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I am Boris Tsirelson, an experienced mathematician and less experienced wikipedian...

Articles I've written (started)[edit]

and also

Articles I've contributed to[edit]

On different wikis (if you like)[edit]

  • Entanglement (physics) [2] = [3]
  • Theory (mathematics) [4] = [5]
  • Plane (geometry) [6] = [7]
  • Probability space [8] = [9]
  • Proof assistant [10] = [11]
  • Measurable space [12]
  • Measure space [13]
  • Standard Borel space [14]
  • Measure algebra (measure theory) [15]

Quantum mechanics is not a physical theory[edit]

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.

— Scott Aaronson, "Quantum computing since Democritus", Cambridge 2013 (p. 110).

Oddities of mathematical terminology[edit]

A linguist would be shocked to learn that if a set is not closed this does not mean that it is open, or again that "E is dense in E" does not mean the same thing as "E is dense in itself".[1]

A set, however, is not a door: it can be neither open or closed, and it can be both open and closed. (Examples?)[2]

Like the alligator pear that is neither an alligator nor a pear and the biologist’s white ant that is neither white nor an ant, the probabilist’s random variable is neither random nor a variable.[3] (Alligator pear = avocado; white ant = termite.)

"Finite measure" is a measure, but "signed measure", "vector measure" and "finitely additive measure" are (generally) not measures. On the other hand, every measure is both a signed measure and a finitely additive measure. That is, "signed" means here "not necessarily unsigned", "vector" means "not necessarily scalar", and "finitely additive" means "not necessarily countably additive". See also Measure (mathematics)#Generalizations.

Unbounded operator on X means "not necessarily bounded operator, not necessarily defined on the whole X".

Dirac delta function is not a function; rather, it is a generalized function.

Constant random variable satisfies the definition of random variable even though it does not appear random in the everyday sense of the word.

Every differential equation is a stochastic differential equation but most stochastic differential equations are not differential equations.

In mathematics, a “red herring” need not, in general, be either red or a herring.[4]

  1. ^ Littlewood, A Mathematician's Miscellany, Chapter 3 "Cross-purposes...", §14 "Verbalities". See also Dense set, Dense-in-itself.
  2. ^ Shurman, "Multivariable calculus" [1], Sect. 5.1.
  3. ^ S. Goldberg “Probability: an introduction”, Dower 1986, p. 160.
  4. ^ nLab; visit that page for more items.

See also "Nonstandard adjectives in mathematics" by Mark Dominus.

Choice versus randomness[edit]

Theorems follow from axioms (and definitions); axioms formalize our intuition. Nowadays, axioms of mathematics are axioms of set theory. Which intuitive idea is thus formalized?

The idea is, infinite mind (called also "ideal thinker", "ideal mathematician", "omnipotent mathematician", "infinite intelligence" etc.) Every mathematical statement is either true or false (even if neither follows from our poor axioms), since the infinite mind can check all special cases at once, no matter how many cases, — finitely many, countably many or even uncountably many. I prefer to say "infinite machine", but it is still the same idea.

We humans are able to write down all subsets of a 10-element set, surely not of a 1000-element set. Nevertheless we human mathematicians are pretty sure that the idea of a finite machine (or mind), able to write down not only 21000 but also 221000 objects, does not lead to any contradiction, in other words, is consistent.

About the infinite machine (or mind) we are less sure. Otherwise Hilbert would not ask for an arithmetical proof of consistency of the set theory, and Goedel would not discover that arithmetic cannot prove even its own consistency.

The infinite machine is able to form (and store in its infinite memory) not only a list of all subsets of the real line, but also a list of pairs (A.x) where A runs over all nonempty sets of reals, and x is an element of A chosen by the machine. This ability is the idea of the famous choice axiom. We cannot instruct the machine how to choose, but still, it can choose. Free will? Not necessarily; maybe the internal representation (of these sets, and whatever) makes it possible.

Being a probabilist, I wonder, what about a random generator? Can the infinite machine produce an infinite array of random bits? A countably infinite array would satisfy me. Alas, this is impossible!

