Parts Catalog Accessories Catalog How To Articles Tech Forums
Call Pelican Parts at 888-280-7799
Shopping Cart Cart | Project List | Order Status | Help



Go Back   PeachParts Mercedes-Benz Forum > General Discussions > Off-Topic Discussion

Reply
 
LinkBack Thread Tools Display Modes
  #1  
Old 12-19-2006, 09:12 PM
Botnst's Avatar
Banned
 
Join Date: Jun 2003
Location: There castle.
Posts: 44,601
What is meant by "proof"?

For me, the word brings to mind Euclid and Aristotle.

Bot
----------------------
Foolproof

Mathematical proof is foolproof, it seems, only in the absence of fools
Brian Hayes

I was a teenage angle trisector. In my first full-time job, fresh out of high school, I trisected angles all day long for $1.75 an hour. My employer was a maker of volt-meters, ammeters and other electrical instruments. This was back in the analog age, when a meter had a slender pointer swinging in an arc across a scale. My job was drawing the scale. A technician would calibrate the meter, recording the pointer's angular deflection at a few key intervals—say 3, 6, 9, 12 and 15 volts. When I drew the scale, using ruler and compass and a fine pen, I would fill in the intermediate divisions by interpolation. That's where the trisection of angles came in. I was also called upon to perform quintisections and various other impossible feats.

click for full image and caption


I joked about this with my coworker and supervisor, Dmytro, who had been drawing meter scales for some years. We should get extra pay, I said, for solving one of the famous unsolvable problems of antiquity. But Dmytro was a skeptic, and he challenged me to prove that trisection is impossible. This was beyond my ability. I did my best to present an outline of a proof (after rereading a Martin Gardner column on the topic), but my grasp of the mathematics was tenuous, my argument was incoherent, and my audience remained unconvinced.

On the other hand, Dmytro himself quickly produced visible evidence that the specific method of trisection we employed—drawing a chord across the angle and dividing it into three equal segments—gave incorrect results when applied to large angles. After that, we made sure all the angles we trisected were small ones. And we agreed that the whole matter was something we needn't discuss with the boss. Our circumspect silence was a little like the Pythagorean conspiracy to conceal the irrationality of ?2.

Looking back on this episode, I am left with vague misgivings about the place of proof in mathematical discourse and in everyday life. Admittedly, my failure to persuade Dmytro was entirely a fault of the prover, not of the proof. Still, if proof is a magic wand that works only in the hands of wizards, what is its utility to the rest of us?

Reading Euclid Backward

Here is how proof is supposed to work, as illustrated by an anecdote in John Aubrey's Brief Lives about the 17th century philosopher Thomas Hobbes:

He was 40 yeares old before he looked on geometry; which happened accidentally. Being in a gentleman's library in..., Euclid's Elements lay open, and 'twas the 47 El. libri I. He read the proposition. "By G—," sayd he (he would now and then sweare, by way of emphasis), "this is impossible!" So he reads the demonstration of it, which referred him back to such a proposition; which proposition he read. That referred him back to another, which he also read. Et sic deinceps, that at last he was demonstratively convinced of that trueth. This made him in love with geometry.
click for full image and caption


What's most remarkable about this tale—whether or not there's any trueth in it—is the way Hobbes is persuaded against his own will. He starts out incredulous, but he can't resist the force of deductive logic. From proposition 47 (which happens to be the Pythagorean theorem), he is swept backward through the book, from conclusions to their premises and eventually to axioms. Though he searches for a flaw, each step of the argument compels assent. This is the power of pure reason.

For many of us, the first exposure to mathematical proof—typically in a geometry class—is rather different from Hobbes's middle-age epiphany. A nearer model comes from another well-worn story, found in Plato's dialogue Meno. Socrates, drawing figures in the sand, undertakes to coach an untutored slave boy, helping him to prove a special case of the Pythagorean theorem. I paraphrase very loosely:

Socrates: Here is a square with sides of length 2 and area equal to 4. If we double the area, to 8 units, what will the length of a side be?
Boy: Umm, 4?

Socrates: Does 4 x 4 = 8?

Boy: Okay, maybe it's 3.

Socrates: Does 3 x 3 = 8?

Boy: I give up.

Socrates: Observe this line from corner to corner, which the erudite among us call a diagonal. If we erect a new square on the diagonal, note that one-half of the original square makes up one-fourth of the new square, and so the total area of the new square must be double that of the original square. Therefore the length of the diagonal is the length we were seeking, is it not?

Boy: Whatever.

click for full image and caption


At this point I trust we are all rooting for the kid. I would like to be able to report that the dialogue continues with the boy taking the initiative, saying something like, "Okay, dude, so what's the length of your erudite diagonal? It's not 4 and it's not 3, so what is it, exactly?" Alas, Plato reports no such challenge from the slave boy.

The problem with the Meno proof is exactly the opposite of the one I faced when I was an untutored wage slave. Whereas I was too inept and intellectually ill-equipped to craft a proof that would persuade my colleague (or even myself, for that matter), Socrates is a figure of such potent authority that the poor kid would surely go along with anything the master said. He would put up no resistance even if Socrates were proving that 1=2. It's hard to believe that the boy will go on to prove theorems of his own.

