TOK RESOURCE.ORG - 2024

AREAS OF KNOWLEDGE:
MATHEMATICS
Why is ethics like math and not like math?

A preposterous question!

At the very beginning of a class in the fall of 2022 some of my students seemed quite agitated. They had just come from IB History class, and were continuing a heated argument about “moral equivalency.” It was in the context of the Cold War. A bold, and previously untried, tactic immediately flew into my head, so I decided to defer my planned activity.

With as much gravitas as I could muster, I wrote “Why is ethics like math?” on the whiteboard. Students were aghast. This was my “Surely you are joking, Mr Brown” moment. I might as well have written “Why is ethics like a banana?” on the board.

Feeling the resistance in the room my first move was to adjust the question to “Why is ethics like math, and not like math?” Then, with some stubborn coaxing from me to break the silence, we jumped into some informal brainstorming. It was hesitant at first, but quickly cascaded. The class activities on this page evolved from these serendipitous, authentic moments.

This class activity works best with fearless open-minded TOK students in their second year. It ends with a poetic, mind-boggling flourish. It could work as a concluding TOK class, after Essays have been completed and uploaded.

Breaking the 4th wall

At first glance “Why is ethics like math, and not like math?” seems like a foolish question. In reality, it is nuanced and fairly accessible. It has been hiding in plain sight all along. The reason is simple. Students have already acquired a high degree of social and political sophistication, infused with some strong ethical intuitions. Some are passionate about issues of global justice large and small. TOK students will have already grappled with Apprenticeship in ethics earlier in the course.

Students are already in the thick of the mathematical action, after more than a decade of formal math classes in school. An important aspect is that they have been immersed in the convention of “show your working.” Stated more precisely, and when performed elegantly, this is “line-by-line communication through the problem, using defensible logic.”

And there you have it. The bottom line. Whether you like it, or not. Whether you are good at it, or not. Math has a cold, unforgiving, abstract beauty that is—even if your own efforts are not—always right!

Now back to our question—comparing math and ethics! Are they two, parallel, incommensurable worlds, or will we encounter meaningful commonalities? Asking, say, “Why is history like the natural sciences, and not like the natural sciences” would have been a more straightforward knowledge question because history and science are both overtly rooted in the messy, real-world. In the History is not what happened unit, students wrestled with the brute fact that history is gone, and cannot repeat itself; precluding controlled experiments. However, there was comfort in identifying the commonalities—both are evidenced-based and seek true explanations.

class ACTIVITY I — BRAINSTORM!

Brainstorming is well named. It can be magical and fun. I prefer not to overuse it—to retain its novelty value; and to ensure that my TOK teaching does not resemble touchy-feely, corporate management training.

Before laying down the ground rules for brainstorming, take a moment to revisit the Establishing common agreements session in Starting TOK. We are all responsible for content in TOK classes. “If you have something to contribute there is a moral imperative to do so” is a key quote from the consensual agreements. This sentiment is critical for any authentic brainstorming session. Here are some other guidelines:

One voice at a time
Keep it succinct
Mostly stay on topic
Allow enough time for the scribes to record the ideas
There is no wrong or silly contribution
The more ideas the better
Be controversial
Don't worry about duplicate ideas
Play devil’s advocate
Modify the question if you like
Feel free to contradict yourself
Moments of awkward silence are normal
Marginal or bizarre ideas may be some of the very best

READY, FIRE… AIM!

Do not reveal the question (or my “breaking the 4th wall” musings above) in advance and have your timer ready. Appoint two student scribes. My preference is brainstorming for precisely 4 minutes. Tell students that things might get chaotic. Their quest is a simple one, but it is challenging. They must trust their smarts, and their lightning fast intuitions, as they venture into unmapped territory.

“Get ready to reveal your spontaneous thinking on:
Why is ethics like math, and not like math?
You have four earth minutes!”

The following generative questions may, or may not, be necessary for ensuring lively class discussion and subsequent collective refinement of the inscriptions on both sides of the chart.

• Is the pure vs. applied math distinction relevant here?

