amavect

I've thought about this before. It gets harder when dealing with >2 inputs and names longer than 1 letter, so I organized it with lines to make it look more tree-like.

  │u│v│x│y
  ├─┴─├─┴─
  │f  │g
  ├───┴─
  │h
  └─

With >2 inputs and longer names.

  │uh│v  │xizzy│yeez
  ├──┴─  ├─────┴────
  │fooey │gar
  ├──────┴───
  │hello
  └─────

The associative property.

  (a+b)+c=a+(b+c)=a+b+c

  │a│b    │b│c
  ├─┴─    ├─┴─
  │+  │c│a│+  │a│b│c
  ├───┴─├─┴─  ├─┴─┴─
  │+    │+    │+
  ├─────┴─────┴─
  │=
  └─

The quadratic formula.

  (-b±sqrt(b^2-4*a*c))/(2*a)
  
    │b│2│4│a│c
    ├─┴─├─┴─┴─
    │^  │*
    ├───┴─
  │b│-    
  ├─├─    
  │-│sqrt│2│a
  ├─┴────├─┴─
  │±     │*
  ├──────┴─
  │/
  └─

Obviously, it takes way more space and time to draw. Tell me if you like it!

>it very quickly learns to work on "it's on adjacent lines therefore it's a third apart" when you read sheet music

Maybe you read sheet music different than how I do, but when I play euphonium or trombone, I translate from note position+features to finger+mouth position without needing to think about the note letter. More like, "I see the note located here with an accidental feature in the context of the clef and key signature therefore I use this finger position and mouth position". I would love to simplify it down to "it's located here therefore I use this finger position and mouth position".

Keep the accidentals to notate alternate finger positions for enharmonics (and microtones), but eliminate the global variables, the clefs and key signature. Heck, keep the key signature to reduce the noise of alternate fingering symbols, but PLEASE eliminate the clefs. I would also love to play treble clefs at the same sight-reading speed as bass clefs. Also, staff notation should visually repeat every octave, but it doesn't.

Wow, I wish I had this when I learned to play. Keys, clefs, and accidentals felt like a hazing ritual, completely accidental (hah!) complexity. I always hated that I couldn't read the treble instrument parts at sight-reading speed simply because the sheet music used a different clef.

Rotational symmetry or L^2 norm only matter in vector formulations. I assumed a scalar formulation.

Conservation of E-field flux certainly implies additive-homomorphism (addition of charges equals addition of forces). But that seems a bit ahistorical because Maxwell would develop field theory 70 years after Coloumb. Either way, the axiom you choose requires empirical justification, and I think a home hobbyist could more easily demonstrate by experiment that adding charges will add the forces.

Or, if you meant F(r,1,1) = K/r^2, then yeah, conservation of E-field flux could give you an inverse square law. But again, that requires an experiment to justify the axiom.

Actually, yes! We can use homogenous-degree-1 (provable from the homomorphism) to prove it.

  F(r,q0,q1)
  = F(r,q0*1,q1*1)
  = q0*q1*F(r,1,1)

Very simple! It remains to show that F(r,1,1) = K/r^2, as intended.

Show by experiment that the force of charge0 against charge1+charge2 equals the force of charge0 against charge1 plus charge0 against charge2. Induce an additive-homomorphic property F(r,q0,q1+q2) = F(r,q0,q1)+F(r,q0,q2). Then, exponent 1 follows.

I figured this out by listing a bunch of mathematical properties. I couldn't see how the author jumps from zero-preserving to multiply-charges, and I still don't know how, but we can call it out of scope lol

  r : distance between p and q
  q0 : charge 0
  q1 : charge 1
  F : coulomb force function
  charge-commutative: F(r,q0,q1) = F(r,q1,q0)
  zero-preserving: 0 = F(r,q0,0)
  additive-homomorphic: F(r,q0,q1+q2) = F(r,q0,q1)+F(r,q0,q2)
  
  homogenous-degree-1: F(r,q0,n*q1) = n*F(r,q0,q1)
  multiplicative-separability: F(r,q0,q1) = K*R(r)*Q(q0,q1)
  multiply-charges: F(r,q0,q1) = K*R(r)*(q0*q1)^a

  Given F(r,q0,q1) = K*R(r)*(q0*q1)^a, charge-commutative, zero-preserving, additive-homomorphic.
  Induction using additive-homomorphic proves homogenous-degree-1. (For example, F(r,q0,2*q1) = F(r,q0,q1+q1) = 2*F(r,q0,q1))
  Equational proof follows from homogenous-degree-1:
  K*R(r)*(q0*n*q1)^a = n*K*R(r)*(q0*q1)^a
  (q0*n*q1)^a = n*(q0*q1)^a
  n^a*(q0*q1)^a = n*(q0*q1)^a
  n^a = n
  n = 0 or a = 1
  n≠0, therefore a=1.

My hidden assumption: I said the set of names must be countable! I assumed you would know that naming means assigning a finite string (in the Ithkuil writing system of course). and don't nitpick further or else I'll have to write a proof in Agda or Rocq lol

I love philosophy Calvinball, so I would counter by asserting that undetectable implies no possession, an immediate contradiction. Or go further and assert that undetectable implies nonexistence. We all possess an immense undetectable nonexistent sphere. No bounds on assumptions means I can make up anything to annoy the interlocutor.

So, you're right. This shows why we should use formal math, so we can agree on the result yet bicker about the interpretation. Some folks point to Cantor's diagonalization theorem to show that some unnameable things exist, when the theorem doesn't say that at all.

guess I won't respond now :( "what is a real number, anyways" is one of my favorite questions, not foolish at all

No surjective function exists from names to real numbers (diagonalization). With any naming scheme, some unnamed real numbers always remain.

On the other hand, given any real number, I can name it. I'll run out of unique names, since no injective function exists from real numbers to names.

So, some unnamed real numbers will always remain (non-constructively), I can make real numbers that escape a naming scheme (constructively), and no unnameable real numbers exist.

You proved that definable implies nameable, and also unnameable implies undefinable. Obviously true. However, the idea of undefinable real numbers closely resembles a modern version of the paradox. No surjective function exists from definitions to real numbers.

Really, the blurb about "seems impossible to verify this by giving positive instances" contains the tension between constructive math and non-constructive math. Does an unnameable (and undefinable) thing actually exist? If a tree falls in a forest, but no one can hear it, does it make a sound?

Thanks for sharing!

I love approval voting the most by pure simplicity and effectiveness. How does score voting work better than STAR? I thought STAR degenerates into score voting when clone candidates run?

I haven't heard much of an argument against proportional voting before. Your article just casts doubt, but doesn't really make a decisive argument against. I actually don't like proportional voting because it reifies parties into the system, but I only support it because I think it can break the party duopoly and step towards better systems. I should rethink that.

Ranked choice is not better. It has the center squeeze effect.