many important arbitrarily complicated functions can be represented as sums of much simpler functions
Check out Integral transforms, there’s a huge class of transforms that solve a bunch of math problems handily, and it comes with that nice explanation of why they seem to work so often.
The trick consists on discovering a twist you can apply to your complicated object which makes it easily separable into pieces that are susceptible to the function you wanted to apply to your object, such that you have some way of also untwisting your function’d twisted object to get your answer.
Here’s an example of Tim Gowers trying to apply the trick to help solve the Erdos Discrepancy Problem, subject of Polymath5 (collaborative mathematics over the internet!). That’s less than 24 hours ago. Yeah, it’s that useful in mathematics. :D (Edit: looking at it again, I’d say it’s a combination of transform and meta, which is a common combo)
Check out Integral transforms, there’s a huge class of transforms that solve a bunch of math problems handily, and it comes with that nice explanation of why they seem to work so often.
The trick consists on discovering a twist you can apply to your complicated object which makes it easily separable into pieces that are susceptible to the function you wanted to apply to your object, such that you have some way of also untwisting your function’d twisted object to get your answer.
Here’s an example of Tim Gowers trying to apply the trick to help solve the Erdos Discrepancy Problem, subject of Polymath5 (collaborative mathematics over the internet!). That’s less than 24 hours ago. Yeah, it’s that useful in mathematics. :D (Edit: looking at it again, I’d say it’s a combination of transform and meta, which is a common combo)
Let’s try applying that to a recent hard problem: The Preference Utilitarian’s Time Inconsistency. (...time passes...) Didn’t see a way. How about a reference class that says “cryonics, singularity, superhuman AI etc. are highly probable”?
Here’s my attempt.