# What could be built on top of a World Digital Math Library?

The title is copy-pasted from the following question by Ingrid Daubechies, on mathforge.org [I added some links]:

Suppose most mathematical research papers were freely accessible online.

Suppose a well-organized platform existed where responsible users could write comments on any paper (linking to its doi, Arxiv number, or other electronic identifier from which it could be retrieved freely), or even “mark it up” (pointing to similar arguments elsewhere, catch and correct mistakes, e.g.), and where you could see others’ comments and mark-ups.

Would this be, or evolve into, a useful tool for mathematical research? What features would be necessary, useful, or to-be-avoided-at-all-costs?

This is not a rhetorical question: a committee of the National Research Council is looking into what could be built on top of a World Digital Math Library, to make it even more useful to the mathematical community than having all the materials available. This study is being funded by the Sloan Foundation.

Input from the mathematical community would be very useful.

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UPDATE:  David Roberts points to the fact that Daubechies asked the same question at mathoverflow before asking at mathforge. The answers are much more welcoming there, interesting read.

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Notice: “a World Digital Math Library”, not “the World Digital Math Library”. Concerning the involvement of the Sloan Foundation, that would be great, let me cite again the Conjecture 4 by Eric Van de Velde, proposed  in   MOOCs teach OA a lesson:

OA is not sufficiently disruptive. Hoping to minimize resistance to OA, OA advocates tend to underemphasize the disruptiveness of OA. Gold and Green OA leave the scholarly-communication system essentially intact. When presented in a minimalist frame, they are minor tweaks that provide open access, shift costs, and bend the cost curve. Such modest, even boring, goals do not capture the imagination of the most effective advocates for change, advocates who have the ears of and who are courted by academic leaders: venture capitalists. This is a constituency that seeks out projects that change the world.

There seems to be two camps in the discussion about comments for articles as a tool of mathematical communication:

• the cons: few, very vocal, are using the straw man argument that comments to articles are like comments in blog, therefore unreliable. It is my interpretation that in fact this is motivated by fear of authority loss. Maybe I am wrong, anyway their argument is blown away by one fact which I shall mention further.
• the pros: they are not disputing the utility of the tool, they would like to have more details instead, about what exactly will be comments for: a kind of online perpetual peer-review, will be them considered as original contributions, where do comments sit in the continuum between the original article and its peer review and, most important, how to motivate mathematicians to seriously participate.

I think the formulation of the question by Ingrid Daubechies is precise and very interesting. Accordingly, mathematicians from both camps could take some moments to think about it.

Is this the kind of disruptive idea which could make the people dream about, concerned about, and also, very important, which could be considered as world-changing? I certainly hope so.

Let me close with the funniest argument (in my opinion) against the idea that comments are bad, because they are like comments in blogs. You see, there is an elephant in the room. Who invented blogs? Why, a mathematician, John Baez with his This Week’s finds. And what exactly is the content of Baez’ first blog in the world? Well, dear naysayers, it is about comments by John Baez of mathematical (and other) scientific articles.

John Baez participates to the discussion initiated by Ingrid Daubechies with this:

I would like some way for me to be able to easily read lots of comments on people’s papers.  Right now to find these comments I either use Google or trackbacks on the arXiv.  But I think there could be something better.
To be honest, I mostly want to read my own comments on people’s papers, because I wrote a lot of them in This Week’s Finds, and nobody else writes nearly enough.  I don’t have much trouble finding my own comments: I use Google, and use keywords that single out This Week’s Finds.  But it’s harder finding comments when I don’t know who wrote them or where they are.

Congratulations John Baez, you are an example for many of us!

In other fields we have PeerJ  and Knowledgeblog.org and BMJ pico, to give only three extremely interesting examples. In mathematics we (shall) have Episciences-Math.

