Science is simply our agreement that when trying to learn about the physical world, we agree to let observations of the physical world be the ultimate mitigator of our arguments, rather than the authority of some powerful individual.
But there are many religions (or factions thereof) that reject the authority of a (politically; socially) powerful individual to mitigate arguments. Even in (some forms of) protestant Christianity recognize only their God (who's individual power is taken as [an interpretation of, and let us not forget that all observations are necessarily interpreted] an observation of the physical world) as the ultimate mitigator of arguments. So by your statement (and I'm not sure if this was intended) certain religions and factions thereof meet your definition of "science".
Consider the totalitarian principle. I'll readily concede that the probability of a given non-forbidden event occurring is non-zero, what take faith is the acceptance of the assertion "if it can (with non-zero probability) happen it will", which is accepted constantly in particle physics.
Hell go on and look at any mathematical discipline (ZF or ZFC set theory anyone? Peano arithmetic? the list is infinite), they are all systems of reasoning that are consistent *from the axioms*. The truth value of the axioms *must* be assumed, and their proof is necessarily outside of the system, hence all these systems require faith in the truth of the axioms as their starting point.
As an interesting side-note consider Clarkian presuppositionalism, which readily acknowledges this from the "other side" of the debate.
The integer factorization problem *can be no more difficult* than NP. Put differently integer factorization is NP-easy. It is trivial to show that integer factorization can be no more difficult than NP (and is in fact in NP), as multiplication can be performed in polytime (giving a polynomial time verifier) and brute force operates in exponential time on the digit length of the input (thus providing an exponentially bounded solving algorithm).
HOWEVER, it has NOT been proven that integer factorization is NP-complete (yes, GP did misspeak, saying "NP completely"). Put differently, there is no generalized polytime conversion from an NP complete problem (such as TSP) to integer factorization. Thus, per current understanding, it would be ENTIRELY POSSIBLE to prove that integer factorization is in, say, P, and even with a constructive proof, not have given a polytime algorithm for TSP and friends.
Thus the GP is incorrect, a proof that P=NP *WILL* break RSA/DSA and friends, but a break of RSA/DSA & co is not a proof that P=NP. This is why the question of NP-completeness is important.
Independently of all this, though, I'd hazard that any constructive proof that integer factorization is (not) in P would likely be a proof that P(!)=NP, as the (seemingly) most likely alternate proof would be a proof that NP-complete(!)=co-NP-complete, which as far as I can tell is less analyzed.
NOTE: Strong possibility that the last statement given above is bullshit as IANAC[omplexity]T[heorist] but IAAC[omputer]S[cientist] and I know several complexity theorists.
I agree MS Equation Editor is not worth mentioning, and I've never had the patience to sit down and teach myself LaTeX. I'm genuinely suprised that no one has mentioned OO.o's MATH though.
Through four years of college OO.o MATH has been the best method I've found to take math notes digitally. The symbol support is reasonable (although certain weirder algebras may necessitate changing character maps), and the markup keywords are simple and intuitive enough, and configurable to boot. While it's not perfect I've definitely found that its very fast (in my case faster than writing it out by hand). I also like the fact that it integrates cleanly in OO.o Writer, which means I can inline any equations with my textual notes as well.
Specifically I found it exceptionally useful in calculus, statistics, cryptography and relational algebra.
Hope that helps.
Slashdot Mirror
Well I just checked, and my homepage is still 123 bytes. Is this like the reverse of overcompensating?
Science is simply our agreement that when trying to learn about the physical world, we agree to let observations of the physical world be the ultimate mitigator of our arguments, rather than the authority of some powerful individual.
But there are many religions (or factions thereof) that reject the authority of a (politically; socially) powerful individual to mitigate arguments. Even in (some forms of) protestant Christianity recognize only their God (who's individual power is taken as [an interpretation of, and let us not forget that all observations are necessarily interpreted] an observation of the physical world) as the ultimate mitigator of arguments. So by your statement (and I'm not sure if this was intended) certain religions and factions thereof meet your definition of "science".
Consider the totalitarian principle. I'll readily concede that the probability of a given non-forbidden event occurring is non-zero, what take faith is the acceptance of the assertion "if it can (with non-zero probability) happen it will", which is accepted constantly in particle physics.
Hell go on and look at any mathematical discipline (ZF or ZFC set theory anyone? Peano arithmetic? the list is infinite), they are all systems of reasoning that are consistent *from the axioms*. The truth value of the axioms *must* be assumed, and their proof is necessarily outside of the system, hence all these systems require faith in the truth of the axioms as their starting point.
As an interesting side-note consider Clarkian presuppositionalism, which readily acknowledges this from the "other side" of the debate.
The integer factorization problem *can be no more difficult* than NP. Put differently integer factorization is NP-easy. It is trivial to show that integer factorization can be no more difficult than NP (and is in fact in NP), as multiplication can be performed in polytime (giving a polynomial time verifier) and brute force operates in exponential time on the digit length of the input (thus providing an exponentially bounded solving algorithm). HOWEVER, it has NOT been proven that integer factorization is NP-complete (yes, GP did misspeak, saying "NP completely"). Put differently, there is no generalized polytime conversion from an NP complete problem (such as TSP) to integer factorization. Thus, per current understanding, it would be ENTIRELY POSSIBLE to prove that integer factorization is in, say, P, and even with a constructive proof, not have given a polytime algorithm for TSP and friends. Thus the GP is incorrect, a proof that P=NP *WILL* break RSA/DSA and friends, but a break of RSA/DSA & co is not a proof that P=NP. This is why the question of NP-completeness is important. Independently of all this, though, I'd hazard that any constructive proof that integer factorization is (not) in P would likely be a proof that P(!)=NP, as the (seemingly) most likely alternate proof would be a proof that NP-complete(!)=co-NP-complete, which as far as I can tell is less analyzed. NOTE: Strong possibility that the last statement given above is bullshit as IANAC[omplexity]T[heorist] but IAAC[omputer]S[cientist] and I know several complexity theorists.
I agree MS Equation Editor is not worth mentioning, and I've never had the patience to sit down and teach myself LaTeX. I'm genuinely suprised that no one has mentioned OO.o's MATH though. Through four years of college OO.o MATH has been the best method I've found to take math notes digitally. The symbol support is reasonable (although certain weirder algebras may necessitate changing character maps), and the markup keywords are simple and intuitive enough, and configurable to boot. While it's not perfect I've definitely found that its very fast (in my case faster than writing it out by hand). I also like the fact that it integrates cleanly in OO.o Writer, which means I can inline any equations with my textual notes as well. Specifically I found it exceptionally useful in calculus, statistics, cryptography and relational algebra. Hope that helps.
Thirdly, hand-off actually works in mobile protocols.
I'll give you this one, however I'd rather have a fully controlled home network and only be at the whim of my phone company while im outside.
And with WiFi you could just have a giant mesh network.
The days when Seymour Cray could design a product which was cutting edge & saleable for a decade are long gone.
Yes, but this is largely because Cray is dead, not because it is impossible for someone similarly gifted to do what he did.