Logic sometimes breeds monsters. Henri Poincaré In math, we often speak of things being “nice,” “well behaved,” “elegant,” or even “beautiful.” The flip side of this is those things in math which often remain in shadow: the unfriendly, the unruly, the awkward, and the ugly. In some cases, inelegant solutions are apparently unavoidable. In other…
Category: Math
Probability estimation vs probability calculation
If I take the bus downtown today, what are the chances I’ll run into my friend (let’s call her Alice)? Suppose I’m on the bus for an hour. According to the local metro, there are approximately 31 people using a single bus per hour of operation. Assume that during the hour I’m on the bus…
The language of balanced parentheses
Grammatically, parentheses must be balanced, meaning every open parenthesis ‘(‘ must eventually be followed by a close parenthesis ‘)’. Parentheses can be nested, meaning there can be parentheses inside parentheses. If we were to take a piece of writing and remove all letters and punctuation except parentheses, we might end up with something like: ()()()(())()(),…
How compression works and why JPEG artifacts look like that
Do I look like I know what a JPEG is? In a computer, all information is represented as bits, strings of zeros and ones. However, bits can more or less be interpreted as generic numerical data written in base 2 instead of as decimals. In other words, in a computer, all information is represented as…
The state of math in the world today
I went to the grocery store today and at the self checkout, one of my items was not recognized by the barcode scanner. It scanned successfully, but it wasn’t in the system and couldn’t give a price. An employee came over to help. He was young, no older than his 20s, though he looked like…
Equality, equity, equivalence, identity, indistinguishability, isomorphism, resemblance, congruence, and similarity
While the title of this post might as well be a thesaurus entry, my purpose here is to investigate the differences between these concepts and how they operate in different contexts. Mathematical equality The reader is almost certainly very familiar with the equals sign or = used in math. Many of us first encounter it…
I like nonsense the best
I have often said that my favorite way to sort objects is alphabetically by color name. For example, if your objects are colored red, yellow, green, blue, and purple, then you would put the blue one first, followed by green, then purple, red, and finally yellow. But why? Partly for humor. I think it’s funny…
The geometry of body plans
Let’s talk about evolution. All living organisms require a way to acquire and metabolize food, a way to respirate, and a way to reproduce. Many single-cell organisms like bacteria can eat by engulfing food within their cell, a process called phagocytosis. Algae and plants are autotrophic, meaning they produce their own food. Some simple animals,…
A plea to teach sig figs in math class
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…
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…
thoughts on mind 