Questions/answers about Gold OA: (please add your answers and other questions)
1. Is the author a customer of a Gold OA publisher?
I think it is.
2. What is the author paying for, as a customer?
I think the author pays for the peer-review service.
3. What offers the Gold OA publisher for the money?
I think it offers only the peer-review service, because
- dissemination can be done by the author by submitting to arxiv, for example,
- +Mike Taylor says that the Gold OA publisher offer the service of assembling an editorial board, but who wants to buy an editorial board? No, the authors pays for the peer-review process, which is managed by the editorial board, true, which is assembled by the publisher. So the end-product is the peer-review and the author pays for that.
- almost 100% automated services, like formatting, citation-web services, hosting the article are very low value services today.
However, it might be argued that the Gold OA publisher offers also the service of satisfying the author’s vanity, as the legacy publishers do.
4. Why no Gold OA publisher present itself as a seller of the peer-review service?
Have no idea.
5. Why is the peer-review service valuable?
- it spares time for the reader, who will select more likely a peer-reviewed paper to read,
- it is a filter for the technical quality of the articles,
- it helps authors to write better articles, as an effect of the referees comments,
- it is also a tool for influencing the opinions of the community, by spinning up some research subjects and downplaying others.
Also on G+ here.
Knowledge is like an irregular blob.
(thanks due to Jon Awbrey for the following quote)
Thus, what looks to us like a sphere of scientific knowledge more accurately should be represented as the inside of a highly irregular and spiky object, like a pincushion or porcupine, with very sharp extensions in certain directions, and virtually no knowledge in immediately adjacent areas. If our intellectual gaze could shift slightly, it would alter each quill’s direction, and suddenly our entire reality would change. [Herbert J. Bernstein, “Idols of Modern Science”]
Researchers usually make dents which advance the knowledge frontier (can anybody point me to the well-known phd comics with represents the phd research as a dent in the knowledge frontier?)
From time to time, some very good researchers achieve a breaktrough:
Breakthroughs are possible when the large scale picture of the knowledge blob is this:
There is also the Grothendieck style of research (extremely rare):
If successful, after some time the global picture looks like this:
i.e. it facilitates multiple breakthroughs!
Democratic changes in OA can be only reactive. That means one step back with respect to active opposition to change, methodically pursued by interests of a small but powerful minority of big players in the publishing game (i.e. publishers themselves and their academic management friends, sometimes overlapping). And even more, one might say that democratic changes are even two steps back with respect to strategic decisions taken by the said big players. It’s only speculation, but for example the admirable DORA could throw us in the future into the arms of the newly acquired Mendeley.
By democratic changes I mean those which are agreed by a significant part of the research community.
So, what else? Privately supported changes. By this I mean support of any potentially viral solution for getting us out from this tarpit war. It’s clear that Gold OA is the immediate future change agreed by the big players, although it’s just as useless as the actual research communication system based on traditional publication. Why waste another 10 years on this bad idea, only to repeat afterwards that it is already technically possible to disseminate knowledge without making the authors (or public funding agencies which support those) pay for nothing?
The advantage of a new dissemination system is already acknowledged, namely it is far more convenient, economically speaking, to profit from the outcomes of low Coase cost research collaborations, than to keep paying a hand of people who offer an obsolete service and don’t want to adapt to the new world of the net.
This point of view is stressed already in my Seven years forecast (i.e. until 2020), part 5:
In seven years all successful changes of the process of dissemination of knowledge will turn out to be among those born from private initiatives,
Wish I have a crystal ball, though I only have some hope.
UPDATE: Oh, yeah, maybe the uber-library idea is not the right thing. Yes, everybody wishes for a world library at a click distance, but that’s not all. That’s like “what can we do with cars? Well, let’s make them like coaches, only without the horse. The rich guys will love them.” And boum! the car concept became a success from the moment they were mass-produced.
UPDATE 2: Maybe relevant for the idea from the first update, Cameron Neylon’s post “The bravery of librarians” ends with the question:
What can we do to create a world where we need to rely less on the bravery of librarians and therefore benefit so much more from it?
