I have a very simple problem when doing mathematics.
I want to write a proof. But I also want to save time. And so I miss nuances and make false assumptions and often think the answer is simpler than it is. It’s almost certainly motivated cognition, rather than inadequate preparation or “stupidity” or any other problem.
I know the answer is “Stop wanting to save time”—but how do you manipulate your own unvoiced desires?
Do you have any ideas, including guesswork, about where your hurry is coming from? For example, are you in a hurry to go do other activities? Are you stressing about how many problems you have left in your problem set? Do you feel as though you’re stupid if you don’t immediately see the answer?
Some strategies that might help, depending:
Block off time, know and visualize that this time is for proof-writing and nothing else (you have this block of time whether you use it or not, and cannot move onto other activities), and visualize that this is the only problem in the world.
Make a plan for the rest of the day (and write your “must hurry to do” activities down on a list, with their own timeslots) so that you can believe the blocked off time in 1. When your brain tells you you have to hurry and do X, remind it that you’ll do X at 4pm (or whenever), that this is the timeslot for proofs, and that focusing slowly will get the most done.
Find a context wherein you have the sort of slow, all-absorbing focus that would be helpful here (whether on proof-writing, conversation, or whatever else). Try to understand the relevant variables/mindset and to set up the outside context similarly, and/or copy your internal frame or stance.
Think of great performers who were utterly absorbed in their tasks, and of the excellence they embodied. Put up their names, photos, or other priming influences. Visualize yourself as embodying that same mindset.
Use “positive self-talk” to prime yourself as you work, by saying things like “I am moving slowly, with full focus. I am noticing every nuance I can notice. My mission is to do well, regardless of speed.”
Do your proof with a friend or a student, while showing them what patience looks like and talking about how you’re learning patient, focussed mindsets.
First I come up with a sketch of the proof and try to formalise it and find holes in it. This is fairly creative and free and fun. After a while I go away feeling great that I might have proven the result.
The next day or so, fear starts to creep in and I go back to the proof with a fresh mind and try to break it in as many ways as possible. What is motivating me is that I know that if I show somebody this half baked proof it’s quite likely that they will point out a major flaw it. That would be really embarrassing. Thus, I imagine that it’s somebody else’s proof and my job is to show why it’s broken.
After a while of my trying to break it, I’ll then show it to somebody kind who won’t laugh at me if it’s wrong, but is pretty careful at checking these things. Then another person… slowly my fear of having screwed up lifts. Then I’m ready to submit to publish.
So in short: I’m motivated to get proofs right (I have yet to have a published proof corrected, not counting blog posts) out of a fear of looking bad. What motivates me to publish at all is the feeling for satisfaction that I draw from the achievement. In my moderate experience of mathematicians, they often seem to have similar emotional forces at work.
I know the answer is “Stop wanting to save time”—but how do you manipulate your own unvoiced desires?
If you think of the brain as having two “programming languages”: the “far” (symbolic) and “near” (experiential), and the “unvoiced desire” as being something that’s running on the “near” system, then what you need to do is translate from the symbolic to the experiential.
In this case, you’d begin by asking what experiences you anticipate will happen if you don’t “save time”, and what your emotional reaction to those experiences is.
Take care, though, to imagine actually experiencing one specific situation (in sensory detail) where you currently want to “save time”, and to anticipate the results in sensory detail as well. Otherwise, you’ll only engage the “far” (symbolic) system, and won’t get any useful information.
Thanks for all the good advice!
I think I’ll try blocking off time (I’ve already started tracking how much time a day I spend actually working and found it was much less than I’d assumed) and also try the two-stage process (first try to get something, then try looking for flaws.)
I want to write a proof. But I also want to save time. And so I miss nuances and make false assumptions and often think the answer is simpler than it is. It’s almost certainly motivated cognition, rather than inadequate preparation or “stupidity” or any other problem.
At least based on personal introspection, the part of my mind that comes up with proofs feels very similar to engaging in motivated cognition. This is in some ways ok because if a proof is valid then counterarguments aren’t something that need to be thought about. But yes, this can lead to the problem of constructing apparently valid proofs that then don’t work. One thing that seems to help is to engage in more or less motivated cognition to make a proof and then go through that proof in close detail looking for flaws. So essentially, use motivated cognition to try to get something good, and then use motivated cognition to try to poke holes in it. If you iterate this enough one will generally have an ok proof.
This is a well-known issue. Basically, a mathematical problem tends to involve several non-trivial steps. If you are too pessimistic, it is impossible to see all these steps (because you get bogged down in proving details and lose track of the point of the problem.) On the other hand, if you are too optimistic, you will take too long to debunk an incorrect sequence of steps, leading to the problem you describe.
One solution is to work with someone else, and take turns being optimistic. (E.g., one person proposes a solution, then the other tries to shoot it down; it’s much easier to be pessimistic about other people’s ideas.) Another solution is what Mr. Weissman proposes: just investigate the problem, look at similar problems, try to falsify the problem, try to prove something stronger, etc.
I’m sure that professional mathematicians deal with this issue all the time, so you might want to ask one of them as well.
Ask yourself what are the thrilling aspects of what you want to prove. Look for what you cannot explain, but feel is true.
I want to write a proof.
Before writing, you should be satisfied with your understanding of the problem.
Try to find holes in it, as if you were a teacher reading some student work.
You should also ask yourself why you want to write a correct proof, and remember that a proof that is wrong is not a proof.
