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Helmholtz decomposition is wrong

Dear members of world mathematical community!

The Fundamental theorem of vector calculus, (Helmholtz decomposition) states that any sufficiently smooth, rapidly decaying vector field in three dimensions ${\mathbf{F}}$ can be constructed with the sum of an irrotational (curl-free) vector field and a solenoidal (divergence-free) vector field (scalar potential $\varphi$ and a vector potential ${\mathbf{A}}$)

${\mathbf{F}} = - \operatorname{grad} \psi + \operatorname{rot} {\mathbf{A}} \Rightarrow {\mathbf{F}} = \operatorname{grad} \varphi + \operatorname{rot} {\mathbf{A}}$ (1)

However, the gradient of scalar function does not form the vector field. As well known from textbook [1, p. 15] « … under co-ordinate change the gradient of function transforms differently from a vector »: hence the theory requiring (1) must be false. The next unpleasant things we can see for such well-known classical rules. In mathematics and physics the rot (or curl) is an operation which takes the vector field ${\mathbf{A}}$ and produces another vector field $\operatorname{rot} {\mathbf{A}}$ . However it is well-known that $\operatorname{rot} {\mathbf{A}}$ is an Antisymmetric Tensor . Therefore under co-ordinate change the tensor $\operatorname{rot} {\mathbf{A}}$ transforms differently from a true vector. For elimination of these contradictions the Fundamental theorem of vector calculus can be written as follows:

$\vec F = \operatorname{grad} \varphi + \operatorname{rot} \operatorname{rot} \vec A$. (2)

This formula completely corresponds to transformed Navier–Stokes equations(NSE) for incompressible fluids ($\operatorname{div} \dot \vec u = 0$)

$\rho \vec F - \operatorname{grad} p + \mu \nabla ^2 \dot \vec u = \rho \ddot \vec u \Rightarrow \rho (\vec F - \ddot \vec u) = \operatorname{grad} p + \operatorname{rotrot} \mu \dot \vec u$ . (3)

Here, $\vec F = \vec F_1 + \vec F_2 + ...$ vectors sum of a given, externally applied forces (e.g. gravity $\vec F_1$ , magnetic $\vec F_2$ and other), $p$- pressure (scalar function), $\dot \vec u$- velocity vector, $\ddot \vec u = d\dot \vec u/dt$ - acceleration vector, $\rho$ - density (const), $\mu$ - viscosity (const), $\nabla ^2$ - Laplace operator.

Equations (3) and (2) are consistent. Hence there is no reason to say that the theory requiring (2) must be false. As we can see from NSE the sum - $\operatorname{grad} p + \mu \nabla ^2 \dot \vec u = - (\operatorname{grad} p + \operatorname{rotrot} \mu \dot \vec u)$ forms the vector field.

Note that we will receive the formula (2) also after similar transformation of the Navier–Stokes for a compressible fluid and after transformation of the Lame equations for an elastic media.

From this brief note follows that Helmholtz decomposition is wrong and demands major revision. This follows from comparison of two articles in Wikipedia (http://en.wikipedia.org/wiki/Navier%E2%80%93Stokes_equations and http://en.wikipedia.org/wiki/Helmholtz_decomposition ).

Therefore let's try to formulate the text for editing of this article: http://en.wikipedia.org/wiki/Helmholtz_decomposition .

1.B. A.; Fomenko, A. T.; Novikov, Sergeĭ Petrovich (1992). Modern Geometry--methods and Applications: The geometry of surfaces, transformation groups, and fields] (2nd ed.) . Springer. (p. 15).ISBN 0387976639.

--Alexandr (talk) 18:35, 7 February 2012 (UTC)

Do you have counterexamples to Wiles' proof of Fermat's last theorem, too? Tkuvho (talk) 18:49, 7 February 2012 (UTC)

To Continuum-paradoxes: Your argument against the Helmholtz decomposition assumes that the decomposition must be invariant under coordinate transformations. However, the decomposition theorem does not claim that the decomposition is so invariant. So your argument fails.

In any case, you must provide a reliable secondary source for any such "fact" which you wish to include in Wikipedia. Your original research is not acceptable. See WP:NOR. JRSpriggs (talk) 07:39, 8 February 2012 (UTC)

On one hand, Alexandr is right (and should not be asked for "counterexamples to Wiles' proof of Fermat's last theorem, too"). The gradient of a function is a covector rather than vector, of course. On the other hand, working in a Euclidean (rather than just linear) space it is possible, and quite usual, to treat its dual space as (another copy of) the same space (see Linear functional#Dual vectors and bilinear forms). Or, in terms of transformations (if you prefer this old language): only orthogonal transformations are relevant. Boris Tsirelson (talk) 08:52, 8 February 2012 (UTC)

Fomenko et al is certainly a standard reference. This is all the more reason that I find their discussion of gradients bizarre in the extreme. The exterior derivative df is the usual notation and terminology for the associated covector. The gradient of a function is almost always taken to be the vector, exploiting the usual identifications as mentioned by Boris. I don't think we need to relate to Fomenko's odd choice of terminology. Tkuvho (talk) 09:14, 8 February 2012 (UTC)

The vector potential in the invariant version of the Helmholtz decomposition is a pseudovector. However most sources do not make this distinction, so I don't think our article should either. If someone is bothered by it, then we can add a remark about it somewhere. Sławomir Biały (talk) 12:33, 8 February 2012 (UTC)

However Fomenko does make a distinction and if people are going to go on to deal with manifolds it seems a good idea so I think a note at the very least is called for. Personally I don't like pseudovectors as it strikes me as a kludge or not quite figured out half way to there kind of idea. Dmcq (talk) 13:13, 8 February 2012 (UTC)

On a manifold, it's the Hodge decomposition (which is invariant by design). The boundary conditions are different. Those for the Helmholtz decomposition only really make sense in Euclidean space. Sławomir Biały (talk) 13:31, 8 February 2012 (UTC)

In quantum mechanics when they have CP violated do they talk about that in terms of pseudovectors or what do they call it when things don't look the same in a mirror thanks? Dmcq (talk) 18:23, 8 February 2012 (UTC)

Chirality? Sławomir Biały (talk) 22:07, 11 February 2012 (UTC)

To Continuum-paradoxes: In my previous comment above, I should have said that the Helmholtz decomposition is invariant under the group of translations and rotations ($\mathbf{R}^{3} \rtimes SO(3) \,$), but not under more general curvilinear coordinate transformations. In the smaller group, there is no difference between the behavior of contravariant vectors (what some call vectors) such as the rate of flow of a fluid and the behavior of covariant vectors (what some call covectors) such as the gradient of temperature, nor between ordinary vectors and axial vectors. JRSpriggs (talk) 11:26, 10 February 2012 (UTC)

It's the other way around: vectors are covariant (go in the same direction), covectors are contravariant (go in the opposite direction). Thus, you push forward a tangent vector, but pull back a differential form. Tkuvho (talk) 20:11, 11 February 2012 (UTC)

I see that the page you referenced says otherwise. There is a problem of terminology here. Tkuvho (talk) 20:14, 11 February 2012 (UTC)

It's an unfortunate abuse of language that I have at various times tried to minimize on Wikipedia: referring to vectors as "contravariant" and covectors as "covariant". What is in fact the case is that the components of a vector in a coordinate system are contravariant and those of a covector are covariant. So it is infinitely preferable to talk about whether the components of some quantity are covariant or contravariant than whether the something itself is. Sławomir Biały (talk) 22:07, 11 February 2012 (UTC)

How would that work in abstact index notation? One is no longer allowed to say whether a tensor is covariant or contravariant? Or do we only talk about covariance and contravariance of "placeholders"? Tkuvho (talk) 12:49, 12 February 2012 (UTC)

Well, it's not really meaningful to talk about covariance and contravariance of abstract tensors. The co/contravariance refers to the behavior under what physicists call passive diffeomorphisms, whereas tensors themselves are actually invariant. In an abstract index setting, I think it's common to refer to the indices themselves as covariant or contravariant. But this is also an abuse of language that should probably be minimized. Sławomir Biały (talk) 13:05, 12 February 2012 (UTC)

This is not how the term is used in category theory. See contravariant functor. Here I am using the term "category theory" in a very loose sense. This usage of "contravariant" and "covariant" has certainly "permeated the fabric of modern mathematics", to quote Carl. Tkuvho (talk) 13:28, 12 February 2012 (UTC)

┌────────────────────────────────────────────────────────────────────────────────────────────────────┘ Actually, it is precisely the same notion as that in category theory, provided that a vector or a covector is defined to be a functor that associates a list of numbers to a frame, where both frames and lists of numbers carry the structure of a GL(n)-torsor. But it is not the same in the category of manifolds and mappings between them. However, I much rather prefer to think of the vector as existing independently of how it is described in coordinates (that is invariant under passive diffeomorphism), so calling a vector "contravariant" because of how its components transform seems to put the cart before the horse. Sławomir Biały (talk) 13:36, 12 February 2012 (UTC)

I agree, it is best not to think of it in terms of coordinates. Thus, if you think of a differential 1-form intrinsically as an assignment of an element of T* at every point, then a diffeomorphism will result in a pullback of the differential form. Therefore differential forms are contravariant according to the definition found at contravariant functor. Tkuvho (talk) 13:41, 12 February 2012 (UTC)

Yes, I agree with this. My point is that it depends on what category you are working in. Some define a tensor as an equivariant function from the frame bundle to a representation of GL(n). The GL(n) action gives morphisms on the frame bundle, and linear maps define morphisms of the representation. From this point of view, a vector is definitely a contravariant functor. If, on the other hand, you consider a vector as a functor on a category of manifolds whose morphisms are local diffeomorphisms, then it is covariant. This is why I find it to be an abuse to call the vector itself contravariant: what we are really talking about is its representation in terms of GL(n) torsors. Sławomir Biały (talk) 13:55, 12 February 2012 (UTC)

Well, I think that you two are dealing in pointless abstractions. I am all for using coordinates and indices which represent numbers designating specific directions in spacetime (the local tangent space). In physics, they begin with equations for each component separately and only later realize that these components can be combined into something like a matrix (i.e. tensors). Thus, to my mind, a tensor is its components as a function of: the event, the choice of coordinate system, and the values of the indices. JRSpriggs (talk) 20:26, 12 February 2012 (UTC)

So, for you a vector is contravariant, period. Things are not necessarily so absolute for the rest of the world, though. It cannot hurt to insist on referring that components transform contravariantly or covariantly, accordingly. Most reliable sources do this already. Sławomir Biały (talk) 22:05, 12 February 2012 (UTC)

Dear Participants of discussion!

Many thanks for your professional comments. Please, pay attention to addition in my message (Notation 1). However I ask (if it is possible) not talk this problem outside of rectangular Cartesian co-ordinates. It can be made later (after consensus for rectangular Cartesian co-ordinates). I ask to apply only short phrase without difficultly translated words. Remember that your comments are reading all over the world by means of computer translators.

Notation 1.

The vector fields cannot be constructed out of scalar fields using the gradient operator. Therefore so-called Laplacian field is not a true vector field. Thus, the requirements $\operatorname{rot} \dot \vec u = 0,\operatorname{div} \dot \vec u = 0$ are inconsistent for true vector fields.

This result confirms the proof about impossibility of irrotational velocity field in this old university textbook [1]p. 100-101. 2. Other unpleasant things we can see for many well-known classical equations in Wikipedia. For example the Euler equations (fluid dynamics) can be written as follows

$- \operatorname{grad} p = \rho (\operatorname{div} \ddot \vec u - \vec F)$

Note that such equations have no sense as exact vector equations because $\operatorname{grad} p$ is not the true vector.

