(The Erdős–Kac Theorem) The number of distinct prime factors of a random integer between and behaves like a normal distribution with mean and variance .
(Heilbronn’s Triangle Problem) What is the maximum for which there exist points in the unit square, no three of which form a triangle with area less than ?
(The Erdős-Rényi Phase Transition) A typical random graph where each edge exists with probability has connected components of size . A typical random graph where each edge exists with probability has a giant component of size linear in .
To be a Nail is to study a single problem from every angle. It is often the case that each technique sheds light on only one side of the problem, and by circumambulating it via the application of many hammers at once, one corners the problem in a deep way. This remains true well past a problem’s resolution – insight can continue to be drawn from it as other methods are applied and more satisfying proofs attained.
The upshot is not that Hammers are better than Nails. Rather, there is a place for both Hammers and Nails, and in particular both mindsets are far superior to the wishy-washy blind meandering that characterizes overwhelmed novices. There may be an endless supply of advice – even great advice – on the internet, and yet any given person should organize their life around systematically applying a few tricks or solving a few problems.
3 Comments to “Hammers and Nails”
The skill to integrate these concepts is valuable indeed, however one that seems far more valuable on the margin (and eludes me all the same) is that of sifting through the concepts available to integrate. After all, you said it yourself, fully realizing an idea is very expensive and trying them all is completely unfeasible. So how does one find the most beneficial concepts when sifting through the mass currently available? You tangentially caught the edge of this point with your posts about babble and specifically pruning. But I’m left wondering what heuristics and techniques would be useful in the simulated annealing of the rationalist concept-space. Especially given evidence about what one’s current situation is like. This seems like it would make an excellent follow up to the last few posts of yours.
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