# User talk:Tsirel

## Zero probability

Hi Tsirel,

I disagree with the statement in the probability density function portion. I recognize the point you are trying to illustrate but the answer is too absolute. I recognize that you are in essence trying to describe: lim(x-->∞) 1/x (except x never reaches infinite, it just gets really really close so we round to zero).

At the same time I dont believe what you are describing exhibits the same behaviour. While the likelihood may be slim in my opinion you must always leave room for a highly improbable event to occur. For example if I flip a coin 2 million times - what is the chance I flip 2 million consecutive heads? Very unlikely but still _possible_. If I flip it 2 trillion times - what is the probability that I flip 2 trillion heads? Very very unlikely but it is still a possible outcome.

The only point I am trying to make is that while it is very very unlikely that any one event occurs at a very specific time it will always remain possible. Pretending that it is an absolute is misleading with respect to how probability works.

Sorry if this is poorly formatted - I am new to all of this!

Bobrossmademedoit (talk) 14:00, 23 January 2016 (UTC)

Hi. I know that many non-mathematicians somehow interpret "infinite" as "very large but still finite". Remarkably, many years ago I was asked by a student: "really, I do not understand, how are the points placed on the continuum; are they exactly next to each other, or are there small gaps between them?" I was confused and have not found an answer. I still can not answer. The mathematical continuum just cannot be discussed in such terms. (This is not about physical continuum; there, no one knows the ultimate truth.) Anyway, probability theory is based on measure theory. A single real number (or rather, the singleton - the set containing only this single number) cannot have any non-zero measure, since for every ${\displaystyle \epsilon >0}$ it is contained in some interval shorter than ${\displaystyle \epsilon .}$ See Almost surely. True, the probability of 2000000 consecutive heads is 2-2000000, not zero. But the probability of the infinite sequence of heads is exactly zero (since it is less than 2-k for all k. You may wonder, how it happens that the total probability 1 is the sum of zeros. The answer (of the measure theory) is that the sum rule applies to finite sums, and to countably infinite sums, but fails for uncountable sums; and the continuum is an uncountable set. Boris Tsirelson (talk) 15:12, 23 January 2016 (UTC)
Additional reason for possible confusion: the limit, used in calculus/analysis, treats infinity as something only approached but never reached. Moreover, during many centuries this was the only approach to infinity in mathematics. But the last century (and a bit more) the actual infinity is allowed, and widely used, in mathematics. In particular, in measure theory, and therefore in probability. An infinite probability model is not an infinite sequence of finite models! Boris Tsirelson (talk) 15:33, 23 January 2016 (UTC)
You may say: if so, then probability theory is detached from reality, since an infinite experiment can be only a thought experiment, never a real experiment. Well... idealization is the way of mathematics. Geometry is about points of zero size, lines of zero width, etc. Does it mean that geometry is detached from reality? Boris Tsirelson (talk) 15:48, 23 January 2016 (UTC)
I think it is correct to say that an outcome having probability zero does not mean the same thing as the outcome being impossible. YohanN7 (talk) 12:36, 25 January 2016 (UTC)
Well, here is my opinion on this interesting point.
We do not ask the Euclidean geometry questions about the physical space: is it Euclidean in the large? in the small? what is the meaning of a curve of zero width? etc. We understand that mathematics is not philosophy; relations between mathematics and reality are a philosophical topic, not mathematical. (Still, mathematicians may actively discuss them; they may also play chess, make music etc. etc.) But the same applies to probability theory! It calculates probabilities. No less, no more. (Well, also expectations, spectral densities and many other things that ultimately boil down to probabilities.) Sometimes it says : this probability is zero. But you cannot ask it, what does this mean in reality. Ask philosophy, not probability theory.
I recall the expression "quantum silence" (means: you cannot ask the quantum theory, how it may be like that...); but similarly we could say "mathematical silence", "probabilistic silence" and so on. Boris Tsirelson (talk) 14:37, 25 January 2016 (UTC)
To be more specific: zero probabilities (and the notion "almost sure") do not occur in discrete probability (just like curves of zero width do not occur in finite geometries). They occur in continuous probability, and are about thought experiments only. Thus, they are not related to reality at all! Well, in some indirect way they are; but not directly. Before asking, whether an outcome of probability zero is possible or not, think, whether the corresponding experiment is feasible, or not. Boris Tsirelson (talk) 14:43, 25 January 2016 (UTC)
In simpler words: before asking "can a monkey type all digits of π at random?" ask "can a monkey type infinitely long at all?" (And even if you answer affirmatively, the next question: "can we observe and estimate the ultimate result, after the endless experiment is completed?") Boris Tsirelson (talk) 15:14, 25 January 2016 (UTC)
And if someone feels able to imagine that we can perform infinite probabilistic experiments and after that (!) collect their results and observe their statistics, then I have an interesting question to such person. Let A be a nonmeasurable(!) subset of [0,1] (its existence is ensured by the choice axiom). Let us choose at random a number on [0,1] and check, whether it belongs to A or not. Let us repeat this experiment 1000000 times. What will we observe, typically?
In some sense, both the choice axiom and the probability theory extend our intuition from finite to infinite. But these two extensions contradict each other! Boris Tsirelson (talk) 17:41, 25 January 2016 (UTC)
When I wrote the above I had in mind that we philosophically accept (for the sake of reasoning) that we actually can, say, throw darts at the unit disk and record the exact result. You partly already answered a question I then meant to ask about non-measurable sets. Another question about murky waters: What should one think about the axiom of symmetry? (I have an opinion, but not a very strong one.) YohanN7 (talk) 10:53, 26 January 2016 (UTC)
Wow! I did not hear about this. Thanks for letting me know. But generally, play with axioms is not my hobby. I am glad that the continuum hypothesis is not called "the continuum axiom" (nor its negation is); and I'd prefer the name "symmetry hypothesis" to "axiom of symmetry".
A physicists likes to play with his "axioms", written on a blackboard that has to be erased every five years (who said so? I do not remember). But a mathematician would prefer axioms to be engraved on the tablets. And no wonder: a physicist can test his axioms against empirical facts; a mathematician cannot.
We would be most happy with an axiom "the infinite behaves like the finite"; alas, this leads to contradictions. We take some special cases of this principle. But doing so we are in the trouble of Buridan's ass. For example, what to choose: the choice axiom, or the axiom "everything is measurable"? Here is my text about this, if you like.
I really enjoy the set theory. But its consistency is for me hardly more than an optimistic hope. I do not think I really understand what it is, the set of all points of the plane (or the disk). Boris Tsirelson (talk) 17:56, 26 January 2016 (UTC)
I think that the axiom of choice breaks the axiom of symmetry as well. At the risk of me blundering: Whatever the cardinality α of the continuum is, wellorder the unit disk with order type ωα. Now throw the first dart. The ordinal corresponding to the first real number then has β predecessors with β < α, and hence is a set of measure zero. The next dart will hit a real number with corresponding ordinal bigger than the first with a probability of 100%. What happens to symmetry here? Am I making a blunder? (If CH holds, the first dart hits a countable ordinal. There are uncountably many countable ordinals bigger than it.) YohanN7 (talk) 11:04, 29 January 2016 (UTC)
"has β predecessors with β < α, and hence is a set of measure zero" — really? is there such theorem? Boris Tsirelson (talk) 12:15, 29 January 2016 (UTC)
I don't know, but I have read on talk pages here that sets of cardinality less than the continuum have Lebesgue measure zero. The post I think about was written by someone I really trust. Don't want to "out him" holding him responsible, because I may remember wrong. I'll try to find the post I am thinking about. YohanN7 (talk) 12:22, 29 January 2016 (UTC)
Found it: Talk:Vitali set#Vitali sets and the Continuum Hypotheses. Incidentally, it was my original question (many years ago, now I believe more or less that non-measurable sets are the norm rather than the exception) if not smaller sets could have badly behaves measures. Trovatore's reply mentions the 2 case at least, but not the general case. I remembered wrong. Buit still, AC breaks symmetry if the continuum is 2 or smaller? (I'll be unable to respond further for a couple of days.) YohanN7 (talk) 12:31, 29 January 2016 (UTC)
I see. But Trovatore does not say this is a theorem; rather, that this does not contradict ZFC. Play with axioms, still. What should we do with all these tempting (separately) but contradicting each other "new axioms"? Boris Tsirelson (talk) 13:07, 29 January 2016 (UTC)
As for me, if two "new axioms" are both tempting but contradict each other, then neither deserves the name "axiom", since neither is "a self-evident or universally recognized truth". Also, if a contradiction was not found during some centuries, this is not a reason to believe that it does not exist. In this sense, mathematics should not be experimental. Boris Tsirelson (talk) 14:42, 29 January 2016 (UTC)
I agree. I am also not entirely sure whether mathematics needs axioms beyond (say) ZFC. But just like physicists must search for dark matter, set theorists must search for new axioms, perhaps even search for, what they believe, are true new axioms. Besides all that, playing with axioms is fun – even for a layman. There are even respected set theorists that play with them to the extent that the continuum hypothesis and Bell's theorem are mentioned (Magidor) on the same page (section 5). YohanN7 (talk) 13:25, 4 February 2016 (UTC)
I suspect that the fun of playing with axioms leads to a kind of The Glass Bead Game. Boris Tsirelson (talk) 11:41, 5 February 2016 (UTC)

(Unindent)

Philosopher Nicolas Malebranche was the first to advance the hypothesis that each embryo could contain even smaller embryos ad infinitum, like a Matryoshka doll. According to Malebranche, "an infinite series of plants and animals were contained within the seed or the egg, but only naturalists with sufficient skill and experience could detect their presence."

(Quoted from Preformationism#Elaboration_of_preformationism.)

Could God create an infinite decreasing sequence of embedded embryos? Well, by definition, He could create anything. But, to this end, He should first create the physical space(-time) that follows the idea of the mathematical continuum. As far as we understand, He did not. Or do you believe that Banach–Tarski paradox tells us something physically meaningful? I do not. It seems to me that some reasonably good subsets of the mathematical continuum may be interpreted as parts of the physical space, but nonmeasurable sets surely cannot (and many measurable sets cannot, too; just think, how to distinguish physically a Borel set from an analytic set in the physical space). For me, the phrase "nonmeasurable set of points in the physical space" is as ridiculous as "nonmeasurable set of directions in the physical space". Thus, for me Pitowsky's idea is at best a good joke. Boris Tsirelson (talk) 15:52, 4 February 2016 (UTC) I tried once to voice my opinion about choice axiom in general and Pitowsky's idea in particular, see Sect. 4, page 33, here. Boris Tsirelson (talk) 19:09, 4 February 2016 (UTC)

Infinite decreasing sequence of embedded embryos would be ruled out by the axiom of regularity if spacetime is (described by) any set?
No, why? A decreasing sequence of embedded intervals surely exists, and is widely used in analysis. Boris Tsirelson (talk) 11:45, 5 February 2016 (UTC)
I confused belonging () and subset () there (the description could be read that way). YohanN7 (talk) 11:49, 5 February 2016 (UTC)
I don't have any idea about the true nature of spacetime, but if it can truly be modeled by 4, then non-measurable sets of spacetime must exist. (Notice I didn't say I believe this is the case, or that there is anything substantial (besides math) in Pitowsky's ideas.) I still don't think the Banach–Tarski paradox means anything physical in this case. For it to mean anything, we'd need a bunch of non-measurable sets to be filed with matter with matter only there. This is ridiculous, since even localization of a single particle to a point is physically impossible. I'd apply the same argument to Borel sets or any sets. But empty spacetime doesn't pose the exact same problems. The sets would be there but totally undetectable and devoid of physical meaning. YohanN7 (talk) 11:34, 5 February 2016 (UTC)
And what about Quantum foam? Boris Tsirelson (talk) 11:51, 5 February 2016 (UTC)
My knowledge about quantum foam is not even at the popular science level. I couldn't say. YohanN7 (talk) 13:35, 8 February 2016 (UTC)
Well, even if I admit existence of a nonmeasurable set of directions in the physical "empty space" (which I am reluctant to do), it does not save Pitowsky's idea. The Stern-Gerlach apparatus is material, and has no more than a finite number of states! (I recall this argument used by Bekenstein: the state space of everything in the given finite volume is finite-dimensional, since too high energy would lead to gravitational collapse...) Its orientation is a coherent combination of its spin states; these are a lot - but finite; and detection probability cannot be a nonmeasurable function of the orientation, since its is polynomial! Boris Tsirelson (talk) 14:37, 5 February 2016 (UTC)
Oops, no, we should not assume any property of nature beyond the local realism! Hmmm... then Pitowsky is right: violation of Bell inequality is possible in "classical" physics that admits nonmeasurable functions in basic interactions... a monstrous assumption... but the alternative is entanglement! Boris Tsirelson (talk) 15:27, 5 February 2016 (UTC)
No, Pitowsky is wrong. His nonmeasurable function explains the experimental fact only if the directions (of the apparata) are fixed ideally (with no error at all). But clearly, they are not, and still, Bell inequality is violated. Boris Tsirelson (talk) 16:18, 5 February 2016 (UTC)

After all, my point is ridiculously simple:
Every feasible sensor has a finite resolution.
Therefore any fine structure of a (mathematical) function on a (mathematical) continuum (including two-valued functions, a.k.a subsets) is physically meaningless.
Boris Tsirelson (talk) 17:24, 5 February 2016 (UTC)

Dark matter has (at least) gravitational effect. The fine structure (discussed above) cannot have any physical effect at all. Never. Much deeper darkness... Boris Tsirelson (talk) 17:30, 5 February 2016 (UTC)

I agree with you. But I also agree to some extent that (from Magidor)
As to be expected we do not have any definite case in which different set theories have an impact on physical theories, but we believe that the possibility that may happen in the future is not as outrageous as it may sound.
Mathematics itself cannot decide on new "true" axioms. There is Gödel's theorem (there are always new axioms), but, in addition, the disagreement among set theorists about which candidates for new axioms are "true", let alone that several "viable" axioms are incompatible. That physics could "decide" is outrageous. But it is not as outrageous as it might sound that physics could act as a tie-breaker. (This is clearly not the case with Pitowsky's theories, but suppose for the sake of reasoning that Pitowsky's theories were sound and that experimental evidence existed. Then this would be a strong argument for card c is not real valued measurable. (It is strong because it is more than nothing and deductive mathematical arguments, in any direction, from ZFC do not even exist.) Note that this is rather physics having an impact on set theories than the other way around.) I assign to it happening a nonzero probability, but nonzero numbers can be very small – hence I am not rushing to the betting shop – and can also not be held responsible for believing that it will happen. I don't. YohanN7 (talk) 13:35, 8 February 2016 (UTC)
Nice. I feel consoled to see that Magidor admits that "it may sound outrageous", and by your reservations. A consensus is reached, I'd say. But here is my reservation. After Einstein we know that, in some reasonable sense, the physical space is non-Euclidean (which was suspected by Gauss...). But it did not lead to any change in the axioms of mathematics. Mathematics provides models for everything (as Anatoly Vershik told me); vector space and Riemann manifold are just two of them. Imagine that, influenced by Einstein's theory, mathematicians switch to different foundations that exclude vector spaces from mathematical objects. Now what? Should Heisenberg describe quantum observables by (nonlinear) transformations of a manifold? Boris Tsirelson (talk) 14:10, 8 February 2016 (UTC)
Interesting and relevant reservation. No, imo physics could never invalidate (even parts of) mathematics because it applies to more than physics. But it could provide a pointer to what a Platonist might accept as true axioms, i.e. some axioms are more equal than others? Without a doubt, there will always be those taking the point of view that anything consistent goes (because it is just symbols scrapped down on a sheet of paper obeying some logic). But then Einstein's general relativity surely did change the direction of mathematics a little bit, by supplying beautiful application fairly early on after Riemann, making it (Riemannian geometry) even more of a subject worth of study. But this is just one example of mathematics and physics influencing each other. YohanN7 (talk) 15:02, 8 February 2016 (UTC)
Just one example? For now, unexpected physical revelations influence the choice of (a) relevant mathematical objects (Riemannian-like 4-dim manifold with indefinite metric? Inf-dim vector space over complex numbers?), and (b) their physical interpretation (general relativity? Copenhagen interpretation?), but not (c) axioms. Should this pattern be changed? When and why? Boris Tsirelson (talk) 15:12, 8 February 2016 (UTC)
Just one example?
That had me puzzled for a while. Now I see. Of course, experimentally verifiable physics having mathematical impact isn't discovered every year. Once per century at current pace? Maybe. Newton's second law is a differential equation, and needed prior development of analysis to be formulated. General relativity mentioned. Quantum mechanics has certainly had impact on mathematics. Figures like Hermann Weyl were physicists as much as mathematicians. Next event? Who knows.
But the current trend of physics at any time does have an impact, whether the physics is real or not. The prime example of the present day would be the continued failure of string theory to produce physics while pumping out massive mathematics, including a fields medal (Witten).
I think (a) and (b), but not (c) is what we will see with probability 1 – ε, ε > 0, ​1ε >> 1. Our intellect will guide (but can take us just that far for (c) (ZFC but not much more is the limit?)), not physics – unless something really unexpected happens. YohanN7 (talk) 11:24, 5 March 2016 (UTC)
Yes.
By the way, about "anything consistent goes": in some sense this is evidently absurd. Everyone can "invent" thousands of games more or less similar to chess but different. But then he can play these games with himself only (maybe, with two or three other eccentric persons). This is just "coagulation" with no real reason. Everyone can "invent" thousands of axiomatic systems whose consistency follows from that of ZFC (say). But his chance to find a partner is even much less than in the chess case. Axiomatics attracts not just by "coagulation"; it reminds us something that happens outside mathematics. When deducing theorems in a "random" axiomatics I would be much weaker than a computer. When deducing theorems in a "meaningful" axiomatics I am much stronger than a computer. This "meaningfullness" is a channel of strong influence of physics on mathematics. Boris Tsirelson (talk) 11:52, 5 March 2016 (UTC)
And here is another explanation why ZFC is (more or less) immune to physics and other influences. See my "Theory (mathematics)" [3] = [4], especially Section "Mathematics is not isolated". ZFC is like a unformatted harddisk. Then you define set inclusion, function, cardinality etc., and get something like a formatted disk. Then you build group theory, topology etc., and now the disk contains something useful. And so on. Some day you forget the *.mp3 file format and use *.mp4 format; but it does not mean you need another hard disk, as long as it is large enough and fast enough. This is the metaphor... Boris Tsirelson (talk) 12:43, 5 March 2016 (UTC)

