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Are Computers Ready to Create Mathematical Proofs?
DoraLives writes "Interesting article in the New York Times regarding the quandary mathematicians are now finding themselves in. In a lovely irony reminiscent of the torture, in days of yore, that students were put through when it came to using, or not using, newfangled calculators in class, the Big Guys are now wrestling with a very similar issue regarding computers: 'Can we trust the darned things?' 'Can we know what we know?' Fascinating stuff."
> 'Can we know what we know?' Fascinating stuff.
Reminds me of Rumsfeld... "Reports that say that something hasn't happened are always interesting to me, because as we know, there are known knowns; there are things we know we know. We also know there are known unknowns; that is to say we know there are some things we do not know. But there are also unknown unknowns -- the ones we don't know we don't know."
...precisely why it's so hard to see that a pyramid of cannonballs is an optimal stack? This seems almost axiomatic.
I guess the obvious Monte Carlo simulation doesn't constitute "proof," but still, what exactly is the big question here?
I'll use a calculator as long as it's easier, screw accuracy.
http://ipod.fresh27.net/
The headline does a slight disservice in describing the article that way: Whether or not computers can create proofs isn't an issue. The problem comes when the resulting proof is too involved to be verified by a human, and so the computer's work has to be trusted.
Depends whether it's a Pentium with an FDIV bug, I imagine...
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I think the issue will become: can we learn anything on our own, we don't want to rely on imposibly long proofs. My calc teacher hates us using calculators, she thinks it will be the end of calculus as we know it.
Well, we certainly can't trust them to let us read the article. Anyone got a registration handy?
The Braying and Neighing of Barnyard Animals Follows.
I mean, we've used computers to prove much of boolean and linear algebra. The most famed result in the field is that of the Robbins Conjecture, proven entirely by computer. The computer produced a very "inhuman" proof...
and there are already programs out that help with this. Here's one for example...
there will be a computer that will automatically post on slashdot.
In Math, Computers Don't Lie. Or Do They?
By KENNETH CHANG
Published: April 6, 2004
leading mathematics journal has finally accepted that one of the longest-standing problems in the field -- the most efficient way to pack oranges -- has been conclusively solved.
That is, if you believe a computer.
The answer is what experts -- and grocers -- have long suspected: stacked as a pyramid. That allows each layer of oranges to sit lower, in the hollows of the layer below, and take up less space than if the oranges sat directly on top of each other.
While that appeared to be the correct answer, no one offered a convincing mathematical proof until 1998 -- and even then people were not entirely convinced.
For six years, mathematicians have pored over hundreds of pages of a paper by Dr. Thomas C. Hales, a professor of mathematics at the University of Pittsburgh.
But Dr. Hales's proof of the problem, known as the Kepler Conjecture, hinges on a complex series of computer calculations, too many and too tedious for mathematicians reviewing his paper to check by hand.
Believing it thus, at some level, requires faith that the computer performed the calculations flawlessly, without any programming bugs. For a field that trades in dispassionate logic and supposedly unambiguous truths and falsehoods, that is an uncomfortably gray in-between.
Because of the ambiguities, the journal, the prestigious Annals of Mathematics, has decided to publish only the theoretical parts of the proof, which have been checked in the traditional manner. A more specialized journal, Discrete and Computational Geometry, will publish the computer sections.
The decision represents a compromise between wholehearted acceptance and rejection of the computer techniques that are becoming more common in mathematics.
The debate over computer-assisted proofs is the high-end version of arguments over using calculators in math classes -- whether technology spurs greater achievements by speeding rote calculations or deprives people of fundamentals.
"I don't like them, because you sort of don't feel you understand what's going on," said Dr. John H. Conway, a math professor at Princeton. But other mathematicians see a major boon: just as the computers of today can beat the grand masters of chess, the computers of tomorrow may be able to discover proofs that have eluded the grandest of mathematicians.
The packing problem dates at least to the 1590's, when Sir Walter Raleigh, stocking his ship for an expedition, wondered if there was a quick way to calculate the number of cannonballs in a stack based on its height. His assistant, Thomas Harriot, came up with the requested equation.
Years later, Harriot mentioned the problem to Johannes Kepler, the astronomer who had deduced the movement of planets. Kepler concluded that the pyramid was most efficient. (An alternative arrangement, with each layer of spheres laid out in a honeycomb pattern, is equally efficient, but not better.) But Kepler offered no proof.
A rigorous proof, a notion first set forth by Euclid around 300 B.C., is a progression of logic, starting from assumptions and arriving at a conclusion. If the chain is correct, the proof is true. If not, it is wrong.
But a proof is sometimes a fuzzy concept, subject to whim and personality. Almost no published proof contains every step; there are just too many.
The Kepler Conjecture is also not the first proof to rely on computers. In 1976, Dr. Wolfgang Haken and Dr. Kenneth Appel of the University of Illinois used computer calculations in a proof of the four-color theorem, which states that any map needs only four colors to ensure that no adjacent regions are the same color.
The work was published -- and mathematicians began finding mistakes in it. In each case, Dr. Haken and Dr. Appel quickly fixed the error. But, "To many mathematicians, this left a very bad taste," said Dr. Robert D. MacPherson, an Annals editor.,
To a
Computers are a human creation...it's not a matter of whether we can trust the computer, but rather a matter of can we trust that the people who built the computer and coded the software it runs knew what they were doing and didn't make any errors. Computers can only do what we tell them to...so really it was humans who indirectly made the proofs by producing a system capable of doing so. All it really boils down to is whether the folks who made the system and it's software knew what they were doing or not and whether they made any errors or not.
do you mean that this picture will be true someday? It does seem a lot friendlier then SKYNET.
no.
q.e.d.
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ping -f 255.255.255.255 # if only
Theorem provers have been around for a long time. A net search should turn up a ton of hits. The key is to implement a system that can be verified by hand, and then build on it.
(S(SKK)(SKK))(S(SKK)(SKK))
20th century mathematics has seen some pretty amazing things, but at the same time, there are very real questions as to what constitutes "proof" any more.
consider this: the hypothesis of the famous Riemann Zeta problem has been tested for trillions of different solutions, and it has held true in every case. (If you want an explanation of the Zeta problem, look elsewhere, I don't have the time)
Now that means that it's *probably* true, but nobody accepts that as mathematical proof.
On the other hand, the classification problem for finite simple groups has been rigorously solved, but the collected proof (done in bits by hundreds of mathematicians working over 30 years) is tens of thousands of pages in many different journals. given the standards of review, it is a virtual certainty that there is an error somewhere in there that hasn't been found. So, again, the solution to this problem is *probably* right, but it has been accepted as solved.
What's the difference between these two cases really? What's the difference between these and relying on computer proofs that are, again, *probably* right?
In this light, the math of the late 19th century and early 20th century was something of a golden age, modern standards of logical rigor were in place, but the big breakthroughs were still using elementary enough techniques that the proofs could be written in only a few pages, and the majority of mathematically literate readers could be expected to follow along. These days proofs run in the hundreds of pages and only a handful of hyper-specialized readers can be expected to understand, much less review them.
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The people who the computer is made, the software that cord/code do those which run the mistake which the computer had done and did not make know, thing we who are compilation of the human can rely on the computer, but on the other hand us you rely on problem it is possible, how, it is not problem?.. with that. The computer so... but as for indirectly making evidence with the compilation of the system which really so can do being the human we saying to those it is possible not to be. Everything where that boils really the system was made, whether or not how people who and that software knew and are made the mistake which is those which have been done or it is not, and.
???
BOOM!
"Cleanup in aisle 10"
I don't know about computers making proofs. I'm no expert, but afaik they only know what we tell them to know. Then again, if we use the same logic processes found in computers in real life, then there should be no problem... until we think up loopier ways of thinking. I don't look foreward to the days when my math teacher will say "Ok, everyone back to addition. We've found a bug". On another note, calculators are awesome. My math teachers yell at me for doing ridiculously easy problems in them (simple addition and the like) but I keep shrugging them off. But, the other day, my math teacher told us that there was a kid (no names mentioned) who got a test problem wrong because there was an addition mistake. There is no arguing that the kid would have benefited from using a calculator on her/his simple addition. Score number 2 for slacking! I rest my long-winded case.
I regularly report MSN spam to the Hotmail admins.
As far as calculator example goes, I believe a person should understand the fundamentals before using the calculator. Don't give a power tool to a kid. Somebody with an understanding of the fundamentals can wield the tool correctly and wisely whereas some cowboy is just dangerous. The old saying comes to mind "know just enough to be dangerous". As far as those oranges in the article, well somebody had better figure out a way to confirm the computer answer, without having to go through the exact same meticulous steps. Don't put your trust in technology without the proof to back it up, because technology built by people is prone to error. Now I'm sure the orange problem won't cause harm to anybody, however I hope I never read such an article about a nuclear power plant!
I haven't read the article, but...
It always seemed to me that the real question here is whether or not we trust mathematics, not computers. After all, if we can prove that theorem provers work correctly, then it follows that the proofs they generate will be correct. Humans write the theorem provers, so that problem is certainly comprehensible.
The question is whether the people setting up to create mathematical proof are ready themselves?
So many times I see people use programs like MAPLE to show something mathematical, and it ends up a disaster.
Problems is the brain on the chair, not brain on machine.
I don't see what the big deal is; not only is the general problem of proving mathematical statements undecidable (even without considering Godel's theorem) but even solvable problems require a lot of human intervention to get solved. Most problems (ie - examples out of math textbooks) aren't going to come up with a proof in any reasonable amount of time by simply dropping it into a theorem prover and pushing "go".
