This is a discussion on t-shirt within the General Discussions forums, part of the Community Boards category; Originally Posted by laserlight I do not see how that is the case. You are essentially finding an easy solution ...
Last edited by Mario F.; 04-02-2010 at 09:53 AM.
Originally Posted by brewbuck:
Reimplementing a large system in another language to get a 25% performance boost is nonsense. It would be cheaper to just get a computer which is 25% faster.
So... how that does support your claim that "by believing in P != NP without the ability to prove it, one is in fact making a strong case for P = NP"?Originally Posted by Mario F.
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Err... maybe I didn't explain myself well.
Because, under a belief, it becomes as easy to provide as is to verify the solution. A belief proves itself by simply existing. There's no need for scientific formulation or testing, other than rudimentary logic.
So by believing in P != NP, one is making a case for P = NP.
Originally Posted by brewbuck:
Reimplementing a large system in another language to get a 25% performance boost is nonsense. It would be cheaper to just get a computer which is 25% faster.
Sorry, but I do not follow your argument. It seems to me that you are jumping to a conclusion. As far as I can tell, believing in a proposition does not make a case for the negation of that proposition; it makes no case at all, since as you say, it does not provide a proof (or a counterexample).Originally Posted by Mario F.
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It does not need to provide a proof. It's a Belief. And it's solely in that context I'm speaking. In more detail:
By supporting a position such as P != NP without proper proof, you proposed your solution to the problem. You solved the equation. It's P != NP.
The solution has no scientific support and you don't feel you need to prove it beyond the formulation of your belief. Your faith in it. When you then look back, you will eventually realize that you are in fact supporting P = NP when you defend P != NP based on faith.
I guess, the implications of this reasoning may look like I'm making an indirect criticism of religion and faith. But that's not my point at this moment. I'm strictly speaking about the curious paradox you get from supporting P != NP without any proof.
EDIT:
Even more paradoxal because, from a purely philosophical view, yet P != NP is a strong support for the existence of God (and by extension, faith). But please, I'm not taking a stab at religion here. Just approaching the problem of belief in the context of the P vs NP problem.
When I support P != NP, because we have no proof yet, I'm actually approaching P = NP
Last edited by Mario F.; 04-02-2010 at 10:42 AM.
Originally Posted by brewbuck:
Reimplementing a large system in another language to get a 25% performance boost is nonsense. It would be cheaper to just get a computer which is 25% faster.
I think that this assertion is not accurate. The current evidence is in favour of P != NP. The evidence does not amount to anything close to a proof, but if I were posed the question of which I think is more likely, I would choose P != NP. If I were posed the question of whether I believed that P != NP, I would say no.Originally Posted by Mario F.
Effectively, you are asking me to believe youOriginally Posted by Mario F.
Sorry, but I am not opposing you because of religious issues. I am opposing you because this "curious paradox" that you contend is stated without proof.Originally Posted by Mario F.
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I have no other way of explaining this to you. You seem to insist in bringing the word "proof" to a context that exactly refuses any proof.
Faith: I believe P != NP. My "proof" is my belief. Thus I equal the solution to the verification of that solution. That means P = NP.
Science: I do not believe in P != NP, nor in P = NP. I believe in P = NP?.
Originally Posted by brewbuck:
Reimplementing a large system in another language to get a 25% performance boost is nonsense. It would be cheaper to just get a computer which is 25% faster.
I do not think so. The way I see it, what you are proposing is tantamount to saying that a flawed proof of conjecture X constitutes support for !X. But the flaw found might just show that X remains a conjecture rather than provide evidence (or even a proof) that !X is a theorem.Originally Posted by Mario F.
EDIT:
I would appreciate it if you elaborated on this sentence as I do not understand what you mean by it.Originally Posted by Mario F.
Last edited by laserlight; 04-02-2010 at 11:25 AM.
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A class that doesn't overload all operators just isn't finished yet. -- SmugCeePlusPlusWeenie
A year spent in artificial intelligence is enough to make one believe in God. -- Alan J. Perlis
No. But that on this case, yes.
The sentence is: "Thus I equal the solution to the verification of that solution." And under the concept of faith, I'm confused what there is to explain that I haven't already. What is that you don't understand?I would appreciate it if you elaborated on this sentence as I do not understand what you mean by it.
How do you verify your faith, if not by the very formulation of the faith itself?