• Are physics and economics (and ethics) just applied math?

• Are statistics, computer programming and AI subsets of math? How, if at all, do they play into ethics and morality?

class ACTIVITY II — PROVE IT!

The first of the follow up activities after the brainstorm is quick and easy. Students watch this derivation of the quadratic formula video and respond informally to the following generative questions:

• What does “derive” mean in the realm of pure mathematics?

• To what extent did the presenter successfully provide line-by-line communication through the problem, using defensible logic?

• What is your own relationship with the quadratic formula?

• Are trolley problems an intersection of ethics and math?

class ACTIVITY III —
INVENTING MATH THOught experiments

In order to further reinforce the notion of the axiomatic nature of pure math, we will revisit Imagining geometry—a thought experiment!

In the Imagining Geometry thought experiment we started from a single, idealized dot. From this self-evident, foundational point we invented, and defined, simple geometric objects, and some emergent relationships between them. We also concocted one, then two, then three dimensions! If we had kept going, and worked more systematically, recording our thoughts on paper; in principle we could have invented the axioms of Euclid, and eventually, perhaps, the entire edifice of geometry!  

Reprise the thought experiment, this time led by a student volunteer with theatrical reading skills.

Next arrange students in pairs. Challenge them to:

• Formulate and write a rough draft of their own thought experiment that will enable participants to imagine counting numbers (integers) and then use them to construct the laws of arithmetic.
  
  Allow a full 15 minutes. They should begin with a single imaginary dot. Then add another dot. They could initially place their dots equidistant on an imaginary number line and/or arrange them in patterns.
  
  🍎🍎🍎🍎🍎🍎
  and/or

  🍎🍎🍎
  🍎🍎🍎

For fun, students could imagine identical emojis instead of dots. After leaking this advice, tell that them “you are now on your own.” In their thought experiments they should attempt to invent/discover:

• Addition and subtraction on a number line

• Zero

• Negative numbers

• Multiplication

• Division

Inventing the concept of division could also open up imaging fractions, square roots—even imaginary numbers! But let’s not get too ambitious. Remind the pairs to end their draft with “Now, open your eyes—welcome home to the real world!” When the dust clears, invite some of the pairs to talk about their drafts, even if they only had time to produce some interesting bullet points.

⇨OPTIONAL extra fun⇩

Ask students to sit in silence, close their eyes, and imagine an empty space. A perfect, infinite, timeless, silent, empty space. Now imagine a straight, flat, horizon extending across that still empty space like the surface of a giant ocean or desert meeting the sky. Now imagine a single rock, suddenly falling from a great height in a straight line, impelled by the force of gravity. Observe it, in your minds eye, zipping vertically and violently through your horizontal horizon.

Congratulations you just invented/discovered the perpendicular x and y axes for coordinate geometry! Now, open your eyes—welcome back to the real world!

• Are the fundamentals of mathematics rooted in the natural world? What are some of the implications of this?

• How could we derive algebra by thought experiment? Try it if you dare!

class ACTIVITY IV —
attempting a calculus of felicity

Moving now from the abstract, into the inherent messiness and fallibility of the human arena, we will briefly engage with the “felicific calculus.” This curiously named algorithm was formulated with cheerful confidence by utilitarian philosopher and social reformer, Jeremy Bentham. It purports to calculate the degree of pleasure or pain that a specific action causes using seven defined variables.

1. Intensity—How strong is the pleasure?
2. Duration—How long will the pleasure last?
3. Certainty—How likely or unlikely is it that the pleasure will occur?
4. Propinquity—How soon will the pleasure occur?
5. Fecundity—The probability that the action will be followed by pleasurable sensations of the same kind.
6. Purity—The probability that it will not be followed by sensations of the opposite kind.
7. Extent—How many sentient beings will be affected?

Bentham also provided this short poem to help his readers to memorize his scheme.

how to proceed

Start this class activity by showing the Calvin and Hobbes cartoon as an icebreaker. Then introduce Jeremy Bentham. Mention his fame as an ahead-of-his-time, fearless reformist for abolishing slavery, the death penalty, and child corporal punishment. He was also advocated for women’s suffrage, decriminalizing homosexual acts, and not banishing Brit criminals to Australia!

