User talk:Quondum

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Gamma 5[edit]

Hi Q!

You removed my note on γ5 forming a Clifford algebra together with the other gammas for five spacetime dimensions. Why?

I think it is relevant, and I can source the removed statement verbatim.

Cheers, YohanN7 (talk) 16:28, 6 November 2013 (UTC)

Hi. No offence intended. For context, this is the bit I removed after you added it:
But it is a particularly appropriate name because the set 01235} provide a set of gamma matrices in five spacetime dimensions.
Even as a quote from a source, it appears to be the opinion of the author you're quoting, and (if you'll forgive the wikilawyering) thus would have to be reported as such, even if said opinion was notable. However, quoting the opinions of individual authors is not what the article is about. And the article does not deal with generalizations to other dimensions at all, aside from the statement "It is also possible to define higher-dimensional gamma matrices" in the lead (duly linked to in another article). Its presentation was only a point of incidental interest. Relevance would only be shown if the name was actually notably used in this particular way, not simply some source saying that it forms a Clifford algebra.
The "appropriateness" of the name is that authors (Weinberg, The Quantum Theory of Fields vol 1) opinion of course (and should go out anyhow). But 01235} giving gamma matrices in 5-d is just a simple fact, notable or not. I think we mention somewhere that the Pauli matrices are 3-d gamma matrices. Why not mention the 5-d? As for spacetime in various dimensions, it is treated in plenty of advanced QFT texts, not to mention string theory. The fact that the article doesn't treat arbitrary dimensions doesn't mean it shouldn't.
You'd also have had to indicate that it would be a real Clifford algebra. There is already enough confusion about whether the Dirac algebra is Cl1,3(R) or its complexification Cl4(C). This algebra is Cl2,3(R), which we shouldn't consider as being of a 5-dimensional "spacetime", since this term would normally be intended to mean having a Lorentzian metric. This is a slightly scatty reply. but I think you get the message that I think it is inappropriate to include this. — Quondum 06:39, 7 November 2013 (UTC)
I don't follow you exactly. I need to read up on this, but from where do you get that the algebra is Cl2,3(R)? I'd say the algebra is Cl4,1(R) (or Cl1,4(R)). YohanN7 (talk) 14:41, 7 November 2013 (UTC)
If the five matrices span the grade 1 space of a real Clifford algebra and also anti-commute, then their squares necessarily give the signature of the space. And here we have that two of the matrices square to +1, and three to −1. As a check on this, we already know that the algebra is isomorphic to M4(C). As a real Clifford algebra over five dimensions, Classification of Clifford algebras#Classification gives three candidates with this ring isomorphism: Cl4,1(R), Cl2,3(R), Cl0,5(R). So while you could argue that though starting with a (1,n−1) convention, we've had to flip to a (n−1,1) convention to match your assertion. But to do so, we've had to replace at least two of the original matrices (e.g. by multiplying 2 or 4 of them by i), to get the signature to be (+ + + + −). That is to say, the five matrices as listed do not span any grade 1 subspace of this Clifford algebra. Which disqualifies them as what we mean by "gammas", even if we leave the rest as a hidden puzzle for the reader to disentangle.
But ignoring the maths, the statement still does not qualify for inclusion. You included it as a comment on the name γ5, which it fails to do in a coherent fashion. The mathematical argument is actually a diversion from this point. — Quondum 15:27, 7 November 2013 (UTC)
I included it mostly for its mathematical message, the name was just an odd twist. Forget that.
On the math: Note that it says 01235}, not 01235}. The Weinberg books are famous for containing extremely few typos, and of course no such gross errors as you hint above. YohanN7 (talk) 16:18, 7 November 2013 (UTC)
I noticed the subscript. But raising the index is equivalent to flipping the sign only if the corresponding element squares to −I. But since (γ5)2 = +I, we have γ5 = +γ5, and it makes no difference whether the index is lower or upper. C'mon, what do you read into the sequence ((γ0)2,(γ1)2,(γ2)2,(γ3)2,(γ5)2) = (+I, –I, –I, –I, +I)?squaring added — Quondum 02:14, 8 November 2013 (UTC)
Since I have now gone into details of the isomorphism, it is clear that I am not contradicting the statement that this can be used as a 5d "spacetime" Clifford algebra (by minor changes to the matrices it gives you a signature (+ + + + −), so depending on how it was presented, there is no deep contradiction. Starting with different gamma matrices that have a (− + + +) signature (just multiply each by i), you get a (− + + + +) signature with the exact construction you give. But it remains esoteric, since this construction is probably confined to only a few cases (i.e. choice of number of dimensions and signature). Add to this that at some level Clifford algebras of opposite signature are equivalent for this use, and you might find that Weinberg et al are simply glossing over this detail? Or perhaps they just use the opposite sign convention? — Quondum 19:13, 7 November 2013 (UTC)