Here Helmholtz_decomposition we can read: “This theorem is of great importance in electrostatics, since Maxwell's equations for the electric and magnetic fields in the static case are of exactly this type.[2]” Thus Maxwell's equations have no sense as exact vector equations. We can to continue a list of similar incorrect mathematical physics equations in Wikipedia.

--Alexandr (talk) 12:05, 14 February 2012 (UTC)

You seem to be very confused. The gradient is defined as a vector (see, for instance, Borisenko and Taparov "Vector and tensor analysis with applications"). There is no problem with the equations you have listed. Sławomir Biały (talk) 12:58, 14 February 2012 (UTC)

I see that Covariance and contravariance of vectors states that the gradient is covariant. Is this consistent with the approach you developed above? Tkuvho (talk) 13:46, 14 February 2012 (UTC)

No. That should probably be clarified somehow. It's true that the partials transform covariantly, but these are the components of the differential. The gradient involves the inverse metric tensor. Sławomir Biały (talk) 14:02, 14 February 2012 (UTC)

It may be a good idea to make a note of it at Talk:Covariance and contravariance of vectors. Tkuvho (talk) 12:33, 15 February 2012 (UTC)

It probably doesn't help matters that in applied mathematics, people talk about "covariant components" and "contravariant components" of a given vector. This could be a big source of the OPs original confusion. Sławomir Biały (talk) 12:59, 15 February 2012 (UTC)

To Sławomir Biały. I have formulated my conclusion on the basis of this university textbook; 1.B. A.; Fomenko, A. T.; Novikov, Sergeĭ Petrovich (1992). Modern Geometry--methods and Applications: The geometry of surfaces, transformation groups, and fields] (2nd ed.) . Springer. (p. 15).ISBN 0387976639. Authors of this textbook – authoritative mathematicians: http://www.mathnet.ru/php/person.phtml?&personid=8368&option_lang=eng http://www.mathnet.ru/php/person.phtml?option_lang=eng&personid=4537 http://www.mathnet.ru/php/person.phtml?option_lang=eng&personid=21899 As well known from this textbook « … under co-ordinate change the gradient of function transforms differently from a vector ». Therefore the gradient of scalar function does not form the vector field. Thus, your objections «You seem to be very confused…. » concern first of all these authors.

I can present other and newer arguments that gradient of scalar function does not form the vector field.

--Alexandr (talk) 11:38, 17 February 2012 (UTC)

You should consult other books. What Fomenko et al define is usually called the differential. The gradient is usually defined to be the vector obtained from the differential by applying the inverse metric. See for instance Definition 2.3.9 of Abraham, Marsden, Raitu "Manifolds, Tensor Analysis, and Applications". Basically any book on tensor analysis agrees with me. Look at James Simmonds "A Brief on Tensor Analysis" (Springer UTM); Arfken and Weber "Mathematical methods for physicists", Chapter 2. Even in any calculus textbook you will see the definition that the gradient of a function is the vector whose magnitude is the greatest rate of change of the function and whose direction is the direction in which that rate of change occurs. (Under this definition, the gradient will transform as a vector.) Sławomir Biały (talk) 12:11, 17 February 2012 (UTC)

The gradient is the list of the coefficients of the differential of the function. The differential is a linear form, and thus, by definition of the dual vector space, is an element of the dual of the working space. A basis of this working space being chosen, the gradient is thus the vector of the coefficients of the differential on the dual basis (the members of the dual basis are the linear forms which send a basis vector to 1 and the others to 0). This being stated, to know if the gradient is a vector field depends on which definition of a vector field you choose (the definition in vector field is ambiguous): If a vector field is a function of the working space into its associated vector space, then the gradient is not a vector field for this definition. If a vector field is a function of the working space into an arbitrary vector space, then the gradient is a vector field for this definition. Whichever definition is chosen, the metric of the working space (dotproduct) induces an isomorphism between the working vector space and its dual. But this isomorphism may not be considered as an identification. This is the reason of the distinction between vectors and covectors, which are both vectors but in different spaces. — D.Lazard (talk) 14:49, 17 February 2012 (UTC)

No, the gradient is a vector, not a covector. The differential is the covector you describe. The gradient is the vector whose inner product with another vector is the differential of the function applied to the vector. This is the definition of the gradient. It is not in the dual space. The gradient has no meaning without a metric. (See also symplectic gradient for the symplectic version of this idea.) Sławomir Biały (talk) 16:06, 17 February 2012 (UTC)

I do not agree with you. You may be right if you consider only the usage of the gradient in physics. In the study of the functions of several variables, and in particular in optimization, the gradient is defined and used independently of any metrics, like in a sentence like the gradient is null at the local extrema of a differentiable function. The covector property of the gradient is clear in the conjugate gradient method, because the direction of the minimum of a quadratic function is not the gradient but its conjugate direction. D.Lazard (talk) 16:52, 17 February 2012 (UTC)

Gradient descent is a good example of what I mean. If it makes sense to "move in the direction of the gradient" then the gradient is a vector. One cannot move in the direction of a covector. Sławomir Biały (talk) 17:14, 17 February 2012 (UTC)

Dear Participants of discussion!

The assumption “vector fields can be constructed out of scalar fields using the gradient operator and so-called Laplacian field is a true vector field” is 100 years old. This assumption have many “Strict proofs”, covered in hundreds of textbooks, and taught each year to many thousands of students. It is difficult to believe that all “Strict proofs” are wrong. Therefore has changed nothing after edition in 1979 of the textbook Modern Geometry in which it is written '« … under co-ordinate change the gradient of function transforms differently from a vector » The convincing counterexample is necessary. Such counterexample I bring to your attention. This counterexample kills mentioned “Strict proofs”. I will specify a source of this counterexample later.

Counterexample. As we well know the divergence of any vector field on Euclidean space is a scalar field. Therefore as an example let's calculate the divergence of an acceleration vector $\operatorname{div} \ddot \vec u$ . The acceleration vector components can be written as

$\ddot u_i = \frac{{d\dot u_i }} {{dt}} = \frac{{\partial \dot u_i }} {{\partial t}} + \dot u_x \frac{{\partial \dot u_i }} {{\partial x}} + \dot u_y \frac{{\partial \dot u_i }} {{\partial y}} + \dot u_z \frac{{\partial \dot u_i }} {{\partial z}},(i = x,y,z)$

After taking an operator div we have

$\operatorname{div} \ddot \vec u = \frac{{\partial \ddot u_x }} {{\partial x}} + \frac{{\partial \ddot u_y }} {{\partial y}} + \frac{{\partial \ddot u_z }} {{\partial z}} = \frac{\partial } {{\partial t}}\operatorname{div} \dot \vec u + \dot u_x \frac{\partial } {{\partial x}}\operatorname{div} \dot \vec u + \dot u_y \frac{\partial } {{\partial y}}\operatorname{div} \dot \vec u + \dot u_z \frac{\partial } {{\partial z}}\operatorname{div} \dot \vec u +$ $+ \left[ {\left( {\frac{{\partial \dot u_x }} {{\partial x}}} \right)^2 + \left( {\frac{{\partial \dot u_y }} {{\partial y}}} \right)^2 + \left( {\frac{{\partial \dot u_z }} {{\partial z}}} \right)^2 + 2\left( {\frac{{\partial \dot u_x }} {{\partial y}}\frac{{\partial \dot u_y }} {{\partial x}} + \frac{{\partial \dot u_y }} {{\partial z}}\frac{{\partial \dot u_z }} {{\partial y}} + \frac{{\partial \dot u_x }} {{\partial z}}\frac{{\partial \dot u_z }} {{\partial x}}} \right)} \right]$ (1)

As we can see this formula can be written as

$\operatorname{div} \ddot \vec u = \frac{d} {{dt}}\operatorname{div} \dot \vec u + (\operatorname{div} \dot \vec u)^2$ (2)

if and only if such equality is true

$\left( {\frac{{\partial \dot u_x }} {{\partial x}}} \right)^2 + \left( {\frac{{\partial \dot u_y }} {{\partial y}}} \right)^2 + \left( {\frac{{\partial \dot u_z }} {{\partial z}}} \right)^2 + 2\left( {\frac{{\partial \dot u_x }} {{\partial y}}\frac{{\partial \dot u_y }} {{\partial x}} + \frac{{\partial \dot u_y }} {{\partial z}}\frac{{\partial \dot u_z }} {{\partial y}} + \frac{{\partial \dot u_x }} {{\partial z}}\frac{{\partial \dot u_z }} {{\partial x}}} \right) = (\operatorname{div} \dot \vec u)^2$ (3)

The realization of (3) require such equality:

$\frac{{\partial \dot u_x }} {{\partial y}}\frac{{\partial \dot u_y }} {{\partial x}} + \frac{{\partial \dot u_y }} {{\partial z}}\frac{{\partial \dot u_z }} {{\partial y}} + \frac{{\partial \dot u_x }} {{\partial z}}\frac{{\partial \dot u_z }} {{\partial x}} = \frac{{\partial \dot u_x }} {{\partial x}}\frac{{\partial \dot u_y }} {{\partial y}} + \frac{{\partial \dot u_y }} {{\partial y}}\frac{{\partial \dot u_z }} {{\partial z}} + \frac{{\partial \dot u_x }} {{\partial x}}\frac{{\partial \dot u_z }} {{\partial z}}$ (4)

Note that equality (3) can make sense only for $\operatorname{rot} \dot \vec u \ne 0$ In the case $\operatorname{rot} \dot \vec u = 0$ all terms in brackets of (3) are positive and $\operatorname{div} \dot \vec u = 0$ is impossible. Thus the requirements $\operatorname{rot} \dot \vec u = 0,\operatorname{div} \dot \vec u = 0$ for vector field are inconsistent. As we well know $\dot \vec u = \operatorname{grad} \varphi ,_{} \nabla ^2 \varphi = 0$ , if $\operatorname{rot} \dot \vec u = 0,\operatorname{div} \dot \vec u = 0$. Therefore the vector fields cannot be constructed out of scalar fields using the gradient operator and so-called Laplacian field is not a true vector field.

--Alexandr (talk) 12:20, 27 February 2012 (UTC)

If this counterexample were true, it would be a counterexample to the chain rule, having nothing to do with the covariance and contravariance of vectors. Since this is clearly ridiculous, you must have made a mistake. Your error seems to be in going from (1) to (2). Sławomir Biały (talk) 13:25, 27 February 2012 (UTC)

Horocycle merged to horosphere

The article horocycle was recently merged to horosphere. I've undone this merger, but was quickly reverted. I'd appreciate outside input at Talk:Horosphere. Sławomir Biały (talk) 13:44, 13 February 2012 (UTC)

Surely one is just the one-dimensional specialisation of the other; or, alternatively, the other is a longstanding multidimensional generalisation of the first. Doesn't it make sense to treat both together? -- if not with horocycle under horosphere, then with horosphere under horocycle? Jheald (talk) 14:31, 13 February 2012 (UTC)

I am not so sure. Thus, the remark that the geometry of the horosphere is euclidean is not really significant for horocycles. The statement that a horocycle has geodesic curvature 1 cannot be made for horospheres, etc. Tkuvho (talk) 14:41, 13 February 2012 (UTC)

We have separate articles on circles and spheres. Surely this is the same distinction? —David Eppstein (talk) 16:12, 13 February 2012 (UTC)

Except that here the article seems to deal with general horo-n-spheres, so the distinction is perhaps different. We have n-sphere and circle, so I'm not saying there shouldn't be a separate article if enough material exists on it to justify this (though there is very little at the moment). — Quondum^☏✎ 17:22, 13 February 2012 (UTC)

There probably should be separate articles but seeing as how both are stubs and on related subjects it's not that surprising that an attempt was made to merge them. Our article on hypercycles is longer but apparently in need of more work, and I couldn't find anything on their higher dimensional analogs.--RDBury (talk) 21:01, 13 February 2012 (UTC)

I just found this discussion here. Circles, horocycles and hypercycles (generalized circles) are the 2D variants of generalized spheres (spheres, horospheres or hyperspheres) in Hyperbolic geometry (balls and disks are just the usual interiors). Actual descriptions vary by model and whether 2 or 3d but arise via the intersection of H^n with an affine plane (or an r-sphere as the intersection with an affine r+1 plane)88.82.206.110 (talk) 14:38, 18 February 2012 (UTC)Selfstudier (talk) 14:39, 18 February 2012 (UTC)

Is Correspondence (mathematics) a useful page?

