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There is definitely more to understanding than compression of knowledge. Here is one thing. Knowledge is often represented as graph, where the links represent f

There is definitely more to understanding than compression of knowledge. Here is one thing. Knowledge is often represented as graph, where the links represent facts that are related in some way - perhaps causally, perhaps because they are analogs in two different scenarios etc. Understanding would include your ability to quickly traverse this graph so you can point out connections between facts which are more than one link apart.

erikpukinskis 7y ago
Wouldn’t more compressed graphs generally be faster to traverse?
- abdullahkhalids 7y ago
  Depends on how you compress them. Some representation are good for storing, others good for processing in different ways.
  - TeMPOraL 7y ago
    Seems to match real-life thinking, too. For extreme case, see memorization: you learn a set of nodes in a graph, probably highly compressed through application of whatever tricks you use to quickly memorize and recall information. But by default, you learn almost no connections between those nodes, you can't really reason from the material you memorized. Building these connections requires further mental effort.
hyperman1 7y ago
I can think of multiple graphs representing the same knowledge, some smaller than others as irrelevant nodes are pruned, paths are shorter, etc.. The smaller ones encode that same knowledge with better understanding.
- abdullahkhalids 7y ago
  Let's consider understanding a mathematical theory - say linear algebra. In principle, the entirety of the knowledge of LA is contained in it's axioms and the standard rules of propositional logic (i.e. can be compressed into them). If you just know those, you can recover any theorem or statement of LA. But this is not how human understanding operates at all. Usually one understands LA when one has memorized many different theorems of LA, have perhaps memorized how to derive some from others, or even the same theorem in several different ways. That doesn't even begin to cover establishing links between LA and other branches of mathematics, all of which a proficient mathematician memorizes individually, and memorizes connections between the links. The mathematicians mental knowledge graph is compressed, but not that much.
  - hyperman1 7y ago
    That's a fair point and an interesting view. I would consider the axioms as necessary but incomplete knowledge. You need more knowledge to apply these axioms. A bit like the axioms are the parts of a car. You can use the car schematic and the parts to create a car. Or you can generate the missing knowledge of the schematic by spending lots of time combining random parts. I am a bit out of my depth here, but maths can be reformulated. I presume like Minkowski did with Einstein's relativity. Or Riemann with integration. The result of the reorganized knowledge is deeper understanding. There is both compression and correction going on, and even correction can be seen as compression by removal of special cases.

erikpukinskis 7y ago

Wouldn’t more compressed graphs generally be faster to traverse?

abdullahkhalids 7y ago
Depends on how you compress them. Some representation are good for storing, others good for processing in different ways.
- TeMPOraL 7y ago
  Seems to match real-life thinking, too. For extreme case, see memorization: you learn a set of nodes in a graph, probably highly compressed through application of whatever tricks you use to quickly memorize and recall information. But by default, you learn almost no connections between those nodes, you can't really reason from the material you memorized. Building these connections requires further mental effort.

hyperman1 7y ago

I can think of multiple graphs representing the same knowledge, some smaller than others as irrelevant nodes are pruned, paths are shorter, etc.. The smaller ones encode that same knowledge with better understanding.

abdullahkhalids 7y ago
Let's consider understanding a mathematical theory - say linear algebra. In principle, the entirety of the knowledge of LA is contained in it's axioms and the standard rules of propositional logic (i.e. can be compressed into them). If you just know those, you can recover any theorem or statement of LA. But this is not how human understanding operates at all. Usually one understands LA when one has memorized many different theorems of LA, have perhaps memorized how to derive some from others, or even the same theorem in several different ways. That doesn't even begin to cover establishing links between LA and other branches of mathematics, all of which a proficient mathematician memorizes individually, and memorizes connections between the links. The mathematicians mental knowledge graph is compressed, but not that much.
- hyperman1 7y ago
  That's a fair point and an interesting view. I would consider the axioms as necessary but incomplete knowledge. You need more knowledge to apply these axioms. A bit like the axioms are the parts of a car. You can use the car schematic and the parts to create a car. Or you can generate the missing knowledge of the schematic by spending lots of time combining random parts. I am a bit out of my depth here, but maths can be reformulated. I presume like Minkowski did with Einstein's relativity. Or Riemann with integration. The result of the reorganized knowledge is deeper understanding. There is both compression and correction going on, and even correction can be seen as compression by removal of special cases.