You are right. Weinberg uses the (− + + +) convention. What we should say (if anything) is that 0123,iγ5} gives a 5-d Clifford algebra. B t w, I checked another reference (that uses (+ - - -)): Respected author, freely available online, page 93. He states that the reason for the terminology is, in fact, that 0123,iγ5} is a set of gamma matrices for five spacetime dimensions. YohanN7 (talk) 20:03, 7 November 2013 (UTC)
With the Weinberg sign convention, I guess someone might have used this as a "reason", but it doesn't quite work for me as a reason with the added i. The source you gave a reference to seems to do as you say: use a (+ − − −) convention, and then adds iγ5 in an attempt to make a five dimensional (+ − − − −) Clifford algebra. The anticommutation relations given in the source and required for a Clifford algebra do hold, but this is a crucial: they are not sufficient. As you can check for yourself, 0123,iγ5} does not work for five dimensions at all: we have the identity iγ5 = −γ0γ1γ2γ3}, which is not R-linearly independent of the space that is generated by the other matrices; it is already present in the Clifford algebra. This collapses the 5-d Clifford algebra to a 4-d one. Weird, but it looks to me to be a genuine mistake to call this a "five-dimensional Clifford algebra". — Quondum 20:47, 7 November 2013 (UTC)
There is nothing about linear independence of the gammas from everything generated by the gammas in the defining condition. In odd spacetime dimension d the totally antisymmetric (AS) tensors of rank n are linearly related to the AS tensors of rank d - n. [The AS tensors are obtained by antisymmetrizing the products of the gammas. This is something I'll put into the Dirac algebra article.] There is a formula expressing this in the Weinberg book (with a million of indices and subscripts/superscripts, too lazy to write it here). This formula implies the identity iγ5 = −γ0γ1γ2γ3, so everything is as it should be. YohanN7 (talk) 10:35, 8 November 2013 (UTC)
You seem to be mixing up the two cases. You cannot transfer Weinberg's mathematical details to the other case. Weinberg uses different gammas from the ones in the article (this must be the case to get the different signature). Also, take care not to confuse R-linear with C-linear. {1, i} is R-linear independent, but C-linear dependent. Remember that the two cases use different matrices throughout, so they must be dealt with independently. For the moment I'm going to use another name (γ4) for one of the matrices, to try to keep things straight.
  • Weinberg's (− + + +) convention, and his proposal 01234) with γ4 = γ5 = iγ0γ1γ2γ3 as a 1-vector basis for the generating space of a real 5-dimensional Clifford algebra: Here we have the relation γ4 = iγ0γ1γ2γ3, so γ4 is R-linearly independent of the product γ0γ1γ2γ3. We also get the desired squaring to (− + + + +).
  • The (+ − − −) convention, and the proposal 01234) with γ4 = iγ5 = −γ0γ1γ2γ3 as a 1-vector basis for the generating space of a real Clifford algebra: Here we have the relation γ4 = −γ0γ1γ2γ3, so γ4 is R-linearly dependent on the product γ0γ1γ2γ3. We get an awkward squaring to (+ − − − +) apparent squaring to (+ − − − −), which is not the case for any Clifford algebra isomorphic to M4(C), denoted C(4) in the classification article.
The confusion possibly comes in because previously we did not have a symbol for the fifth basis element, and were using the an expression in its place. One of the properties of a Clifford algebra of a vector space of dimension n over a field F (often referred to as an n-dimensional Clifford algebra, even though it has a higher dimension over F as a vector space in its own right) is that its dimension is 2n over F. That is to say, a basis of the algebra has 2n F-linearly independent elements. If this is not what we find, we know something went wrong. — Quondum 15:34, 8 November 2013 (UTC)
Weinberg writes that there are 2n independent elements in even spacetime dimensions, and 2n-1 in odd spacetime dimensions. Edit: No, he doesn't exactly. He is talking about antisymmetrized products, and I can't say immeduatly that it is the same thing. That aside, I find it hard to believe that the metric signature (exactly one + or exactly one -) has any real significance for anything we are discussing. YohanN7 (talk) 16:43, 8 November 2013 (UTC)
If we have a set of (linearly independent!) gamma matrices squaring and anticommuting the right way, I'd say we have a Clifford algebra. The article Clifford algebra says that the dimension of the algebra is 2n (no exceptions mentioned), that's true. But the definition of a Clifford algebra doesn't say anything about dimension. There is a problem somewhere. YohanN7 (talk) 17:31, 8 November 2013 (UTC)
Perhaps we could continue the discussion here: User:YohanN7/Gamma matrices. I wrote a program to confirm anticommutation/squaring. The matrices with signature (+ − − −) certainly behave as advertised. YohanN7 (talk) 19:14, 8 November 2013 (UTC)