Instead of setting out to prove a proposition, investigate whether or not it is true. Perhaps genuine curiosity will override your desire to save time.
I like the writing here: very clear and useful.
I have a very simple problem when doing mathematics.
I want to write a proof. But I also want to save time. And so I miss nuances and make false assumptions and often think the answer is simpler than it is. It’s almost certainly motivated cognition, rather than inadequate preparation or “stupidity” or any other problem.
I know the answer is “Stop wanting to save time”—but how do you manipulate your own unvoiced desires?
Do you have any ideas, including guesswork, about where your hurry is coming from? For example, are you in a hurry to go do other activities? Are you stressing about how many problems you have left in your problem set? Do you feel as though you’re stupid if you don’t immediately see the answer?
Some strategies that might help, depending:
Block off time, know and visualize that this time is for proof-writing and nothing else (you have this block of time whether you use it or not, and cannot move onto other activities), and visualize that this is the only problem in the world.
Make a plan for the rest of the day (and write your “must hurry to do” activities down on a list, with their own timeslots) so that you can believe the blocked off time in 1. When your brain tells you you have to hurry and do X, remind it that you’ll do X at 4pm (or whenever), that this is the timeslot for proofs, and that focusing slowly will get the most done.
Find a context wherein you have the sort of slow, all-absorbing focus that would be helpful here (whether on proof-writing, conversation, or whatever else). Try to understand the relevant variables/mindset and to set up the outside context similarly, and/or copy your internal frame or stance.
Think of great performers who were utterly absorbed in their tasks, and of the excellence they embodied. Put up their names, photos, or other priming influences. Visualize yourself as embodying that same mindset.
Use “positive self-talk” to prime yourself as you work, by saying things like “I am moving slowly, with full focus. I am noticing every nuance I can notice. My mission is to do well, regardless of speed.”
Do your proof with a friend or a student, while showing them what patience looks like and talking about how you’re learning patient, focussed mindsets.
The way it works for me is this:
First I come up with a sketch of the proof and try to formalise it and find holes in it. This is fairly creative and free and fun. After a while I go away feeling great that I might have proven the result.
The next day or so, fear starts to creep in and I go back to the proof with a fresh mind and try to break it in as many ways as possible. What is motivating me is that I know that if I show somebody this half baked proof it’s quite likely that they will point out a major flaw it. That would be really embarrassing. Thus, I imagine that it’s somebody else’s proof and my job is to show why it’s broken.
After a while of my trying to break it, I’ll then show it to somebody kind who won’t laugh at me if it’s wrong, but is pretty careful at checking these things. Then another person… slowly my fear of having screwed up lifts. Then I’m ready to submit to publish.
So in short: I’m motivated to get proofs right (I have yet to have a published proof corrected, not counting blog posts) out of a fear of looking bad. What motivates me to publish at all is the feeling for satisfaction that I draw from the achievement. In my moderate experience of mathematicians, they often seem to have similar emotional forces at work.
If you think of the brain as having two “programming languages”: the “far” (symbolic) and “near” (experiential), and the “unvoiced desire” as being something that’s running on the “near” system, then what you need to do is translate from the symbolic to the experiential.
In this case, you’d begin by asking what experiences you anticipate will happen if you don’t “save time”, and what your emotional reaction to those experiences is.
Take care, though, to imagine actually experiencing one specific situation (in sensory detail) where you currently want to “save time”, and to anticipate the results in sensory detail as well. Otherwise, you’ll only engage the “far” (symbolic) system, and won’t get any useful information.
Thanks for all the good advice! I think I’ll try blocking off time (I’ve already started tracking how much time a day I spend actually working and found it was much less than I’d assumed) and also try the two-stage process (first try to get something, then try looking for flaws.)
At least based on personal introspection, the part of my mind that comes up with proofs feels very similar to engaging in motivated cognition. This is in some ways ok because if a proof is valid then counterarguments aren’t something that need to be thought about. But yes, this can lead to the problem of constructing apparently valid proofs that then don’t work. One thing that seems to help is to engage in more or less motivated cognition to make a proof and then go through that proof in close detail looking for flaws. So essentially, use motivated cognition to try to get something good, and then use motivated cognition to try to poke holes in it. If you iterate this enough one will generally have an ok proof.
This is a well-known issue. Basically, a mathematical problem tends to involve several non-trivial steps. If you are too pessimistic, it is impossible to see all these steps (because you get bogged down in proving details and lose track of the point of the problem.) On the other hand, if you are too optimistic, you will take too long to debunk an incorrect sequence of steps, leading to the problem you describe.
One solution is to work with someone else, and take turns being optimistic. (E.g., one person proposes a solution, then the other tries to shoot it down; it’s much easier to be pessimistic about other people’s ideas.) Another solution is what Mr. Weissman proposes: just investigate the problem, look at similar problems, try to falsify the problem, try to prove something stronger, etc.
I’m sure that professional mathematicians deal with this issue all the time, so you might want to ask one of them as well.
Get someone to pay you by the hour, but not so much that the money swamps your desire to write the proof?
Ask yourself what are the thrilling aspects of what you want to prove. Look for what you cannot explain, but feel is true.
Before writing, you should be satisfied with your understanding of the problem. Try to find holes in it, as if you were a teacher reading some student work.
You should also ask yourself why you want to write a correct proof, and remember that a proof that is wrong is not a proof.
Instead of setting out to prove a proposition, investigate whether or not it is true. Perhaps genuine curiosity will override your desire to save time.