Nice straw man. What you said was, and I quote: The axioms on which the theorems are built are explicitly part of the theorems, leading to a logically self-consistent system.
Godel says that you cannot show the truth of that claim, ergo the claim is not necessarily true. Indeed, I'm fairly sure it's a simple matter to come up with a set of axioms that is inconsistent. You are mistaken.
You misunderstand --- Godel did not say that no statements can be proven.
Essentially, a mathematical system is consistent if there exists no theorem which states both P and NOT P are true. Then he showed that in ALL logical/mathematical systems capable of supporting integer arithmetic, there exists no proof that the system is consistent. We all tacitly assume that our maths are consistent, but the maths themselves cannot prove this. Godel leaves open the possibility that a system may prove its own inconsistency.
Also, from philosophy standpoint, math relies on lots of nontrivial physical observations that humans take for granted (e.g. of paper, of mental states, etc).
What physical observations led to Galois Theory, Lie Groups, Hilbert Spaces, Zorn's Lemma, and a host of others? Nowadays, the bulk of mathematics is highly abstract and far removed from physical phenomena.
Actually, math is completely self-consistent given a set of axioms.
That may well be, but Godel showed that any logical system strong enough to support the usual arithmetic on the integers cannot establish its own consistency. That is, it is entirely possible that a mathematical system may not be self-consistent.
... I must insist that God has harmed us by not disclosing this information. We need it to fight the diseases which God allows to plague us, and to heal injuries that God allows to happen to us. This information qualifies as critical, need-to-know information, and if God won't give it to us, then we have no choice but to figure it out for ourselves.
In this hypothetical situation, wouldn't we be guilty of violating the will of God? What type of response would that provoke? I seem to recall a tale from the Old Testament regarding God's disposition towards men who had gotten to big for their pants. Maybe God or Mother Nature invented human disease for a reason. Have you considered the consequences of completely eliminating disease? If no one were to die of disease or genetic abnormality, more people would probably die of starvation or war. Unchecked population growth would demand new and creative ways to thin the herd. Logan's Run anyone?
Do we act in a manner to minimize loss of life? For if that is the case, we could ban automobiles and subsidize health care for everyone. I don't see that happening, which leads me to the conclusion that protecting peoples' lives is not the highest priority. The government spins it that way, but their actions say otherwise.
How about ensuring that our legal and civil rights are not trampled upon by big government? Isn't that one of the reasons trumpeted for entering into conflict? We must protect the American Way of Life! How many wars/police actions has the U.S. entered into for the publicized sake of spreading/preserving democracy? And yet, in our own country, laws like the Patriot Act are passed that allow the enforcers to wiretap our phones and enter our homes without a court order --- without due process. What are we fighting for again?
Where is the guarantee that these wiretaps will protect and serve the people? The war on terrorism will fail just like the war on drugs failed. Yet we continue to throw good money after bad, and are oh so willing to burn the Constitution in the process. The terrorists have already won.
This was about 9 years ago. I feel sorry for the poor souls who have to fork over $150 for a beginning algebra textbook today. It's outrageous to pay that much for material which is a) in the public domain, and b) hasn't substantially changed in hundreds of years.
Tell me about it. Once while in grad school I paid $85 for a shiny new textbook for a topology course. After thumbing through it for a couple of hours, I realized that it had less information than a Dover paperback topology text I had previously purchased for $7. I returned that new piece of crap for a full refund, and copied the homework problems from a classmate. I passed with an A.
You are dodging the question. I never said that automobile accidents weren't a big deal. But, the fact remains that if they were outlawed then traffic fatalities would be zero. Why don't we do this? Because COMMERCE > HUMAN LIFE. That's why. So, when people drag out the corpses and the grieving families, that doesn't convince me that their motive is to save lives. It doesn't wash.
So, is it the loss of life --- yours and your loved ones --- that warrants your fear? Because if that is the case, the probability of being killed by terrorists is tiny when compared to say the probability of dying in an automobile accident. For some strange reason, I don't hear masses of people calling for the destruction of the Ford Motor company. Why could that be? Maybe, just maybe, loss of life and limb isn't the reason behind all the noise.
I doubt it. I would be willing to venture that nearly everyone has something to hide. Be it a list of medications they take, or which type of porn they like. This "terrorist threat" is so blown out of proportion as to almost be comical. In all seriousness, let me ask this: Why are we concerned about terrorists?
For most students, college is just a phase of life that must be endured on the way towards the working world.
When did things take this sorry turn?
For people to make enough money to buy the lifestyle they want, businesses want a college degree as a minimum. High school used to be enough. College is the new minimum. The businesses will know what to expect from an undergrad student.