The presentation of the project Episciences-Math, as given here:

The editorial process envisioned for the Episciences-Maths epijournals is quite standard: authors submit their articles after making them available in arXiv or in HAL, and provide the ID of their e-print to a specified epijournal of their choice. The Editorial board of that epijournal handles the submission exactly as for a traditional scientific journal, appointing referees, and deciding to publish – or not – when the report is received. If the article is accepted after suitable corrections have been made, it is subsequently listed on the web page of the journal as a link to the actual file, the final version of which is stored solely in the open archive. At some point in the future, the Episciences platform might also allow the publication of additional contents attached to each article (review by a reporter or by the editorial boards of epijournals, additional data provided by the author: source codes, lecture notes, presentations …)

The Episciences-Maths initiative will be supervised by an “Epicommittee” composed of leading mathematicians. Its role is to stimulate the constitution of editorial boards willing to create new epijournals, especially thematic epijournals in areas not yet covered, to manage possible takeovers of existing journals, and finally to treat any ethical and professional issues. Members of the Epicommittee may or may not themselves take responsibility of an epijournal.

This is the project announced in the “Good guys” post by Gowers.  Many mathematicians are looking forward to see the details.

Several posts on this blog  witness the desire to see that  epijournals become reality. Don’t get me wrong, therefore, if I make some comments about some aspects which worry me a little:

• The public presence of this project is very low. Am I wrong about this? Please send me links to relevant places where this project is explained and …
• … discussed! Is there any public discussion about it, besides the fact that almost everybody who cares to comment wishes the best to the project? Yes, the creators of the project may say “it’s out project, be patient”, but that would be plain wrong. That is because it does not matter whose project is, provided that it is a successful one, or, in order for the project to have success, they need us, those who are waiting to see what is this really about.
• The third point is that, just by looking at the presentation, I don’t get what is new in this project, excepting the fact that the final versions of the articles will be hosted by HAL or arXiv. Annals of Mathematics did this, why do the Episciences think they will succeed?
• I get that they hope to create the SEED of a journal, but platforms for journals exist already. The problem of scientific publication is not technical, it is psychological. I don’t get how they want to address this.
• What about all the features which many people expect? Comments, peer-review, multiple journals “publishing” the same article (i.e. independent, multiple, peer-reviews by different journals for the same article, according to different communities interests, like Andrew Stacey suggests on G+), who will review the articles, what incentives will have mathematicians to publish in an epijournal, knowing that hiring committees and moronic bureaucratic organisms are still pushing authors to publish in traditional ways?

I invite anybody to discuss, here or anywhere. This satisfied silence, after the bad cop – good cop pair of posts by Gowers, looks to me as if our mathematics community is a bit sedated. Or maybe many mathematicians just think our field does not need to change publication practices, even if every other scientific field does it (I am mean, but really, that’s the truth.)

# Another parable of academic publishing: the fall of 19th century academic art

I was impressed by Mike Taylor’s  parable of the farmers and the Teleporting Duplicator.    I sketched my own one in Boring mathematics, artistes pompiers and impressionists. Motivated by Mike Taylor’s post, I try here to gather evidence for the fact that the actual development of academic practices parallel the old ones of the 19th century’s   Académie des beaux-arts .

Everybody knows who won that battle.

Further are excerpts from various wiki pages on the subject. They serve as evidence for  parallels between academies, between the practice of journal publishing and classifications versus exhibiting in the Paris Salon,   between arxiv and the Salon d’Automne.

By reading these excerpts, all of this becomes obvious.

Accademia di San Luca later served as the model for the Académie royale de peinture et de sculpture founded in France in 1648, and which later became the Académie des beaux-arts. The Académie royale de peinture et de sculpture was founded in an effort to distinguish artists “who were gentlemen practicing a liberal art” from craftsmen, who were engaged in manual labor. This emphasis on the intellectual component of artmaking had a considerable impact on the subjects and styles of academic art.

Towards the end of the 19th century, academic art had saturated European society. Exhibitions were held often, and the most popular exhibition was the Paris Salon and beginning in 1903, the Salon d’Automne. These salons were sensational events that attracted crowds of visitors, both native and foreign. As much a social affair as an artistic one, 50,000 people might visit on a single Sunday, and as many as 500,000 could see the exhibition during its two-month run. Thousands of pictures were displayed, hung from just below eye level all the way up to the ceiling in a manner now known as “Salon style.”