I've just stumbled across the page Correspondence (mathematics). As well as being completely unsourced, it strikes me as a collection of things that really don't belong together—each definition could better be placed on the page for the appropriate topic. Also, the first definition contradicts the definition given at relation (mathematics). Links such as those from the first sentence of function (mathematics) to correspondence (mathematics) only serve to muddy the waters further. Does anyone see this page as worth keeping? Jowa fan (talk) 23:23, 13 February 2012 (UTC)

Maybe a bit slimmed down as a {{mathdab}}. But other articles should be linking to the articles listed at Correspondence (mathematics) rather than linking to Correspondence (mathematics) itself. —David Eppstein (talk) 23:35, 13 February 2012 (UTC)

BTW there is something like mathdab at ru:Отображение (значения) (Russian: Отображение = map). Other interlanguage links apparently correspond to map (mathematics) or are redirected, so the "article" is apparently a PoV fork. Incnis Mrsi (talk) 09:01, 14 February 2012 (UTC)

I see no contradiction between the first definition and the definition at binary relation that it refers to. It would be better I think to distribute the meanings and make it a proper disambiguation page I think, so yes the page would be slimmed down a bit. I see no evidence of any POV and don't know where that idea came from and the other interlanguage links do not support what is said as far a I can see. I wouldn't depend on how things are translated for very much anyway though I do like sometimes to look at other languages to get some ideas. Dmcq (talk) 09:23, 14 February 2012 (UTC)

The page Correspondence (mathematics) says correspondence is an alternative term for a relation between two sets, whereas binary relation says A correspondence: a binary relation that is both left-total and surjective. Jowa fan (talk) 11:32, 14 February 2012 (UTC)

I think the page should not be trimmed down. The many incarnations of correspondences all have something in common: "a point gets mapped to many points". A good article would make this clear. I have added two references. Jakob.scholbach (talk) 12:37, 14 February 2012 (UTC)

I wonder where the binary relation page got that definition of correspondence from. In page 1331 of Encyclopedic dictionary of Mathematics the definition of a correspondence is just any relation plus the two sets with no restriction about being left and right total. Dmcq (talk) 13:18, 14 February 2012 (UTC)

A recent change to function (mathematics) now defines a function as a correspondence. The discussion above is clear enough indication that the term is ambiguous, confusing, and unhelpful. The far better term "rule" is strongly supported by three editors but has met resistance on what seems like pedantic, if not bourbakist, grounds. Further input would be appreciated. Tkuvho (talk) 08:57, 15 February 2012 (UTC)

Well it would be if we can get a citation for the definition of correspondence in the article about binary relation. There's a citation in the article showing its use in function is perfectly correct and the dictionary entry I pointed to supports that as well. Dmcq (talk) 09:07, 15 February 2012 (UTC)

Tkuvho - the term "rule" will, for most people, imply a finite and deterministic rule or algorithm. As I expect you know, not every function can be defined by a finite rule, otherwise every function would be a computable function. The existance of incomputable functions is so fundamental that glossing over it in the article's lead would be a gross over-simplification. For me, the current opening sentence "In mathematics, a function is a correspondence that associates each input with exactly one output" strikes the right balance between accuracy, brevity and clarity. Gandalf61 (talk) 09:38, 15 February 2012 (UTC)

I responded at Talk:Function_(mathematics)#summary_of_correspondence_vs_rule. Tkuvho (talk) 07:48, 16 February 2012 (UTC)

In my opinion the debate follows that, in mathematics, "correspondence" has frequently its usual, non formal, English meaning, as in article function. But it may also have a technical meaning, subject to a formal definition, as in relation (mathematics). Thus I propose to modify the beginning of correspondence (mathematics) as

In mathematics and mathematical economics, correspondence may be used informally with its usual English meaning. It may also have a technical meaning subject to a formal definition.

• In the theory of relations, a correspondence is a relation between two sets, such that every element of each set is related to at least one element of the other.

(For the item, I have kept the definition of Relation (mathematics), but the item must be changed when the definition in Relation (mathematics) will change. D.Lazard (talk) 11:16, 15 February 2012 (UTC)

Have you a citation for that? I've put a citation needed into binary relation because as described above I found a dictionary giving something quite different - and which in fact works very well in the lead of function. I found no definition like the one in that article. Dmcq (talk) 19:15, 15 February 2012 (UTC)

No, I have no citation, but like you I am not sure that this definition is not OR. It is the reason of my comment in parentheses. My feeling is that this first item could be suppressed, the non formal meaning being sufficient for the definition of a function. D.Lazard (talk) 21:26, 15 February 2012 (UTC)

Conditional statement – a solution needed, not a smoldering edit war

At WP: Articles for deletion/Conditional statement (logic) I proposed to make a conceptdab article, but there was not any movement in this direction. Later, I asked help at WikiProject Logic, to be ignored. Now, at Conditional statement (logic) (edit|talk|history|links|watch|logs) users Artur Rubin and History2007 try to redirect this to material conditional, which I consider as inappropriate. On the other hand, Hanlon1755 (talk · contribs) pushes his own ideas about what is logical condition. Please, help to put the end to redirects' jumble and make a valid disambiguation of the term "conditional" in logic, programming and linguistics. Preferably, as a WP:CONCEPTDAB article. Incnis Mrsi (talk) 13:21, 14 February 2012 (UTC)

The dab page you refer to exists at Conditional statement. -- 202.124.72.200 (talk) 10:19, 15 February 2012 (UTC)

I could, although reluctantly, remember one (static) IP address, but it is not polite to prompt Wikipedia users to waste time for whois and reverse DNS queries and comparisons of persistently changing IPs. Please, register yourself; this is the last time I reply to an IP posting from this ISP. Incnis Mrsi (talk) 18:18, 15 February 2012 (UTC)

---

The war progressed for another 5 days. Now I self-proclaimed a mediation at Talk:Conditional statement (logic)#Conditions for acceptable solution and ask the WikiProject for support. Please, provide an explicit output. Don't give just a silent agreement, because warriors can disrespect my self-imposed conditions. Please, express some will to end the edit war even if the cause and exact conditions seems not so important. Incnis Mrsi (talk) 09:02, 20 February 2012 (UTC)

Proposed changes to mathematical categories

Good Olfactory (talk · contribs) has proposed renaming Category:Triangulation to Category:Triangulation (geometry). Discuss at Wikipedia:Categories for discussion/Log/2012 February 14. The same discussion page also contains a proposal to delete Category:Polyhedra rest category and merge it into its parent category. And there are also quite a few renames of mathematical categories proposed at Wikipedia:Categories for discussion/Speedy, e.g. Category:Logical symbols to Category:Logic symbols, Category:Tiling to Category:Tessellation, etc. —David Eppstein (talk) 22:50, 14 February 2012 (UTC)

Charles Wells (mathematician) at AfD

Charles Wells (mathematician) is up for deletion.  --Lambiam 02:44, 16 February 2012 (UTC)

External links at Tangram

There is a small dispute as to whether certain links should be included in the External links section. Additional opinions at Talk:Tangram#EL links removed will be appreciated.--RDBury (talk) 04:51, 17 February 2012 (UTC)

Projective resolutions and free resolutions

Projective resolutions and free resolutions is a new article. Projective resolution and Free resolution redirect elsewhere. Should the redirects be altered or should some articles get merged or what? No other articles linked to the new article until a moment ago when I added a cross-reference. If it is not merged into other articles, then some things should link to it. Michael Hardy (talk) 17:31, 17 February 2012 (UTC)

Both redirects are to Projective module#Projective resolution. Thus the new article is useful. I'll redirect Projective resolution and Free resolution to the new article and move the new article to Projective resolution and free resolution. This will add a lot of links to the new article. D.Lazard (talk) 17:52, 17 February 2012 (UTC)

I like the idea of having pages dedicated to homological resolutions, but this page title leaves us in an awkward situation of where to place the flat and injective resolutions. Is four names too many for a page title? Rschwieb (talk) 18:48, 17 February 2012 (UTC)

For the moment Flat resolution and Injective resolution link to sections in Flat module and Injective module. By the way Flat module has a red link to Resolution of a module. I suggest, first, to add a "see also" section in Projective resolution, linked to Flat resolution and Injective resolution. A second step could be to expand Projective resolution and free resolution in order to move it to Resolution of a module. D.Lazard (talk) 20:14, 17 February 2012 (UTC)

I think a good name would be Resolution (homological algebra). This should cover projective (and thus free) resolutions, injective, flabby, flat, acyclic resolutions. The title "Projective resolution and free resolution" is too long and focussing on just these two resolutions is also awkward from a content-point of view. Jakob.scholbach (talk) 15:01, 21 February 2012 (UTC)

I agree that "Projective resolution and free resolution" is too long. But I am not sure that Resolution (homological algebra) is better than Resolution of a module. In fact, in ring and module theory, resolutions are frequently used independently of the consideration of any homology. This occurs especially in the computational theory of polynomial ideals, where the degrees which occurs in a minimal free resolution (Castelnuovo-Mumford regularity) are strongly related to the complexity of computing a Gröbner basis and to the Hilbert series. Most people interested in these questions do not know nothing of homological algebra and the name Resolution (homological algebra) could lead them to miss the page which is relevant to them. In any case, whichever name is chosen, the other should be a redirect. D.Lazard (talk) 16:00, 21 February 2012 (UTC)

I believe that Resolution of a module would be the right scope for this situation. The full blown categorical concept would probably best be tackled in its own article. Rschwieb (talk) 21:30, 21 February 2012 (UTC)

What about resolution (algebra)? I am concerned that resolution of a module is too restrictive a title, as it does not include resolutions in general abelian categories (such as categories of sheaves or categories of complexes of modules). Surely we need an article about those. Ozob (talk) 21:38, 21 February 2012 (UTC)

Well, that's a surprise. Resolution (algebra) already exists and is about just this topic. Ozob (talk) 21:40, 21 February 2012 (UTC)

Good catch, I suppose that'll be a better starting point for this material! Rschwieb (talk) 00:49, 22 February 2012 (UTC)

(unindent) I have tried my hand at resolution (algebra)--improve it if you can! Jakob.scholbach (talk) 14:10, 22 February 2012 (UTC)