## Law of a stochastic process

Hi Tsirel,

you reverted my edit to add an example of how to evaluate the probability measure of a single (possible) trajectory in the article about the law of a stochastic process. I think that example makes the article clearer for people without experience in mathematical formalities. I would like to restore that example, maybe separate it from the definition. Niout (talk) 12:36, 20 February 2016 (UTC)

But did you read the reason (on my edit summary)? "No, this formula makes sense when a single function has a positive chance, which is atypical." Your comment, please. Boris Tsirelson (talk) 15:12, 20 February 2016 (UTC)

## Article needs help!

Stochastic interpretation could perhaps use your particular expertise. 15:13, 5 March 2016 (UTC)

Why? It is already seen by User:Headbomb. Boris Tsirelson (talk) 16:20, 5 March 2016 (UTC)
It doesn't offer much of a clue about the current state-of-the-art. 16:46, 5 March 2016 (UTC)
I was never interested in that approach. Just now, after a look at Tsekov 2009 in arXiv (version of 2015), I did not find there anything able to produce an interpretation of quantum mechanics. A vague hint (in the Wikipedia article) about nonlocality could refer to microscopic wormholes able to violate Bell inequality. But Tsekov does not mention anything like that. Current state-of-the-art? What is it, really? The only non-Tsekov ref from the 2015 version of Tsekov to anything after 2000 is [4] that is hardly relevant to wormholes. Thus, I am still not interested. But, being not a physicist, I do not want to fight against it, given that Headbomb does not (and I do not know, why). Boris Tsirelson (talk) 17:49, 5 March 2016 (UTC)
According to Who'sWho, Roumen Tsekov is a chemistry educator. Hmmm... Boris Tsirelson (talk) 17:59, 5 March 2016 (UTC)
His ORCID data: [5]. Boris Tsirelson (talk) 18:08, 5 March 2016 (UTC)

## Shall we continue somewhere else?

Evidently I'm rather bitter about Wikipedia (and the mathematics section of it) I'd love someone to play devil's advocate for me. 90.199.52.141 (talk) 19:30, 17 March 2016 (UTC)

Clearly you are the (anonymous) author of this pearl.
Well, I am bitter, too. But, it seems, I understand that this is inevitable. In the "real world" (outside wikipedia) we have a lot of literature on every (notable) matter. Why? Since we readers are very different. Tensors are of interest for mathematicians, physicists, engineers, the last time even health professionals. Clearly no one can satisfy them all with a single text. (And moreover, no one can satisfy most students of mathematics with a single textbook for each course.) Wikipedia refuses the idea of having parallel articles for different readers. (I do not support this refusal; but anyway, I cannot fight it.) This is one of the inherent drawbacks of Wikipedia. If you have ideas how to overcome it, I'll be glad to hear. For now, every reader must suffer from the presence of other, very different readers. Such a communal apartment. Just a recent example: yesterday, someone marked "Gambler's ruin" as "too technical" (really?). Boris Tsirelson (talk) 19:45, 17 March 2016 (UTC)
I had prepared a long slight rant of an answer, after venting that there are clearly two cases. We have things where 1 word means different things to different groups (see any disambiguation page for that) and we have cases where something is used by groups in-the-same-way-even-if-they-don't-know-it-kindof. Obviously there is a point where we switch from separate pages to unified pages. I cannot hope to give you a specific set of rules for this. As I'm imagining you know too, we're neural networks, we're naff at remembering explicit lists of rules and checking something matches them but we are quite good at learning patterns and even if we cannot reconstruct explicit rules can still apply their sum.
The same applies to articles, I think (and have an example of a starting point) that expectation is on the "1 for all" side of "expectation, as in of a random variable" - I include my old answer below:

I meant better than edit and indent. But sure I do! This might give away who I am and I don't really want that (it's no one famous, or that you'd have heard of) but there are some doctrines I try to follow, the idea being that it is useful then to whichever level of reader (first year or beyond) ends up seeing it. Some things have to be "first year friendly" like sequences, or functions (like putting the set-view (relations) of functions before the lambda, but mentioning the lambda in a see-also section) and so forth. Expectation I give you is less clear-cut because the countably infinite (and finite, that's just one with a load of 0s) and on ${\displaystyle \mathbb {R} ^{n}}$ are two huge distinct chunks and it isn't the article's place to be like "Yeah these are specific cases of a general concept of integration" so I would start with the general and move down. We have sub-headings! That page is scattered all over the place.
I'm not making much sense and I really don't want to blow my cover.
What a page might look like, using indents as subheadings

(intro, __TOC__ maybe a nice infobox....)
Definition
Given a random variable, ${\displaystyle X}$ (a measurable map from a probability space ${\displaystyle (\Omega ,\Sigma ,\mathbb {P} )}$ (this is what I was looking for: to what? Borel sigma-algebra, naturals.... - there has to be some structure here; totally ordered?)) we define the expectation(references here) as:

• ${\displaystyle \mathbb {E} [X]:=\int _{\Omega }X\mathrm {d} \mathbb {P} }$

Where (that integral) denotes the Lebesgue integral (it is NOT THAT HARD to get a reader to have some not-formal-but-I-can-see-how-that'd-work understanding of integrating this way - I can prove this claim)
Notable special cases

The integers and countably infinite AKA DRV case
CRV / real case
A vector of CRVs is isomorphic (perhaps wrong word) to a CRV of vectors (you know what I mean)

Immediate results
It's linear, that stuff
Intuitive understanding
I'd use the "expected value of 1/n" (being n) example here. Maybe another example like a game/distribution with 0.25 for ${\displaystyle \mathbb {P} (X=1)}$ 0.5 for 2, 0.25 for 3 and be like "if you did this 1 million" times
Examples
Why not, Maybe some non-trivial examples.

The reader will scroll down and at some point lock onto a familiar definition. If you want to prove it's linear for the general definition, DRVs and CRVs - add that. I like collapsible boxes for these.
• May I apologise for the weird format of this comment, by the time I realised how strange it was I was already committed. I went surprisingly far with statistics before I went all formal (and knew measure theory was a thing) so I do see what you mean by accessibility, and evidently I'm only now doing it formally, but that's how I'd start the page. I've found that if people are "spooked" by the top of a page often they'll scroll down until they find something they can latch onto. It'd be a good place to put "but if you think about conceptually, there are a large number of parallels between expectation of CRVs and DRVs" but I digress.

Regarding bitterness, I mean Wikipedia's way of doing things. If someone is going to judge what I wrote or say something isn't notable and should be deleted, they better be from a circle in which if it was notable they've heard about it! Then you have these rules (this isn't a personal thing, I found this from reading about "duck typing") where you ignore votes that say basically the same thing. but on the voting to delete page it's like "but also if someone seems too contradictory we ignore that vote too" it's absurd.
I get deleting crap like when someone creates a page for their dog, but why would you put any weight on the option of deleting a page with some content and redirecting it somewhere barely related! EVEN IF you delete Mediawiki stores the deleted version just MOST PEOPLE cannot see it any more! No bytes are saved!
"We have a cleanup project" is code for "I just let my eyes loose focus and if I don't see enough little blue squares (citations) I put a vote-to-delete template on there and remove it from the work list" - I sometimes think for a "joke" I should build up a reputable account then when it comes to "cleaning up" training a neural network along similar lines to those I just mentioned.
THEN! When someone without an account sees this and thinks "this is crap" the same the program would go find people who are involved with polar opposite areas of Wikipedia to talk to the newbie and link to various internal wikipedia pages on said IP's talk page. The best troll doesn't dirty his or her hands.
Not entirely sure what I hope to gain from this and I do wish I'd chosen a better example 90.199.52.141 (talk) 20:35, 17 March 2016 (UTC)
I understand if it's TL;DR but do scan 90.199.52.141 (talk) 20:35, 17 March 2016 (UTC)
You express many thoughts; I understand only few, mostly because I seldom go outside math articles, and in addition, English is not my native language. Anyway, looking at your sketch of the article I see that you want it to be "Most general definition first, most accessible explanations last". Personally, I have no reason to object. But I know that Wikipedia is driven by wide audience, not by experts. And there is absolutely no chance that it will switch from "Most accessible first" to the opposite. In fact, I tried Citizendium (driven by experts), as you can see on my userpage. And it appears to be much worse than Wikipedia. It seems, "Wikipedia is the worst form of free information, except for all the others". Such is the life. Use it whenever it helps you, and use textbooks whenever it does not. There you can choose a presentation that fits your needs this time; here you cannot. Boris Tsirelson (talk) 21:48, 17 March 2016 (UTC)
(EDIT CONFLICT) Specifically, about the values of a random variable: generally it may be just a measurable space; but if expectation is used, then it has to be (at least) a linear topological space. This is not specific to probability; this is what is needed for integration in general. Though, integration of vector-functions (with inf-dim values) involves additional troubles; whether you need these this time, or not, depends on your needs. Boris Tsirelson (talk) 22:04, 17 March 2016 (UTC)
(RE CONFLICT) with some sort of (pointwise) multiplication. It's a job that requires a scrap of paper, I tried to take a short cut with wikipedia and here we are 90.199.52.141 (talk) 22:10, 17 March 2016 (UTC)
I guess we agree, and I love the "worst form of free information" bit, I've spent most of the last year (when I can) carefully referencing definitions, proving "differing" definitions the same and creating something I'd quite like your opinion on actually. Got a cyphered email address you'd be kind enough to put here?
I created a subdomain for one of my sites and said "if after a year it's still an active project, I'll make it permanent" 90.199.52.141 (talk) 21:57, 17 March 2016 (UTC)
Sorry, your English is again not so accessible to me (I mean "Got a cyphered email address..."). Anyway, my email address is public on my (professional) homepage that you can find easily. Alternatively, use "email this user" on Wikipedia. Boris Tsirelson (talk) 22:09, 17 March 2016 (UTC)
Sent, please pretend 90.199.52.141 never happened. 90.199.52.141 (talk) 22:52, 17 March 2016 (UTC)
Received and replied. Boris Tsirelson (talk) 15:41, 18 March 2016 (UTC)

## Fortifying wikiquanta

Two years ago the article "Affine space" was attacked by a non-expert. His position: the notion of affine space (like any other) must have just one definition treated literally; not only the structure, but also its encoding in the set theory must be fixed once and for all; otherwise mathematics is not rigorous. The attack was repulsed, but, bothered by the vulnerability, I built a bastion against possible attacks of this kind.

During almost two years, several articles in quantum mechanics were attacked by a non-expert. His position (as far as I understand): many authors use uncritically the mathematical formalism of quantum mechanics; one must follow literally Dirac's ideas about oven, anti-oven, analyzer etc., otherwise one gets some mathematics of doubtful physical meaning and relevance; in particular, most of the theory of entangled states is very unreliable. The attack is now repulsed, but, bothered by the vulnerability, I urge to erect a bastion against possible attacks of this kind.

It would be nice to have an article, or better several articles, that show how firm is the quantum theory, how reliable is its mathematics. With emphasis on successful experimental verification already done. We have "Quantum information science" and there on the bottom (footer template) "Physical implementations: Quantum optics, Ultracold atoms, Spin-based, Superconducting quantum computing." These articles are professional (which is good) and hardly accessible to non-physicists (which is worse). Their references (almost all) are articles, not books, in spite of existence of a number of books on these topics.

Various complicated entangled states of several qubits are prepared, then changed coherently by quantum gates, then measured, and results correspond to the theory. This holds uniformly for different physical realizations of the qubits, and holds both when all these qubits are situated together within a microscopically small volume, and when they (or at least some of them) are separated macroscopically. Joint probabilities of outcomes of measurements on different qubits are observed, and conform to the theory.

Therefore there cannot be any reasonable doubt that the quantum description of such states (generally highly entangled, and approximately pure) is successful. Boris Tsirelson (talk) 10:50, 14 April 2016 (UTC)

### Discussion

Looks like a good cause, but no memorable book jumps at me... I have my hands full at the moment, but will think about your quest. In some sense, I am a bit blase' about this... I had Feynman's volume III in college in the spring of 1971, and have not succeeded in questioning any of his pedagogical schemes and assertions there since... so have spent almost half a century scratching my head as to why people seem so metaphysically conflicted about QM... and what the fuss is about, really. Following the math, I have always thought these issues were settled before Acton's or EPR type experiments, Bell's inequalities, etc... (even though I am delighted by clever new schemes confirming orthodoxy like the Delayed choice quantum eraser.) Quantum computing types are breathing new life into this, but the issues have been settled in my mind for so long that they have become dull for me... almost as dull as arguing about the round earth... Will try to stay in the loop, making small, salutary changes when and where I could. Thanks, Cuzkatzimhut (talk) 19:14, 15 April 2016 (UTC)

Interesting! Are you a counterexample to the "rule" that those who are not shocked by the quantum theory did not understand it? Boris Tsirelson (talk) 20:22, 15 April 2016 (UTC)
I guess I am... I am happy with the part I do understand, and I see no reason to seek trouble. I mean, after 90 years of unremitting success and clearly bogus, harebrained, or worse, pseudo paradoxes, one would expect spirits to have calmed down— they actually largely have, in the professional community of users of QM. Since I am not interested in philosophy or psychology I would not get into the mode of how people keep on getting confused by Schr's cat nonsense and misconceptions... There are deeper mysteries to understand out there. Cuzkatzimhut (talk) 21:35, 15 April 2016 (UTC)

Hi Boris, this is really an appreciable aim. That articles are indeed rudimentary at best and just full of citations. Not so useful indeed. I cannot promise anything for sure but as I have some time I will approach some of these unsatisfactory articles. My current view about quantum optics, since the time were I was a contributor, is that is completely oriented to quantum computation. A respectable goal indeed but there are a lot of open questions yet that experiments like those by Serge Haroche could help to clarify, mostly for many-body systems. On the other side, all this mania about interpretations does not heat me up at all. My view is that this is just wasting precious time and resources. I hope to be helpful to your program.--Pra1998 (talk) 19:49, 15 April 2016 (UTC)

Thank you. Yes, I understand your feeling about interpretations. But this time the theory was attacked, not interpretation! It was claimed that most predictions for entangled states are not really predictions, but rather an abuse of the quantum theory. This is provably wrong, as we both know. But we are here on Wikipedia in order to (try to) spread our understanding, aren't we? Boris Tsirelson (talk) 20:15, 15 April 2016 (UTC)

I can see your point but I'm having a hard time imagining what the articles you say should be written should actually look like! A vast swathe of quantum mechanical articles have sources for experimental verifications of the predictions of quantum theory. Porphyro (talk) 10:42, 21 April 2016 (UTC)

To your first phrase: Yes; indeed, I feel the same.
To your second phrase: "verifications of the predictions", sure; but are they about entangled states?
The feature of both attacks (on affine spaces and quantum mechanics) is that the attacker is not stupid. He likes the theory! He objects to a careless overuse of the good theory (as he understands it).
The "quantum attacker" likes the wave function and all that. His red line is, an entangled state of a composite system whose subsystems are addressed separately. This case was indeed not experimentally available to the founding fathers. But now it is available; its "verifications of the predictions" are mentioned mostly in our articles on quantum information, and these are hardly accessible to non-experts, and do not emphasize this aspect: not only technological progress, but also verification of entanglement theory.
Now, back to your first phrase: maybe, a new section in the "Quantum information science" article? Boris Tsirelson (talk) 13:17, 21 April 2016 (UTC)