Before proving this negative answer, let me comment on it. For me, our idea of the infinite machine is thus questionable. We want to endow the machine with all our basic abilities, extended to the infinite; but we cannot. Either the choice ability, or the random generator, but not both. The set theory stipulates the choice and sacrifices the randomness. This fact bothers me.

Here is why the choice and the randomness cannot coexist[edit]

Let the infinite machine do the following.

It considers all infinite sequences of bits (not a harder job than all reals...) and groups them into equivalence classes; here two sequences are called equivalent if they differ only in finitely many positions (that is, xn=yn for all n large enough). (Only a continuum of equivalence classes, — much less than all sets of reals...)

It chooses (and stores in memory) one sequence in each equivalence class, — call it the representative of this class.

Now it generates at random an infinite sequence of bits, finds its equivalence class, picks up the representative of this class, and compares the random sequence and the representative via the bit-wise XOR ("exclusive OR") operation. It gets a random element of the zero equivalence class (a sequence with only finitely many "one" bits).

Mind it: a random element of a countably infinite set! Distributed uniformly, that is, with equal probabilities for all elements! This is incompatible with any reasonable probability theory for many reasons. Here is my favorite reason. If X and Y are two independent, uniformly distributed random integers, then XY with probability 1, since for every y we have Xy with probability 1. But similarly, YX with probability 1, — a contradiction.

This is why the choice and the randomness cannot coexist.

Why not a textbook[edit]

I observe repeatedly that good faith editors, striving to make Wikipedia more useful for students, conflict with WP:NOTTEXTBOOK. Naturally, I ask myself: but really, why not textbook? Here is my answer. There is only one Wikipedia, and a lot of textbooks (on the same topic, I mean). Why not a single optimal textbook? Just because a textbook cannot be universally optimal. Different students need different textbooks. Quite different, indeed! As long as content forking is disallowed, WP cannot provide textbook(s). No textbook on tensors could satisfy mathematicians, physicists, engineers and biologists. (Can an encyclopedic article satisfy them all? A good question. Maybe not. But for textbooks the problem is much harder.)

Notes for myself[edit]


Stochastic cellular automaton

Gravitational wave observation

User talk:Sheila Nirenberg

Red herring principle

my sandbox + User:Tsirel/Frustrating discussions + Special:PrefixIndex/User:Tsirel/

Expert involvement 2011 survey

Special:OldReviewedPages + Who Writes Wikipedia?

Wikipedia:WikiProject Wikipedia reliability + User:History2007/Content protection

Lectures on Probability, Statistics and Econometrics

User:Mathbot/Changes to mathlists + List of probability topics + Lists of mathematics topics + List of integration and measure theory topics + List of geometry topics + Wikipedia:WikiProject Probability

Wikipedia:WikiProject Mathematics/missing mathematicians

Template:Disputeabout + Template:Citizendium + Gamma function#External links

Template:Onlinesource + Wikipedia:Wikipedia as an academic source + Wikipedia:Public domain image resources + Interface metaphor

User:CataBotTsirel + User talk:CataBotTsirel + Special:Contributions/CataBotTsirel + Catalog of articles in probability theory + Talk:Catalog of articles in probability theory + Wikipedia:Editnotice

User:Citation bot/use

Wikipedia:Manual of Style (mathematics) + Wikipedia:WikiProject Mathematics/Conventions + Template:Ref + Wikipedia:Citing sources + Cite.php (about "ref") + WP:Footnotes#Naming a ref tag so it can be used more than once + Wikipedia_talk:Footnotes#Mark-up would be better than encouraging people to remove reference information + Wikipedia:Line break handling + Wikipedia:How to edit a page + Help:Formula + Wikipedia:WikiProject Mathematics/Typography + User:KSmrq/Chars + Wikipedia:Footnotes + visits counter (thanks to User:Henrik) + encyclopaedia of math + Wikipedia:database download + Wikipedia:Pages needing attention/Mathematics + Wikipedia:Expert editors + Wikipedia:Expert retention + Wikipedia:Flagged revisions + Wikipedia:Template messages + Template:Main + Wikipedia:Text editor support + User:Cacycle/wikEd + User:Cacycle/wikEd help + Wikipedia:Tools/Editing tools + Template:TOClimit + Wikipedia talk:WikiProject Mathematics/Proofs + Wikipedia talk:WikiProject Mathematics/Proofs/Archive 1 + Wikipedia talk:WikiProject Mathematics/Proofs/Archive 2 + Wikipedia_talk:WikiProject_Mathematics/Archive_46#Proofs + Wikipedia_talk:WikiProject_Mathematics/Archive_46#Connected_space/Proofs + The pernicious influence of mathematics on science

