taeto

I got encouraged, I think, to open a new thread on a topic that was sub-sub-topic in an already closed thread. I was just very curious about the historical facts and how people think about questions like this. It is well-known that a publication surfaced in 1931, in Germany, supposedly containing the writings of 100 people slash scientists(?) each of which attempted to criticize Einstein slash RT slash GR. Einstein replied, sensibly, that if anything is actually wrong with relativity, then a single opponent with a valid objection should be quite sufficient. This seems to not have swayed the Einstein cranks one bit, as they do actually still have a penchant to reference those same contributions, as if their seemingly common occurrence shows valid evidence against GR. The question is what motivated the 100 supposed authors to make their contributions to this publication. I heard for the first time the suggestion that they would do this as a result of pressure from the Nazi party slash individual Nazi party members. So even though only the period 1933-1945 really qualifies as the "Nazi Germany" rule, it would have very much applied also a few years before, in 1931. In fact, one of the authors of the publication was Emanuel Lasker, a long-year world champion of chess and a respected professional mathematician at the time. Lasker published his own objections to relativity at least as far back as 1928 as well, though mostly philosophically inspired. Again it was suggested, to me surprisedly, that he did this only through pressure from the nazis. This gets very interesting. Both Einstein and Lasker were hugely prominent figures in Germany. They lived nearby from each other in Berlin and were good friends. It was only in Bavaria that the nazis had anything like moderate influence. The Nazi party had about 1% representation in the German parliament, and no measurable support in Berlin before 1929. It would be historically interesting if they were already in 1928 in a position to influence the publications of reputable scientists. What would be the mechanism by which this is possible?

Okay then, but thanks! I will just open a new thread, I guess that would be the default. I got genuinely curious about the unresolved discussion that was running in the closed thread.

But what have you got as a recourse when a thread in which you try to participate suddenly gets closed? You should open a new thread to continue the discussion? It seems unnatural and awkward. I was in a thread quite recently, in which a poster attempted to discredit the OP by making up stuff with extraordinary claims seemingly out of the blue. To me it seemed unfair to force the OP to react to made up claims. In addition the claims were, possibly unintentionally but nonetheless, of an insulting nature towards a no longer living scientist, well mathematician, but I tend to make no great distinctions. The thread got closed before I could object to the unfair treatment of the OP and the inappropriate personal attacks.

Very unlikely in 1931. Moreover, already in 1928 Lasker published "Die Kultur in Gefahr", in which he wrote as his concluding statement: Die Relativitäts-Theorie als Ganzes, als ein System der Erklärung der Wirklichkeit, ist irrig sowohl in ihren Methoden wie in ihren Ergebnissen. The relativity theory as a whole, as a system of explanation of reality, is erroneous both in its methods and in its results. Before the start of the Great Depression in 1929, there were few people who took the Nazi Party seriously as a threat. It is quite improbable that a celebrity such as Lasker could have been under any pressure from that or any other side.

According to a well-known crackpot theory based on some unfortunately ill thought-out suggestions by Halton Arp. It is doubtful that any physicist with a valid "science license" believes in it.

So I can throw away my official chess arbiter license card now? Nazi Germany had not been invented yet in 1931 though. It is the name used for Germany 1933-45.

There clearly is an infinite spectrum of wavelengths received by the receptory organs. Perception is a completely separate issue. For some organisms it is sufficient to distinguish between just "light" and "no light". Such organisms "could potentially distinguish between an infinite amount of colors", except they don't, because they are not equipped with a sensory apparatus to do so.

Yes, they do not add, because the potential energy involved in their electromagnetic bonds. Brilliant! But is it also the best way to explain it to a high school student? ( I keep imagining that an explanation on such a level would be best suited for our OP. ) Could the OP not say, well, to pull the neutron apart and make a free electron and a free proton, naturally you have to do some work, since they are bound by electromagnetic forces. And therefore you expect the neutron to have a larger mass-energy than the sum of the masses of the electron and proton? After doing the work, you will have less energy left to distribute between the two.

Presumably the binding of an electron to a proton increases the energy to an amount larger than their sums.

I imagine to have to answer a high school student who asks "why neutrons in the nucleus, and not just protons and electrons?" Do you have evidence to say that the OP has an understanding of physics which substantially exceeds that of said high school student? If not, can you present any evidence to suggest that "We have models and data for fusion and beta decay (and nuclear structure and stability, and many other things - including spin) which are based on the presence of neutrons in the atom" would be a nearly optimal answer to the question? By forum rules you have to answer this question

I understand the OP as saying that free neutrons are okay. But somehow in the nucleus you do not have any neutrons, only some amount of electrons and a larger amount of protons. And if an electron and a proton escape together from the nucleus, a free neutron is formed. Conversely, if a nucleus captures a free neutron, it somehow gets converted into an electron and a proton. What you are saying does not immediately contradict this possibility. I guess that preservation of spin might.

