taeto

Could the connection not be named after Planck instead of after Strange? The difference between pi and phi is exactly h .

That is more or less how I learned it back in the day. Most recently I lectured "Abstract Algebra", and all that got skipped over by the textbook, basically because there are now so many applications of algebra to teach that there isn't enough time for the nitty-gritty

Indeed. So we should be more precise. Let a pseudo-group be a structure that satisfies the group axioms, except the neutral element axiom is replaced by the axiom [math] \exists e : ee = e [/math]. The purpose being to eliminate the impredicative axiom \(\exists e \forall x : xe=ex=x\), so that we obtain a predicative definition of a group. Certainly any group is a pseudo-group; just take \(e\) to be the neutral element. So we are half-way there. Unfortunately \( (\{0,1\},\cdot) \) is a pseudo-group, but not a group. Choose \(e = 0\), then \(e\) satisfies the new axiom, since \(0\cdot 0 = 0.\) The associativity axiom is trivial, since multiplication is associative on all \(\mathbb{R}\). The axiom of the existence of an inverse is satisfied, because \(1\cdot 0 = 0\cdot 1 = 0,\) hence \(0\) and \(1\) are mutual inverses. I found math.stackexchange.com/questions/1744856/group-theory-vs-type-theory with a definition of a group in type theory. But to me it just looks like the exact same definition as we are used to. In particular the neutral element axioms states that there exists an element \(e\) for which \(ae=ea=a\) for all \(a\). So I do not know why the usual definition is supposed to be impredicative, when it looks the exact same as the type theory definition .

With the new axioms, each element does have an inverse under multiplication .

No, just the set of two elements \(0\) and \(1\). Yes, I know it is not a group. The point is that if you replace the group axiom which says that there is a neutral element by another axiom which says that there is an element \(e\) for which \(ee=e,\) then the set \(\{0,1\}\) satisfies this new axiom under usual multiplication, as well as the other axioms. Hence the suggested fix does not work. At least Russell would then have a definition of a group that is predicative, so that is a nice pointer, thanks! I still do not see how induction is not predicative.

A definition of an object \(x\) is called impredicative if the expression used to explain the value of \(x\) contains mention of \(x\) itself. The example that I want to ask about is perhaps illustrative enough: In Group Theory, \(e\) is a neutral element for \(G\) if \(ex = xe = x\) holds for every element \(x\) of \(G\). Since \(e\) is itself an element of \(G\), this is a typical example of an impredicative definition. This statement is the axiom that a group has a neutral element. Therefore I began wondering if perhaps Group Theory is an impredicative theory. My original motivation comes from a nice booklet written by Edward Nelson titled Predicative Arithmetic: web.math.princeton.edu/~nelson/books/pa.pdf The title of Chapter 1, which has a length of about one printed page, is "The impredicativity of induction". Superficially a single axiom of the axiom schema of induction appears to have a similar structure to the example from Group Theory. If \(P\) is a predicate for which both \(P(0)\) and \(P(n) \rightarrow P(n+1)\) hold, then \(P(n)\) is true for all \(n.\) This axiom does again contain a universal quantifier. But if you look for an element which is defined and which is also similarly quantified by the quantifier, you are harder pressed than in the group example above, at least in my limited understanding. Nelson must have meant something by this. Unfortunately he passed on some years ago. Maybe someone can provide an insight? I honestly believe that my understanding is not up to par with this. The point of it all is that a theory that contains impredicative definitions is somehow reasonably deemed deficient. A little similar to having circular definitions. I did ask around for an explanation for why Group Theory is not equally considered deficient. Now I am not mentioning names, but I got the explanation from a very knowledgeable mathematician that the point is that you may substitute the above definition of the neutral element \(e\) by the equivalent definition "there exists a unique element \(e\) in \(G\) with the property \(ee = e.\)" This is indeed the case. The problem now is that to say the \(e\) is `unique' means to say \(\forall x : xx = x \rightarrow x = e,\) and now you have a universal quantifier which has \(e\) in its scope, and it makes the definition impredicative. If you do not include the `unique' part, I leave it as an exercise to show that \((\{ 0,1 \},\cdot)\) becomes a group, where \(\cdot\) is usual multiplication. My question here is whether there is some clear argument to say that the definition of a group is not necessarily impredicative?