Semimartingale + Itō_calculus + Kolmogorov extension theorem + Wasserstein metric + Transportation theory + Coupling (probability) + Product measure + Disintegration theorem + Regular conditional probability + There is no infinite-dimensional Lebesgue measure

Dvoretzky's theorem + Quotient of subspace theorem + Duality (mathematics)

Surreal number + Stone–von Neumann theorem + Heisenberg group + Infinite-dimensional holomorphy + Self-adjoint operator + Extensions of symmetric operators + Smooth infinitesimal analysis + Where Mathematics Comes From + Philosophy of mathematics + Mizar system + Frank Morgan (mathematician) + Entropy in thermodynamics and information theory + Entropy#Entropy and Information theory + Linear response function + Green–Kubo relations + Fluctuation theorem + Dispersion relation + Fluctuation dissipation theorem + Dynamical billiards + Schwarzschild coordinates + Quantum electrodynamics + Precision tests of QED + Kepler problem in general relativity

Alexander horned sphere + Antoine's necklace + Long line (topology) + Garden of Eden (cellular automaton) + Ham sandwich theorem + Knot group + Knot (mathematics) + Gordon-Luecke theorem + Boy's surface

Distribution (mathematics) + Sensor + Generalized_function

Fundamental concepts of geometry + Primitive notion

Šahbazov + Miller + Stafford

User:Sullivan.t.j + User:Roboquant + User:R.e.b. + User:Gala.martin + User:GaborPete + User:Jmath666 + User:hirak 99 + User:Grubb257 + User:Michael Hardy + User:Trovatore + User:Thenub314 + User:Ptrf + User:3mta3 + User:Flavio Guitian + User:McKay + User:Jakob.scholbach + User:Plclark + User:Arthur Rubin + User:John Baez + User:Dradler + User:C_S + User:Hans Adler + User:JonAWellner + User:Linas + User:Pym1507 + User talk:DoronZeilberger + User_talk:Pwm86 + User:OdedSchramm

Wikipedia:Wikipedians with articles

Saint Petersburg Lyceum 239 + Wikipedia:The Core Contest + MediaWiki + Millennium Prize Problems + Bogdanov Affair


  • — <math> (\Omega,\mathcal{F},P) \,</math>, or (Ω, ℱ, ''P'')
  • — <math> \mathbb{R} \,</math> or
  • n = −1 — ''n'' = &minus;1
  • E ( P ( A | X ) ) = P ( A ).{{nowrap begin}}E ( P ( ''A'' | ''X'' ) ) = P ( ''A'' ).{{nowrap end}}
  • f ′ — f&nbsp;′
  • ∈ ∉ ⊆ ⊇ ∅ ± ∞ ℓ
  • {{disputeabout|'''inaccurate formulation'''|Not any basis is orthogonal|date=April 2009}}
  • {{Fact|reason=I know this is true, but where is it written?|date=April 2009}}

Table of mathematical symbols


∫ ∑ ∏ √ − ± ∞
≈ ∝ ≡ ≠ ≤ ≥
× · ÷ ∂ ′ ″
∇ ‰ ° ∴ ℵ ø
∈ ∉ ∩ ∪ ⊂ ⊃ ⊆ ⊇
¬ ∧ ∨ ∃ ∀
⇒ ⇐ ⇓ ⇑ ⇔
→ ↓ ↑ ← ↔