"Emanuel Lasker was undoubtedly one of the most interesting people I came to know in my later years. We must be thankful to those who have penned the story of his life for this and succeeding generations. For there are few men who have had a warm interest in all the great human problems and at the same time kept their personality so uniquely independent." (Albert Einstein, 1952, in a foreword to a biography of Lasker) Suspecting the OP is not referring to Lasker solely in his capacity as a chess player, rather as friend of Einstein and an excellent mathematician with an interest in physics. Also: Argumentum ad hominem. Would he not simply think, well, there could be \(n+1\) protons and \(n\) electrons in the nucleus, and a single electron in orbit? Do you not have to know about nuclear spin to reject this suggestion? Or is the point to explain why higher isotopes than tritium are highly unstable?

It isn't a Lemma if it is not useful for anything further

Okay then. But I like to use \(\gcd\) (greatest common divisor) instead of GCF, since that is the standard, and otherwise I will invariably make typos. Unfortunately "factor" is a little ambiguous. Assume that we have the information for positive integers \(x,y,\) that \(x^2y^2\) and \(xy^3\) have \(\gcd(x^2y^2,xy^3)=g\) as their GCF, for some prescribed positive integer \(g.\) The question is what \(y\) can possibly be? Theorem 1 (studiot's theorem). \(y=1\) is always possible. Proof. Let \(x=g.\) Then for \(y=1\) we get \(\gcd(x^2y^2,xy^3) = \gcd(g^2,g) = g.\) So \(y=1\) is possible. Theorem 2. \(y= 2\) is possible if and only if \(g\) is either \(4\) times an odd number or \(8\) times an even number. Proof. Assume \(y=2.\) Then for each \(x\) we have \(\gcd(x^2y^2,xy^3) = \gcd(4x^2,8x) = 4x\gcd(x,2).\) If \(x\) is odd then \(\gcd(x,2)=1,\) which implies \(g = \gcd(x^2y^2,xy^3)= 4x,\) which is \(4\) times an odd number. If \(x\) is even, then \(\gcd(x,2)=2,\) so \(g = \gcd(x^2y^2,xy^3)= 8x,\) eight times an even number. The converses follow by setting \(x=g/4\) if \(g\) is \(4\) times an odd number, and setting \(x=g/8\) if \(g\) is \(8\) times an even number. End of proof. Who will volunteer to do Theorems 3,4,5,...?

This is not an example of what I wrote . It also makes sense, but I cannot recall having seen it before.

That would make it worse...

It seems like if there had not been any list of possibilities given beforehand, then \(y=1\) would be the unique solution?

Likely referring to Turritopsis dohrnii, the immortal jellyfish, is a species of small, biologically immortal jellyfish found in the Mediterranean Sea and in the waters of Japan. It is one of the few known cases of animals capable of reverting completely to a sexually immature, colonial stage after having reached sexual maturity as a solitary individual. Theoretically, this process can go on indefinitely, effectively rendering the jellyfish biologically immortal, although in practice individuals can still die. In nature, most Turritopsis are likely to succumb to predation or disease in the medusa stage, without reverting to the polyp form.

Was it Robert Heinlein who wrote a profetic SciFi story about a reality in which machines have taken over to do practically all tasks, and humans are left with having to consume goods? The people worst off would be forced to consume exorbitant amounts of food and luxury items, while only a few lucky ones would be allowed to live in a simple living space, eat just a sufficient amount of food, and maybe even go to work now and then. It was scary to read as an impressionable teenager, and it was written so as to not sound really that appealing.

You are not missing anything. Where does the question come from though?

That is a very interesting discussion, thanks! I will look at the links. I suspect that a "random subset" can be explained so that if \(t\) is any real number between 0 and 1, and if \(n\) is any positive natural number, then from random and independent binary values \(b(t,n)\in \{0,1\} \) we get a random real \(x(t)\) between 0 and 1, which has \(b(t,n)\) as its \(n\)'th binary digit. The set \( X = \{ x(t) : t \in [0,1] \} \) should be the random set. If this does not work, then it is a little difficult to find a relationship with the typing monkeys question. I find it hard to imagine that the random subset \(X\) is Lebesgue measurable with positive probability in ZFC, since the Choice Axiom produces a lot of non-measurable subsets, likely enough to drown out the contribution from the measurable ones. But in ZF it is at least consistent that \(X\) is always measurable, due to the Solovay model. And it is reasonable to expect that further axioms are needed to make \(X\) provably measurable, and the question makes sense whether \(\pi-3\) is in \(X\) with some probability.

I am just a monkey typing away Maybe there are other forum members reading this, who happen to be unfamiliar with Solovay. I do not know why you think that by pointing to this relevant model, it shows that I am not giving thought. Where should I pause to think? I am already aware of all the facts that you presented so far. You are able to combine them better than I can, but once I get the drift, I feel up to date and ready to comment back. Now you make me feel that you already have the answer, and you are just holding back on it to see how I will improve. But would you mind to skip that, please?

Solovay's model is well-known. I can provide other references if you like, but most are also on the bottom of the wiki page.

That is a very astute remark, thanks! It seems silly to assume that if monkeys are typing away each one randomly, then together they will somehow produce a measurable subset of the real numbers between 0 and 1, do you agree? It forces the assumption that all subsets are measurable. As in Solovay's model of the real numbers en.wikipedia.org/wiki/Solovay_model. The Choice Axiom is false in this model, so one has to be careful.

That sounds like an excellent representation! And it makes it seem like the answer is indeed that pi will belong to the set with probability zero. Good point.