Gaining Approval: Insights From “How To Prove It”

[Note: Th­ese are in­sights that sur­prised/​con­nected the dots for me per­son­ally. I feel dumb ad­mit­ting some of these, but my hope is that this’ll be a good data point for oth­ers]

How to write a math book:

1. It’s fun to prove things wrong. Ex­am­ples such as “What’s wrong with this proof” re­ally worked for me.

2. A great pat­tern for teach­ing: a solid ex­pla­na­tion of new con­cept, 3-5 ex­am­ple prob­lems to work through, and solu­tions & com­men­tary im­me­di­ately af­ter! It had a short feed­back loop that I felt my­self want­ing to be a part of.

3. Con­cepts built upon each other in quick suc­ces­sion, so you don’t have to flip back and re­view.

Proofs:

1. Proofs aren’t made to show how they made the proof, just what they’re prov­ing in a suc­cinct way. Since this is the case, it’s okay to be con­fused even if the proof sounds like ev­ery­thing of course fol­lows ev­ery­thing else, and you’re an idiot for not im­me­di­ately fol­low­ing. The proof took hours/​weeks/​years/​cen­turies to con­dense in that form, so it’s rea­son­able if it takes a few hours & math stack ex­change to un­der­stand it.

2. There are tips and tricks for writ­ing proofs de­pend­ing on the prop­er­ties of what you’re prov­ing. For in­stance, it’s gen­er­ally eas­ier to prove a pos­i­tive, so if you’re prov­ing a nega­tive, make it a pos­i­tive and do a proof by con­tra­dic­tion on it!

Math:

1. A ⟷ B is mak­ing two claims: A ⟶ B & A ⟵ B.

2. In­tu­ition for ma­te­rial im­pli­ca­tion (rule of in­fer­ence):

  • Either Bob did it or Alice did it. If it’s not Bob, then it’s Alice.

  • It’s ei­ther A or B. If it’s not A, then it’s B.

  • A ∨ B ⟷ ¬A ⟶ B

3. In­tu­ition for con­tra­pos­i­tive of im­pli­ca­tion:

  • ¬A ⟶ B ⟷ A ∨ B (rule of in­fer­ence above)

  • A ∨ B ⟷ B ∨ A (Bob or Alice = Alice or Bob)

  • B ∨ A ⟷ ¬B ⟶ A (rule of in­fer­ence above)

  • ¬B ⟶ A ⟷ ¬A ⟶ B (Bam! Con­tra­pos­i­tive!)

4. Com­bin­ing in­duc­tion with func­tions gets you re­cur­sion.

5. X = Ø is a nega­tive state­ment: ∀y(y ∉ X)

6. X is in­finite is a nega­tive state­ment: not finite.

Math, Danc­ing, and Mu­sic:

I im­prov dance & pi­ano, and to hit on some­thing in­ter­est­ing, you have to com­bine differ­ent re­la­tion­ships with differ­ent ob­jects. Let’s hone in on just nega­tion.

In math, negate “a sub­set of” to get “dis­joint”, “in­ter­sec­tion of to get “sym­met­ric differ­ence”, “for all x, P(x)” to get “there ex­ist an x where ¬P(x)” (plus the 3-4 men­tioned above).

In dance, you can dou­ble your reper­toire by sim­ply ap­ply­ing nega­tion to all of your dance moves. Even re­vers­ing walk­ing or turn­ing your head looks smooth (re­sults may vary). The only proper name dance re­ver­sals I can think of is the run­ning man & the jerk.

In mu­sic, re­vers­ing your melody, chord pro­gres­sion or­der, or me­ter can also pro­duce in­ter­est­ing re­sults. Ex­am­ple: play heart-and-soul’s melody back­wards. Chords to go with it: Amin, F, Dmin, Emin (though you will have to shift a cou­ple notes if you use those chords to make it sound right).

...that’s just nega­tion, too! This idea re­minds me of alk­jash’s Ham­mer and Nails with nega­tion be­ing the ham­mer and the above men­tioned be­ing the nails. One pos­si­ble im­pli­ca­tion of this is that any new re­la­tion­ship you learn in math, dance, mu­sic, etc. could be ap­plied to all ob­jects/​prop­er­ties you have in math, dance, mu­sic, etc. to pro­duce some­thing in­ter­est­ing. I would like to ex­plore this idea in a much greater depth in the fu­ture.

Ti­tle and re­view in­spired by TurnTrout