──────────────────────────────────────────────────────────────────────────────────────────────────── (ec: posting this here, but will continue discussion on your page when I get back to it later.)

As I've said before, linear independence and the commutation relations of a basis for a vector space is not a sufficient condition for producing a full Clifford algebra. If this were the case, we could construct higher-dimensional Clifford algebras with very limited representations. Take for example the quaternions H ≈ Cℓ0,2(R), which is a "two-dimensional" Clifford algebra, with the algebra having 22 = 4 dimensions of the algebra. (If you want a strictly real full matrix representation example, we could use Cℓ1,1(R) ≈ M2(R) instead, which can be used to illustrate the same point.) Now take the three elements i, j, k, which are R-linearly independent, and which satisfy the necessary commutation relations. Yet, no amount of finding products is going to build a Clifford algebra over three dimensions (i.e. is going to construct Cℓ0,3(R) with an algebra dimension of 8), for the simple reason that we have a predefined relation ij = k, which would normally be a new, linearly independent basis element of the algebra in the construction of a Clifford algebra. The relation ij = k is an additional equivalence that collapses the attempted construction back to H. This is exactly the problem that is occurring in our example with gamma matrices. Or, to put it more succinctly, M4(C) (or any subalgebra of it) is not algebra isomorphic with Cℓ1,4(R); it simply cannot be used as a representation of the Clifford algebra Cℓ1,4(R), even though it is a representation of Cℓ4,1(R).

Your difficulty in accepting that switching the signature should have any effect is shared by many people, but it is a fact of Clifford algebras: this switch often produces a completely nonisomorphic algebra. However, in physics we are usually interested only in specific subspaces of the algebra, for example scalars, 1-vectors, bivectors, spinors/rotors, pseudovectors, etc., and the full algebra never comes into play. Because of this, the sign convention (and the fact that we would be working in nonequivalent algebras) has no effect: we always get the same result, regardless of the sign convention. Which will hopefully allay the feeling that this contradicts your intuition. But it does not change my assertion that the source got it wrong. — Quondum 19:24, 8 November 2013 (UTC)

If you two don't mind me cutting in, I just found a very nice pdf: [1] on the Gamma matrices. It looks relevant and useful for RQM articles. Unfortunately, the pdf alone is not reliable as a source, and there could be errors in it, but at least it has a clear presentation style. M∧Ŝc2ħεИτlk 20:45, 8 November 2013 (UTC)



I often emphasize text a little bit to much. When I go back it might not read well. Here in Talk:Symmetry in quantum mechanics the phrase "Somebody knowledgeable of the subject might know" reads like an insult. It was unintentional and the italics is removed. YohanN7 (talk) 03:28, 12 November 2013 (UTC)

