Those three words,

that show a feeling

has endured

beyond believing,

that put ends

to arguments,

that suspend

all discontent,

show I’ve grasped

the thing I sought at

times now past.

And if I thought that

I was wrong,

my inkling broken,

how I’d long

to hear them spoken

stating it

was meant to be,

as if writ

by destiny,

as if not

just something random:

“Quod”, “Erat”

and “Demonstrandum”

# Moser's Circle Problem

Take two points on a circle,

and draw a line straight through.

The space that was encircled

is divided into two.

To these points add a third one,

which gives us two more chords.

The space through which these lines run

has been fissured into four.

Continue with a fourth point,

and three more lines drawn straight.

The new number of disjoint

regions sums, in all, to eight.

A fifth point and its four lines

support this pattern gleaned.

Counting sections, one divines

that there are now sixteen.

This pattern here of doubling

does seems a sturdy one.

But one more step is troubling

as the sixth gives thirty one.

# Primes

The primes,

through times,

mystified

those who pried.

One fact answers why

they’re simple yet sly:

Layers of abstraction yield

complex forms when pierced and peeled.

Addition lies under multiplication,

defining him as repeated summation.

Then he defines primes as the atoms of integers,

for when multiplied, they give numbers their signatures.

But when we breach the layer between these two distinct operations,

asking about how primes add and subtract, there are endless frustrations.

Even innocuous questions, “what are all their sums?”, or “how often do they differ by two?”,

stump every man who has ever lived, with progress made only quite recently by just a few.

However, to recruit for and progress math we need to have such questions which can be phrased simply and remain unsolved.

What child does not hear such conjectures and dream, if only for a moment, that they’ll be the one to see them resolved?

For otherwise the once vigorous curiosity of a child towards math’s patterns, as they grow older, tends to grow tame,

just as the rhyme and rhythm of the primes seems to fade as numbers grow, though in both the underlying patterns remain the same.

# All Numbers are Interesting

Is any number that we count banal,

and lacking in a feature one might note?

Suppose some are, and gather up them all.

The smallest one has quite the cause to gloat.

It is precisely how far one must count

before the numbers get uninteresting.

But this would be a curious amount,

which contradicts how we defined the thing!

So every number has something to show,

including ones whom we will never know.

Let’s see if this joke proof applies as well

to real numbers; do they all stand out?

Suppose some don’t, and find one who rebels

by being meaningful beyond a doubt.

Before it was the smallest who amused,

but real sets might lack a minimum.

Will there be a special one to choose

from subsets of the whole continuum?

It isn’t clear where such a choice comes from,

yet some treat “choice” like it’s an axiom.

# Invention vs Discovery

A lurking question, old as Greece, does ask:

Is math invention or discovery?

The answer’s both, but here the harder task

is knowing how it switches. So you see,

most truths are found with constructs, which in turn

are built from truths like chickens and their eggs.

One needs to know, should this fact raise concern,

that constructs can be daft; just logic’s dregs.

The truths discovered tell how to define

the structures that make math and world align.

Pythagoras’s theorem about lengths,

is one such truth that he (and others) found.

Observed, discovered, and through mental strength,

it was then proved with pictures that astound.

Much later mathematicians, minds alit,

would formalize, as sets, both “length” and “space”.

These constructs were defined so that by writ

Pythagoras’s truth remains the case.

That is, this “theorem” now seems just defined.

As such, one might then lose their peace of mind.

So in this world where "space" is formalized,

do all those pretty proofs now lose their charm?

Of course not! We all ought to realize

how “length” could change its meaning without harm

to mathematical consistency

in all the theory giving facts of space.

However, then these “facts” would hardly be

related to the nature we embrace.

And hence, a truth has told how to define

a structure that makes math and world align.

# 2π

Fixed poorly in notation with that two,

you shine so loud that you deserve a name.

Late though we are, to make a change it's true,

We can extol you ‘til you have pi's fame.

One might object, "Conventions matter not!

Great formulae cast truths transcending names."

I've noticed, though, how language molds my thoughts;

the natural terms make heart and head the same.

So lose the two inside your autograph,

then guide our thoughts without you "better" half.

Wonders math imparts become so neat

when phrased with you, and pi remains off-screen.

Sine and exp both cycle to your beat.

Jive with Fourier, and forms are clean.

“Wait! Area of circles”, pi would say,

“sticks oddly to one half when tau’s preferred.”

More to you then! For write it in this way,

then links to triangles can be inferred.

Nix pi, then all within geometry

shines clean and clear, as if by poetry.

# 1-1+1-1+...

When one takes one from one

plus one from one plus one

and on and on but ends

anon then starts again,

then some sums sum to one,

to zero other ones.

One wonders who'd have won

had stopping not been done;

had he summed every bit

until the infinite.

Lest you should think that such

less well-known sums are much

ado about nonsense

I do give these two cents:

The universe has got

an answer which is not

what most would first surmise,

it is a compromise,

and though it seems a laugh

the universe gives “half”.

# Defining Math

In math, we strive and stickle with a pride,

to wash away all ambiguity.

And yet, definers cannot all decide

what “math” itself means. What an irony!

Perhaps a science, solving mysteries,

but happy to pursue what seems absurd.

Perhaps a brother of philosophies,

but with a rigid rigour in each word.

Computer science might be suitable,

but most of math is not computable.

Perhaps it is a language of its own,

which, as to english, has the most false friends.

Perhaps religion, where the truth alone

is worshiped, giving means as well as ends.

Perhaps a sister of creative arts,

but one arousing reason, not a sense.

Indeed it shares a number of its parts

with almost every study in some sense.

But it’s an orphan, floating in abstract,

and it alone encloses what’s exact

# Euler's Formula

Famously

start with e,

raise to π

with an i,

we've been taught

by a lot

that you've got

minus one.

Can we glean

what it means?

For such words

are absurd.

How to treat

the repeat

of a feat

πi times?

This is bound

to confound

'til your mind

redefines

these amounts

one can't count

which surmount

our friend e.

Numbers act

as abstract

functions which

slide the rich

2d space

in its place

with a grace

when they sum.

Multiplied,

they don’t slide,

acting a

second way.

They rotate,

and dilate,

but keep straight

that same plane.

Now what we

write as e

to the x

won’t perplex

when you know

it’s for show

that “x” goes

up and right.

It does not,

as you thought,

repeat e

product e.

It functions

with gumption

on functions

of the plane.

It turns slides

side to side

into growths

and shrinks both.

Up and downs

come around

as turns round,

which is key!

This is why

π times i,

which slides north

is brought forth

and returned,

we have learned,

as a turn

halfway round.

Minus one,

matched by none,

turns this way,

hence we’re done.

# Ode To Infinity

Philosophers have claimed you are not "real",

but you are no less true than 3, in fact.

For what is "three"? Some thinking will reveal,

a term that points to triplets in abstract.

And so are you, a pointer in this light,

but aimed instead at families without size.

From here it took great Cantor's mental sight

to see these sets disjoin before his eyes.

For that's the tricky thing about you, friend,

what you are is something without end!

Aside from all the masks you wear as sets,

you permeate all math like pi and e.

Without you series, sequences and nets,

would vanish, as would all geometry.

To cook the real numbers all it takes,

is you and all the fractions to react.

When added as a simple point you make

both arguments and spaces more compact.

When patterns, lists and odes drag on and on,

just say "infinity", and woes are gone.