Number theory

I wonder how the numbers feel, packed together head to heel.

 

Some say it takes

a long, long time

to walk full down

the number line¹.

 

That's what they say,

but I say "No!

It just depends

how fast you go!²"

 

If you can choose³

and do it right,

all numbers flash by

in a single night.

 

But going that fast

there isn't time

to greet each one⁴

...and some are prime!⁵

 

"There are too many,"

they just say,

"to meet them all

anyway!⁶"

 

"Too many to count⁷,

for most are real⁸,

being irrational⁹

their great appeal!"

 

Defeated then,

I turn to go.
Georg Cantor¹⁰,

you’ve caused much woe.

 

With numbers dense¹¹

the line's paved they say,

but I'll ponder this,

another day¹². 

 

Notes
 
¹The number line contains an "uncountable" number of numbers.  "Countable" numbers have a "cardinality" equal to the "counting numbers", i.e., the positive integers.  Uncountable numbers, like the real numbers, have a cardinality larger than this.  So it would take a "long, long time, / to walk full down / the number line".
 
²The real numbers can not be listed even in an infinite amount of time with the fastest possible computer.  So to run through all of the real numbers, you must go very fast indeed.
 
³Running through the real numbers requires the "Axiom of Choice", a postulate logically independent of Set Theory, which stipulates that there is a way to "choose" an element from any set.  This sounds simple, but consider the set of real numbers for which we have no names.  Now, try and pick a member of this set.
 
⁴Since there are "uncountable" real numbers, one cannot list them.  Hence there is no time to "greet each one".
 
⁵Prime numbers are integers divisible only by 1 and themselves.  The ancient Greeks knew that there were an infinite ("countable") number of prime numbers.  This is a play on words between the mathematical and normal definition of “prime”.
 
⁶The real numbers are “uncountable”.
 
⁷They are "uncountable".
 
⁸Real numbers include numbers beyond the integers and rational numbers (numbers which can be expressed as a fraction).
 
⁹Irrational numbers dominate the real numbers.  Georg Cantor’s “Continuum Hypothesis” postulated that the cardinality of the real numbers is the second largest infinity.  Later Kurt Godel proved that the "Continuum Hypothesis" is unprovable from the axioms of Set Theory plus the "Axiom of Choice".
 
¹⁰Georg Cantor is the father of transfinite mathematics and was an outcast in his time with mathematicians of the day discrediting his work.  Even Poincaré referred to Cantor's ideas as a "grave disease" infecting the discipline of mathematics.  His work, however, is now considered inspired and far beyond his time.
 
¹¹“Dense” is a mathematical term.  The rational numbers are "dense" on the real number line because between any two distinct numbers taken from the number line there is at least one rational number between them.
 
¹²Can you believe how complicated numbers are!?  Let's get back to this tomorrow!
 
 
 
©2008, Richard Puetter

Reviews

[JCharo]

Very interesting concept and nice rhyme to the poem. I've never really seen footnotes in a poem before and, while I think that a poem is something that should exist by itself (without explanation) I understand and appreciate your style and will look out for more from you.

[Rick Puetter]

Dear JCharo,

Thank you for your kind review. And yes, the issue of footnotes. A good fraction of my poetry has footnotes (e.g., "Byzantium", "The Mahabharata", "Once We Were Giants", "Helium", "Tiny Specks", etc.) as I often write on technical topics, historical topics, or topics in which the context and details are not widely known by the general reader. But you are not the first to express reservations about footnotes. With the number of comments I get on the footnotes it's generally running 2-1 for the footnotes, i.e., there are twice as many positive comments as negative comments. And footnotes aren't unique to me. Have a look at some of T.S. Eliot's poetry ("The Waste Land" comes to mind). Again, a complicated and detailed poem with lots of footnotes.

My very best regards,

Rick

[Ravyne Hawke]

It's been a while since I've been in a mathematical setting, but I was surprised at how many of the terms I understood without looking at the footnotes. I think the title would tell someone to get ready for mathiness poem, rendering the footnotes obsolete. Or you could put the notes in the Author's Notes box instead. Great job Rick.

[Ken Simm.]

Weaves. Numbers may be complicated but this write (without taking into account the footnotes) is multilayered. Full of plural energies that make the, at first glance, complicated, lie within reach and all the more wonderful for that.

[Marie]

Well, I enjoyed the footnotes more than the poem--at least they taught meomething.

[Christina May Shanaberg]

I really didn't read all of the footnotes; I have an accounting background. However, I enjoyed the piece in itself. Hopefully, the footnotes may help some to understand your humor. Nice write!

[deeps]

Cute, creative and clever. Enjoyable read.

[Ruth]

Numbers are baffling love! They scare me! lol
This is a clever poem, love it!
xx

[alesuriko]

Well although I really enjoyed the poem... I really didn't like the footnotes. Sure it's helpful to point out what you mean with those words... but that takes away the whole point of poems and the illusions that go with them. Not to mention, it's also slightly patronizing.

But it's a great poem, funny with a great rhythm and rhyme scheme. I really loved the line "... and some are prime!" It just sounded to sarcastic and satirical in my mind as I read it. Nicely done.

[Vain Apocalypse]

This poem begs to be read aloud, and I found myself unwittingly doing just that by the fourth stanza. It has a remarkably charming tone, sprinkled with bits of mathematical insight and the depth of content that's characteristic of the author.

[Nancy Lee Shrader]

Congrats on your great poem. Well Done Rick!