Yes, graphic lambda calculus has a freedom sector. Which means in that sector you can do anything you like (modulo some garbage, though). It’s yet not clear to me if this means a kind of universality property of graphic lambda calculus.
The starting point is the procedure of packing arrows explained in this post. This procedure can be seen in the following way:
Here, the left and right void circles with the respective arrows represent: the one from the left is a generic out arrow which exits from a gate and the one from the right is a generic in arrow which enters in a gate.
This gives the following idea: replace the inputs and the outputs of the gates from graphic lambda calculus by the following graphs (the green wiggly arrow means “replace”):
For example, look how it’s done for the graph. Technically we define new macros, one for each elementary gate. Let’s call these macros “the free gates”.
These free gates define the free sector of the graphic lambda calculus, which consists all graphs made by free gates, along with the move of cutting or gluing arrows.
The free sector has inside a copy of the whole graphic lambda calculus, with the condition of adding a local move of elimination of garbage, which is the local move of elimination (goes only one way, not both) of any graph which is not made by free gates with at most, say, 100 arrows + gates. This move is needed, for example, for the case of emulating the graphic beta move with free gates, where we are left with some garbage consisting of one gate and one gate, seen as disconnected graphs.
What happens in the real world, the one of the powers that be, as concerns open access, peer-review and communication of research results? Let’s see.
Funding agencies, institutions that employ scientists, and scientists themselves, all have a desire, and need, to assess the quality and impact of scientific outputs. It is thus imperative that scientific output is measured accurately and evaluated wisely. [...]
A number of themes run through these recommendations:
- the need to eliminate the use of journal-based metrics, such as Journal Impact Factors, in funding, appointment, and promotion considerations;
- the need to assess research on its own merits rather than on the basis of the journal in which the research is published; and
- the need to capitalize on the opportunities provided by online publication (such as relaxing unnecessary limits on the number of words, figures, and references in articles, and exploring new indicators of significance and impact).
Read it. Disseminate it. Sign it.
The bad: The apparatus of research assessment is driven by the academic publishing industry and has become entirely self-serving. In this article by Peter Coles you find:
The involvement of a company like Elsevier in this system just demonstrates the extent to which the machinery of research assessment is driven by the academic publishing industry. The REF is now pretty much the only reason why we have to use traditional journals. It would be better for research, better for public accountability and better economically if we all published our research free of charge in open archives. It wouldn’t be good for academic publishing houses, however, so they’re naturally very keen to keep things just the way they are. The saddest thing is that we’re all so cowed by the system that we see no alternative but to participate in this scam.
The iawful: From this g+ post by Peter Suber we find out that:
Elsevier, NewsCorp, Facebook, and Yahoo are some of the major players in NetChoice, an industry group “promoting convenience, choice, and commerce on the net.”
NetChoice has a watch list for bad legislation that it calls iAWFUL (Internet Advocates’ Watchlist for Ugly Laws). The latest version of iAWFUL includes the White House OA directive plus the state-level OA bills in California, Illinois, and North Dakota. (Yes, there was a bill in ND, and no, NetChoice doesn’t seem to know about the OA bill in NY.)
Insofar as NetChoice has an argument for opposing these OA initiatives, it’s a crude bolus of false assertions and assumptions. I haven’t seen this kind of motivated distortion since the days of PRISM and the Research Works Act.
UPDATE (23.05.2013): “Elsevier distances itself from open-access article“
The publisher Elsevier has disassociated itself from an article by a trade association it belongs to that condemns proposed open-access mandates in several US states.
So, things happen … eventually. But slowly. I bet many of us, not entangled with the high politics or management in academia, wish for a faster pace. For my part, I would rather play the Game. It has a very low Coase cost, you know?