### Sources (tentative)

• Goong Chen, David A. Church, Berthold-Georg Englert, Carsten Henkel, Bernd Rohwedder, Marlan O. Scully, M. Suhail Zubairy "Quantum computing devices: principles, designs, and analysis", Chapman & Hall/CRC 2007.
A wonderful book! Implementation of qubits and real experiments are discussed expertly; general conclusions are articulated (and the math of quantum computing is here, of course).
• Our own writing has benefitted from many online resources. For example, Wikipedia, the free internet encyclopedia (http://en.wikipedia.org), is a constant source of ready help and valuable information. (p. xvii)   :-)
• But seeds for quantum computing were planted more than half a century earlier. The following three profound events actually constitute the most important preludes that paved the way for the development of a modern quantum computer (QC):
(1) The Stern-Gerlach experiment (1920s);
(2) The Einstein-Podolsky-Rosen (EPR) paper [8] (1935) and its reply from Schrödinger [26];
(3) The Landauer principle on information erasure [19] (1961). (pp. 1–2)
• There is no mechanism that decides whether the particle is spin-up or spin-down. Rather, it is a truly probabilistic phenomenon.
As another consequence, we have deduced that there are nonlocal correlations in the real world. Hence there are no local hidden variable theories.
(p. 16)
Sect. 2.2 "Quantum mechanical systems; basics of atoms and molecules": no "axioms"; entanglement is introduced first, and the second is the Schrödinger equation. (Then, Sect. 2.3 "Hilbert spaces".)
Chapter 3 "Two-level atoms and cavity QED".
Chapter 5 "Quantum computation using cold, confined atomic ions".
• N. Chandra, R. Ghosh "Quantum entanglement in electron optics", Springer 2013.
• Quantum entanglement is one of the most intriguing phenomena in nature. (p. ix, Foreword by Uwe Becker)
• Entanglement, like many others (e.g., wave-particle duality of light and matter, uncertainty principle), is a purely quantum phenomenon which defies all classical intuitions. (p. xii)
• Availability of two or more qubits with entanglement (i.e., nonlocal correlation) is an essential ingredient for any quantum information related studies. These qubits can be of any kind of particles possessing two independent and simultaneously accessible states. Some of the well-known examples of such qubits are [17]: an electron or any other spin-½ particle; a two-level atom/ion; a photon with negative (left) and positive (right) helicities (circular polarizations), or with horizontal and vertical linear polarizations; states of two photons entangled with respect to their phase and momentum [19], or energy and momentum [20], etc. (p. 2)
• <...> measurements on the entangled states of spatially separated subsystems allow physicists to test fundamental notions about the nature of the physical world in general, quantum theory in particular. The two kinds of of systems whose entangled states have hitherto been investigated [103] are those in which one can study properties of individual particles, or wherein collective measurements are possible. The later type includes, for example, cold clouds of 107 atoms(e.g, [104]), optical lattices consisting of 105 two-level atoms (e.g. [105]), etc. These ([104,105]) and other (e.g., [106, 107] etc.) collective measurements have shown [103] that multiparticle entanglement is capable of influencing macroscopi thermodynamical properties (e.g., magnetic susceptibility, heat capacity, etc) of solids. (p. 30)
• Dieter Heiss (Ed) "Fundamentals of quantum information: quantum computation, communication, decoherence and all that", Springer 2002.
• Modern experimental techniques have provided convincing evidence about various aspects of "quantum weirdness" in that, for instance, entanglement or teleportation are established as physical realities. Also, the mysterious collapse of the wave function is now replaced by a thorough understanding of the dynamical process which is decoherence. (Preface)
• We accept the laws of quantum physics, mysterious and beautiful as they are, and try to explore them to design surprising and potentially useful devices. By doing so, it turns out that we do gain more and more insight into the working of quantum physics. (Bouwmeester, Howell, Lamas-Linares, p. 150)
• Other principles, like the ones related to the measurement process and the superposition principle, have only recently become important in some applications. In particular, they form the basis of what is called quantum communication and quantum computation. These two fields have been strongly developed during the last few years, and they may well give rise to a technological revolution in the fields of communication and computation [1]. (Cirac, p. 199)
• Entanglement plays an important role in most of the applications in the field of Quantum Information [1,2]. <...> Although for pure states of two systems, entanglement is well understood, ... (Cirac, p. 200)
• For the moment, experimentalist have been able to perform certain quantum gates, and to entangle 3 or 4 atoms [19,12]. (Cirac, p. 202)
• Generally, in Physics we associate physical quantities and situations with mathematical concepts. These mathematical concepts can then be processed using a series of rules (or axioms), which allows us to make predictions back on the physical systems. In particular, in Quantum Mechanics we associate different situations (states) of a physical system with the elements of a complex Hilbert space H. <...> The consequences of the mathematical structure of Quantum Mechanics are even more intriguing when we have a composite system. (Cirac, p. 210)
• The existence of correlations, by itself, is not a property of entangled states. <...> However, the correlations carried by entangled states are, in some sense, different <...> we can perform things that are not possible using classical correlations. (Cirac, p. 212)
• Implementation of an electron-spin entangler that make use of the s-wave (spin singlet) nature of conventional semiconductors were proposed in [13,14]. (Burkard, Engel, Loss p. 243)
• Yoshihisa Yamamoto, Kouichi Semba (Eds) "Principles and methods of quantum information technologies", Springer Japan 2016 (Lect. Notes Phys. 911).
Basic rules of quantum mechanics (p. 3-10); generalized measurements and quantum operations (p. 15-23).
Cryptography: Quantum Key Distribution (p. 67) We have achieved QKD over 200 km of fiber <...> QKD is the first quantum information technology that can be put to practical use.
• Jozef Gruska "Quantum computing", McGraw-Hill 1999.
Implementation of qubits and real experiments are (almost) not considered.
Sect. 2.2 Quantum entanglement
One of the most specific and also most important concepts for quantum computing and quantum information theory is that of quantum entanglement—also one of the most puzzling concepts of quantum physics. (p. 73)
Sect. 9 (Appendix)
Sect. 9.1.3 Quantum theory versus physical reality
• Of course, there are attempts to assign physical reality to such concepts as quantum state, quantum systems and quantum measurement. However, they lead to hard-to-accept mysteries and so-called paradoxes. <...> the attempts to derive these theoretical concepts and principles only from the physical reality and to assign them physical meaning have not worked well. (p. 350)
Sect. 9.2 Hilbert space framework for quantum computing
(including density matrices, superoperators, and generalized measurements)
• Frank Gaitan "Quantum error correction and fault tolerant quantum computing", CRC Press 2008.
Appendix B "Quantum mechanics" lists "axioms" of quantum mechanics. Implementation of qubits and real experiments are not considered.
• Jonathan A. Jones, Dieter Jaksch "Quantum information, computation and communication", Cambridge 2012.
This book is aimed squarely at undergraduate physics students who want a brief but reasonably thorough introduction to the exciting ideas of quantum information, including its applications in computation and communication. <...> As this text is aimed at physics undergraduates, we believe that it is vital to cover experimental techniques, rather than merely presenting quantum information as a series of abstract quantum operations. We have, however, concentrated on the basic ideas underlying each approach, rather than worrying about particular experimental details. (p.1)
There are two ways to teach quantum mechanics. The first way — which for most physicists today is still the only way — follows the historical order in which the ideas were discovered. So, you start with classical mechanics and electrodynamics, solving lots of grueling differential equations at every step. Then, you learn about the "blackbody paradox" and various strange experimental results, and the great crisis these things posed for physics. Next you learn a complicated patchwork of ideas that physicists invented between 1900 and 1926 to try to make the crisis go away. Then, if you're lucky, after years of study you finally get around to the central conceptual point: that nature is described not by probabilities (which are always nonnegative), but by numbers called amplitudes that can be positive, negative, or even complex.
Look, obviously the physicists had their reasons for teaching quantum mechanics that way, and it works great for a certain kind of student. But the "historical" approach also has disadvantages, which in the quantum information age are becoming increasingly apparent. For example, I've had experts in quantum field theory — people who've spent years calculating path integrals of mind-boggling complexity — ask me to explain the Bell inequality to them, or other simple conceptual things like Grover's algorithm. I felt as if Andrew Wiles had asking me to explain the Pythagorean Theorem.
<...>
So, what is quantum mechanics? Even though it was discovered by physicists, it's not a physical theory in the same sense as electromagnetism or general relativity. In the usual "hierarchy of sciences" — with biology at the top, then chemistry, then physics, then math — quantum mechanics sits at a level between math and physics that I don't know a good name for. Basically, quantum mechanics is the operating system that other physical theories run on as application software (with the exception of general relativity, which hasn't yet been successfully ported to this particular OS). There's even a word for taking a physical theory and porting it to this OS: "to quantize."
But if quantum mechanics isn't physics in the usual sense — if it's not about matter, or energy, or waves, or particles — then what is it about? From my perspective, it's about information and probabilities and observables, and how they relate to each other. (pp. 109–110)
• Nicolas Gisin "Quantum chance: nonlocality, teleportation and other quantum marvels", Springer 2014.
• But does that mean that physicists must abandon all their endeavours to understand nature? It always surprises me that many physicists do not appear much concerned about this question. They seem satisfied by being able to do the necessary calculations. Perhaps these physicists would say that computers understand nature? <...> And yet science has always been characterised by the quest for good explanations. (p. 105)
• Today, violation of a Bell inequality is the very signature of the quantum world. <...> We could usefully begin by dropping the old-fashioned term ‘quantum mechanics’ and replacing it systematically by ‘quantum physics’. There is nothing mechanical about this particular branch of physics! (p. 107)
• <...>the distance we have come since Einstein, Schrödinger, and Bell. In those days, the question was: Do the nonlocal correlations predicted by quantum theory really exist? Today, no physicist could doubt this. (p. 109)
• Lucien Hardy "Quantum Theory From Five Reasonable Axioms", arXiv:quant-ph/0101012.
• A.M. Zagoskin "Quantum engineering: theory and design of quantum coherent structures", Cambridge 2011.
• Lajos Diósi "A short course in quantum information theory", Springer 2007.

### Annecdote and comment

A primary source: the Alain Aspect experiments in 1985/1986 confirm that entanglement persists even when the measurements are made at space-like separations. Before this, faster-than-light was considered a loop-hole in various debates.

An anecdote for the "historical teaching": I was taught QM by Ugo Fano, from a textbook he wrote quite late in life: type-written, with hand drawings. It stepped through each of the historical experiments, and at each step, provided the simplest semi-classical explanation of the results of that experiment. It was quite difficult, and the class was a struggle. Then one day, a girl in the class approached me (later, it turned out she was ranked first in the graduating class; wish I remembered her name): she had gone to the stacks, and found OTHER books by Fano, from the 1950's. These books were ... night and day: they were ... standard textbooks, on glossy heavy-weight paper, excellent typography, top-notch graphics: first-rate textbooks. What's more, the explanations were straight-forward and EASY to understand! We pored over these, and, for the first time in a long time, started to understand what was going on. She and I looked at each other with a WTF? Why isn't he teaching from this book?

We left the question unanswered, but years later, I developed the idea that he was deeply unsatisfied with the state of QM, and had decided to go back to the beginning, to those very first experiments, and see if somehow, something had been missed or overlooked, hunting perhaps for some alternative explanation. This hunt lead him to write the new textbook. So I believe. And so you can put him into the "shocked by QM" category.

Here's another, much milder annecdote: in grad school, the chair was Nobel laureate C. N. Yang and he never taught class, but apparently, one year, someone shamed him into it. He only taught half a semester before he begged out and someone else took over. But ... he choose to lecture on the neutron interferometer. This was remarkable, because the upper and lower branches of the interferometer are entangled, and must be waves; yet neutrons have mass and "must be" particles, and so the question was: can entangled states of waves feel gravitational forces? The answer, from those experiments is clearly "yes", and the theory works out just fine. What was remarkable was the choice of topic to lecture on: of the zillions of things he could have talked about, e.g. reminiscing about TD Lee or talking about gauge fields ... no ... he talked about a basic experiment probing the deep fundamentals of QM. Call me sensationalist, but I think he was another that was secretly, privately shocked by QM.

The latest in this vein seems to be the proposed pigeonhole principle experiment from Aharonov. QM makes clear predictions, and I'm sure QM will carry the day, but its still an interesting experiment, if it could be done with charged particles, as charges repel, and reveal the (lack-of) pigeon-hole-ness taking place. 67.198.37.16 (talk) 01:53, 23 April 2016 (UTC)

p.s. reviewing the above, and looking a bit into the choygame disruption: I feel compelled to point out: entanglement is not just some aspect of QM limited to modern quantum computing or to Bell states or spin states: its rather deeply embedded into *all* of QM. Essentially, its due to the use of Hilbert spaces, in general, whether finite or infinite dimensional, and that essentially all state preps and measurements amount to a choice of basis for Hilbert space; entanglement is "nothing more" than a change-of-basis. Thus there are zillions of experiments over 100 years that confirm entanglement; you can't honestly single out a handful of modern ones. That said, the part about entanglement that gets most people all worked up is precisely this: the decomposition of products of representations of Lie algebras into direct sums of the irreducible reps. Specifically, for the Lie algebra of su(2), and when the decomposition is performed at spatially-separated locations. The math says "it must be so", the experiments confirm this; its just very difficult to visualize this, even when you have the requisite training. That there's a continuing crisis is undeniable: some of the weak measurement results are just plain bizzarre, and the firewall paradox has lead the leading figures to suggest that QM entanglement might be modelled by wormholes in spacetime, e.g. some variation on the black hole electron. However, no one knows how to write down e.g. the Kerr solution, and connect the two ends so that they look like above-mentioned direct sum of irreducible reps of su(2) (... and obey the myriad other properties needed viz path integral formulation, etc). Entanglement in QM is deeply and firmly established experimentally and theoretically in a zillion ways; yet basic concepts like "what is mass" "what is time" "what is space" remain in a deep crisis of not being understood and thoroughly befuddling.

67.198.37.16 (talk) 03:49, 23 April 2016 (UTC)

Thank you. (Why not register to Wikipedia and participate more systematically?)
It is interesting to see a similarity between Ugo Fano and our attacker (even though you put the former into the "shocked by QM" category, while the latter emphatically denies such affiliation).
What exactly shocks? I join the opinion of Scott Aaronson "Can Quantum Computing Reveal the True Meaning of Quantum Mechanics?":
<...> one of the wisest replies <...>: "a quantum possibility is more real than a classical possibility, but less real than a classical reality." In other words, this is a new ontological category, one that our pre-quantum intuitions simply don’t have a good slot for.
This is the shock: new ontological category! Not just a new form of matter (or even space-time). This is why "quantum mechanics sits at a level between math and physics that I don't know a good name for" (Scott Aaronson; already quoted above).
I'd say, Scott Aaronson shows that such "opposite" interpretations as Bohmian mechanics and many-worlds, even if enlightening for a while, are ultimately futile in the same way as the flywheel analogy in Maxwell's 1865 paper on electrodynamics. We really need fields (rather than flywheels) and the new ontological category (rather than a new combination of old ontological categories, including particles, worlds, wormholes etc).
More from Scott Aaronson:
David Deutsch, who’s considered one of the two founders of quantum computing (along with Feynman), is a diehard proponent of the Many Worlds interpretation, and saw quantum computing as a way to convince the world (at least, this world!) of the truth of Many Worlds.
Yes, the neutron interferometry impressed me too. However, zillions of interferometric experiments, or zillions of spin experiments, leave some wulnerability, closed by the quantum information experiments.
Indeed, zillions of spin experiments have lead Joy Christian (another entanglement denier) to the idea that a nontrivial connection (parallelism) on a sphere is the key. Probably, zillions of interferometric experiments may lead another entanglement denier to some interferometer-specific idea.
In contrast, less than a hundred of quantum-information experiments can show that the shoking feature is purely informational. Being formulated in the language of qubits, quite diverse quantum predictions are confirmed using quite different qubits. Therefore it is futile to seek a key in specific details of a single physical realization of qubits. Boris Tsirelson (talk) 16:24, 23 April 2016 (UTC)

## More off-topic remarks

I've recently come to a conclusion about wave-function collapse that others (e.g. penrose) have been saying, but didn't make sense to me until just now: gravitation is essential. In a nutshell: QM and MWI holds true for assemblages of less than about 10^18 atoms, above that its classical. The number 10^18 is the planck mass. The idea is that when 10^18 atoms are involved, then you can no longer ignore the gravitational field: the gravitational field of a dead cat is different than that of a live cat. Notice that we have no quantum operators that can rotate the phase of a wave-function of |dead cat> + |live cat> -- formally, you can write that vector and call it a superposition, but there is no technology to alter the phase. Next: wave function collapse does NOT occur when you run photons through polarizers, slits, etc. and that is why quantum erasers work (you can still rotate the phase of a wave function after its gone through a birefringent crystal, etc.. Wave-function collapse *does* occur when photons hit silver nitrate in a piece of film, or a photomultiplier tube, or bubble chamber or cloud chamber because of the 10^18 atoms rule. (Note that chlorophyll is smaller than that, which is why biologists are seeing one-photon entanglement in chlorophyll)

Per Deutsch, the transactional interpretation seems to work: when a w.f. has not collapsed, you need to use the two-state vector formalism (TSVF) -- this is required by modern weak measurement experiments, and is implicitly in all the quantum computing work, in the guise of positive operator valued measure (POVM) that the computing guys like to use. Note that the TSVF has both a forward-time and a back-ward-time part: in essence, future measurements can go back in time to alter the past (this is the transactional interpretation) but the ONLY thing it can alter in the past are the qubits and NOT the classical bits. The alteration of qubits is exactly what you need for entanglement, and the backwards-time resolves the issues with e.g. space-like separated entanglement measurements.

Basically, the hard part is a change of focus: rather than saying, like in QM, a particle is in two places at once, one instead must say that there is a single particle, and it has, as its fiber, multiple space-time locations. That is, the base space are the particles themselves, and the (non-local) space-time locations of the particle are in the fiber. Particles in the base space are entangled in the usual sense: (I'm thinking entanglement is just the usual Lie-algebra decomposition of large representations into sums of irreducible ones; that's what is in the base space -- the base space is some big giant Lie algebra rep of say dimension 10^18 or something crazy like that).

So I find this gravity+TSVF relatively satisfying, and the new ER=EPR lends it new credence and weight. Things are moving in the right direction.

Next up: what is the mechanism of wave-function collapse? So: MWI is firmly anchored in the Feynmann functional integral, so if you want to explain collapse, you have to somehow argue that some of the trajectories in that integral vanish or go to (exactly) zero measure or something. Once upon a time, I used to think that this was the right approach, and so hunted around anything that could point there: e.g. groping through differences between measurable and non-measurable sets (thus stumbling over your work) and wondering if a quantum measurement caused some fraction of that Feynmann integral go non-measureable or something crazy. Or maybe some Hauptvermutung or something like that. Recently, I decided that cannot possibly work, because it does not explain why its 10^18 is the magic number.

Here's what can work: QCD confinement. Here's the idea: any non-abelian Yang-Mills field appears to confine fermions (although this is an unsolved millenium prize problem but lets assume we can prove it). The standard Einstein action is "just" a Yang-Mills field on the frame bundle, but essentially its got the same non-abelian terms (three-gluon vertexes, four-gluon vertexes, three-graviton vertexes, four graviton vertexes). Now, the gravitational coupling is about 10^40 weaker or about 10^18 atomic mass-scale weaker than the gluon coupling: there is a gravitational "confinement" preventing wave-functions of more than 10^18 atoms being coherent. Its the same non-linearity. So, for example, there's work by Dan Freed showing how spin connections can be rotated into baryonic solitons, and all this chiral bag stuff from QCD. I think some of that can be ported over to gravity, to give a characteristic size for the max size of a coherent wave function.

Here's some supporting evidence: experiments at RHIC show that the quark-gluon plasma behaves like a superfluid. Now, take a look at the BCS theory: it requires long-range entanglement of fermion pairs. So I envision the uncollapsed quantum system as being a kind-of "superfluid" of MWI wave-function phases floating about. There's an order parameter too: a kind of ratio between qubits and classical bits, e.g. see the article SIC-POVM. There's even some recent work that suggests that the surfaces of black holes behave is if they are superfluids (i.e. if ER=EPR then the superfluidity is an important part of it -- it explains why entanglement seems to be kind-of-ish conserved). The black-hole-superfluid stuff comes from some AdS/CFT formulation. Note, BTW, that AdS/CFT can also be used to simplify certain completely standard QCD amplitude calculations too: there's more than enough overlap in these areas.