Now let’s dive into the calculus of felicity. After a quick silent reading, a student volunteer should read aloud the Bentham doggerel, milking the clever, but not very elegant, rhymes. Next, have the same student read each of the seven single word variables; one at a time, each time pointing to random classmate, who responds by cheerfully reading Bentham’s corresponding diagnostic question.

DISCUSSION QUESTION

Jeremy Bentham lived a life of public service and activism. He was a virtuous fellow who clearly wanted to make the world a better place for the good of all. He was a consequentialist—believing in the utilitarian doctrine of “the greatest happiness for the greatest number.”

• To what extent is it possible to measure happiness meaningfully and effectively using the Bentham’s felicific calculus?

class ACTIVITY V —
the Golden mean is not THE arithmetic mean

We encountered Aristotle’s “golden mean” in Virtue ethics. Aristotle didn’t call it that, but the name persisted throughout the ages. In this activity students will first perform silent close readings of two famous passages from Aristotle’s Nicomachean Ethics (335-322 BCE). After the readings they will share their thoughts and respond to the two generative questions in randomly assigned groups of three.

Silent readings

In NE: p.94 Aristotle defines his “cardinal rule”:

Right conduct is incompatible with excess or deficiency in feelings and actions…

He continues by detailing his “doctrine of the mean”:

The man who shuns and fears everything and stands up to nothing becomes a coward; the man who is afraid of nothing at all, but marches up to every danger, becomes foolhardy, Similarly the man who indulges in every pleasure and refrains from none becomes licentious; but if a man behaves like a boor and turns his back on every pleasure, he is a case of insensibility. Thus temperance and courage are destroyed by excess and deficiency and preserved by the mean. 

A few pages later, in NE: pp.100-102, he offers two important qualifications:

…If ten is "many' and two 'few' of some quantity, six is the mean… by arithmetical reckoning. But the mean in relation to us is not to be obtained in this way. Supposing that ten pounds of food is a large and two pounds a small allowance for an athlete, it does not follow that the trainer will prescribe six pounds; for even this is perhaps too much or too little for the person who is to receive it… 

It is possible, for example, to feel fear, confidence, desire, anger, pity, and pleasure and pain generally, too much or too little; and both of these are wrong. But to have these feelings at the right times on the right grounds towards the right people for the right motive and in the right way is to feel them to an intermediate, that is to the best, degree; and this is the mark of virtue…

But not every action or feeling admits of a mean; because some have names that directly connote depravity, such as malice, shamelessness and envy, and among actions adultery, theft and murder. All these, and more like them, are so called as being evil in themselves; it is not the excess or deficiency of them that is evil. 

Generative questions

• We read that Aristotle’s doctrine of the mean is not the exact arithmetic mean that we calculate in math class; how would you characterize the mark of virtue that he offers instead?

• To what extent are virtue ethics and utilitarianism like math and not like math?

class ACTIVITY VI —
a self-evident, intuitive,
AXIOMATIC FOUNDATION FOR ALL OF ETHICS?

REVISITING CULTURAL RELATIVISM

If the class has not already done so, complete the Exorcising cultural relativism activity (located in the Knowledge and Indigenous Societies optional theme). In this unit we learn that a moral relativist is:

…anyone who rejects the view that moral rules and principles are absolute and universal, applying to all persons, in all places, and at all times.

Which is not the same as taking the crude position that “anything goes.” Outright subjectivism, or extreme cultural relativism, is very quickly exorcized by, for example, consideration of the nationalist and totalitarian genocides of recent history.

In the relativism class activity students encountered extreme, historic, culturally-specific customs like foot-binding, female genital mutilation, honor killing, and sati—the Hindu rite of the widow throw herself on the husband's funeral pyre! These are all extreme cases. But the fact remains that, all around the world, there are competing incompatible, culturally-embedded moral positions that collide with each other and remain irreconcilable.

a one sentence foundation

What if—and this is “a monumental if”—it might be possible to pinpoint an ethical position that is absolute and universal, applying to all persons, in all places, and at all times?

