We do math in order to understand what has happened and what is happening, and one reason we want to understand those things is so we can make good guesses about what’s going to happen.
I’m Jordan Ellenberg, a math professor at the University of Wisconsin-Madison. I study number theory, algebraic geometry and topology, which basically means I study very old questions about numbers using very new methods developed in the last few decades. I’m also a writer; I’ve written articles about math for Slate, the New York Times, the Wall Street Journal, Wired, and a bunch of other publications… plus two books. The most recent, How Not To Be Wrong: The Power of Mathematical Thinking, is about the ways mathematics is wrapped up with everything we do and think about, from elections to poems to religious reveries to Supreme Court decisions to baseball games.
If you want to find me on Twitter, I'm at https://twitter.com/JSEllenberg
Here are a few things I’ve written lately:
The war on gerrymandering, and how math is fighting on both sides: https://www.nytimes.com/2017/10/06/opinion/sunday/computers-gerrymandering-wisconsin.html
Are we paying too much attention to child math prodigies? https://www.wsj.com/articles/the-wrong-way-to-treat-child-geniuses-1401484790
The amazing, autotuning sandpile: http://nautil.us/issue/23/dominoes/the-amazing-autotuning-sandpile
I’m featured in NOVA’s latest episode, “Prediction by the Numbers,” which asks what math can and can't tell us about the future. The show is now available for streaming online. I’m here now to take questions about the math on the show, or anything else mathematical you want to talk about!
I took your number theory class at Wisconsin in 2011, and u/allibishop's algorithms class at Columbia in 2017. Just in case you needed a reminder of how long you've been teaching.. your former students are teaching your former students.
I'd answer this, but Matlock is about to come on.
Terry Tao writes a lot about the differences between competition and research mathematics. Do you think competition math does a good job building the skills necessary to have success as a research mathematician? If not, what can high school/undergraduate students do to start developing those skills?
I was very into competitions as a kid; then when I was starting out as a mathematician I sort of looked down on competitions because they are so different from research math. (Short version: competitions value speed while research math values depth; competitions are about questions to which somebody already knows the answer, questions which are often designed "from the answer backwards," while research math is about questions to which nobody knows the answer, and often isn't really about well-defined questions at all but rather about building a correct understanding of a mathematical landscape.
They ARE really different; but now I would say I really do see the value of contests, because for a lot of kids they're very motivating and a good way into the subject. But there are a LOT of good ways into the subject. It would be a mistake to think that the teen math olympians of today are necessarily the same people who will be doing great mathematics a decade from now.
I recently graduated with a degree in mathematics and one of my final courses was in Topology. We used Munkres' textbook and only made it a quarter of the way through if I remember correctly. Throughout the whole course I felt lost and was not understanding what most of the proofs were getting at. Real analysis and abstract algebra were easier as subjects because we've been familiar with those concepts since grade school.
What would you say are the fundamentals of algebraic topology and how does it differ from the core mathematical subjects students have been familiar with for years?
I had the same response in that course. "You just spent two weeks of class proving that a closed curve in the plane cuts the plane into an inside and an outside -- That is obvious, why the hell are you wasting my time?!?" Honestly, I think point-set topology requires a bit more mathematical maturity than I had when I first learned it, or tried to learn it. It takes some time to appreciate why the things you prove in topology require proof! Starting with algebra makes more sense to me. But then again, I'm algebraist -- maybe a topologist would give a different answer!
I am working on a graduate degree in industrial engineering, so needless to say I like math a numbers a lot, especially statistics.
Topics like "Prediction by the Numbers" always make me thing of this book I read called "Weapons of Math Destruction". The book was about how many algorithms that are in use today aren't necessarily fair. What is your experience with unfair algorithms/prediction methods? How do you combat them?
Also, do you have any book recommendations for mathematical prediction methods? Besides your own book, of course :)
Cathy O'Neil, who wrote "Weapons of Math Destruction," is an old friend of mine (we got our Ph.D.s at the same time with the same advisor) and most of what I know about that issue I know from her.
The instance of algorithmic unfairness I'm thinking about most right now is legislative districting
Here, I think the best way to combat unfairness algorithms is with unfairness-detecting algorithms. Lots of really interesting work in this area. The resources at the Metric Geometry and Gerrymandering Group are a good place to start:
As a number theorist how did you get interested in machine learning? Do you stay up to date on the latest research? How did you approach learning a new field that's different from your area of expertise? Were there any resources you found particularly helpful?
So in academia we have a sabbatical system where every seven years you get a year off teaching and committee work to expand your academic practice. I had my first one in 2011-12 and I used it to write How Not To Be Wrong and to start learning about machine learning. The subject is growing too fast for there to be a well-defined teaching corpus, but for a truly mathematical take on that subject I highly recommend Ben Recht's blog: http://www.argmin.net/
Hi, thank you for doing AMA , i have a few questons for you.