No sweat. It seems to have at least attracted an opinion from someone else, who may have more experience in the field. I welcome anyone with the knowledge and experience to talk with some confidence; I am guilty of speaking above my confidence level (even if that is just to express doubt). You at least have a number of years of study in the general field to go on. —Quondum 05:53, 12 November 2013 (UTC)

Nontrivial idempotent[edit]

Hi Quondam,

At zero divisor, I think it is probably safer to say "idempotent e \ne 0,1" instead of "nontrivial idempotent e" since I think that some people might interpret "nontrivial" as meaning nonzero. In any case, the idempotent needs to be named e, for the sake of the equations at the end of the sentence. Anyway, this is a small matter, so I leave it up to you on how to edit it. Ebony Jackson (talk) 22:47, 16 November 2013 (UTC)

Thanks. I've edited it, settling on a reduced restriction (we do not need to exclude 0).
A different point: in the next edit did you not mean "nontrivial ring" where you put "nonzero ring"? (Zero ring has a different meaning, which makes "nonzero ring" really confusing.) —Quondum 23:05, 16 November 2013 (UTC)
You are right; no need to exclude 0! As for "nontrivial ring" vs. "nonzero ring": the standard name (in modern notable textbooks such as Artin, Atiyah-Macdonald, Bourbaki, Hartshorne, Lang, etc.) for the ring with 0=1 is the zero ring. I'm going to try to persuade Wikipedia to reflect this. Ebony Jackson (talk) 23:16, 16 November 2013 (UTC)
I saw your suggestion at Trivial ring. A book search suggests that "trivial ring" occurs in more publications than "zero ring" does. So determination of the "best" (most notable) term is liable to be tricky (or contested). This is well outside my sphere of knowledge.
I see that "nonzero ring" is used (notwithstanding that I find it strange, but I guess one can get used to that). —Quondum 23:27, 16 November 2013 (UTC)

Pauli exclusion principle[edit]

I've added some of the reasons I think the article should state all electrons rather than all electrons in the same atom on the talk page here. I'd be grateful if you reviewed them regarding the wording of the article. Wolfmankurd (talk) 21:14, 24 November 2013 (UTC)

I've added a comment. This is a tricky point. The Pauli exclusion principle applies to all electrons, as you say. But the example should perhaps remain worded (for simplicity) to consider a system consisting of a single atom, in which all the electrons are in the atom's orbitals. As you left it, the wording could suggest that the state with the same quantum numbers in a different atom is the same state; this is not so, because the other atom is in a different place. An electron here is in a different state from an electron there, even if the energy levels are mathematically identical. —Quondum 00:44, 25 November 2013 (UTC)

Group structure and the axiom of choice[edit]

Hi Q!

Thanks for cleaning the article. (I didn't know what to do with it really, had it for a year or so, wrote it most for fun. User:Arthur Rubin had a look at it, said it was correct, but probably didn't belong in main space. Well, instead of tossing it, I submitted, expecting it to take ages for an assessment (it said it would). It took 2 hours;)) YohanN7 (talk) 12:13, 20 December 2013 (UTC)

Sure, just my OCD getting the better of me, and it was already pretty cleanly presented, notwithstanding that it is way over my head. Actually, it has the feel of being such a fundamental but somehow surprising result (theorem, whatever), that it "just belongs", even if the title might need a bit of thought. You've had some mathematicians make tweaks without much comment, which suggests that my uneducated gut feel is not way off. It will be good to have it exposed to mathematician editors in main space to see what comes of it. It might end up merged somewhere, but until then, no harm leaving it as its own article. Perhaps Arthur meant that it does not deserve a main space article to itself, but that does not mean that it does not belong in some form in main space. Perhaps it should be linked under Axiom of choice#Equivalents? —Quondum 15:57, 20 December 2013 (UTC)
It is linked from there.
You are right about "fundamental but somehow surprising result", especially group -> choice. Over at mathworld mathoverflow, where I found most stuff to begin with, they were pretty enthusiastic about it. Follow the link from the articles talk page, it's worth a read. YohanN7 (talk) 16:18, 20 December 2013 (UTC)

Quasigroups and cancellative magmas[edit]

