in the history of science thus far–ancient or modern, we’ve mostly chosen rules that match the most accurate data at that point in time and that’s why it appears as though we can modify the “laws” of physics as we see fit; the “laws” change as technology and techniques improve/change. there are some examples of laws that are closer to what i think of as laws (inviolable/unchangeable) e.g. the law of conservation of energy.

to me, it’s much harder to envision the violation of concrete physical laws than mathematical ones. unlike mathematical axioms that one can create at will using, sometimes, vague or arbitrary definitions, physical phenomena are what they are, and the laws are what they are, and we can only try to understand them whereas with math, we can choose which axioms we like or fit what we’re trying to accomplish and explore without limits (other than those axioms).

in other words, it’s the maths that’s untrustworthy–it has no constraints! you can begin your definition at a point of your choosing and there are no limits to how far you can stretch.

the laws of thermodynamics, the uncertainty principle (which you often mention or allude to in this blog or your own work)…they’re necessarily useful because they already let us know where the ceiling is and we don’t try to do things like build 100% (let alone more) efficient machines etc. and can use them to tell whether a paper/article is worth reading or not.

with maths, you find one geometry too restrictive? go to any of the other infinite ones (or create a new one). you’ve hit the wall with the currently-defined numbers? define new numbers for your research.

“You can easily use math to describe physical laws totally different from the laws we observe.”

we don’t prove things physically just by using maths. just because it’s mathematically valid doesn’t mean it’s physically valid, let alone a physical law otherwise there’d be plenty of physical laws now coming from string theories.

the reason people thought that you could just compose velocities in the classical way regardless of how fast the objects were moving was based on mathematical induction (type of) reasoning.

“You should trust the laws of math.”

maybe; to do mathematics or to build upon it, but not necessarily in other disciplines.

since this post is long enough, i’ll just apologise for the length and abruptly end with a quote from eddington (on probably my favourite law):

“…if your theory is found to be against the second law of thermodynamics I can give you no hope; there is nothing for it but to collapse in deepest humiliation”

]]>Haha, great, I was looking for a list of non trivial problem instances to test. Perfect. ]]>

…where can I join that “P=NP Anonymous”?

If there’s enough interest in the Greater Boston area, I’ll finally start a chapter at MIT. But serious inquiries only… after all, the first step is admitting you have a problem. 😉

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P.S. Of course, you could argue you don’t have a problem and you have you P = NP “hobby” under control if in addition to any “proof” of P = NP you have, you provide proof your algorithm can at least do the modest task of solving at least 1 of the “unsolved instances” from the SAT 2013 Competition, which are databased along with all the correctly solved instances in loving detail at

http://edacc4.informatik.uni-ulm.de/SC13/experiments/

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P.P.S. Solving an “unsolved instance” from the SAT 2013 Competition is a much, much more modest task than, say, providing the prime factors of all the remaining RSA Challenge Numbers or providing the keys that join plaintexts and ciphertexts encoded under a full 10 rounds of AES-256 or any other good stuff that’ll instantly cause you to be a person of interest to the NSA even if your algorithm isn’t a true solution to P = NP in the rigorous, asymptotic sense.

Namely, an “unsolved instance” from the SAT 2013 competition is merely one that couldn’t be solved within 5,000 seconds (i.e., 1 hr, 23 mins, 20 secs) by any of the algorithms in the competition, all of which were run on 2.83 GHz Intel Xeon E5440 processors, either singly in the “Sequential” track or on 8 of ’em in the “Parallel” track.

In case you’re wondering, the smallest such “unsolved instance” that is (almost assuredly) in fact satisfiable is an instance of Random 7-SAT on 93 variables with 8164 7-CNF clauses. (The clause/variable ratio of 87.7 is right below the apparent SAT-UNSAT threshold.)

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P.P.P.S. If you don’t understand either of the last two postscripts, yet you think you’ve proved P = NP, you’re almost assuredly a crackpot, and until you’re ready to admit you have a problem we ask you not come to “P=NP Anonymous” meetings. You’ll just upset the people there who are doing the hard work of doing the 12 Steps. 😉

]]>I can’t think of a precedent where a math discovery by a single individual would have such an impact.

Just reading up about PGP from the wiki:

“Shortly after its release, PGP encryption found its way outside the United States, and in February 1993 Zimmermann became the formal target of a criminal investigation by the US Government for “munitions export without a license””

P=NP would be the digital equivalent of “discovering” and “building” the a-bomb all on your own in your garage.

Proving P!=NP would be way safer!

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