"Automated theorem proving" is less an automatic process (like you'd get with an automated production line) and more of a mechanical assistance to the job (like using a fork-lift to move heavy things faster than you could by hand).
It not only takes work to convert a problem into a good representation, but then you have to structure the problem statement in such a way that a theorem prover can make optimal use of it. Often times, you're forced to, upon following the output, prove lemmas (sub-proofs).
Then, when you finally get a proof, you get the joy of trying to simplify it to something that -can- be understood by a person; again, this is part of the process that can't really be automated well.
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Change the title to: "Are Computers Able to Verify Mathematical Proofs Beyond All Doubt?"
Doesn't the Godel incompleteness theorem say they can't?
YES
The answer is left as an exercise for the reader.
That's ok, Jesus likes me anyway.
Though Dr. Hoare danced around that question a little, presumably that aspect of the project would have to be done by hand, a monumental task to say the least.
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Mathematics is just Symbol Manipulation. I suspect computers are pretty good at that.
;)
Also, chess is just Pattern Matching... I don't know if humans have the edge there or not.
http://www.nytimes.com/2004/04/06/science/06MATH.h tml regfree link
My father sent this to me first thing this morning. I told him that I didn't think that rigorous mathematical proofs should be based on software either in whole or in part. All software of any complexity is inherently buggy. That doesn't mean that rigorous mathematical proofs are flawless by nature. That's why they have peer review. But peer review on software still isn't always sufficient.
Another issue is that you're then excluding any mathematicians who aren't also fairly adept programmers, from really understanding your proof.
All of this said, computers are necessary to do math these days and I think mathematicians should make use of them. I just don't believe we've reached a level of maturity in software development that meets the stringent requirements of mathematical proofs.
..if you can prove the program. I know that a lot of people look down on axiomatic semantics and model checkers; but I also know some people that started in that area a long time ago that still believe in it, and even more that are trying to get faculty appointments out of this rebounding field. If you can prove a program does what you expect it to and it, it turn, can be shown to prove what you really wanted, I don't see the problem. Maybe some of our theoretical mathematicians just need a dose of practical computer science.
No reg link
most insightful comment i have seen for days.
The article says that published proofs generally skip steps. I'd bet the humans make more mistakes than the computer. Actually, Wiles's proof of Fermat's Last Theorem origanally had a mistake which was fixed later.
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The obvious solution is to have the computer create a new proof that shows that the algorithm it used to create the original proof is, in fact correct.
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Rumsfeld is the only senior staff member that seems to have a pulse. He is always lively and joking in his press briefings. I may not agree with some of his decisions or policies, but at least I can listen to him without falling alseep. I wish more politicians were not robots. Every time I hear a snippet of a Bush speech it sounds so fucking robotic, almost on the same level as Gore. Rumsfeld is always trying to squirm his way out of a direct answer to pretty much every question the press asks him. It's hillarious.
This story and this issue are not about whether or not the mathematical community trusts a computer created proof. The issue is whether or not the community can trust the human behind the computer to create a computer program/system that is "flawless enough". Issues and bugs may arise, and the community can't trust that these issues will 1. be found and 2. be severe enough to affect the validity of the proof.
Wasn't the 4 color map problem solved in a similar fashion, in the late 1970's, by Ken Appel? This is nothing new, there are just some problems you are not going to solve by hand.
"To those who are overly cautious, everything is impossible. "
The problem with computer generated proofs is that in order to trust the result of the computer, you have to trust:
And of course, our understanding of the hardware depends on how accurate our understanding of the laws of physics is. Any mistake in either the source code, compiler, or hardware, and potentially the proof produced is incorrect. That's an awful amount of stuff you have to check just to make absolutely sure the computer is correct. Then, consider how many bug-free pieces of software you've encountered. ... Yeah, I can see why mathematicians would not trust computer generated proofs.
Of course, people are not infallible either, but that's well known and expected. It's all about how much uncertainty people are willing to accept.
Math is constantly limited by the continuous need to be rigorous. It's not a bad thing, but imagine the potential. A mathematician (or even, a semi-informed layman) would be able to input a logical statement and have the computer determine the validity, special cases, problems, and so forth.
The hard part would become formulating the statements. For science, this result would be immeasurable. Ask the computer if it can be done, and get an answer. Knowledge could increase exponentially! (No, really -- once upon a time, only advanced mathematicians knew calculus, but now we learn it in high school. Just wait until warp theory is an entry level college engineering course)
as a graduate in the fields of mathematics, i spent a large portion of my five undergraduate years doing proofs. there are a great many ways to prove things, sometimes applicable sometimes not. (e.g. using inductive proofs for numeric theorems is all well and good but completely useless for any sort of ring-theory or spatial proofs)
there are also several levels of depth for proofs, ranging from "i've found a counter example so i can write the whole thing off as garbage" to "i have exhaustively and rigorously proved this starting with the basic axioms of number theory and worked my way on up"
the latter is really the only acceptable way to prove anything seriously. sometimes when you are reworking an already- done proof to illustrate a point, other mathematicians will allow a bit of latitude when it comes to cutting corners, but for a proof as far-reaching as the one in the article, i would only be interested in a "rigorous" proof, that is, one that started with the foundational tenets of mathematics and combined those to form and prove other postulates, etc. very much a form of abstraction, not unlike large development projects.
the problem arises when one (or several) humans have to be able to objectively check the whole thing. to use my prior example of a large development project, no one developer at microsoft understand the whole of windows. it's too big for a single human to understand. each developer knows what he needs to do to complete his part, and so on and so forth.
traditionally, for proofs, a single mathematician (or a small group) would hammer out the whole proof, so the level of complexity remained at a human-understandable level. (even if tedious) my concern, as a mathematician, with using an automated solution would be the rapidly growing order of complexity needed to properly back up increasingly complex proofs. as stated in the article, it's like trying to proofread a phonebook. (only, you must also consider that for every element of the proof (a particular listing) there are several branching layers of complexity (fundamental, underlying proofs many layers deep) underneath. this gets more complicated in an exponential fashion) obviously this approach will only remain human-checkable for relatively small problems. (programmers: think of some horrible nondeterministic- polynomial problem like the "traveling salesman" problem. systems, like humans, are a finite resource, but if you increase the size of the problem, the complexity will quickly grow far beyond your ability to compute. large proofs suffer from the same difficulties, if not quite as concrete and pronounced as NP algorithms)
in closing, i would have to agree that proofs, no matter the effort and computing time put into them, really should not be viewed as being as rigorous as those provable "by hand" and human- understandable, even if the computer has arrived at a satisfactory conclusion, because we have no way of KNOWING if the computer has built up the proof correctly, except that it says it has.
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Verification of mathematical proofs is left up to the referees of a paper. No one (at the moment) is suggesting that computers are able to perform the verification job of the referees; all the computer was able to do in this case was essentially trot out a (big) number of cases and verify certain computable propositions. It was the checking of those verifications that stumped the referees, but, as those verifications formed an essential part of the proof of Kepler's Conjecture itself, their removal made the proof incomplete. Perhaps an even more appropriate title would be "Are Computers Ready to Assist in Proving Mathematical Theorems?"
Aren't you dead?
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We trust everyone, Trust but verify. Why should machines be any differnt. If I ask you to do a math problem do I trust you with the correct answer, sure I do, but I still verify. I preform the calculation my self, and allow others to preform the calculation. If we all come up with the same result we most likly have the correct answer. We must be careful not to allow the problem to define the solution method as that allows for us all to preform the same mistakes. Just like the Seti@home and distributed crack peopel found out that not everyone can be trusted that the only thing todo is have many people test the same key space no one should trust only one answer but rather develop consenous
I think the "as we know it" is the key word here. It creates a new juggling act that teachers have to deal with. On one hand, calculators are extremely useful for the things at which humans are error prone. If students can use calculators or Mathematica or something, they can check their arithmetic much more reliably, which is great for isolating problems with learning concepts from basic mathematical errors. Granted, some people don't see this as a bonus since it's good to be able to do arithmetic reliably without aid. For smaller numbers I agree, but in the real world people use calculators for arithmetic with numbers that have a lot of sig figs. Making students do this arithmetic by hand is just distracting them from learning the concepts they are supposed to be learning.
On the other hand, many students are prone to using these devices and applications as crutches and try to get away with doing things like using their calculator's implementation of Newton's Method instead of solving the problem themselves.
Some professors have found solutions to this problem, others havent. When I was at college, I think our math department had achieved a pretty good level of harmony with Mathematica - we were expected to do a lot of stuff by hand or in our heads - Gaussian elimination, for example - but in order to make the math seem useful, we were also exptected to be able to solve real-world problems with the stuff we learned. Not contrived "real-world" problems from your high-school textbook, but stuff like interpreting large and dirty scientific datasets where the specific technique we would have to use to solve the problem was something we could figure out, but not something that had been explicitly laid out by the textbook or in lecture. We had to apply the concepts we had learned to figure out the problem, but there was no way we were going to chug out that arithmetic by hand - when was the last time you tried to work on a 16x16 matrix using a pencil and paper? How about a 100x100 one?
Mathematics is one of the most intensely human of human endeavours. Everything in it is a production of the human mind entirely. Yes, the real world can sometime lead us into an interesting area of inquiry, but at its core the uncoverings of truth from axioms is a human endeavour.
