Looking into the guts of things often reveals a very important perspective. Let me elaborate what I mean, via this anecdote:

Many machine learning practitioners will make a mistake of the following form:

"I ran a K-means clustering algorithm on my data, for k = 2, and it didnt show me anything interesting. Therefore, I conclude there isn't a good 2-clustering of my data."

There's a big problem here: if these practitioners knew what k-means does, they would realize the conclusion was off. K-means only works when the Euclidean metric is meaningful on their data. But, said practitioners have never looked into the guts of this method.

This example is from a 2nd year PhD student in Machine Learning at M.I.T., studying under a competent professor. So, such mistakes are not restricted to amateurs.

I posit that the general form of this mistake is: not looking into the guts of things you use every day.

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Examples:

(A) Brains: Most humans go about their daily life using their brain all the time, and never bother to look deeply and scientifically into its function. Lesswrong writ large makes a very solid attempt to look into the guts of how the brain works.

(B) Things: Every day objects are marvels of material science, chemistry, physics, and modern engineering. Very few people can tell you remotely how a table is built.

(D) Math: Terence Tao has a concept called: "Pre-rigor, rigor, and post-rigor". The concept is here, and it is excellent: https://terrytao.wordpress.com/career-advice/theres-more-to-mathematics-than-rigour-and-proofs/. This is a must-read for anyone interested in mathematics.

This can be restated in terms of "looking into the guts of things."

Pre-rigor is the stage a mathematician is in when they have not looked deep into the guts of the mathematics they are manipulating.
Rigor is the act of looking into the messy guts of math (often compiling a lot of analysis down to Epsilon Delta arguments, or re-deriving set theory from Zermelo Frankel Axios).
Post-rigor is the act of summarizing the guts of what they see, in a compact form that allows them to finely and precisely manipulate it's contents.

Many amateur mathematicians deal with high-level summaries of deep mathematics, in the "pre-rigorous" stage. Any student who has understood calculus but not real analysis is in this phase. These students will wave the...

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