Resolution (algebra) is looking pretty good. Earlier I sorted some redirects for consistency and weeded out a few circular redirects caused by the recent activity. (The resolution pages all point toward resolution, the dimension pages point toward the correct sections in injective, projective, and flat articles.) Rschwieb (talk) 02:54, 23 February 2012 (UTC)

One more thing: the article mentions projective resolutions are unique up to chain homotopy. I would guess the same can be said for injective and flat resolutions, but I'm not familiar enough with the subject matter to verify. Is this the case? Thanks. Rschwieb (talk) 14:24, 23 February 2012 (UTC)

I am not sure, but, as far I remember, this uniqueness is a consequence of the fact that one may pass from a projective module to another one by a chain of operations consisting in adding (direct sum) a projective module or removing a direct factor. This is certainly not true for flat modules and probably not for injective modules (I am not familiar with them). D.Lazard (talk) 18:29, 23 February 2012 (UTC)

OK then the answer is at least not obvious. I'll leave it as it is until someone certain comes along. Rschwieb (talk) 21:25, 23 February 2012 (UTC)

covering set

I always thought that covering set was another term for cover (topology). I was surprised that the covering set article is purely about a number-theoretic meaning. Now I'm not sure whether to add an xref. Can someone take a look? Thanks. 67.117.145.9 (talk) 04:03, 18 February 2012 (UTC)

It may also be confused with covering space. I have added a disambiguation hatnote in covering set. D.Lazard (talk) 08:16, 18 February 2012 (UTC)

Ayme's theorem

WP:OR? Michael Hardy (talk) 03:18, 19 February 2012 (UTC)

Yes. I'd be very skeptical of anything claimed to be a significant new result in Euclidean plane geometry, new and significant being pretty much mutually exclusive at this point in the history of the subject.--RDBury (talk) 05:13, 19 February 2012 (UTC).

Notability and secondary sources (lack thereof) could disqualify this for WP, regardless of "significance". Is there not some WP-related project for this kind of non-encyclopedic result where it does not have to adhere to quite the same criteria as for Wikipedia? It feels though there should be some middle ground between Youtube and Wikipedia. — Quondum^☏✎ 05:59, 19 February 2012 (UTC)

PlanetMath? Though in this case checking against the known centers in the Encyclopedia of Triangle Centers and, if unknown, adding it there looks appropriate. —David Eppstein (talk) 07:33, 19 February 2012 (UTC)

To RDBury. This result is certainly not significant. It is a special case of a general theorem which is certainly well known, even if I have never seen it explicitly written.

Theorem: Let ABC be a triangle. Let Sa, Sb, Sc (Ayme's notation) be three points constructed from the triangle. If these points are permuted by every permutation of A, B, C which leaves fixed the other choices implied by the construction, then the three lines ASa, BSb, and CSc are concurrent.

Proof: The matrix of the coefficients of the barycentric equations of the lines is antisymmetric, and thus has a null determinant.

It is thus easy to construct billions of similar theorems. D.Lazard (talk) 13:47, 19 February 2012 (UTC)

Ayme can post it up on ProofWiki himself, if he likes, and it will then probably (eventually) be categorised as an example of the aforementioned theorem on the coefficents of barycentric equations. But for once I'm in agreement: this is not significant enough for Wikipedia. --Matt Westwood 14:29, 19 February 2012 (UTC)

manifold destiny

There is a discussion at Talk:Manifold Destiny#Birman of Joan Birman's comments concerning Yau. My personal opinion is that the comment is not only incorrect but borders on slanderous, and should not be included. Tkuvho (talk) 07:59, 20 February 2012 (UTC)

An editor thinks it's censhorship here. Please comment. Tkuvho (talk) 12:38, 22 February 2012 (UTC)

Some extra eyes on matrix (mathematics)

A known problematic editor has set his sights on matrix (mathematics). I have no intention of further engaging with this particular editor. It would be helpful if some project members could keep an eye.TR 07:34, 21 February 2012 (UTC)

Upon being extended an invitation to edit cooperatively to address the concerns we raised, he gave up. This suggests the user was not interested in anything short of the reinstatement of the deleted text. The same tactic might shorten future skirmishes with the same user. Rschwieb (talk) 14:09, 24 February 2012 (UTC)

From d to d

An IP has been straightening out all the italic "d"s in dy/dx at fundamental theorem of calculus. I had the impression the consensus in an earlier discussion was otherwise. Tkuvho (talk) 14:04, 21 February 2012 (UTC)

I have undone those changes, in addition to making many other typographical fixes. Ozob (talk) 21:46, 21 February 2012 (UTC)

I don't think it was actually a consensus but more of an agreement that in general people shouldn't change whatever is being used in an article, nothing was put in MOSMATH about it though.--RDBury (talk) 23:15, 22 February 2012 (UTC)

This seems a sensible thing to at least make a note about in WP:MOSMATH. If there is general agreement about what to say. Perhaps under WP:MOSMATH#Notational conventions a bullet stating that since both notations are used in the literature, consistent use of either d or d within an article should not be modified? Opinions? — Quondum^☏✎ 08:27, 23 February 2012 (UTC)

Sounds like a good idea to me. Leonxlin (talk) 04:59, 28 February 2012 (UTC)

Agree. Paul August ☎ 13:25, 1 March 2012 (UTC)

Euler Archive

I was trying to track down a reference for the article on Euler's criterion when I found this: http://www.math.dartmouth.edu/~euler/

Is there a special convention or template for citing it?

Virginia-American (talk) 21:51, 21 February 2012 (UTC)

I don't think there is so just use the cite-web template or use cite-book with the url.--RDBury (talk) 23:22, 22 February 2012 (UTC)

Category move proposals

Wikipedia:Categories_for_discussion/Log/2012_February_15

There are four proposals to move categories in the logic department. I think some people who have actually done some study on the subject should take a look. Please do drop in. Greg Bard (talk) 02:19, 22 February 2012 (UTC)

Actually I agree with Greg Bard on not renaming these but haven't any real stake in the matter. I'd probably rename logical syntax to logic syntax rather than using parenthesis and logic symbol instead of logical symbol makes me think of drag and dropping a symbol in designing a circuit. Dmcq (talk) 13:03, 22 February 2012 (UTC)

Merge help

Please discuss and do the merge at Talk:Arithmetic complexity of the discrete Fourier transform. I don't understand this math at all and can't do it myself. D O N D E groovily Talk to me 04:29, 22 February 2012 (UTC)

Solved by redirecting to Arithmetic_complexity_of_the_discrete_Fourier_transform#Bounds_on_complexity_and_operation_counts where one may find the relevant content of this page. The remainder of the page consisted in awful formulas lacking of any explanation, which have not their place in Wikipedia. D.Lazard (talk) 16:03, 24 February 2012 (UTC)

EoM again

I saw that there was an old thread about the Encyclopedia of Mathematics and its new wiki form at http://www.encyclopediaofmath.org. Of the points raised, I'm much the most concerned about the broken links, at present. Do we have a plan of action for fixing them? And, if we can decide about how that could be sorted out, where are we on MathJax or indeed any long-term solution for formulae? Charles Matthews (talk) 16:33, 22 February 2012 (UTC)

On the first issue, afaik there is no concerted effort but I hope people are fixing the broken links when they find them. There are probably more articles where there should be a link to EoM but there is none, broken or not. It seems like one of those tasks that seem like a good idea in principle but are too tedious to garner volunteers to actually get them done.--RDBury (talk) 23:38, 22 February 2012 (UTC)

From experience, this is rather boring and time-consuming. I have already suggested the following patch: a script that looks for eom links with id of the form ?/*, and replaces the id with the title field (space -> underscore). Of course, this can be only semi-automatic (although theoretically the script can also ping the page to check that it exists). Script experts, is this a feasible task? Sasha (talk) 02:01, 23 February 2012 (UTC)

Someone please do this! Sławomir Biały (talk) 13:11, 23 February 2012 (UTC)

Didn't we have a template for EoM?--Kmhkmh (talk) 14:25, 24 February 2012 (UTC)

the issue is as follows. EoM (later renamed springer, and later, SpringerEOM) is a template for EOM. However, the encyclopaedia changed its format a few months ago. Therefore the old links no longer work. The template itself has been fixed; what we are discussing is a way to fix the links (more precisely, change the "id" field from the format A/123456 to the new format Abc_equation). Sasha (talk) 15:26, 24 February 2012 (UTC)

There are 746 transclusions of the template {{SpringerEOM}} to fix. Bulk edit like this are probably done by a semi-automatic tool like WP:AWB.--Salix (talk): 16:42, 24 February 2012 (UTC)

how complicated is it? Does AWB have a ping option? Sasha (talk) 17:02, 24 February 2012 (UTC)

It fairly simple. You need to run it from windows and you start it by giving a list of files you want to edit, its smart enough to find out all the pages which transclude {{SpringerEOM}}. It then goes through the pages 1 by 1 and it can do some substitutions and allow you to adjust the edit. You need to OK each edit. You can go through many pages quickly about 4 a minute.--Salix (talk): 17:49, 24 February 2012 (UTC)

The news from the External link finder is actually not that bad. Just now there were 368 hits: but I took out those not from actual articles, and the number came down to 178. And some of those are multiple uses of the same link. So this could get done, I suggest. Charles Matthews (talk) 22:02, 24 February 2012 (UTC)

(this probably has to do mainly with my stupidity, but) I had trouble using the AWB regexp (and I did not find anything similar to a ping feature). If someone volunteers to post a working AWB script here, I am ready to share the load of running it. Sasha (talk) 22:14, 24 February 2012 (UTC)

You might not be registered to use it, see the AWB page for detail of registration.--Salix (talk): 14:31, 27 February 2012 (UTC)

I've made a simple converter[[2]] which can take the citation from the bottom of the springer page and produces the complete template with parameter for our reference. --Salix (talk): 14:31, 27 February 2012 (UTC)

thanks! (actually, I am AWB-registered, the problem was indeed with my stupidity). Sasha (talk) 15:58, 27 February 2012 (UTC)

Interviews about categories

When I saw this Wikipedia:Village pump (miscellaneous)#University research project on categories seeks interviewees I immediately thought of this project.Can't for the life of me say why ;-) Dmcq (talk) 17:57, 22 February 2012 (UTC)

Where did "labelled enumeration theorem" go?