It seems to me that quasigroups and cancellative magmas are in one-to-one correspondence. The cancellative magmas are just the corresponding quasigroups stripped off of right and left division? YohanN7 (talk) 04:45, 9 February 2014 (UTC)

Consider the monoid of non-negative integers with addition. If a + x = b and a + y = b, this implies that x = y, so this is a cancellative magma. There is no element a such that a + 5 = 3, thus it is not a quasigroup. So we have an example of a cancellative magma that is not a quasigroup. We can't even speak of a "corresponding quasigroup" to relate it to. Rather counterintuitive at first; I wrestled with this a bit. —Quondum 05:55, 9 February 2014 (UTC)
Yep, you are right. (See article talk page too, I think i blundered there as well and striked out some). It is possible that we don't have a Latin square, but we have a subset of a Latin square, and that's what is important, so that ax ≠ bx and xa ≠ xb whenever a ≠ b. This fortunately follows directly from the cancellation properties. I'm to quick to draw conclusions, especially since a few hours ago, I had no clue about what a quasigroup or a cancellative magma is... YohanN7 (talk) 06:29, 9 February 2014 (UTC)
I've been doing some rapid learning about them too. Quasigroups I'd seen a while ago, but cancellation is new to me. I've been monkeying around with "groupes" (groups plus the empty groupe) defined as associative quasigroups, so that I can produce a consistent definition of the field with one element. And succeeded, I think. —Quondum 06:44, 9 February 2014 (UTC)


Problematic sentence from the article:

There are models of ZF with sets having properties (1)–(3). One such is obtained from the standard Cohen model with countably many adjoined Cohen reals[1] by passing to a certain symmetric submodel[2].[3]

Ref 1 applies to "standard Cohen model with countably many adjoined Cohen reals", ref 2 to "symmetric submodel", while ref 3 applies to the whole sentence. I argue that this is correct, but it I'm in an edit war with a robot claiming that both of the two last refs should go after the punctuation mark.


YohanN7 (talk) 00:08, 16 February 2014 (UTC)

My sympathies. Bots are implacable. The rules are as ridiculous as those of US punctuation itself, and I suspect that the bot will end up getting affirmed: there is apparently only one punctuation/citation ordering permitted.
I'd suggest either giving up and recognizing that in WP a citation after a period may apply to the preceding phrase or sentence, or re-ordering the sentence so that the citation is not adjacent to a punctuation mark.
It'd be nice if citations allowed highlighting of the text to which it applies, as you'd expect from an electronic medium such as this. Why has no-one thought of this? I think I've seen something like this with templates for tagging text for clarification/inaccuracy. —Quondum 00:36, 16 February 2014 (UTC)
How ammusing, Wikipedia:REFPUNC gives me right, even if it doesn't explicitly give an example where a ref should go immediately before punctuation. Logically, it should be as it stands now in the article. (I emphasized the keywords.) I talked to the bots owner. He's under another impression of course. (I have some sympathy for the poor bastard trying to implement a bot getting this right haha.) YohanN7 (talk) 03:01, 16 February 2014 (UTC)
Gives you the right? Not the greatest wording, but I believe it is intended to mean that the references go after punctuation (other than dashes), regardless. And I'm pretty sure you'll get the same interpretation from others.
Take a gander at {{Citation needed span}}. Perhaps it can be used for ideas on how to create a template that will highlight text when your cursor hovers over it? —Quondum 03:23, 16 February 2014 (UTC)
I'm pretty sure I have the correct interpretation. It says The ref tags should immediately follow the text to which the footnote applies, including any punctuation (see exceptions below), with no intervening space. Note the first part of the sentence, the "the text to which the footnote applies". It need not apply to a complete sentence. Moreover, there is an example,
  • Example: Flightless birds have a reduced keel[10] and smaller wing bones than flying birds of similar size.[11][12],
partly confirming what I claim. Here there is a ref in the middle of the sentence. Logically then, if a ref applies to a keyword that just happens to be the last in the sentence, then it should follow right away, before punctuation. To argue further, placing the footnote in question after the punctuation mark is misleading, whereas placing it before is not.
But don't worry, I'm not going to make a case out of this.
I'll take a look at {{Citation needed span}}, but I haven't done anything with templates, so I doubt that I could come up with something useful. Cheers! YohanN7 (talk) 04:40, 16 February 2014 (UTC)
Refs in the middle of sentences is okay, [11] could apply to only what comes after [10] and [12] to the whole sentence, so that does not disambiguate it. But no matter. —Quondum 04:53, 16 February 2014 (UTC)
Uhm... Refs [11] and [12] are on the same footing in the example. If [12] applies tho the whole sentence, so does [11]. YohanN7
Nope, that's my point. Take this phrase: "He said 'Yes.'" In this example, each of the quoted word and the quoted phrase may or may not have ended a sentence; since the period at the end of an entire sentence terminated by a quote always goes inside the quote marks (in the US), that information is always lost. The same rule applies to references on WP as to quotes: a comma or period always goes before the reference, and the information about what the ref applies to is simply lost and the reader is expected to make the necessary inferences. Abominable, silly and nonsystematic, but that's the way it is. —Quondum 05:34, 16 February 2014 (UTC)
Ok, I believe you, but Wikipedia:REFPUNC is ambiguous. YohanN7 (talk) 05:49, 16 February 2014 (UTC)
Yes, I did say "Not the greatest wording" above, by which I meant that it was grammatically ambiguous, and could do with a copyedit. —Quondum 05:54, 16 February 2014 (UTC)

