"Why understand? What does it mean to understand?". Two posts ago, I talked about compact expression, and that is part of the answer. Today I try a different angle.

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Certain things "bear thinking about". Which things those are would depend on who you are. How funny/recursive though, to have thoughts about the utility of further thinking.

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The answer to a "where?" question is the name of a place, while "who?"s are answered by giving the names of people. I was thinking about "why" and "how" when this became apparent:

We manage our knowledge by classifying it under various levels of fundamentality/depth. Shallow statements are easily seen to be true or false, but are often quite specific and hence limited in applicability. Deep statements are abstract and cannot be applied directly, but can be used to construct various shallow statements. The construction process takes time and is often error-prone. The deep is induced from the shallow, and the shallow is deduced from the deep.

When we ask "why?", or "how?", what we expect is some elucidation of the connection between the shallow phenomena in the question with some deeper rules. For an answer to be satisfying, two conditions are required.

Firstly, the connection has to be logically sound. We are actually able to verify the correctness of the reasoning without an appeal to authority, i.e. it is possible for an answer to make no sense and hence be rejected. This works because verifying logic is easier than searching for logic.

Secondly, a link between different levels of depth must be made. Since the classification of depth is a priori, I have difficulty coming up with a simple example of this - the concept of inanity is very context dependent. A co-worker hypothesized the other day that cats slept so much because they needed to conserve energy, which in turn was due to them having high body temperature. That I consider an inane explanation because "sleep a lot" and "high body temperature" are at the same depth of truth and do not answer "why?". A potentially acceptable answer would have appealed to the desert origins of cats and evolutionary biology.

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It is important to take into account the conceptions of depth that one's audience has in order to provide satisfying answers and also ask many of the more important questions in life.

Topics that "bear thinking" are those which are anticipated to provide the most satisfying connections in answer - those which relate vastly different depths of reality.

The rain splattered in vain on the glass, insignificant and ignored - my eyes following well-crafted sentences, my mind filled with images of Trantor and thoughts of interstellar politics.

Lost in another world while hurtling home on the bus in rainy weather... it makes me feel really smug.

This post was inspired by "Who Can Name the Bigger Number?".

"oooooooooooo", "1100", "12" and "1.2e1" are different ways of representing the same number.

In coming up with a new representation, the first step is to invent a process by which all members of the old representation can be written in the new one. For example, to convert the "ooo" form to "11" (binary), one would do the following procedure:

1) "ooo" - ""
2) "oo" - "1"
3) "o" - "10"
4) "" - "11"

Addition and Multiplication are defined for the old rep, and to do the same on the new rep, one can convert to the old rep, do the operation, and then convert back. However, most useful reps also have direct ways of doing the same operation. In fact, the usefulness of a rep is often determined by how efficient the operations in it are. (Transforms are used to the same effect.)

Also, in principle the different reps describe the same objects, but this is assuming we have infinite memory and time. For example, to write "1e100" in the "o"-form is impossible, since that would require more "o"s than there are particles in the universe (~1e79).

Representations neglect information that we do not need. Scientific notation, for example, lists only the leading digits of a number and allows us to discard inaccurate digits early in a computation and thus reap the results of error propagation by compacting the expression.

The equivalence of an operation in a higher and lower rep must be proven using mathematics, and not by going through all possibilities, since the higher rep is capable of expressing values that the lower rep is not.

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In music, the representation used to describe frequency is the pitch class followed by the octave. This is actually just the frequency in base 2 scientific notation, with the octave being the exponent. When you double the frequency of a pitch, it remains in the same pitch class but just one octave higher. This notation acknowledges the physics behind musical instruments - frequencies of a same note are those which are physically possible given the same boundary conditions (i.e. fingers). The octave usually refers to the lowest frequency in the note family generated. Higher octaves of the same note (harmonics) are all mixed in to some degree and are responsible for the same note played on two instruments sounding different.