I’m working this year on a specific area of math, but it can also be helpful to browse around and see what math ideas are out there. Last week, I wrote about the kinds of answers that pop up on Quora’s general math topic. Another math destination is Mathematics Stack Exchange. That site works a bit differently.
How Stack Exchange Works
As I discussed last week, Quora’s strategy relies on high question volume plus filtering. The high volume part is understandable. Every website owner would love more page views and user engagement. But Stack Exchange enforces policies that reduce the number of questions and answers on their site.
To maintain content quality, both Quora and Stack Exchange use employee moderators, signals from users (upvotes, downvotes, etc.), and algorithms. The key difference at Stack Exchange is the cultural norm that every user is responsible for the quality of the questions and answers on the site. This philosophy affects how Stack Exchange builds its software and grows its community.
At a high level, both Quora and Stack Exchange work like typical user-driven web content companies: Users submit questions and answers and have some ability to moderate the content through voting and reporting. But Quora relies more on faceless moderators and bots. At Stack Exchange, though the company makes the final decisions, there’s more of a sense of community ownership. Volunteer moderators are elected by the community. Users get increasing influence and access to moderation tools as their reputation increases. Employees and users hash out changes to the site in public, on the Meta sites. When a question is put on hold or closed, the users who voted to take action are clearly visible in the user interface.
The Stack Exchange design is based on the philosophy that people who ask and answer questions are part of a community that collectively owns content quality. The Stack Exchange company provides hosting resources, community-building experience, and some ground rules. But it’s mainly up to the community to make the site what they want it to be. This is theoretically also true on Quora, but site design and company behavior work against a true sense of ownership.
When a question on Stack Exchange doesn’t follow site rules (e.g., it asks for opinions, is off-topic, or is unclear), users swing into action. Often these questions get deleted. There’s an initial period of discussion in the question comments, sometimes users vote to put the question on hold, and if there are enough votes to delete, the question disappears. The more egregious the violations of site rules, the faster this process happens. If a question is a duplicate of an existing question but is otherwise appropriate, it quickly gets marked as such and users can no longer answer it.
This moderation process drives most of the criticism of Stack Exchange from those who prefer the Quora model and say that Stack Exchange is unwelcoming to new users. The company acknowledges that it has room to improve in this area, and has recently launched changes like a question wizard and a visual indicator that a question author is new to the site. But these changes don’t affect the core principle that a Q&A site should enforce rules about the kinds of questions it allows.
Math on Stack Exchange
There are a few math-oriented sites on the Stack Exchange network, but the two that offer general math Q&A are Mathematics and MathOverflow. MathOverflow is only for professional mathematicians, and since I am not one of those, I will focus on Mathematics Stack Exchange.
The Stack Exchange equivalent of the Quora topic feed appears on the home page of each Stack Exchange site. Only questions appear in the list; you have to click through to see answers. There are a few tabs that change which questions appear. To find some interesting questions, I clicked the Month tab. So here is a selection of questions from the Mathematics Stack Exchange Top Questions for the current month:
In early algebra classes, everyone learns how to find solutions to simultaneous linear equations (e.g., given two equations involving $x$ and $y$, find the values of $x$ and $y$ that satisfy both equations). There are many methods for solving systems of linear equations, but one I hadn’t heard of is Cramer’s rule, which uses determinants to specify a formula for the value of each variable.
While a cone might seem fancier than a cube, the accepted answer to this question argues that a cone is actually “simpler” than a cube because
it is an “algebraic object” that can be defined by a simple polynomial identity ($x^2 + y^2 – z^2 = 0$). Taking cross-sections preserves this algebraic nature (since an infinite plane is also algebraic) so we end up with a quadratic curve in two variables, which is a fairly nice object. In some sense these are the “simplest” possible shapes beyond straight lines.
This question came from a new user, has stayed open for a few days, has several good answers, and has more upvotes than downvotes. This is evidence that the Math Stack Exchange community is receptive to puzzle questions, as long as they have a math component and show some effort.
I learned from one answer that there’s a language called MiniZinc that can be used to solve “constraint satisfaction and optimization problems” like this one. However, the accepted answer uses a proof instead.
Not every Math Stack Exchange question has to look like a textbook problem. This one is about mathematical writing. The single (accepted) answer is mainly prose, with math notation just used in an example.
Quora Math vs. Stack Exchange Math
The best math answers on Quora are just as good quality as those on Math Stack Exchange. Talented math writers have found both sites to be a rewarding place to write, despite the proliferation of low-quality questions (as described last week) on Quora. Here’s why Alon Amit says he writes more on Quora than on Math Stack Exchange:
For whatever reasons – the format, the audience, the nature of the questions – my writing here has had more impact, and with more impact came greater motivation, and more writing, and so on. That’s how these things go. MSE is very busy and the vast majority of answers get very little exposure. That’s fine, it’s how the system is designed, but it doesn’t lure me back in.
One big difference between Quora questions and Math Stack Exchange questions is that MSE questions can be mini essays, just like MSE answers. In fact, MSE question writers are required to demonstrate that they have thought about their question. The results of that thought process must appear in the question description.
On Quora, question writers might as well be prohibited from showing their thought process. The current Quora question format provides a question title and a question URL, so in theory you could link to another web page and write your thoughts there. But that would be working against the platform. Quora is looking for canonical questions, and has designed their platform to encourage them. Math Stack Exchange starts with the requirement that question writers bring a real problem that they have worked on, which may as a side effect turn out to be a canonical question.
If you’re studying math, here are a few ways to approach these two popular Q&A sites:
- If you have a question that you can express in one or two sentences, you can ask it on Quora.
- If you have done some work on a question, you can ask it on Math Stack Exchange, even if it takes multiple paragraphs to explain what you’re asking.
- If you’re just browsing to read about interesting math topics, both the Quora math topic feed and the Math Stack Exchange Top Questions feed are useful resources.