Etymology of magma[edit]

Hi Quondum, I've just seen that you have removed my addition to the etymology of the term magma in algebra. When we are referring to the etymology of a word we must give the first meaning, its root. I understand that the article want to give the meaning that the Bourdaki group wanted and what they might have in mind, but then the title should not be "Etymology" but something like "Meaning" or "History of the term" etc. Also i don't find it bad to add the origin of the word and coexist with all the other meanings, as many other articles in wikipedia do the same thing. Waiting for you opinion! Happy to discuss it!--Papxr (talk) 11:55, 2 March 2014 (UTC)

Any mention of the etymology in this context (mathematics) would be to aid us think about the use of the word, and why it came to be used in the sense being discussed, and then only if it differs from a readily available etymology, as the longer-term linguistic history (i.e. its evolution before any non-mathematical use) being of interest would be found in a dictionary. Remember that, unlike a dictionary, a WP article is about a concept, not about a word, and distinct meanings will not be discussed in the same article. You say that many other articles in WP do the same thing? Could you give examples? Preferably of mathematical terms, but the same principle applies to all articles. I would find it very strange if the original etymology of words such as ring, group etc. were so much as mentioned in the articles. I'm happy to get involved in a discussion, though at Wikipedia Talk:Wikiproject Mathematics you'd get a broader cross-section of opinions. —Quondum 21:20, 2 March 2014 (UTC)

Hi again, thanks for your reply. Well, an encyclopedia like WP is more than a dictionary. So it has all of a dictionary and much more. As I said before I got your point but I think the term "etymology" is misleading and I believe it should be changed. As far as the examples are concerned, take the word school --> Etymology: The word school derives from Greek σχολή (scholē), originally meaning "leisure" and also "that in which leisure is employed", but later "a group to whom lectures were given, school". So we have the original meaning which is different from the current use. Also you can take a look at the etymology of the word "Mathematics" in WP. In addition, many terms that didn't exist in the antiquity and formed later come from greek words. For example take the term Homomorphism. We surely will not say that the one who introduced the term was thinking about the meaning of the word morph in english which means shape but we just say that morph comes from ancient greek μορφή (morphe) meaning "shape"(see homomorphism). So in the same way we shall say that magma comes from greek and its metaphorical meaning is a mixture of various things. Lastly (that's beside the point), do you know why they had chosen the name "Burbaki" which is a Greek name? Is there any "tale" or it was completely random. Happy again to have this conversation! --Papxr (talk) 01:02, 3 March 2014 (UTC)