Besides the millenium questions, what question or theorem if answered/proved will change math the most.
Who is your favorite mathematician, alive or dead?
Topology, as i understand, is now THE thing in math, what field will be the next big thing?
Thanks for answering.
The developments that change math the most are often not answers to questions that were precisely articulated before the development took place.
In other words, I have no idea!
What book/resource would you recommend to teach combinatorics/probability to elementary schoolers, ages 8-10 say?
I think the best resources out there for younger kids are the materials at Art of Problem Solving
I don't know if they do probability specifically, but their courses go very deep, introducing students to topics that never get touched in a typical K-12 school curriculum, and they're really well-designed. They have more stuff for middle-school-aged kids, but their Beast Academy books are great for younger elementary school students.
While the sciences mostly involve things that most people are aware of, very few of the concepts, objects and phenomenon that math deals with are known to normal people (compare, for example, the math questions asked on reddit’s r/askscience to the science ones).
This makes it very hard to popularize even the most exciting research happening in the various fields of mathematics (such as Scholze’s work in your own field of expertise to take an example). A troubling consequence of all this is that even people who take an active interest in science do not know what mathematicians primarily concern themselves with.1
This clearly can’t be a healthy state of affairs for the subject and its practitioners. So, my question to you is how can mathematicians honestly go about spreading awareness of what they study instead of the usual ‘useful for cryptography, string theory etc.’ narrative.
1 Apparently not just lay people but even scientists, given the trouble that Mumford and Tate had with Grothendieck's obit for Nature.
Great question. I don't think it's realistic for the average newspaper reader to truly know what Scholze is doing. I think what we can do with important new results is find something about the result that you can really convey to the public. I tried to do that with Yitang Zhang's work on bounded gaps:
Obviously, someone reading this isn't going to thereby understand Zhang's proof! But they will understand something real about the problem, I hope.
Worth noting: like a lot of pure mathematicians, I have the idea that people mostly want "news they can use" and don't want to hear about research developments in pure math. But in fact this was one of the most popular pieces I ever had on Slate. I think people really are hungry to know what's going on in math, and the challenge for us is to find the right stories to tell.
Economics is a field where models have done a notoriously poor job from predictive standpoint, and yet economists have been slow to adopt modern methods of machine learning and data mining. Do you think this will change in the future? Is economics an area where you think ML would be effective?
Of course I think people should try. I can't say I see the chance of success as very high. When we ask "can ML recognize a photo of a cat?" we are asking about a task that a biological machine definitely can do. When we talk about predicting the weather, we're talking about a task humans can't do well but which we know is governed by differential equations we can write down.
For economic predictions we have neither of these advantages. So I feel less optimistic about it. But people should still try!
I have a theory that too many kids walk into their first math class preprogrammed to fail by their parents who told them, "Math is hard! I couldn't do it". So being that parents are roughly equivalent to God in the child's life they assume they won't be able to "do math" either. Any thoughts on this idea?
I don't like lying to kids, but I tell parents who hate and fear math that it's OK to lie to their kids about that.
What's the most exciting project you're involved with at the Wisconsin Institute for Discovery?
I'm not directly involved, but one of the coolest math projects there is Rob Nowak's work on judging the New Yorker cartoon caption contest: https://www.cnet.com/news/how-new-yorker-cartoons-could-teach-computers-to-be-funny/ http://papers.nips.cc/paper/7171-a-kl-lucb-algorithm-for-large-scale-crowdsourcing
Can advanced mathematicians be successful on their own or do they need peers to continually push them along? What is the role of mentors/teachers in the process of stronger theorists or are prodigies predetermined? How much of an island are you?
In principle it should be possible to learn math completely on your own. But in practice, it's very rare. It seems to be a communal activity. Being in a room with other mathematicians is an incredibly supercharged way to learn. (Especially if the room has a blackboard.)
Hi Jordan! I'm a huge fan of "How Not to be Wrong", in fact I finished it about a month ago.
I have been looking for something similar to your book but can't find anything that is simultaneously as entertaining and "insightful". Do you mind giving me a few suggestions?
I especially enjoyed the parts where geometry is used as a tool for intuitively thinking about math. So I would like to learn about the geometrical origins of some more mathematical concepts.
Thanks in advance, and thanks for your great book!
I wrote my book, in part, to try to reproduce the excitement I had when I was a kid and I read Douglas Hofstadter's "Godel, Escher, Bach." That book's old now but it still has lots of richness to offer anyone who thrills to math. I'd say the same about Martin Gardner's books. For contemporary stuff, you would probably really like Steve Strogatz's "The Joy of X" or Eugenia Cheng's eccentric but exciting books about category theory and the infinite.