Phew. That's it. Yes, I'm aware that the last few paragraphs sound totally nutty and insane, but the first few should be fairly convincing. I'm currently boiling the ocean to see if I can find some non-hand-waving way of expressing the above, but its very hard. I'm trying to avoid committing to some pre-existing theory of quantum gravity or to string theory, but for things like spin connections on bundles, ones hand might be forced. 67.198.37.16 (talk) 20:08, 7 May 2016 (UTC)

Well... this is much more than I could digest in a reasonable time (or at all). Basically, I understand that you treat gravity as the inherently non-quantal part of the nature, and accordingly, gravity is the ultimate decoherence. Thus, I guess, you believe that gravitons do not exist. Well... given that the classical gravitational wave is on the wedge of detection for now, I do not expect to hear an experimental confirmation or refutation of this idea. But wait; what do you think about "quantum gravity", Planck foam etc? If gravity (=space-time) is non-quantal at all, then there is no reason for its quantum fluctuations in the small. Or do you mean that the quantum-classical boundary separates weak gravitation field from strong one? Boris Tsirelson (talk) 20:48, 7 May 2016 (UTC)
A more practical question.
Cold crystal of mass 10-9 kg, electrically neutral, rests in zero gravity in a deep vacuum. I try to measure its momentum with an accuracy of 10-28 kg⋅m/s, thus creating for the center-of-mass a wave packet of width about 0.5⋅10-6 m. What happens? Does gravitation prevent formation of this wave packet? Or does it destroy it gradually? If so, how fast? Does the packet survive (that is, approximately follows the Schrodinger equation) during 1 sec? Boris Tsirelson (talk) 05:53, 8 May 2016 (UTC)

## acknowledgement your contributions to a submitted article

I was writing a draft for a Wikipedia article when the dean requested that I attempt to publish a paper. I have tenure and feel that my Wikimedia efforts have higher long-term value, but decided to humor him by submitting the draft to AJP. Your contribution involved the space-time figure and superdeterminsm. Let me know if you do not wish me to acknowledge you in the article The draft is at User:Guy vandegrift/AJP--Guy vandegrift (talk) 13:32, 9 May 2016 (UTC)

Nice; but indeed, I do not think that my misunderstanding about backward light cones is something to be acknowledged. Boris Tsirelson (talk) 15:38, 9 May 2016 (UTC)
I cannot disagree with that comment. I will remove the acknowledgement.--Guy vandegrift (talk) 16:56, 9 May 2016 (UTC)

## Off topic personal attack

Although I was attacked from david eppstein for a half year.--Takahiro4 (talk) 20:18, 7 June 2016 (UTC)

## Mathematical point of view

Hi, Tsirel! How do you see from a mathematical point of view (including a combinatorial context) the aspects discussed at talk:apparent molar property and user talk:Dirac66#Possibilities re the definitions and derivation of formulae for apparent properties in multicomponent solutions and as a consequence of this perspective the restoration of some content in the article removed in 28th of January this year? Your input is very valuable by coming from a professional mathematician.--5.2.200.163 (talk) 15:03, 19 July 2016 (UTC)

Oops, sorry, I cannot. I am not acquainted with these notions. I do not know, how to translate them into math language. Boris Tsirelson (talk) 18:29, 19 July 2016 (UTC)
Of course, some additional specifications are very useful to underline the mathematical background and problem formulation.--5.2.200.163 (talk) 12:37, 20 July 2016 (UTC)

The problem formulation starts with enumerating some definitions:

• the definition of an ideal solution or mixture characterized by addivity of volumes of components :${\displaystyle V=\sum {V_{i}}}$ and heats of mixing,
• the definition of a non-ideal solution where these ideal additivities do not hold and requires the introduction of partial molar property :${\displaystyle V=\sum {\bar {V_{i}}}^{0}}$which are additive and characterized by Euler homogenous function theorem,
• the definition of an apparent molar property of a component which has the purpose of isolating each component's contribution to the non-ideality of the mixture. In the case of binary mixture the situation is somewhat simple; it gets more complex or more degrees of freedom of definition starting from the ternary case where the grouping of components by subsets or combinations from all possible subcombinations of components from the powerset associated with the cardinality (number of components) of the mixture leading to the definition of pseudobinary, pseudoternary,...mixtures.--5.2.200.163 (talk) 13:28, 20 July 2016 (UTC)

For further steps of reasoning deployment a source (Apelblat) has been found that deals with pseudobinaries along the lines of mathematical combinatorical intuition which assigns partial differences to submixtures in order to isolate the contribution of each component to total non-additivity by analysing partial non-additivity in mixing when forming, for instance, a ternary mixture from two binaries with a common component.

The main question is to what extent mathematical derivations and notations of formulae and/or expressions in the apparent property article can be allowed according to WP:CALC without being necessarily sourced (in contrast with political sciences or biography articles where controversial statements density requiring sourcing is far greater)?--5.2.200.163 (talk) 13:44, 20 July 2016 (UTC)

I see. It reminds me interaction potential for three and more bodies in the general relativity; in some approximation it is the sum of two-body potentials, but in the next approximation it involves three-body potentials, and so on. Also in the theory of Gibbs measures I recall a similar situation.
However, I do not think that a similar argument may be used in chemistry just according to WP:CALC, unless it was already used this way in some reliable sources. Boris Tsirelson (talk) 19:16, 20 July 2016 (UTC)
Perhaps there are some sources somewhere that partially address the issue (by the way, interesting comments about (zero) probability and other mathematical remarks on your talk page sections above) but considering the rather low probability of locating them and the less stringent need to source derivations based on followings from definitions I consider that mathematical reasoning can be deployed without too much anxiety of sources. Sources can be explored to extract particular data to exemplify some derivations.--5.2.200.163 (talk) 12:36, 21 July 2016 (UTC)
One can notice the unifying capacity of mathematical reasoning in various domain of science mentioned by you. From the point of view of applied mathematics it is very important to apply a consistent modus operandi in all scientific aspects, regardless of different affiliations like chemistry, celestial mechanics, statistical mechanics, etc.--5.2.200.163 (talk) 12:49, 21 July 2016 (UTC)
A very useful mathematical application to the mixing of solutions and associated non-additivity is to analyze what examples of (partial) canceling of non-ideality as function of composition occur when two binary mixtures with a common component, one having negative deviation from ideality and another with positive deviation, are mixed. This case surely requires data from sources.--5.2.200.163 (talk) 12:59, 21 July 2016 (UTC)
Speaking of two and three-body potentials and iterative reduction, are they applicable in ordinary celestial mechanics?--5.2.200.163 (talk) 13:35, 21 July 2016 (UTC)
About the latter: no, in Newtonian mechanics the two-body potentials exhaust all the gravitational interaction.
About other matter: well, in principle I like your approach... but I am afraid that Wikipedia is too conservative for accepting it. You know, all editors are equal here; no one is treated as expert; and because of this, no one is entitled to publish his/her "original research". Boris Tsirelson (talk) 14:14, 21 July 2016 (UTC)
Perhaps there is a potential misunderstanding of the concept of "original research" which could be adjusted by mentioning the improvement of articles and/or WP:IAR. No one need to boast with "expert" labels. The content and style of scientific seminaries could be adopted when dealing the level of presumed originality.--5.2.200.163 (talk) 14:29, 21 July 2016 (UTC)

About two-body potentials exhaust in Newtonian mechanics compared to GR, to which feature of GR is this exhaust due? Are the details mentioned somewhere on w'pedia?--5.2.200.163 (talk) 14:43, 21 July 2016 (UTC)

Basically, nonlinearity of the field equations.
In Newton gravity (as well as electrostatics) all possible space-time configurations ("histories") of the gravitational (or electrostatic) field are a vector space. That is, a linear combination of possible configurations is also a possible configuration. And moreover, the energy is a quadratic form on this vector space.
Thus, given a three-body system, the sum of the three "individual" potentials is the potential of the system. And the energy is the sum over pairs.
Nothing like that holds in general relativity. Boris Tsirelson (talk) 17:32, 21 July 2016 (UTC)

### Newtonian gravitation extensions

Since both GR and NG have been mentioned in the above lines, how do you view from a MPV (Mathematical Point of View) the extensions for classical gravitation law mentioned in Newtonian gravitation#Extensions (arXiv article link https://arxiv.org/abs/hep-ph/0608346 mentioning some experimental backup/tests of theoretical concepts), started by Newton himself?--5.2.200.163 (talk) 11:36, 22 July 2016 (UTC)

I am not interested in these; I'd leave them in the Newton era. Though, I say so from my personal taste (rather than Mathematical Point of View). Boris Tsirelson (talk) 15:26, 22 July 2016 (UTC)

### Classical distribution of charge (and mass) density hypotheses

Also from a MPV, how do you view classical models of charge and mass distributions in elementary particles like electron (as mentioned in talk:electron magnetic moment#section11)?--5.2.200.163 (talk) 11:54, 22 July 2016 (UTC)

I am not interested in these; I'd leave them in the pre-quantum era. Boris Tsirelson (talk) 15:29, 22 July 2016 (UTC)

### Classical atomic model

Ho do you view (also from MPV) the following classical free-fall atomic model?--5.2.200.163 (talk) 13:15, 22 July 2016 (UTC)

I am not interested in these; I'd leave them in the pre-quantum era. Boris Tsirelson (talk) 15:31, 22 July 2016 (UTC)

## Extremely small probabilities and rare events

Seeing the section above about zero probability, I want to ask you how do you see the connection between extremely small probabilities and some (possibly problematic) assumptions encountered in applications of probability theory to financial risk management? How do rare events and black swan theory concepts should influence the application of probability to risk management?--5.2.200.163 (talk) 13:49, 21 July 2016 (UTC)

How do the fat-tailed distributions and heavy-tailed distributions influence the creation of more accurate mathematical models of risk management?--5.2.200.163 (talk) 14:01, 21 July 2016 (UTC)

I do not know. In particular, I do not know, what do you mean here by "extremely small probabilities"? Something like 10-10? Or 10-20? Or 10-30? These are different things. Also, some of our models are, hopefully, close to reality up to 10-30, but some - hardly up to 10-10. Boris Tsirelson (talk) 14:23, 21 July 2016 (UTC)
Of course, further context details are required in analysis. I'll look for them.--5.2.200.163 (talk) 14:31, 21 July 2016 (UTC)
An important aspect of this issue which is criticized is the very frequent tacit assumption and use of the normal distribution in the field of risk management. Practical consequences of very rare, but catastrophic and very hard to predict events and their avoidance or mitigation in case of occurring are very important, as well as expert error in handling/steering large systems as the social ones.--5.2.200.163 (talk) 11:05, 22 July 2016 (UTC)
Yes. When I was young, normality of distributions was a usual assumption that simplifies analysis a lot, but of course need not be satisfied in reality; some statistical procedures were known to be more sensitive to non-normality, others less sensitive. But recently I was quite astonished by the independent component analysis; there, non-normality is the crucial resource rather than a pesky annoyance. Boris Tsirelson (talk) 15:22, 22 July 2016 (UTC)
Can usual assumptions be tricky and a form of subtle ideological infiltration?--5.2.200.163 (talk) 08:28, 25 July 2016 (UTC)
They are not invented for this goal; but maybe sometimes they can be used for it, I did not think in this direction. Boris Tsirelson (talk) 15:44, 25 July 2016 (UTC)

## Moore method

How do you view the (advantages) of Moore method style of mathematics education and teaching?--5.2.200.163 (talk) 14:34, 21 July 2016 (UTC)

Oh! I did not know it is Moore method, but most my understanding of mathematics, I owe it to this method! Just see the first paragraph of my reminiscences. Boris Tsirelson (talk) 16:48, 21 July 2016 (UTC)
I see that that the name differed (Youth School..) but the method and guiding lines were similar. I see also that you mention in last paragraphs the ideological intervention and labeling from competent bodies which is less likely (comparison of probabilities would be interesting in connection to a section above) to be encountered in an imperialist system that ruled the non-Soviet world.--5.2.200.163 (talk) 10:50, 22 July 2016 (UTC)
Sure, sure. The youth school was volunteered by mathematicians, not competent bodies. And, yes, I definitely prefer the non-Soviet world. Boris Tsirelson (talk) 15:12, 22 July 2016 (UTC)
Coming across the biography of Andrei Kolmogorov I've noticed his involvement in Luzin affair and also the participation of Ernst Kolman described as ideological watchdog in Soviet science.--5.2.200.163 (talk) 09:11, 25 July 2016 (UTC)
Maybe; I am not acquainted with that. Many weird things happened during stalinism. Boris Tsirelson (talk) 15:49, 25 July 2016 (UTC)

It can be said that these mentioned teaching methods can activate the mathematical spirit (in re to a section above) more than the rather traditional methods of mathematics education who seems to spend more 3/4 of the time insisting on algorithms of hand calculations and less on conceptual creativity. I've encountered very recently a site http://computerbasedmath.org/case-for-computer-based-math-education.php which mentions these drawbacks of traditional math ed curricula which can be a source of mathematical functional scientific literacy. I was very surprised initially when hearing an interview with Solomon Marcus saying this tough assertion about traditional math curricula and math illiteracy.--5.2.200.163 (talk) 12:58, 30 September 2016 (UTC)

Hmmm... Now I got puzzled: should I say that I always did computer based math education? I did not mention computers (if only rarely); but algorithms of hand calculations definitely occupy less than 1/4 of the time on my courses (available on my site). Boris Tsirelson (talk) 14:00, 30 September 2016 (UTC)
This is something very good to be heard re time fraction of hand algorithms in your courses. It can be said that the style of yours (similar to what is now called comp-based math edu) is very important in promoting mathematical understanding spirit.--5.2.200.163 (talk) 14:13, 30 September 2016 (UTC)
Of course things are not so fortunate in pre-university world-wide education science and math (less intelligentlly designed) curricula, about which someone said they are a time-filling of children until 18-19 years. Children activities up to this age needs to be supervised by some adults who happens to be high-school teachers. There hasn't been to much interest until now in offering an interesting passtime curricula based on reasoning and mathematical modelling in contrast with excessive memorization of info that can be found in books an other publications waiting to be explored.--5.2.200.163 (talk) 14:25, 30 September 2016 (UTC)
Indeed, that is not my merit, but my good luck. I teach to students of math, on a good university; this is why I can afford to teach notions, proofs etc; and of course I note that it is too hard for about 1/3 of my students. I could not afford it in a worse environment. Boris Tsirelson (talk) 14:37, 30 September 2016 (UTC)
Indeed good placement is very important. It would be interesting to know more details about the specific difficulties for the third of students mentioned! On the other hand, have you ever taught to engineering students?--5.2.200.163 (talk) 14:51, 30 September 2016 (UTC)
Аbout the specific difficulties for the third of students mentioned? What can I say? They are not ready to such work. Not prepared, not willing. But I happen to know personally one like that: my distant relative. No, he was never my student; but I was asked (by his father) to help him. He started repeatedly in different universities (father had money...), able to get grade 60..65 (on math after one year), but need 80+ in order to be accepted to computer science (harder competition there). Well, I wanted to explain him something; but he refused to hear! He insisted that I'll solve exam problems found by him --- and he'll look in the hope to get the know-how... Well, I did so. And, interestingly, it appeared that our grading system (in all our universities) is nicely calibrated: such a student can get 60+ but never 80+. Boris Tsirelson (talk) 15:47, 30 September 2016 (UTC)
Have I ever taught to engineering students? Yes. Exactly once. "Probability and statistics". I asked colleagues: which failure rate is OK? 1/3, as on math? They answered: no; rather, 1/6, something like that. Well, I am very proud: on the very new environment, I managed to get 1/6! Needless to say, I took a level much, much lower than that for math students; of course, no serious proofs etc.; and was happy to get 1/6. But then I got unhappy: each one of these 1/6 came to me, not to complain, just to ask in confusion: "what to do?? I was ready to start working as an engineer; I succeeded on all needed engineering courses; this was my last semester; "probability and statistics" is of little importance; but I cannot get my diploma!" Boris Tsirelson (talk) 15:47, 30 September 2016 (UTC)
Interesting misleading evaluation of importance! What is the probability that such a person could be a good engineer?(a rethorical question perhaps? in the context of applied probability)--5.2.200.163 (talk) 16:31, 30 September 2016 (UTC)

It is intriguiging to notice in this context two wikiarticles mathematical maturity and mathematical knowledge management. I mention these as premises in after formal education development of math (and engineering) graduates. An important question is about the impact or use of mathematical knowledge in real world post-education. How is this related to the level of mathematical maturity and capability of use and develop new mathematical models by each graduates of math and also engineering?--5.2.200.163 (talk) 12:08, 7 October 2016 (UTC)

In relation to the above I want to mention some aspects or impressions from my graduate engineering math use. I can say now that I'm somewhat dissatisfied by the rather insufficient mathematical applications and exercises in the engineering thermodynamics seminars in which I was involved. I was expecting something a level of applications promoting mathematical competence similar to the recently mentioned paper by Nev A Gokcen in Journal of Physical Chemistry 1960. I am puzzled by the lack/insufficient level of such seminar applications given that good thermodynamics textbook and treatises by established authors had existed in Romanian language long before I were in engineering undergrad studies about 15 years ago. Was this sitauation due to the fact that thermodynamics is a also an experimental science beside theoretical and the number of laboratory hours allocation problem should have been considered?--5.2.200.163 (talk) 12:26, 7 October 2016 (UTC)

I also think that this situation is somewhat due to the lack of some guiding courses like philosophy of science and logic of scientific investigation that provide an overall image of scientific investigation beyond individual exams to each academic discipline in the curricula! I certainly did not expect to follow and wait some 5 years curriculum to arrive at some aspects that had to be emphasized from the beginning to avoid wasting so much time and pursue real scientific research initiatives. Also there was some lack of awareness to some implicit aspects very recently encountered (like reading a very good 1970 book called Introduction to Scientific Documenting which explicity undelines the use of scientific journal article use and spotting as a important component in real research logistics and also the 1955 The Art of Scientific Investigation by William Ian Beardmore Beveridge).--5.2.200.163 (talk) 12:50, 7 October 2016 (UTC)

Could these shortcomimgs be circumvented by educational acceleration? I have encountered at least two example of persons, with wikiarticles here, who have enrolled in a PhD program in math without undergratuate studies. It seems that these cases are an extreme form of educational acceleration.--5.2.200.163 (talk) 13:13, 7 October 2016 (UTC)

My education was not accelerated in the sense that I did not skip any stage. But it was accelerated in the sense that, being formally a pupil, I was in fact learning graduate-level math, and got enough of mathematical maturity. About mathematical knowledge management, I do not observe any real success in this direction. Is it possible, at all? Computers are successful in playing chess, but not in checking proofs (the more so, not in finding proofs). Why? Probably, because math can express quite a lot (if not everything), as we know after Kurt Gödel; chess cannot. Maybe, only a human can understand humans' math; for chess this is not so. Boris Tsirelson (talk) 21:18, 7 October 2016 (UTC)

## Combinatorial aspects in game theory

Can the above aspects mentioned about binary to ternary be present also in game theory, involving the reduction of a n-player game to a number of subgames of (n-1), (n-2), ... 2-player games?--5.2.200.163 (talk) 08:35, 25 July 2016 (UTC) (I see that this glossary of game theory defines complements : ${\displaystyle \sigma \ _{-i}}$ an element of ${\displaystyle \Sigma \ ^{-i}=\prod _{j\in \mathrm {N} ,j\neq i}\Sigma \ ^{j}}$, is a tuple of strategies for all players other than i.)--5.2.200.163 (talk) 08:41, 25 July 2016 (UTC)

A similar aspect could be defined for mixture when considering volumes of submixtures from a n-ary mixture like:V23, Vij, V123, Vijk and complements like c1V=V-V23, c2V=V-V13, etc.--5.2.200.163 (talk) 08:51, 25 July 2016 (UTC)

About chemical mixture I do not know. About game: if the strategy of one player is fixed, and moreover, is a common knowledge, then the rest of the game is a game of (n-1) players. So what? Boris Tsirelson (talk) 16:50, 25 July 2016 (UTC)