The page symbolic combinatorics contains several redlinks to labelled enumeration theorem. It looks like the latter page existed in 2009, since it has been copied at http://citizendia.org/Labelled_enumeration_theorem, but I can't find a deletion discussion for it. Does anyone know what happened here? Jowa fan (talk) 03:51, 23 February 2012 (UTC)

The deletion log of labelled enumeration theorem says "Expired PROD, concern was: this is unsourced junk". PROD's don't have deletion discussions. Follow the link to see how they work. As an administrator I can see the deleted page history. It was prodded with "this is unsourced junk" in 2010 by User:Zahlentheorie who had created the article in 2006. PrimeHunter (talk) 04:04, 23 February 2012 (UTC)

From a brief look at the citizendia version it appears to be a variation on the Polya enumeration theorem. But is it an known theorem with a made up name, original research, or just junk? In retrospect an AfD might have been useful to determine which.--RDBury (talk) 23:15, 23 February 2012 (UTC)

I restored the article so that everyone can look at the content. I recommend adding additional references if the article is kept, to demonstrate that the topic passes the inclusion/notability criteria. This is just a pro forma undeletion, I have no opinion about whether the article should be deleted again. — Carl (CBM · talk) 16:43, 24 February 2012 (UTC)

Just browsing around and following some links, I just noticed that Proofs involving the totient function was PROD-ded in July 2010 ("concern was: wikipedia is not a directory of proofs, but rather of theorems"), while under its old title Totient function/Proofs it was AfD-ded in 2007 : Wikipedia:Articles for deletion/Totient function/Proofs and the result was Keep. Maybe some events I did not notice took place between 2007 and 2010, but might it not be a similar dubious Prod-ding ? French Tourist (talk) 09:45, 25 February 2012 (UTC)

A few minutes later, I notice the strange Wikipedia_talk:WikiProject_Mathematics/Archive_51#Strangest_edit_war_I.27ve_ever_seen. Might it be related with this PROD-ding ? French Tourist (talk) 10:03, 25 February 2012 (UTC)

There is a no double jeopardy for PRODs, see Wikipedia:Proposed deletion#Nominating under 'Before nomination'. So technically the PROD was improper, not that I'd request that it be restored.--RDBury (talk) 15:35, 25 February 2012 (UTC)

Proofs by Dooooot

I don't think any of these "proofs" should be given, for the following reasons:

1. Almost all of the rules for which he provides a "proof" are considered primitive rules in some system
2. The selection of rules used seems arbitrary, and needs a source
3. The proofs do not fall under WP:CALC, so they need sources.

— Arthur Rubin (talk) 03:42, 25 February 2012 (UTC)

You didn't specify which proofs, but I assume one of them is the one given in Hypothetical syllogism. In that case at least I agree and I'd question whether is it encyclopedic as well, generally if the proof is easy enough to assign as an exercise then we don't need to have it here. Another issue is that proofs should be in prose, not a table of symbols. Also, turquoise? This might be raised at the logic project as well.--RDBury (talk) 08:06, 25 February 2012 (UTC)

There's other wikis that go in for that sort of stuff. Proofs here should be either short and unobtrusive or have some element of notability in the literature. I don't think Wikipedia should become a repository for every proof in Mizar for example, they should refer to a source for this sort of thing. Dmcq (talk) 09:42, 25 February 2012 (UTC)

More to the point, you can't provide a proof of particularly this sort of PropLog theorem without first stating exactly which axioms you are working from. In order for the HS proof to be of any worth at all, at least one ought to allow Modus Ponendo Ponens as an axiom (or previously proved). Instead a great pile of other theorems (Transposition, Constructive Dilemma, etc.) are used instead, making this ridiculously overcomplicated. Besides, it uses LEM which unnecessarily removes it from the domain of constructivist proofs. So this proof is IMO seriously unworthy of WP. If all the rest of this user's proofs are like that, then I concur. --Matt Westwood 07:56, 26 February 2012 (UTC)

Oh, and even more to the point, the proofs are circular. To prove absorbtion, conjunction is used. To prove conjunction, absorption is used. As proofs go, these are complete and utter piffle.

Aha, I just see someone's already made that point below. --Matt Westwood 08:00, 26 February 2012 (UTC)

(ec) I was referring to the ones he seems so proud of at User:Dooooot#Proofs I've Written, although there may be others. WikiBooks and WikiVersity (if it's still open) seem better venues. — Arthur Rubin (talk) 09:56, 25 February 2012 (UTC)

Yes they seem reasonable places for someone who wants to go in for this sort of thing. Dmcq (talk) 10:16, 25 February 2012 (UTC)

More to the point, my guess is that the "proofs" are formally invalid, because they confuse levels of the system, normally kept separate by using distinct symbols, e.g. ⊢ versus →. If the "proofs" were valid, we'd call them "theorems of inference". — Quondum^☏✎ 10:39, 25 February 2012 (UTC)

I notice a certain circularity: Modus ponens is used to "prove" Disjunctive syllogism and vice versa. — Quondum^☏✎ 11:18, 25 February 2012 (UTC)

I don't think we should have "proofs" of these rules. I would not object to a truth table in some cases, though. Sławomir Biały (talk) 14:23, 25 February 2012 (UTC)

It does seem a bit funny to prove the inference rule modus ponens. Dmcq (talk) 14:29, 25 February 2012 (UTC)

Such "proof" as Hypothetical syllogism#Proof unlikely is useful, but is misleading in its use of negation and equivalences valid in classical logic only. If you did not get yet why is it incorrect, I resort to the analogy with algebra. Consider an identity $2x^2 + 2(x+1)^2 = (2x+1)^2 + 1$ and a proof deriving it from $(a+b)^2 + (a-b)^2 = 2a^2 + 2b^2$:

$2x^2 + 2(x+1)^2 = 2((x+\frac{1}{2}-\frac{1}{2})^2 + (x+\frac{1}{2}+\frac{1}{2})^2) = 4((x+\frac{1}{2})^2 + \frac{1}{4}) = (2(x+\frac{1}{2}))^2 + 1 = (2x+1)^2 + 1$

Yes, a proof is valid for real, complex and rational numbers, but it is invalid for any field of characteristic 2 (there is no such number as ½) and it is inapplicable to unital rings (because there is no division at all), although the identity still holds. The "universally correct" proof should be:

$2x^2 + 2(x+1)^2 = 4x^2 + 4x + 2 = (4x^2 + 4x + 1) + 1 = (2x+1)^2 + 1$

I am convinced that a proof may be useful only if it is valid in the most strict theory where a statement in question is a theorem. Incnis Mrsi (talk) 16:54, 25 February 2012 (UTC)

I think a lot of what you're saying is covered in point 2 in the original post. In general it's one of the pitfalls of doing proofs in an encyclopedia rather than than a text book that you don't get to formally establish which axioms are being used and which theorems are "known" and can be used without raising questions of circularity. In many cases you can make reasonable assumptions as to what should be considered common knowledge for the purposes of a proof, but for symbolic logic there are many equivalent formulations of the axioms and rules of inference and having a proof without knowing which formulation is being used doesn't make much sense. In fact in most of these cases the proposition may just as easily be taken as a axiom so giving a proof is unnecessary. I like the idea above of using truth tables as justification rather than formal proofs, they should be enough to convince casual readers, who probably compose a large segment of our readers anyway. Truth tables would be especially useful in cases where the result may be counterintuitive, $\neg p \to (p \to q)$ for example.--RDBury (talk) 04:53, 26 February 2012 (UTC)

If one still did not get the point, I repeat: the problem with some concrete formula, which Wikipedia attempts to present as a theorem (logical one, algebraic or else), is not only in an arbitrary selection of rules (Arthur Rubin) or even axioms. The problem is that Wikipedia, unlike a textbook, must obey WP:NPOV, which means that a description of P→Q, Q→R ⊧ P→R from classical PoV without mentioning other propositional calculi contradicts to the Wikipedia policy. On the other hand, I do not see a grave heresy in proofs of axioms and unordered graphs of proofs. I know what is circulus vitiosus, but if the article on the "axiom" C′ says A, B, C ⊢ C′ and the one about C says A, B, C′ ⊢ C, this does not mean a logical flaw, but only equivalence of {A, B, C} and {A, B, C′}. Incnis Mrsi (talk) 07:57, 26 February 2012 (UTC)

The proofs should be allowed, with qualifying language. Any given proof isn't "the" proof but rather "a" proof. I do understand Rubin's points though. Even the transformation rules template gives a particular set of rules. However, that isn't intended to represent any particular system, but rather gives common rules used in various systems. At some point I think this kind of information (i.e. Dooot) is expected in a comprehensive encyclopedia article on particular rules of inference. Greg Bard (talk) 19:25, 26 February 2012 (UTC)

I think they need to be cited or deleted. If there really is some sort of subtle problem introduced by a logic aficianado, it would be fuel for confusion. Secondly, has anyone figured out why there are two "proofs" here? To all appearances the second seems like a less efficient duplicate of the first. Rschwieb (talk) 20:53, 27 February 2012 (UTC)

False (logic), contradiction and principle of explosion

A help needed from experts in logic. I recently wrote a small article about the false (I complained about its absence since 2009, and nobody else made it), but I probably failed to explain a subtle difference between false and contradiction. We all understand the difference between logical truth and a theorem, and there should be the same on the dark side of the logic. So, it would be nice to clarify the terminology in "contradiction" and "principle of explosion". What is contradiction: an occurrence or use of the false in proofs? A proof-theoretical interpretation of the false distinguished for historical reasons from, say, truth-functional one? And what terminology ("false", "contradiction", both as synonyms, or distinction) to use in "principle of explosion"? Incnis Mrsi (talk) 18:37, 26 February 2012 (UTC)

I don't really think people understand the relationship between logical truth and theorems at all. All we can agree on with consensus is that they "are related." You should take a look at the history and talk pages of rule of inference, theorem, and logical truth. I recently made a description at Rules of inference,Theorems, and Arguments to address one of the issues which you bring up. In many cases we have popularly used or named theorems which are also rules of inference, and also argument forms, etcetera. So in some cases we have one and not the other. It's not consistent. Greg Bard (talk) 19:25, 26 February 2012 (UTC)

I think one of the main problems lies with the part about contradiction, especially this sentence: "Contradiction means that a statement is proven to be false". In my opinion, this is not correct. A contradiction is a statement that is false for any possible evaluation. That does not necessarily mean the same as "proven to be false". Take for example the sentence "People watch TV" opposed to "People watch TV and People don't watch TV"; the first one can be proven false, but it is not necessarily always false. However, the second sentence is always false because it is an inherent contradiction. So basically what I'm trying to say is that there is a difference between proven something to be false and having something that is always false (a contradiction). I think this contributes in great part to the confusion, as well as the sentences surrounding the one I mentioned. Other than that, I agree with the commenter above that the relationship between logical truth and theorems is probably not clear to everyone. Mythio (talk) 19:31, 26 February 2012 (UTC)

Some people try to convince me what the contradiction is not. But what the hell it is, indeed? Look at the history of False (logic) (edit|talk|history|links|watch|logs) and to its talk page. Initially it was a redirect to Logic. In mid-2009 I changed it to "contradiction" because I felt that it is the closest target of all articles. After a half year it was reverted; more exactly, a user changed it again to Logic, may be independently of the past history. Then I complained to WikiProject Logic about a bad redirect, to no avail. And now, when there is already a small article, no one can explain why "contradiction" was not a possible target for a redirect! Incnis Mrsi (talk) 20:31, 26 February 2012 (UTC)

Simply put a contradiction is a statement that is false in all possible interpretations. False is not the same as a contradiction, since something can be false in some cases and not in all. With this in mind, I hope its clearer why, imo, a redirect to contradiction is not the correct course of action. A possible solution could be Hans Adlers proposal in here. On another note, Its not clear to me what your aiming for with this article when reading your reply; what are you trying to make this article about? For instance the sentence you start with is "false is the opposite to logical truth". The argument could be made that this is incorrect, because reading the definition of logical truth on its page, the opposite would be a contradiction (and hence the article already exists). Could you clarify a bit further what you understand to be the concept of false perhaps? Mythio (talk) 20:56, 26 February 2012 (UTC)

I do not take Hans Adler's proposal seriously first because he eventually did not even attempt anything, and second, because he argued for a merger of all possible targets to Truth value, which is "not the correct course of action" for reasons mentioned in this discussion and also in the discussion just above (I had not so strong feeling yet in 2009). IMHO the logical truth article is a bit confusing in an ambiguity and lack of distinction between (the abstract) logical truth and a property of a statement to be necessary true. It explains in details, what means "necessary", but does not explain, what means "true". What I understand to be the concept of false? First, a proposition which is a priori opposite to the truth, such as "⊥" nullary connective in those versions of propositional calculus which have it. This is not exactly symmetric to "⊤", if we use only material conditional, conjunction and disjunction. We easily can (re)define "⊤" as p→p but cannot define "⊥" if we have not a negation yet. Second, the false as a truth value, which is different from the truth, and which is always assigned to "⊥". Incnis Mrsi (talk) 22:23, 26 February 2012 (UTC)