Hi Jordan (or do you prefer Prof. Ellenberg?) , I really enjoyed your book. I've been wondering this for a while, what does a mathematician do on a daily basis? Also, what are you currently working on?
Good question. Of course it depends what your job is. I'm a professor so a lot of what I do on a daily basis is stuff related to the university (figuring out who to hire and what graduate students to admit, talking to current students about what they're working on, planning and giving courses, etc.) So if you watched me at work you might think I spend a lot of time answering email! The research work is a mix -- some of it looks like me sitting in front of my notebook writing with a pen, some of it looks like me actually writing in LaTeX, some of it is kind of invisible because I'm walking down the street or taking a shower or lying in bed and letting the ideas turn over in my mind.
What up Sir! First off, you're the man!
This is selfish question, but it's be on the forefront of my thoughts lately.
I'm a veteran, and I got out of the Air Force about a year ago. I really enjoy math, and I'm back in school, currently in Calc 2. I want to major in math, but I'm intimidated by the prospect of it. Is there room for a guy with an average IQ and a good work ethic in a graduate math program? Is it a smart major? Am I Just gonna trampled and left in the dust by those brilliant kids when I get to more in-depth analysis classes. I know I belong in school, but i'm older now, and I feel like a fish out of water and over my head sometimes.
Also, math rules, and i'm not struggling, but I just don't even know man haha. Anyways, I don't really expect a response to this, but felt like writing it anyways. Thanks for doing the AMA, i'll definitely check out your new book.
Yes, there is room. Yes, it's a smart major. Not every math major becomes a mathematician, and that's good; I truly believe it would be better if way more people out in the world were math majors. I know from talking to employers that they crave people with math training.
Are you going to be trampled and left in the dust by brilliant kids? Well, we all are, eventually, right? In math, there is always someone stronger/faster/more advanced than you. That is not a reason not to do it. If it were, there would only be one mathematician. That would stink.
Here's something I wrote about this in the Wall Street Journal (possibly behind paywall, sorry) https://www.wsj.com/articles/the-wrong-way-to-treat-child-geniuses-1401484790
I study very old questions about numbers using very new methods developed in the last few decades.
Could you expand on that? Which old questions? What new methods?
Well for instance one of the first problems I worked on was the generalized Fermat equation
A4 + B2 = Cp.
You could ask hundreds of years ago (and people did!) what the solutions to this equation were. But there was no real hope of making progress until the work of Wiles, Taylor, and many others in the 1990s, which involved concepts (Galois representations, modular forms, etc.) which simply didn't exist until recently. That's one of the beautiful things about math; the old questions somehow just keep spurring new ideas. At least, it's beautiful from my point of view. Another person might find it frustrating and dispiriting that we stay confused about the same questions for hundreds of years at a stretch!
Hey Jordan. I can’t believe someone wrote a book about a question that many of my professors haven’t been able to answer. Will math make me smarter? (Read better, more logical decision maker)
I came to love math 10 folds when I took my first discrete mathematics class. Why Discrete math? Generic, high school math has always escaped me. I was never particularly gifted in it. I could grasp the underlying ideas to pass, but the fine details and actual workings never seemed to click. I need to be able to get an image in my head in order to work out any sort of system.
In discrete math, I found that many of the different ideas and solutions were easy for me to visualize in a physical form. I was able to picture the shapes, objects or groups in my head and then move and manipulate them allowing me to follow along with what was being described. This is the only time I felt a class made me smarter. I expected as I studied more logic, I'd become a more logical person (read a better decision maker). I will be forthright, with you: I do feel smarter because of DM but not necessarily better decision maker and that's fine. I think philosophical logic is the solution. I just wanted to hear your opionion.
TL;DR: Sorry for the long rant. I love math. I guess a better way to phrase my question is, what will getting a math degree as double majoring along Comp Sci add to me as a person? (It’ll be a sacrifice to GPA and almost nonexistent social life, no extra money)
Like forget careers, resumes, professional whatever. Just when I reflect on myself and improve along the journey of life.
I'm a big believer in people doing math alongside another major. There's almost nothing people think about that doesn't have some mathematical component. So I think a doctor who knows some math is going to be able to think more fully and deeply about medicine, a lawyer who knows some math is going to be able to think more fully and deeply about law, etc.
Discrete math is a great course. My favorite undergraduate course to teach. I wish more students took it as first-years instead of taking the path of least resistance and taking another calculus course just because it feels like "the next thing after the thing I did last year."
Hello Dr. Ellenberg, thank you for doing this AMA! Your book „How not to be wrong“ was a deciding factor for my decision to pursue a degree in statistics and I can’t thank you enough for that.