When did businesses start requiring a college degree as a minimum level of education? Maybe when high schools stopped doing their job and began releasing legions of illiterates into the workforce? Hmmm. I wonder. And when the businesses cry that the new crop of high school graduates can't read or follow basic instructions, the dum-dums get sent to college for re-training. What are the colleges to do? On the one hand, most of these kids can't be re-trained --- they are used to doing inadequate work and sliding through the system anyway. The colleges could fail them. BUT, there is a lot of money to be made. So, the kids' grades get inflated once more, and the dum-dums go back to the interview with their nice, shiny college degree. And they still can't read or write. How long will it be before a M.S. or Ph.D. is the minimum level of eduction required for entry into the workforce?
The term "African American" has been in common usage in the United States since the late 1980s, when greater numbers of African Americans began to adopt the term self-referentially. Malcolm X favored the term "African American" over "Negro" and used the term at an OAAU (Organization of Afro American Unity) meeting in the early 1960s, saying, "Twenty-two million African-Americans - that's what we are - Africans who are in America." Former NBA player/coach Lenny Wilkens is another who used the term as a teenager when filling a job application. Many Blacks began to abandon the term "Afro-American", which had become popular in the 1960s and '70s, for "African-American," because they desired an unabbreviated expression of their African heritage that could not be mistaken or derided as an allusion to the afro hairstyle. The term became increasingly popular, and by the 1980s, Jesse Jackson and others pressed for its adoption and acceptance.
I'm out of my right mind --- I pay my state's use tax. As long as no individual purchase exceeds a certain amount, and I don't keep all my receipts, I can check a box to withold a tiny fraction of my income to cover the use tax.
If you don't have $$$$$$, you can't even play the political game, not at the national level anyway. Our society is so addicted to information from TV, radio, & newspaper, that if you can't afford to plaster your voice/image throughout the media, you can forget about winning any elections.
Slashdot Mirror
The axioms on which the theorems are built are explicitly part of the theorems, leading to a logically self-consistent system.
Counterexample --- Add to the axioms of ZFC the following axiom:
There exists a set B such that for all ordinals x, x is an element of B.
There is a proof in ZFC that that statement is false. Therefore, this mathematical system is not self-consistent. Go back to school.
Nice straw man. What you said was, and I quote: The axioms on which the theorems are built are explicitly part of the theorems, leading to a logically self-consistent system.
Godel says that you cannot show the truth of that claim, ergo the claim is not necessarily true. Indeed, I'm fairly sure it's a simple matter to come up with a set of axioms that is inconsistent. You are mistaken.
You misunderstand --- Godel did not say that no statements can be proven.
Essentially, a mathematical system is consistent if there exists no theorem which states both P and NOT P are true. Then he showed that in ALL logical/mathematical systems capable of supporting integer arithmetic, there exists no proof that the system is consistent. We all tacitly assume that our maths are consistent, but the maths themselves cannot prove this. Godel leaves open the possibility that a system may prove its own inconsistency.
Also, from philosophy standpoint, math relies on lots of nontrivial physical observations that humans take for granted (e.g. of paper, of mental states, etc).
What physical observations led to Galois Theory, Lie Groups, Hilbert Spaces, Zorn's Lemma, and a host of others? Nowadays, the bulk of mathematics is highly abstract and far removed from physical phenomena.
Also, what is an observation of a mental state?
Actually, math is completely self-consistent given a set of axioms.
That may well be, but Godel showed that any logical system strong enough to support the usual arithmetic on the integers cannot establish its own consistency. That is, it is entirely possible that a mathematical system may not be self-consistent.
You should read what Kurt Godel has to say about logical systems and consistency before you start spouting off this non-Informative nonsense.
... I must insist that God has harmed us by not disclosing this information. We need it to fight the diseases which God allows to plague us, and to heal injuries that God allows to happen to us. This information qualifies as critical, need-to-know information, and if God won't give it to us, then we have no choice but to figure it out for ourselves.
In this hypothetical situation, wouldn't we be guilty of violating the will of God? What type of response would that provoke? I seem to recall a tale from the Old Testament regarding God's disposition towards men who had gotten to big for their pants. Maybe God or Mother Nature invented human disease for a reason. Have you considered the consequences of completely eliminating disease? If no one were to die of disease or genetic abnormality, more people would probably die of starvation or war. Unchecked population growth would demand new and creative ways to thin the herd. Logan's Run anyone?
How best to serve and protect the people?
Do we act in a manner to minimize loss of life? For if that is the case, we could ban automobiles and subsidize health care for everyone. I don't see that happening, which leads me to the conclusion that protecting peoples' lives is not the highest priority. The government spins it that way, but their actions say otherwise.