A computer can be a useful tool (I'll be doing computational graph theory this summer), but it is not human. It does not have the ability to hold the possiblities of ideal forms within it and understand. It does not think.
The use of numeric methods to solve applied problems, or symbolic methods to pure problems is good and useful, but it does not constitute proof.
A human being, given an understanding of the underlying mathematics, must be able to go through the proof step by step, and see that, from the givens, the conclusion is inevitable.
I don't accept the Four-Colour theorem as proven true. I strongly suspect it to be so, but my suspicion does not truth make.
The Riemann hypothesis, on the other hand, is much, much further from being proved then the Four-Colour Theorem. Yes, millions of zeroes have been checked...but there are infinitely many zeroes, and all it takes for it to be false is for ONE of those zeroes to fall off the Re=1/2 part of the complex plane.
If I were giving odds, then millions divided by infinity is awfully close to zero.
Insanity is contagious. - Yossarian
I professor showed me the Robbins Algebra proof a while ago. I was going to link here, but first I searched the page for (Score:5, Informative), and there you were :)
Here's an excerpt:
In 1933, E. V. Huntington presented [1,2] the following basis for Boolean algebra:
x + y = y + x. [commutativity]
(x + y) + z = x + (y + z). [associativity]
n(n(x) + y) + n(n(x) + n(y)) = x. [Huntington equation]
Shortly thereafter, Herbert Robbins conjectured that the Huntington equation can be replaced with a simpler one [5]:
n(n(x + y) + n(x + n(y))) = x. [Robbins equation]
Robbins and Huntington could not find a proof. The theorem was proved automatically by EQP, a theorem proving program developed at Argonne National Laboratory.
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No matter what the proof, don't we still have to accept blindly on faith that A=A?
Sure, it seems highly probable but.. I just.. I guess I'm a skeptic. All of logic seems like a joke to me, as long as this one little potentially huge loophole looms in the background...
Spoon not. Fork, or fork not. There is no spoon.
I just decided to take a break from my Epistemology (the study of knowledge and 'how do we know that we know'?) and look at Slashdot to find this as headline!!!! Now I know this is a sign from God and those Iraqi weapons of mass destruction to get back to work!
Your CPU is not doing anything else, at least do something.
Its a machine. I'll trust it to obey the laws of physics every time. If someone doesn't design the machine to behave correctly, writes a buggy or flawed piece of code, builds a flawed model, or feeds it bad data, then its the person you can't trust.
Machines don't make mistakes. People make mistakes.
*singing*
You canno' change the laws of physics, laws of physics, laws of physics!
There's klingons on the starboard bow, starboard bow, starboard bow, Jim.
We come in peace. Shoot to kill! Shoot to kill! Shoot to kill men!
These posts express my own personal views, not those of my employer
I don't know about computers in general, but as for my computer, I don't know it's been doing chores regularly now for a couple months, and I think it's reached a level of maturity where it can take on a few extra responsibilities. Besides what could it hurt, heck I had a paper route when I was its age.
"The human mind will never be replaced."
640k ought to be enough for anybody.
Hmmm....
Wow, how ironic. Talk about putting faith into science. I never thought us humans would out-think ourselves. Then again, at the rate of human progress maybe we are overdue for this. Perhaps, another layer of conceptual management is needed.
Life is not for the lazy.
once upon a time, only advanced mathematicians knew calculus, but now we learn it in high school. Just wait until warp theory is an entry level college engineering course
Once upon a time, the majority of adult males knew how to trap a rabbit (or similar creature), gut it, skin it, start up a fire, cook it, and eat it.
I don't.
Heck, I couldn't even look up at the sky at night and tell you which way was north.
Once upon a time, most people could.
All I'm saying is that the amount of knowledge and skills the average human being can possess will not increase expontentially over time (barring artificial manipulation). We gain new skills as a population and lose old ones.
Just have another computer that proves that the first proof was generated using methods that can themselves be mathematically proven...wait...I'm already confused.
Here's a link to the Flyspeck Project, briefly mentioned at the end of the article, which aims to give a formal proof of the theorem.
I asked a maths university lecturer once "how do you know when something has been proved?" and he laughed and replied "When someone else agrees that the proof is correct. The problem is there is no definition of a proof."
was waiting for someone to say that. I hope 'paranoia' like this (worried whether or not we should "believe" computers) doesn't stop advancement of any area of mathematics that computers can be of aid. computers were invented as tools to help people.. not lowerclass 'beings' to believe or not believe at a whim of uncertainty. just like guns don't kill people, people kill people - computer algorithms dont have bugs, people create bugs.
"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."
(PM 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)
When I was a young boy, I awoke every morning to the delicious smell of pancakes. My mother, and father's dojo contained within it a hot griddle perfect for making pancakes, waffles, and a multitude of other pancake-like breakfast pastries. I remember them well -- The pleasant, care-free days of my childhood in the dojo were often spent peering into the kitchen with eager anticipation as my mother prepared pancakes my family.
.. the ultimate pancake. My journey took me to the many islands of my homeland, many days away from my dojo. My hunger for pancakes became my teacher, and foolishly I let it control the path that I walked upon. My feet, sore from travel, ached as my heart and stomach did, until I came to a realization. My duty was clear. I needed to take a stand and accept my love for the art of the ninja AND my love for pancakes. It was not wrong for me to love both. I love one as a dear friend, and one as a lover. Yes--My mission was clear--I must become a ninja, a secret assassin hired by the imperial family BUT I MUST ALSO ENJOY THE OCCASIONAL PANCAKE.
As I grew older, and began my journey to spiritual enlightenment, the memories of my pancake-eating youth filled my heart and dreams with warm, fluffy goodness....Ahhh, yes..the sweet, sweet memories... The day I ate 10 pancakes... The day I placed a warm pancake between my fleshy loins and performed the forbidden dance... The day pressed a pancake to my buttocks and encouraged my dog to come eat.. Indeed, much of my childhood was spent in pure innocence -- An innocence only pancakes can provide. It was heaven. A heaven, filled with pancakes, where I sat at the throne of God, with my hand-maidens Aunt Jemimah and Mrs. Butterworth seated beside me. An indestructible triumvirate made of flour, eggs, sugar, milk, water, and love.
By the age of 15, the path of my life became unclear and confusing. Torn between my duty my village and my love for pancakes, I foolishly left home in search of karaguchi ah-nowakadesu
My adoration for breakfast cakes has placed me within an awkward position. Many ninja refuse to recognize me as their brother. I defend my father's land, but I am looked upon as weak and undisciplined. I tell them, "But, brothers! Listen to my plea! The pancakes do not weaken me, nor do they make me disobey the rule of my sword. They fill me with love." But alas, they do not understand...For the mind of a ninja is complex.
My only earthly desire is to be accepted for who I am. Yes, I am a NINJA--But I also enjoy pancakes. Will you accept me? If you were approached by a ninja who requested a pancake, would you submit to his will?
5770
When I was a young boy, I awoke every morning to the delicious smell of pancakes. My mother, and father's dojo contained within it a hot griddle perfect for making pancakes, waffles, and a multitude of other pancake-like breakfast pastries. I remember them well -- The pleasant, care-free days of my childhood in the dojo were often spent peering into the kitchen with eager anticipation as my mother prepared pancakes my family.
.. the ultimate pancake. My journey took me to the many islands of my homeland, many days away from my dojo. My hunger for pancakes became my teacher, and foolishly I let it control the path that I walked upon. My feet, sore from travel, ached as my heart and stomach did, until I came to a realization. My duty was clear. I needed to take a stand and accept my love for the art of the ninja AND my love for pancakes. It was not wrong for me to love both. I love one as a dear friend, and one as a lover. Yes--My mission was clear--I must become a ninja, a secret assassin hired by the imperial family BUT I MUST ALSO ENJOY THE OCCASIONAL PANCAKE.
As I grew older, and began my journey to spiritual enlightenment, the memories of my pancake-eating youth filled my heart and dreams with warm, fluffy goodness....Ahhh, yes..the sweet, sweet memories... The day I ate 10 pancakes... The day I placed a warm pancake between my fleshy loins and performed the forbidden dance... The day pressed a pancake to my buttocks and encouraged my dog to come eat.. Indeed, much of my childhood was spent in pure innocence -- An innocence only pancakes can provide. It was heaven. A heaven, filled with pancakes, where I sat at the throne of God, with my hand-maidens Aunt Jemimah and Mrs. Butterworth seated beside me. An indestructible triumvirate made of flour, eggs, sugar, milk, water, and love.
By the age of 15, the path of my life became unclear and confusing. Torn between my duty my village and my love for pancakes, I foolishly left home in search of karaguchi ah-nowakadesu
My adoration for breakfast cakes has placed me within an awkward position. Many ninja refuse to recognize me as their brother. I defend my father's land, but I am looked upon as weak and undisciplined. I tell them, "But, brothers! Listen to my plea! The pancakes do not weaken me, nor do they make me disobey the rule of my sword. They fill me with love." But alas, they do not understand...For the mind of a ninja is complex.
My only earthly desire is to be accepted for who I am. Yes, I am a NINJA--But I also enjoy pancakes. Will you accept me? If you were approached by a ninja who requested a pancake, would you submit to his will?
4892
42!
0 00)
What else is there to know?
I doubt I'm the only one who read that as 42-factorial...
(BTW: 42! = 1405006117752879898543142606244511569936384000000
This looks good until you factor in the pisshead, alzheimer's, or "I changed my mind" factors.