I remember you saying in your book, that you only really understand new mathematical concepts once you can interpret them geometrically/ can visualize them. (I don’t recall the exact term that was used)
I am sometimes stuck with little to no real interpretation and deeper understanding of some concepts during the semester, which leads to me forgetting the details of these concepts.
What’s your approach to tackle new mathematical concepts, to be able to fully understand them and to develop a visual feel for them? Do you wrangle with problems regarding the subject until you „get it“? Do you read up about it in various kinds of literature?
Thanks a lot for your answer in advance.
I think the best way to understand a general concept is to work out examples. Often multiple examples. In math you learn by doing. As a teacher, of course I try to make my lectures as clear and explanatory as possible, but I also try to make it very explicit that nobody can learn a semester's worth of mathematics just by listening to me talk three hours a week! My more important role in the classroom is to try to convince students it's going to be worth it to spend the ~10 hours a week on their own that it actually takes to learn the material. As teachers we are salespeople for the subject; it's part of the job.
Hi Jordan. I have a 6 year old son that fits your self description in your child math prodigy article (reading before 3, multiplying and squaring 2 digit numbers at 4). Your article suggests avoiding praise and focusing on developing work ethic and grit. What resources do you suggest to find the appropriate challenge level and instructor for very mathematically inclined children?
Hello Jordan. Good to see that you are doing well since Churchill. I went to High School with you. Thanks for bringing the grading curve up hehe. Now that your older do you feel like for child prodigies such as yourself that being great in one thing like Math in your case has to come at the expense of something else? Like social skills. Is that something you have recognized and tried to deal with.
Being advanced in math isn't like being advanced in figure skating or piano. It doesn't take 8 hours a day. So in my view it doesn't really have to take away from other things. In particular, in my experience the distribution of social skills among mathematicians is pretty similar to that in the general population!
I'm an aspiring engineer with an interest in math. What was your favorite math class you ever took and why?
Halfway through Mathematical Thinking and how to not be wrong (would be finished but left it in my dorm room over break </3). It's a really enjoyable read. The voice is written in is uncomfortably close to my Calc 3 professor's.
"Linear representations of finite groups," which I took from Benedict Gross when I was in college. I can't think of another subject that so magically builds beauty and structure out of what feels like something.
I’m still reading through, but thus far, I love your book.
Anyway, what is you favorite mathematical series/sequence, and why?
I don't know if it's truly my favorite but it cracks me up every time: https://en.wikipedia.org/wiki/Look-and-say_sequence
Where are the time travellers hiding Jordan?
They're not hiding me anywhere; I'm right here. Also, let's eat Grandma.
What's your vision on the future of mathematics? Will there be another Pythagoras, Fourier, Lagrange, more big mathematical revelations that change the way we do something or the other - or have we discovered most of the important things, with future discoveries now to simply become gradually more niche and obscure?
On the contrary, mathematical progress is faster now than at any other time in human history. I hope it keeps up!
What are your feelings about the state of mathematics today?
It is the greatest it has ever been in the history of the world, and I worry that we won't be able to sustain it.
Wow! I don't have a question but just wanted to say I picked up the book in an airport and I love it! Thanks for the great read!
Glad you liked it! I've been told by a lot of airport bookstore folks that it's a good seller there, by the way! I wonder why. Maybe my book goes really well with tomato juice and small packs of peanuts.
How do you improve your math skills where subjects such as calc becomes easy to understand. Im just tired of everyone around me telling me how easy derivatives are and how stupid i look when i tell them its hard for ms
Math isn't easy to understand. It's hard to understand. There's a reason it took thousands of years of mathematical progress to get to calculus!
The worst thing a teacher can say is "This is simple." It isn't simple! But it's doable, with work.
The problem isn't you, it's the people around you.
Hello Jordan! I'm in grad school studying symplectic geometry and representation theory. I just had a daughter last November and intend to homeschool her.
Do you think it would be helpful or harmful to teach her abstract things like set theory and algebra at a very young age, say 4 or 5? (That is, assuming abstract thinking is accessible to her in a meaningful way by then.)
Helpful! Just don't expect her to see it the same way an older kid would. I actually think set theory is much more conceptually basic than algebra. When I talk about algebraic ideas with my younger kid, I talk about "mystery numbers" -- "OK, there's a mystery number, when I double it and add two I get 14, what's the mystery number"?
I mean or you could just start right in with symplectic geometry....
Tell me all you know about the relationship between one, zero, and infinity.
Infinity is biggest, zero is smallest, one is sort of in the middle.
Does the sum of all positive numbers equal -1/12 ?
It depends what you mean by "sum"! There's a bunch about this in my book, which I excerpted in Slate:
That article doesn't write about the divergent series 1+2+3+... specifically, but the ideas are the same. "What is that sum?" is the wrong question. "What, if anything, should we define that sum to be?" is a better one.
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