&int; &sum; &prod; &radic; &minus; &plusmn; &infin;
&asymp; &prop; &equiv; &ne; &le; &ge;
&times; &middot; &divide; &part; &prime; &Prime;
&nabla; &permil; &deg; &there4; &alefsym; &oslash;
&isin; &notin; &cap; &cup; &sub; &sup; &sube; &supe;
&not; &and; &or; &exist; &forall; 
&rArr; &lArr; &dArr; &uArr; &hArr;
&rarr; &darr; &uarr; &larr; &harr;

Unicode character SCRIPT CAPITAL F (U+2131) can be typed as &#x2131; — ℱ

Category:Mathematical formatting templates

<s>cross out</s> cross out


== See also ==
* [[Wikipedia:How to edit a page]]
* [[Wikipedia:Manual of Style]]



==Further reading==

==External links==

References 1[edit]

  • <ref>{{citation|last1=Blyth|first1=Colin R.|last2=Pathak|first2=Pramod K. |year=1986|title=A note on easy proofs of Stirling's theorem|journal=American Mathematical Monthly|volume=93|issue=5|pages=376&ndash;379|url=}}.</ref> <ref>{{citation|last=Gordon|first=Louis|year=1994 |title=A stochastic approach to the gamma function |journal=American Mathematical Monthly |volume=101|issue=9|pages=858&ndash;865|url=}}.</ref>
  • <ref name="RY">{{citation|last1=Revuz|first1=Daniel|last2=Yor|first2=Marc|year=1994|title=Continuous martingales and Brownian motion|edition=2nd|publisher=Springer}} (see Exercise (2.17) in Section V.2, page 187).</ref>
  • <ref name="RY">{{citation|last1=Revuz|first1=Daniel|last2=Yor|first2=Marc|year=1994|title=Continuous martingales and Brownian motion|edition=2nd|publisher=Springer}} (see Exercise (2.17) in Section V.2, page 187).</ref>
  • <ref>{{citation|last=Durrett|first=Richard|author-link=Rick Durrett|year=1984|title=Brownian motion and martingales in analysis|place=California|publisher=Wadsworth|isbn=0-534-03065-3}}.</ref>
  • <ref>{{citation|last=Fulman|first=Jason|year=2001|title=A probabilistic proof of the Rogers–Ramanujan identities|journal=Bulletin of the London Mathematical Society|volume=33|issue=4|pages=397&ndash;407 |url=|doi=10.1017/S0024609301008207}}. Also [ arXiv:math.CO/0001078].</ref>
  • ==Notes==
  • <references />

References 2[edit]

  • <ref>{{harvnb|Pollard|2002|loc=Sect. 5.5, Example 17 on page 122}}</ref>
  • <ref>{{harvnb|Durrett|1996|loc=Sect. 4.1(a), Example 1.6 on page 224}}</ref>
  • <ref name="Pollard-5.5-122">{{harvnb|Pollard|2002|loc=Sect. 5.5, page 122}}</ref>
  • quoted in <ref name="Pollard-5.5-122">{{harvnb|Pollard|2002|loc=Sect. 5.5, page 122}}.</ref>
  • ==Notes==
  • <references />
  • ==References==
  • *{{citation|last=Durrett|first=Richard|author-link=Rick Durrett|title=Probability: theory and examples|edition=Second|year=1996}}
  • *{{citation|last=Pollard|first=David|title=A user's guide to measure theoretic probability|year=2002|publisher=Cambridge University Press}}

References 3[edit]

