was like a traveling salesman of math, and so he became really prolific. He published thousands of papers. There’s actually an Erdős number, which is similar to a Bacon number. Have you heard of the Bacon number? It’s basically how many degrees away you are from Kevin Bacon in a movie. So if you were in a movie with Kevin Bacon, your Bacon number is zero. If you were in a movie with someone who was in a movie with Kevin Bacon, it’s one. There is an Erdős number too, which is if you co-authored a paper with Paul Erdős, you’re zero. And if you co-authored a paper with someone who co-authored with Paul Erdős, you’re one. There’s even an Erdős–Bacon number, where you add the two. So then you have people that have co-authored a paper with Paul Erdős and starred in a movie with Kevin Bacon. Someone has an Erdős–Bacon number of three because they were in Good Will Hunting with a mathematician who co-authored a paper with Paul Erdős. Anyway, there are these Erdős problems. There are about 1,100 of them. About half of them are still unsolved to this day, so half of them are open. And now with Aristotle, people are doing this systematically. They’re going through them, and there was one that Aristotle solved fully autonomously, and then a bunch of others where Aristotle assisted in solving or converting them into formal math language. So that was quite a bit faster than I thought people would actually be using the product to accelerate math. And then actually last week, we announced a pretty cool result. There’s a William Lowell math competition. If you’ve seen A Beautiful Mind, they mention this math competition, because John Nash, who was a Nobel winner, never did well on the Putnam. So he always sort of had a bit of a chip on his shoulder, and he was like, “Oh, that person’s really smart. They took the Putnam in this year and they did really well.” Well, Aristotle—which is the name of the math model that the company builds in a consumer form that people can use freely—got 10 out of the 12 problems on the Putnam correct, which is a lot more than I got on the Putnam. I got a (BLEEP - watch the video) Actually, I didn’t even check. I might have gotten some points, but I knew I did so poorly that I didn’t even check my result. But yeah, it’s like the child has surpassed the parent in mathematical ability, and then things get weird." MO: "Okay, so is Harmonic going to solve time travel?" Vlad: "I hope so. If there is a solution" Well, you could imagine traveling forward in time is much easier, in a sense, because you just have to freeze yourself. Traveling backward in time might actually violate fundamental laws of physics. Fun. Yeah, so that one’s harder. Faster-than-light travel I think is somehow a little bit more tractable, even though that’s also very hard and might be impossible."