Sadly, Hobbes didn't get much more benefit from his own geometry lesson. He became a notorious mathematical crank, claiming to have solved all the most famous problems of classical geometry, including the trisection of the angle, the squaring of the circle and the doubling of the cube. These claims were a little less foolish in the 17th century than they would be now, since the impossibility of the tasks had not yet been firmly established. Nevertheless, Hobbes's contemporaries had no trouble spotting the gaffes in his proofs.

Enormous Theorems, Unwieldy Proofs

In recent years proof has become a surprisingly contentious topic. One thread of discord began with the 1976 proof of the four-color-map theorem by Kenneth Appel, Wolfgang Haken and John Koch of the University of Illinois at Urbana-Champaign. They showed that if you want to color a map so that no two adjacent countries share a color, four crayons are all you'll ever need. The proof relied on computer programs to check thousands of map configurations. This intrusion of the computer into pure mathematics was greeted with suspicion and even disgust. Haken and Appel reported a friend's comment: "God would never permit the best proof of such a beautiful theorem to be so ugly." Apart from such emotional and aesthetic reactions, there was the nagging question of verification: How can we ever be sure the computer didn't make a mistake?

Some of the same issues have come up again with the proof of the Kepler conjecture by Thomas C. Hales of the University of Pittsburgh (with contributions by his student Samuel P. Ferguson). The Kepler conjecture—or is it now the Hales-Ferguson theorem?—says that the pyramid of oranges on a grocer's shelf is packed as densely as possible. Computations play a major part in the proof. Although this reliance on technology has not evoked the same kind of revulsion expressed three decades ago, worries about correctness have not gone away.

Hales announced his proof in 1998, submitting six papers for publication in Annals of Mathematics. The journal enlisted a dozen referees to examine the papers and their supporting computer programs, but in the end the reviewers were defeated by the task. They found nothing wrong, but the computations were so vast and formless that exhaustive checking was impractical, and the referees felt they could not certify the entire proof to be error-free. This was a troubling impasse. Eventually the Annals published "the human part of the proof," excluding some of the computational work; the full proof was published last summer in Discrete and Computational Geometry. Interestingly, Hales has turned his attention to computer-assisted methods of checking proofs.

In the case of another famously problematic proof, we can't put the blame on computers. An effort to classify the mathematical objects known as finite simple groups began in the 1950s; the classification amounts to a proof that no such groups exist outside of five known categories. By the early 1980s the organizers of the project believed the proof was essentially complete, but it was scattered across 500 publications totaling at least 10,000 pages. Three senior authors undertook to revise and simplify the proof, bringing together the major ideas in one series of publications. The process, still unfinished, is testing the limits of the human attention span—and lifespan. (One of the leaders of the revision program died in 1992.)

If some proofs are too long to comprehend, others are too terse and cryptic. Four years ago the Russian mathematician Grigory Perelman announced a proof of the Poincaré conjecture. This result says—here I paraphrase Christina Sormani of Lehman College—that if a blob of alien goo can ooze its way out of any lasso you throw around it, then the blob must be nothing more than a deformed sphere, without holes or handles. Everyday experience testifies to this fact for two-dimensional surfaces embedded in three-dimensional space, and the conjecture was proved some time ago for surfaces (or "manifolds") of four or more dimensions. The hard case was the three-dimensional manifold, which Perelman solved by proving a more-general result called the geometrization conjecture.



More at: http://www.americanscientist.org/template/AssetDetail/assetid/54428?&print=yes

Reply With Quote
  #2  
Old 12-19-2006, 09:56 PM
Kuan's Avatar
unband
 
Join Date: Jan 2001
Location: At the Birkebeiner
Posts: 3,841
Do you recognize proof by reductio ad absurdum?

You would like to prove S. You assume not-S, and if not-S implies a contradiction of the form A and not-A, then S is true.
__________________
You don't need a weatherman to know which way the wind blows - Robert A. Zimmerman
Reply With Quote
  #3  
Old 12-19-2006, 10:01 PM
Botnst's Avatar
Banned
 
Join Date: Jun 2003
Location: There castle.
Posts: 44,601
Quote:
Originally Posted by Kuan View Post
Do you recognize proof by reductio ad absurdum?

You would like to prove S. You assume not-S, and if not-S implies a contradiction of the form A and not-A, then S is true.
I always get the ad absurdum part. It's the reductio that eludes me.
Reply With Quote
  #4  
Old 12-19-2006, 10:08 PM
Larry Delor's Avatar
What, Me Worry?
 