I disagree on "it has all of a dictionary and much more". It is an explicit principle to avoid the functions of a dictionary, although this is not always implemented strictly. The use of the term "etymology" might be better restricted to a dictionary, and yes, I think it should not be used in WP except where it is of historical relevance to the subject, and in general a paragraph under History could be devoted to it, not a top-level section on itself. Your example of the etymology of school has some relevance to the history of the concept, but I'll stick with what I've just said, and I don't feel that even there a separate section is appropriate. In the article Mathematics, it is a subsection as I'd expect. In both cases, the aspects of the etymology mentioned relate to the origin of the word in the meaning of the article, at least as a progression; this does not appear to apply to the "magma" and numerous other mathematical terms. Further, in your examples, the topics are fairly broad, so the articles are descriptive rather than acting as references for the detail of the topic, and can include more of general interest and in particular history. You are presumably aware that mathematics co-opts words for various uses, a bit like we use variable names: the general meaning of the word and its mathematical use are often only tenuously connected, so to draw attention to its other uses prior to it having been (nearly arbitrarily) co-opted is simply a distraction. There are instances where compounds are formed, so homo-, auto-, morphism etc. have specific meanings, which are worth knowing about, as this helps to quickly understand new compounds that incorporate from the same roots and modifiers. Again, origins are to be considered in the historical context. (As an aside, one of my pet hates is the way the same word may be used in several sub-disciplines of mathematics, but often with confusingly overlapping but distinct meaning; what's worse is that there will be many arguments precipitated where this is not recognized.)
I have no idea why Nicolas Bourbaki was chosen as a pseudonym for the group (I'm not well-read, so I don't know whether this is even discussed in the literature); the reason for using a pseudonym at all would be more interesting than the inspiration for the choice of name. Perhaps the members put names into a hat and that was drawn? —Quondum 01:52, 3 March 2014 (UTC)

It is interesting to note that the removed passage seems to be undisputable facts (unsourced), and what remains is disputable speculation (It is likely that ..., equally unsourced). Sorry Q, couldn't resist. YohanN7 (talk) 03:36, 3 March 2014 (UTC)

Yeah, I was tempted to remove that as well, and do think that it should go (it is even phrased as speculation); I just felt less confident removing that as it would have some relevance. However, now that you point it out... —Quondum 05:17, 3 March 2014 (UTC)

Well I was a little bit exaggerating when I said "it has all of a dictionary and much more"... wanted to say that it has same common ground like the basics of an etymology and so it's not completely "foreign". I agree with what you said and I am happy that we came to a conclusion and deposited our thoughts! --Papxr (talk) 12:23, 3 March 2014 (UTC)

Regarding that reference desk question just now, as we know, magmas have peculiar implications. The presence of a cancellative one implies that the underlying set is wellorderable, the presence of one on every set implies the axiom of choice. I think that is kind of cute (and remarkable). YohanN7 (talk) 22:22, 11 April 2014 (UTC)
But no, that is only partly correct. The presence of a magma doesn't mean the set is wellorderable. When I thought about what I wrote in the post above, I realized that if it is true, then I, in passing, had solved the second part of Hilbert's first problem (can the reals be wellordered?, CH was the first part). :D YohanN7 (talk) 22:33, 11 April 2014 (UTC)
Yeah, this is dabbling in things that stretch my mind. I'm liable to tie myself in knots on the subtleties of this. —Quondum 04:26, 12 April 2014 (UTC)

Interval (mathematics)#Dyadic intervals[edit]

Thank you for your recent correction! While I do not strongly object the correction itself, you might be interested to know that the term "remarkable" is used in mathematical literature to denote exactly the properties that are peculiar and deserve being noticed. Whether such properties are "useful" is a separate issue. The term "useful" is more applicable in the context of a theorem proof, where certain properties can assist with the task at hand. Without a context of application, any peculiarity is just "remarkable", not "useful". Nyq (talk) 15:25, 31 March 2014 (UTC)

Thanks for the info. Terminology usage is always a challenge as it varies from context to context. Here it is doubly challenging, since an encyclopaedia does not qualify (IMO) as "mathematical literature" due to the different audience: the reader cannot be assumed to be familiar with quirky usage. To avoid potential subtleties of this nature, I've rephrased it again to avoid the term "useful" as well, hopefully without detracting from the reference value in any way. —Quondum 16:33, 31 March 2014 (UTC)


I noticed on a few occasions you mention your access to references may be limited. In the following I'm probably stating the obvious, and you're more than welcome to delete this thread, but in case it's useful...

