The Depth of Knowledge

25 Jan 2020

"Knowledge of history is stored somewhere. There is a record of everything that took place. There are forces which keep those records in a form that can be interpereted by some consciousness: so that it can understand the events that took place." - Thomas Grothe (October 3 2019)

Knowledge’s breadth is greater that it’s depth.

It doesn’t take long to ask successive “whys” until you reach a limit of human knowledge. A child needs to ask you about 6 “whys" that depend on one another until you don’t know how to answer to her question.

“Why is the sky blue?”

“Because the particles in the air scatter blue light more than the other colors.”


“Because blue colors travel as short waves.”


After about 3 more “whys”, you’ll have to explain quantum theory and after that you’ll have to explain why quantum theory is the way it is; a question we don’t have an answer to. In Wikipedia, it takes at most 6 clicks to get from one article to another. In 6 degrees of separation, all people are at most 6 social connections away from each other; the “small world” observation (1).

An explanation for the observed shallowness of knowledge is Emergence; a phenomenon where a complex system has properties the parts that make it don’t have (2). Life emerges out of inanimate matter. Love is not just chemicals in your brain; it’s an emergent property of your environment and body. Sampling in Hip hop music illustrates how entirely new sounds can be created by copying pieces of pre-existing sounds. A lot of complexity can be created from simple rules to a point where it’s difficult or impossible to realize which part of a rule created which part of a system.

An abstract representation of a computer is simple. Called a Turing machine; It’s an infinitely long tape with a head that writes symbols on the tape and instructions on how to operate the machine. The complexity that can arise out of this is machine is tremendous. We can create write books, play games, socialize with people on it .. the list is endless. Since programming a Turing machine is tedious because we have to deal with every detail of the program, we define subroutines which we can use to build other subroutines which allows us to build complicated programs. This type of abstraction is how we are able to compound our knowledge because we don’t have to worry about the details that are irrelevant to the problem we are trying to solve. We couldn't make progress without abstractions because we would have to start from scratch every time.

Even though it's difficult to create knowledge, it's easy to verify it. It’s easy to recognize a great album or a piece of art but most people would be disoriented if they were tasked to create something similar. This is analogous to the P versus NP problem; where a computer can't solve some classes of problems efficiently but can efficiently determine that a solution is correct. The combination and thus complexity of work, skills, emotional states and other variables your favorite artist possesses that made him/her create music you like took exponentially more time than for you to appreciate that music. Emergence takes a lot of time. It's costly for complicated things to create simpler things; where the simpler things are easy to recognize.

Some computer scientists like Avi Wigderson believe that: “If any problem is solved by the brain, then it was an efficiently solvable problem” (3). This is misleading because it’s not a brain that solved it. It was lots of brains; and not just brains; bodies and environments took part since knowledge is not only passed through genetics, but through human language as well. The amount of knowledge that it took to solve that problem was compounding and abstracting since the beginning of the universe; and now that one brain got the right combination of everything that was designed through billions of years of evolution. Avi seems to be making a comparison of the human brain with Turing machines, but Turing machines start solving most of their problems from scratch. The programs that run human minds have been running for longer and have accumulated a lot of knowledge to be able to solve problems some classes of problems faster than computers. The spontaneous order of emergence is the cause for the sporadic instances of creativity. Once an efficient algorithm is found for a class of problems, then it is easy to solve those problems because there is a formula. It took billions of years for the first airplane to be created. Now an airplane can be built in less than a week.

One major hindrance to coming up with new ideas is different fields having different cultures which have different notations and terminology. To create ideas that are deemed “original”, one has to spend a lot of time internalizing different representations of different ideas enough to merge them to a point where the combination makes sense. The more disparate the notation for the researcher, the harder it is to cause different ideas to have a happy marriage. Since knowledge is created at an exponential rate, an effort to abstract different classes knowledge into the same language is important because it's less cognitive load for the person who’s trying to create knowledge.

Category theory is a general theory of functions that studies abstract relationships between functions (4). It’s recently been able to connect dissimilar areas like biology, computer science, electrical engineering, topology and chemistry under a common language. This and other attempts to get seemingly different fields under the same mathematical theory will be good for knowledge creation because it will dampen specialization; it will be easier for a biologist to take her idea and apply it to an electrical engineering problem without learning the details of circuit theory.

Complex systems theory; which study how different systems interact with each other is meant to address questions of emergence and abstraction. It can be formalized by the mathematics of automata. Realizing these automata in the physical world is hard because we have to deal with the laws of physics. Like the Turing machine which won’t be able to solve arbitrarily hard classes of problems efficiently, the universal constructor won’t be able to make arbitrary things efficiently. It will take decades for us to build a constructor that constructs things we want. There are many more ways to take advantage of how easily things connect with each other; of the shallowness of knowledge

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(1) Six degrees of separation

(2) Emergence

(3) Wigderson, Avi “Knowledge, creativity, and P VS NP”, 2009.

(4) Category theory

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