How about ensuring that our legal and civil rights are not trampled upon by big government? Isn't that one of the reasons trumpeted for entering into conflict? We must protect the American Way of Life! How many wars/police actions has the U.S. entered into for the publicized sake of spreading/preserving democracy? And yet, in our own country, laws like the Patriot Act are passed that allow the enforcers to wiretap our phones and enter our homes without a court order --- without due process. What are we fighting for again?
Where is the guarantee that these wiretaps will protect and serve the people? The war on terrorism will fail just like the war on drugs failed. Yet we continue to throw good money after bad, and are oh so willing to burn the Constitution in the process. The terrorists have already won.
Introduction to General Topology by George L. Cain. I took the course in 1997.
This was about 9 years ago. I feel sorry for the poor souls who have to fork over $150 for a beginning algebra textbook today. It's outrageous to pay that much for material which is a) in the public domain, and b) hasn't substantially changed in hundreds of years.
Tell me about it. Once while in grad school I paid $85 for a shiny new textbook for a topology course. After thumbing through it for a couple of hours, I realized that it had less information than a Dover paperback topology text I had previously purchased for $7. I returned that new piece of crap for a full refund, and copied the homework problems from a classmate. I passed with an A.
I have absolutely no problem with paying representatives well, as that is the only way to ensure that the most talented people seek the job.
Kindly explain how that follows? Is the goal of being a representative to serve yourself (make money) or serve the people?
You are dodging the question. I never said that automobile accidents weren't a big deal. But, the fact remains that if they were outlawed then traffic fatalities would be zero. Why don't we do this? Because COMMERCE > HUMAN LIFE. That's why. So, when people drag out the corpses and the grieving families, that doesn't convince me that their motive is to save lives. It doesn't wash.
So, is it the loss of life --- yours and your loved ones --- that warrants your fear? Because if that is the case, the probability of being killed by terrorists is tiny when compared to say the probability of dying in an automobile accident. For some strange reason, I don't hear masses of people calling for the destruction of the Ford Motor company. Why could that be? Maybe, just maybe, loss of life and limb isn't the reason behind all the noise.
Try again.
I doubt it. I would be willing to venture that nearly everyone has something to hide. Be it a list of medications they take, or which type of porn they like. This "terrorist threat" is so blown out of proportion as to almost be comical. In all seriousness, let me ask this: Why are we concerned about terrorists?
For most students, college is just a phase of life that must be endured on the way towards the working world.
When did things take this sorry turn?
For people to make enough money to buy the lifestyle they want, businesses want a college degree as a minimum. High school used to be enough. College is the new minimum. The businesses will know what to expect from an undergrad student.
When did businesses start requiring a college degree as a minimum level of education? Maybe when high schools stopped doing their job and began releasing legions of illiterates into the workforce? Hmmm. I wonder. And when the businesses cry that the new crop of high school graduates can't read or follow basic instructions, the dum-dums get sent to college for re-training. What are the colleges to do? On the one hand, most of these kids can't be re-trained --- they are used to doing inadequate work and sliding through the system anyway. The colleges could fail them. BUT, there is a lot of money to be made. So, the kids' grades get inflated once more, and the dum-dums go back to the interview with their nice, shiny college degree. And they still can't read or write. How long will it be before a M.S. or Ph.D. is the minimum level of eduction required for entry into the workforce?
There's a joke that circulates among math professors and graduate students: Calculus is where you learn College Algebra.
Yeah. It's a good thing that bad guys have no way to obtain official ID.
From Wikipedia
The term "African American" has been in common usage in the United States since the late 1980s, when greater numbers of African Americans began to adopt the term self-referentially. Malcolm X favored the term "African American" over "Negro" and used the term at an OAAU (Organization of Afro American Unity) meeting in the early 1960s, saying, "Twenty-two million African-Americans - that's what we are - Africans who are in America." Former NBA player/coach Lenny Wilkens is another who used the term as a teenager when filling a job application. Many Blacks began to abandon the term "Afro-American", which had become popular in the 1960s and '70s, for "African-American," because they desired an unabbreviated expression of their African heritage that could not be mistaken or derided as an allusion to the afro hairstyle. The term became increasingly popular, and by the 1980s, Jesse Jackson and others pressed for its adoption and acceptance.
Nope, it's called having a taste of their own medicine. What's good for the goose is good for the gander.
Bingo. If I had mod points, I'd mod you up.
I'm out of my right mind --- I pay my state's use tax. As long as no individual purchase exceeds a certain amount, and I don't keep all my receipts, I can check a box to withold a tiny fraction of my income to cover the use tax.
Of course, most pr0n films don't have a good script. In fact, a good script would probably get in the way.
The problem is that my choices are:
[] Tweedle-dee
[] Tweedle-dum
If you don't have $$$$$$, you can't even play the political game, not at the national level anyway. Our society is so addicted to information from TV, radio, & newspaper, that if you can't afford to plaster your voice/image throughout the media, you can forget about winning any elections.