  • see {{harv|Durrett|1996|loc=Sect. 7.7(c), Theorem (7.8)}};
  • *{{citation|last=Durrett|first=Richard|author-link=Rick Durrett|title=Probability: theory and examples|edition=Second|year=1996}}
  • '''Theorem''' {{harv|Barany|Vu|2007|loc=Theorem 1.1}}.
  • *{{citation|last1=Barany|first1=Imre|last2=Vu|first2=Van|year=2007|title=Central limit theorems for Gaussian polytopes|journal=The Annals of Probability|publisher=Institute of Mathematical Statistics|volume=35|issue=4|pages=1593&ndash;1621|doi=10.1214/009117906000000791}}. Also [ arXiv].
  • as proven in [[#CITEREFArBaBaNa2004|Artstein, Ball, Barthe and Naor (2004)]].
  • * <cite id=CITEREFArBaBaNa2004> S. Artstein, K. Ball, F. Barthe and A. Naor (2004), [ "Solution of Shannon's Problem on the Monotonicity of Entropy"], ''Journal of the American Mathematical Society'' '''17''', 975&ndash;982. Also [ author's site].</cite>
  • '''Theorem''' ([[#CITEREFKlartag2007|Klartag 2007]], Theorem 1.2).
  • * <cite id=CITEREFKlartag2007>[[European_Mathematical_Society#2008 prizes|Klartag]], Bo'az (2007), [ "A central limit theorem for convex sets"], ''Inventiones Mathematicae'' '''168''', 91&ndash;131. Also [ arXiv].</cite>
  • <ref>[[European_Mathematical_Society#2008 prizes|Boaz Klartag]], " A central limit theorem for convex sets", Inventiones Mathematicae 168:1, 91&ndash;131. []</ref>

References 4[edit]


and have a label somewhere else ("equation (1)")

:<cite id="equation1" style="float:right;margin-right:2.5em">(1)</cite> <math>a = 0</math> and have a label somewhere else ("equation [[#equation1|(1)]]")

[derivation 1]

:<math>\frac{n(n + 1)}{2} + (n+1) = \frac{(n+1)((n+1) + 1)}{2}\,,</math><ref group="derivation"> Derivation of induction formula for summing consecutive positive integers: :<math> \begin{align} \frac{n(n + 1)}{2} + (n+1) & = (n+1)\left( \frac n 2 + 1 \right) \\ & = (n+1)\left( \frac n 2 + \frac 2 2 \right) \dots \end{align} </math></ref>

  1. ^ Derivation of induction formula for summing consecutive positive integers:

<references group="derivation" /> .


It's "a Euclidean" because it's pronounced "yoo-", not "oy-" like in German. Same reason for "a European" instead of "an European". But would still be "an Eulerian" though.

Usage of articles in math English[edit]

Should I write "if x and y are positive then a number z=x+y is positive" or rather, "if x and y are positive then the number z=x+y is positive"? On one hand, this z appears anew, it was not introduced before. On the other hand, given x and y, there exists only one z defined by that phrase. Boris Tsirelson (talk) 18:11, 12 October 2014 (UTC)

Using the lost grammatical adage "when in doubt, rephrase", I'd use "if x and y are positive then z=x+y is positive". But it forced to pick from one of the two choices I vote for the second one. --RDBury (talk) 18:57, 12 October 2014 (UTC)
Thank you. Yes, I feel forced to choose, since really I write more complicated texts :-) such as "...then the/a function f defined by...satisfies..." (and instead of "function" it could be a longer noun group, like "separable reflexive Banach space" etc). Or do you think I can always rephrase? How? Really, I also feel more comfortable writing "the", but I got unsure, being confused by opposite opinions, like this: but when you introduce me someone, you say "a friend of mine" even though he is uniquely defined, if not by your words then by your gestures. Boris Tsirelson (talk) 19:54, 12 October 2014 (UTC)
It may sound strange, but I don't mind if slightly different styles (as long as they are "correct") are used in a longer text. It can help avoiding monotonic repetition, of which there is plenty anyway in mathematical texts. But in the present case, the second option gets my vote too. YohanN7 (talk) 20:49, 12 October 2014 (UTC)
Thank you. Having vote 2:0 (or even 3:0, counting myself) I get more sure. No, I do not find it strange... I also like some variations; but I face this case quite often. But wait, do you say that these "a" and "the" are both correct, or not? Boris Tsirelson (talk) 20:57, 12 October 2014 (UTC)
I'm not a native English speaker (I'm Swedish), but I'd say, as a guess, that both options are correct, but the first option seems unusual, it doesn't seem to fit. Perhaps it would fit in a bigger example with more ingredients. YohanN7 (talk) 21:07, 12 October 2014 (UTC)
It depends a bit on how you "read it out loud inside your head" when reading. An equation like z = x+y can "be pronounced differently" when given a context. YohanN7 (talk) 21:17, 12 October 2014 (UTC)
The Language Reference Desk may be a better place to post this question. As a native English speaker I am sure that it should be the and not a, because as you said there exists only one z defined by that phrase. --catslash (talk) 23:18, 12 October 2014 (UTC)
Thank you; if you are sure, also I am. Yes, I understand it is a language question; but sometimes math jargon differs from usual English. Boris Tsirelson (talk) 05:57, 13 October 2014 (UTC)