┌────────────────────────────────────────────────────────────────────────────────────────────────────┘Perhaps something should be said about how classical logic can get away with treating all non-truths as equivalent to each other? JRSpriggs (talk) 10:27, 27 February 2012 (UTC)

I think the best answer is simply to delete false (logic), given that it's ambiguous between the truth value "false" and the notion of "logical falsehood" (that is, logically necessary falsehood, which is very close to if not the same as contradiction). If it were actually a useful link, I suppose you could set up a disambig page, but I do not understand what is the rationale for having such a link at all. False (logic) is an unlikely search term and a bad internal link; having it around seems to do nothing but encourage overlinking. There is rarely if ever going to be any good reason to link the word false at all. --Trovatore (talk) 03:39, 29 February 2012 (UTC)

I would strongly support keeping the article, which is linked from the dab page False. It logically covers the truth value and "⊥". There is plenty of material around to improve the article with. -- 202.124.75.236 (talk) 12:26, 1 March 2012 (UTC)

I support keeping the article. There is tautology which is the negation of a contradiction; there is theorem, but no nontheorem (which is not the same as the negation of a theorem, and could easily be an article of its own). I don't see why we can't have logical truth and logical falsehood. However False (logic) should be about the truth value, not about contradiction, and not about logical falsehood which are types of sentence or proposition. Greg Bard (talk) 21:35, 1 March 2012 (UTC)

Projective range

I came up on the page Projective range. It lacks of a formal definition that I can understand. Should it be expanded or deleted? D.Lazard (talk) 22:47, 26 February 2012 (UTC)

This is standard classical terminology in projective geometry. Several collections of specially related objects are referred to as pencils (a pencil of lines through a point in a plane, a pencil of hyperplanes through a codimension 2 space, a pencil of conics, etc.). In all these cases the dual concept to a pencil is called a range. The page in question is certainly not well written and lacks clarity, but I would say that it needs to be expanded rather than deleted. I don't see myself doing that for at least a couple of weeks. An alternative might be a merge into Duality (projective geometry), where the originator of Projective range has placed an out of context sentence containing a link to the new page. Bill Cherowitzo (talk) 06:06, 27 February 2012 (UTC)

My understanding is that the generic term is "projective form" which would include projective pencils, projective sheaves, etc. So I'd suggest changing the name covering all of these in the article.--RDBury (talk) 07:15, 27 February 2012 (UTC)

Proposing stubs: Zero algebra, Trivial algebra, Flexible algebra

The following "trivial" or "uninteresting" cases seem to have encyclopedic value at least inasmuch as they provide clarity for someone seeking to understand their status. Unlike a field with one element, they appear to be uncontrovertial:

• Zero algebra is used to mean an algebra over a field (or ring) in which the product is given by the map a × b ↦ 0. These are usually uninteresting, but appear to be mentioned in some sources. I imagine it sees use in the classification of algebras.
• Trivial algebra is an algebra over a set of one element (see nLab, mentioned in Quasivariety). It is a category covering several cases such as singleton set, trivial group and trivial ring.
• Trivial algebra over a field (or ring) is an obvious example of a trivial algebra, on par with trivial group and trivial ring. It would seem appropriate for completeness, but I've not seen reference to this specifically.

I've come across another term in a few places and papers, enough to warrant a stub:

• Flexible algebra, defined as an algebra with the property (ab)a = a(ba) for all a and b. (See Planetmath.)

Any opinions on the creation of these stubs? — Quondum^☏✎ 07:21, 27 February 2012 (UTC)

In general I'd oppose creating a stub unless there is reasonable expectation that the article can be expanded beyond that or it would awkward to put the material in another article. For example for zero algebra I's suggest adding it as section of Algebra over a field, then creating a redirect to that section. It would be reasonable to include the zero algebra there anyway as an example. The fact that articles for similar objects should not carry any weight here, see WP:OTHERSTUFF.--RDBury (talk) 09:42, 27 February 2012 (UTC)

Some comments about "zero algebra": There are two notions of zero algebra. The non unital one that you consider, which is, in fact, another name for "module", and the unital one which is the direct sum of the basis ring and a module, with null product for two elements of the module. This is not an "uninteresting" notion, at it allows frequently to extend straightforwardly to modules and submodules some notions which are primarily defined for rings and ideals. For example, this allows, not only to extend the theory of Gröbner bases to modules and submodules, but also to use for submodules any implementation of a Gröbner basis algorithm for ideals. Thus the question is what is the right place for "zero algebra", in module (mathematics) or in algebra (ring theory) or in both. D.Lazard (talk) 10:38, 27 February 2012 (UTC)

Thanks for the input. I have followed RDBury's guideline and avoided stubs (and not even created redirects), but inserted sections for zero algebra in Algebra over a field#Kinds of algebras and examples and a point on flexible algebras in Algebra over a field#Non-associative algebras. I am having difficulty reconciling an algebra with an identically zero product with a module, so I'm leaving that as beyond my present understanding. The rest I've left alone. — Quondum^☏✎ 15:24, 27 February 2012 (UTC)

I came too late for "zero algebra", but put in my 5коп for "trivial algebra". In Wikipedia, it may be referred as a "trivial algebra", although such link lies on the edge of WP:EGG. Zero-dimensional space just has no dimensions at all. Any possible linear structure is not relevant, because may be either trivial or not existent. Incnis Mrsi (talk) 15:50, 27 February 2012 (UTC)

Got wrong, not yet and probably not such. I expected an article about 0d linear space, but it is about a topological one. Something to be disambiguated and fixed. Incnis Mrsi (talk) 15:57, 27 February 2012 (UTC)

Now all is in order, trivial algebra is linkable. By the way, I discovered and fixed a severe mistake. One user thought that "trivial module" is something like "zero algebra" from this topic, which is (according to MathWorld) not true. All three 0-dimensional algebraic entities are now in one article. Incnis Mrsi (talk) 16:47, 27 February 2012 (UTC)

It's probably not so much a severe mistake as it is a difference in convention. A google search immediately returns several instances of "trivial module" meaning any module for which mr=0 for any choice of elements m in the module, r in the ring. Both the concept you put in and this one are "trivial" in some sense so it is natural different people use them different ways. I think we should modify it a bit to reflect this, as is done with zero/trivial ring. It's a little arbitrary to declare one sense correct, here. It only perpetuates the confusion of usage. Rschwieb (talk) 18:06, 27 February 2012 (UTC)

So, I can propose to rename it to zero space to wipe an ambiguity out ultimately. This name has an advantage to omit mentioning of a ground object (either a ring or a field). Happily, I have a technical possibility to kick the current redirect off. Certainly, if there was no objections here to this move. Incnis Mrsi (talk) 18:35, 27 February 2012 (UTC)

I think renaming should depend on which usage dominates. If the trivial product version mr=0 usage is in the minority, I would suggest keeping the name as is, but mention this as an alternative interpretation of the term within the article. Technically, a zero vector space is still over a base ring/field, and thus does not really get rid of mentioning which. One could just as easily omit this mention in the case of a trivial algebra/module. — Quondum^☏✎ 18:54, 27 February 2012 (UTC)

Yes, I agree, it should be based on which usage is predominant among workers in the field. Not being one of them, I'm not sure which one that would be. But I want to urge very strongly that the decision not be based on MathWorld. MW has some uses, but a reliable source for nomenclature it is not. --Trovatore (talk) 00:17, 28 February 2012 (UTC)

Okay, I see how the unital zero algebra works. I generally don't know what is meant by "direct sum", since its meanings can be so different. This would imply that dual numbers constitute a unital zero algebra over the reals. Thanks for the assistance. — Quondum^☏✎ 17:46, 27 February 2012 (UTC)

Exactly, the dual numbers are the unital zero algebra build up from a real vector space of dimension one. I'll add this example in algebra over a field. D.Lazard (talk) 19:00, 27 February 2012 (UTC)

Serif/sans-serif for math expressions in running text

I could put this on the talk page of trivial module but we all seem to be here and it might come up in the other articles, so here goes: Quondam's latest change involved using the {{math}} template to boldface some zeroes. Personally I do not feel this is an improvement. (I concede that my feeling on this may be influenced by the fact that I loathe that template in general, mostly for its imposition of serif fonts in running sans-serif text rather than for the boldface). I would prefer to remove that part of Quondam's edit, while keeping the copyedits. Thoughts? --Trovatore (talk) 19:07, 27 February 2012 (UTC)

AFAIK the template does not bold anything. The intent is consistency of the font of the symbols (including numerals) throughout the text (as distinct from stand-alone lines) within each article. This is a reasonable forum to discuss the idea of a more uniform serif/sans-serif choice or guideline for symbols/math globally across math articles. Replacing {{math}} with {{nowrap}} is easy if consensus is for a sans-serif font. I find the serif font aids interpretation by visually distinguishing math and text without the formatting problems of the <math> tags, but I'll be happy to go with consensus. — Quondum^☏✎ 14:54, 28 February 2012 (UTC)

I prefer serif to sans-serif for math expressions, so I am in favor of {{math}} but at this point I think that changing over to it is somewhat of a waste of effort. What we should be doing instead is pushing harder to get MathJax support made more standard in Wikipedia, relative to its current experimental status, and then using <math>. —David Eppstein (talk) 17:14, 28 February 2012 (UTC)

In isolation serif is better. It's the mixture of serif with sans-serif that I object to. If {{math}} continues with serif, then it should be used only displayed, the same as <math>. --Trovatore (talk) 19:17, 28 February 2012 (UTC)

It is just imitating what <math> does with TeX. The target is to use Tex eventually but at the moment the results can be ghastly inline. If you're really worried by this then you should stick in some private css that ensures {{math}} uses a non-serif font. In the long term you'll need to ensure that Mathjax has some option to do the same sort of thing because many more things will use TeX when it works inline okay. Dmcq (talk) 00:08, 29 February 2012 (UTC)

In my view one of the things that's "ghastly" about inline TeX in the current implementation is the serif/sans mixing. It's not as bad as the mixed sizes, but it's still pretty bad. MathJax is still useful even if we use it only displayed, because it just renders so much better than PNGs. --Trovatore (talk) 00:55, 29 February 2012 (UTC)

I've just started running with MathJax enabled and I'm very happy with its inline capabilities though I'll be using the math template for a while because of problems with the current PNGs and how long I know it will b before it is generally supported. You really do need to have a way of setting the font selected by MathJax and the math template if you want to avoid serif fonts inline. What you want will never be the general default so you'll need an option. Dmcq (talk) 01:12, 29 February 2012 (UTC)

It's not for me. Mixing serif and sans looks bad. We shouldn't do it, for anyone. --Trovatore (talk) 01:16, 29 February 2012 (UTC)

I happen to like it mixed like that. It makes the math variables more clearly variables rather than one-letter words. —David Eppstein (talk) 02:10, 29 February 2012 (UTC)

You know, that might be OK if the typefaces were somehow designed together, like a serif version and a sans version of the same typeface. But I have never actually heard of such a font, and in my estimation the ones that actually render look awkwardly jammed together. The overall effect is something like one of those stereotypical ransom notes with letters cut out of magazines — not so extreme, of course, but along those lines. To me it comes across as distracting and unprofessional.