The two sets of votes will NEVER match up owing to people who voted two different ways, nevermind any machine-generated irregularities (power surges, water drips onto the circuitry from a leaky ceiling, cosmic ray strikes, younameit).
Pfft, the whole thing goes right out the window. Nailing this sort of thing down isn't too far removed from nailing the uncertainties of the original article down. Seems trivial at first glance, but when it comes to guaranteeing results, then all of a sudden all this crap comes out of the woodwork and lands squarely in your lap.
Is it fascism yet?
When I was a young boy, I awoke every morning to the delicious smell of pancakes. My mother, and father's dojo contained within it a hot griddle perfect for making pancakes, waffles, and a multitude of other pancake-like breakfast pastries. I remember them well -- The pleasant, care-free days of my childhood in the dojo were often spent peering into the kitchen with eager anticipation as my mother prepared pancakes my family.
.. the ultimate pancake. My journey took me to the many islands of my homeland, many days away from my dojo. My hunger for pancakes became my teacher, and foolishly I let it control the path that I walked upon. My feet, sore from travel, ached as my heart and stomach did, until I came to a realization. My duty was clear. I needed to take a stand and accept my love for the art of the ninja AND my love for pancakes. It was not wrong for me to love both. I love one as a dear friend, and one as a lover. Yes--My mission was clear--I must become a ninja, a secret assassin hired by the imperial family BUT I MUST ALSO ENJOY THE OCCASIONAL PANCAKE.
As I grew older, and began my journey to spiritual enlightenment, the memories of my pancake-eating youth filled my heart and dreams with warm, fluffy goodness....Ahhh, yes..the sweet, sweet memories... The day I ate 10 pancakes... The day I placed a warm pancake between my fleshy loins and performed the forbidden dance... The day pressed a pancake to my buttocks and encouraged my dog to come eat.. Indeed, much of my childhood was spent in pure innocence -- An innocence only pancakes can provide. It was heaven. A heaven, filled with pancakes, where I sat at the throne of God, with my hand-maidens Aunt Jemimah and Mrs. Butterworth seated beside me. An indestructible triumvirate made of flour, eggs, sugar, milk, water, and love.
By the age of 15, the path of my life became unclear and confusing. Torn between my duty my village and my love for pancakes, I foolishly left home in search of karaguchi ah-nowakadesu
My adoration for breakfast cakes has placed me within an awkward position. Many ninja refuse to recognize me as their brother. I defend my father's land, but I am looked upon as weak and undisciplined. I tell them, "But, brothers! Listen to my plea! The pancakes do not weaken me, nor do they make me disobey the rule of my sword. They fill me with love." But alas, they do not understand...For the mind of a ninja is complex.
My only earthly desire is to be accepted for who I am. Yes, I am a NINJA--But I also enjoy pancakes. Will you accept me? If you were approached by a ninja who requested a pancake, would you submit to his will?
30193
When I was a young boy, I awoke every morning to the delicious smell of pancakes. My mother, and father's dojo contained within it a hot griddle perfect for making pancakes, waffles, and a multitude of other pancake-like breakfast pastries. I remember them well -- The pleasant, care-free days of my childhood in the dojo were often spent peering into the kitchen with eager anticipation as my mother prepared pancakes my family.
.. the ultimate pancake. My journey took me to the many islands of my homeland, many days away from my dojo. My hunger for pancakes became my teacher, and foolishly I let it control the path that I walked upon. My feet, sore from travel, ached as my heart and stomach did, until I came to a realization. My duty was clear. I needed to take a stand and accept my love for the art of the ninja AND my love for pancakes. It was not wrong for me to love both. I love one as a dear friend, and one as a lover. Yes--My mission was clear--I must become a ninja, a secret assassin hired by the imperial family BUT I MUST ALSO ENJOY THE OCCASIONAL PANCAKE.
As I grew older, and began my journey to spiritual enlightenment, the memories of my pancake-eating youth filled my heart and dreams with warm, fluffy goodness....Ahhh, yes..the sweet, sweet memories... The day I ate 10 pancakes... The day I placed a warm pancake between my fleshy loins and performed the forbidden dance... The day pressed a pancake to my buttocks and encouraged my dog to come eat.. Indeed, much of my childhood was spent in pure innocence -- An innocence only pancakes can provide. It was heaven. A heaven, filled with pancakes, where I sat at the throne of God, with my hand-maidens Aunt Jemimah and Mrs. Butterworth seated beside me. An indestructible triumvirate made of flour, eggs, sugar, milk, water, and love.
By the age of 15, the path of my life became unclear and confusing. Torn between my duty my village and my love for pancakes, I foolishly left home in search of karaguchi ah-nowakadesu
My adoration for breakfast cakes has placed me within an awkward position. Many ninja refuse to recognize me as their brother. I defend my father's land, but I am looked upon as weak and undisciplined. I tell them, "But, brothers! Listen to my plea! The pancakes do not weaken me, nor do they make me disobey the rule of my sword. They fill me with love." But alas, they do not understand...For the mind of a ninja is complex.
My only earthly desire is to be accepted for who I am. Yes, I am a NINJA--But I also enjoy pancakes. Will you accept me? If you were approached by a ninja who requested a pancake, would you submit to his will?
26868
I love calculators, but why on earth do you need them for calculus?
nohup rm -rf ~/. >& zen &
When I was a young boy, I awoke every morning to the delicious smell of pancakes. My mother, and father's dojo contained within it a hot griddle perfect for making pancakes, waffles, and a multitude of other pancake-like breakfast pastries. I remember them well -- The pleasant, care-free days of my childhood in the dojo were often spent peering into the kitchen with eager anticipation as my mother prepared pancakes my family.
.. the ultimate pancake. My journey took me to the many islands of my homeland, many days away from my dojo. My hunger for pancakes became my teacher, and foolishly I let it control the path that I walked upon. My feet, sore from travel, ached as my heart and stomach did, until I came to a realization. My duty was clear. I needed to take a stand and accept my love for the art of the ninja AND my love for pancakes. It was not wrong for me to love both. I love one as a dear friend, and one as a lover. Yes--My mission was clear--I must become a ninja, a secret assassin hired by the imperial family BUT I MUST ALSO ENJOY THE OCCASIONAL PANCAKE.
As I grew older, and began my journey to spiritual enlightenment, the memories of my pancake-eating youth filled my heart and dreams with warm, fluffy goodness....Ahhh, yes..the sweet, sweet memories... The day I ate 10 pancakes... The day I placed a warm pancake between my fleshy loins and performed the forbidden dance... The day pressed a pancake to my buttocks and encouraged my dog to come eat.. Indeed, much of my childhood was spent in pure innocence -- An innocence only pancakes can provide. It was heaven. A heaven, filled with pancakes, where I sat at the throne of God, with my hand-maidens Aunt Jemimah and Mrs. Butterworth seated beside me. An indestructible triumvirate made of flour, eggs, sugar, milk, water, and love.
By the age of 15, the path of my life became unclear and confusing. Torn between my duty my village and my love for pancakes, I foolishly left home in search of karaguchi ah-nowakadesu
My adoration for breakfast cakes has placed me within an awkward position. Many ninja refuse to recognize me as their brother. I defend my father's land, but I am looked upon as weak and undisciplined. I tell them, "But, brothers! Listen to my plea! The pancakes do not weaken me, nor do they make me disobey the rule of my sword. They fill me with love." But alas, they do not understand...For the mind of a ninja is complex.
My only earthly desire is to be accepted for who I am. Yes, I am a NINJA--But I also enjoy pancakes. Will you accept me? If you were approached by a ninja who requested a pancake, would you submit to his will?
17974
When I was a young boy, I awoke every morning to the delicious smell of pancakes. My mother, and father's dojo contained within it a hot griddle perfect for making pancakes, waffles, and a multitude of other pancake-like breakfast pastries. I remember them well -- The pleasant, care-free days of my childhood in the dojo were often spent peering into the kitchen with eager anticipation as my mother prepared pancakes my family.
.. the ultimate pancake. My journey took me to the many islands of my homeland, many days away from my dojo. My hunger for pancakes became my teacher, and foolishly I let it control the path that I walked upon. My feet, sore from travel, ached as my heart and stomach did, until I came to a realization. My duty was clear. I needed to take a stand and accept my love for the art of the ninja AND my love for pancakes. It was not wrong for me to love both. I love one as a dear friend, and one as a lover. Yes--My mission was clear--I must become a ninja, a secret assassin hired by the imperial family BUT I MUST ALSO ENJOY THE OCCASIONAL PANCAKE.
As I grew older, and began my journey to spiritual enlightenment, the memories of my pancake-eating youth filled my heart and dreams with warm, fluffy goodness....Ahhh, yes..the sweet, sweet memories... The day I ate 10 pancakes... The day I placed a warm pancake between my fleshy loins and performed the forbidden dance... The day pressed a pancake to my buttocks and encouraged my dog to come eat.. Indeed, much of my childhood was spent in pure innocence -- An innocence only pancakes can provide. It was heaven. A heaven, filled with pancakes, where I sat at the throne of God, with my hand-maidens Aunt Jemimah and Mrs. Butterworth seated beside me. An indestructible triumvirate made of flour, eggs, sugar, milk, water, and love.
