Never Forget Recursion

All problems are solved through recursion. Through proper recursion, one can see how to derive problems.

Lets say that one starts logic by suggesting that knowledge results from applying deduction to sensation. Through sensation, one is presented with axioms. Through deduction, one can use the axioms. Simple, right.

No. The problem arises when one develops a postulate relating how to develop postulates. This is the first recursion. This makes thinking hard. To make empirical judgments, one must use a method that is also based on empirical judgments. This produces a mathematical situation where probabilities are based off of probabilities, which in turn are also based off of other probabilities, ad infinitum.

Mathematics comes down to representation

There are a fair amount of difficulties with people learning some math at one point. I've certainly experienced it. It comes down to symbolism just like learning a vocabulary word. This is a topic to be explained in greater detail later on, but for now, lets look at it from a very rudimentary perspective. One instinctively knows that he can group objects into sets. However, in order to characterize the set he needs numbers. This idea, that one can take two objects to be identical in a certain respect, is the origin of all mathematics. So if one has a lot of objects and each object looks like an apple, tastes like an apple, smells like an apple, etc, he deems them all to be apples. He might perhaps, put the apples in a barrel. This is a set. Now how does he describe one barrel of apples as opposed to a different barrel of apples? Well, if he only cares about the number of apples he's got, he will surely describe them in terms of the quantity of apples. So, he invents numbers. He uses symbols to describe numbers(0,1,2,3,4,5,6,7,8,9,F'for full'). Then, he notices that each barrel of apples fills at ten apples. Then, for the eleventh apple, he needs another barrel. So, to keep track of how many apples he has, he should also keep track of how many full barrels he has. So, he can count full barrels with the same symbols that he counts apples(1,2,3,etc.) Here's the really interesting part that I consider the first great innovation in mathematics. The guy realizes that he can use the number of barrels he has to keep track of the number of apples he has! So if each barrel holds ten apples, and he has four barrels, each of them completely full, then he can say he has 4 full barrels. This represents forty apples. To make it easier to write, he decides to make a sequence of digits. The last digit at the right represents the number of apples in an unfilled barrel. The digit to the left of that represents the number of full barrels. Say he had two barrels. One barrel had ten apples and the other had zero. Rather than writing 0F, he can eliminate the F(for ten) digit altogether and make ten a two digit number, 10. The one representing the number of full barrels, and the zero representing how many apples are in the unfilled barrel.

There is a key assumption here that is related to the "Pigeonhole Principle." Since all of the apples go into the same barrel until it is full, only one barrel is ever allowed to be full.

That is the foundation for our entire society. Simple, yet important.