In his first proof-based mathematics course, a student asks, “Is the identity element in a group unique? It’s not one of the axioms.”
I respond, “It’s not one of the axioms, but you can prove it from them. Try.”
As far as I can tell, what starts happening in the student’s mind is something along the lines of figuring out why having multiple identities would not be useful, or persuading himself that the word “identity” connotes uniqueness already, or being mystified by how even such a thing could be proved.
What is definitely NOT happening is a search for a symbolic proof such as .
I’ve been playing a game called Slay The Spire, the lovechild of FTL: Faster than Light and Hearthstone’s Arena Mode. The goal is to win a series of randomly-generated card-based combats, where the reward of each combat is a choice of one of three cards to add to your deck.
The random and single-player nature of the game allows the designers to print some truly preposterous cards, such as the card “Seek: 0 mana, draw two cards of your choice from your deck.” Note that one of those cards can be Seek. Note that there is no limit to the number of copies of a given card you can have. Anyway, the absurdity of Seek is a story for another post.
I had the following series of thoughts the other day after yet another aborted run.
I wish there was a tier list for Slay the Spire cards.
I bet there is a tier list online.
Maybe I can make a tier list.
I spent the next couple hours copy pasting data into a spreadsheet and making up fake conversion rates from every card effect to damage numbers: “1 block is worth about 1.2 damage, 1 mana is worth about 6 damage, card draw about 3.” After all, if it’s worth doing, it’s worth doing with made-up numbers.
The resulting sheet was extraordinarily enlightening. I learned that several common cards are truly overpowered. I learned that a number of the flashy rare cards are truly abysmal – at least two are worth negative points in my system. This exercise didn’t feel like making magic math and receiving new wisdom from on high. It felt more like synthesizing and explicitly stating knowledge my gut already knew.
The point of these two anecdotes, insofar as anecdotes have points (which they shouldn’t), is a sort of article of faith in the power of articulation, the ability to represent one’s fuzzy feelings as symbols: words or numbers.
I used to have an instinct that putting words or numbers to fuzzy intuitions would distort and flatten them. I think this instinct is essentially wrong, or at least incomplete. When one’s ideas are put forth onto the page, something is lost, but the clarity created in the process more than makes up for it.
So whatever illegible intuition you are currently gripped by, remember that there are symbols for it too.