Theoretical math often revolves around the idea of proving some conjecture. E.g. Bob has a gut feeling that a particular mathematical structure has a particular interesting property, so he sets about to prove that (or disprove it). Sometimes the conjectures are based on real-world applications, sometimes they’re not.
Mathematical topics comprise a spectrum from very pure to very applied (and shade off into other sciences such as physics and computer science at the applied end). There is no clear demarcation between “pure” and “applied” in mathematics.
That said, there are definitely some mathematical topics that are conventionally considered more “pure” or more “applied” than others—but sometimes applications surface unexpectedly in topics previously considered “pure”.
For example, number theory, or the study of the properties of integers, has long been considered a pure math field with few if any real-world applications. Then public-key cryptography was developed, which relied on mathematical problems like factoring products of large prime numbers in order to securely encode data, and suddenly there were some very practical applied reasons to investigate certain subjects in number theory.
It’s a question of your motivation. As a pure mathematician, you might be interested in algebraic topology for its own sake. As a researcher in electrical engineering, you might be interested in the same material for its applications in sensor networks. So even though the two disciplines use the same methods and look at a lot of the same problems, the endgames are pretty different.
You always have some specific problem in mind, either because it’s your dissertation topic, or because you need to have enough publications to get a promotion. There are more long-term goals, but they have more to do with developing a deep understanding of something and not so much with what a layperson might think of as a goal.
(For the sake of disclosure, I’m a statistician, not a mathematician. I think what I wrote above is a pretty reasonable description, but it definitely reflects my perspective, which may not be the same as that of a pure mathematician.)
From my admittedly mostly non-mathematical viewpoint, mathematicians work from both ends of the subject.
Sometimes a mathematicians will essentially be conducting experiments to see what the results are. They start puttering away at some numbers to see what the results are and see if any unforeseen patterns emerge.
Other times they’re very goal-oriented. They have a target they want to reach - proving a specific proposition is correct. And they have a starting point - the field of already proven theorems. They then work on building a connection from the proven theorems to their proposition.
First, you got the vowel wrong. It’s Erdős, with a double-acute accent over the o, not a dieresis.
Second, Erdős only said this when he was quoting someone else, namely Alfréd Rényi, one of his colleagues.
(Also, in the original, it was “Ein Mathematiker ist eine Maschine, die Kaffee in Sätze verwandelt”, which means “A mathematician is a machine for turning coffee into [Sätze can mean either ‘theorems’ or ‘the residue from coffee’]”, the pun being completely lost in English.)
“Staring off into space” is likely high on the list.
From what I’ve heard, this is accurate. And not just of mathematicians, but of scientists, artists, etc. Some of their work consists of exploring, experimenting, playing around to see what they can come up with; while sometime they have a definite goal: a specific thing they want to make or problem they want to solve.
Not a Mathematician, a former Computer Science prof. But I did a lot of research that involved finding algorithms, proofs of: correctness, impossibility, resource bounds, etc.
Sometimes you just work and work on a problem until you solve it. Sometimes you just happen across something.
E.g., I was working with a friend on something and I noticed that the proof was awfully similar to something else I did. I then realized that we also had another proof that went the same way. I pointed this out to my friend who figured out how there was a general system for these kinds of proofs. The result was our best paper together. We didn’t go looking for it, it just was suddenly there.
But usually, you just kept trying A, then B, then C, …
My point of view is summed up in a Douglas Adams HHGTTG quote: "… we’ll be saying a big hello to all intelligent life forms everywhere … and to everyone else out there, the secret is to bang the rocks together, guys. "
That’s what I did. I banged rocks together. All day every day. Even in bed at night. You just hope you eventually get a rock with a really interesting edge.
The real trick is knowing which rocks to bang together and how to do it.
For some math ‘research’, I’ve often wondered whether the researcher’s motivation (and/or approach to the solution) is based on some intuition (“a picture” or image or, perhaps, through analogy with something already known) versus simply grinding away at manipulation of symbols according to rules. These two paradigms are, of course, not mutually exclusive.
I used to think that most of it must surely be ‘intuitive’ but, having looked at (but not understood, not even close) some more advanced proofs, I now wonder whether the latter prevails.
Manipulating symbols won’t tell you what questions to ask. In a lot of cases you need to do a few pages of symbol manipulation to fill in the details of an idea, but that’s never going to get you very far in and of itself. We really do have to understand this stuff.
I also don’t really know why you put research in quotes, but that doesn’t strike me as a fair thing to do.
I recently read about a mathematician who wondered the same thing you did. He “thought” about mathematics using pictures and analogies and never formal symbols. He wondered if he was “doing it wrong” so he sent letters to his colleagues asking them to describe their thought process. To his surprise, they all responded that they think in terms of pictures, etc.
Unfortunately, I don’t remember who the mathematician was. I think that anecdote was embedded in a larger book that wasn’t specifically about mathematics. Maybe it was Moonwalking with Einstein by Joshua Foer. I don’t remember for sure but I’m thinking it was one of those pop science type of books.
Not gonna make the edit window on this one, so let me expand on this a little bit more. A large part of what you have to do as a mathematical researcher is to develop an intuition about whatever you’re studying. You are going to spend a little bit of time crapping around with symbols at first, but as time goes on, you have insights, and that starts to drive your investigations. In a lot of cases, you’ll end up rewriting what you did early on to reflect what you now know.
I saw a great metaphor for this once that described mathematical research as trying to describe the furniture in a room where the lights are off. Sure, you might get a little glimpse of what’s going on without finding the light switch, but once you do find it, the difference is just night and day.
Here’s some trivia that’s coincidental with your username:
Gauss’ writing style was terse, polished, and devoid of motivation. Abel said, He is like the fox, who effaces his tracks in the sand with his tail'. Gauss, in defense of his style, said, no self-respecting architect leaves the scaffolding in place after completing the building’.
… so if some mathematicians do rely on imagery they’re not necessarily going to admit it.
What do we mean by “understanding” something? We can imagine that this complicated array of moving things which constitutes “the world” is something like a great chess game being played by the gods, and we are observers of the game. We do not know what the rules of the game are; all we are allowed to do is to watch the playing. Of course, if we watch long enough, we may eventually catch on to a few of the rules. The rules of the game are what we mean by fundamental physics. Even if we knew every rule, however, we might not be able to understand why a particular move is made in the game, merely because it is too complicated and our minds are limited. If you play chess you must know that it is easy to learn all the rules, and yet it is often very hard to select the best move or to understand why a player moves as he does. So it is in nature, only much more so; but we may be able at least to find all the rules. Actually, we do not have all the rules now. (Every once in a while something like castling is going on that we still do not understand.) Aside from not knowing all of the rules, what we really can explain in terms of those rules is very limited, because almost all situations are so enormously complicated that we cannot follow the plays of the game using the rules, much less tell what is going to happen next. We must, therefore, limit ourselves to the more basic question of the rules of the game. If we know the rules, we consider that we “understand” the world.
Neither do I, really. I think I was trying to indicate (to myself, at least) research that was devoid of physical motivation, inspiration, or application.
Thank you (all) for your extremely thoughtful responses!
ETA: And, in terms of manipulating symbols, I wasn’t intending to be pejorative. I wondered, in part, whether the pattern (symmetries) and appearances of the symbols themselves, may suggest an approach and/or solution (and possibly even other research questions).