## Mathematical research orchestration

What is your personal perception as professional mathematician re how mathematical research programs are developed and orchestrated, perhaps in comparison with other fields of science? What is the role of a conductor or director (Pyotr Kapitsa's metaphor) of scientific research in collective research programs? How do the intuitions of individual mathematicians in exploring the unknown contribute to establishing research programs? What is the role of unexpected in spotting new interesting research directions?--5.2.200.163 (talk) 09:32, 25 July 2016 (UTC)

I am too much individualistic for answering such questions. Being young I was quite independent of any adviser. Being mature I was quite a bad adviser. I never organized a conference.
I only remember that once I tried to convince our Electrical Engineering division to hire a young expert in quantum cryptography, and their answer was quite unexpected to me: no, we cannot, since we have no quantum cryptography laboratory. I was puzzled: if so, how at all can you start a new direction? They answered: in principle, we can hire a whole laboratory at once, but this is a rare event, of course. (In Russian this is called научный десант.)
In contrast, in our Math division we usually hire the strongest applicant, be his/her research close to that of some existing member(s) or not. In this sense, math is unusually individualistic, I'd say. Boris Tsirelson (talk) 16:46, 25 July 2016 (UTC)
I see (interesting aspect about the informal(?) term scientific desant). Individualistic in the sense of rejecting or minimizing subordination required by non-scientific factors such as those required by managerial constraints and research funding. There is no real subordination in science, said Ernest Rutherford, in other words there is a so-called scientific democracy. Mathematics investigation therefore belong to Small Science in contrast with Big Science done by large collectives and tight cost management constraints. Math research is the least costy compared to experimental sciences that usually fit in Big Science.--5.2.200.163 (talk) 10:20, 26 July 2016 (UTC)
Ho do you view in this context the role of scientific collaborations between (applied) mathematicians and non-mathematicians as a source for scientific exploration and knowledge growth and also for the induction of rigorous mathematical reasoning spirit or style to non-mathematicians who might be less aware of subtle reasoning glitches?--5.2.200.163 (talk) 10:32, 26 July 2016 (UTC)
Also in this context of enhanced knowledge development, how do you view the mathematical collaborations between individual mathematicians, such as the collaborations by Paul Erdos or Bourbaki?--5.2.200.163 (talk) 10:40, 26 July 2016 (UTC)
Wow... I did not hear the terms "Small Science" and "Big Science". Anyway, my science was always small, and I have nearly nothing to say about Big science. About "collaborations between (applied) mathematicians and non-mathematicians": I only know that I was unable to collaborate with applied mathematicians, and non-mathematicians such as economists; and I heard from other pure mathematicians that this is typical: applied scientists keep a perimeter defense against pure mathematicians. (However, just now I got a co-author of a paper in NeuroImaging.) About Erdös or Bourbaki, ask someone closer to them. Boris Tsirelson (talk) 20:31, 27 July 2016 (UTC)
Very interesting this term perimeter defense against pure mathematicians! How does it manifest more concretely? What category of non-mathematicians is the most affected by it? How do physicists and engineers stand in this regard? Is this somehow due to the so-called specificity towards a domain of the non-mathematician (and the implied apology of the domain by the non-mathematician) and inability to see things in a wider perspective (perhaps due to some cognitive or epistemic limitation)?--5.2.200.163 (talk) 07:21, 28 July 2016 (UTC)
Given that you have developed the Tsirelson bound, can you be considered, partially at least, a physical mathematician or an applied probabilist?(I've just come across Little law) What is in fact the border between pure mathematics and applied mathematics and methodological differences?--5.2.200.163 (talk) 07:28, 28 July 2016 (UTC)
How can developments in pure math be influenced by contact with applied math and non-math? (It seems that the theory of stochastic processes has been developed by contact with biology. I wonder how and where queueing theory and its applications can be further encountered in other fields.) --5.2.200.163 (talk) 07:38, 28 July 2016 (UTC)
About perimeter defense against pure mathematicians. First, some facts. My collaboration with a professor of economics resulted in several manuscripts; some of them are self-published, and all are rejected from journals. My collaboration with a professor of mathematics and another professor of economics resulted in a manuscript self-published and rejected from journals. Second, interpretations of these facts. Of course, it may mean that all these manuscripts are too weak. But I do not think they are. I recall sporadic discussions with some colleagues. One told me: "probably, some non-simple mathematics appears there?" My reply: "Sure; after all, for this reason I was welcome to the collaborations". He: "Oh, then, no chance to get it published; such journals never accept such articles". Boris Tsirelson (talk) 06:36, 6 August 2016 (UTC)
I think the situation of those journals you mention is very bizarre if they do not accept articles with non-simple mathematical content for publication. Are they low quality journals? Or more inclined to qualitative or non-rigorous research? It seems so.--5.2.200.163 (talk) 10:45, 26 August 2016 (UTC)
These are more or less the journals cited by us: Journal of Economic Theory, Econometrica, etc. Low quality? Hmmm... depends on your definition of quality... usually not called low quality. Inclined to qualitative or non-rigorous research? Look there, and you'll find a lot of rigorous quantitative results, formulated as mathematical theorems (explicitly); but proofs are often relegated to appendices, in contrast to math journals. I guess that formulations of our main results are accessible to their typical readers, but (some) proofs are not. Thus, I'm afraid, they prefer to restrict their scope (and ultimately the knowledge), not to enlarge the class of their authors. Boris Tsirelson (talk) 11:22, 29 August 2016 (UTC)
If proof are relegated often to appendices, then what forms the main body of such articles? (a legitimate questions about articles structure and filling words). (I'd say that at least one of this journal (for instance Journal of Economic Theory) is relegated to the category love-to-hate-able by a well known probabilist, author and thinker.) Regarding the procedure or habit of relegating proofs to appendices, it is also very interesting, I'd say, to imagine, counterfactually of course, what chemistry journal in the XIX-th century would have published the works of Gibbs on phase equilibria without resorting to appendices instead of the relatively obscure and generalist Transactions of the Connecticut Academy of Art and Sciences where they have been published.--5.2.200.163 (talk) 15:31, 5 September 2016 (UTC)
The main body of such articles? Well... introduction, informal discussion, formulations of the theorems, discussion of their meaning/interpretation, etc. Boris Tsirelson (talk) 16:12, 5 September 2016 (UTC)
I recall encountering a recent example of a(n) (rather weakly argumentative) article in a chemistry journal about the theoretical premises behind an electrochemical device with (a rather weak) proof in the appendices. The conclusions of the article depended on the existence of an assumed temperature gradient between the device and the environment. The numerical value of this quantity is not specified in the text of the article.--5.2.200.163 (talk) 10:20, 6 September 2016 (UTC)
Now I recall an instructive anecdotal story. My collaborator economist told me once a ridiculous answer of a lecturer to the question "what is this theorem good for?"; the answer was "it helps in proving other theorems". I was puzzled, even shocked: what is ridiculous here? But after a short meditation I realized that indeed, this is a (or "the"?) distinction: in pure math, a theorem (if at all useful) helps in proving other theorems; in applications, it does not, it is a terminal vertex, a mean of consumption, not production. Another related distinction: in pure math the class of readers (of journals) is nearly the same as the class of writers; in applications, the former class is much larger. And readers that are not writers are not interested in proofs. Boris Tsirelson (talk) 16:59, 5 September 2016 (UTC)
I think that lack of (at least partial) interest in proofs is rather disturbing from the point of view of scientific reasoning. This is equivalent to lacks in the formative aspect of traditional higher education which does not succeed (or perhaps it does not even have as explicit objective) to form and exercise the logical-mathematical scientific spirit which goes beyond simple memorization of facts info to construct a network of notions. I've encountered recently a procedural recommendation re logical dependencies identification by Terrence Tao on his blog.--5.2.200.163 (talk) 10:12, 6 September 2016 (UTC)
How about engineering journals, what have you noticed, are they more wellcoming to proofs?--5.2.200.163 (talk) 09:49, 6 September 2016 (UTC)
Another colleague gave me the following explanation (I'd prefer it to be wrong, but...). Before the 20 century, pure math and applied math were (usually) created by the same persons. In the 20 century it appeared that the society needs much more applied mathematicians (than pure mathematicians). As a result, those who (being students) got grades (nearly) 100, are pure mathematicians, and those who got grades (nearly) 60, are applied mathematicians. Understandably, the latter do not want the former to join... Boris Tsirelson (talk) 06:57, 6 August 2016 (UTC)
A notable example of collaboration before the 20th century that can be mentioned is that of Laplace and Lavoisier regarding a scientific statement in thermochemistry. Another interesting example (of a single person) is that of J. W. Gibbs who betwen 1875-1890 developed the mathematical theory of phase equilibria based on the concept of chemical potential not in collaboration. Afterwards there were some mathematicians like G. H. Hardy more inclined to pure math due to less potential to be used in war-related applications. As collaborations examples in the 20th centuries can be mentioned that of Thomas Hakon Gronwall who collaborated with some physical chemists on the theory of electrolytes (involving the solution to an integral equation).--5.2.200.163 (talk) 11:11, 26 August 2016 (UTC)
Now, a bit of good news: my collaboration with numerous medicine, engineering and other experts resulted in this published paper. Boris Tsirelson (talk) 07:03, 6 August 2016 (UTC)
How was the atmosphere when collaborating with these experts on this published article? Was the perimeter defense less obvious to non-existent?--5.2.200.163 (talk) 10:49, 26 August 2016 (UTC)
Perimeter defense, I never felt it from coauthors (otherwise it would not be a collaboration at all), but only from journals. This time my contribution is invisible to readers (it is a detail of the algorithm of data processing, not at all detailed in the text), thus, it cannot provoke defense; I do not know, what would happen otherwise. Boris Tsirelson (talk) 11:31, 29 August 2016 (UTC)
I encountered some physical chemistry articles in some journals like Journal of American Chemical Society and Journal of Physical Chemistry that have a very rich mathematical content (and even mathematical spirit detected) that can be mentioned below. The JACS article authors thank to a third autor mathematician S. Ito from the same university (of Minnesota) who had a great contribution in heavy mathematical proof involving integral equations inserted in long appendixes. The JPC 1960 author Nev A Gokcen developed a full mathematical article with geometric content and pde re thermodynamics of mixtures and Gibbs-Duhem equation.--5.2.200.163 (talk) 15:27, 22 September 2016 (UTC)
Links to mentioned articles: JPC, JACS.--5.2.200.163 (talk) 15:37, 22 September 2016 (UTC)
Good news... Probably I should keep closer to hard sciences, away from humanities... but I was too much impressed by emergence of differential equations in the theory of auctions. Given that auction is discrete in space and time, what to describe by differential equations?.. Boris Tsirelson (talk) 16:49, 22 September 2016 (UTC)
Interesting situation you mention about auction theory, which I see is included in wikicateg game theory. Is therefore game theory a humanities math branch? It seems that in the so-called humanities the presence of probabilistic phenomena is more obvious to perceive. The distinction between hard sciences and humanities can be said to be somehow connected with this presence of randomness, hard sciences giving the appearance of lower to zero presence of randomness. The high level of (quasi)-randomness in humanities makes them more difficult to predict, unlike hard sciences. Non-hard/behavioural sciences are based on the actions of intelligent agents which allow the occurrence of randomness. Time as a physical entity can be perceived by the sequences of events generated by (human) agents. Therefore the prediction in humanities is about prediction of actions by agents who have the possibility/degrees of freedom of free action which affects prediction attempts. The wikiarticle scientific prediction I see it has a section about finance.--5.2.200.163 (talk) 12:18, 29 September 2016 (UTC)
Perhaps the curious situation can be linked to hybrid discrete continuous time events and time scale calculus. I've encountered an interesting link about diff eqs in auction theory: http://math.stackexchange.com/questions/1385728/system-of-differential-equations-asymmetric-first-price-auction.--5.2.200.163 (talk) 12:33, 29 September 2016 (UTC)
No, no! Physics now, in the quantum era, is harder a science than ever, but it elevates randomness to the rank of an objective immanent essence!
“A philosopher once said, "It is necessary for the very existence of science that the same conditions always produce the same results." Well, they don't!” ― Richard Feynman.   Boris Tsirelson (talk) 12:44, 29 September 2016 (UTC)
At least, before quantum era, that it was the classical distinction, approximately valid in the range of macroscopic human perception scale. Of course, it could be that some macroscopic physical phenomena have a less reproducible character due to randomness at microscopic scale.--5.2.200.163 (talk) 14:06, 29 September 2016 (UTC)
Or maybe you mean free will rather than randomness. I recall, philosophers say that determinism excludes free will, but randomness excludes it, too. Boris Tsirelson (talk) 18:30, 29 September 2016 (UTC)
Interesting aspect about randomness excluding free will and also rather paradoxical.--5.2.200.163 (talk) 10:39, 30 September 2016 (UTC)
For a random sample of 2000 "yes/no" answers we may be pretty sure that the frequency of "yes" in the first 1000 answers is close up to 10% to that in the other 1000. Should free will behave like that? Boris Tsirelson (talk) 12:48, 30 September 2016 (UTC)
Free will is tricky and slippery in connection with randomness and determinism. Of course there may be a bound on possible choices of a free will agent.--5.2.200.163 (talk) 13:19, 30 September 2016 (UTC)
Interesting aspects those you mentioned about sampling types properties.--5.2.200.163 (talk) 13:33, 30 September 2016 (UTC)

## Constructibility

How do you think of the article Set-theoretic definition of natural numbers? Shouldn't it be renamed using construction instead of definition to emphasize constructibility like in construction of real numbers? (Natural numbers are primitive notions like that of set, can't be defined by genus differentia, aren't they?)--5.2.200.163 (talk) 11:24, 25 July 2016 (UTC)

My attitude to natural numbers is voiced in Sect. 1 of "Equivalent definitions of mathematical structures". Yes, I'd prefer construction to definition. Boris Tsirelson (talk) 15:54, 25 July 2016 (UTC)

## Wikiquote

Hi, I would like to invite you to contribute to our sister project Wikiquote. Some articles there that may use more informative quotes include:

• q:Functional analysis: Needs quotes on the duo Sobolev and Schwartz and other contributions after them (i.e. all after the Banach-Hilbert-Riesz trio).
• q:Harmonic analysis: Needs quotes after the 1920s-30s development extending today to the Calderón–Zygmund–Stein-Fefferman-Tao clan.
• q:Continuous-time Markov chain: Needs quotes on other branches of probability theory except this one, especially the various theoretical/applied school of thoughts.

Do let me know if you are interested in this. Solomon7968 04:44, 21 August 2016 (UTC)

Thank you for the invitation, but I never contributed to Wikiquote, and I am astonished to see these long (and not famous) quotes about such rather technical matters. In rare cases I take a quote, but only something like Einstein's or Feynman's opinion on the essence of nature... No, I do not thing I could help this way, sorry. Boris Tsirelson (talk) 13:46, 22 August 2016 (UTC)

## Mathematical spirit (and reasoning)

I open a new section re a very important aspect mentioned only tangentially in previous sections. How do you view the importance of mathematical spirit in guiding Modern Science taking into consideration that the Scientific Revolution has been driven mainly by math? How can this logical mathematical spirit be formed and imposed (a rather unfortunate word) or better said formed/developed and exercised to as many people as possible? What educational methods would serve this purpose?--5.2.200.163 (talk) 11:10, 6 September 2016 (UTC)

Sorry, this is too global problem for my narrow vision. I have only a slight idea of the Scientific Revolution, educational methods etc. Boris Tsirelson (talk) 09:38, 9 September 2016 (UTC)
No matter how small, an input from a professional mathematician is a valuable step forward.--5.2.200.163 (talk) 14:50, 22 September 2016 (UTC)

## Math reference desk

Boris, I am not allowed to post on the math reference desk; could you repost this for me? Re renormalization, there are slow attacks being made on it, by mathematicians, see for example this recent work by Alain Connes: http://arxiv.org/abs/hep-th/9912092 which, I quote: "This paper gives a complete selfcontained proof of our result announced in hep-th/9909126 showing that renormalization in quantum field theory is a special instance of a general mathematical procedure of extraction of finite values based on the Riemann-Hilbert problem. We shall first show that for any quantum field theory, the combinatorics of Feynman graphs gives rise to a Hopf algebra $\Hc$ which is commutative as an algebra. ..." Its quite interesting. And some general background: the reason for the operator product expansion in QFT is pursued in part to get around the issues of renormalization, many of which arise due to the operator-valued-ness of the fields-- that is, replace operator-valued fields by operator-valued measurables. Anyway, the initial post by whomever is really kind-of wrong, there's a huge amount of QFT that can be made rigorous, but it involves some really deep understanding of differential eqns -- for example Yang Baxter opens the door for a connection between diff eq and qft. which is why its in Connes paper. The issue is that teaching thisstuff would take yet another 5-10 years of education, so is not practical for PhD students... its only available to people like use with excess free time :-) 67.198.37.16 (talk) 14:06, 19 September 2016 (UTC)

Or maybe never mind, I posted directly on User talk:YohanN7 who asked the initial question. 67.198.37.16 (talk) 14:23, 19 September 2016 (UTC)
Thank you for the information! Boris Tsirelson (talk) 14:47, 19 September 2016 (UTC)

## Mathematical literacy vs numeracy

I've just noticed the wikiarticle mathematical literacy being redirected to numeracy. It seems like a misleading redirect, the two concepts should not be confused. How do you consider this situation?--5.2.200.163 (talk) 11:35, 7 October 2016 (UTC)

I have no idea about either of the two concepts. But if you know they differ (and have supporting sources), just fix... Boris Tsirelson (talk) 20:40, 7 October 2016 (UTC)

## Pure and applied

Given the discussion about pure and applied math, I think that most of the traditional applied math like mathematical chemistry, mathematical physics, biomathematics can be further divided into the pure aspect of pursuing the reasoning and knowledge development in its own right like in pure math and the application of knowledge. I think that given the highly interconnected nature of mathematical concepts (undelined in math maturity or math knowl managem wíkiarticles), following the mathematical lead in any other so-called discipline emphasizes also the interconnectedness of nature and reality which seems to be overlooked by the traditional division of knowledge by discipline which overemphasizes the memorization of info regarding the discipline instead of creative thinking and finding pleasure in proof as underlined by some excerpts from a dialogue between G. H. Hardy and Bertrand Russell about finding pleasure in proofs.--5.2.200.163 (talk) 13:06, 7 October 2016 (UTC)

