User talk:Prime.mover

Future Proofs Guidance
I am just about ready to tackle the theorem of Transfinite Induction and, a little later, the slightly more difficult proof of Transfinite Recursion. Neither should be tough, because the proof has already been worked out in Takeuti/Zaring. Simultaneously, I will be proving the Peano Axioms for a subset of the ordinals. Notice that three of these are theorems which have multiple ways of stating them. For example, transfinite induction could be stated as $\forall y \in \operatorname{On}: ( \forall x \in y \phi ( x ) \implies \phi ( y ) ) \implies \forall y \in \operatorname{On}: \phi ( y )$ or by giving a specific set of conditions for a base case, successor case, and limit case. How should I go about this theorem -- I am assuming I should create just one page and stuff all the theorems on there. It would be nice if I could have some guidance, considering that I am still not too familiar with all the details of proofwiki... Andrew Salmon 00:42, 29 November 2011 (CST)


 * I apologise, but I have been told I do not have the authority to make decisions about the direction of ProofWiki. I have been reminded that ProofWiki is a wiki, and not "just prime.mover's personal blog". I am no longer allowed to answer questions as to what should or should not be in ProofWiki. This is completely up to the individual contributors, and editing pages of others and telling others what pages to put up should not be done by me. --prime mover 00:47, 29 November 2011 (CST)


 * To respond on-topic: I think you'd best put it up there with a Template:Tidy tag so that we can adapt it and guide you from there. As to the direction of the site, I think coherence and uniformity are very important for a pleasant reading experience. This is what the house style was developed for, and anything that does not conform to it should be put in appropriate form. At all times, when editing work of non-experienced authors, it is probably a good habit to drop a note on their talk page explaining where they violated the house style and how this can be prevented in the future. And I think you, prime.mover, are perfectly entitled to perform such corrections, despite what reactions have come on that part in the past. --Lord_Farin 11:55, 29 November 2011 (CST)


 * What he said. --Joe (talk) 12:14, 29 November 2011 (CST)


 * Ultimately comes down to, if we advertise this as a wiki, the rules of it being a wiki ought to be adhered to. And I'm not sure I'm contributing positively. Best if I stand down and let those who are actually legally qualified to take over. --prime mover 14:33, 29 November 2011 (CST)
 * Well, the rules of this wiki are whatever we make them to be. Saying that I believe that any rule can be changed, so long as a majority of active users think it's best. As our body of content grows, I think that a consistant (as possible) style of presenting said body will be essential. Right now I think that Prime.mover has the best idea of what the general style should be. If anyone has a problem with a particular part of the style, then that should be addressed by a vote (or something) from the current active users. People who want to post shouldn't be deterred by the house style and should post whatever they can, however they can. I for one dont' know that current style, so others shouldn't feel discouraged to  post. Leave it some one else to fix up later. Right now we are still in a growing stage and need as much content as possible. Most importantly, this is a wiki that allows anyone to edit and so to quote this statement again:  If you do not want your writing to be edited mercilessly and redistributed at will, then do not submit it here. Just so long as you are making changes to improve the content, not edit the style to fit your own personal preferences. I hope this made sense, was on topic and didn't have too many gramatical errors. --Joe (talk) 15:25, 29 November 2011 (CST)
 * It wasn't so much that contributions are discouraged (i dont tink ive discouraged much) so much as objection is made to tinkering. If a user comes in with a completely different general style of presentation (possbly becase of differnt cultures proidicung books with different language) then do we amend it (e.g. change $\mathcal T$ to $\tau$ to denote a topology)? Quetionable, but tempting as lots of oher topology pages have $\tau$. Otoh if a different symbol is used (unexplained) on the grounds that "everyone knows what that symbol means" (except they dont' always, specally if its me) you either say "where :**--! means the boggityboo of humdingersland" or you convert it. Then you got where a mathematical truth itself is challenged (e.g. the symbol $\infty$ is at best a fiction whose meaning is interpreted by limits, but someone insists that it is not necessary to point this out because "all modern mathematicians accept this") and that's where I got to stand down because I don't know what I'm talking about, all I know is what i read. --prime mover 15:48, 29 November 2011 (CST)

