Thinking Outside The Sphere
I think this has the makings of a potential top level post, but at the moment it seems to lack any kind of real conclusion.
A lot of satire has been raised on the subject of “thinking outside the box”, but if I were to describe a habit-of-mind common to the most instrumentally competent people I can think of, I’d have to call it “thinking outside the sphere”.
A sphere is a minimum surface solid; it has the smallest possible surface area of any shape with the same volume. This seems like an innocuous fact until you remember that we live on a sphere which is running out of usable surface area. The Earth is only spherical because of an accident of gravity, not because it’s useful. In a lot of ways living on the outside surface of a solid sphere is actively undesirable, and if we were to design a new habitat for human life, we could be a lot more creative.
In computer security there is a concept known as the attack surface, and it’s comprised of everything a potential attacker could have access to. The most potentially worrying attacks are the ones you can’t predict or plan for, and the only way to defend against them is to minimise the total attack surface of the system by giving potential attackers as little to work with as possible. Could an attacker exploit this unused service? No idea, but if you take it away it’s one less thing to worry about.
I regularly work on (admittedly quite menial) real-world problems with no immediate solution but well-defined success criteria, and something that I’ve realised is that they all have an attack surface. The size of that surface is dependent on their shape, and the shape of a problem is determined by how you think about it. Especially powerful techniques for dealing with a type of problem force them into a shape with a larger attack surface, and make them vulnerable to more potential solutions.
The trouble is that simply describing a problem can give it a certain shape. If you describe (as a completely non-controversial example) climate change in the context of collective personal energy consumption, it becomes a messy social and political problem with a highly fragmented attack surface. If you describe it in the context of the distribution of carbon in the biosphere, all of a sudden it’s an engineering problem, and you have a whole host of additional tools at your disposal. The overall volume (search space) of the problem is the same, but you’ve changed its attack surface, and even if the new surface isn’t that much greater (although in this case I would argue it is), it’s at least a more contiguous surface; one vector of assault may miss its intended target, but still have an impact on the overall problem.
This very avenue of thought is, in its own way, attempting to force a problem into a shape with a more pliable attack surface. I want to be able to describe why certain methods are more generally useful problem-solving tools than others, and in doing that I would like to find general properties of those methods which would let me more easily identify them. Information-theoretic concepts are probably quite useful here. Ways of describing problems with relatively low Kolmogrov Complexity (a fractal as opposed to a sphere within the confines of the metaphor), present greater attack surfaces, and provide more information about the problem. An example of a powerful technique with this property would be recursion.
Within the confines of the attack surface metaphor, what other properties of large surface area solids might be analogous to useful properties of powerful problem solving methods?
One conclusion I drew from this: In order to solve a difficult problem, reframe it in as many different ways as possible since each framing has a different attack surface, i.e., each framing suggests different potential solutions.
Well, suggests. Problems are problems and solutions are solutions. I find it more helpful to think of everything at once, rather than focusing on ideas one at a time. Like how in a math problem, when you’re stuck you should start writing down all the related equations you can think of.
This strikes me as doing something similar, or maybe the same thing. For example, in probability theory, often we care about whether a sequence of random variables converges in distribution to another random variable. If you work straight from the definition of convergence in distribution, you’ll write down the CDF of a variable in the sequence and then try to show that it converges to the other random variable (except at discontinuity points). This can often be messy, but thinking in terms of characteristic functions suggests a different way of approaching the problem: show that the characteristic function of the sequence converges to the characteristic function of the other random variable. This method is often easier.
Writing down all related equations and theorems would allow you do see this other possible avenue of attack (there’s a theorem: X_n converges in dist to X iff the characteristic function of X_n converges to that of X). As soon as you start trying to solve the problem in terms of characteristic functions, you have reframed the problem. Writing down all those related equations just allowed you to see how you could reframe it.
I was a little over-general. But basically, you can try a solution without only thinking about the problem in that one way. If the problem is complicated, yeah, one might need to simplify it in order to work on it, and if that’s the case you can get better results by comparing multiple simplifications. But it’s better not to simplify at all, when you can.
For example, I don’t need to simplify the problem of global warming. But it’s reasonable that other people might need to, so yeah, in that sort of case go ahead and reframe.
Self demonstrating article and an entertaining, if a bit stretched, metaphor. Feels like the kind of thing that might be good for you to read in ways I’m not sure why. Upvoted.
The more I think about it, the less sure I am it’s just a metaphor.
It is obviously a metaphor in the sense that I meant it as one. I use surface area metaphors all the time because I find them conceptually useful. (I often think of plans as having “fuck-up targets”, which is the union of the probabilities of all major modes of failure, and then think what I’d have to do to minimise the target of any plan I’m implementing).
I’m thinking it might not be a metaphor in the sense that the properties of large surface area solids might be isomorphic with the properties of search spaces that are easy to systematically search. For example, for an object made up of a finite number of points, the maximal surface area is in effect a binary tree.
I am now no longer sure whether I’ve come up with a useful way of thinking about problem-solving, or if I’m just digging myself into a tautological hole.
The most obvious reason, to me, why it’s just a metaphor is that there’s no rigorous definition of what the space actually is. So surface area and volume of problems are in fact not defined. If we let volume be unique to a problem as you seem to be thinking, then we could say that it has units of knowledge. And then surface area has units of knowledge^(2/3). So… I would guess not.
I assumed it was obvious that the volume of a problem is its solution space and the attack surface is the affordances given to it by whatever method of analysis you’re using.
I guess if that isn’t obvious, it probably reads very differently.
Ah, so the units of surface area are solutions^(2/3) then.
Why would you assume a solution space to be 3-dimensional? That said, I don’t see how the possible approaches could be regarded as the surface area even to an n-sphere (even with noninteger n). A triangulation might be more accurate.
Fair enough. You get my point, though. The units don’t work for a literal interpretation. Instead, what seems to be talked about as attack angles is simply which solutions have a decent chance of working. But there’s no reason why high-likelihood avenues should correspond to the edges—in fact, they should tend to represent the center.
Yes, I agree with that.