UPDATE: if you still wonder about Gold OA, is it good? is it bad?, here is a tweet about Elsevier and iawful:
Suppose that we want to group together three arrows in graphic lambda calculus. We have this:
We want to group them together such that later, by performing graphic beta moves, the first arrow available to be 11′, then 22′, then 33′. Moreover, we want to group the arrows such that we don’t have to make choices concerning the order of the graphic beta moves, i.e. such that there is only one way to unpack the arrows. The solution is to “pack” the arrows into a variant of a list. Lists have been defined here, in relation to currying.
Basically we take a zipper and we close it. Further we see how to unpack this list.
The dashed red curve encircles the only place where we can use a graphic beta move. The first move frees the 11′ arrow and then there is only one place where we can do a graphic beta move, which frees the 22′ arrow and finally a last move frees the 33′ arrow and produces a loop which can be eliminated.
The uniqueness of the order of moves is true, in fact, if we accept as valid beta moves only those from left to right (i.e. those which eliminate gates). Otherwise we can go back and forth with a beta move as long as we want.
There is another way to pack the three arrows, under the form of another graph, which could aptly be called a “set”. This time we need a graph with the property that we can extract any arrow we want from it, by one graphic beta move. Here is the solution:
Indeed, in the next figure we see that we have three places, one for each arrow, which can be independently used for extraction of the arrow of choice.
In between these extremes, there are other possibilities. In the next figure is a graph which packs the three arrows such that: there are three places where a graphic beta move can be performed, as in the case of the set graph, but once a beta move is performed, the symmetry is broken. The performed beta move does not free any arrow, but now we have the choice between the other two possible beta moves. Any such choice frees only one arrow, and the last possible beta move frees the remaining two arrows simultaneously.
Here is the figure:
The graph from the left hand side is not a list, nor a set, although it is as symmetric as a set graph. There are possible ways to unpack the graph. So this graph encodes all lists of two arrows out of the three arrows.
… so that we can freely play the game of research. Because is a game, i.e. it is driven by curiosity, desire to learn, does not depend on goals and tasks, it is an extension of a child attitude, lost by the majority of adults. Let the vanity aside and just play and interact with other researchers, on equal foot. Let the creativity manifest freely.
Two Three Four examples:
Rap Genius is a very well-loved and well-used online tool for annotating rap songs. Only, not so surprisingly, people are starting to use it to annotate other things. Like scientific papers.
- Olivier Charbonneau writes
Actually, that’s an interesting take on mass data visualization – imagine creating an algorithm that could parse a dataset of bibliographic information into minecraft (for example) – what would that research “world” look like?
- Hermann Hesse’s Das Glasperlenspiel (aka Magister Ludi)
- Timothy Gowers, some time ago, in this post, writes:
What I think could work is something like a cross between the arXiv, a social networking site, Amazon book reviews, and Mathoverflow.
I want to make a bit more clear one of the goals of the research on graphic lambda calculus, which are reported on this blog. I stress that this is one of the goals and that this is live research, in the making, explained here in order to attract, or invite others to join, or use this exploration for their purposes.
More precisely, further I present several justifications for two series of posts
- Emergent algebras as combinatory logic part I, part II, part III, part IV,
- Towards qubits part I, part II,
which have as common goal the application of graphic lambda calculus to some form of quantum programming (probably some version of a quantum lambda calculus). I use the informative linked wiki page on quantum programming for citing. Please click on the links to go where the real information is.
Efforts are underway to develop functional programming languages for quantum computing. Examples include Selinger’s QPL, and the Haskell-like language QML by Altenkirch and Grattage. Higher-order quantum programming languages, based on lambda calculus, have been proposed by van Tonder, Selinger and Valiron  and by Arrighi and Dowek.
Simon Gay’s Quantum Programming Languages Survey has more information on the current state of research and a comprehensive bibliography of resources.
I hope that in some finite time I can prove that there is a “quantum lambda calculus” sector in graphic lambda calculus. Let me explain why.
Basically, leaving much detail aside, quantum computation needs a mix of at least two ingredients:
- some algebraic structure, which contains objects like complex vector spaces, real projective spaces, unitary transformations, projections, etc,
- some logical structure overarching the algebraic one (purists may say that in principle a lambda calculus would do).
