One strategy for proving a mathematical theorem is a "proof by contradiction." In such a proof one begins by supposing the opposite of what is to be proved -- and then reasons logically to obtain a statement that contradicts a known truth. This contradiction verifies that our opposite-assumption was wrong and that our original statement-to-be-proved is indeed correct. (An easily-read introduction to "proof-by-contradiction" is given here.)
Peggy Shumaker is an Alaskan poet whom I had the pleasure of meeting at a reading at Bloomsburg University where I was a math professor a few years ago. Her poem, "What to Count On," below, has a beautiful surprise after a sequence of negations -- and reminds me of the structure of a proof-by-contradiction.
What to Count On by Peggy Shumaker
Not one star, not even the half moon
on the night you were born
Not the flash of salmon
nor ridges on blue snow
Not the flicker of raven’s
never-still eye
Showing posts with label contradiction. Show all posts
Showing posts with label contradiction. Show all posts
Monday, June 6, 2016
Saturday, July 11, 2015
Math fun with song lyrics
Song-writer Bill Calhoun is a faculty member in the Department of Mathematics, Computer Science and Statistics at Pennsylvania's Bloomsburg University (where I also hung out for many years). He belongs, along with colleagues Erik Wynters and Kevin Ferland, to a band called "The Derivatives." And Bill has granted permission for me to include several of his math lyrics (parodies) here. (In this previous post, we consider the connection between song parodies and mathematical isomorphism.) My first Calhoun selection deals with difficult mathematical questions concerning classification of infinite sets and decidability. Following that, later lyrics consider proving theorems and finding derivatives.
Questions You Can’t Ever Decide* by Bill Calhoun
Picture yourself in a world filled with numbers,
But the numbers are really just words in disguise.
Gödel says “How can you prove you’re consistent,
If you can’t tell that this is a lie?”
Questions You Can’t Ever Decide* by Bill Calhoun
(These lyrics match the tune of "Lucy in the Sky with Diamonds" by Lennon and McCartney.)
Picture yourself in a world filled with numbers,
But the numbers are really just words in disguise.
Gödel says “How can you prove you’re consistent,
If you can’t tell that this is a lie?”
Labels:
Bill Calhoun,
cardinal,
compact,
contradiction,
decide,
derivative,
induction,
infinite,
lyrics,
parody,
proof,
real number,
slope,
theorem,
uncountable
Thursday, September 22, 2011
The wealth of ambiguity
When we read these lines by Robert Burns (1759-1796),
Oh my luv is like a red, red rose,
That's newly sprung in June . . .
we don't know whether he compares a woman he loves to a flower or whether it is his own emotion he describes. And the multiplicity of meanings is a good and pleasing thing. Similarly, when we read the problem,
Solve the equation, x² + 4 = 0
Oh my luv is like a red, red rose,
That's newly sprung in June . . .
we don't know whether he compares a woman he loves to a flower or whether it is his own emotion he describes. And the multiplicity of meanings is a good and pleasing thing. Similarly, when we read the problem,
Solve the equation, x² + 4 = 0
Labels:
ambiguity,
contradiction,
logic,
Michael Palmer,
principle,
proposition
Thursday, November 11, 2010
Theorem-proof / Cut-up / poems
For mathematicians, reading a well-crafted proof that turns toward its conclusion with elegance and perhaps surprise -- this mirrors an encounter with poetry. But can one have that poetry-math experience without being fluent in the language of mathematics? Below I offer a proof (a version of Euclid's proof of the infinitude of primes) and a "cut-up" produced from that proof-- and I invite readers (both mathematical and non-mathematical) to consider them as poems.
Labels:
Brion Gysin,
contradiction,
cut-up,
Euclid,
finite,
infinite,
mathematics,
poem,
poetry,
prime,
proof,
theorem,
William Burroughs
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