A successful showing at the salon was a seal of approval for an artist, making his work saleable to the growing ranks of private collectors. Bouguereau, Alexandre Cabanel and Jean-Léon Gérôme were leading figures of this art world.

As noted, a successful showing at the Salon was a seal of approval for an artist. The ultimate achievement for the professional artist was election to membership in the Académie française and the right to be known as an academician. Artists petitioned the hanging committee for optimal placement “on the line,” or at eye level. After the exhibition opened, artists complained if their works were “skyed,” or hung too high.

Young artists spent four years in rigorous training. In France, only students who passed an exam and carried a letter of reference from a noted professor of art were accepted at the academy’s school, the École des Beaux-Arts. Drawings and paintings of the nude, called “académies”, were the basic building blocks of academic art and the procedure for learning to make them was clearly defined. First, students copied prints after classical sculptures, becoming familiar with the principles of contour, light, and shade. The copy was believed crucial to the academic education; from copying works of past artists one would assimilate their methods of art making. To advance to the next step, and every successive one, students presented drawings for evaluation.

The most famous art competition for students was the Prix de Rome. The winner of the Prix de Rome was awarded a fellowship to study at the Académie française’s school at the Villa Medici in Rome for up to five years. To compete, an artist had to be of French nationality, male, under 30 years of age, and single. He had to have met the entrance requirements of the École and have the support of a well-known art teacher. The competition was grueling, involving several stages before the final one, in which 10 competitors were sequestered in studios for 72 days to paint their final history paintings. The winner was essentially assured a successful professional career. [source]

What happened eventually?

As modern art and its avant-garde gained more power, academic art was further denigrated, and seen as sentimental, clichéd, conservative, non-innovative, bourgeois, and “styleless”. The French referred derisively to the style of academic art as L’art Pompier (pompier means “fireman”) alluding to the paintings of Jacques-Louis David (who was held in esteem by the academy) which often depicted soldiers wearing fireman-like helmets. The paintings were called “grandes machines” which were said to have manufactured false emotion through contrivances and tricks.

This denigration of academic art reached its peak through the writings of art critic Clement Greenberg who stated that all academic art is “kitsch“. References to academic art were gradually removed from histories of art and textbooks by modernists […]  For most of the 20th century, academic art was completely obscured, only brought up rarely, and when brought up, done so for the purpose of ridiculing it and the bourgeois society which supported it, laying a groundwork for the importance of modernism. [source]

What was the initial course of action?

In 1725, the Salon was held in the Palace of the Louvre, when it became known as Salon or Salon de Paris. In 1737, the exhibitions became public and were held, at first, annually, and then biannually in odd number years. They would start on the feast day of St. Louis (25 August) and run for some weeks. Once made regular and public, the Salon’s status was “never seriously in doubt” (Crow, 1987). In 1748 a jury of awarded artists was introduced. From this time forward, the influence of the Salon was undisputed.

In the 19th century the idea of a public Salon extended to an annual government-sponsored juried exhibition of new painting and sculpture, held in large commercial halls, to which the ticket-bearing public was invited. The vernissage (varnishing) of opening night was a grand social occasion, and a crush that gave subject matter to newspaper caricaturists like Honoré Daumier. Charles Baudelaire, Denis Diderot and others wrote reviews of the Salons.

The 1848 revolution liberalized the Salon. The amount of refused works was greatly reduced. In 1849 medals were introduced.

The increasingly conservative and academic juries were not receptive to the Impressionist painters, whose works were usually rejected, or poorly placed if accepted. The Salon opposed the shift away from traditional painting styles espoused by the Impressionists. In 1863 the Salon jury turned away an unusually high number of the submitted paintings. An uproar resulted, particularly from regular exhibitors who had been rejected. In order to prove that the Salons were democratic, Napoleon III instituted the Salon des Refusés, containing a selection of the works that the Salon had rejected that year. It opened on 17 May 1863, marking the birth of the avant-garde. The Impressionists held their own independent exhibitions in 1874, 1876, 1877, 1879, 1880, 1881, 1882 and 1886.

