AREAS OF KNOWLEDGE:
MATHEMATICS
IDEAL GAS LAW COMPARED TO THE EULER RELATION
The Euler relation is often shown to students of Higher Level mathematics as a kind of pinnacle for the initiated. The actual derivation is university level mathematics, but most TOK students will have acquired enough mathematics to have an inkling of why the equation is so surprising and charismatic.
The generic questions will have initiated class discussion. Typically, students will be quick to identify the fundamental mathematical objects: 1, 0, π, i and e, and will admire how they―“magically”―come together. This is all the more astonishing when we consider that π and e are both "irrational" numbers; and i is "imaginary"!
CLASS ACTIVITY: PURE AND APPLIED MATHEMATICS
Students will be alternating between collaborative work in pairs and reporting back to generate class discussion. They will tackle series of Knowledge Questions that should build to a counterintuitive and discombobulating position. Engineer the pairings to include a strong mathematician and/or chemistry or physics student in each.
Provide a strictly timed 5 minutes for students to grapple with the following Knowledge Questions:
What are the essential differences between equations like the Euler formulation and the applied mathematics of physics and engineering?
Based on your own mathematical education so far, do you think that Euler invented or discovered his equation?
It has often been said that the Euler relation is the most beautiful equation in all of mathematics. What is meant by beauty and elegance in mathematics?
Solicit volunteers to report back, in order to generate some whole class discussion. Before the conversation peaks project the following image:
Then, abruptly send students back to their pairs for more collaborative discourse. This set of questions will merits a full 12 minutes. Tell students that the first 6 minutes are for discussion purposes. Raise the bar by telling them that the second 6 minutes is to write solo a one paragraph reflection on questions 5, 6 and 7 combined. This will be collected and graded.
Tell students that question 4 is just for fun, and is for "hard science geeks" only!
1. What are the similarities and differences between these two equations?
2. Are both equations examples of pure mathematics? Are both firmly in the domain of abstract, analytical knowledge? Do they both convey certainty?
3. The first equation is called Euler's relation and the second equation is known as the Ideal Gas Law. Does knowing the names of the equations change your responses to the previous two questions?
4. For the Ideal Gas Law what do each of the symbols represent? Why do physicists use lower case letters for p and n? What are the SI units for each of the components?
5. What is “ideal” about the Ideal Gas Law?
6. When chemists do real world experiments based on the Ideal Gas Law, actual results come close to predicted results, but they are never identical. Why?
7. What has this got to do with the idea that “The Map is Not the Territory”?
Ensure that students are given a one minute warning before the writing portion begins. When the activity ends and the paragraphs are collected unleash this final image. This time address the questions as a whole class.
Now things have escalated. We have placed a couplet from William Blake’s great poem, Auguries of Innocence (1863) alongside the Ideal Gas Law the Euler Relation. No doubt students will recognize the Blake lines from the Allegory of the Cave: Truman Show unit from the beginning of the course.
Can we agree that the Blake couplet has much more in common with the Ideal Gas Law than the Ideal Gas Law does with the Euler Relation (despite the superficial resemblance of being expressed in mathematical symbols)? Whether you agree or disagree, you must argue your case.
Can we say that the Ideal Gas Law, the Krebs' Cycle, the definition of Osmosis, the Law of Supply and Demand and, for that matter, all other models in the natural and human sciences, have more in common with a Shakespeare soliloquy or a Rembrandt self-portrait than they do with the Euler Relation, the Theorem of Pythagoras or the Sine Rule?
Printable pdf. of the equations, the Blake quote and questions
“A Robin Redbreast in a Cage
Puts all Heaven in a Rage.
[L]et no one tell you that this quotation is only a particular statement. It derives its general appeal to us all from its high specificity, and that is the miracle of this kind of remark; but it is a statement which says something about the human situation and not just about a robin or a cage.”
Rembrandt (1665-69) Self Portrait with two circles. Kenwood House, London.
Also see: The Rembrandt self portraits in the Arts as a Way of Knowing.
THE PATH FROM METAPHOR TO ALGORITHM
“When Newton saw the moon as a ball that had been thrown round the earth, he was initiating a gigantic metaphor. And when it finished up… it was an algorithm (a formula with which you can calculate).”
Newton's original thought experiment is a powerful metaphor... which he formalized to an algorithm that we now call "Newton's inverse square law of gravitation"... in formal prose the algorithm states that:
The gravitational attraction between two massive bodies is proportional to the product of those two masses and inversely proportional to the square of the distance between them:
Or, to put it even more succinctly, using only mathematical symbols:
By introducing the Gravitational Constant (G) we get an equation containing only mathematical symbols; looking just like equations from pure math. At this level the techniques and symbolic conventions of abstract math are being used as a tool for describing the real world.
Glistening Euler Identity (2023)—Oil painting by the author
Ideal Gas Law (2024) Oil painting by the author