While there are advantages of serif fonts in mathematics, I think it's worth noting that the Beamer class uses sans, and this has not impeded its widespread adoption. --Trovatore (talk) 02:36, 29 February 2012 (UTC)

┌────────────────────────────────────────────────────────────────────────────────────────────────────┘ Before the debate goes too far, I think it should be noted that this topic has been covered before (e.g. here), and it seems clear to me that consensus on this question will not emerge here. In particular, there is enough support for inline {{math}} use that no recommendation to the contrary will be accepted. I think the inadequacy of the fonts in the context of math and browsers should be addressed as a broader WP issue, not at the template level. So I think the only principles that will emerge are already in place:

• The original writers of an article are free to set their own style
• One may fix inconsistencies of format style in individual articles
• Don't switch the style of an entire article without consensus.

— Quondum^☏✎ 06:46, 29 February 2012 (UTC)

There are a bunch of people who really don't like it when two fonts that aren't extremely similar are used together. Some cry out in pain if they see a poster just using comic sans. For those people there needs to be facilities for changing the font in the math template, in MathML and when MathJax gets used. They are not going to get he default changed, the Tex font is accepted in professional maths books, and in the future there will be even more of it in the running text, so they have to just accept that and move on to what can be done to fix the situation as far as they're concerned. Dmcq (talk) 09:15, 29 February 2012 (UTC)

No, you're quite wrong here. Of course the TeX font is accepted in professional math books — they're written in that font! So it looks fine. It looks horrible when mixed with sans. Having the mixture as a default is unacceptable. --Trovatore (talk) 17:48, 29 February 2012 (UTC)

Do you or do you not accept that you're talking to the wrong people and the math template is the wrong target for doing anything about fixing what you want fixed? That in fact the math template is the easiest one for you to fix for your own use by setting your private css but if you don't engage with MathJax any effort with the math template will eventually be totally wasted? Dmcq (talk) 18:39, 29 February 2012 (UTC)

How am I talking to the wrong people? WikiProject Math is where standards for mathematical articles get discussed. My private use is entirely irrelevant here — I'm arguing to improve the professionalism of the display of our articles. --Trovatore (talk) 20:11, 29 February 2012 (UTC)

The people here have very little control over technical issues on Wikipedia such as the use of MathJax as a default. WP:VPT may be a better choice for that. —David Eppstein (talk) 20:32, 29 February 2012 (UTC)

But what we can discuss is whether it ought to be used inline. --Trovatore (talk) 20:34, 29 February 2012 (UTC)

You're not seriously suggesting we try banning all inline maths? Dmcq (talk) 22:16, 29 February 2012 (UTC)

Not a hard ban, no. But I am saying that if MathJax uses a distractingly different font from the rest of the article, we should try hard to avoid inline uses of it where reasonably feasible, and HTML may still be better than MathJax for unavoidable inlines. --Trovatore (talk) 22:46, 29 February 2012 (UTC)

I think we should avoid using the {{math}} template (and its cousins like {{var}}, etc.) This makes the code more difficult to edit by hand, and it is looking very likely that MathJax will give a much better solution very soon. I think it's time we start to deprecate html (and these funny templates). Sławomir Biały (talk) 11:48, 29 February 2012 (UTC)

The normal Tex to html or png converter could have been fixed ages ago to do most of the things {{math}} patches over like being able to put ≥ in the inline text without getting some png messing it up. $a \geq 0$ or a ≥ 0 anybody? How soon is soon when we couldn'r fix something like that in years? Dmcq (talk) 14:05, 29 February 2012 (UTC)

What might be worth developing in time for Mathjax is a special tool which would search for {{math}} templates and automate converting them to <math> format. That way we needn't deprecate anything for the moment and do a better job both now and eventually. Dmcq (talk) 14:13, 29 February 2012 (UTC)

Just to clear some illusions about MathJax... It too wil use a serif font, even when used inline. It uses webfonts for rendering, so it has the potential to clash with whatever font a user has set for running text. With regard to matching fonts, I know of only one that matches serif and sans-serif in design and size: DejaVu. I you already use DejaVu Sans as the default font, use this CSS to display formulae in {{math}} in a matching serif:

span.texhtml {
  font-family: 'DejaVu Serif', serif;
  font-size: 100%;
}

— Edokter (talk) — 16:47, 29 February 2012 (UTC)

See e.g. this edit for why sans-serif inline math is bad. If we can't distinguish the capital vowel I from the lowercase consonent l from the digit 1 from the vertical bar |, we have a problem. (In the font I use, the digit is clearly distinguishable from the other three, but obviously even that's not true for everyone.) —David Eppstein (talk) 21:47, 1 March 2012 (UTC)

Honestly I like serif better than sans in general. I wish Wikipedia as a whole were written in a serif font (see e.g. a recent discussion at talk:Iago). But the "ransom note" effect is real, and distracting, and detracts from our image of professionalism. The workaround for the issue you bring up is to strain to avoid using those letters as variables. --Trovatore (talk) 21:54, 1 March 2012 (UTC)

I think there are reasons that sans-serif tends to work better with screens than on paper which are very different media, something to do with interlacing and resolutions. MathJax does have some support for sans-serif fonts, but its limited to only the standard letter and numbers. If you look at many of the other symbols in maths $\sum\int \theta$ the fonts used have more in common with serif fonts: varying line widths and little serifs. Going completely over to sans wound end up with a very mixed typography within equations. Better to mix sans text with serif maths, at least those are used for completely different things.--Salix (talk): 01:31, 2 March 2012 (UTC)

I can buy that it may not be workable to make MathJax do sans. But in that case I think MathJax is not really a solution to the inline problem (so basically we still don't have, and may never have, a solution to the inline problem). And if that is the case, then we should continue to avoid inline mathematics to the extent feasible. --Trovatore (talk) 02:11, 2 March 2012 (UTC)

I'm pretty certain there's some straightforward way of getting the normal running text in Wikipedia to use any font you like so you could use a serif font for that if you like. I'm sure someone on the help desk could do that fairly easily. That would probably be quite an easy option to put in the appearance preferences for general use and might be quite popular. Dmcq (talk) 09:40, 2 March 2012 (UTC)

Please quit talking about what I would like for my personal use. That's entirely beside the point. My objection is to a problem that detracts from the professionalism of the appearance of our articles. --Trovatore (talk) 22:18, 2 March 2012 (UTC)

I think this suggests that some thought should be given to coordinating the default fonts (perhaps only for maths articles?) for the main text, {{math}}, <math> and MathJax, much as is done in professional texts. This would mean some override of the browser's default font choice for serif and sans-serif, and carries with it the pitfall that these may not be installed fonts for a large enough base. If serif is to be avoided due to display problems, and widely installed suitable matched serif and sans-serif fonts cannot be found, this problem is probably going to be around for longer than we'd like. — Quondum^☏✎ 13:12, 2 March 2012 (UTC)

It will only be people who are worried by this sort of thing who would set a preference and we can have a help page link in the preferences page we direct them to show them how to download fonts if they have problems. Fonts on the web have a good fallback facility so the main problem is if a font is used that has bad looking characters in it. Dmcq (talk) 23:27, 2 March 2012 (UTC)

The problem is a problem for everybody, not just for people who set preferences. I am completely opposed to shunting it off into preferences. --Trovatore (talk) 23:35, 2 March 2012 (UTC)

I concur with Travatore – in principle. But until we can identify suitable fonts for setting as the Wikipedia defaults with most browsers and typically available fonts, this does not seem achievable. Is there a suitable set of fonts typically available with all browsers that is suited? (DejaVu is not generally installed, and has other drawbacks IMO.) Alternately, is it necessary/desirable to remain with sans-serif in the body of the article, or could we switch to serif (for maths articles)? I personally like the serif/sans-serif contrast between math and text, but jarring discontinuity as would occur with badly mismatched fonts should preferably not occur in the typical default setup. — Quondum^☏✎ 07:49, 3 March 2012 (UTC)

See WP:Typography. There really is not much choice. When I created {{math}}, I took into account what most users would be seeing on screen, with default fonts installed, at default sizes, and tried to match up as best I can. It is impossible to take every deviation into account, presicely because teh lack of standards in web typography. What prevailed in my mind is the legibility of math, which suffers badly in a sans-serif font. — Edokter (talk) — 10:15, 3 March 2012 (UTC)

Jean-Claude Sikorav

Jean-Claude Sikorav is a new article by Tkuvho, which will probably soon be listed on AfD. Comments/improvements are welcome. Sasha (talk) 19:52, 27 February 2012 (UTC)

It has just been the topic of a somehow frantic (and sometimes badly informed) deletion review on :fr (the result was "Keep"). The article had been created there by a group of two juvenile students of École Normale Supérieure de Lyon, where JCS teaches, and I am not yet sure whether they were serious or if they were playing a funny joke with Wikipedia. You can have a look at the (quite unhealthy) debate at fr:Discussion:Jean-Claude Sikorav/Suppression. French Tourist (talk) 21:17, 27 February 2012 (UTC)

I agree that, unless there is some clearer sign that the article passes WP:PROF, it is likely to be a subject of a deletion discussion, particularly because it was already deleted once under CSD and undeleted by me. My undeletion was only to give more time to add content. I think the current content of the article does not make for a clear case that the notability guidelines are satisfied. — Carl (CBM · talk) 22:18, 27 February 2012 (UTC)

I think the citations in Google scholar and the award are enough that he would probably pass an AfD. (I'm not convinced that calling out the citation count explicitly within the article is a good idea, though.) —David Eppstein (talk) 23:33, 27 February 2012 (UTC)

Category:Theorems in number theory

Are there any suggestions of subcategories for this category in order to improve the usefulness of this category? At the moment there is around 80 articles in it which may be daunting to somebody looking through the theorems of number theory or maybe not. At the moment there are a few subcats; a couple which are more specific branches of number theory and another relating to the prime numbers, but perhaps there are more that could be added for ease of the user. Perhaps related - where does number theory and algebraic geometry intersect? Brad7777 (talk) 00:45, 28 February 2012 (UTC)

I would suggest "theorems in additive number theory", "theorems in the geometry of numbers", and "theorems on discrepancy" (or perhaps "theorems on equidistribution", a subcat of "theorems in analytic nt") -- if we can populate them in a reasonable way. Also, many of the theorems now in "theorems in number theory" can be moved to either algebraic or analytic n.t. (e.g. Mordell-Weil -> algebraic n.t., Turan-Kubilius -> analytic n.t., ...) Sasha (talk) 05:29, 28 February 2012 (UTC)

Maths rating template name

There is a suggestion at Template talk:Maths rating to rename the template. — Carl (CBM · talk) 11:56, 28 February 2012 (UTC)

Administrator requested

Wikipedia:Sockpuppet investigations/119.154.67.223 notes disruptive editing by three neighboring IPs.

Blocks for disruptive editing are warranted, regardless of the SPI issue.  Kiefer.Wolfowitz 12:12, 28 February 2012 (UTC)

Make that FOUR neighboring IPs.

Please help relieve poor Melcombe from vandal fighting against this Lahore-based terminator/Energizer Bunny from hell.  Kiefer.Wolfowitz 12:18, 28 February 2012 (UTC)

There seems to be about 8 users from this Lahore prefix, perhaps one of whom is not a disruptive editor.