Under the assumption of valid references (that is, books and the like) I think it would be most consistent to amend indeed with the 'where' clause, and also make sure the definition pages cover possible differences in notation. In case of very obscure notation or possible confusion, retracting notation is justified, but again I stipulate all occurring notation for a concept that is encountered so far should be mentioned on the definition page. In general, it might help if we expanded on the Help namespace; it seems somewhat... insufficient. --Lord_Farin 15:56, 29 November 2011 (CST)
 * I was actually going to ask for a ruling on $\infty$ on PW because it's used haphazardly in many books. I don't really care what all modern mathematicians know or don't know, isn't PW a way for people who don't know something yet to learn it? --GFauxPas 16:02, 29 November 2011 (CST)
 * In my opinion, the lemniscate $\infty$ is a symbol which needs strict definition in every context. Otherwise its use is just too ambiguous. --Lord_Farin 16:08, 29 November 2011 (CST)
 * LF: I know, i got to do that thing but I get tired. I have a plan to put a section in (but I need to guess up a title) for concepts which have more than one notation but this site favours one over the others. I do what you say, if there's more than one notation or name for a concept (or even if there's more than one definition) we put them all up, say "alternative symbol is this". I keep offering the suggestion that it's a good idea to be consistent but if the message keeps coming back that another person prefers to use another notation so you can't win and it's difficult not to call them a prick.
 * This subject got raised at Nijmegen and I was asked what I did about it so I said, it doesn't matter because nobody stays around for longer than a few weeks and once they're gone I change their pages to house style and nobody's the wiser. Everybody laughed.
 * Mind there's still pages from 3 years ago that I never got round to tidying up because they are just so colossally mungy having been vomited from twitipedia in all their 40kBness.
 * GFP: That's what I would have thought. The page on infinity what I put together is what I know. Because there's more to infinity than just the symbol $\infty$ (countable, uncountable, inaccessible, bloody knows whatible) my view is that $\infty$ should only really be used as a shorthand for "tending to the limit" of "greater than any actual real number", with the added proviso that the "extended real number line" and "extended complex plane" have a point / points "at infinity" BUT THEY ARE ONLY FICTIONS. Trouble is, applied mathematicians and other sorts of people like that insist on treating it as an accepted concept that can be treated as a number like any other, and because they tend only to be doing stuff with real-world applications they can get away with it. But in pure maths you can't. --prime mover 16:15, 29 November 2011 (CST)

A note on arrays
Hello, could you please read my notes here and let me know what you think, and whether this sounds as original or it has been published before in Maths literature. I am going to study LaTeX and how to properly write on this site very soon, for now, please accept my New Year gift. Happy 2012, cheers. --Dr Who 16:46, 31 December 2011 (CST)
 * You might want to look at : Chapter $2.2.6$ "Arrays and Orthogonal Lists". (This is considered by many to be the seminal work on computer architecture from a programming perspective.) It describes the technique of expressing a 2-dimensional (or more) object as a one-dimensional one (in this case, an ordered tuple of ordered tuples).
 * In my erstwhile career as a code-monkey I did quite a lot of array management in this way in Java, by using Lists of Lists (or Maps of Maps, and whatever). The fun game is in converting the 2-d $i, j$ element location $a_{ij}$ into the index into the one-dimensional array location in the sequence of ordered tuples.
 * Have fun ... --prime mover 16:56, 31 December 2011 (CST)
 * Thank you for everything :) --Dr Who 19:07, 31 December 2011 (CST)

Subcategory template
I tried to add the Category:Definitions/Hilbert Spaces category to Category:Definitions/Vector Spaces today, but found the template Template:Subcategory not being up to it. The Category:Definitions/Vector Spaces circumvented this by a rather ugly trick using some code present in Template:Subcategory. It might be useful to extend the functionality of Template:Subcategory to deal with multiple supercategories. --Lord_Farin 06:25, 16 January 2012 (EST)


 * Yeah probably, there are instances of other definition categories with multiple supercategories - I see you used the same technique (also see e.g. Definitions/Topology). Yes it needs to be expanded, but I never got round to bending a brain cell to it. --prime mover 15:58, 16 January 2012 (EST)

Tarski's stuff
You said to remind you to make a template Journals/The Bulletin of Symbolic Logic. In case I forget, please remind me to remind you.

