Richard Jernigan -> RE: Black Hole eats sun (Jan. 16 2013 6:03:53)
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ORIGINAL: BarkellWH If there were aliens in some distant world in the universe who had developed sufficiently to question the fabric of the Cosmos and were at a high enough intellectual level to run experiments and do the math, they would have to have an understanding of the same mathematics and physical laws of General and Special Relativity, as well as Quantum Mechanics, as we Earthlings if they were to understand it. It could not be otherwise, because they and their world, as a part of the same space-time framework of the universe as we and ours, would be subject to the same laws of physics, including Relativity and Quantum Mechanics, as we are. Cheers, Bill Ummm....I beg to differ. Even among humans there are radically different approaches to certain mathematical concepts. For example, Leibniz embraced the use of infinitesimals--qantities less in magnitude than any real number, but still not zero. Newton flirted with them, but abandoned them for fluxions, the limits of varying quantities. Cardinal Newman bitterly opposed Newton, accusing him of promoting atheism. After Newton was dead, no longr able to defend himself, Newman ridiculed infinitesimals as "the ghosts of departed quantities". Newman purported to demonstrate contradictions using infinitesimals. Newton never fell into this error. But mathematicians were stung by Newman's criticism and abandoned infinitesimals. Cauchy is credited with the first logicaly precise definition of limits for numbers, published in the 19th century. Absent infinitesimals, the limit concept has been the foundation of mathematical analysis. But, lo and behold, in the 1960s comes Abraham Robinson, who demonstrated that there is a logically consistent mathematics employing the real numbers and infinitiesimals. In Robinson's work you can add up infinitely many infinitesimals and get a finite number, just as Leibniz did. But nobody uses Robinson's stuff, since so much work was already based on the other approach. Who knows where Robinson's infinitesimals may have led, if as much work had been devoted to them by aliens? In another branching, the Dutch mathematician Brouwer, in the early 20th century, founded a school called Intuitionism. Brouwer objected to such things as proving the solution to a differential equation exists, without giving a procedure for calculationg it. Unfortunately this approach eliminates large swaths of otherwise accepted mathematics. What if the aliens were Intuitionists, and found ways to work around the difficulties this introduces? Another ambiguity arises from the proof, again in the 1960s, that the Axiom of Choice is independent of the Zermelo-Frankel axioms of logic. This gets a little technical, but just as you can have Euclidean and non-Euclidean geometries, you can have logic with or without the Axiom of Choice. Adopting the Axiom of choice leads to some bizarre results. See the Banach-Tarski Paradox. Before the independence of the Axiom of Choice was known, its appeal was so strong that such a great mathematician as R. L. Moore referred to it as "that fact". Much of Moore's work in the foundations of topology depended upon the axiom of choice. Which road would the aliens take? A fundamental unsolved problem of Earthling physics is that quantum mechanics and gravitation are incompatible in an essential way. You can use General Relativity to investigate gravitational effects, and you can use quantum mechanics to investigate subatomic and other phenomena, ignoring gravity, but you can't combine the two in a mathematically compatible way. Some new fundamental insight is required. String theory can unite gravity and particle physics--- in theory. But its predictions are untestable within the limits on the energy available to our most powerful tool, the Large Hadron Collider. The best hope is that at full power the LHC may give evidence of the massive partners of the particles of the Standard Model predicted by supersymmetry. If it does not, string theory remains unconstrained and unverifed by experiment. What if the aliens go straight to the unified theory of gravitation and subatomic particles, and it turns out not to look very much like either quantum mechanics or General Relativity? Moore's method of teaching mathematics was devoid of lectures and the reading of reference works. It consists of a brilliantly constructed sequence of problems that students were expected to work out on their own. Amyone who has employed this method in their own teaching will have been impressed by the originality of their students. Valid proofs of well-known theorems differ wildly among students who are thrown upon their own resources. Mathematics is a creative art. The path that humanity has followed in trying to understand nature is not the only one we could have taken, nor is it one that an alien civilization is required to follow. RNJ
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