JSH: Depressing reality, prime reality
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From: Jim Ferry on 9 Mar 2010 10:20 On Mar 8, 7:43 pm, JSH <jst...(a)gmail.com> wrote: > On Mar 7, 6:54 pm, Jim Ferry <corkleb...(a)hotmail.com> wrote: > > > > > > > On Mar 5, 9:08 pm, JSH <jst...(a)gmail.com> wrote: > > > > There is a lot of satisfaction with having my own axiom, which I had > > > the honor of naming as I'm the discoverer, which is of course, the > > > prime residue axiom, and yes, posters can reply in the negative or > > > derisively, but there you see the difference between finding something > > > and talk. > > > Axiom 1: Axioms are never named by their formulators. > > Axiom 2: Axioms are never named after their formulators. > > > Does this axiomatic system model the practice of mathematics > > accurately? Yes! I assert that it is so, willfully ignore any > > counterexamples, and denounce those who disagree with me as corrupt > > imbeciles. > > > So now that that's settled, I hereby name the prime residue axiom, > > "Musatov's Axiom #19". And it's too early to say, but it could be > > Mustov's greatest legacy. > > You're deluded. Google your try, and then Google: prime residue axiom > > Or go to any web search engine. Here it does not have to be Google. > > The world listens to me. It does not listen to you. > > Doubt me? Try the searches then. I say you're deluded but I wonder > if you'll acknowledge reality or continue to mouth off your defiance > against it. > > The latter is a little more interesting. > > James Harris All right, I'll google it. The top hit for "Musatov's axiom" is ... this thread! Woo hoo! A little lower down is something called "Musatov's Axiom of Truth". So it looks like I could be correct about "Musatov's Axiom #19" being his greatest legacy. Which is not surprising since pretty much anything Harris does has deeper mathematical content than anything Musatov does himself. (I won't include the results Musatov steals from others.) Okay, clearly I'm just joking. Google hits forsooth! The axiom is called Musatov's Axiom #19 not because of some kind of consensus of the herd, but simply because I have willed it to be so. That's how I roll, James. If I prove some groundbreaking mathematical result, I don't need my ego validated by publishing it in one of your "peer- reviewed" journals. I don't need someone to tell me whether I have proved Fermat's Last Theorem: I know whether I've proved it. What does Zarathustra care about the mooing of your herd? Oh, I know what you'll say. That the very existence of a field called mathematics is predicated on the ability of human beings to converse in a shared language. That something which cannot be communicated (i.e., written up in a way that practitioners of mathematics can understand) is, by definition, not mathematics. Etc. Etc. Now watch me brush aside this argument with a disdainful wave of my hand. I decide what is and is not mathematics. That is the hero's way. In fact, I have defined mathematical truth. I am not the first to do so, but I am the first to do so correctly. Mathematical truth is: whatever I say it is.
From: Arturo Magidin on 10 Mar 2010 00:25 On Mar 9, 11:08 pm, Transfer Principle <lwal...(a)lausd.net> wrote: > This, of course, settles everything: > > -- So it's not Dirichlet's Theorem, but Chebotarev's that applies. > -- According to the theorem, {keN | (P(k) == 1) mod 4} does have > _asymptotic_ density 1/2, but by _another_ density measure (the > "certain" density measure mentioned by Magadin), it differs from 1/2. > > Now Magadin doesn't recall whether it's 1 mod 4 or -1 mod 4 that has > more primes by this other measure, but in another thread, someone > pointed out that it's the quadratic nonresidues that have more primes. The "other measures" are not 'densities' in the usual sense. See: http://groups.google.com/group/sci.math/msg/843495f52ff54f83?dmode=source Rather, they are a kind of measure of how "often" there are more primes of one type than of the other. (This is rather rough, since there is no probability distribution over the natural numbers, of course). From there: -- Begin insert -- (1/N)#{ x<=N | there are more primes of the form 4n+3 up to x than of the form 4n+1} does not have a limit as N-->oo. But that Rubinsten and Sarnak proved in 1994 that (1/ln(N))*(Sum_{x<= N; there are more primes 4n+3 than 4n+1 up to x} 1/ x) goes to .9959.... as N-->oo. (This limit they call the "logarithmic measure" of the set) So you have that about 99.5% of the time, there are "more" 4n+3 primes than 4n+1. Now that I have the article in front of me, I see that if you look at primes of the form qn+a vs primes of the form qn+b, then if we assume the Generlized Riemann Hypothesis plus a few more technical assumptions, then if b is not a square modulo q and a is a square modulo q, then there is "usually" more primes up to N of the form qn+b than of the form qn+a. If both a and b are squares modulo q, or both a and b are nonsquares modulo q, then the primes of the form qn+b up to x are 'ahead' exactly half the time. -- End Insert -- -- Arturo Magidin
