Measurement is an essential part of math. Quantities encountered in the real world are usually one of two things: a count or a measurement. Counting is what we learn first about math, before doing anything else with numbers. Counting is a way of measuring a quantity of discrete objects, but it is different from other…
Category: Math
The paradox of incentive
If something is incentivized too strongly, it becomes disincentivized. Extrinsic motivation has important social utility. It can both drive prosocial behavior and inhibit antisocial behavior. In many cases, extrinsic rewards are natural. Praise is a natural response to someone doing something we want them to do, for example, as is expressing anger towards someone for…
On learning by mimicry
Consider sports, music, and foreign language. Most people grow up being exposed to what these things are, and in particular often observe experts in action: professional athletes and musicians and native speakers of foreign languages. People also often have opportunities to engage in learning these things from a young age, and their education or training…
Feelin’ the vibes (literally)
Has this ever happened to you? You’re driving your car and pull up to a stoplight, and another car pulls up beside you absolutely blasting the bass from their sound system. That’s got to be an aftermarket subwoofer, you think to yourself. Even though the sound is muffled and you can’t hear a melody, the…
Ways of scrambling a message: Permutation polynomials over the integers mod n through the lens of monoalphabetic substitution ciphers
When Julius Caesar was on his military campaign in Gaul, he sent coded messages by courier back to Rome using a simple cipher. The so-called Caesar cipher is a letter shift by three places. In other words, every instance of the letter A gets replaced with the letter D, B gets replaced with E, C…
I wrote a poem: “A first research experience in mathematics”
A dark fog surrounds me.I see that something is there,but I can’t make it out.I squint and the edges begin to sharpen,but then are lost again to the darkness.My eyes are beginning to adjust now.For a moment I seem to recognize the shape,but then it is gone again.I reach out for it.It’s too far away.As…
How do you graph something in 4D?
This post is a brief explanation for non-experts of a graphing technique you might have seen but not completely understood. As always, we’ll go over some necessary concepts first. Complex numbers If we want to solve an equation like x2+1=0, we get stuck because we need to take the square root of -1, but no…
Why do unlikely things happen?
I find that people often seem to have trouble understanding how an unlikely event can occur. I have heard people ascribe supernatural causes to such events, and I have also heard people express incredulity at the event occurring at all. This is one of many ways in which probability can be counterintuitive. We usually expect…
(A little bit of) the math of AI
Artificial intelligence is a vague term for computer applications that carry out some kind of decision making. In general, we’re more inclined to call something AI the more sophisticated and independent its decision making capabilities are. AI are usually designed for a single task or a small collection of related tasks. Many tasks we would…