In December 1890, the leader of the Société des Artistes Français, William-Adolphe Bouguereau, propagated the idea that Salon should be an exhibition of young, yet not awarded, artists.

In 1903, in response to what many artists at the time felt was a bureaucratic and conservative organization, a group of painters and sculptors led by Pierre-Auguste Renoir and Auguste Rodin organized the Salon d’Automne.

The tipping point:

In 1903, the first Salon d’Automne (Autumn Salon) was organized by Georges Rouault, André Derain, Henri Matisse and Albert Marquet as a reaction to the conservative policies of the official Paris Salon. This massive exhibition almost immediately became the showpiece of developments and innovations in 20th-century painting and sculpture. Jacques Villon was one of the artists who helped organize the drawing section of the first salon. Villon later would help the Puteaux Group gain recognition with showings at the Salon des Indépendants. During the Salon’s early years, established artists such as Pierre-Auguste Renoir threw their support behind the new exhibition and even Auguste Rodin displayed several drawings. Since its inception, artists such as Paul Cézanne, Henri Matisse, Paul Gauguin, Jean Metzinger, Albert Gleizes and Marcel Duchamp have been shown here. In addition to the 1903 inaugural exhibition, three other important dates remain historically significant for the Salon d’Automne: 1905, bore witness to the birth of Fauvism; 1910 witnessed the launch Cubism; and 1912 resulted in a xenophobe and anti-modernist quarrel in the National Assembly (France).

The rest is history. There are many lessons to be learned from this, by the new “impressionists”, as well as by the “pompiers”, what do you think?

# Towards qubits: graphic lambda calculus over conical groups and the barycentric move

In this post I want to pave the way to the application of graphic lambda calculus to the realm of quantum computation. It is not a short, nor too lengthy way, which will be explained in several posts. Also, some experimentation is to be expected.

Disclaimer: For the moment it is not very clear to me which are the exact relations between the approach I am going to explain and linear lambda calculus or the  lambda calculus for quantum computation.  I expect a certain overlapping, but maybe not as much as expected (by the specialist in the field). The reason is that the instruments and goals which I have come from fields apparently far away from quantum computation, as for example sub-riemannian geometry, which is my main field of interest (however, for an interaction between sub-riemannian geometry and computation  see  L_p metrics on the Heisenberg group and the Goemans-Linial conjecture, by James R. Lee and Asaf Naor). Therefore, I feel the need to issue such a disclaimer for the narrow specialist.

Background for his post:

• The page Graphic lambda calculus
•  [1] Infinitesimal affine geometry of metric spaces endowed with a dilatation structure, Houston J.  Math., 36, 1 (2010), 91-136, arXiv:0804.0135.
• [2]  On graphic lambda calculus and the dual of the graphic beta move, arXiv:1302.0778.

Affine conical spaces. In the article [1] they appear under the name “normed affine group spaces”, definition 3. We may use the same type of arguments as the ones from emergent algebras   in order to get rid of the need to have a norm on such spaces.  Instead of anorm we shall put an uniformity on such a space, such that the topology associated to the uniformity makes the space to be locally compact.

Theorem 2.2 [1] characterizes affine conical spaces as self-distributive emergent algebras. The relations satisfied by self-distributive emergent algebras, if graphically represented by gates in graphic lambda calculus, are the following:

Notice that I don’t want to use the dual of the graphic beta move ([2], section 8), which is simply too powerful in this context (see [2] section 10). That is why I use instead the move R3a (which is a composite of dual beta moves). Another instance of this choice will be explained in a future post, having to do with the distributivity of the emergent algebra operations with respect to the application and lambda gates.

The barycentric move.   In order to obtain usual affine spaces instead of their more general,  noncommutative versions (i.e. affine conical spaces), we have to add the barycentric condition. This condition appears as (Af3) in Theorem 2.2 [1].  I shall transform this condition into a move in graphic lambda calculus.