From: Arturo Magidin on 10 Mar 2010 00:28 On Mar 9, 11:08 pm, Transfer Principle <lwal...(a)lausd.net> wrote: > On the other hand, suppose Wiles had proved that FLT is actually > _undecidable_ in ZFC. If FLT were undecidable in ZFC, then it would be true in the standard model, (Like Goldbach's Conjecture and other similar statements), since the existence of a counterexample would be "checkable". > Then one may have been justified in adding > FLT as an axiom, especially if FLT leads to other results. But > FLT doesn't really imply other useful results, and since Wiles > actually _proved_ it FLT in ZFC, this is a moot point. Actually, if you could prove FLT (or Goldbach's) was undecidable, I suspect most people would find that as a good argument for saying it had been proven "true" (in the standard model), so that would be a reason for 'adding [it] as an axiom'. -- Arturo Magidin
From: Transfer Principle on 10 Mar 2010 00:38 On Mar 9, 12:43 am, "Ostap S. B. M. Bender Jr." <ostap_bender_1...(a)hotmail.com> wrote: > On Mar 9, 12:30 am, "Ostap S. B. M. Bender Jr." > > > if someday in the future > > > someone proves the Twin Primes Conjecture, but the proof requires > > > JSH's claim, and that claim has been proved undecidable in ZFC, > > > then number theorists might choose to work in ZFC+JSH rather than > > > ZFC, thus vindicating JSH's axiom once and for all. > I am too lazy to check, but wouldn't the proof (using only ZFC) that > JSH doesn't contradict ZFC, be actually a proof that JSH is true in > ZFC? It could be the case that _neither_ JSH _nor_ its negation contradicts ZFC, so that JSH would be _undecidable_ in ZFC. Actually, since Magadin established that JSH isn't really an "axiom" at all, let's use Riemann's hypothesis intead. In other words, it could be the case that _neither_ RH _nor_ its negation contradicts ZFC, so that RH would be _undecidable_ in ZFC. Obviously, ZFC doesn't prove both RH _and_ its negation (unless ZFC is inconsistent). There have been a few threads here at sci.math which discuss the possibility that RH is undecidable in ZFC, or that Goldbach's Conjecture is undecidable in ZFC or PA. Some standard theorists wonder, if RH or GC is undecidable in ZFC, does this mean that RH or GC are "true"?
From: Ostap S. B. M. Bender Jr. on 10 Mar 2010 06:12
On Mar 9, 6:00 am, mstem...(a)walkabout.empros.com (Michael Stemper) wrote: > In article <7fbe8630-0467-497d-9ee4-d169927af...(a)k6g2000prg.googlegroups.com>, Arthur <nabikov1...(a)yahoo.com> writes: > > > > >On Mar 5, 6:08=A0pm, JSH <jst...(a)gmail.com> wrote: > >> There is a lot of satisfaction with having my own axiom, which I had > >> the honor of naming as I'm the discoverer, which is of course, the > >> prime residue axiom, and yes, posters can reply in the negative or > >> derisively, but there you see the difference between finding something > >> and talk. > >You seem to say that you have discovered a new "axiom" in number > >theory. For it to be a new axiom, it is necessary to: > > >1. Prove that it cannot be derived from the already existing axioms as > >a theorem > > >2. Explain why you believe that this axiom/fact is correct, and there > >exist no counterexamples to it. > > >What seems to have happened is that you were trying to prove some > >theorem but couldn't. So, you declared it a "new axiom". That's a very > >lazy way of approaching science. What if every time a teacher gives a > >homework to high school or colege students to prove something, the > >students declare that no proofs are needed because these homework > >problems are "new axioms"? > > Think of the improvement in productivity possible with this approach. > If the mathematician's guild, with its medieval insistence that every > theorem be worked out from axioms dating back fifty, one hundred, two > hundred years old, can be overthrown, new methods of production could > be put in place. > > Replace the individual mathematician, working alone with only a white > board, with huge teams of axiom-discoverers. Each one could prove scores > of theorems per day, > No, no, no. There will be no theorem-proving any longer. All theorems will be proved by declaring them "new axioms". In fact, they will add both these theorems AND their negations to the set of axioms. The more - the merrier! > > resolving many long-standing conundra by the simple > expedient of discovering new axioms. After clearing away all of this > backlog, vast new vistas of mathematics can then be explored. > > I believe that your modest proposal would be a good first step in > vocational training for the new breed of axiom-discoverers. > > -- > Michael F. Stemper > #include <Standard_Disclaimer> > "Writing about jazz is like dancing about architecture" - Thelonious Monk |