Imagine that you are floating weightless in space, and holding on to one corner of a large cube-shaped structure. This cube has only corners and struts between adjacent corners; the interior and faces are empty. Now imagine that you want to travel to the opposite corner of this cube. The safe thing to do would be to pull yourself along a strut to an adjacent corner, always keeping at least one hand on a strut, and then repeat that process two more times. If you are in a hurry you might be tempted to just launch yourself through the middle of the cube. But if you don’t get the direction right, you risk sailing past the opposite corner on into open space.
Empirically, interior methods CAN work pretty well in high-dimensional optimization problems. https://en.wikipedia.org/wi...
It's like a news story today about a Kentucky valedictorian who offered a quote from Donald J. Trump. The crowd loved it. Then he revealed it was a quote from Donald J. Trump. The crowd went silent and someone booed. Same idea. Same endpoint. Your cube does not model that. We are not dealing here with a difference between sloppy and careful thinking.
Well, no. But if our desired endpoints are irrational, and our justifications for choosing them mere rationalization of decisions we made for merely partisan reasons, why should we care whether we think our way towards them sloppily or carefully?
Is it really so hard to imagine different people moving different directions on the cube?
This model makes a giant assumption that does not fit political life. It assumes we are at one point and agree upon a destination or endpoint. This is almost never the case in politics. For one thing, we have genuine disagreements over values that affect our desired endpoints. So, for example, some believe it would be best if everyone were given health insurance, while others believe it would be best if everyone had to earn health insurance. The latter will be happy if some lazy people lack it; the former want everyone to have it.
But that doesn't even capture the extent of our disagreement about endpoints. Ask people today how important it is for the president to tell the truth. Compare the result to the answer you would have gotten 4 years ago. You will find that many people who deeply desired a truthful president 4 years ago have utterly abandoned that as a declared end today. Our partisan desires shape our declared ends, even when those ends seem self-evidently obvious.
What you would call "sloppy thinking" in most cases is actually very careful thinking—but thinking in the service not of movement toward a stated endpoint, but of rationalizing decisions already made for unstated reasons.
The curse of dimensionality in nature is there are no peaks, at least for long, only meandering in the plane, for the lattice is always changing, and if you attempt to cling to a single vertex, you are already lost because the lattice shifts under you. Instead you need to judge where the vertices will be rather than where they are and grow large and acquire as many vertices as you can in the hope some will remain in your grasp. In this, nature prefers abundance to economy.
To expand on this point a bit, Robin said that in the 1000-dimensional case, "Slow and steady wins this race," but even then it depends to some degree on incentives. If the only thing that matters is who gets to the opposite corner first, then even a high probability of getting lost in space may be a risk worth taking, if there are enough other participants in the race: as the number of them willing to launch themselves through the interior increases, the probability approaches 1 that one of them will get lucky, guaranteeing that any slower competitors cautiously hugging the struts will lose. An example of this sort of dynamic would be startups competing to rapidly scale up and capture market share in a natural monopoly niche, e.g. social networks. "Move fast and break things," rack up vast amounts of debt, of both the financial and technical varieties, because it's winner-take-all.
Of course, determining to what extent one is in this type of race rather than the slow and steady type can be a difficult problem in its own right, as it depends on game-theoretic reasoning about one's competitors' strategies. To the extent that both dynamics are found in academic research, I wonder how much of researchers' strategies are determined by competitive pressures, as opposed to ingrained personality traits. Do bold conjecturers make their intuitive leaps of faith because it's more fun for them than sweating out the details, or is it that they feel their colleagues breathing down their necks, and fear being scooped more than they fear having to retract a mistaken claim?
I have thought about a concept related to this many times. Depth is not supported in the way that it builds something, but it is always worth it. There is also the aspect I have recently been thinking about, in that long-term thinking is both beneficial in the long-term(clearly) and the short-term(because self and others can see that it has more to it than the current moment). That concept is not that far off from some of what you are describing with the cube metaphor.
Also, as far as visual thought patterns, I regularly have one where I imagine risk-taking in terms of jumping from hill to larger hill in the Monte Carlo method of falling into energy funnels. Getting caught up in one hill seems fine, but it takes jumps of risk to reach another hill that has a higher peak, which could only be seen through a large enough jump.
Even within academia there is a division of labor between sloppy interior thinkers and careful border thinkers. E.g., in lots of physics, some physicists make sloppy arguments using a combination of intuition and convenient approximations. Then, once these arguments are seen as plausible, or confirmed using experiment, mathematical physicists kick-in and try to find a careful, rigorous, border-based approach to come to the same conclusion. Similarly, I think there might be something useful in having both kinds of thinkers in the community. Sloppy public conversations might hit upon an important corner by accident, which then careful border academics can approach using their reliable tools. E.g., I see Ayn Rand as a sloppy interior thinker who hit upon some interesting insights, which later careful border thinkers like Robert Nozick arrived at more rigorously.
There is also the obverse problem, which occurs when you think you're living in a 3D space but in fact you are living in a 1000D space. You arrive at a point where you think you have found the answer, but you are in fact at a local minimum (energetically speaking). You are unaware that there are better minima to be found if you could escape the "flatland" you are privy to and explore the other 997 dimensions.
This strikes me as a better metaphor for human effort, and folly. "There are more things in heaven and earth, Horatio, Than are dreamt of in your philosophy." We tend to solve problems from the perspective our own philosophy, or experience – to look where the light cast by our own searchlight is brightest.