Have you access to books on mathematics by Dover Publications? Many (not all) great works are published by them at a very affordable price (approx equivalent of £ 10 = US $ 16.64), about the same as a Schaum's outlines book. Some examples include

(Unfortunately not all the classics seem to be available, e.g. Gibb's vector analysis, Schouten and Courant's treatise on Ricci calculus, Courant's Calculus and analysis, etc. which tend to get published in the high quality and ridiculously expensive Springer and Wiley or so on).

Two often good places to buy books online (at least in my experience) are abebooks or waterstones, rather than amazon, ebay, etc.

Best, M∧Ŝc2ħεИτlk 21:00, 3 April 2014 (UTC)

Thank you – this is very useful. These are prices that I can live with. I should start a list of "intended purchases". So far, I've only looked at the new book market, which is really silly of me. —Quondum 23:50, 3 April 2014 (UTC)
Willard's General Topology - much recommended. YohanN7 (talk) 02:24, 4 April 2014 (UTC)
Cool. You may also like:
  • Kreyszig's Differential geometry,
  • Flanders' Differential forms with application to the physical sciences,
  • Rucker's Geometry, relativity, and the fourth dimension, (qualitative review like Penrose rather than a textbook)
  • Pauli's Theory of relativity, (historically a standard reference, today the real drawback is his use of ict..., anyway covers SR and GR, including relativistic EM, some fluid mechanics, and thermodynamics),
  • Tolman's Relativity, thermodynamics, and cosmology (similar to Pauli but more detail and better),
  • March, Young, Sampanthar's The many body problem in quantum mechanics (from the 1960s, but good for historical perspective).
I have the books listed here (plus a few more on waves, fluid mechanics, and thermodynamics), but out of the ones above, only Cartan's theory of spinors. M∧Ŝc2ħεИτlk 12:29, 4 April 2014 (UTC)
Others I don't have (and hope to get soon), and which you may be interested in, include:
  • Weyl's Theory of groups and quantum mechanics,
  • Riesz and Nagy's Functional analysis,
  • Knopp's Theory of functions,
  • Silverman's
    • Introductory Complex Analysis,
    • Essential Calculus with Applications,
    • Complex Analysis with Applications,
  • Coxeter's Regular polytopes
I'll stop here. M∧Ŝc2ħεИτlk 12:55, 4 April 2014 (UTC)

Okay, I'm compiling a list below. —Quondum 04:56, 4 April 2014 (UTC)


  • Sergei L. Sobolev, Partial Differential Equations of Mathematical Physics: $19.95
  • James Clark Maxwell, A Treatise on Electricity and Magnetism, Vol. 1: $19.95
  • James Clark Maxwell, A Treatise on Electricity and Magnetism, Vol. 2: $22.95
  • Élie Cartan, The Theory of Spinors: $16.45 (paperback + ebook)
  • Tullio Levi-Civita, The Absolute Differential Calculus (Calculus of Tensors): $30.75 (paperback + ebook)
  • Stephen Willard, General Topology: $24.95
  • — other books (I haven't transcribed all those on tensors above yet)


  • — to be compiled


Hi, what do you meant with "..without more context so early in the lead, this could misleadingly imply a more direct connection than is the case: it depends on what else is kept constant" -what context are you missing exactly? Prokaryotes (talk) 17:12, 6 April 2014 (UTC)

The bald statement "The entropy of a system increases or decreases with temperature" is confusing and apparently not generally true. Consider an ideal gas as an adiabatic process: it is an isentropic process, yet its temperature increases and decreases. (I'm no expert, but I cannot see how one could escape this conclusion.) —Quondum 17:28, 6 April 2014 (UTC)
I changed the part to "Systems which are not isolated may decrease in entropy.". Prokaryotes (talk) 18:11, 6 April 2014 (UTC)
Please keep in mind that the lead of an article is for introducing the concept, not for explaining details. —Quondum 19:34, 6 April 2014 (UTC)

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