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Can you summarize any explanation? Question
When explaining a piece of knowledge, sometimes the explanation is short and simple and other times it is long and complex.
Some people put the burden on the explainer, and demand that they make it shorter or more detailed, which implies the explainer chooses how long the explanation is to be.
On the other hand, explainers can object that the explanation cannot be made any simpler or any more in depth. Of course, it could be that the topic is complex and the explainer simply has limited knowledge, so claiming inability to make an explanation longer and more complex is a moot matter. My interest is in simplifying. It seems like if you are able to produce a complex explanation, there should be nothing stopping you from a producing a simpler one.
I am aware of concepts like Kolmogorov complexity, which obviously put a floor on the simplification of complex things. However, these refer to lossless simplification. I assume that a summary, by its nature, is lossy - the purpose of it is to provide a superficial description, and leave breadcrumbs which the audience can research if they are interested in learning more.
Obviously you can always produce a summary that loses more or less information. However, are there cases where the information loss is catastrophic? Is there ever a case where summarizing a topic into something with half the length will leave you with vastly less than half the information content? Or does the information content scale linearly with length, regardless of topic?
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[Does] information content scale linearly with length, regardless of topic?
Information content pretty clearly doesn't scale linearly with length. I used to joke that The Scarlet Letter author wrote everything at least three times, so you only needed to read 30% of the words to understand what was happening.[1]
Consider this extremely contrived "information:"
AAAAAAAAAAAAAAAAAAAA
Rewriting it as "20 As" loses no information, and it's lots shorter. But only the context determines what the effect of shortening it further to "As" might be.
Can you summarize any explanation?
No. At some point, there will be a case "where the information loss is catastrophic" as you put it. Determining the catastrophic case is the tricky part. Catastrophic for what?
If you're making a tire for a wheel that is 1 meter in diameter, the tire's inner edge needs to be π meters, or 3.14159… meters. How important the extra digits are depends on your tools, the stretchiness of your tire material, and so on.
“…the supreme goal of all theory is to make the irreducible basic elements as simple and as few as possible without having to surrender the adequate representation of a single datum of experience.” Albert Einstein, 1933
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Literature fans: Please don't hate me. Instead, suggest a Unicode character to wear in penance. ↩︎
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Consider Shamir's secret sharing. The idea is you need $k$ out of $n$ parts of a key to recover some secret $S$. Then, if you "summarize" a key and encrypted text without providing at least $k$ parts of the key, you effectively lose all of the information in the encrypted text.
But this example is quite convoluted and I'm not quite sure if it's what you're looking for.
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