I enjoyed alkjash’s recent Babble and Prune posts on Less Wrong, and it reminded me of a favourite quote of mine, Feynman’s description of science in The Character of Physical Law:
What we need is imagination, but imagination in a terrible strait-jacket. We have to find a new view of the world that has to agree with everything that is known, but disagree in its predictions somewhere, otherwise it is not interesting.
Imagination here corresponds quite well to Babbling, and the strait-jacket is the Pruning you do afterwards to see if it actually makes any sense.
For my tastes at least, early Less Wrong was generally too focussed on building out the strait-jacket to remember to put the imagination in it. An unfair stereotype would be something like this:
‘I’ve been working on being better calibrated, and I put error bars on all my time estimates to take the planning fallacy into account, and I’ve rearranged my desk more logically, and I’ve developed a really good system to keep track of all the tasks I do and rank them in terms of priority… hang on, why haven’t I had any good ideas??’
I’m poking fun here, but I really shouldn’t, because I have the opposite problem. I tend to go wrong in this sort of way:
‘I’ve cleared out my schedule so I can Think Important Thoughts, and I’ve got that vague idea about that toy model that it would be good to flesh out some time, and I can sort of see how Topic X and Topic Y might be connected if you kind of squint the right way, and it might be worth developing that a bit further, but like I wouldn’t want to force anything, Inspiration Is Mysterious And Shouldn’t Be Rushed… hang on, why have I been reading crap on the internet for the last five days??’
I think this trap is more common among noob writers and artists than noob scientists and programmers, but I managed to fall into it anyway despite studying maths and physics. (I’ve always relied heavily on intuition in both, and that takes you in a very different direction to someone who leans more on formal reasoning.) I’m quite a late convert to systems and planning and organisation, and now I finally get the point I’m fascinated by them and find them extremely useful.
One particular way I tend to fail is that my over-reliance on intuition leads me to think too highly of any old random thoughts that come into my head. And I’ve now come to the (in retrospect obvious) conclusion that a lot of them are transitory and really just plain stupid, and not worth listening to.
As a simple example, I’ve trained myself to get up straight away when the alarm goes off, and every morning my brain fabricates a bullshit explanation for why today is special and actually I can stay in bed, and it’s quite compelling for half a minute or so. I’ve got things set up so I can ignore it and keep doing things, though, and pretty quickly it just goes away and I never wish that I’d listened to it.
On the other hand, I wouldn’t want to tighten things up so much that I completely stopped having the random stream of bullshit thoughts, because that’s where the good ideas bubble up from too. For now I’m going with the following rule of thumb for resolving the tension:
Thoughts can be herded and corralled by systems, and fed and dammed and diverted by them, but don’t take well to being manipulated individually by systems.
So when I get up, for example, I don’t have a system in place where I try to directly engage with the bullshit explanation du jour and come up with clever countertheories for why I actually shouldn’t go back to bed. I just follow a series of habitual getting-up steps, and then after a few minutes my thoughts are diverted to a more useful track, and then I get on with my day.
A more interesting example is the common writers’ strategy of having a set routine (there’s a whole website devoted to these). Maybe they work at the same time each day, or always work in the same place. This is a system, but it’s not a system that dictates the actual content of the writing directly. You just sit and write, and sometimes it’s good, and sometimes it’s awful, and on rare occasions it’s genuinely inspired, and if you keep plugging on those rare occasions hopefully become more frequent. I do something similar with making time to learn physics now and it works nicely.
This post is also a small application of the rule itself! I was on an internet diet for a couple of months, and was expecting to generate a few blog post drafts in that time, and was surprised that basically nothing came out in the absence of my usual internet immersion. I thought writing had finally become a pretty freestanding habit for me, but actually it’s still more fragile and tied to a social context that I expected. So this is a deliberate attempt to get the writing flywheel spun up again with something short and straightforward.
Would you be willing to explain how I could have invented the Laplace transform? It makes sense why e^-t makes things integrate to a finite sum but what is the “s” doing there? Why is it “Integral e^-st dt”?
Hmm, my intuition for the Laplace transform is not the best unfortunately… I’ve used the Fourier series and transform a lot more and have a lot more feeling for them than the full Laplace transform. I guess I think of it something like ‘Fourier decomposes your functions into oscillating sinusoidal functions (so s is purely imaginary), Laplace also adds a real exponential part so that you can find oscillation modes that are damped or blowing up.’ That makes it clear (to me at least) why I’d want such a thing, but might not be enough for your ‘explain how I could have invented it’ criterion?
I dug around a bit and found this SE question: https://math.stackexchange.com/questions/6661/laplace-transformations-for-dummies
…which led to the following video: https://ocw.mit.edu/courses/mathematics/18-03-differential-equations-spring-2010/video-lectures/lecture-19-introduction-to-the-laplace-transform/
The first 15 minutes or so of that does a pretty nice job of explaining how the s gets in there… a lot better than what I got at undergrad anyway.
If none of that’s really what you’re asking, I may have misunderstood your question, so let me know!
Thank you, that was very helpful. Your explanation as the Fourier transform with a real exponential part too, makes sense. And the video nicely explained that we can see the Laplace transform as a continuous version of a power series projected onto the interval from 0 to 1 rather than from 0 to infinity (I used to do that geometrically when I was in middle school). So if the base of our power series is x then e^s = e^ln(x) = x