I have been running an open source MathJax-based mathematics pastebin[1] since 2012 and I had received 2 donations for it until recently. Then a mathematician tweeted[2] about it convincing others to send some donations. After that kind gesture by him, I received 3 more donations. So pretty sure I can't make a living out of donations. :-) But I still appreciate them because they help in covering a portion of the hosting cost.
Also, I never seriously expected donations for this project because I don't work much on it these days apart from cleaning up spam from time to time, complying with legal notices and occasional maintenance. I added a donation button only to see if someone would use it. Apart from covering hosting cost, I think an important side effect of the donations is that it provides some additional motivation to continue working on the project and develop it further which I indeed plan to do as soon as I can find the time for it.
What kind of legal notices are you routinely dealing with? That's a frightening aspect of maintaining an open source side project I hadn't really considered.
The notices are usually takedown notices to remove content violating various regulations that spammers often post to the website. This issue occurs if my spam filter or I do not detect and remove such content before someone else finds it and submits a complaint about it.
In my experience, this additional overhead comes with hosting a live service that allows user-generated content. Other types of projects where the tool is run offline or where the users take the tool and host it themselves are fine.
Economists are not generally mathematicians, any more than physicists are. I draw that analogy intentionally because in both cases there are subsets of the areas where the lines get a little bit fuzzy and individuals are doing a bit of both.
Second to your question. The shortest answer is that mathematicians create mathematics.
There are significant distinctions in how they typically think about world (e.g. discrete vs continuous) or what motivates/justifies the work (i.e. pure vs. applied) but underlying it all at the core is the act of creation, and of understanding those creations and how they relate to other things.
Many non-mathematicians use some mathematics routinely as a means to an end, but for mathematicians it is much of the end itself.
A couple of people have commented on the ways in which it is like art, and aesthetics is important. Hardy said "Beauty is the first test: there is no permanent place in the world for ugly mathematics." , and I can't think of mathematician who would disagree. Applied mathematicians tend to take motivation from a problem from somewhere else, but also value aesthetics of the solution.
Indeed! There is a popular saying, "Arithmetic is to mathematics as spelling is to writing." Of course, a mathematician is often good at arithmetic just like a writer is often good at spelling. However, someone merely good at arithmetic is no more a mathematician than someone good at spelling is a writer.
> There is also a running joke that mathematicians tend to be bad at mental calculations. The story about Grothendieck prime comes to my mind: In a mathematical conversation, someone suggested to Grothendieck that they should consider a particular prime number. "You mean an actual number?" Grothendieck asked. The other person replied, yes, an actual prime number. Grothendieck suggested, "All right, take 57."
I've seen questions of this kind a lot on HN, and I think they really deserve a better answer than what I can come with. Here's my best shot at an answer:
Mathematics is a bit like art. There is no functional purpose to art, people just do it for the sake of itself, and for the intellectual stimulation. Likewise, mathematics deals with building theory on top of previous theory, with no real purpose other than for the sake of building theory and getting intellectual stimulation.
Think of this: Where did that previous theory come from? It was built on top of previous theory, etc. Mathematics is pretty much a never-ending stack of things built on top of each other for the sake of building things on top of each other.
Every now and then (quite often actually), there is some really useful stuff that can be done with this enormous stack of theory, with applications in practical fields like computer science or engineering. In fact the computer was invented as a result of a lot of merely theoretical contemplations about the nature of logic. But for a "pure" mathematicians, the applications are not the main focus of mathematics. For a forum like HN where a lot of folks are driven by doing stuff, this idea might seem rather strange, but it is the same motivation that drives artists and probably many other things.
Then there's of course the applied mathematicians who try to use everything they know about math to work on some real-world problem. Theoretical computer science is often a good example, where we build algorithms that try to improve performance for some practical problem by using mathematical insights into the properties of the problem. But there's also less applied computer science, where algorithms are built merely because they are interesting or beautiful or whatever, and then maybe someone finds a need to apply them, or not.
For those without any formal education in mathematics, here's a little problem that will help you understand what a mathematical proof actually is: Consider some set S containing some elements (anything really, just some stuff). For any elements a, b in S, we define an operation: a + b yields some other element of S (+ here is just a symbol, not necessarily addition). The operation + has only the following properties:
- (0) a + b is another element of S
- (1) For any c in S, we have (a + b) + c = a + (b + c)
- (2) There exists a special element in S, written as 0, such that a + 0 = a, for any a in S
Note that we did not assume that b + a = a + b. In particular, 0 + a might not be a. Assume that there exists an element 0' such that 0' + a = a = a + 0. Use only the properties stated above to argue that 0 = 0'. If it helps, replace "+" with any other symbol.
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Solution: This structure (a set with an associative operation and a neutral element) is called a monoid. We are proving that the neutral element of a monoid is both "left-neutral" and "right-neutral". The proof proceeds as follows:
By (1), we have 0' + 0 = 0', but also 0' + 0 = by our assumption of 0. Hence 0' + 0 = 0' = 0, which was to be shown.
This solution is fairly straightforward. The next step is to try your hand at something a little more complicated: Define a second operation (-a) which takes an element of S and returns an element of S called the inverse of a, with the property that a + (-a) = (-a) + a = 0, for any a in S. Prove that (-(-a)) = a.
To elaborate on the idea that Math is like art, others have pointed out that despite mathematicians best efforts to avoid practical applications, it often ends up being useful anyways:
I don't think it's accurate to say mathematics is always art. Two theoretical fields in particular: fourier analysis, which came from wanting to solve heat flow equations, and probability, which came from wanting to win at games of chance, come to mind as applied math migrating to theoretical
There's a handful of extremely accomplished professors, but beyond that contributions come from all over the place. Anyone with a bright idea can contribute. As for getting paid to do it, well, you are probably right, but for many people it is probably also just a bit of a hobby.
I can't remember which TV-series I saw it in, some kind of murder or "who dun' it?" show. But I remember that they had a pretty genuine and true role for a mathematician in that episode.
The mathematician was tasked with calculating the areas of manhattan that would be hit during stormy weather and give a rough estimation of the required drainage for the water to seep away.
That's the rough details, can't remember exactly. However, it always struck me as a job where the "mathematician as a job" fit really well.
Not sure I agree. That task was "quantity surveying, estimating, and arithmetic" not mathematics.
A mathematician would have proved for all cities and terrain topologies there's a critical viscosity of liquid above which it is impossible to effectively drain them, along with an estimate of the lower bound of that viscosity.
It was the spam filter at work. I have tuned it just now. Would you please try once more and see if you can use it now?
By the way, this spam filter is one of the things I want to fix when I resume work on this project again. Currently, it blacklists IP addresses whereas ideally it should be blocking certain type of content only.
Also, I never seriously expected donations for this project because I don't work much on it these days apart from cleaning up spam from time to time, complying with legal notices and occasional maintenance. I added a donation button only to see if someone would use it. Apart from covering hosting cost, I think an important side effect of the donations is that it provides some additional motivation to continue working on the project and develop it further which I indeed plan to do as soon as I can find the time for it.
[1] http://mathb.in/
[2] https://twitter.com/daveinstpaul/status/1345082256361193473