The Liar paradox is a conundrum about the nature of truth arising when a person, the Liar, announces “I am lying.” Is the Liar's pronouncement true or false? If the Liar speaks truly, then truly he is lying; but if he is lying, then he spoke truly. Either case thus leads to contradiction. And so the laughing Liar lights logic's law afire, laughing while the world burns.

According to Tarski's disquotational theory of truth, a sentence is true when the proposition it asserts is the case. Thus, “snow is white” is true if and only if snow is white—truth merely removes the quotation marks. The Liar challenges this deflationary theory for a self-applicable truth predicate, one applying to assertions using that predicate, as in the Liar. The Visser-Yablo paradox aims to avoid self-reference with an infinite list, each statement asserting the falsity of all subsequent statements. If any is true, then all the following are false, and so they should be true; and so none are true, which means that all must be true. Crazytown!

Music and video by Hannah Hoffman; lyrics by Joel David Hamkins. Read more about the Liar paradox in my essay Infinite Liars.

Lyrics

A merry prankster is the Liar,
Lighting logic's law afire.
Reason's ruin, chaos eyeing,
With three words only: I am lying.

The liar's sentence, “I am lying”
Has neither True nor False applying,
For if he's truthful, then he lies,
But if lying, then Truth flies.
Either case is contradiction,
Truth here has no jurisdiction.

I am lying,
Falsifying,
Self denying,
Truth untying!

The laughing Liar's paradox
Unlocks a logic tinderbox.
Liar's fire climbing higher,
Havoc stocks and Truth defrocks.

For if he lies, logic dies,
And if true, untruth is due.
There's no avoiding contradiction,
Truth enjoys no benediction.

He is lying,
Falsifying,
Self denying,
Truth untying!

If he's truthful, Liar lies,
But if lying, reason cries.
The proof is clear, in contradiction;
Truth's arrears, in dereliction!

Tarski's Truth is disquotational,
His Truth scheme truly inspirational.
He says that “snow is white” is true,
In case the powder's bright for you...

A self-applying predicate,
For Truth is truly delicate.

I am lying,
Self decrying,
Reason trying,
Logic frying!

What a jokester, whimsy's sire,
Laughing Liar's dumpster fire.
Reason ruined, in a mire,
Truth thus tried by Liar's ire.

A solemn choir sings at the pyre,
A funeral for Truth expired.
All logic, reason, lit afire,
Thanks to prankster laughing Liar.

A cue to Visser-Yablo's due;
Review their endless Liars queue.
Each softly coos a clue anew:
“No sentence subsequent is true...”
“No sentence subsequent is true...”
“No sentence subsequent is true...”

Bonus choral verses:

He is lying!
uncomplying!
disqualifying!
Are you buying?!

He is lying
stupefying
trouble-spying
Fools are flying!

I am lying
self belying
ill supplying
Truth defying

I am lying
self decrying
reason trying
logic frying

I am lying
misapplying
mystifying
mortifying

He is lying
crucifying
Truth is crying
truly dying!

Plenitudinous Primes

Lyrics by Joel David Hamkins, Music by Hannah Hoffman

Let's prove the primes' infinitude
they do exist in multitude

of quite astounding magnitude
we count the primes in plenitude!

'Twas proved by Euclid way back when
in Elements he argued then

He proved it with exactitude
in his Book IX he did conclude

for any given finite list
there will be primes the list has missed

to prove this gem let number N
be when you multiply them then

multiply the list for fun,
multiply, and then add one

If into N we shall divide
the listed primes seen in that guide

remainder one each leaves behind
so none as factors could we find

but every number has some prime factor
with our N take such an actor

since it can't be on the list
we thus deduce more primes exist

No finite list can be complete
and so the claim is in retreat

By iterating this construction
primes do flow in vast deduction

as many as we'd like to see
so infinite the primes must be

Thus proved the primes' infinitude
in quite enormous multitude

of arb'trarly great magnitude
we count the primes in plenitude!

'Twas known by Euclid way back then
he proved it in the Elements when

in modern times we know some more
math's never done, there's new folklore:

For Bertrand, Chebyshev had said
about the puzzle in his head

and Erdős said it once again:
there's always a prime between n and 2n.

We proved the primes' infinitude
they do exist in multitude

of inconceiv'ble magnitude
we count the primes in plentitude!

The primes are plenitudinous
they're truly multitudinous!

This is one of several mathematical/philosophical music videos I have undertaken with Hannah Hoffman, all of which will eventually be posted here on Infinitely More—look for them in the Math Music Videos section.

Of course, I had set off immediately to answer the challenge. My wife Barbara Gail Montero and our daughter Hypatia and I spent a day in Oxford thinking, writing, revising, rewriting, rethinking, rewriting, and eventually we had a set lyrics providing the proof, in rhyme and meter. We had wanted specifically to highlight not only the logic of the proof, but also to tell the fateful story of Hippasus, credited with the discovery, afterwards drowned at sea. Was it punishment from the Gods?

Hannah used our lyrics to create the amazing musical version. Enjoy!

Ode to Hippasus

Lyrics by Joel David Hamkins, Barbara Gail Montero & their daughter Hypatia

Music by Hannah Hoffman

The diagonal of a square is incommensurable with its side
an astounding fact the Pythagoreans did hide

but Hippasus rebelled and spoke the truth
making his point with irrefutable proof

it's absurd to suppose that the root of two
is rational, namely, p over q

square both sides and you will see
that twice q squared is the square of p

since p squared is even, then p is as well
now, if p as 2k you alternatively spell

2q squared will to 4k squared equate
revealing, when halved, q's even fate

thus, root two as fraction, p over q
must have numerator and denomerator with factors of two

to lowest terms, therefore, it can't be reduced
root two is irrational, Hippasus deduced

as retribution for revealing this irrationality
Hippasus, it is said, was drowned in the sea

but his proof lives on for the whole world to admire
a truth of elegance that will ever inspire.

This is the first of several mathematical/philosophical music videos I have undertaken with Hannah Hoffman, all of which will eventually be posted here on Infinitely More—look for them in the music-videos section, linked on the main menu.