Every proof rests upon axioms, and if we accept the axioms then we must accept a logically correct proof. However, we may also reject a proof on the basis of those axioms. For example, much of analysis rests upon the axiom of choice, a seemingly innocent statement with some absolutely bizarre consequences (refer to the link). In the past, some mathematicians would be inclined to reject a proof which used the axiom of choice, simply because they didn't want to accept the other consequences of the axiom. In this sense, mathematics is entirely subjective.
It is inevitable if you believe that human progress will continue indefinitely. Good science develops in many different directions simultaneously. Someone will always be taking a fresh look at something, and they may develop a new approach. At the very least, the human race is forced to consider new ideas because people die and we are constantly have to teach replacements.
Whether or not something is a proof is entirely our distinction to make. We choose the axioms on which the proof is based.
To paraphrase Bill Klem (a famous umpire): when asked whether a pitch was a ball or a strike, "It isn't anything until I call it".
Without humans to drive the computers doing the work in the right directions, it could take a long time before a computer would be able to get its proof - it simply doesn't know what it is looking for.
To some extent that's true, except in areas where human understanding has reduced mathematical proof to a mechanical process. For example, verifying algebraic identities, or even geometric proofs. A more advanced example is the Risch algorithm for elementary integration. It amounts to a proof that an integral either is or is not expressible in terms of elementary functions. Eventually we come to understand an area to such an extent that we can implement mechanical algorithms and move on. The proper role of the computer is to carry out these algorithms, so that we can use them to discover something else.
I believe that Feigenbaum's constant was discovered with the aid of a programmable pocket calculator. He noticed something, and used the computer to check it out. I think that of all the things that computers can do for mathematicians, this is the most valuable. You can ask "what about this" and the computer will do the grunt work. It's made a grad student's life much easier I'm sure:)
If it can't be checked, it's not really a "proof" is it ? Now granted, that leaves a lot of room for leaway. For example, one could construct an algorithm for the problem and then prove that the algorithm is correct. There are other more inventive possibilities too.
Well at least in regards to math, I stongly doubt that this will ever be the case. Mathematics is developed over decades and centuries. With a few notable exceptions, it doesn't just fall out of the sky in textbook form. Most areas of math started out as a giagantic mess (ex; calculus, linear algebra, even geometry), and it has taken the work of countless researchers, authors, and teachers to distill and refine it. This process will continue, and it is inevitable that the subjects which baffle us today will be hammered out and taught to grade school students eventually. Well developed theory makes mathematics easier, and this in turn fuels new discoveries.
Are you serious ? The company purchased the item. The company is entitled to the rebate. As an employee, you have bought the item on behalf of your company, you are not entitled to take a "cut". What if people in the payroll department were taking a cut off your wages ? ie: the company's books say you make 20% more than you really do, but someone is just taking that money because they have the opportunity. Do you see it now ?
I write programs for algebra (in Maple). I never use assertions. The programs must work correctly for all valid input. Invalid input is caught with a type check, and an appropriate error is returned. Assertion failures can only frustrate users, who typically do not understand what is going wrong.
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Every proof rests upon axioms, and if we accept the axioms then we must accept a logically correct proof. However, we may also reject a proof on the basis of those axioms. For example, much of analysis rests upon the axiom of choice, a seemingly innocent statement with some absolutely bizarre consequences (refer to the link). In the past, some mathematicians would be inclined to reject a proof which used the axiom of choice, simply because they didn't want to accept the other consequences of the axiom. In this sense, mathematics is entirely subjective.
It is inevitable if you believe that human progress will continue indefinitely. Good science develops in many different directions simultaneously. Someone will always be taking a fresh look at something, and they may develop a new approach. At the very least, the human race is forced to consider new ideas because people die and we are constantly have to teach replacements.
Whether or not something is a proof is entirely our distinction to make. We choose the axioms on which the proof is based. To paraphrase Bill Klem (a famous umpire): when asked whether a pitch was a ball or a strike, "It isn't anything until I call it".
Without humans to drive the computers doing the work in the right directions, it could take a long time before a computer would be able to get its proof - it simply doesn't know what it is looking for.
To some extent that's true, except in areas where human understanding has reduced mathematical proof to a mechanical process. For example, verifying algebraic identities, or even geometric proofs. A more advanced example is the Risch algorithm for elementary integration. It amounts to a proof that an integral either is or is not expressible in terms of elementary functions. Eventually we come to understand an area to such an extent that we can implement mechanical algorithms and move on. The proper role of the computer is to carry out these algorithms, so that we can use them to discover something else.
I believe that Feigenbaum's constant was discovered with the aid of a programmable pocket calculator. He noticed something, and used the computer to check it out. I think that of all the things that computers can do for mathematicians, this is the most valuable. You can ask "what about this" and the computer will do the grunt work. It's made a grad student's life much easier I'm sure :)
If it can't be checked, it's not really a "proof" is it ? Now granted, that leaves a lot of room for leaway. For example, one could construct an algorithm for the problem and then prove that the algorithm is correct. There are other more inventive possibilities too.
Well at least in regards to math, I stongly doubt that this will ever be the case. Mathematics is developed over decades and centuries. With a few notable exceptions, it doesn't just fall out of the sky in textbook form. Most areas of math started out as a giagantic mess (ex; calculus, linear algebra, even geometry), and it has taken the work of countless researchers, authors, and teachers to distill and refine it. This process will continue, and it is inevitable that the subjects which baffle us today will be hammered out and taught to grade school students eventually. Well developed theory makes mathematics easier, and this in turn fuels new discoveries.
No, it would be "New Critical IE Vulnerability" and it would be on the front page...
Hence, we can't interact with them too well.
I leave that to my nets...
Something must be off with your configuration. It should not take that long to boot or to login. Do you have a lot of programs loading at startup ?
Are you serious ? The company purchased the item. The company is entitled to the rebate. As an employee, you have bought the item on behalf of your company, you are not entitled to take a "cut". What if people in the payroll department were taking a cut off your wages ? ie: the company's books say you make 20% more than you really do, but someone is just taking that money because they have the opportunity. Do you see it now ?
It is totally unethical - it is a form of fraud. link.
The Panasonic Toughbook beats any Apple notebook - half the weight and double the price :)
Damn, before reading what you actually quoted, I thought you were referring to the pigs.
I'd rather just install Windows. It's a great threat for old machines.
And we wod them down for original thought!
...damn :)
Tom and Jerry, The Simpsons, and shopping on Sunday ?
Maybe that's the true reason for his killing spree: his town has a stupid name.
The mods appearently think this doesn't happen, even though I know it does from first hand experience. Be careful changing your profile.
I write programs for algebra (in Maple). I never use assertions. The programs must work correctly for all valid input. Invalid input is caught with a type check, and an appropriate error is returned. Assertion failures can only frustrate users, who typically do not understand what is going wrong.
I've always wanted to try Xfce, but it seems like underkill on a modern machine. What are your impressions of it ?
That's fine for a dedicated machine, but for a general purpose desktop it's a nightmare. I honestly think Gentoo is best suited to hobbyists.
I'm reinstalling Gentoo after some time away from it. Is KDE 3.4 in the default tree yet ?
Then watch your system break if you were using a much older profile.