recreational group theory

Stream: general

[Johan Commelin]

Exercise A.1.22. The fundamental group of English is generated by the 26 letters a, . . . ,z, subject to all relations w=u where w and u are letter sequences that are pronounced identically in at least some contexts. E.g., gh is trivial since it is silent in some words, and ir = er because of the words “bird” and “her”.

Prove that this group is trivial.

[https://www.math.uni-bielefeld.de/~bux/texts/ggt_notes.pdf]

[Mario Carneiro]

I suppose the goal is to prove that the group is trivial?

[Mario Carneiro]

I wouldn't want to formalize the question though

[Johan Commelin]

No, but you could still prove formally that the group is trivial

[Johan Commelin]

I imagine it would push us towards creating some tools

[Mario Carneiro]

you could prove formally that any group with at least some chosen key subset of relations is trivial

[Mario Carneiro]

but the entire description of the group is quite large

[Johan Commelin]

Well, if we prove that:

G/H is trivial, and H < H', then G/H' is trivial
then I consider that part settled.

[Mario Carneiro]

That still doesn't constitute a formal proof. At the very least you have to write down all the relations and note that the ones you care about are in the list

[Mario Carneiro]

I believe that it is possible to make a convincing argument that there exists in theory a description of a fundamental group of English, and any such description must necessarily contain such and such relations therefore QED, but that's more like a proof of provable statement existence than a proof

[Johan Commelin]

I agree.

[Johan Commelin]

So... the challenge would be: write down an explicit list of relations (like "er" = "ir", with witnesses in the comments), and prove that the free group on 26 letters of the alphabet modulo these relations is trivial.
That is something we can do in principal, but is still nontrivial.

[Mario Carneiro]

that's true

[Mario Carneiro]

I guess I would try to set up the problem such that I need to prove that a = 1, b = 1, ...., z = 1 and conclude

[Mario Carneiro]

with the equalities being regular equalities of elements in a group, so that I can apply cancellation rules as normal

[Kevin Buzzard]

One could also prove the arguably more famous theorem of Zagier at al, where two words are the same if they're pronounced the same way. They proved that the homophonic group was trivial for English and French IIRC

[Kevin Buzzard]

https://projecteuclid.org/euclid.em/1062620828

[Kevin Buzzard]

This is the same journal that will have a maths formalisation issue next year

[Scott Morrison]

"One could give a proof with a more topological flavor of the triviality of f using cerf = serre."

[Kevin Buzzard]

I remember this caused quite a stir when it came out

[Mario Carneiro]

huh, there seem to be a lot of missing theorems here around free_group and presented_group

[Mario Carneiro]

it isn't even proven that the relations in a presented group are trivialized in the group

[Mario Carneiro]

also universal properties in terms of membership in a subgroup

[Kevin Buzzard]

I think formalising this theorem would (a) be easy (b) be interesting and (c) teach us about holes in mathlib

[Kevin Buzzard]

It would probably also be something which undergraduates could do and send to Exp Math :-)

[Mario Carneiro]

import group_theory.presented_group

namespace free_group

theorem subgroup_induction {α} {p : set (free_group α)} [is_subgroup p]
  (h : ∀ a, of a ∈ p) (x : free_group α) : x ∈ p :=
free_group.induction_on x is_submonoid.one_mem h
  (λ _, is_subgroup.inv_mem) (λ _ _, is_submonoid.mul_mem)

end free_group

namespace english

inductive alphabet
| a | b | c | d | e | f | g | h | i | j | k | l | m
| n | o | p | q | r | s | t | u | v | w | x | y | z

section
open alphabet
open free_group (of)

inductive rel : free_group alphabet → free_group alphabet → Prop
| er_ir : rel (of e * of r) (of i * of r)
| gh_ : rel (of g * of h) 1
-- add relations here
end

@[derive group]
def G := presented_group {z | ∃ x y, rel x y ∧ z = x⁻¹ * y}
instance : has_coe alphabet G := ⟨presented_group.of⟩

theorem to_G {a b : free_group alphabet} (h : rel a b) :
  (quotient_group.mk a : G) = quotient_group.mk b :=
quotient_group.eq.2 (group.subset_normal_closure ⟨a, b, h, rfl⟩)

open alphabet
theorem foo (γ : G) : γ = 1 :=
begin
  suffices : ∀ α : alphabet, (α : G) = 1,
  { refine quotient_group.induction_on' γ (λ z, _),
    refine (@is_group_hom.mem_ker _ _ _ quotient_group.mk z).1 _,
    refine free_group.subgroup_induction _ z,
    intro, rw is_group_hom.mem_ker, exact this _ },
  clear γ, intro α,
  have : (e * r : G) = i * r := to_G rel.er_ir,
  have : (g * h : G) = 1 := to_G rel.gh_,
  cases α,
  -- have fun
end

end english

[Kenny Lau]

[Kenny Lau]

til "halfpenny" is pronounced /ˈheɪp(ə)ni/ ???

[Mario Carneiro]

ha-penny

[Sebastien Gouezel]

What is fivepence, and how is it pronounced?

[Mario Carneiro]

fi-pence I think

[Kenny Lau]

@Kevin Buzzard please fix your language

[Sebastien Gouezel]

Come on, they are even more crazy than I thought.