I looked up the word "cult". It seems a reasonable fit. Exceptions to the sensible "stay away from them" policy do suggest themselves. Such as when videos full of made up nonsense get labeled in the category "Education" on Youtube. How does that even happen? I hope that reporting will help, but I have no positive evidence so far. And obviously when the muppets distribute defamatory emails to a number of one's colleagues. Which technically is criminal, though authorities might not care much.

Something like Skewes' numbers perhaps? sites.google.com/site/largenumbers/home/2-3/skewes

That is a nice list, good find! Thanks, and happy New Year! The entry alone on Stephen Crothers is gold. I add it here just for lulz: "Stephen J. Crothers: Stephen is probably the most able scholar in Einsteinian type general relativity. He has produced many definitive refutations of big bang, black holes and other fallacies and errors of Einsteinian general relativity."

In combinatorics we often have to look at all functions from a set X to a set Y. For each element x of X there are |Y| possibilities for a y in Y that x can map to, where |Y| is the number of elements in the set Y. It means that the number of such functions is |Y|^|X|, where |X| is the number of elements in X, since for each x we can choose any one of the y for x to be mapped to y by a function. The equality 0^0 = 1 then just follows from the general definition. Your reasoning is wrong because there is no such thing as 0^-1. It would be a number x for which 0*x = 1. That does not exist.

You want Euclid's Algorithm: en.wikipedia.org/wiki/Euclidean_algorithm E.g. you have 1.05, which is 105/100. Euclid gives that 5 is the greatest common divisor of 105 and 100. Therefore 105/100 = (105/5)/(100/5) = 21/20.

What is known about this organization principia-scientific.org ? On the extreme end of being anti scientific, so far as I can see. I wrote a book review on Amazon of a book apparently written by one of the members or affiliates of this organization: www.amazon.de/Mathematik-für-die-ersten-Semester/dp/3110377330/ref=sr_1_1?s=books&sr=1-1#customerReviews My entire mathematics department immediately got attacked by spams from one of the members of PSI, Klaus D. Witzel, and the review itself got angry replies also from John Gabriel, a well-known extreme racist anti-semitic anti-science crackpot, as well as Stephen Crothers himself, who needs no further introduction. Has anyone else got any similar experience, and if so, is there any known recourse. It has been a while ago now that this happened, but I have some similar concerns building up now, due to my critiques of a crank poster of youtube videos, whom I know to have connections to the same organization.

That is an interesting comment. I cannot come up with many examples of "opponents" of relativity from the first half of the 20'th century who were obviously anti-semitic. It did not really kick in as a popular item until late. I could volunteer numerous examples that I have encountered and/or engaged of current day anti-Einstein cranks who are very openly anti-semitic (as well as anti-Cantor cranks, for that matter). These days, it seems a vastly more popular thing. Admittedly, it is usually easier to identify a crank in our time, because it has to deny a hugely larger compound of observational evidence than was present 75 years ago. In previous times, they seemed somehow more timid and less attracted to public exposure. So that might skew the statistics.

Now I suspect that you are just trolling me. The point is, that in the pre-1930 discussion there might be present a concept of a kind of "pure vacuum", really literally with nothing in it, no permittivity, no permeability, no stuff whatsoever. Just like school children are taught today everywhere. Which we well know nowadays is not the case, but I want to keep it open that at least some people at that time might have believed this to be the case. Einstein definitely had lots of evidence to support that the actual vacuum has nonzero permittivity and permeability. He obviously had no experimental evidence to support what a hypothetical "pure vacuum" without QM phenomena happening would behave like, because such a thing did not exist then either.

Is "an ideal vacuum" a perfect vacuum in some classical sense, or in a QM sense? Those are distinct, no?

The problem with that, I think, is that there is no "uncontaminated" vacuum between the plates. If there were, the experiment might come out differently than what it actually does. Now, in 2019? Why is that relevant?

