Prev: Al Gore's key constituency was felons
Next: Question about maximality principles, lattices, AC and its equivalents
From: Archimedes Plutonium on 25 Jul 2010 03:25 Now I was asking Lwalk in a prior post, and hope he does not ignore the question. The question was that since Euclid Infinitude of Primes proof, both direct and indirect yield the infinitude of Regular primes and since the twin primes are a subset. One has to wonder, logically, that logic demands the same apparatus of proof mechanism "multiply the lot and add, subtract 1" that this same mechanism must yield a proof of twin primes. Sounds logical, sounds reasonable. And has math already proven in Algebra theory that if you have a proof of the larger set, then the same mechanism must prove the small set. Any thoughts on this LWalk? The Galois Group theory should have some say on this. Another thing I want to point out about the misleading or failing of our education system on the proof by Contradiction. I heard it in college in a Freshman Logic class and several times in math classes that in the contradiction method, any contradiction will do. These are one of those statements that is half wrong and half right, and so very misleading. Every math proof is a focused line of arguement with many constraints. So the trouble with the statement that "any contradiction will do" is typified by Iain Davidson error filled Euclid Indirect IP. He has the erroneous assumption that just about any contradiction in Infinitude of Primes will do. And so, he, like thousands, millions of others who heard it say in classrooms around the world "any contradiction will do". And so these impressionable minds when encountering a proof by contradiction, have this false idea that like a menu list, they can pick and choose from a wide assortment of attacks and have a wide assortment of contradictions waiting for each math proof. I think one of the reasons this misleading or falsity of method is preached in college, is because, the professor was preached that same idea, and like parrots handing down to the next generation of parrots. Or, the professor wanted something more to say, rather than a blank mind in class, harps on "any contradiction will do." What Euclid IP shows us, is that the mechanism is "multiply the lot and add 1." So now, let us examine the proof logically. Like, proving the proof. Would anyone think that if you formed "multiply the lot and add 1" that you had a menu of chooses for a contradiction? Would it not seem highly logical that if the mechanism is "multiply the lot and add 1" that the contradiction is narrowed down to only one contradiction that gives one and only one valid proof? Seems logical. And that the mechanism, "multiply the lot and add 1" would only entail a contradiction of W+1 being a larger prime than the supposed assumed finite list largest prime p_k. In other words, when some bloke like Iain Davidson goes running around looking for an alternative contradiction with his "prime divisors", does not he miss the entire point of the proof, that if you construct W+1 the contradiction is uniquely revolving around the fact that W+1 as prime is larger than the p_k prime. So I think our education system spreads alot of bogus ideas. It spreads the bogus idea that Euclid's proof was by contradiction when all along it was a Direct method/ construction proof. And the bogus idea that any contradiction will do. What was meant by that is all contradictions in the reductio ad absurdum reverse the assumption step. But when taught to impressionable students, they pick that up as meaning there is a menu of contradictions to choose from in a proof. And even Hardy/Woodgold/and Davis in the Mathematical Intelligencer are spreading more bogus ideas when they spoke about every construction proof turned into a contradiction proof and vice versa, or words to that effect in their Fall 2009 magazine. Here is where the authors and editor are spreading more bogus that future impressionable minds will pick that up as meaning, the Construction is the same as the Contradiction proof, or almost identical after the first few steps. And nothing is further from the truth, because as we can see, the Twin Primes Infinitude comes easily out of the Contradiction method whereas impossible in the construction method. And many proofs in mathematics have only one method and never the other. So when Hardy/Woodgold ask the question at the end of the article as to where the bogus error that Euclid's proof was reductio ad absurdum got started, one can easily ask how much bogus misleading ideas were in that article for which Hardy/Woodgold are to be blame for a new generation of bogus ideas. Archimedes Plutonium http://www.iw.net/~a_plutonium/ whole entire Universe is just one big atom where dots of the electron-dot-cloud are galaxies
From: sttscitrans on 25 Jul 2010 03:59
On 25 July, 08:25, Archimedes Plutonium <plutonium.archime...(a)gmail.com> wrote: If the theorem "Every natural >1 has at least one prime divisor" is false as you claim, you should be able to state an n>1 that does not have a single prime divisor. (other mindless ramblings deleted) |