You are often confident that something is true long before you have an airtight proof for it (this happens especially often in geometry). The main reason is that you have a large catalogue of connections between concepts, and you can quickly intuit that if X were to be false, that would create tensions with other things you know to be true, so you are inclined to believe X is probably true to maintain the harmony of the conceptual space. It’s not so much that you can imagine the situation perfectly, but you can quickly imagine many other things that are logically connected to it.
.. One of the main skills of research scientists of any type is knowing how to work comfortably and productively in a state of confusion.
.. you do not spend much time straining to visualize those things that do not have obvious analogues in two and three dimensions. (Violating this principle is a huge source of frustration for beginning maths students who don’t know that they shouldn’t be straining to visualize things for which they don’t seem to have the visualizing machinery.) Instead…
.. you automatically focus on very simple examples that are easy to think about, and then you leverage intuition about the examples into more impressive insights
.. To me, the biggest misconception that non-mathematicians have about how mathematicians work is that there is some mysterious mental faculty that is used to crack a research problem all at once
.. by the time a problem gets to be a research problem, it’s almost guaranteed that simple pattern matching won’t finish it. So in one’s professional work, the process is piecemeal: you think a few moves ahead, trying out possible attacks from your arsenal on simple examples relating to the problem, trying to establish partial results, or looking to make analogies with other ideas you understand
.. in calculus classes you think about functions or curves. In functional analysis or algebraic geometry, you think of spaces whose pointsare functions or curves — that is, you “zoom out” so that every function is just a point in a space, surrounded by many other “nearby” functions
.. Indeed, you tend to choose problems motivated by how likely it is that there will be some “clean” insight in them, as opposed to a detailed but ultimately unenlightening proof
.. You develop a strong aesthetic preference for powerful and general ideas that connect hundreds of difficult questions, as opposed to resolutions of particular puzzles. .. Mathematicians don’t really care about “the answer” to any particular question; even the most sought-after theorems, like Fermat’s Last Theorem, are only tantalizing because their difficulty tells us that we have to develop very good tools and understand very new things to have a shot at proving them. It is what we get in the process, and not the answer per se, that is the valuable thing.
.. structuring things in a modular way allows you to spend several days thinking about something you do not understand without feeling lost or frustrated.
.. You stumble around bumping into the furniture, but gradually you learn where each piece of furniture is. Finally, after six months or so, you find the light switch, you turn it on, and suddenly it’s all illuminated.
.. You can sometimes make statements you know are true and have good intuition for, without understanding all the details.
.. You are good at generating your own definitions and your own questions in thinking about some new kind of abstraction.
.. “[After learning to think rigorously, comes the] ‘post-rigorous’ stage, in which one has grown comfortable with all the rigorous foundations of one’s chosen field, and is now ready to revisit
.. They are not typically intellectual superheroes. For instance, they often become resistant to new ideas and uncomfortable with ways of thinking (even about mathematics) that are not their own. They can be defensive about intellectual turf, dismissive of others, or petty in their disputes.