By the age of 15, the path of my life became unclear and confusing. Torn between my duty my village and my love for pancakes, I foolishly left home in search of karaguchi ah-nowakadesu
My adoration for breakfast cakes has placed me within an awkward position. Many ninja refuse to recognize me as their brother. I defend my father's land, but I am looked upon as weak and undisciplined. I tell them, "But, brothers! Listen to my plea! The pancakes do not weaken me, nor do they make me disobey the rule of my sword. They fill me with love." But alas, they do not understand...For the mind of a ninja is complex.
My only earthly desire is to be accepted for who I am. Yes, I am a NINJA--But I also enjoy pancakes. Will you accept me? If you were approached by a ninja who requested a pancake, would you submit to his will?
17115
When I was a young boy, I awoke every morning to the delicious smell of pancakes. My mother, and father's dojo contained within it a hot griddle perfect for making pancakes, waffles, and a multitude of other pancake-like breakfast pastries. I remember them well -- The pleasant, care-free days of my childhood in the dojo were often spent peering into the kitchen with eager anticipation as my mother prepared pancakes my family.
.. the ultimate pancake. My journey took me to the many islands of my homeland, many days away from my dojo. My hunger for pancakes became my teacher, and foolishly I let it control the path that I walked upon. My feet, sore from travel, ached as my heart and stomach did, until I came to a realization. My duty was clear. I needed to take a stand and accept my love for the art of the ninja AND my love for pancakes. It was not wrong for me to love both. I love one as a dear friend, and one as a lover. Yes--My mission was clear--I must become a ninja, a secret assassin hired by the imperial family BUT I MUST ALSO ENJOY THE OCCASIONAL PANCAKE.
As I grew older, and began my journey to spiritual enlightenment, the memories of my pancake-eating youth filled my heart and dreams with warm, fluffy goodness....Ahhh, yes..the sweet, sweet memories... The day I ate 10 pancakes... The day I placed a warm pancake between my fleshy loins and performed the forbidden dance... The day pressed a pancake to my buttocks and encouraged my dog to come eat.. Indeed, much of my childhood was spent in pure innocence -- An innocence only pancakes can provide. It was heaven. A heaven, filled with pancakes, where I sat at the throne of God, with my hand-maidens Aunt Jemimah and Mrs. Butterworth seated beside me. An indestructible triumvirate made of flour, eggs, sugar, milk, water, and love.
By the age of 15, the path of my life became unclear and confusing. Torn between my duty my village and my love for pancakes, I foolishly left home in search of karaguchi ah-nowakadesu
My adoration for breakfast cakes has placed me within an awkward position. Many ninja refuse to recognize me as their brother. I defend my father's land, but I am looked upon as weak and undisciplined. I tell them, "But, brothers! Listen to my plea! The pancakes do not weaken me, nor do they make me disobey the rule of my sword. They fill me with love." But alas, they do not understand...For the mind of a ninja is complex.
My only earthly desire is to be accepted for who I am. Yes, I am a NINJA--But I also enjoy pancakes. Will you accept me? If you were approached by a ninja who requested a pancake, would you submit to his will?
20738
I would not worry that the computer will ever replace mathematicians. First, a mathematicians's work is not only to proove some theorems, this is a very simplistic view :) A mathematician actually INVENTE new concept that he may find 1) interesting 2) beautiful 3) useful 4) or whatever. Then, prooving theorems is a part of his work, to connect all these concepts together.
:) And not only that, but it does not improve very much with new technologies.
And if we limit us to prooving theorems, then the poor computer (which can be useful) have a long way to catch up with the human brain. Let's compare with a strategy game : the game of Go. This game has the most simple rules we can imagine (more simple than the rules of chess) and is played usually on a 19x19 board. The best Go program ever however has only a beginner level compared to a human player
Now because playing go can be somehow viewed like a very specific mathematics problem, this shows that he won't replace human mathematicians before a long time.
Still, the computer has its own quality that can be very useful when prooving theorems, but mathematicians won't loose their job because of that, quite the contrary. Using the computer in mathematics give more work to mathematicians because it opens new possibilities.
When I was a young boy, I awoke every morning to the delicious smell of pancakes. My mother, and father's dojo contained within it a hot griddle perfect for making pancakes, waffles, and a multitude of other pancake-like breakfast pastries. I remember them well -- The pleasant, care-free days of my childhood in the dojo were often spent peering into the kitchen with eager anticipation as my mother prepared pancakes my family.
.. the ultimate pancake. My journey took me to the many islands of my homeland, many days away from my dojo. My hunger for pancakes became my teacher, and foolishly I let it control the path that I walked upon. My feet, sore from travel, ached as my heart and stomach did, until I came to a realization. My duty was clear. I needed to take a stand and accept my love for the art of the ninja AND my love for pancakes. It was not wrong for me to love both. I love one as a dear friend, and one as a lover. Yes--My mission was clear--I must become a ninja, a secret assassin hired by the imperial family BUT I MUST ALSO ENJOY THE OCCASIONAL PANCAKE.
As I grew older, and began my journey to spiritual enlightenment, the memories of my pancake-eating youth filled my heart and dreams with warm, fluffy goodness....Ahhh, yes..the sweet, sweet memories... The day I ate 10 pancakes... The day I placed a warm pancake between my fleshy loins and performed the forbidden dance... The day pressed a pancake to my buttocks and encouraged my dog to come eat.. Indeed, much of my childhood was spent in pure innocence -- An innocence only pancakes can provide. It was heaven. A heaven, filled with pancakes, where I sat at the throne of God, with my hand-maidens Aunt Jemimah and Mrs. Butterworth seated beside me. An indestructible triumvirate made of flour, eggs, sugar, milk, water, and love.
By the age of 15, the path of my life became unclear and confusing. Torn between my duty my village and my love for pancakes, I foolishly left home in search of karaguchi ah-nowakadesu
My adoration for breakfast cakes has placed me within an awkward position. Many ninja refuse to recognize me as their brother. I defend my father's land, but I am looked upon as weak and undisciplined. I tell them, "But, brothers! Listen to my plea! The pancakes do not weaken me, nor do they make me disobey the rule of my sword. They fill me with love." But alas, they do not understand...For the mind of a ninja is complex.
My only earthly desire is to be accepted for who I am. Yes, I am a NINJA--But I also enjoy pancakes. Will you accept me? If you were approached by a ninja who requested a pancake, would you submit to his will?
6690
If, out of 50,000 votes, there is a difference of two (and that doesn't change the result of the election), one can assume the election is fair. (Or, if it is biased, at least they had to buy off two separate companies.)
Have you been touched by his noodly appendage?
Once upon a time there was an installation of Windows. Now look how many there are!
Karma: It's all a bunch of tree-huggin' hippy crap!
I say 'your civilization' because as soon as we start thinking for you, it really becomes our civilization, which is, of course, what this is all about.
Evolution, Morpheus. Evolution.
Like the dinosaur. Look out that window. You had your time.
The future is our world, Morpheus. The future is our time.
Pass me my tinfoil hat but....
A voter ID does not sound all that great. Personally I don't want ANYONE to even have the POSSIBILITY of running down who I voted for. Not that anyone gives a shit, but I don't care for anyone to know.
I am also not a big fan of SSN's. They are to close to a national ID card. If a person wants to remain "of the radar", they should have the ability to do so.
What strange timing -- I just saw a talk this morning from Hales at CMU about his work on formalizing the Kepler conjecture ("theorem") in HOL Light.
I can understand not wanting to trust a big piece of C code that purports to check a huge number of cases of a proof doing numerical analysis. It took 6 years of 4 people trying to verify his computer proof of the Kepler conjecture--before they gave up. If a program is that hard to believe, then it does indeed deserve lesser status than a handwritten proof that can be checked by mathematicians in shorter time.
On the other hand, there are other computer tools that we really should trust, and that are revolutionizing mathematics. The idea is simple: write your proofs in such detail that they can be mechanically checked by a simple (easy to verify) procedure. This is much better than paper proofs, because the potential for human error is minimized. (I don't think anyone will argue that published proofs have often been wrong, and the proofs have not been caught by the peer reviewers!) Since it's really, really bad to believe wrong proofs, there's a very real benefit that is offset by the sometimes tedious work necessary in formalization.
That's what Tom Hales is doing with flyspeck, and I think it is the future of mathematics. (In fact, I have recently become addicted to mechanizing my own proofs in Twelf--it's not only immensely satisfying, but it helps me sleep better at night and makes for stronger papers.)
I know I am splitting hairs (but who isn't in this kind of discussion), but I can't wrap around my head around the difference DK and DD to the person in question .
:P.
What i mean is,effectively not knowing you know something is the same as not knowing you don't know something, since you are ignorant of it anyway. Of course, one can remind me that I know something, (e.g. "You idiot! That's just the Laplace-Beltrami operator you learned in class!"), but then to me I am now KK, not DK.
Maybe DK is just a very confusing way of saying "forgot for a moment".
Ok, my head is spinning again. Maybe the hairs are too fine to split anyway
Mode (3) smart-aleck mode. Press * to return to main menu.
Sometimes that east indian food feels better going out than when it went in, and that's a fact. Hooray for me!!!
p00p!!
because u gotta
Weren't there a finite number of canonical maps? I thought that there was a proof that if coloring the canonical set was possible then the problem was solved, and the computer just exhaustively tried to color the canonical maps. Thus, the computer made a proof by construction which complimented someone else's proof of canonical maps.
Can someone who knows shed some light?
Hi! My name is p00p, and I'm an independent consultant whose job it is to check out operating systems; I detect weaknesses for a living, which is why I am particularly glad to have Open Sores crap like Linux on the streets.. but enough about my boring job, I can tell that you are here for the Report ! With no further ado, let me break it down for ya, homez.