The barycentric move BAR is described by the following figure and explanation. We take the commutative group $\Gamma$, which is used to label the emergent algebra gates, as $\Gamma = K^{*}$, where $K$ is a field. (Therefore $K = \Gamma \cup \left\{ 0 \right\}$.) We have then two operations on the field $K$: multiplication $\varepsilon, \mu) \mapsto \varepsilon \mu$ and addition $(\varepsilon, \mu) \mapsto \varepsilon + \mu$. Because $K$ contains also the element $0$, the neutral element for addition, we add a new gate $\bar{0}$.  With these preparations, the BAR move is the following:

Notice that when $\varepsilon = 1$ at the left hand side of the figure appears the gate $\bar{0}$. This gate corresponds, in the particular case of a vector space, to the usual dilation of coefficient $0$. We don’t need to put this as a sort of an axiom, because we can obtain it as a combination of the BAR move and ext2 moves. Indeed:

Knowing this, we can extend the emergent algebra moves R1a, R1b and R2 to the case $\varepsilon = 0$. Here is the proof. For R1a we do this:

The move R1b, for the degenerate case $\varepsilon = 0$, is this:

Finally, for the move R2 we have two cases, corresponding to $0 \, \varepsilon = 0$ and $\varepsilon \, 0 = 0$. The first case is this:

The second case is this:

Final remark: The move BAR can be seen as analogous of an infinite sequence of moves R3 (but there is no rigorous sense for this in graphic lambda calculus). Indeed, this is related to the fact that $\frac{1}{1-\varepsilon} = \sum^{\infty}_{0} \varepsilon^{k}$.  See [1] section 8 “Noncommutative affine geometry” for the dilation structures correspondent of this equality and also see the post Menelaus theorem by way of Reidemeister move 3.

# Unlimited detail challenge: the most easy formulation

Thanks to Dave H. for noticing the new Euclideon site!

Now, I propose you to think about  the most easy formulation of unlimited detail.

You live in a 2D world, you have a 1D screen which has 4 pixels. You look at the world, through the screen, by using a 90deg frustrum. The 2D world is finite, with diameter N, measured in the unit lengths of the pixels (so the screen has length 4 and your eye is at distance 2 from the screen). The world contains atoms which are located on integer coordinates in the plan. There are no two atoms in the same place. Each atom has attached to it at most P bits, representing its colour. Your position is given by a pair of integer coordinates and the screen points towards  N, S, E, W only.

Challenge: give a greedy algorithm which, given your position and the screen direction,  it chooses 4 atoms from the world which are visible through the screen, in at most O(log N) steps.

Hints:

• think about what “visible” means in this setting
• use creatively numbers written in base 2, as words.

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The Teaser 2D UD might help.

I’m not giving this challenge because I am a secretive …. but because I enjoy collaborations.

# Fixed points in graphic lambda calculus

Background: the page Graphic lambda calculus.

Let $A$ be a graph in $GRAPH$ with one input and one output. For any  graph $B$ with one output, we denote by $A(B)$ the graph obtained  by grafting the output of $B$ to the input of $A$.

Problem:  Given $A$, find $B$ such that $A(B) \leftrightarrow B$, where $\leftrightarrow$ means any finite sequence of moves in graphic lambda calculus.

The solution is in principle the same as in usual lambda calculus, can you recognize it? Here is it. We start from the following:

That’s almost done. It suffices to do this:

to get a good graph $B$:

# An interface for graphic lambda calculus?

I realized I would need a (java?) interface for graphic lambda calculus. It should be able to:

• represent graphs in $GRAPH$
• allow to pass from one embedding of a graph in $\mathbb{R}^{2}$ to another, i.e. to allow to move the nodes in the plane while preserving the connectivity
• to mark a group of nodes and arrows of interest
• to recognize local patterns, in particular those which appear in the moves
• to allow to give names to patterns, i.e. to define macros
• to perform the local moves, maybe even to propose a list of possible local moves
• to construct the graph associated to a lambda term, as explained here.
• to keep a sequence of moves into a graphic form (for example as a sequence of graphs).

Any help/advice would be appreciated.