Remember we are talking pre 1930. Did Einstein at that time have reason to believe in a "pure" vacuum, of the really empty kind, or in a "contaminated" vacuum, of the QM type? And if he was certain of which, then what is the experimental evidence which he would have had?

I thought that permittivity and permeability would be observed physical quantities. How would you gather mathematical evidence of their quantitative values? Anyway, Eise already presented mathematical evidence, to argue that if either magnetic permeability or electric permittivity vanishes, then the speed of light becomes infinite.

I suspect that you miss the point. The actual vacuum has positive magnetic permeability and electric permittivity. Which we can both measure. The imagined "perfect" vacuum presumably has no magnetic permeability nor electric permittivity, hence the speed of light is infinite therein. If you for some reason believe that a vacuum could conceivably act like that, then you would have to admit that there would be no absolute bound to the speed of light. The suggestion seems to be that the actually observed vacuum is a "contaminated" version of such an idealized vacuum. The actual vacuum has stuff in it which slows down light to a finite speed. We do actually observe such a "contamination" in the shape of QM effects.

Thank you very much Eise, for a very nice account! Your conclusion here is consistent with what I would have expected. Which is why I raised the topic, since I was very surprised to learn that there were knowledgeably people even in this forum who thought differently. Of course I also appreciate their thoughts on this question. And yet: Not all authors would have thought that Einstein's idea of the invariance of the speed of light only goes back to the Michelson-Morley experiment. I mentioned that Emanuel Lasker was in continuous contact with Einstein, so this would not have been an issue for him. In the same vein, misunderstandings about SR, or strict adhesion to Kantian principles, can be ruled out on the same grounds for that particular author. And I do not think that the positivism issue comes up strongly when you are in a discussion with a pure-bred mathematician, as in this case. Although it could be debatable in special cases, but I do not think in this one. I have met mathematicians who seem a little quirky. Einstein and Lasker would have had hours and hours of friendly chat about relativity. I will try to make a guess of what a central point might have been, and maybe it would not have been a trivial one for either of them. I suspect that the statement "light travels with speed c through vacuum with respect to every inertial frame of reference" would have the same meaning to both of them, in terms of experimental validity. It is what you see and measure. I also suspect that Lasker thought that Einstein expects that vacuum is a complete void, literally with no content whatsoever. And Lasker expected that light would move with "infinite speed" through a complete nothing. Today we are in a comfortable position to say that if Lasker was right about what Einstein thought about a vacuum, then Einstein would be wrong, because of the effects of QM. I am not sure whether we can say that Lasker was wrong about assuming how fast light would move if there is "nothing" in place of an actual vacuum, he might have been right?! If my suspicion is right, then Lasker would be wholly justified, based on the explanation that were available to him given by Einstein himself, to be critical of relativity. It comes down to Einstein's level of agreement or disagreement with QM, so far as I can see. But always happy to be corrected.