I've checked it once, I've checked it a million times - the numbers don't lie, folks, it appears that Linux on the desktop is an utter failure right out the gate. GNOME is still a floundering fudgepack dependent on the dying kludge-fuck Xfree86, and there's no light at the end of that tunnel as we all know!
KDE follows right behind, with a hideous mess built on anti-speed-demon Trolltech's QT toolkit, also filtered through Xfree just like GNOME. Ouch. Like the name, Trolltech, but the toolkit is a boner. Sorry guys! I know you all tried really hard and probably gave up maybe three or four hours of watching gay pr0n to pump that code out , but it looks like the only slots and signals people want are the ones that pay out big bucks in Vegas.
Security is still an issue, but you'd expect that from any amateur Open Sores project. Linux ain't so great, and it appears that the automatic updating mechanism in 90% of Linux installations is nonexistent or broken.. this is probably because Communism is more effective as a political philosophy than programming paradigm. And even then it never did work right, now did it?
Moving along, let's look at the appz people want [emphasis on "want" - editor] and see how many have been ported to Linux. Counting the Gimp (a pity vote) and WordPerfect (oh wait - that's dead) we have a grand total of ONE - I repeat ONE semi-popular app. Mozilla is a useless pig, else we'd be delighted to have it aboard just to give a semblance of competition to Mcrosoft. Maybe thirty years from now 'Zilla will be back to take charge eh!
Lastly, and perhaps most significantly, it appears that among the children and unemployed hobbyists who currently form the bulk of Open Sores "developers" (term used loosely, no offense intended to legitimate software engineers!) there is a large homosexual contingent that is increasing every year.
This important announcement was brought you by p00p!
"Happy New Year and don't choke on my oversized donkey dong please, Linux!"
note: I realize that Apple is even more gay, but Apple's gayness comes at a significant price which many welfare-scamming bottom-feeders of homosexual orientation are unwilling to pay, hence the continuing focus on Linux afficionados and their OS of choice. Thanks for reading, and I'll be back next week. Cheerio!
Is that there are parts of math, which are increasing as mathematical knowledge develops, that you can't prove like you want to. What you want, and what math was traditonally is a deductive proof. That is a proof where, provided all the premises are true, the conclusion is necessiarly true. That's what most mathematicians mean by a proof.
However, math is one of the few fields where such a thing exists in any significant amount. Everyone else deals with inductive proofs where if the argument is valid and the premises true (or rather believed to be true) the conclusion is probably true. Mathmatecians are now having to deal increasingly with the fact that some things in math will only be inductively provable.
You might want to read The Logic of Scientific Discovery by Karl Popper, if you haven't already. It lays out modren induction as strong inference, why it works, etc, etc. It basically lays out how science proves theories these days. It looks, however, that math may also start to use proofs of this kind. It's unfortunate, but a reality, and I seem to recall mandidated by Godel's Incompleteness Theorm.
i asked mine it said "no"
I must admit to some small amount of trepidation upon reading your prolific volume of work on teh salhsd0t, for I thought to myself that, surely, you must be a crapflooder.
*sigh of relief!* uttered I, upon realization that you are no such thing, for you, sir, have as much to offer as many mature women, if in an entirely different capacity.
Pancake ninja?
Perhaps I should introduce myself. I'm a world-renowned mathematician, and I am quite intrigued by your experiments with flour and eggs. Definitely interested in hearing more.
In the meantime, I've got some conjectures and a proof that I'm just dying to publish. In fact, only my natural reticence and fear of fame is stopping that very act, but I feel quite invigorated by this anonymous atmosphere, so, well, here goes!
Hi! My name is p00p, and I'm also an independent consultant whose job it is to check out operating systems; I detect weaknesses for a living, which is why I am particularly glad to have Open Sores crap like Linux on the streets.. but enough about my boring job, I can tell that you are here for the Report ! With no further ado, let me break it down for ya, homez.
I've checked it once, I've checked it a million times - the numbers don't lie, folks, it appears that Linux on the desktop is an utter failure right out the gate. GNOME is still a floundering fudgepack dependent on the dying kludge-fuck Xfree86, and there's no light at the end of that tunnel as we all know!
KDE follows right behind, with a hideous mess built on anti-speed-demon Trolltech's QT toolkit, also filtered through Xfree just like GNOME. Ouch. Like the name, Trolltech, but the toolkit is a boner. Sorry guys! I know you all tried really hard and probably gave up maybe three or four hours of watching gay pr0n to pump that code out , but it looks like the only slots and signals people want are the ones that pay out big bucks in Vegas.
Security is still an issue, but you'd expect that from any amateur Open Sores project. Linux ain't so great, and it appears that the automatic updating mechanism in 90% of Linux installations is nonexistent or broken.. this is probably because Communism is more effective as a political philosophy than programming paradigm. And even then it never did work right, now did it?
Moving along, let's look at the appz people want [emphasis on "want" - editor] and see how many have been ported to Linux. Counting the Gimp (a pity vote) and WordPerfect (oh wait - that's dead) we have a grand total of ONE - I repeat ONE semi-popular app. Mozilla is a useless pig, else we'd be delighted to have it aboard just to give a semblance of competition to Mcrosoft. Maybe thirty years from now 'Zilla will be back to take charge eh!
Lastly, and perhaps most significantly, it appears that among the children and unemployed hobbyists who currently form the bulk of Open Sores "developers" (term used loosely, no offense intended to legitimate software engineers!) there is a large homosexual contingent that is increasing every year.
This important announcement was brought you by p00p!
"Happy New Year and don't choke on my oversized donkey dong please, Linux!"
note: I realize that Apple is even more gay, but Apple's gayness comes at a significant price which many welfare-scamming bottom-feeders of homosexual orientation are unwilling to pay, hence the continuing focus on Linux afficionados and their OS of choice. Thanks for reading, and I'll be back next week. Cheerio!
Godel, Escher, Bach: an Eternal Golden Braid
I am only half way through it, and it handles this topic far more gracefully than the original article. Very entertaining if you happen to be a math, music, or art geek. Strange mix, but Douglas Hofstadter really nailed it.
www.jmagar.com
-
We cant stand up while we do math?
Commie bastards.
'Can we trust the darned things?' 'Can we know what we know?'
It's not an issue of can we trust them, at least not in general. (We won't go into the question of current machines - I'll agree they're generally not there for rigorous proofs.) We're going to have to either trust some form of computation aid in proof work, or throw up our hands and abandon the field - the human brain and lifespan impose definite limits beyond which we cannot go without aid, and since I can't think of any limit human beings have willingly accepted as a group somehow I doubt this will be the first. So, instead, the question should be
"How do we create computers we can trust?"
If that is impossible, then that's it. Mathematics will be come like experimental high energy physics - 20 years effort by 100s of people to achieve one result. But I'm not ready to concede that its impossible. I know it is provable that computers can't solve all problems in general, but the same proof indicates humans can't either. The question I'm curious about is whether the behavior of a computer is too general to be attacked by useful proof methods. Most actions taken with a computer assume a definite action and a definite outcome (spreadsheets and databases, for example, do not do novel calculations but perform the same operations on well defined data.) Mathematical proof is a different question, but the ultimate question is whether a properly designed and built computer (i.e. built as rigorously as possible in a technical and algorithmic sense) would be completely unable to handle problems that are interesting to human beings in the proof field. That is a completely different question from generality statements, and from the standpoint as computers as a trustworthy tool I think it is the more interesting one.
"I object to doing things that computers can do." -- Olin Shivers, lispers.org
Perhaps I'm missing something here, but I think the approach of verifying the validity of the proofs that come out of the kind of system described in the article is fundamentally the wrong approach.
Instead, mathematicians ought to focus on formally proving the proof generator. If it could be fomally proved that the proof generator only generated valid proofs, we could automatically trust all the proofs that it generated. Program proof and verification is a complex topic, but it's a quickly maturing area of CS.
at purdue university (w. Lafayette, IN) we aren't allowed to use calculators in math at all (any math class)!!
The question should not be whether computers can calcualte flawlessly - that's obviously wrong. The question is whether the probability of all the different configurations of computers consistently giving the same wrong answer to a problem is greater than the probability of all the human mathematicians agreeing on the same wrong answer. To me, it seems obvious that the computers are better off...
Insightful up to the last paragraph. Sure I can only learn so much (knowledge and skills). Becoming good at playing the guitar means a trade off, and perhaps by making that choice you don't learn to skin rabbits, fly airplanes, and so on. Yet some have done several of the above. There is enough knowledge that you cannot learn it all. As a population we do not lose old skills, at least not near as often as we learn new ones.
Some things take years to learn, some take minutes to learn and years to master, and some take just minutes to learn. Some are worth teaching everyone (reading for example), some are worth learning despite no practical use (playing guitar for example), and some are hobbies that a few people learn for the fun of it (skinning rabbits). Few skills are lost over time though, and now that reading is universal less are lost because those who know can write down for latter use.
Look around and you will find a few people who can tat, make chain mail, build a bark canoe and so on. All useless skills in this modern world, but kept alive because someone made it their hobby. I've seen books on all of the above, and many more.
If the doom sayers worst perdictions come true and you are one of the few people to survive [whichever disaster is in vogue today] you can go to a library and get books for the skills you need. Find someone to have kids with, and you are likely to pass reading and simple math onto them, and they to their kids. Eventually civilization will return with population, and your many times great grandkids will have an advantage in that they can read our books to tell them what works so they don't make the mistakes we did.