The algebraic structure is not needed entirely, i.e. the needed part is the web of relations between the various algebraic operations. For example, the vector space operations are needed and not the points of the vector space. Likewise, we need “linearity”, “unitarity” and not particular linear or unitary transformations. Enough is to know how linearity and unitarity interact with the algebraic operations.
In the same way, as concerns the logic part, we need (say, if we are interested in a quantum lambda calculus) an abstraction an an application operations (like in lambda calculus) which interact well with the algebraic structure. Right?
There is one more ingredient needed: some form of evaluation procedure. There we can see a difference between a quantum and a classical lambda calculus. A quantum lambda calculus is more geometrical, less commutative than a classical one. One has to take care of phases, of the order of evaluations more than in the classical one.
Graphic lambda calculus seems to be a welcoming host for all these demands. Indeed, let’s see.
Graphic lambda calculus encodes algebraic structures in the barest way, by using only one gate: the emergent algebra gate , with the parameter in a commutative group. This models “scale”, it is usually taken in or in . However, phase is a kind of scale, i.e. the formalism works well with the choice of the commutative group of scales to be . Any algebraic operation and any algebraic computation in complex vector spaces, or in real projective spaces, may be expressed into graphic lambda calculus by the intermediary of the emergent algebra gate. Moreover, even some of the differential calculus (needed but not mentioned previously) can be embedded into graphic lambda calculus, in a kind of constructive way. This is the “emergent algebra” point of view, introduced in arXiv:0907.1520 .
So, shortly said, in graphic lambda calculus we have the algebraic structure needed. It “emerges” from the gate, when we take the scale parameter to be in . With the barycentric move BAR from Towards qubits part I we get the algebraic structures of vector spaces (see how to get projective spaces in part II, work in progress). More interesting, without the barycentric move we get Carnot groups, i.e. non-commutative vector spaces.
Question 1. What we obtain if in the formalism of quantum mechanics we renounce at complex vector spaces and we replace with their non-commutative version, the Carnot groups?
For the logic part, we know that graphic lambda calculus has a sector which corresponds to untyped lambda calculus. In quantum programming it would be interesting to find a quantum version of the lambda calculus which interacts well with the algebraic structure. But in graphic lambda calculus are allowed interactions between the lambda calculus gates, (or logical gates) of abstraction and application, and the algebraic gates. We don’t need more, that is what I shall try to convince you eventually. Indeed, probably obscured behind the lambda scale calculus (which is a first, non-graphical version of the graphic lambda calculus), this was already explored in section 4 “Relative scaled calculus” of arXiv:1205.0139, where we see that to any scale parameter is associated a relative lambda calculus. This was done in whole generality, but for the needs of a quantum lambda calculus ”linearity moves” like in the “Towards geometric Plunnecke graphs” series could be applied selectively, i.e. only with respect to the part of , thus obtaining a relative lambda calculus which is phase-dependent.
Question 2. What would a relative scaled lambda calculus look like in graphic lambda calculus?
Finally, for the evaluation procedures which are adapted to quantum world, in this respect, for the moment, I have only results which indicate how to get usual evaluation procedures in graphic lambda calculus by destroying it’s geometrical nature (that’s what I call the “cartesian disease“, if you care), which are explained in some detail in Packing and unpacking arrows in graphic lambda calculus and Packing arrows (II) and the strand networks sector.
Question 3. Design evaluation procedures in graphic lambda calculus which are geometrical, in the sense that, at least when applied to the yet vague quantum lambda sector of the graphic lambda calculus, they give evaluation procedures which are useful for quantum programming.
So, that’s it, I hope it helps a bit the understanding. You are welcome to join, to contradict me or to contribute constructively!
I don’t get it, therefore I ask, with the hope of your input. It looks that the Gamifying peer-review post has found some attentive ears, but the Game on the knowledge frontier not. It is very puzzling for me, because:
- the game on the frontier seems feasible in the immediate future,
- it has two ingredients – visual input instead of bonus points and peer-review as a “conquest” strategy – which have not been tried before and I consider them potentially very powerful,
- the game on the frontier idea is more than a proposal for peer-review.