Join Date: Jun 1999
Location: Sarasota, Fl.
Posts: 3,114
And here I thought the proof was in the pudding.
__________________
It is a truism that almost any sect, cult, or religion will legislate its creed into law if it acquires the political power to do so. Robert A. Heinlein


09 Jetta TDI
1985 300D
Reply With Quote
  #5  
Old 12-19-2006, 10:10 PM
Botnst's Avatar
Banned
 
Join Date: Jun 2003
Location: There castle.
Posts: 44,601
Quote:
Originally Posted by Larry Delor View Post
And here I thought the proof was in the pudding.
Rum pudding?
Reply With Quote
  #6  
Old 12-19-2006, 10:11 PM
Kuan's Avatar
unband
 
Join Date: Jan 2001
Location: At the Birkebeiner
Posts: 3,841
Quote:
Originally Posted by Botnst View Post
I always get the ad absurdum part. It's the reductio that eludes me.
It's OK, just bypass that part.

It's funny how many of us place our faith in those 2-3 people who claim to actually understand these groundbreaking proofs. A division of scientific labor I like to say.
__________________
You don't need a weatherman to know which way the wind blows - Robert A. Zimmerman
Reply With Quote
  #7  
Old 12-20-2006, 12:35 AM
LaRondo's Avatar
Rondissimo
 
Join Date: Oct 2006
Location: West Coast
Posts: 162
For some, proof in the sense of evidence may still be unrecognized even while starring at it
__________________
Reply With Quote
  #8  
Old 12-20-2006, 04:56 AM
dacia's Avatar
Member of the board
 
Join Date: Mar 2000
Posts: 315
"I don't know, a proof is a proof. What kind of a proof is a proof? A proof is a proof and when you have a good proof it's because it's proven."

(EX-Prime Minister Jean Chretien, when asked what kind of proof he would need of weapons of mass destruction in Iraq before deciding to send Canadians along on the Bush invasion-September 5th on CTV news)

Last edited by dacia; 12-20-2006 at 05:02 AM.
Reply With Quote
  #9  
Old 12-20-2006, 06:35 AM
LaRondo's Avatar
Rondissimo
 
Join Date: Oct 2006
Location: West Coast
Posts: 162
Quote:
Originally Posted by dacia View Post
"I don't know, a proof is a proof. What kind of a proof is a proof? A proof is a proof and when you have a good proof it's because it's proven."

(EX-Prime Minister Jean Chretien, when asked what kind of proof he would need of weapons of mass destruction in Iraq before deciding to send Canadians along on the Bush invasion-September 5th on CTV news)
And how does the story continue?
__________________
Reply With Quote
  #10  
Old 12-20-2006, 07:36 AM
dacia's Avatar
Member of the board
 
Join Date: Mar 2000
Posts: 315
Well, if I remember correctly he didn't send any ground troops.

Alex
Reply With Quote
  #11  
Old 12-20-2006, 09:01 AM
jlomon's Avatar
Registered User
 
Join Date: Aug 2002
Location: Toronto, Ontario
Posts: 310
Isn't a "proof" equal to one-half percent per volume? 80 proof liquor is 40% alcohol per volume.

Maybe I drink too much.
__________________
Jonathan

2011 Mazda2
2000 E320 4Matic Wagon
1994 C280 (retired)
Reply With Quote
  #12  
Old 12-20-2006, 09:05 AM
Botnst's Avatar
Banned
 
Join Date: Jun 2003
Location: There castle.
Posts: 44,601
"I don't know, a proof is a proof. What kind of a proof is a proof? A proof is a proof and when you have a good proof it's because it's proven."

Unless of course, the horse is Mr Ed.
Reply With Quote
  #13  
Old 12-20-2006, 10:06 AM
Kuan's Avatar
unband
 
Join Date: Jan 2001
Location: At the Birkebeiner
Posts: 3,841
You ask a stupid question, you get a stupid answer.
__________________
You don't need a weatherman to know which way the wind blows - Robert A. Zimmerman
Reply With Quote
  #14  
Old 12-20-2006, 10:41 AM
Registered User
 
Join Date: Oct 2005
Posts: 4,263
Quote:
Originally Posted by Kuan View Post
Do you recognize proof by reductio ad absurdum?
These are certainly recognized, but not preferred as they show an indirect conclusion.
Reply With Quote
  #15  
Old 12-20-2006, 11:32 AM
Botnst's Avatar
Banned
 
Join Date: Jun 2003
Location: There castle.
Posts: 44,601
Quote:
Originally Posted by Matt L View Post
These are certainly recognized, but not preferred as they show an indirect conclusion.
Damn! Paid attention in class.

Reply With Quote
Reply

Bookmarks

Thread Tools
Display Modes

Posting Rules
You may not post new threads
You may not post replies
You may not post attachments
You may not edit your posts

BB code is On
Smilies are On
[IMG] code is On
HTML code is On
Trackbacks are On
Pingbacks are On
Refbacks are On




All times are GMT -4. The time now is 06:50 AM.


Powered by vBulletin® Version 3.8.7
Copyright ©2000 - 2024, vBulletin Solutions, Inc.
Search Engine Optimization by vBSEO 3.6.0
Copyright 2024 Pelican Parts, LLC - Posts may be archived for display on the Peach Parts or Pelican Parts Website -    DMCA Registered Agent Contact Page