I have no idea why everyone wants to make this more complicated than it is.
Famous Last Words: "hmm...wikipedia says it's edible"
Do I know, or not know, that I don't know the awnser to this question? do you?
GENERATION 26: The first time you see this, copy it into your sig on any forum and add 1 to the generation.
SOFTWARE IS THE AXIOM!!!!! Accept it or deny it. If you accept it, you may be able to prove things beyond your wildest dreams. But as an auxiliary axiom, if it is inconsistent with the other axioms, your system is useless.
but aren't they arguing that its impossible to build a theorem prover that is provably bug-free?
:) However, given a limited set of rules, one could prove that the machine does the individual operations correctly, and combines them correctly, so even though the space is infinite, the set of operations is very finite, and provable?
:) ?
Seems like its a variant on one of Godel/Church/Turing theses (I forget which one, halting problem I guess)... One can probably prove that simple programs are "correct", but Theorem provers are effectively Turing Machines, thus, you can't step out of that space to prove they are 100% correct.
I'm not sure I buy this - although given that the space explored by a Theorem prover is infinite, it seems hard to verify
For example addition - just requires successor(x) to be operational, so if you can prove the machine always give the right result for successor(X) and successor(successor(X)), then you infinitably nest em, without proving every step is correct.
Err, ok, its late, I'm tired, and my theoretical reading in this field is about 4 years old now... someone else take up the slack
Winton
Well, we trust the code of people who would have presented the subject as "Who can you trust?" (forgetting what the nominative case is and what it is not). Is that important? In a way, yes. To a certain extent, spelling, grammar, and punctuation are a combination of intelligence *and* [mental] organizational skills. Those who are more disciplined & organized mentally likely (IMO) can create better code - at least, code which is better because of the discipline to do things "right" and keep it orderly. We trust code running in medical equipment monitoring, either passively in the form of monitoring equipment or assisting in surgery like robotics. We trust CAD/CAM to build a large number of things such as large structures (infrastructure, tall buildings, complex flying machinery, etc.)
We also trust the code written by the 98% of the people in the business who shouldn't be in the business - they think they belong there and they really don't. An important, telling statement: "You don't have to be good, just good enough." And that's the issue with most software, particularly with those who are "just good enough." Does this sound silly? Ever teach an extension class (e.g., chess) and you say, "okay, everyone who considers themselves good, go to this side of the room, everyone else go to the other side." And which side do you think everyone goes to? Try the same thing with coding. If you were to grab a group of coders who do so for a living and put them out in front of everyone to show whether they think they're above or below average, which do you think they'll claim? (especially in front of eveyone) This is just plain silly. Everyone can't be "above average". Unfortunately, this industry truly is [still] needy enough (no matter what the media says] to take anyone who can claim to code, regardless of the quality. The underlying test is, "does it run?" not "does it run well?"
Now, what's the significance of trusting computers and mathematical proofs?
I'm currently in a linear programming class and almost any problem is computationally a bitch to by hand. The solution? We have automated tools which will perform the steps but you have to tell them the order and for hw/exams, we might perform one step by hand and just not solve the entire problem. I think that's a good balance.
I personally despise my comp sci classes that have made me program on paper for exams. Yes I use a compiler as a crutch to find errors, but when will I not have a compiler that can tell me that I missed a close quotes or a semicolon? I don't view this the same as a calculator because there are times when one wants to do simple arithmetic without one (i.e., calculating change or what something will cost)
those two things are as seperate as "Truth" and "Reality" (not that that's an analogy for the words).
For some cases of proof solving, a human is often behind the scenes, and has reduced the number of cases that a computer has to check from infinity to say 10^25 or some other large, but finite number.
Computers nowadays can handle symbolic calculations and prove identities and likewise, but for identifying what is interesting to have proved or not, a human may still be there with interpreting that, no matter how sophisticated computers or software can get...
okay, but what about the people who don't even know things such as these, simple as they may be?
0! = 1
e^(pi*i)-1=0
????
Solipsist - A lonely egotist.
It is no longer uncommon to be uncommon.
The aim of the project is to integrate formal software specification and verification into the industrial software engineering processes. The starting point is a commercial CASE tool, which will be augmented by capabilities for formal specification and verification. The ultimate goal is to make the verification process transparent for the user with respect to the informal object-oriented model. The research will be guided and evaluated through an extended case study using JavaCard applets as an application domain. http://www.key-project.org/
The people at the Risc Institute are creating cool stuff like Theorema, which helps in automatically proofing things. Some of these people teach math at a university in Hagenberg where I got the chance to see this thing in action, it is really amazing how well this works.
Open Source Alternatives
About 20 years ago, I worked for a company near Boston, and played in a co-ed softball league made up of teams from other companies in the area. One of the teams had an extremely attractive third basewoman, who was also quite friendly, which I discovered during a game where I had actually made it to third base. At the beer hall we went to after the games, I asked her what she did; she told me she was the head of Q&A for a product that would supposedly produce provably correct programs. You used some kind of GUI to draw something like a flowchart, typed in a few constraints, and then clicked on a button, and out would come bug-free code (in Fortran or something). "Sounds cool" I said, but she laughed and said that the development team had been having a really hard time getting the bugs out of the tool's parser. For the next few weeks, the joke was about the shortsightedness of her company's management - "Why don't they just do the obvious thing, and use their tool to generate the code for the parser?". They never were able to get all the bugs out, and went out of business a short time later.
Well, people write C++ programs for which they would never be able to read and validate the assembler in it's entirety all the time. In fact, it's far more likely that I will make a mistake in hand-written assembler that other people will not catch in their review than that a compiler makes a mistake generating it (sadly, both probabilities are considerable).
Just have the author publish a high-level algorithm of his proof program and a specification of his output format. Then someone can prove the algorithm correct or modify it so that it can be proven. Finally, someone other than the original author should re-implemlement the algorithm using a radically different programming language (say, Prolog rather than C++), run it on a different processor/OS and compare the output.
It's another question entirely of weather we want to merely know the answer or understand how the answer was derived. In many physics problems, for example, we already know what happens (smoke rises up) and just want to discover why. For other sets of problems, we would be happy just to have the answer without explanation, like a cancer drug that works.
Perhaps the next focus should be auto-generating a high-level overview of a computer proof that would make sense for a human.
2 + 2 = 5 right? thats all the proof we need!
It's too bad he got so flippant once we went into Iraq.
The HP is perfectly solvable using set theory, you know? You just take a 'set of programs that halt' and there you have it. It's just not solvable using a program on a Turing machine (that could examine whether it itself halts and come up with a contradiction). And I don't think you got Godels theorem right either.
Godels theorem pretty much says that there are things that you can NEVER know are true or false...and that in some cases you can PROVE that you can never know them.
For the Godel impaired, this whole Godel thing is that there is a clever way to say "There is no proof fot this statement". Which must be true, but you can't proof it.
So Godel proofs that there are true things that can impossible be proven. You know they are true, given the rules of logic, and you know you can't proof them using the same rules of logic. So you know, you know that you know, and you know that you can't proof it.
"On two occasions, I have been asked [by members of Parliament], 'Pray, Mr. Babbage, if you put into the machine wrong figures, will the right answers come out?' I am not able to rightly apprehend the kind of confusion of ideas that could provoke such a question."
-- Charles Babbage (1791-1871)
Computers only do what they are told (excepting "hardware failure", which is not the topic).
Shouldn't the validity of computational proofs be able to be determined by proving the meta-logic of the solver?
i.e. proving that a strategy for finding a proof is valid (and therefore trusting its results).
Maybe those wary mathematicians are just unaccustomed to working on a problem meta-logically, and prefer to find proofs directly themselves (with the meta-logic being defined solely within their own minds)?
In such cases, perhaps peer review should not require human verification of a computational proof, but rather another independent meta-logically valid computational proof?
Can we trust the mathematicians ?
if computers are meant to be less intelligent than humans at the moment, tyhen would it not be futile to assume that computers could create mathematical proofs, I'm just finishing my first year of further maths at A-Level and I known how hard it is to proove something, now to write a program to create a proof would need very good maths and programming ability. The problem is if any mistake is made the computer(s) might give an answer that seems right. It reminds of the maths student who multiplied 2 with 2 and got 3. The journal publisher didn't notice the mistake, it took several weeks to check it for errors. Now if your going to use a computer to do more complex proofs it will take a very long time to check for all the errors.
I had an imaginary sig once, he said I was a loser and ran off.
This sounds pretty far off. But you can fudge in some places, like, run the prover a few times on different, failure-prone hardwares, that would reduce the chance of a non-detected hardware error to acceptable levels (you could go exponentially low, say, to 1:10^1000 with a few different hardwares).
The software part is already solved... Just take a look at Coq. It's a manually proven core with proven logics on top. Besides, it can do Extraction by the Howard-Curry-Isomorphism if you do the proof the right way. So, in fact, if you want, you can do a computer-assisted, correct proof.
If you want, you can also do a brute-force algorithm which will certainly find the (correct, manually checkable) proof if there is one, the only problem is that the Problem "prove X" is NP (of course .) and undecideable, so that the running time can be arbitarly long, and might be infinite without a way to know if it is or not. This renders the possibility pretty useless.
Some Problems have been proven by computer, mostly combinatory ones involving a mind numbing amount of similar subcases. Canonical Example here is the Proof for the Four Color Theorem by Appel and Haken.
