Category: Engagement

Predicate Logic and Popular Culture (Part 279): My Chemical Romance

Let p be the statement “It looks like I’m laughing,” and let q be the statement “I’m really just asking to leave this alone.” Translate the logical statement

p \Longrightarrow q

This matches a line from “The Sharpest Lives” by My Chemical Romance.

Context: Part of a discrete mathematics course includes an introduction to predicate and propositional logic for our math majors. As you can probably guess from their names, students tend to think these concepts are dry and uninteresting even though they’re very important for their development as math majors.

In an effort to making these topics more appealing, I spent a few days mining the depths of popular culture in a (likely futile) attempt to make these ideas more interesting to my students. In this series, I’d like to share what I found. Naturally, the sources that I found have varying levels of complexity, which is appropriate for students who are first learning prepositional and predicate logic.

When I actually presented these in class, I either presented the logical statement and had my class guess the statement in actual English, or I gave my students the famous quote and them translate it into predicate logic. However, for the purposes of this series, I’ll just present the statement in predicate logic first.

Predicate Logic and Popular Culture (Part 278): The Dark Knight

Let M be the set of all men, and let W(x) be the statement “x likes to watch the world burn.” Translate the logical statement

exists x \in M (W(x))

This matches a line from Alfred Pennyworth in the 2008 film “The Dark Knight.”

Context: Part of a discrete mathematics course includes an introduction to predicate and propositional logic for our math majors. As you can probably guess from their names, students tend to think these concepts are dry and uninteresting even though they’re very important for their development as math majors.

In an effort to making these topics more appealing, I spent a few days mining the depths of popular culture in a (likely futile) attempt to make these ideas more interesting to my students. In this series, I’d like to share what I found. Naturally, the sources that I found have varying levels of complexity, which is appropriate for students who are first learning prepositional and predicate logic.

When I actually presented these in class, I either presented the logical statement and had my class guess the statement in actual English, or I gave my students the famous quote and them translate it into predicate logic. However, for the purposes of this series, I’ll just present the statement in predicate logic first.

Polynomial Long Division and Megan Moroney

A brief clip from Megan Moroney’s video “I’m Not Pretty” correctly uses polynomial long division to establish that 2x+3 is a factor of 2x^4+5x^3+7x^2+16x+15. Even more amazingly, the fact that the remainder is 0 actually fits artistically with the video.

And while I have her music on my mind, I can’t resist sharing her masterpiece “Tennessee Orange” and its playful commentary on the passion of college football fans.

Predicate Logic and Popular Culture (Part 277): Kellie Pickler

Let T be the set of all times, and let G(t) measure how good day t is. Translate the logical statement

\exists t_1 < 0 \exists t_2 < 0 \forall t \in T ( (t \ne t_1 \land t \ne t_2) \Longrightarrow (G(t) < G(t_1) \land G(t) < G(t_2)),

where time 0 is today.

This matches the chorus of “Best Days of Your Life” by Kellie Pickler, co-written by and featuring Taylor Swift.

Context: Part of a discrete mathematics course includes an introduction to predicate and propositional logic for our math majors. As you can probably guess from their names, students tend to think these concepts are dry and uninteresting even though they’re very important for their development as math majors.

In an effort to making these topics more appealing, I spent a few days mining the depths of popular culture in a (likely futile) attempt to make these ideas more interesting to my students. In this series, I’d like to share what I found. Naturally, the sources that I found have varying levels of complexity, which is appropriate for students who are first learning prepositional and predicate logic.

When I actually presented these in class, I either presented the logical statement and had my class guess the statement in actual English, or I gave my students the famous quote and them translate it into predicate logic. However, for the purposes of this series, I’ll just present the statement in predicate logic first.

Predicate Logic and Popular Culture (Part 276): Heart

Let T be the set of all times, and let G(t) be the statement “I got by on my own at time t.” Translate the logical statement

\forall t \in T ( ((t<0) \longrightarrow G(t) ) \land (t \ge 0) \longrightarrow \sim G(t)),

where time 0 is today.

