4 ms·

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 o

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.

4 ms·

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 o

by hyperman1 7y ago

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.