No. We just don't know yet. It hasn't be proven yet. We can feel more inclined that way, we may accept that as the most probable answer. But we don't know yet. And trying to pretend we know is not science.Originally Posted by Yarin
Originally Posted by brewbuck:
Reimplementing a large system in another language to get a 25% performance boost is nonsense. It would be cheaper to just get a computer which is 25% faster.
I can understand that part, but then from what I understand, your conclusion does not follow. How do you manage to conclude P = NP from a flawed proof of P != NP? (EDIT: Or are you saying that from a contradiction, we can derive anything... but there is no clear contradiction since this is an open problem, and even if there was, "anything" includes supporting P != NP.)Originally Posted by Mario F.
I think Yarin was trying to be pedantic about the difference in rigour in mathematical proofs and scientific theories.Originally Posted by Mario F.
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Ah, I think I can understand the source of our confusion, now.
I'm not concluding that proves P = NP. I'm merely saying that by their own admission of faith they are in fact making their case for P = NP. Because both the solution and the verification of that solution become equal in complexity.
Originally Posted by brewbuck:
Reimplementing a large system in another language to get a 25% performance boost is nonsense. It would be cheaper to just get a computer which is 25% faster.
I do not understand what you mean by this, but...Originally Posted by Mario F.
I can more or less accept this, if you mean that this "admission of faith" constitutes admission that this "proof" of P != NP is flawed (i.e., effectively stating that X is a theorem while acknowledging that the proposed proof is not a proof; the fact that it is not a proof means that !X might be the case, so in a sense this makes a case for !X). I will still object to the phrase "strong case" though, since that comes closer to indicating that this admission is a purported proof of P = NP.Originally Posted by Mario F.
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Not exactly. But that the method employed by them to prove P != NP results in P = NP. They will be in fact, unwillingly, making a case for P = NP.
Both the solution (P != NP) and the validation of that solution end up becoming the same thing. Their belief. Thus the Solution and the Validation of that solution have equal complexity.
It's thus paradoxical because trying to prove P != NP through a profession of faith, not only invalidates P != NP a priori, but also ends up having them validating P = NP by their own admission.
Agree entirely. It's not. I wouldn't dare dream of it. I'm actually doing my best to show it doesn't. Instead, I'm exactly trying to expose the weaknesses of trying to come up with a "proof" based on belief.I will still object to the phrase "strong case" though, since that comes closer to indicating that this admission is a purported proof of P = NP.
Originally Posted by brewbuck:
Reimplementing a large system in another language to get a 25% performance boost is nonsense. It would be cheaper to just get a computer which is 25% faster.
Has anyone here said or even suggested that "you don't feel you need to prove it beyond the formulation of your belief"? Certainly not I.By supporting a position such as P != NP without proper proof, you proposed your solution to the problem. You solved the equation. It's P != NP.
The solution has no scientific support and you don't feel you need to prove it beyond the formulation of your belief.
I of course also considered the more cautious formulation as a question but preferred to state an hypothesis in the form of a claim (to be proved or disproved). Such hypotheses are pretty much always the starting point for proofs because until it's proved one way or another, you don't really know for sure. They give you some direction for exploring proof strategies. And if you keep trying and failing to prove some proposition p, you may then want to try proving not p for a while.
Obviously the issue isn't settled in mathematics until the proof one way or the other has been provided. But before getting there, one works with hypotheses about the problem at hand in order to develop strategies for what a proof might look like.
Mario, while I still don't entirely understand why even the dogmatic belief that P != NP would suggest that P == NP, I think you're not drawing a sufficiently strict line between "working hypotheses" and "evidence for" on the one hand and rigorous mathematical proof on the other.
Just one example of your formulations:
You seem to be confused about the difference between "results in" (suggestive of proof and in this case obviously refuting a method which would then result in the logical contradiction P != NP and P == NP) and "making a case for," which is done in mathematics before a proposition is proven. Once proven, you don't have to worry about the much weaker "making a case." The issue is then truly decided.... the method employed by them to prove P != NP results in P = NP. They will be in fact, unwillingly, making a case for P = NP.
In any case, the front of the t-shirt is on my view (obviously) only intended to say that I share the mainstream hypothesis that P != NP. 15+ years ago it was also possible to hypothesize (as many did without being in possession of a proof) something like x**n + y**n = z**n => (n > 2 || x = y = z = 0)
It's also well-known (and provable) that there are true statements that aren't provable...