Thanks studiot, and also strange, BeeCee, swansont, Eise, MigL, for many thoughtful comments and information! Lasker: practically a neighbor to Einstein in Berlin. They were fairly close friends for nearly a decade, and both being jewish, they fled Germany as soon as they could in 1933 after Hitler came to power. Surely Lasker would be very far from being "one of those opposed to Einstein" in an obvious sense. That in my view makes Lasker's contribution to the pamphlet interesting, because he is not obviously one who would go Einstein-bashing just because it was a free-for-all at that time, or with racist/religious motivation. And it is hard to dismiss him as an idiot, like you probably could do now with some of the other contributors. I am interested in the fact that he was a close friend of Einstein, and he contributed to a collective criticism of Einstein. Which is what you would expect a serious scientific mind to do, I am sure you agree? Disagreement or not about science should be independent of friendship. The timing was a bit iffy, admittedly. The threat was perceived differently by the potential victims. Some got out in time, like Lasker and Einstein, who were among those with the best resources. Still, the both waited until late 1933. If already 2-3 years before, or even earlier, they were hard under pressure, why would they wait until Nazi Germany came into full effect? Especially if the thugs were already preventing them to publish their own thoughts, and instead forcing them to write ridiculous pamphlets in support of nazi ideas? The nazis certainly had quickly built a reputation for violent and criminal behavior. Hitler himself spent time in prison even before the 20's had started. The story is that Lasker was quite open about not liking the constant speed of light in vacuum, because of the implications to the relativity of simultaneity in particular. That is why I call it a "philosophical" problem for him, it seemed to him an unnatural consequence, however much supported by the mathematics, with which of course he had to agree. For possible comparison, Hubble did not like the ideas of expansion of the universe, and tried to ridicule "the big bang". I just imagine that the idea made him feel uneasy. But I would not call him a "crank" for that reason. Despite there likely being more evidence in his disfavor than there was available earlier to Lasker (who I imagine would also be well-informed, as he was in constant dialogue with Einstein). Fair enough. I am actually the one who speak in favor of the accepted explanation in the case of Lasker's contribution, that he was genuinely worried about some of the consequences of relativity, and therefore he chose to write his small piece. It was BeeCee who came up with a fairly extraordinary claim, with no extraordinary evidence, that he had to make up something to make Einstein sound bad, or there would be consequences, because Nazi Germany. Some of the other 99 contributors probably had ulterior motives, I am not disputing it. I should have expanded some more. There were 100 authors to this publication, and I believe that Einstein acknowledged this. One of them was a close friend and fellow scientist, also jewish, and a prominent figure in his day. This seems special. Maybe some authors were more than willing to chime in, maybe driven by animosity, or because the times were so that it was convenient to bash the jews, or just because they didn't really care or understand the physics anyway. But Lasker would seem the most unlikely to fall in any of those categories. But was he really under tough pressure to make an attack on Einstein. Or did he just voice his honest opinion. I find it interesting, because if Lasker actually didn't believe in relativity despite having had nine years to his avail of discussions and explanations by Einstein himself, then maybe there was or is something about relativity that is genuinely difficult.

Fine. So now you require that I trace down for every one of the 100 contributions to this anti-Einstein pamphlet, who wrote it and why, otherwise the question is not even qualified to be considered in this forum? I chose to focus on one contribution, which was clearly thoughtful and made by a fairly clever chap, who was unfortunately unaware of observations that were forthcoming about 30 years ahead of time. If such a question is really abhorred in the forum then I request that this thread also be closed down as swiftly as the other one.

I oppose rather strongly to the above characterization of a "crank". If there is no or a very little experimental evidence, it is not reasonable to deem someone a "crank" who somehow has a philosophical or other reason to suggest a different interpretation of the evidence which is available at that point in time. I have encountered a number of actual cranks, and they are either ignorant of experimental facts or attribute their knowledge of said facts to pure conspiracy. None of that needs to be present in a criticism of relativity between 1905-1931 which could only be resolved into the 1970's by experiment. In the particular case in question, the contribution to the publication was authored by Einstein's good friend Emanuel Lasker, who, as his wife, was also jewish. Both Einstein and Lasker fled Germany in 1933, soon after Hitler came to power. Both Lasker and Einstein were eminently prominent figures. If anti-semitism was so strongly prominent in pre-Nazi Germany, stretching back to 1931 and even before then, enough that the nazis could coerce Lasker into writing some kind of nonsense crank criticism of relativity that he did not believe in himself, why could the nazis not, even simpler, try to make Einstein himself repent his relativistic jewish sins in writing? Yes, Lasker had a wife, true, that might help for repression. But Einstein had a daughter too, right?

Yes, that was the thread. No, it actually seemed to me that someone else than the OP was making stuff up, and the moderator did not intervene with the made up stuff, even though it was pointed out. The OP certainly had some quirks in that thread, so I cannot disagree with the closing of it. I just wondered how to continue the discussion when you are cut off like that. Because obviously if you have to start a new thread, it becomes a lot of work to repeat all the context and the arguments that were present in the closed thread. Though in this case it was a fierce opponent of the OP against another opponent to the OP kind of debate.

I believe that you are ahead of Euler by now. He had hundreds of theorems and lemmas, but no "laws".