## Quantum Nonlocality

Good afternoon, One of the articles I keep tabs on is Quantum nonlocality and I noticed you made an edit where you reverted the re-addition of a couple of sentences about the relationship between entanglement and nonlocality. I was thinking of reverting this, with a couple of additional comments/changes, but didn't want to until I understood the rationale for your change. It seems like the removed material is helpful, and the addition of the word "pure" an unnecessary restriction. The edit is at https://en.wikipedia.org/w/index.php?title=Quantum_nonlocality&type=revision&diff=745917222&oldid=745892529

Porphyro (talk) 13:48, 26 October 2016 (UTC)

"Reverted the re-addition"? No, I do not see it this way. I see that Tal Mor added the word "pure", and the phrase "and there exist non-entangled (separable) states that do produce some type of non-local behaviour"; he did not delete anything, and therefore CAPTAIN RAJU could not re-add anything, right? In addition, I was disturbed by his (CAPTAIN RAJU's) summary "test/vandalism"; Tal Mor is a well-known expert in the field, and his edit is not at all "test/vandalism" (as I noted in my summary). Some misunderstanding? Boris Tsirelson (talk) 15:06, 26 October 2016 (UTC)
And moreover, I just have found the following on User talk:CAPTAIN RAJU page: "Dear CAPTAIN RAJU, My name is Tal Mor, a new wikipedian but a researcher in the field of quantum physics and quantum information for many years. My modifications to quantum nonlocality were not a test and are not vandalism, but important modifications to clarify that there is also nonlocality without entanglement. You may search google scholar to see that topic and if you wish - also to see my involvement in the field. I did not add citations to nonlocality without entanglement - but of course such citations are important to add. But I cannot add them because Wikipedia does not allow self-citations. Best Regards Tal Mor (talk) 18:11, 24 October 2016 (UTC)" Why is this not answered by CAPTAIN RAJU? Boris Tsirelson (talk) 15:13, 26 October 2016 (UTC)
Hi Boris- I realise I have made a mistake! I thought that Tal Mor had removed the sentence beginning "a well known example of this", when in fact he had not. I also fully agree with you that the categorisation of that edit as vandalism or a test is totally erroneous and I see now why you reverted the change. However, I'm not sure that, with the article written how it is now, the edit made by Tal Mor is tenable. It seems to me that the article's definition of quantum nonlocality is stricly there in the sense of Bell: that a quantum state is "nonlocal" if it gives rise to probability distributions that could not be simulated by a local hidden variable model. With the discussion couched in this definition, I think that stating that separable states can produce non-local behaviour is misleading. I would support the re-addition of the edit made by Tal Mor if the beginning of the article were reworked to allow for subtler notions of nonlocality. Would you say that would be fair? Porphyro (talk) 15:39, 26 October 2016 (UTC)
I see. I think, the best option is, to discuss the matter with Tal Mor on Talk:Quantum nonlocality. It would be nicer to do so from the start, but this was not done, to my regret. Hopefully, Tal Mor did not leave us, still. Boris Tsirelson (talk) 20:11, 26 October 2016 (UTC)
When I posted my comment above, I also left a message on Tal Mor's talk page. I think that tomorrow I will revert the edit until I or someone else has found the time to give the article the necessary changes. Porphyro (talk) 22:27, 26 October 2016 (UTC)
Thank you Boris. I am not sure yet (as I am a new Wikipedian) where to discuss this topic, so in the meanwhile, I answer only here, in Boris' Talk page. Thank you Porphyro for mentioning that already the first sentence, defining quantum nonlocality, is not sufficiently general. Can you or Boris suggest how to modify that first sentence? Changing the FIRST sentence of this article is quite challenging, hence before doing that, let us try to agree on the modification, for example, based on the definition of the paper on quantum nonlocality without entanglement.
Tal Mor (talk) 21:53, 1 November 2016 (UTC)
Understandably, you feel uncomfortable with self-citations, but here is a workaround. You may propose appropriate changes on talk:Quantum nonlocality, and hopefully, others will copy them to the article. Boris Tsirelson (talk) 06:03, 2 November 2016 (UTC)
Thanks. It is a possibility. I am not sure if it is by far more legitimate though. Are you following in your watchlist the Talk page of Porphyro? I wrote there that he is correct about the incosistency that exits now between the FIRST sentence describing quantum nonlocality and the sentence that I added. However, in my eyes, the inconsistency is due to a too limited (or too specific) definition in the FIRST sentence. Changing the first sentence seems more challenging, hence I suggested we first discuss it among the three of us. Please see his Talk page. Tal Mor (talk) 06:31, 2 November 2016 (UTC)
Yes, it is legitimate. Completely. (Many years ago I added this way my photo :-) to the "Boris Tsirelson" page.) Moreover, self-citation is not quite prohibited, it is rather discouraged ("because it is too difficult to be neutral toward oneself...", something like this). But, I repeat, it is OK to propose and let others decide.
I agree that the first sentence should be changed. But all that is not a private business of the three of us. Very probably, more wikipedians watch "Quantum nonlocality" (and its talk page, automatically); the consensus should be reached among all, not just three. This is why it is better to go there for the discussion. Boris Tsirelson (talk) 07:40, 2 November 2016 (UTC)
If it is agreed on a talk page to add a citation, and then someone participating in the discussion (not necessarily an author) adds it, it is not even half suspect. This way is better than if the author adds it for the reason that talk pages get archived, and it may look suspicious to someone later on. I'll add the pages to my watch list, and I will be happy to do the adding if there is talk page consensus. YohanN7 (talk) 10:42, 2 November 2016 (UTC)
I agree that the best place for this discussion would be here so that others may join in the discussion. I think we are all on the same "side" here so to speak and I've kicked the discussion off with a summary of my position. Porphyro (talk) 15:15, 2 November 2016 (UTC)
Nice!   :-)   Boris Tsirelson (talk) 16:11, 2 November 2016 (UTC)
Hi Boris, what about openning a new article for "quantum nonlocality of a single quantum state" that will contain what is now (kind-of-wrongly) contained in "quantum nonlocality"? Tal Mor (talk) 16:42, 5 November 2016 (UTC)
OK with me. On WP jargon this is called 'move "quantum nonlocality" to "quantum nonlocality of a single quantum state"'. I never did a move, and do not know the procedure. Boris Tsirelson (talk) 19:23, 5 November 2016 (UTC)
Alternatively - what about adding a note right after the title of the current page, saying something like "This article focuses on quantum nonlocality of a single quantum state. For other types of quantum nonlocality see ..... Tal Mor (talk) 16:42, 5 November 2016 (UTC)
OK with me, too... but wait, where to go for other types? Or do you mean, to first write the general article? Boris Tsirelson (talk) 19:23, 5 November 2016 (UTC)
I do not think talking about Turing, hypercomputing, etc., is useful for the current discussion, and might lead to a (potentially long) side-track, that might keep the error in the current article for a long time, hence I moved the discussion to here - to your Talk page. For a while at least Tal Mor (talk) 16:42, 5 November 2016 (UTC)
Yes, this is problematic, I understand. Do it as you like. But for me it is rather boring, just to correct "the error in the current article". The challenge is, to explain to the reader, what is meant by "other types of quantum nonlocality". As long as this remains unexplained, the non-expert reader will feel puzzled, I think so. Boris Tsirelson (talk) 19:23, 5 November 2016 (UTC)
I agree the goal of course is to properly define "quantum nonlocality".

I am just saying that first let's correct the error, or discuss various ways to correct the errors (even if this is boring). Because otherwise - it might stay there for a very long time.Tal Mor (talk) 21:03, 5 November 2016 (UTC)

## GA-nomination?

Hi Boris!

I have plans (yet somewhat vague, especially when it comes to a time table) to try to bring Representation theory of the Lorentz group to GA-status. Before nominating it, I'd like to hear some opinion from others, with different viewpoints than mine (that of an amateur math/physics nerd). If you have some time to spare, I'd surely like to hear your opinion on whether it is worthwhile, and if so, what kind of improvements are the most necessary ones.

Most of the present version is mostly written by me. The major exceptions are the sections Principal series, Complementary series and Plancherel theorem on the general theory in the infinite-dimensional case. These are hardboiled mathematics, probably impeccable since their author knows his stuff. The rest have a slight (sometimes decided) physics flavor, though I tried to avoid getting too "physical" when it could be presented in a neutral style.

The article is fairly long, and I realize that you are busy IRL, but if you could parse it through and have a closer look at some sections of choice, I'd highly appreciate it. I'll also ask a couple of other editors for advice whether to go/not go. YohanN7 (talk) 11:06, 1 December 2016 (UTC)

Thanks for the invitation; but I do not fit for this role, for two reasons. First, I never participated in GA process (and do not feel interest in that). Second, my knowledge about representation theory of the Lorentz group is rudimentary. Boris Tsirelson (talk) 07:19, 2 December 2016 (UTC)
I understand. My intent was to collect opinions on whether it is worth an attempt to nominate the article for GA, not to begin the actual process for now, since GA-status is very unusual for a technical article. YohanN7 (talk) 09:34, 2 December 2016 (UTC)

## Nomination of Catalog of articles in probability theory for deletion

A discussion is taking place as to whether the article Catalog of articles in probability theory is suitable for inclusion in Wikipedia according to Wikipedia's policies and guidelines or whether it should be deleted.

The article will be discussed at Wikipedia:Articles for deletion/Catalog of articles in probability theory (2nd nomination) until a consensus is reached, and anyone is welcome to contribute to the discussion. The nomination will explain the policies and guidelines which are of concern. The discussion focuses on high-quality evidence and our policies and guidelines.

Users may edit the article during the discussion, including to improve the article to address concerns raised in the discussion. However, do not remove the article-for-deletion notice from the top of the article. Marcocapelle (talk) 10:14, 9 December 2016 (UTC)

## Article on stochastic process completely re-written

Hi Boris, I saw you had made a number of comments on the talk page of the stochastic process article. I just thought you would like to know that I completely re-wrote the article. I hope I didn't delete any of you contributions, if you had any. I would also be open to suggestions of improving the current article. Thanks for your time. Improbable keeler (talk) 18:33, 4 January 2017 (UTC)

Thank you for the excellent work! Remarks (if any) go to that talk page. Boris Tsirelson (talk) 19:25, 4 January 2017 (UTC)

## Negative prime numbers

Hi!I see that you are a professional mathematician and I want to ask how do you view in this regard the issue of the existence of negative prime numbers which I notice (from Talk:Prime number) that is somewhat controversial. Please comment there (also)!--82.79.114.105 (talk) 17:50, 23 February 2017 (UTC)

What can I say? All is already said on that talk page. Well, I can add that Wikipedia does not (try to) propose a new terminology (even if better than the existing mainstream terminology). Boris Tsirelson (talk) 21:25, 23 February 2017 (UTC)
I recall someone did mention some NON-ENG math sources (perhaps Russian ones?) which say that the cardinal of the set of divisors of integer prime numbers a is 4, that is the set of divisors consists in a, -a, 1, -1. I think they could be cited, couldn't they?--82.79.114.239 (talk) 15:45, 24 February 2017 (UTC)
Maybe, if found; but I do not recall seeing it. Boris Tsirelson (talk) 15:51, 24 February 2017 (UTC)

## Polyhedra

Just a quick thank you for your constant even-handedness, politeness and humanity, however this discussion eventually works out. I'll be off-wiki for a day or two now. — Cheers, Steelpillow (Talk) 20:05, 2 March 2017 (UTC)

Thanks for the compliment(s). By the way, I took a short look at your "polyhedra" page outside Wikipedia; however, polyhedra are not my hobby. Boris Tsirelson (talk) 20:18, 2 March 2017 (UTC)

## Mixed states optional?

I speculate (do not claim) this: Mixed states are optional. I.e., if a description as a mixed state is necessary, then there is always a larger system in a pure state containing our system as a subsystem. (This is something I took for granted in a discussion with our dear Chjoaygame. Landau & Lifshitz (Landau gets credit for independent discovery of mixed states) introduce mixed states only as states of subsystems of larger isolated systems in their QM book. (There are two entire volumes in their series on quantum statistics that likely treats things differently.))

The speculation would be implied by; an isolated system is in a pure state. (I have, without thinking about it really, always taken this for granted.)

As an aside, I have seen corresponding lines of thought in conjunction with the "measurement problem" (wave function collapse, etc), i.e. mixed states afford a convenient consistent description, but they are not a logical necessity.

Do I get this completely wrong? I usually take the sound approach (I just keep quiet and compute). YohanN7 (talk) 12:40, 8 March 2017 (UTC)

Yes, you are right, as least, in the mathematical sense. It is a well-known theorem that every mixed state (of a system) is a marginal of some pure state of some larger system.
About "an isolated system is in a pure state": on one hand I see no good physical reason why this must be true; on the other hand, this is probably a convenient harmless assumption (not falsifiable).
Boris Tsirelson (talk) 22:25, 8 March 2017 (UTC)
Thanks for the comment. I'll see what my books say on "an isolated system is in a pure state". I actually doubt that any of them will be clear to the point. YohanN7 (talk) 10:54, 9 March 2017 (UTC)
Let me add: whenever I face a mixed state (in physics, not math), I always know (or at least guess), what is another system correlated with the given system, such that the mixed state is the marginal.
Also, the formulation is a bit strange. A system can be isolated (now) and still correlated with another system. You may say that in this case it should be called subsystem rather than system. Well, but I'd emphasize this in the formulation. Boris Tsirelson (talk) 19:00, 9 March 2017 (UTC)
Yes, you are right. The terminology is from L&L - or so I thought; they actually say "closed system". But the meaning must in any case be the above - no correlation with any other system. YohanN7 (talk) 08:05, 10 March 2017 (UTC)
Worse, I recall I saw somewhere in old quantum textbook(s), that a closed system must be in an eigenstate of the Hamiltonian! (Counterexamples: Larmor precession, coherent states.) Boris Tsirelson (talk) 18:47, 10 March 2017 (UTC)
The theorem mentioned above should be related to Stinespring factorization theorem. Regretfully, this is not explicit there. Boris Tsirelson (talk) 12:44, 10 March 2017 (UTC)
But never mind, here is a simple elementary proof. A density matrix ${\displaystyle W,}$ being a compact Hermitian operator, is of the form ${\displaystyle x\mapsto Wx=\sum _{n}p_{n}\langle e_{n}|x\rangle e_{n}}$ where ${\displaystyle e_{n}}$ are orthonormal. Also, ${\displaystyle p_{n}>0,}$ and ${\displaystyle \sum _{n}p_{n}=1.}$ Now consider such a state vector of the bipartite system: ${\displaystyle \sum _{n}{\sqrt {p_{n}}}e_{n}\otimes e_{n}}$ and check that it does the job. Boris Tsirelson (talk) 13:50, 10 March 2017 (UTC)
See also Schmidt decomposition. Boris Tsirelson (talk) 18:31, 10 March 2017 (UTC)

## Apologies

Dr. Tsirelson, I apologize for whatever I atttributed inappropriately to you. It seemed at the time that that was the case, but there could be any number of reasons for why I was so confused. I hope you will forgive me.Daqu (talk) 21:57, 28 March 2017 (UTC)

Please don't let it happen again. Boris Tsirelson (talk) 05:55, 29 March 2017 (UTC)

## Nonlocality again

Hi again Boris, I've proposed some substansive changes to the article Quantum nonlocality; if you have time I'd appreciate your thoughts and comments on the talk page Porphyro (talk) 12:36, 12 April 2017 (UTC)

Nice; I did. Boris Tsirelson (talk) 13:57, 12 April 2017 (UTC)

I think the input of another analyst would be helpful at Talk:Pi#Eigenvalue. Sławomir Biały (talk) 12:52, 14 April 2017 (UTC)

## A barnstar for you!

 The Original Barnstar I appreciate your work. In particular the one on "Fortifying wikiquanta"! Oakwood (talk) 14:57, 13 May 2017 (UTC)

Thank you! But, as far as I know, that project was unsuccessful. Or did I miss a recent progress? Boris Tsirelson (talk) 16:15, 13 May 2017 (UTC)

No, you didn't miss anything. I was just reading about it, and I thought to show my appreciation, at least for the idea :) --Oakwood (talk) 16:30, 13 May 2017 (UTC)

## A barnstar for you!

 The Random Acts of Kindness Barnstar I thank you, Prof. Tsirelson, for your kindness. I'm not a mathematician, I only took some courses in analysis when I studied physics at university, so I can just content myself with specyfing short historical notes. Like this one on Paolo Ruffini, whose rule enthralled me during the first year of high school. Thank you again. Raoul Bertorello (talk) 15:28, 8 August 2017 (UTC)
Nice. Probably I should not thank you for thanking me for thanking you, or else we could enter an endless loop!   :-)   Boris Tsirelson (talk) 18:23, 8 August 2017 (UTC)
Yeah, :)) ! Anyway, Prof. Tsirelson, feel free to give me assignements on historical issues that could bother you: I'll be happy to help. Only know that, just like you or everybody else I guess, I don't login on a daily basis to Wikipedia, so it could pass some time before I read anything new written to me. Have a nice weekend. — Preceding unsigned comment added by Raoul Bertorello (talkcontribs) 15:49, 12 August 2017 (UTC)

## 2017a

Thanks for the "thanks." On a non-Wikipedian topic, I was discussing recently with a colleague the measurability of giant components in the Boolean model. He was quite concerned how it's generally (not) treated (I don't concern myself with such matters). He went searching on the internet and found some lectures or a talk discussing it. Of course, the lectures were in fact written by you. The small world of probability. My colleague is still concerned, and I suggested for him to contact you (if that's OK?) to discuss the matter furtherImprobable keeler (talk) 11:42, 13 March 2018 (UTC)

Why not? Surely he may (try to) contact me. Being retired, I do not start new projects, but hopefully I still understand my (old) courses. I guess, this one is meant. Boris Tsirelson (talk) 19:55, 13 March 2018 (UTC)
Yes, I suspect those notes. And yes, I told him that I suspect you were retired. I think he just wants to bounce some ideas around, not actually start a project. I'll suggest to him to contact you. Thanks. Improbable keeler (talk) 09:32, 16 March 2018 (UTC)

## Structures and transportability

Thank you for the appendix on transport of structures. I found it interesting. There is something that puzzles me and perhaps you can help. Naively, I consider that the usefulness of having an isomorphism between two structures is that, somehow, if you know that two structures are isomorphic, then every theorem that respects a simple criteria and is respected by one structure is also respected by the other structure. What I see from Bourbaki requirement of transportability is that we don't have that. The definition of transportability in Bourbaki is not such a criteria. It only states what we want: we want that the property can be transported. It's the goal. It's not a way to achieve the goal.