My question is: why is the game on the frontier idea less attractive?
Looking forward for your open comments. Suggestions for improvement of such ideas are also especially welcomed.
UPDATE: Olivier Charbonneau writes:
Actually, that’s an interesting take on mass data visualization – imagine creating an algorithm that could parse a dataset of bibliographic information into minecraft (for example) – what would that research “world” look like?
I continue from Parallel transport in spaces with dilations, I. Recall that we have a set , which could be see as the complete directed graph . By a construction using binary decorated trees, with leaves in , we obtain first a set of finite trees , then we put an equivalence relation on this set, namely two finite trees and are close if is a finite tree. The class of finite points is formed by the equivalence classes of finite trees with respect to the closeness relation .
Notice that the equality relation is , in this world. This equality relation is generated by the “oriented Reidemeister moves” R1a and R2a, which appear also as moves in graphic lambda calculus. (By the way, this construction can be made in graphic lambda calculus, which has the moves R1a and R2a. In this way we obtain a higher level of abstraction, because in the process we eliminate the set . Graphic lambda calculus does not need variables. More about this at a future time.) If you are not comfortable with this equality relation than you can just factorize with it and replace it by equality.
It is clear that to any “point” is associated a finite point . Immediate questions jump into the mind:
- (Q1) Is the function injective? Otherwise said, can you prove that if then is not a finite tree?
- (Q2) What is the cardinality of ? Say, if is finite is then infinite ?
Along with these questions, a third one is almost immediate. To any two finite trees and is associated the function defined by
The function is well defined: for any we have , by definition. Therefore , because .
Consider now the groupoid with the set of objects and the set of arrows generated by the arrows from to . The third question is:
- (Q3) What is the isotropy group of a finite point (in particular ) in this groupoid? Call this isotropy group and remark that because the groupoid is connected, it follows that the isotropy groupoid does not depend on the object (finite point), in particular is the same at any point (seen of course as ).
In a future post I shall explain the answers to these questions, which I think they are the following:
- Q1: yes.
- Q2: infinite.
- Q3: a kind of free nilpotent group.
I think we can use the social nature of the web in order to physically construct the knowledge boundary. (In 21st century “physical” means into the web.)
Most interesting things happen at the boundary. Life on earth is concentrated at it’s surface, a thin boundary between the planet and the void. Most people live near a body of water. Researchers are citizens of the boundary between what is known and the unknown. Contrary to the image of knowledge as the interior of a sphere, with an ever increasing interface (boundary) where active research is located, no, knowledge, old or new, is always on the boundary, evolving like life is, into deeply interconnected, fractal like niches.
All this for saying that we need an interesting boundary where we, researchers, can live, not impeded by physical or commercial constraints. We need to build the knowledge boundary into the web, at least as much the real Earth was rebuilt into the google earth.
Game seems to be a way. Because game is both social and an instrument of exploration. We all love games, especially researchers. Despite the folklore describing nerds as socially inept, we were the first adopters of Role Playing Games, later evolved into virtual worlds of the Massively Multiplayer Online Role Playing Games. Why not make the knowledge frontier into one of these virtual worlds?
It looks doable, we almost have all we need. Keywords of research areas could be the countries, places. The physics of this world is ruled by forces with articles citation lists as force-carrying bosons. Once the physics is done, we could populate this world and play a game of conquest and exploration. A massively multiplayer online game. Peer-reviews of articles decide which units of places are wild and which ones are tamed. Claim your land (by peer-reviewing articles), it’s up for grabs. Organize yourselves by interacting with others, delegating peer-reviews for better management of your kingdoms, collaborating for the exploration of new lands.