Announce to the class that—as crazy and hubristic and ambitious, as it seems—we are going to attempt this today! The task is huge, but we are going to adopt a down-to-earth approach. Here it is:

1. Take a few of these Post-It Notes

2. Working solo and in silence, write a draft single sentence on your Post-It that encapsulates an objective, axiomatic foundation for the whole of ethics!
   (Only a suggestion: include the word “flourishing”)

3. Post your Post-It with all the other Post-Its on the whiteboard.

Of course this is thinly disguised, workshop brainstorming again. Allow a timed two minutes. When the dust clears invite students to informally walk up to the whiteboard and do a close reading of all the drafts. When the the students have all returned to their seats again, ask them:

• What are your thoughts? What just happened?

Next merge the pairs to contrive groups of 4 (with some groups of 3, if the class size is uneven). Have some half-letter-sized paper ready so that the newly formed quartets and trios can refine their drafts collectively. Their new drafts will build on their own first attempts, having now seen the first efforts of the entire class, and, perhaps, some out of-the-box thinking! Allow another timed 2 minutes.

Finally, tell the groups to turn over their papers. Ask them to redraft one last time, after considering the multifaceted generative question below. Mention that “considering the question” does not mean you have to include every element. Indeed, you are free to reject it in its entirety!

Did your single sentence, axiomatic foundation for the whole of ethics encompass:

• The notion of “flourishing,” “well-being,” or an equivalent sentiment?

• All of current humanity"?

• Future generations of humanity?

• Other sentient creatures?

• Conscious alien lifeforms?

• Self-aware AI (if it eventually emerges)?

• Other category—fill in the blank?
  

  
  DISTILLATION AND consensus

At this point, the trios and quartets are seated with their third iterations of “a single sentence, axiomatic foundation for the whole of ethics.” We are almost there!

Next, invite a public reading of each draft by a volunteer spokesperson. Allow appreciation, but hold off on comments for now. Next, repeat the readings, this time with a volunteer scribe from each group writing the draft on the whiteboard.

Tell the scribes that they have been “chosen” to collaborate on one final task. They must distill the third drafts down to a single, definitive, consensual, grammatical sentence!

The scribes can solicit feedback from their class mates as they fine-edit. The teacher should stand back, stay quiet, and see what happens.

Interlude—HERE WE ARE!

Producing a tentative one sentence foundation for the whole of ethics was no easy task. Take a well deserved five minute break, preferably walking around outside the classroom (in the open air if possible).

Before unleashing the final activity, reestablish a mathematical ambiance with this Mandelbrot Zoom sequence video.

GOING DEEPER—SUMMATIVE CONVERSATIONS

According to Wikipedia:

An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. In Ancient Greek it referred to “that which is thought worthy or fit” or “that which commends itself as evident.

Read this definition aloud and display it as a starting point for a full class, “summative” discussion about our quest to formulate a self-evident, intuitive, axiomatic, single sentence foundation for all of ethics. Structure the conversation using some, or all, of the smorgasbord of Knowledge Questions below.

GENERATIVE QUESTIONS

• Was writing an axiomatic foundation for ethics helped or hindered by the single sentence constraint?
  

• Would it be possible to resolve a complex ethical problem, taking our axiomatic foundation for ethics as the starting point, and then communicating through the its intricacies systematically—using line-by-line, defensible logic?
  

• Does our axiomatic foundation for ethics hold up as a Kantian Categorical Imperative?
  

• Is our axiom for all of ethics rock bottom and true—as arithmetic and Euclidean geometry are in mathematics?

• Did our our axiomatic foundation for ethics feel like it was invented or discovered?
  

• In a real-life ethical situation can we use our axiomatic foundation as is, or must we use our judgment to act on it “at the right times, on the right grounds, towards the right people, for the right motive, and in the right way”?

bonus question

• Is a rock solid foundation for ethics a best guess; or something immutable, eternal, perfect and abstract (like the Platonic view of pure mathematics)?