For example, let say we have proven that ${\displaystyle \phi }$ is an isomorphism between two structures ${\displaystyle (N_{1},0_{1},S_{1})}$ and ${\displaystyle (N_{2},0_{2},S_{2})}$. The first structure has a property that corresponds to the first Peano axiom, but the fact that it is a Piano axiom is not important here. Because, you have an isomorphism you would like to conclude that the corresponding axiom in terms of the other base set, etc. is also a property of the other structure, without having to prove it directly. Bourbaki says that the criteria is that the property must be transportable. But, the definition of transportability is that you have to check explicitly that the property on the other structure is equivalent, which is exactly what you did not want to do directly.

As far as I am concerned, this is very puzzling. It almost tells me that an isomorphism means nothing. In what way are the two structures the same if one can respect a property while the other does not? It would not be that bad, if we had a simple criteria to determine which properties are transportable, but none is provided. Your appendix is a good illustration of that fact. A paper of Marshall and al. is often mentioned that says that the properties that are transportable are the sentences of Type theory. I went quickly over that paper and it's not a very easy paper to read. It contrasts with the sentence from Bourbaki: It is easy to recognize, in the usual cases, whether a relation is transportable (for a certain typification). (my translation). It makes me feel that some simple criteria to concretize this sentence from Bourbaki is missing.

I am hoping that you can shed some light on this issue. Dominic Mayers (talk) 17:44, 16 March 2018 (UTC)

On the other hand I also feel that a better isomorphism theorem can be formulated and proved in ZFC (not mentioning model theory, nor meta-theory); I tried a bit in my essay (mentioned on the talk page there); and probably something in the spirit of definability could be used. Boris Tsirelson (talk) 19:03, 16 March 2018 (UTC)

I am still puzzled. It seems that you argue that there is a simple criteria, which is to avoid "evil" sentences. If there is such a simple criteria, why in your appendix don't you use it to show, for example, that Peano axioms are transportable? Why would such an argument cannot be made rigorous? If it can be made rigorous, why don't we use it? If it is because it is complicated, why? Dominic Mayers (talk) 22:34, 16 March 2018 (UTC)

More precisely, my concern is with this sentence: «In the absence of “evil”, transportability is not an issue; it always holds, and is easy to prove.» If it always hold in absence of evil, why can't we make it simple to check that there is no evil and then, that's it, nothing else to be proven. Dominic Mayers (talk) 23:51, 16 March 2018 (UTC)

This sentence is useful: «Such “evil” never appears in mathematical practice, since a mathematician always treats elements of a principal base set as points with no internal structure.» How comes it is difficult, if it is, to translate this into a criteria that is practical and easy to use? Dominic Mayers (talk) 01:21, 17 March 2018 (UTC)

Another aspect is that it is also "evil" to contradict the axioms of the theory itself with a statement such as ${\displaystyle X_{1}=X_{2}}$ where ${\displaystyle X_{1}}$ and ${\displaystyle X_{2}}$ are principal base sets. This is the example of a non transportable property given in Bourbaki. Unless I am mistaken, the principal base sets are, by definition of principal base sets, required to be disjoint. Otherwise, consider a species on

${\displaystyle P(X_{1})\times P(X_{2})\times P(X_{3}).}$
A structure is a triplet of sets ${\displaystyle (s_{1},s_{2},s_{3})}$. The simple axiom that there is a bijection from ${\displaystyle s_{1}}$ to ${\displaystyle s_{2}\cup s_{3}}$, which has no evil and is transportable when the principal base sets are disjoint is not transportable anymore when the base sets are not required to be disjoint, because when ${\displaystyle s'_{2}=s'_{3}}$ on the new structure, but not on the original structure, which is possible, despite the individual bijections, we have that ${\displaystyle |s'_{2}\cup s'_{3}|\neq |s_{2}\cup s_{3}|}$ is possible. Dominic Mayers (talk) 01:37, 17 March 2018 (UTC)

Perhaps, if we added that there must also be a global bijection on the union of all the principal base sets, then we could relax the requirement that they are disjoint. Then ${\displaystyle X_{1}=X_{2}}$ might be transportable. Something along these lines might work to make ${\displaystyle X_{1}=X_{2}}$ transportable. It's not important to make it transportable, but it is useful to see that, if it is not transportable, it is only because of a technical aspect of the theory, different from the other "evil" aspect. Dominic Mayers (talk) 02:10, 17 March 2018 (UTC)

"If there is such a simple criteria, why in your appendix don't you use it to show, for example, that Peano axioms are transportable?" — just because I did not prove the simple criterion; moreover, I did not formulate it; you see, my text was written for undergraduate students, not for you...
"why can't we make it simple to check" — just do it!! I like that matter; now I see you like it too; but, as far as I understand, during decades "bourbakism" was disliked; I do not know why exactly; maybe because "category theory is beautiful, bourbakism is dull". In addition, another obstacle: "usual math" does not include model theory (nor meta-math); it is possible to reformulate the criterion (and all the "bourbakism") without these, but for now no one did it.
"the principal base sets are, by definition of principal base sets, required to be disjoint" — No. Rather, it is evil to ask whether they are disjoint or not. Dealing with ${\displaystyle x_{1}\in X_{1}}$ and ${\displaystyle x_{2}\in X_{2}}$ note that "${\displaystyle x_{1}}$" and "${\displaystyle x_{2}}$" are variables of different types, thus, the equality relation ${\displaystyle x_{1}=x_{2}}$ is evil. Note that the scale of sets does not stipulate unions (nor intersections). And indeed, in the "normal" math, do you ever consider something like ${\displaystyle \mathbb {R} \cup L_{2}[0,1]}$? Do you ever ask whether a set of natural numbers can be equal to a natural number, or not? Whether a set of real numbers can be equal to a complex number, or not? All that is evil.
Boris Tsirelson (talk) 06:36, 17 March 2018 (UTC)

You misunderstood my point. The example in Bourbaki is not ${\displaystyle x_{1}=x_{2}}$, but ${\displaystyle X_{1}=X_{2}}$. The confusion comes perhaps from the fact that (the group) Bourbaki uses small letters ${\displaystyle x_{i}}$ to represent principal base sets, whereas we use capital letters. They are very explicit on the fact that the ${\displaystyle x_{i}}$ represent principal base sets and then give ${\displaystyle x_{1}=x_{2}}$ as an example of a non transportable property. There is no evil, not in the way you meant it, in this property. There is something evil in it, which is that it contradicts the definition of principal base sets. In fact, we could say that it is always false and thus transportable. Perhaps you think that there is a typo in Bourbaki and their example was meant to be the evil property ${\displaystyle x_{1}=x_{2}}$, where ${\displaystyle x_{1}\in X_{1}}$ and ${\displaystyle x_{2}\in X_{2}}$. In both cases, I think it could be a source of confusion for many.

Why don't we do it together, writing the simple criteria in a way that can be used?

Dominic Mayers (talk) 08:22, 17 March 2018 (UTC)

I still think that ${\displaystyle X_{1}=X_{2}}$ is evil (though indeed my explanation was flawed). This means ${\displaystyle \forall x_{1}\in X_{1}\;x_{1}\in X_{2}}$ (and the other half...), which is evil, since the type of the variable "${\displaystyle x_{1}}$" disallows the relation ${\displaystyle x_{1}\in X_{2}.}$ Boris Tsirelson (talk) 09:30, 17 March 2018 (UTC)

Below is an extract from the original (in French) of Bourbaki. You can certainly use ${\displaystyle \forall x\in s}$ when the structure ${\displaystyle s}$ is a set.

3) ${\displaystyle {\cal {F}}}$ étant toujours la théorie des ensembles, considérons l'espèce de structure sans ensemble de base auxiliaire, comportant un ensemble de base principal A, la caractérisation typique ${\displaystyle V\in {\cal {B}}({\cal {B}}(A))}$, et ayant pour axiome la relation transportable

${\displaystyle (\forall V')((V'\subset V)\Rightarrow ((\bigcup _{X\in V'}X)\in V){\text{ et }}A\in V}$
${\displaystyle {\text{et }}(\forall X)(\forall Y)((X\in V{\text{ et }}Y\in V)\Rightarrow (X\cap Y)\in V).}$

Cette espèce de structure est appelée espace de structure topologique. Une structure de cette espèce est aussi appelée topologie, et la relation ${\displaystyle X\in V}$ s'exprime en disant que ${\displaystyle X}$ est ouvert pour la topologie ${\displaystyle V}$ (TG, I, ${\displaystyle \S }$ 1).

Dominic Mayers (talk) 15:22, 17 March 2018 (UTC)

Moreover, there is no more ${\displaystyle \in }$ in ${\displaystyle X_{1}=X_{2}}$ than there is one in ${\displaystyle s_{1}=s_{2}}$, where ${\displaystyle s_{1}}$ and ${\displaystyle s_{2}}$ are structures in ${\displaystyle P(X_{1})}$ and ${\displaystyle P(X_{2})}$, respectively. In both cases, we have an equality between sets and the latter is certainly transportable. Dominic Mayers (talk) 15:29, 17 March 2018 (UTC)

Oups, I think I understood what you meant. You see ${\displaystyle X_{1}}$ and ${\displaystyle X_{2}}$ as types, which disallow ${\displaystyle x\in X_{2}}$ when we have ${\displaystyle x\in X_{1}}$. But, Bourbaki is based on ZFC and thus there is no such restriction unless ${\displaystyle X_{1}}$ and ${\displaystyle X_{2}}$ are disjoint. This comes back to exactly what I told you: it contradicts the theory, which must state somewhere that principal base sets are disjoints. However, this does not make the sentence non transportable. It only makes it always false.

Dominic Mayers (talk) 15:42, 17 March 2018 (UTC)

As far as I understand the (relevant part of) the type theory, the types correspond to the sets belonging to the scale of sets. In our case, there is a type "variable that runs over ${\displaystyle X_{1}}$", another type "variable that runs over ${\displaystyle X_{2}}$"; then "variable that runs over ${\displaystyle X_{1}\times X_{2}}$", "variable that runs over ${\displaystyle P(X_{1})}$", and so on. There is no type "variable that runs over ${\displaystyle \{X_{1},X_{2}\}}$" (and no constants ${\displaystyle X_{1},}$ ${\displaystyle X_{2}}$ of any type), thus, you cannot use the equality ${\displaystyle X_{1}=X_{2}}$ "as is". In ZFC you can, of course; but it need not be transportable. Boris Tsirelson (talk) 18:50, 17 March 2018 (UTC)

As you know, type theories were developed separately from ZFC, as an approach to deal with Russel paradox. Type theories did not become as popular among mathematicians as ZFC. I guess the main reason is that ZFC has less boundaries within its own bubble, the universe of sets. In any case, Bourbaki has adopted ZFC as its basis, not a type theory. They do not make any connection with any Type theory. This can easily be seen, because their chapter on structures is self contained and use no other requirement than the theory of sets. It's nice that Marshall made a connection with a type theory, but within the framework of Bourbaki, ${\displaystyle X_{1}=X_{2}}$ is a perfectly legitimate statement and ${\displaystyle X_{1}}$ and ${\displaystyle X_{2}}$ are not apriori disjoint. Therefore, in the definition of base sets, it must be required that they are disjoint, if we want them to be as in a Type theory. As I have shown, we do want that, because, otherwise, even the simple non evil axiom ${\displaystyle |s_{1}|=|s_{2}\cup s_{3}|}$, where ${\displaystyle s_{i}\in P(X_{i})}$ and ${\displaystyle |A|}$ means the size of the set ${\displaystyle A}$, is not transportable.

Dominic Mayers (talk) 20:04, 17 March 2018 (UTC)

### Definition of base sets

I started a new section, because a large section is hard to edit and it seems that we have pin down the issue on the definition of base sets. Well, I have read again the chapter on Structures and Bourbaki do not explicitly require the base sets to be disjoint. It has to be an omission and they meant them to be disjoint. I cannot see any other explanation. It makes no sense to assume that they had a type theory in mind, because they don't say a word about Type theory. Dominic Mayers (talk) 20:30, 17 March 2018 (UTC)

Well, do as you like... Really, I do not bother much about the original text of Bourbaki. They did the first try, and did not really use this technique afterwards. I see many drawbacks in their presentation. Probably the matter will be rectified now, if at all this approach will be used more actively. "Pioneer work is clumsy" (Littlewood, "Miscellany").
I still do not feel any need to require the base sets to be disjoint. Rather I feel that such question is evil. Of course, you are free to try your version of this theory. Tastes differ... As for me, "stange unions" like ${\displaystyle \mathbb {R} \cup L_{2}[0,1]}$ (even worse, ${\displaystyle 5\cup 6}$) are not needed in mathematical practice. Indeed, ZFC has too few boundaries! And in fact, type theories are now actively investigated as potentially new and better basis for math (and especially for structures); see Homotopy type theory, Higher-order logic etc. Boris Tsirelson (talk) 20:41, 17 March 2018 (UTC)

Well, there are two options. The first is that you write axioms from the start that are not limited by any boundaries, no type, no scale sets, thus no concept of transportability is needed. This option excludes Bourbaki approach as well as a type theory approach. These are part of the second option, where transportability needs to be shown. It may be that the first option is what I need. However, now that I have met this second option where the notion of transportability is required, a question was raised that troubles me. How comes there is no simple criteria to determine whether a sentence is transportable? I don't care either about Bourbaki. As soon as transportability is required, this question is raised, independently of Bourbaki. You argue in favor of type theories, which support an intuitive notion of evil, but this does not provide any simple criteria. Therefore, perhaps you should keep an open mind to another approach where we start from ZFC, not a type theory.

In any case, I feel, in both cases, type theories or ZFC, the key ingredient is to see how we can make the link with an approach that adopts the first option, that is, with no boundaries from the start. In the ZFC approach to structures, even though ZFC does not have boundaries in itself, boundaries exist between the different choices of base sets, because somehow every axiom only applies to one set of base sets and one scale of sets. This is the key feature of the second option and this is why we need transportability when we pick this option. The idea of transportability is that we want that the axioms apply in the same way to any choice of base sets under isomorphisms. This is what I mean by we need to see how we can relate the second option to the first option where we do not bother to restrict axioms inside some boundaries, right from the start. I feel there is more chance of success within a ZFC approach to make the link between these two options.

Dominic Mayers (talk) 21:36, 17 March 2018 (UTC)

Finally, my hope is that the answer is in the paper of Marshall et al. I was expecting something way simpler, but it might not be that complicated. At the end, if verifying that a sentence is transportable consists only in verifying that it belongs to a language and the language is reasonable, then it is exactly what one should expect. However, if that is the case, then it would be very interesting that, in your appendix, you apply that verification on different properties, including the Peano axioms. Verifying that these short expressions belong to a given language cannot be that complicated.

Dominic Mayers (talk) 01:27, 18 March 2018 (UTC)

About your first option: do you say "forget the idea of isomorphism"?? If so, then I answer: no, never. For me this blatant anti-univalence is not an option. Boris Tsirelson (talk) 05:39, 18 March 2018 (UTC)
I did not say "forget the idea of isomorphism", of course not. Dominic Mayers (talk) 14:53, 18 March 2018 (UTC)
About your second option: today you require (in addition) ${\displaystyle X_{1}\cap X_{2}=\emptyset ;}$ tomorrow you'll require (in addition) ${\displaystyle X_{1}\cap (X_{2}\times X_{3})=\emptyset ;}$ then also ${\displaystyle X_{1}\cap (X_{1}\times X_{1})=\emptyset ,}$ ${\displaystyle X_{1}\cap P(X_{1})=\emptyset }$ etc. Then you'll observe that finite ordinals are no more an implementation of natural numbers. Boris Tsirelson (talk) 05:39, 18 March 2018 (UTC)
You are right that I might have to require more, but I don't see why I would require the last two ones, because they use the same base set. Not having them should not create an issue. But you are right, I might have to require that any two scale sets in distinct scales of sets are disjoint or something along these lines. Dominic Mayers (talk) 14:53, 18 March 2018 (UTC)
My opinion: the union ${\displaystyle X_{1}\cup X_{2}}$ is evil; the disjoint union (in other words, marked union) ${\displaystyle X_{1}\sqcup X_{2}}$ is acceptable (but rarely needed). Boris Tsirelson (talk) 06:16, 18 March 2018 (UTC)
It is not evil. We are just adding an axiom to ZFC and, in ZFC, even with an extra axiom, ${\displaystyle X_{1}\cup X_{2}}$ is not evil. However, given that I need an extra axiom, I understand now why Marshall and al. did not use the exact formulation of Bourbaki. As long as we agree that there is a need to modify Bourbaki, then we have to be open that there will more than one way to do that. I am confident that there is a way to remain within ZFC and I think it is useful to do so. You are right that I haven't pin down what is the extra axiom. However, I don't think Bourbaki meant for ${\displaystyle X_{1}\cup X_{2}}$ to be evil, because they clearly remain within ZFC and, in ZFC, it is not evil. I also, again, think that it is possible and useful to remain within ZFC. Dominic Mayers (talk) 14:53, 18 March 2018 (UTC)

I have responded within your last post.
Dominic Mayers (talk) 14:53, 18 March 2018 (UTC)

We need to go back to the original request. I use the term "request" instead of "question", because, if it is a question, then it is a question that cannot be clear until you have answered it. This is like when a person requests a new kind of software or a new feature in an existing software. The request is not clear until after you have answered it. I don't know the history of the basic mathematical structures, but I guess it was the same thing. At some point, there must have been a request for these structures, but until after we had found them, we did not know for sure what they were. The request is for a better theory of structures and isomorphisms so that we have a simple criteria for transportability. I believe that you agreed that it was a valid request, because you said that you are looking for a better isomorphism theorem. However, I am not sure that you had the same view on that as me. It is not easy to share a same view in a context of a request that is not, as is often the case with requests, not very clear. Questions are often important contributions in such a context. My question for you is why didn't you use the criteria of Marshall et al. to show the transportability of the Peano axioms? I mean, if it is complicated, why? Is it because the language in which the axioms are stated is a different one, a different way to represent species of structures? In other words, the question is whether Marshall et al. is the answer to the request and, if not, why?

I suspect Marshall isn't the answer and that it will be easier to find the answer to that request if we remain within ZFC, but I might be wrong, but the first step is to try to have a common view on the request itself.

Dominic Mayers (talk) 14:53, 18 March 2018 (UTC)

Regarding the first option vs the second option, they are not meant to be so different. I will try to explain better what I meant. If we had a simple criteria that is only a verification that a sentence belong to a language, then when we would write the axioms, knowing the criteria, they will be automatically transportable. Who would be so stupid as not writing from the start axioms within the language? In that case, these axioms have, by definition, no boundaries, they apply to all scale of sets without having to prove their transportability. This is the first option. It's not that transportability totally disappear, but you can forget about it as long as you use the language, because it has already been proven once for all. In that sense, what we want to do is to make a link between these two options.