Also, I'm almost done putting up all the Tarski axioms. As such, we will be able to resolve:

a) Whether the proof for Real Number Line holds in ZFC without anything in Category:Axioms/Tarski's Axioms

b) Whether Tarski's definition of singleton:

$S \ \text{is a singleton} := \left({S \ne \varnothing}\right) \land \left({a \in S \land b \in S \implies a = b}\right)$

is equivalent to the def'n PW already has --GFauxPas 07:37, 26 January 2012 (EST)


 * Thx, may be this weekend but I can't promise. --prime mover 07:45, 26 January 2012 (EST)


 * PM, can I please have a template along the lines of what I put on the "also see" at Axiom:Euclid%27s_Fifth_Postulate? I'm likely going to need it 24 times, once for each axiom. Or do you have a better suggestion than a template? cf. also Axiom:Uniqueness of Triangle Construction, what's the best way to do that?

Maybe a template in general "this is a thing in this system. See that, an analogue of this in some other system". Kind of like the about template. Or is it exactly the about template I'm asking for? --GFauxPas 03:58, 29 January 2012 (EST)
 * Not the "about" template, that's for disambiguation. Leave it with me, I'll take a look. Only just got up. --prime mover 04:00, 29 January 2012 (EST)
 * Done. --prime mover 05:17, 29 January 2012 (EST)

Limit Theorems Real Numbers
For the reals, there appear to be no results like $a_n < b_n \implies \lim_{n\to\infty} a_n \le \lim_{n\to\infty} b_n$ and the like (or I couldn't find them). Is it me or are there indeed some holes in elementary real analysis? I'm using it in the final part of the gargantuan (IMO) Hilbert Space Direct Sum is Hilbert Space. --Lord_Farin 09:07, 26 January 2012 (EST)
 * There may indeed be some holes. --prime mover 10:45, 26 January 2012 (EST)


 * The one I mentioned is tackled (including proof). Might be an idea to make some redirects to Lower and Upper Bounds for Sequences, to make it easier to find it. --Lord_Farin 11:17, 26 January 2012 (EST)


 * Okay ... not quite sure where you recommend the redirects from, unless that's what you already added. Feel free to take it on. --prime mover 12:20, 26 January 2012 (EST)

Bretschneider's Formula
Hello, I made some changes in Bretschneider's Formula. Could you please check if it is OK now or if not, tell me what other changes should be done? Regards, – Wooden Goat 14:35, 4 February 2012 (EST)


 * Plenty needed to bring it up to house style. Don't worry, others will take care of it. The "tidy" tag is just so we can keep tags on what needs work. It's a good result. --prime mover 17:33, 4 February 2012 (EST)

I answered on my talk page. – Wooden Goat 05:23, 5 February 2012 (EST)

Intuitionist/Classical thought
PM, I'd like some more clarification on what you said:


 * Your last comment highlights the different philosophical schools of thought between the intuitionist and classical schools of mathematics. Intuitionists do not accept the existence of any object they can not construct or demonstrate a means of construction of. If something is proven not to be false, an intuitionist would still not accept that it is true. For my own part, I am prepared to accept the existence of classes, if there is an axiom system that declares that existence. --prime mover

Can you elaborate? If I say:

Let the universe of discourse be the Physical Universe.

The loch ness monster is a sea creature inhabiting the loch ness lake.

Now, does that imply $\exists x : x \ \text {is the loch ness monster}$? Does it matter? Does my personal belief in the existence or not of the loch ness monster make this definition more or less legitimate? --GFauxPas 10:48, 10 February 2012 (EST)


 * My advice is to google for it and read around the subject. Or perhaps look for "Intuitionist" on this website.