Instead of getting bonus points, as mathoverflow works, grab some piece of virtual land that you can see! Cultivate it, by linking your articles to it or by peer-reviewing other articles. See the boundaries of your small or big kingdom. Maybe you would like to trespass them, to go into a near place? You are welcome as a trader. You can establish trade with other near kingdoms by throwing bridges between the land, i.e. by writing interdisciplinary articles, with keywords of both lands. Others will follow (or not) and they will populate the new boundary land you created.
After some time, you may be living in complex, multiply-connected kingdom cities, because you are doing peer-reviewed research in an established, rich in knowledge field. On the fringes of such rich kingdoms a strange variety of creatures live. Some are crackpots, living in the wild territory, which grows wilder with the passage of time. Others are explorers, living between your respectable realm and wild, but evolving into tamer territory. From time to time some explorer (or some crackpot, sometimes is not easy to tell one from another) makes a break and suddenly a bright avenue connects two far kingdoms. By the tectonic plate movement of this world, ruled by citations, these kingdoms are now one near the other. Claim new land! Trade new bridges! During this process some previously rich, lively, kingdoms might become derelict. Few people pass by, but there’s nothing lost: like happened in Rome, the marble of ancient temples was used later for building cathedrals.
If you are not a professional researcher, nevermind, you may visit this world and contribute. Or understand more, by seeing how complex, how alive research is, how everything is interwoven. Because an image speaks a thousand words, you can really walk around and make an idea of your own about the subject you are curious about.
Thinking more about peer-reviews, which are like property documents, as in real life some are good and some are disputable. Some are like spells: “I feel that the article is not compelling enough …”. Some are frivolous nonsense: “I find it off-putting when an author does not use quotation marks as I am used to”. Some are rock-solid: “there’s a gap in the proof” or “I have not been able to find the error in the proof, but here is a counter-example to the author’s theorem 1.2″.
So, how can it be done? We (for example by a common effort at github) could start from what is available, like keywords and citations freely available or easy to harvest, from tools like google scholar profiles, mathscinet, you name it. The physics has to be written, the project could be initially hosted for almost nothing, we could ask for sponsors. We could join efforts with established international organisms which intend to pursue somehow similar projects. The more difficult part will be the tuning of interactions, so that the game starts to have more and more adopters.
After that, as I said, the knowledge frontier will be up for grabs. Many will love it and some will hate it.
Context: The richness of knowledge comes from this web of interactions between human minds, across time and space. This knowledge is not reserved to the statistically few people doing research. We grow with it, during school, we live within, no matter what we do as adults, we talk about and we are curious about it. Even more, immensely more after knowledge has been liberated by the web.
In a short lapse of time (at the scale of history) it has become obvious that research itself needs to be liberated from outdated habits. Imagine a researcher, before the web. She was a dual creature: physically placed somewhere on the physical earth, living in some moment in time, but mentally interconnected with other researchers all over the world, anytime in the history. However, the physical place of living impeded or helped the researcher to reach further in the knowledge world, depending on the availability of virtual connections: books, other physically near researchers, local traditions. We can’t even speculate about how many curious minds did not accessed the knowledge web, due to the physical place and moment in time where they lived, or due to society customs. How many women, for example?
But now we have the web, and we use it, as researchers. It is, in some sense, a physical structure which could support the virtual knowledge web. The www appeared in the research world, we are the first citizens of it. The most surprising effect of the web was not to allow everybody to access the knowledge boundary. Instead, the most powerful effect was to enhance the access of everybody to everybody else. The web has become social. Much less the research world.
Due to old habits, we loose the pace. We are still chained by physical demands. Being dual creatures, we have to support our physical living. For example, we are forced by outdated customs to accept the hiding behind paywalls of the results of our research. The more younger we are, the more is the pressure to “sell” what we do, or to pervert the direction of our work in order to increase our chances of success in the physical world. In order to get access to physical means, like career advancements and grant money.
Old customs die hard. Some time ago a peasant’s child with a brilliant mind had to renounce learning because he needed to help his family, his sister was seen as a burden, not even in principle considered for eventual higher education. Now young brilliant minds, bored or constrained by the local research overlord or local fashion, rather go doing something rewarding for their minds outside academia, than slicing a tasteless salami into the atoms of publishable units, as their masters (used to) advice them.