This matches the opening line of the fabulous power ballad “Alone” by Heart.

And while I’ve got this song in mind, here’s the breakout performance by a young unknown Carrie Underwood on American Idol.

Context: Part of a discrete mathematics course includes an introduction to predicate and propositional logic for our math majors. As you can probably guess from their names, students tend to think these concepts are dry and uninteresting even though they’re very important for their development as math majors.

In an effort to making these topics more appealing, I spent a few days mining the depths of popular culture in a (likely futile) attempt to make these ideas more interesting to my students. In this series, I’d like to share what I found. Naturally, the sources that I found have varying levels of complexity, which is appropriate for students who are first learning prepositional and predicate logic.

When I actually presented these in class, I either presented the logical statement and had my class guess the statement in actual English, or I gave my students the famous quote and them translate it into predicate logic. However, for the purposes of this series, I’ll just present the statement in predicate logic first.

My Mathematical Magic Show: Index

I’m doing something that I should have done a long time ago: collecting a series of posts into one single post. The links below show the mathematical magic show that I’ll perform from time to time.

Part 1: Introduction.

Part 2a, Part 2b, and Part 2c: The 1089 trick.

Part 3a, Part 3b, and Part 3c: A geometric magic trick.

Part 4a: Part 4b, Part 4c, and Part 4d: A trick using binary numbers.

Part 5a, Part 5b, Part 5c, and Part 5d: A trick using the rule for checking if a number is a multiple of 9.

Part 7: The Fitch-Cheney card trick, which is perhaps the slickest mathematical card trick ever devised.

Part 8a, Part 8b, and Part 8c: A trick using Pascal’s triangle.

Part 9: Mentally computing n given n^5 if 10 \le n \le 99.

Part 10: A mathematical optical illusion.

Part 11: The 27-card trick, which requires representing numbers in base 3.

Part 6: The Grand Finale.

And, for the sake of completeness, here’s a picture of me just before I performed an abbreviated version of this show for UNT’s Preview Day for high school students thinking about enrolling at my university.

magician

Mathematical Allusions in Shantaram (Part 4)

I recently finished the novel Shantaram, by Gregory David Roberts. As I’m not a professional book reviewer, let me instead quote from the Amazon review:

Crime and punishment, passion and loyalty, betrayal and redemption are only a few of the ingredients in Shantaram, a massive, over-the-top, mostly autobiographical novel. Shantaram is the name given Mr. Lindsay, or Linbaba, the larger-than-life hero. It means “man of God’s peace,” which is what the Indian people know of Lin. What they do not know is that prior to his arrival in Bombay he escaped from an Australian prison where he had begun serving a 19-year sentence. He served two years and leaped over the wall. He was imprisoned for a string of armed robberies performed to support his heroin addiction, which started when his marriage fell apart and he lost custody of his daughter. All of that is enough for several lifetimes, but for Greg Roberts, that’s only the beginning.

He arrives in Bombay with little money, an assumed name, false papers, an untellable past, and no plans for the future. Fortunately, he meets Prabaker right away, a sweet, smiling man who is a street guide. He takes to Lin immediately, eventually introducing him to his home village, where they end up living for six months. When they return to Bombay, they take up residence in a sprawling illegal slum of 25,000 people and Linbaba becomes the resident “doctor.” With a prison knowledge of first aid and whatever medicines he can cadge from doing trades with the local Mafia, he sets up a practice and is regarded as heaven-sent by these poor people who have nothing but illness, rat bites, dysentery, and anemia. He also meets Karla, an enigmatic Swiss-American woman, with whom he falls in love. Theirs is a complicated relationship, and Karla’s connections are murky from the outset.

While it was a cracking good read, what struck me particularly were the surprising mathematical allusions that the author used throughout the novel. In this mini-series, I’d like to explore the ones that I found.