 * I also point you towards There exist irrational a and b such that a^b is rational, the second proof of which relies upon exactly such reasoning as intuitionist philosophy denies. We do not need to know whether $\sqrt 2^{\sqrt 2}$ is rational or irrational, says the classical mathematician. It is either rational or it is not. On the other hand, the intuitionist will say: because I have not established whether it is rational or irrational, it is unjustifiable to make any deductions about its behaviour based on a belief that it must be either one or the other.
 * My personal view is that the intuitionist viewpoint as such is little more than an interesting curiosity, a quaint dead-end backwater, but within it lie the seeds of something worthwhile: it opens the door to tristate and fuzzy logic. On the other hand, all mathematics is at root an arbitrary result of the axioms which we choose to base it on. All is arbitrary, but with an adequate selection of axioms (I hang my hat on Aristotelian logic and ZFC) you can create something which is more-or-less useful (i.e. a fairly accurate model of the "real world"). --prime mover 13:13, 10 February 2012 (EST)

Oh, and non-existence of the Loch Ness Monster has not been "proven to be false". That means, intuition or not, its existence has not been "proven to be true". If you had direct evidence of its existence (footprints, fishy stinking breath and a "honk!" next to your ear, etc.) but not actually seen it, then maybe intuitionists would still balk at it. But no evidence for LNM has been found (a lot of plausible hoaxes, yes), so the question does not arise. --prime mover 13:17, 10 February 2012 (EST)

Corollaries
Is the header style demonstrated at Bijection iff Left and Right Inverse/Proof of Corollary (which might as well be named 'Bijection iff Left and Right Inverse/Corollary' instead, for brevity) to be enforced as house style?


 * Don't know yet. It just made sense to me at the time. --prime mover 01:17, 2 March 2012 (EST)

Tons of other questions and loads of work appear in my mind pertaining the extension I am developing (though it is in rudimentary stage atm) and the Help:Editing page which is seriously getting outdated and needs to be expanded. But that's not really for now. --Lord_Farin 18:22, 1 March 2012 (EST)

Differentiation of Power Series
What I meant was that it is to be shown that the radius of convergence is left unchanged by formal termwise differentiation (note the 'formal'). This subsequently implies that the desired termwise differentiation is allowed. --Lord_Farin 18:17, 7 March 2012 (EST)


 * The question merits being raised on the talk page. I would need to do research, I'm not good in this area. --prime mover 01:14, 8 March 2012 (EST)

Complex analysis
I have marked all edits (by abcxyz) I have looked over as 'patrolled'. Thus, you can restrict your reviewing to the ones with the exclamation marks. --Lord_Farin 10:58, 9 March 2012 (EST)


 * Cool. Gradually working through. I'm on domestic duties this weekend so I may not be able to spend much time here ... will see. --prime mover 14:14, 9 March 2012 (EST)

Real-valued function
Category:Measure Theory is littered with the following phrase:


 * 'Let $f: \mathcal A \to \overline{\R}$ be a real-valued function.'

However, this immediately makes the extended reals superfluous as a real-valued function (you wouldn't guess it) takes real values; not values in the extended reals. For the definition of measure, I have already changed it, but I thought I should ask about your intentions with the phrase first before obliterating it from PW. See also Definition talk:Measure (Measure Theory). --Lord_Farin 08:16, 16 March 2012 (EDT)


 * Good call. We could either raise the definition "Extended Real-Valued Function" and similar for complex nos, or dispose of the term "real-valued function" in this case completely. I'm undecided. I'm not sure about the usefulness of "Real-Valued Function" anyway, as "$\to \R$" says all that is needed, but some users seem to like it. And as it's in the literature, it makes sense to document it. --prime mover 08:42, 16 March 2012 (EDT)


 * I'd say dispose, because it would be unclear if the function or the reals are extended. It's all about the invisible brackets ;). Also, I've never encountered the term 'extended real-valued function'; it a tad long for what it's expressing. On another note, it might be interesting to investigate and document the extended reals as a two-point compactification of $\R$ (not sure if it isn't already up) in a similar way the Riemann sphere is obtained from $\C$. --Lord_Farin 09:24, 16 March 2012 (EDT)


 * Two-point compactification: I know exactly what that is without having heard of it before and it just feels neat. So feel free to go to that thing - and expunge all those pointless "real-valued function" invocations when you find them. --prime mover 15:28, 16 March 2012 (EDT)


 * I have deleted the occurrences of 'real-valued function' in the measure theory section where this terminology was erroneous. --Lord_Farin 05:22, 17 March 2012 (EDT)