I intended to call this series of posts “What group is this?”, but I switched to this more precise, albeit more bland name. In this first post of the series I take again, in more generality, the construction explained in the post Towards geometric Plünnecke graphs.
The construction starts in the same way, almost. After I give this first part of the construction, an interpretation in term sof groupoids is provided. We consider only the moves R1a and R2a, like in the post “A roadmap to computing with space“:
(The names “R1a”, “R2a” come from the names of oriented Reidemeister moves, see arXiv:0908.3127 by M. Polyak.)
Definition 1. The moves R1a, R2a act on the set of binary trees with nodes decorated with two colours (black and white) and leaves decorated with elements of a set of “variable names” which has at least two elements. I shall denote by … such trees and by … elements of .
The edges of the trees are oriented upward. We admit to be a subset of , thinking about as an edge pointing upwards which is also a leaf decorated with .
The moves are local, i.e. they can be used for any portion of a tree from which looks like one of the patterns from the moves, with the understanding that the rest of the respective tree is left unchanged.
We denote by the fact that the can be transformed into by a finite sequence of moves.
Definition 2. The class of finite trees is the smallest subset of with the properties:
- if then , where is the tree
- if then and , where is the tree
and is the tree
- if and we can pass from to (i.e. ) by one of the moves then .
Definition 3. Two graphs are close, denoted by , if there is such that can be moved into .
Notice that then .
Proposition 1. The closeness relation is an equivalence.
Proof. I start with the remark that if and only if , where is the tree
Indeed, if there is such that . Then
Finally, suppose that , . Then by the previous reasoning. Then there are such that and . It follows that , therefore , which proves that .
Definition 4. The class of finite points of is is the set of equivalence classes w.r.t .
Same construction, with groupoids. We may see as being an equivalence relation. Let be the set of equivalence classes w.r.t . We can define on the operations and (because the moves R1a, R2a are local). Then is the free left idempotent right quasigroup generated by the set .
Idempotent right quasigroups are the focus of the article arXiv:0907.1520, where emergent algebras are introduced as deformations of such objects. An idempotent right quasigroup is a non-empty set endowed with two operations, such that
- (idempotence) for any ,
- (right quasigroup) for any
Let be the trivial (pair) groupoid over . This is the groupoid with objects which are elements of and arrows of the form . Equivalently, we see to be the set of it’s arrows, we identify objects with their identity arrows (in this case we identify with it’s identity arrow ). Seen like this, the trivial groupoid is just the set , with the partially defined operation (composition of arrows)
and with the unary inverse operation
Remark that the function defined by is a bijection of the set of arrows and moreover
- it preserves the objects ,
- the inverse has the expression .
Define the groupoid by declaring to be an isomorphism of groupoids. This means to be the set of arrows , with the partially defined composition of arrows given by
for any pair of arrows such that can be composed in with , and unary inverse operation given by
The groupoid has then the composition operation
the unary inverse operation
and the set of objects .
Consider the set , seen as a subset of arrows of the groupoid .
The class of finite trees appears in the following way. First define to be the set of equivalence classes w.r.t of elements in .
Remark that is a sub-groupoid of , which moreover it contains and is closed w.r.t. the application of , seen this time as a function (which is not a morphism) from to itself. In fact is the smallest subset of with this property. Let’s give to the groupoid the name , seen as a sub-groupoid of .
Moreover is a sub-groupoid of the trivial groupoid , with set of objects . But sub-groupoids of the trivial groupoid are the same thing as equivalence relations. In this particular case if and only if and .
Next time you’ll see some groups (which are associated to parallel transport in dilation structures) which are in some sense universal, but I don’t know (yet) what structure they have. “What group is this?” I shall ask next time.
Do you remark at which stage of this construction the map becomes the territory, thus creating points out of abstract nonsense?
To get a sense of this, replace the set of arrows with a graph with nodes in .