In this fourth and final installment, the narrator has a lengthy conversation with his mentor (a mafia don) about his mentor’s philosophy of life.

[The mafia don said,] “I will use the analogy of the way we measure length, because it is very relevant to our time. You will agree, I think, that there is a need to define a common measure of length, yes?”

“You mean, in yards and metres, and like that?”

“Precisely. If we have no commonly agreed criterion for measuring length, we will never agree about how much land is yours, and how much is mine, or how to cut lengths of wood when we build a house. There would be chaos. We would fight over the land, and the houses would fall down. Throughout history, we have always tried to agree on a common way to measure length. Are you with me, once more, on this little journey of the mind?”

“I’m still with you,” I replied, laughing, and wondering where the mafia don’s argument was taking me.

“Well, after the revolution in France, the scientists and government officials decided to put some sense into the system of measuring and weighing things. They introduced a decimal system based on a unit of length that they called the metre, from the Greek word metron, which has the meaning of a measure.”

“Okay…”

“And the first way they decided to measure the length of a metre was to make it one ten-millionth of the distance between the equator and the North Pole. But their calculations were based on the idea that the Earth was a perfect sphere, and the Earth, as we now know, is not a perfect sphere. They had to abandon that way of measuring a metre, and they decided, instead, to call it the distance between two very fine lines on a bar of platinum-iridium alloy.”

“Platinum…”

“Iridium. Yes. But platinum-iridium alloy bars decay and shrink, very slowly — even though they are very hard — and the unit of measure was constantly changing. In more recent times, scientists realised that the platinum-iridium bar they had been using as a measure would be a very different size in, say, a thousand years, than it is today.”

“And… that was a problem?”

“Not for the building of houses and bridges,” [the mafia don] said, taking my point more seriously than I’d intended it to be.

“But not nearly accurate enough for the scientists,” I offered, more soberly.

“No. They wanted an unchanging criterion again which to measure all other things. And after a few other attempts, using different techniques, the international standard for a metre was fixed, only last year, as the distance that a photon of light travels in a vacuum during, roughly, one three-hundred-thousandth of a second. Now, of course, this begs the question of how it came to be that a second is agreed upon as a measure of time. It is an equally fascinating story — I can tell it to you, if you would like, before we continue with the point about the metre?”

“I’m… happy to stay with the metre right now,” I demurred, laughing again in spite of myself.

“Very well. I think that you can see my point here — we avoid chaos, in building houses and dividing land and so forth, by having an agreed standard for the measure of a unit of length. We call it a metre and, after many attempts, we decide upon a way to establish the length of that basic unit.”

Shantaram, Chapter 23

After this back-and-forth, the mafia don then described how his philosophy of life can be likened to the need to redefine a basic unit, like the meter, based on our ability to make more accurate measurements with the passage of time.

For the purposes of this blog post, I won’t go into the worldview of a fictional mafia don, but I will discuss the history of the meter, which is accurately described in the above conversation. The definition of the meter has indeed changed over the years with our ability to measure things more accurately.

Initially, in the aftermath of the French revolution, the meter was defined so that the distance between the North Pole and the equator along the longitude through Paris would be exactly 10,000 kilometers. (Since that distance is a quarter-circle, the circumference of the Earth is approximately 40,000 kilometers.)

Later, in 1889, the meter was defined as the length of a certain prototype made of platinum and iridium.

In 1960, the meter was redefined in terms of the wavelength of a certain type of radiation from the krypton-86 atom.

In 1983, the meter was redefined so that the speed of light would be exactly 299,792,458 meters per second. (Incidentally, after 1967, a second was defined to be 9,192,631,770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the cesium 133 atom.) Regarding the novel, the above conversation happened in 1984, one year after the meter’s new definition.

These definitions of the meter and second were reiterated in the latest standards, which were released in 2018. This latest revision finally defined the kilogram without the need of a physical prototype.