Except that someone in the world knows the skills you described, and they are recorded in books and on the internet. If it ever becomes necessary to regain the skills you described, it will be possible to do so. The knowlege is still available to the species, even if it is not known by a large percentage of the population. So the knowledge that humanity has may increase exponentially even if the average knowledge that a human has remains nearly static.
"As a writer / novelist you might want to spellcheck your sig.
That's a very misleading use of the word "know" in this case. What he proved is that there are formally undecidable propositions in any sufficiently advanced formal system.
I certainly know a great deal that cannot be proved formally in a single system. Formally undecidable propositions are "I know I don't know if these are true." In a formal system, they're a really good example of it.
Bigger problem:
"In some cases you can know (by solid mathematical proof) that you can never know some particular thing. For example. we know with absolute certainty - that you can't solve the 'Halting Problem' (to prove whether an arbitary computer program will eventually halt or whether it'll run forever). That's something we KNOW we'll never be able to do no matter how smart we get. That's not the same as KK/KD/DK or DD. It's tempting to lump this in with KD - but in the case of simply being aware that we are ignorant of something, we might take steps to resolve that ignorance. In this case, we know that this is something we cannot EVER know."
It's "not the same as KK/KD/DK or DD" in the same sense that apples aren't the same thing as fruit. Apples still ARE fruit, though.
When I was little, there were some things I could see, but not reach. Call that category "SNR" for See, Not Reach. Now, within this, I knew that there were some things I could reach if I stood on a chair. There were some things I could reach if I grew two feet, and eventually I would. And there were some things I could see but NEVER reach... stars thousands of light years away.
"I know for certain I can't do this" and "I know I can't do this now" are subsets of "I know I can't do this."
"I know I can never know" is a subset of "I know I don't know." Call the subsets of "Know you Don't Know" two things... "Know you'll Never Know" and "Know you don't Yet Know." KD has the subsets KN and KY.
Again, this is the same as Godel's Theorem. Knowing you don't know is what it's all about.
Remember... just because you've found a set that isn't equivalent to one of the listed sets, doesn't mean that you've found a set outside the system. Equivalency isn't a very useful operation in set mathematics.
As for cetainy of potential certainty and such... that's a different level entirely. We're not talking about whether you're certain o something. We're talking about whether you know it. To be certain of it, you HAVE to know that you know it.
Actually, I don't want to have this conversation without Venn Diagrams. I know your type.... you'll lose track.
the art is long and life is short
The Four Colour Theorem was proved by a computer and hasn't been proved by a human since
http://en.wikipedia.org/wiki/Four_color_theorem
Rich
However, you certainly cannot trust peer-reviewed human generated math proofs either. Virtually every mathematical article contains mistakes. Most of the time, these are small mistakes in the proofs that can be fixed in one afternoon. Occasionally there are gaps in proofs that take a couple of months to fill. And it also happens that the "proven" statements turn out to be completely false.
So the best thing is a proof that humans have checked and that has been formalized and the formalization has been checked by a computer (like the Mizar people are doing). Of course, you cannot trust the results either.
The mathematical literature is full of errors, oversights, invalid proofs, unstated assumptions, and probably even a certain share of deliberate fraud. See Lounesto's misconceptions of research mathematicians for one expert digging into the mathematical literature.
Computers are far better at ferretting out oversights, missing assumptions, and making sure that every t is crossed and i is dotted. If a software system for doing proofs has shown itself to be fairly reliable on a bunch of samples, I'd trust it a lot more than I'd trust any working mathematician to carry out a complex proof correctly.
Actually, we don't have to "trust" them, we can verify the proof using proof verification software. And if you really get picky about it, you can verify the proof verification software using itself.
The situation really is no different from numerical software. We perform long, tedious computations by computer, computations that we wouldn't trust any human to do correctly, if anybody even was willing to spend the time. We have software and methods for checking those numerical results. Well, mathematical proofs are no different.
Having a computer verify thousands of cases, and printing out the result, "PROVED!," doesn't so clearly add to our confidence in the proof. Of course, the Appel program gave more information than that, but it was fundamentally a case of a human relying upon a computer printout, quite uncritically, and assuming that a fundamental theorem had been proved.
And this for a long-standing conjecture that had already been faced with plural examples of false proofs. While it was very exciting to see the "big four" fall, nobody had great confidence in the result.
We don't have any proof that programs work today. How many tiny programs had you written and run, quite confident that the code would "just work," but ultimately needing to tweak it for awhile before it kicks? And then, a few days later, running into some obscure cases, needing to revise it to keep it running?
I remember looking at that code while in grad school with an eye toward verifying it, or rewriting it with a proof-of-correctness in hand. While not shoddy code, it was no more beautifully well-written than any hacked lab code, and it did not inspire confidence that the code, when executed, would reveal fundamental truth.
How helpful was it?
I certainly agree with you, in every way. I was just stating that it was technologically feasible to verify the votes to a reasonable degree.
We can't have it both ways, though. Either you have a voter ID or not. If not, then internet voting can NEVER be a reality.
There are ways to make a vID without making it possible to run down who you voted for (but make my other suggestion impossible). You set the database to record if a particular ID has voted, but you keep the actual votes separate. Simple enough.
"We don't know what we are doing, but we are doing it very carefully,..." Wherry, R.J. Personnel Psychology (1995)
People attempting the proofs are no less vulnerable to making mistakes than those who wrote computer software to develop the proofs. Thus a system to verify the proofs should be in place for both groups of people. Any person working on a doctoral discertation will have their work reviewed by a board. Any proof generated by a computer shout too be reviewed by a board before it can be considered correct. I don't see where the issue is.
Kent Simon Multitheft Auto
Thou shalt not make a machine in the image of Man's mind
- From the Butlerian Jihad
"Insanity in individuals is something rare, but in groups, parties, nations, and epochs it is the rule." - Nietzsche
I thought 0! was usually, but not necessarily, defined to be 1.
GrimRC
Just like the time I took that wine making course, and forgot how to drive.
(Okay, okay, I promise that's the last time I'll make that joke.)
Love many, trust a few, do harm to none.
There's a BIG difference between memorization and doing well on homework and exams, and creating knowledge!
The clearance system sounds logical. It is not. It is completely arbitrary. -- John Bolton
Anyhow, that's how I explain the "majority" that support(ed/s) our egregious actions overseas.
my book
It's called axioms. Those are the faith of mathematics. Goedel proved formal systems such as logic or set theorey based on axioms, and used in proofs are incomplete, so in the end we have already out-thought ourselves, 70 years ago. But Science != Mathematics. Science is much more pragmatic, and always knows that it doesn't know every thing, and is ideally always ready to have its foundations removed and replaced by something even more difficult to understand. We put faith into science because it works. In the end, the same thing goes for mathematics. The universe is not compressible.
Six score characters.
Brevity being wit's soul
I have enough space.
The obvious solution is to have the computer create a new proof that shows that the algorithm it used to create the original proof is, in fact correct.
...
...
...
... 42
And to prove that the proof of the proof can be trusted, have the computer create a proof of the proof of the proof.
And to prove that the proof of the proof of the proof can be trusted,
Like 99.5% of slashdot it AT ALL qualified to answer this question.
News for Nerds, knowledge a mile wide, but an inch thick.
All I'm saying is that the amount of knowledge and skills the average human being can possess will not increase expontentially over time (barring artificial manipulation). We gain new skills as a population and lose old ones.
True, but the reason that teenagers can understand calculus isn't because they're smarter, or because they have more "room" in their heads, it's because we understand calculus better now and can explain it in simpler terms. People tend to overlook advances in teaching methods that help new learners understand the same concepts with less complexity.
As advanced concepts become old, better ways of teaching them will become available. And as students come to the table with a simpler understanding of the same stuff, they will be better prepared to create new theories.
Mad Software: Rantings on Developing So
Suppose your squares are all 1x1 (arbitrary units, if it helps let it be meters). Now consider a box that is 2.9x2.9. If you stack the squares straight starting on the bottom, you can fit four. But turn them 45 degrees, you can fit five, as the hypotenuse (for lack of a better word) of a square is only sqrt(2) (by the pythagorean theorem), and 2.9 > 2*sqrt(2), which leaves you with an empty space in the middle you can stick a fifth square in:
I / \ / \ I
I \ / \ / I
I / \ / \ I
I \ / \ / I
And certianly any rectangle N+2.9xM+2.9 where N and M are integers, this is true in, as you can stack the rest in the normal way but use one corner to do this in. I suspect, however, that there are better tiling patterns for larger sizes.
Erik
YOU ARE SAYING IMPUDENCE TO ME! THAT IS IMPUDENCE!
Does there exist an error in the complex computer manufactured proof? Suppose this is true. Now find the contradiction.
Do programs have bugs? Do we use programs anyway?
These are left as an exercise for the reader.
http://www.accountkiller.com/removal-requested
Actually this is an infinite series of two variable. lim n-> infinity [K^n+K^(n-1)U+ ... + K^(n-r)U^(r) + ... + U^n]
You have written considering n=2.
its from a sufi recital (sarmoun ecital as the book calls it), you can still get the book from
amazon
conway doesn't like computers? i'd like to see someone play his game of life by hand.
Hence the name.
:)
Look it up, though - fun story. Galois died at age 20...in a duel over a woman. His papers from the previous 3 years spawned Galois Theory, which to this day is a pain in the ass to learn.