[Mario Carneiro]

I question the monsieur example though

[Mario Carneiro]

I get something like n = õ - o

[Kenny Lau]

how about "hymn"

[Kenny Lau]

there's no nasalization in "monsieur" though

[Kenny Lau]

not even in French

[Chris Hughes]

$aar = ar$ because of aardvark and car, so $a = 1$
$mb = m$ because of plumb and plum so $b = 1$
$ck = k$ because of back and book so $c = 1$
$d = 1$
$ea = a$ because of bread and bed so $e = 1$
$f = 1$ because of Ffion and Fiona
$g = 1$ because it is silent in Gnasher
$h = 1$ because it is silent in hour
$i = ee$ because of ski and bee, so $i=1$
$j=1$ because $g=1$ and page and Japan
$$k = 1$ since it is silent in knife
$l=1$ because of nil and
$m=1$ because it is silent in mnemonic
$n=1$ because it is silent in autumn
$oo = o$ because of tomb and boom
$p=1$ because it is silent in pneumatic
$q = c$ because of quarter and cat, so $q=1$
$ar = a$ because of grass and car so $r = 1$ (South East England specific)
$s=c$ because of ceiling, so $s=1$
$t=1$ because it is silent in castle
$u=oo$ because of flu and boo sso $u=1$
$v=1$
$ow=oe$ because of tow and toe, so $w=1$
$x = cks$ because of fox and kicks, so $x = 1$
$y=ee$ because of quickly and bee, so $y=1$
$z = x$ because of xylophone so $z=1$

[Kenny Lau]

@Sebastien Gouezel can confirm

[Chris Hughes]

I was stuck on $d$ and $v$

[Patrick Massot]

The French pronunciation of monsieur is completely random, as often happen with very common words (they are used a lot, hence in bad shape)

[Johan Commelin]

Is rijsttafel an English word?

[Patrick Massot]

sounds like Dutch to me

[Mario Carneiro]

aha, as in it sounds like "misier"

[Mario Carneiro]

it is definitely dutch

[Mario Carneiro]

well, it's a dutch word for an indonesian meal

[Chris Hughes]

fivepence is cheating. There must be a better one for $v$

[Kenny Lau]

revving?

[Kenny Lau]

for d how about add

[Chris Hughes]

I feel like double letters in the middle of words makes it too easy.

[Kenny Lau]

all vowels are equivalent because they all get reduced

[Sebastien Gouezel]

Chris Hughes said:

$p=1$ because it is silent in pneumatic

What?

[Kenny Lau]

yeah it's also silent in pterodactyl

[Kenny Lau]

and psychology

[Kevin Buzzard]

I have never heard of the word fivepence and have no idea about this missing v

[Sebastien Gouezel]

I knew about psychology, I guess the other ones are less common.

[Kenny Lau]

how are you supposed to pronounce "pn" in french

[Kevin Buzzard]

But "hay-p'ny" for halfpenny was common when that coin existed (it disappeared in the mid 80s)

[Kenny Lau]

D is shirking its auditory duties in handkerchief and mostly doing the same in handsome. Its appearance in Wednesday can only be seen as some kind of cruel joke.

[Patrick Massot]

In French pneumatique uses both the p and n

[Kenny Lau]

is "Wednesday" cheating?

[Kenny Lau]

how about "Godmanchester" (pronounced "Gumster")

[Patrick Massot]

Wednesday seems totally fair

[Kenny Lau]

there's a silent d, silent n, silent ch

[Chris Hughes]

I knew someone who lived in Godmanchester and he didn't say Gumster

[Patrick Massot]

City names are cheating

[Kenny Lau]

V is at this point the only letter that refuses to be unheard in any established word of the language. And yet a dark cloud gathers on the horizon: in late May 2017 a much-followed and likely sleep-addled Twitter user tweeted out what was clearly a partially developed composition. The Internet seized on the enigmatic final word and discussed it ad nauseam. Of the myriad pronunciations suggested for this non-word, several of the strongest contenders had a silent v.

[Kenny Lau]

so Merriam Webster doesn't have any better for v

[Kenny Lau]

that just means it can't be silent

[Kenny Lau]

it can still be related via non-trivial means

[Chris Hughes]

But I can't think of words where any other letter is pronounced $v$ either.

[Kenny Lau]

I feel like the closest would be some German borrowings where the "v" is pronounced "f"

[Amelia Livingston]

of?

[Kenny Lau]

Some of you may be happy to know that we have at this point only one English word in which the j is silent: marijuana.

[Kenny Lau]

@Chris Hughes your example for e is invalid: it only shows ea=e, i.e. a=1

[Chris Hughes]

Oops.

[Kenny Lau]

how about meat = meet

[Chris Hughes]

That works.

[Chris Hughes]

Maybe a better example for F would be good as well.

[Kenny Lau]

and for f how about cough

[Rob Lewis]

Mario Carneiro said:

well, it's a dutch word for an indonesian meal

It's a Dutch word for Indonesian food served in a style unique to the Netherlands, or at least I've been told. You all ate this at Lean Together 2019 ;)

[Kenny Lau]

(largely based on Chris' answer)
a: bread / bred
b: plumb
c: scene / seen
d: Wednesday
e: meat / meet
f: cough
g: gnash
h: hour
i: business
j: Japan / page
k: knife
l: salmon
m: mnemonic
n: autumn
o: people
p: psychology
q: pique / peek
r: forecastle (pronounced "foe-ksel"???)
s: seed / cede
t: listen
u: guard
v: of
w: write / rite
x: xylophone (z)
y: sky / tie
z: disaster (s)

[Chris Hughes]

Never heard of forecastle

[Kenny Lau]

The forecastle (/ˈfoʊksəl/ FOHK-səl; abbreviated fo'c'sle or fo'c's'le) is the upper deck of a sailing ship forward of the foremast, or the forward part of a ship with the sailors' living quarters.

[Chris Hughes]

That'll be why I haven't heard of it. I'm not an 18th century sailor.

[Patrick Stevens]

In a similar vein you can get "rollock" and "rowlock" which are pronounced the same way

[Patrick Stevens]

Lots of good nautical words