Dominic Mayers (talk) 14:53, 18 March 2018 (UTC)

Also, when I say that we have to add an axiom to ZFC, I do not mean that we will base the theory of structures on a different theory of sets. We don't modify the theory of sets. The axiom is only an axiom that applies to base sets in the definition of a structure. It is an axiom of a theory of structures.

Dominic Mayers (talk) 15:39, 18 March 2018 (UTC)

Now my feeling is that we are able to agree (at some moment I was in doubt).
Yes, better do not call any definition "axiom" (in order to avoid misunderstanding). Yes, axioms of ZFC are intact, anyway. The definition of structure may be tweaked.
"My question for you is why didn't you use the criteria of Marshall et al. to show the transportability of the Peano axioms?" — did not I answer already? My text was for undergraduate students; and the course was not devoted to structures; my Appendix was already too much for that course (and so, I wrote explicitly that it is not a part of the course); I did not formulate (and prove) the criterion since the students are not mature enough for this, and not devoted enough to this, and not ready to metamatematical notions (like "formula of ZFC"). In fact I think it is better to reformulate the criterion to avoid metamathematics (such technique is known, see b:User:Tsirel/sandbox); but this is not done for now, and (the more so) not ready to undergraduate students. (Did I answer this your question now?) Boris Tsirelson (talk) 18:12, 18 March 2018 (UTC)
"If we had a simple criteria that is only a verification that a sentence belong to a language, then when we would write the axioms, knowing the criteria, they will be automatically transportable. Who would be so stupid as not writing from the start axioms within the language? In that case, these axioms have, by definition, no boundaries, they apply to all scale of sets without having to prove their transportability. — Nice. As for me, this goal is already reached! The next goal is rather to explain this fact to wider mathematical audience. To this end it helps to reformulate the criterion. This is what I meant writing "just do it". But, it seems, you are not sure what to do: reformulate the criterion, or discover a better one. Please recall that the criterion is already necessary and sufficient. And my feeling is that it rather codifies the preexisting social norm. Boris Tsirelson (talk) 18:21, 18 March 2018 (UTC)
But wait; not quite so. Not quite "verification that a sentence belongs to a language", but rather, verification that the sentence is equivalent to some (at least one) sentence of a different language. For now. Hopefully in a near future metamatematics will be eliminated from this matter, and then the word "sentence" will be also eliminated. Boris Tsirelson (talk) 18:27, 18 March 2018 (UTC)

Yes, the "not quite so" is exactly what is my concern. It might be more complicated to show an equivalence between two sentences in different languages than to directly show that two properties on different scale sets are equivalent. May be not. May be it is easy. So let me rephrase the question. Why don't you show to me (and to any people reading it eventually) how you would apply Marshall et al criteria to prove that Peano axioms are transportable? I admit that I did not go through Marshall et al. yet and it would be nice if I could see how the main result can be applied. If it is complicated in the case of these axioms, then I would like to get an idea why. It could be, then, that the criteria itself can be improved, by staying within ZFC. The intuition is that, if we rely on the basic fundamental and well accepted language of set theory to state and prove the criteria, then there would less need to worry about equivalence between two languages, because it is easy to see any axiom as a statement in the language of set theory. I admit that things are a bit fuzzy in my mind at this point. It is a vague intuition and it might be misguided and things can be very simple using Marshall et al. criteria.

Dominic Mayers (talk) 19:40, 18 March 2018 (UTC)

Also, it becomes relevant that I bring again the point that I believe that it should be possible to generalize the definition of isomorphism to require that there is also a global bijection that forces ${\displaystyle X_{1}=X_{2}}$ to be transportable. You may feel "why complicating things more than they are" by generalizing the current definition of isomorphism. The motivation is that I believe that this lack of generality, where ${\displaystyle X_{1}=X_{2}}$ is not transportable is not natural. If you are very much caught into type theory and you cannot see ${\displaystyle X_{1}=X_{2}}$ in any other way than being evil, then I don't know what to say. You maybe right that we should think only in terms of type theory, but I am far from being convinced. Again, then, it would help very much that you show to me that there is no issue at the least to apply Marshall et al criteria to Peano axioms. My feeling is that, on the contrary, we should think in terms of ZFC and remove any superficial (from a ZFC view point) constraint such as ${\displaystyle X_{1}=X_{2}}$ being not transportable. Well, I am aware that it would be always false according to the definition, but other statements such as ${\displaystyle |s_{1}\cup s_{2}|=|s_{3}|}$ will not always be false.

Dominic Mayers (talk) 20:36, 18 March 2018 (UTC)

Ah, well, I'll do (with Peano). Not pedantic, but hopefully clear and formalizable (rather than formal).
Below, variables ${\displaystyle x,x_{1},x_{2},y,y_{1},y_{2}}$ run over ${\displaystyle N,}$ and variable ${\displaystyle K}$ runs over ${\displaystyle P(N).}$ Also, the constant ${\displaystyle 0}$ belongs to ${\displaystyle N,}$ and the constant ${\displaystyle S}$ belongs to ${\displaystyle P(N\times N).}$
${\displaystyle \forall x\;\exists y\;\;(x,y)\in S;}$
${\displaystyle \forall x,y_{1},y_{2}\;\;{\big (}(x,y_{1})\in S\land (x,y_{2})\in S\Longrightarrow y_{1}=y_{2}{\big )};}$
${\displaystyle \forall x_{1},x_{2},y\;\;{\big (}(x_{1},y)\in S\land (x_{2},y)\in S\Longrightarrow x_{1}=x_{2}{\big )};}$
${\displaystyle \forall x\;\;(x,0)\notin S;}$
${\displaystyle \forall K\;\;{\Big (}0\in K\land \forall x,y\;{\big (}(x,y)\in S\land x\in K\Longrightarrow y\in K{\big )}\Longrightarrow \forall x\;x\in K{\Big )}.}$
That is it. The point is, no type conflict. I did not write something like ${\displaystyle x\in y,}$ or ${\displaystyle x=K,}$ or ${\displaystyle (x_{1},x_{2})=y,}$ etc. Each term has its type, and every relation obeys these types.
Boris Tsirelson (talk) 21:43, 18 March 2018 (UTC)

Yes, after you said that you will do it, I thought about it and expected such an answer and thought "oh well", it's going to be simple. The other simple proof that I also need to see is the proof that any such a sentence is transportable. The proof that I have in mind is very easy: in the sentence it is not hard to see that you can replace say ${\displaystyle x\in S}$ with ${\displaystyle \phi (x)\in \phi (S)}$ and vice-versa where ${\displaystyle \phi }$ is the isomorphism, without changing the truth value of the sentence. But I would like to see the formalization of that anyway. Do you have any book to suggest that cover the basic here, including a proof of the transportability of such sentences and, ideally, a bit more about the proof in Marshall et al. that shows the converse, that is, only these sentences are transportable. It's not only the proof itself that I need to see, but the context, some intuition, etc. The paper of Marshall et al. is a bit too direct. Dominic Mayers (talk) 00:41, 19 March 2018 (UTC)

### The importance of the type theory aspect

I created a new section only because it is simpler to edit, no other reason. I just want to say that you overstate, I suspect, the importance of type theory in the argument. I want to look at the proof that any such sentence is transportable, but from what I see the type aspect plays almost no role in the proof. It is only needed so that the overall mapping is a bijection. For example, if we have ${\displaystyle X_{1}=X_{2}}$ for the source base sets and not for the target base sets, then there is no global bijection, even though there is a bijection for each individual base set. If we require a global bijection, by some axiom, then my understanding is that there is no need for type theory. Dominic Mayers (talk) 01:32, 19 March 2018 (UTC)

Similarly, I don't believe that type theory should play an important role in the converse statement. I am not saying that Marshall et al are wrong. They must be correct given their hypothesis. However, it should be possible to have different, perhaps less restrictive and, in a way, more natural hypotheses, so that there exist sentences outside type theory that are transportable. In that sense, if I am right, the essential idea of the statement and of its proof has not much to do with type theory. This essential idea is what I want and, in their paper, it gets loss in the details of type theory. This is an example of what I mean when I say that I need more context and more intuition.

Dominic Mayers (talk) 02:29, 19 March 2018 (UTC)

As you know already, logic was always my hobby; I never was an expert in it. (For more, see my reminiscences, and this, and this.) I have nothing specific to recommend you as textbook; just try something. And I must confess that I did not really read that paper. I took a look and felt that the result is believable, and the easy direction should be no problem to prove. That was enough for me. If something is wrong with details there, I am not responsible. Boris Tsirelson (talk) 05:34, 19 March 2018 (UTC)
"If we require a global bijection..." – I do not like the idea of such global bijection. For me, this is inappropriate (and I can agrue more, if needed). And I am not quite understanding your words about the type theory. As for me, "no type conflict" is exactly the point. You probably can say it in other words; but can you say something really different? Boris Tsirelson (talk) 05:41, 19 March 2018 (UTC)

Yes, I think we can do something very different without type theory. There will be common features such as 'disjointness' of the base sets, but even the way to state that 'common' feature will be different. I feel that the distinctions will be important. The key point in my mind now is the fact that there are two axioms for a species: the typification T and the relation R. The fact that we focus more on R and look at T as a framework in which R is stated, makes us lose the symmetry in the axioms T and R taken together. This lost of the symmetry is even more important with type theory because T takes a different form as a part of type theory.

If we use the symmetry, we might be able to make a connection with the work of Joyal et al. Yes, you are right that they focus on combinatorics but the connection can still be useful.

Dominic Dominic Mayers (talk) 09:24, 19 March 2018 (UTC)

Hmmm... I'll be waiting... Boris Tsirelson (talk) 15:49, 19 March 2018 (UTC)
Yep, you are right. I spoke too fast regarding the connection with with their work, but I still have the feeling that it may be useful to take advantage of the fact that ${\displaystyle T}$ and ${\displaystyle R}$ together are invariant under symmetries on the base sets. Can we define transportability as invariance under symmetries on the base sets? Maybe the answer is obviously yes, but it does not help. Well, conceptually, for me anyway, it would help. Somehow, what I am looking for must be in a textbook ! I just don't know where to look and I don't want to take any book in logic or in general algebra, etc. and hope that it will be there. Dominic Mayers (talk) 16:01, 19 March 2018 (UTC)
Not knowing what you are thinking about, I cannot know whether or not the following example is a counterexample for you, but I want to show you the following example anyway.
Some implementations of natural numbers (that is, Peano axioms) satisfy ${\displaystyle \exists x,y\in N\;\;x\in y;}$ other do not; but this predicate is (trivially) invariant under permutations of the points of ${\displaystyle N}$ and, of course, all the implementations are isomorphic. Boris Tsirelson (talk) 17:36, 19 March 2018 (UTC)

I was not clear myself about the details, but I realize now that it's not invariance under a permutation that is the key, but invariance under a change of implementation, and an isomorphism is the formalization of what we mean by change of implementation. It should have been obvious, now that I think about it. I guess that I am looking for an alternative criteria to tell if a property is transportable, but a criteria that is independent of the details of the language. Invariance under any isomorphism is such a criteria. I should be happy with it, but I was looking for another one that does not involve other implementations and not the details of the language also. Perhaps that such a criteria cannot exist, because, fundamentally, transportability is a property of a formal sentence. Dominic Mayers (talk) 19:32, 19 March 2018 (UTC)

Aha, now our positions are close. "Invariance under any isomorphism is such a criteria"? Rather, the definition. "Such a criteria cannot exist, because, fundamentally, transportability is a property of a formal sentence"? Here I am a bit more optimistic, since (as I wrote before) a technique that eliminates formal sentences is available. Boris Tsirelson (talk) 20:47, 19 March 2018 (UTC)
I like very much this metaphor: Thus, in practice a topology on a set is treated like an abstract data type that provides all needed notions (and constructors) but hides the distinction between "primary" and "secondary" notions. The same applies to other kinds of mathematical structures. Interestingly, the formalization of structures in set theory is a similar task as the formalization of structures for computers. (Quoted from Equivalent definitions of mathematical structures#Mathematical practice.) A pity that the notion "abstract data type" is still outside mathematics. The goal of the "no type conflict" requirement is to hide implementation details from users. Also, for me the ZFC universe is like an empty hard disk, while a structure is like a file. A (say) sound file may be converted (say) from .wav to .mp3; this is a change of implementation. It is evil for an application to ask, is the 123-th bit of the file 0 or 1 (and the more so, to ask the cylinder, track and sector containing the bit). Boris Tsirelson (talk) 21:06, 19 March 2018 (UTC)
And now I can say why I prefer separate bijections for (possibly non-disjoint) base sets. As for me, implementation of one base set may be changed independently of implementation of another base set. Nonempty intersection of base sets is a ridiculous implementation detail. Similarly, a finite sequence of bits may implement both an integer and a string. Or even, both a sound file and a graphics file (well, very unlikely; and impossible if magic numbers are stipulated; but if this happens, it should have no consequences). Boris Tsirelson (talk) 21:23, 19 March 2018 (UTC)

But even among programmers, some like typed languages, others don't. I don't think that that there is a universal agreement, far from it, that typed languages are better. In the case of programming languages, it's a question of how you work to prevent and discover bugs and, if you like the little tricks that are possible with the flexibility of non typed languages. There is also a lost of efficiency with typed languages, but it is only relevant with dynamic type checking in systems that must absolutely communicate in real time with the environment. There is nothing fundamental there. But, yes, I have progressed. Now, I also see that type theory is similar to the use of a typed language. As far as I am concerned, however, I don't feel there is an important need to move to type theory to do mathematics. It will work fine. You will get some advantages out of it, but it seems to me that, for mathematics, the issue of bugs created by using the wrong type because of a typo is less important and that is one of the main advantage of a typed language in computer science. Also, the notion of abstract data type exist in both typed and non typed languages. You can define abstract classes, interfaces, etc., which set the abstract behavior independently of the implementation, in a non typed language. PHP is not typed - there is no restriction at all: any variable can be used anywhere and PHP will cast the value, but still there are abstract classes, interfaces, etc. In the same way, with ZFC, you can define higher level structures and define concepts of isomorphism and transportability, etc, to move away from implementation details. I don't think there is an absolute need for type theory to achieve that.

Dominic Mayers (talk) 23:41, 19 March 2018 (UTC)

So... try to invent a PHP-style theory of mathematical structures... Boris Tsirelson (talk) 06:53, 20 March 2018 (UTC)

I feel the same way, I would feel if I was discussing a new application with someone and the big issue was in which language it should be written. I believe the important new concepts in new applications transcend the choice of the language that is used to implement them. Of course, if I speak Chinese and you speak Russian, we have a problem, but we have already realized that Type theory and ZFC are not so far away. In fact, Bourbaki is in ZFC and Marshall and others have immediately translated Bourbaki in terms of Type theory.
Dominic Mayers (talk) 10:38, 20 March 2018 (UTC)

Back to Wikipedia: for now your contribution to "Equivalent definitions of mathematical structures" is start-class instead of class C. Boris Tsirelson (talk) 05:04, 21 March 2018 (UTC)

I don't think that user talk pages are meant to be classified as if they were articles. This is only between me and you and those who are curious about what we say, but to them I say "don’t look a gift horse in the mouth". As far as the communication between us goes, you have provided no argument in favor of typed theory. I can see that this is the big thing for you, but it's not based on practical criteria that I can follow. It's very easy to create a few rules and then study them. It's valuable to do that, but it does not mean that they should become the basis for mathematics. The same thing can be said for the rules of set theory, but the difference is that they are much more accepted as a basis than the rules of type theory.

Dominic Mayers (talk) 12:18, 21 March 2018 (UTC)

Happy editing. Boris Tsirelson (talk) 19:22, 21 March 2018 (UTC)

I wrote this https://en.wikipedia.org/wiki/User_talk:Dominic_Mayers#About_transportability. I would like to have your feedback. Dominic Mayers (talk) 22:24, 26 March 2018 (UTC)

### Your analogy with a programming language

I read a lot on the subject since last time and, now, I totally agree with your analogy with programming languages, but I would extend on the way you explain it. It might not be an extension in the following sense that you perhaps had this general picture, just did not mention it.

I learned about universal algebra, first order logic, many-sorted logic, second order logic with the Henkin semantic, etc. These are different languages that are interpreted with models defined in set theory. With the little understanding that I had at the time, I did not like the idea of getting rid of set theory, because I interpreted it as if it meant that the structures would not be defined in set theory.

Now, I am much more flexible in two ways. First, I love universal algebra, first order logic, etc. and these can be seen as higher level languages that are interpreted in set theory, I mean, the structures or models are constructed in set theory. In that way, I totally agree with you that it is wise to use an higher level language. Second, I think it can be useful mathematically to consider the interpretation of a higher level language into another higher level language, just as we do in computer science, but I only see a theoretical interest to this and I don't think people have done this.

With this larger picture in mind, my problem is now to find the proper higher level language that I need for the particular application that I have. If I could do it with universal algebra, I would certainly do it in that way, because universal algebra has been so much studied. Unfortunately, I need more elaborated structures. I don't want to use Bourbaki structures, because it is way too close to set theory. It is almost as if we interpreted set theory within set theory as a way to define structures - almost impossible to get interesting theorems. I agree that type theory is a bit higher level, but it might be still too close to the machine. Dominic Mayers (talk) 17:14, 11 June 2018 (UTC)

## That undo

Because I apparently read the diff backwards. Thanks for double checking me. –Deacon Vorbis (carbon • videos) 17:39, 26 March 2018 (UTC)

Thanks for the explanation; I was a bit puzzled, why you reverted it; I know that you could not treat that ${\displaystyle a}$ is the summation variable. Boris Tsirelson (talk) 19:04, 26 March 2018 (UTC)

## “Party” / “particle” in Quantum entanglement

I see you have only reverted one of two recent changes by https://en.wikipedia.org/wiki/Special:Contributions/108.20.79.247 to Quantum entanglement; I suspect you may want to revert the other as well; I wonder if there is some way of formulating it so as to make clearer to the layman that “party” is correct. PJTraill (talk) 20:18, 20 June 2018 (UTC)

Yes, but I see this is already made by User:Porphyro 4-5 hours ago. Boris Tsirelson (talk) 21:13, 20 June 2018 (UTC)
I also inadvertently dealt with at least part of User:PJTraill's issue by changing an instance of "two-party state" to "state shared by two parties". Porphyro (talk) 10:28, 21 June 2018 (UTC)
Nice. I usually prefer "subsystems", but tastes differ. Boris Tsirelson (talk) 14:10, 21 June 2018 (UTC)
I'd be happy with subsystems as an alternative, I guess I was prejudiced by parties already being present in the text. Porphyro (talk) 16:56, 21 June 2018 (UTC)

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