MathJax update, thanks to Nageh. Just to let you know, I have updated my mathJax user script to recent version 1.1 of MathJax. Notable change is the support for webfonts via CDN (i.e., no local font installation requirements). Details at the user script documentation page. Feedback welcome.

How to make SVG diagrams, thanks to Ryan Reich. This question sometimes comes up and it bears answering as often as possible, since a lot of people have never heard that we should be using SVG, and of those who have, few seem to have an easy way of actually accomplishing it. This is addressed at Help:Displaying a formula#Convert to SVG, but their proposed solution relies on a somewhat arcane and arbitrary invocation of two different utilities, followed by a roundabout filtration through two major software packages, which is necessitated by one of them (pstoedit) requiring a costly proprietary plugin to work properly. And the end result is still unusable if your diagram has diagonal lines. Here's the right way:

pdflatex file.tex
pdfcrop --clip file.pdf tmp.pdf
pdf2svg tmp.pdf file.svg
(rm tmp.pdf at the end)

Both pdfcrop and pdf2svg are small, free (if new and somewhat alpha) programs that work properly. I advocate pdflatex since with the alternative, you might be tempted to go the route of latex→dvips→pstopdf before vectorizing, and that runs into a problem with fonts that has to be corrected with one of the arcane invocations above. (There is a correct route, which is to replace that chain with dvipdfm, that I have never seen anyone suggest. Somehow, the existence of this useful one-step solution to getting PDFs from plain latex is always ignored.)

It has been road-tested on, most notably (for the complexity of its images) Triangulated category and found to work quite well.

I think for art like this it would be preferable to use .svg (a vector format) for the graphics instead of .jpg (a bitmap format), if possible. I use Adobe Illustrator for that but it's kind of expensive; the most popular free alternative seems to be Inkscape. —David Eppstein

"Editors making a challenge should have reason to believe the material is contentious, false, or otherwise inappropriate", according to Wikipedia:When to cite#Challenging another user's edits.

Per WP:POV fork, "The generally accepted policy is that all facts and major points of view on a certain subject should be treated in one article. As Wikipedia does not view article forking as an acceptable solution to disagreements between contributors, such forks may be merged, or nominated for deletion."


Mathematics (use of sources)

11.4) If editors disagree on how to express a problem and/or solution in mathematics, citations to reliable published sources that both are directly related to the topic of the article and directly support the material as presented must be supplied by the editor(s) who wishes to include the material. Novel derivations, applications or conclusions that cannot be supported by sources are likely to constitute original research within the definition used by the English Wikipedia.

From WPM talk:

I think that the OR rule together with the Copyright law make coverage of mathematics (or any other subject) impossible. You have to think (commit 'original research') to do mathematics. The only alternative is to blindly copy from 'reliable' sources which violates copyright. Of course, such copying and the verification that the source is indeed reliable also require thought (OR). So the rule against OR is an absurdity which should be repealed.
The reason we have a rule against OR is to try to avoid disputes about what is correct reasoning by appealing to an outside source. Notice that in mathematics, this is usually only necessary when one or more of the disputing parties is a crank or troll. However, refusing to allow an edit on grounds that it is OR is ultimately just an excuse for rejecting what we think is false without having to get the agreement of a crank or troll. JRSpriggs (talk) 03:05, 14 March 2011 (UTC)

This one is fairly complicated. I don't think it true that, outside of mathematics, OR and copyright makes coverage impossible. The problem is that an allowable rephrasing in most fields becomes OR in mathematics, as even a change in notation does not fall in the "routine arithmetic calculation" exemption in Principles 11.