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

[C. Porter]

Omigod, I love this poem! The music of it astounded me, and the conclusion wad perfect. KUDOSx10!

[Ellen Hammond]

The apparent simplicity of this poem I saw in my first reading of it was allowed me to enjoy the flow of rhyme and meter. But, having already read some of your more recent poems, I knew there was complexity in the simplicity. And your notes at the bottom proved me right, so I had to go back and reread it a few more times in a new light. You have a unique gift for combining math and/or science with poetry. It is amazing how one does not detract from the other. It would seem you a bit of a perfectionist, perhaps dominated by the left brain hemisphere, yet it is tempered and balanced well by the visualization from the right hemisphere. Thus, all you do you so well.

[Samuel Ferris]

Sweet poem.

[R J Askew]

A most enjoyable read, very stimulating and delightfully executed...8 verses of 4 lines of mostly 4 syllabs. I can see the number roads and lanes stretching all over the place like bundles or nerves or electrical wiring, different colours etc. I like the idea of number travel, getting into them literally, being part of the concept of 10,368 for example...getting to know it.

[Legion]

lol And this is exactly the reason(s) that I don't like math at all. Great write. Love the notes as well. Kudos.

[THE LOST ROSE]

A different read for me. I enjoyed this style. I don't think I could write like this? I don't think I can think like this? What i do know that this was a gifted piece and I am glad to have come across you after accepting my friends request. I plan on reading some other pieces and I know I will be just as entertained. Peace.

[Jason Henry]

this is a really neat poem! i can see this being used in schools to help children learn about the number line and its theory...very good write! thanks for sharing!

[Emily Rose]

Much better! Your update fits much better with the general scheme (both rhyming and metaphorical) and the reference even creates a double meaning with the lines that precede it, because presumably Cantor was also at one time "defeated" since, as you say in your note, he was something of an outcast to the mathematical world. His work must've been quite complicated too, which explains why he's causing so much woe. However small, it's a wonderful revision - it makes the poem much tighter and no words are used in excess. Excellent job!

[Emily Rose]

How wonderfully clever! I bow to your expertise at mixing math and poetry; you certainly know what you're talking about! The plays on words that you use throughout are delightful (and the notes are wonderfully interesting and really help to clarify it for us English majors, although the poem makes a lot of sense even without the notes. Your words are so well-chosen; you obviously spent a lot of time working on this and that really shows. The conclusion that you draw at the end is wonderful too - a metaphysical throwing of hands up into the air in exasperation. "I'll ponder this/another day!" you cry in frustration, moving on to do something far less complex. You really get the idea across that no one should ever take numbers for granted - they're far more complex than people give them credit for.

The only critique that I have is that the lines "not seeing Pi/I stub my toe" seem a little random and out of place. Are you stubbing your toe as you walk down the number line? It seems a bit contrived to fit the rhyme scheme, so you may want to change it. I'm quite sure that you can come up with something much more rational to put it its place. The second line in the last stanza also seems a bit out of sync with the rest of the poem - one of your abrupt meter changes. I'm not quite sure what you would do to fix it, but you could probably tweak it a bit to make it fit a little better.

This is wonderful! I can really see the mathmatician in you :-)

[Liam]

Well done, I loved "...and some are prime!5", as well as the numbers you put in. Very original. Thanks for the criticism, too.