Knowledge as a weapon
March 15, 2022
We have all heard that knowledge is power. Here are couple of simple maths example of how it can be wielded as a weapon by those unconcerned by ethics.
Although we don't usually consider this perspective, the knowledge of laws of nature gives you a superpower. That superpower can be used as a weapon against those who don't have that knowledge and rely more on intuition or common sense. That is because natural laws are often not intuitive. Here are some simple example to illustrate this point.
Example from Algebra
Here is a common number game children play to amuse and impress their friends. Think of a number, borrow the same amount from your friend and add it to your number, I give you 4 (some arbitrary number picked on the spot), throw away half of what you have, then return the amount you borrowed from your friend. I can tell how is left. It is 2 (it will always be half of what the puzzle presenter gave).
A number puzzle like the above can surprise common sense expectations of kids (and some adults). But underneath its really basic algebra. The puzzle presenter is making you do this algebraic calculation: (x + x + y)/2 - x. Now this expression will always evaluate to y/2. In algebraic language (x + x + y)/2 - x = y/2 is an Identity and there is no x term on the right side.
However, this simple algebraic fact can be used to give an impression that the puzzle presenter can read your thoughts.
Example from Probability
Probability can also be used as such. Consider some examples: (A) what is the probability that two dices thrown together have the same number? (B) what is the probability that two dices thrown together, first one has greater than second.
It is difficult for some untrained to guess the answers simply on intuition. Turns out the first answer is 1/6 - same as probability of getting a particular number but second is 5/12 slightly less than 1/2 and hence much greater than 1/6. The fact that the second answer is so much greater than the first is not obvious. This fact can be exploited to build a betting game since to common sense probabilities of question A and B look closer than they actually are.
Example from Algebra
Here is a common number game children play to amuse and impress their friends. Think of a number, borrow the same amount from your friend and add it to your number, I give you 4 (some arbitrary number picked on the spot), throw away half of what you have, then return the amount you borrowed from your friend. I can tell how is left. It is 2 (it will always be half of what the puzzle presenter gave).
A number puzzle like the above can surprise common sense expectations of kids (and some adults). But underneath its really basic algebra. The puzzle presenter is making you do this algebraic calculation: (x + x + y)/2 - x. Now this expression will always evaluate to y/2. In algebraic language (x + x + y)/2 - x = y/2 is an Identity and there is no x term on the right side.
However, this simple algebraic fact can be used to give an impression that the puzzle presenter can read your thoughts.
Example from Probability
Probability can also be used as such. Consider some examples: (A) what is the probability that two dices thrown together have the same number? (B) what is the probability that two dices thrown together, first one has greater than second.
It is difficult for some untrained to guess the answers simply on intuition. Turns out the first answer is 1/6 - same as probability of getting a particular number but second is 5/12 slightly less than 1/2 and hence much greater than 1/6. The fact that the second answer is so much greater than the first is not obvious. This fact can be exploited to build a betting game since to common sense probabilities of question A and B look closer than they actually are.