Please protect the pages mathematician, statistics, etc. from IP editors.  Kiefer.Wolfowitz 12:21, 28 February 2012 (UTC)

It looks like the changes are being reverted OK at the moment, and there is some presumption against protection as long as reverts are effective. But, if the problem grows particularly large or particularly long-lived, contact me and I can protect the pages. — Carl (CBM · talk) 14:25, 28 February 2012 (UTC)

The placement of articles on fields of mathematics into Category:Fields of mathematics

There are many articles on fields of mathematics, examples include algebra, geometry etc. There also exists a category Fields of mathematics. This category at the moment contains a few selected eponymous categories aswell a few articles related to the term "fields of mathematics". The category Fields of mathematics is a set category, i.e a category named after a class so I think it would be logical to include all articles that fit this class, i.e all fields of mathematics like algebra, geometry etc. I have brought this idea up before but as I was a new editor I was not able to explain it. I hope those who saw my previous effort now understand what I mean. I think not only this logical, it is also very practical for a user-browser of Wikipedia, particularly those with interests across the scope of mathematics, who do not necessarily want to have to dig deep through the subcats. Of course, this should't be done without consesus, so views? Brad7777 (talk) 20:29, 28 February 2012 (UTC)

So you would like to move more field categories? There are about 20 subcats now, but it seems like that might go way up depending on what you are aiming to include. Do you have a concrete list of additions to that category or at least an estimate on how many things would go in? Rschwieb (talk) 14:09, 1 March 2012 (UTC)

I mean adding the articles to this category. These would not be taken from their current categories, they would have just an additional category Category:Fields of mathematics. There is an estimate on Glossary of areas of mathematics. Brad7777 (talk) 17:53, 1 March 2012 (UTC)

Sorry, I knew you just meant adding but since I never mess with categories I might have misspoke. The glossary appears to have hundreds of items, but I don't know if they should all go in. It would certainly seem logical to have areas of mathematics categorized under Fields of mathematics. Will anyone else contribute their 2 cents? Rschwieb (talk) 20:19, 1 March 2012 (UTC)

{{Maths rating}}

template:Maths rating is under discussion, please see template talk:Maths rating

70.24.251.71 (talk) 05:35, 29 February 2012 (UTC)

'New' math rendering options

Check your preferences: with MW 1.9 the math rendering options are down to two: always PNG and display TeX. See the RfC here: [3]. Renders some of the Math MOS redundant, such as MOS:MATH#Forcing output to be an image and MOS:MATH#Very simple formulae. Probably the whole section needs rewriting.JohnBlackburne^words_deeds 02:19, 1 March 2012 (UTC)

You mean MW 1.19 which has just been rolled out. PrimeHunter (talk) 03:00, 1 March 2012 (UTC)

That's really really stupid if it means what I think it means. Anything like that should have been delayed until MathJax or another way of displaying stuff properly inline had been implemented properly. Dmcq (talk) 13:02, 1 March 2012 (UTC)

Yep it is what I thought. I was testing MathJax so I didn't see it before. Luckily or unluckily people seem to have tried avoiding <math> inline in lots of places so the effect isn't as bad as it might be. I really do think it would be a good idea to have a conversion tool to help change {{math}} uses to <math> when MathJax comes along, this latest business is going to force even more instances of <math> to be turned into {{math}} to avoid the ugliness of the inline PNGs. Dmcq (talk) 13:17, 1 March 2012 (UTC)

At some point soon (i.e. maybe during 2012), mathJax should become the default for everyone. Developers are working on it. Does this latest roll-out have anything to do with progress in that direction? Michael Hardy (talk) 16:17, 1 March 2012 (UTC)

The big problem now is the baseline problem with $x^a$ being positioned too low. Brion said he was going to investigate this in the RfC, but it looks like it hasn't happened.--Salix (talk): 16:59, 1 March 2012 (UTC)

The relevant bugs are the following:

• Bug 31406 - Set $wgMathUseMathJax = true on Wikimedia wikis
  • This depends on/blocks other bugs
• Bug 32694 - Use baseline shift when positioning inline math PNGs

Helder 17:18, 1 March 2012 (UTC)

I've now responded to the former bug, it looks like MathJax might be just round the corner.--Salix (talk): 18:38, 1 March 2012 (UTC)

What is a product?

A mathematician may say "I have proved that the following products are both equal to 5:

$\prod_{i\in A} x_i,$

$\prod_{j=1}^{10^{90}} y_j.$

If they're both equal to the same number, and a product is the value that results from multiplying, and these are both equal to 5, then these are not two products, but one. Our article titled product (mathematics) says:

a product is the result of multiplying, or an expression that identifies factors to be multiplied.

The latter usage occurs in such expressions as the title of a book called Table of Integrals, Series, and Products or articles titled "Proof that a Product Considered by Schriemann Diverges to Zero". Yet it seems many sources say only that a product is the value of a multiplication operation. A non-logged-in user has been arguing on my talk page that we should therefore omit the "expression" characterization from the definition given in the article.

Opinions of this proposal? Michael Hardy (talk) 16:24, 1 March 2012 (UTC)

The duality of the meanings of terms such as "product" (and this obviously is not limited to product, but includes limit, etc) is one of the most active areas in math education. Certainly we should keep both meanings on the product page and avoid excessive formalism at all cost. Tkuvho (talk) 16:30, 1 March 2012 (UTC)

This is the usual intensional vs. extensional equality issue that appears everywhere. Is $\lim_{x \to 0} x/x$ the same as $\lim_{x \to 5} 1$? Is $2 \cdot 15$ the same product as $6 \cdot 5$? The same issue arises with derivatives and integrals that have equal values, and with whether two groups are the same if they are isomorphic, and with many other objects. The real point is that when mathematicians say "equal" or "same" they can mean many things depending on context. Regarding products specifically, the current language looks good: sometimes the product is identified with its value, some time it is not. — Carl (CBM · talk) 16:31, 1 March 2012 (UTC)

One of the most active areas in maths education? Sheesh. So someone with maths skills has to both want and like to teach children but also be willing and able to put up with this sort of stuff being thrust at them as being the way to teach maths? Explains a few things. Dmcq (talk) 17:07, 1 March 2012 (UTC)

@Dmcq : I doubt that he was proposing to actually tell children about this stuff. Michael Hardy (talk) 22:52, 2 March 2012 (UTC)

Meriam Webster has both meanings: http://www.merriam-webster.com/dictionary/product . -- Jitse Niesen (talk) 17:14, 1 March 2012 (UTC)

Surely the non-ultra-formalist way of saying this would be "...are two expressions with the same value", or "two different formulations for what can be regarded as the same object"? -- The Anome (talk) 17:18, 1 March 2012 (UTC)

Both usages are important. It is counterproductive to favor one over the other. Rschwieb (talk) 17:45, 1 March 2012 (UTC)

This is a special case of the problem of making the proper distinction between an expression, a function, the value of a function for a given value of its argument and the evaluation of an expression. I would say, for the example given by Michal Hardy, "I have proved that the two following expressions both evaluate to 5". In other words, a product is an expression whose leading operator is a multiplication, and saying that two products are equal is an abuse of language and a shortcut for "when evaluating the functions and operators appearing in the two expressions, we get the same result". Thus I would suggest for the article product (mathematics):

"A product is an expression whose leftmost operator is a multiplication. By abuse of language product denotes also the result of the evaluation of the operations appearing in such an expression"

All of this is not WP:OR. I may not cite any math book for this, but it is the basis for any computerization of the mathematics and appears in some way in the manuals and tutorials of every computer algebra system, like Maple (software).

D.Lazard (talk) 18:42, 1 March 2012 (UTC)

Seconding Tkuvho and Rschwieb. D.Lazard, your proposed alternative, whatever its advantages, will do much to make the article unimpenetrable for many readers. (One possible confusion that it will create: it appears to assert that $x \cdot y + z$ is a product.) The current wording is clear, correct, easy to understand, and should be kept as-is. --Joel B. Lewis (talk) 19:37, 1 March 2012 (UTC)

"Unimpenetrable"? Maybe impenetrable? Rschwieb (talk) 20:13, 1 March 2012 (UTC)

Isn't it about the mathematical counterpart of "Sense and reference"? Boris Tsirelson (talk) 19:44, 1 March 2012 (UTC)

I agree that my formulation is not convenient for the level of the readers and that the current wording is convenient. About "leftmost" : "outmost" or "top" or something like that would be better. But the point is that we have to have this kind of things in mind when looking for the best formulation. Another example of today: The first sentence of discriminant was "the discriminant of a polynomial is an expression ..." which is incorrect. I have replaced this by "the discriminant of a polynomial is an element of the ring generated by its coefficients", which is correct but has been reverted as too technical. After discussing with the author of the reversion, the formulation is now "the resultant of a polynomial is a function of its coefficients", which is sufficiently correct (it is not a function, but the value of a function), easy to understand and contents more information that the previouus formulation. D.Lazard (talk) 20:47, 1 March 2012 (UTC)

I have to agree that that way of using the word "leftmost" will be---um------"unimpenetrable" to almost everyone. I understood it here only because of the context of this present discussion. And I think other aspects of that proposed opening sentence are objectionable on grounds almost as cogent as that. Michael Hardy (talk) 22:55, 2 March 2012 (UTC)

Hello to all,

I am that non-logged-in user (but I learned how to log-in now!) who asked the question from Michael.

First let talk about the difference between multiplication and product in Natural numbers (ℕ). Multiplication in ℕis a binary operation which is a function from ℕ×ℕ to ℕ, so it gets two Natural numbers as input and the result or output of this function is another Natural number. The mathematical symbol for multiplication function is ×, so in function notation we can write: ×(3,4)= 12. In infix notation we can put the operation (here ×) between two operands (here factors) and use the notation 3×4=12. But we know a function is a set of single valued ordered pairs. In this point of view multiplication is a set like ×={((1,1), 1), ((1,2), 2), …} and one of its elements is ((3,4), 12) and the output or value or result 12 associated with the pair (3,4).

By the present definition, product refers just to the result (and result can be a number or an expression like Meriam Webster but still refers just to the result). The difference between product and multiplication is like as the difference between element and set.

There is another close example. When you say the function f(x)=x² actually you omit two important parts, domain and codomain. This function is not one-to-one from ℝ toℝ but it is one-to-one from ℕ to ℕ, so you can omit the details if there is no ambiguity.

I agree both usages of the product are important, so it seems we need to change the definition of the product, but how? By inserting in Wikipedia? I think this is not a good idea because a divergence will appear between Wikipedia and other references. It is better to think for a better way. — Preceding unsigned comment added by Sohrab.Rahbar (talk • contribs) 03:09, 3 March 2012 (UTC)

Mathematical formulas in the lead section of an article

WP:MOSINTRO used to say "Mathematical equations and formulas should not be used except in mathematics articles." So in that state it wasn't really relevant to this project, since it was only about other articles. But in this edit a month ago, an editor (intending to broaden it to allow formulas in technical but non-mathematical articles such as Joule) changed it to instead say "Mathematical equations and formulas should only be used when absolutely necessary." Today this has led to an editor on golden ratio attempting to take all the math out of the lead section there, because math articles are no longer exempt and he didn't see why it was necessary. So anyway, this is just a heads up: discussion on the issue has started at Talk:Golden ratio for the specific editing concerns there, and Wikipedia talk:Manual of Style/Lead section for what the MOS should actually say about this. —David Eppstein (talk) 05:30, 2 March 2012 (UTC)