Motivated by a g+ mention of two posts of mine, I think I need to explain a little bit the purpose of such posts, also by putting them in the context of my experience. (I don’t know how to avoid this appeal to experience, because it is not at all an authority argument. Authority arguments, I believe, are outside of the research realm, they should be ignored in totality.)
Despite my attraction to physics and painting, I was turned to become a mathematician by a very special kind of professor. When I was little there was the habit of taking private preparatory classes for increasing the chances of admission in a good college. So, at some point, although I claimed not to need such classes, one day when I came back home after a soccer game I met a strange old guy, who was speaking in an extremely lively and polite way with my parents. I was wearing my school uniform which was full of dust gathered in the schoolyard and I was not at all in the mood to speak with old, strange persons. He explained to my parents that he is going to give me one problem to solve, for him to decide whether to accept me or not as a student. He gave me an inequality to prove, then I spent a half hour in my room and found a solution, which I wrote. I gave the solution to the professor, he looked at it and started: “Marius, a normal kid would solve this inequality like that (he explained it to me). A clever kid would prove the inequality like this (a shorter, more elegant solution). A genius kid would do like this (one line proof). Now, your proof is none of the above, so I take you.” It was an amazing experience to learn, especially geometry, from him. At some point he announced my parents that he is willing to do the classes with no pay, with the condition that he could come at any time (with a half hour notice). We did mathematics at strange hours for me, like midnight or 5 in the morning, or whenever he wanted. Especially when geometry was concerned, he was never letting me write anything until I could explain with words the idea of the solution, then I could start writing the proof. An amazing professor, a math artist, in the dark of a communist country. I have never met anybody as fascinating since.
If someone would had come to tell me that doing research exclusively means to dig one narrow area in order to write as big as possible a NUMBER of articles in ISI journals, then I would have thought that’s a disgusting perversion of a lovely quest. Then I would have switched to painting, because at least in that field (as old, no, older than mathematics) creativity won against vanity since a long time.
I was young then and I wanted to do research in as many areas as I see fit. There was no internet at the time, therefore I was filling notebooks with my work. Most of it it’s just lost, mostly because of not having anybody around to share my thoughts with, to learn from and to grow into a real researcher in a welcoming environment (with one exception, the undergraduate experience was a disappointment). I was not willing in fact to show what I do because it was much more rewarding to find out some more about some subject than to loose time to explain it to somebody, moreover now I know to trust my intuition which was telling me that there was no point to waste time for this.
The next important moment in my life as a researcher was the contact with the www, which happened in 1994 at Ecole Polytechnique from Paris, when I was doing a master. I was not interested in the courses, because I already had (a bit better, due to the mentioned exception) ones back home, but, OMG, the www! At that point, after having only one article published (The topological substratum of the derivative) — can you imagine? — which was written at a typewriter, with horrific handmade underlines and other physical constraints of the epoch — so I decided that’s have to be the future of doing research and I completely lost interest into the contrived way of communicating research by articles.
I had to write articles, and I did, only that very frequently I had problems concerning their publication, because I hold the opinion that an interesting article should combine at least three fields and it should open more questions than those solved. Foolish, really, you may say. But most of all I am still amazed how much time it took me to start to express my viewpoints publicly, through the net.
Which I am finally doing now, in this blog.
In this context, I use the personal experience as a tool in order to stress the obvious belief that www is changing the (research) world much more, much faster, than the printing press. I don’t complain about the mean reviewers, but I offer examples which support claims as: the future of peer-review is one which is technical (correct or not?), is open to anybody to contribute constructively, not based on unscientific opinions and authority arguments, separated from “publication” (whatever this means today) and perpetually subjected to change and improvement with the passage of time.
More on open peer-review in this blog here.
Fact is: there are lots of articles on arXiv and only about a third published traditionally (according to their statistics). Contrary to biology and medical science, where researchers are way more advanced in new publishing models (like PLoS and PeerJ, the second being almost green in flavour), in math and physics we don’t have any other option than arXiv, which is great, the greatest in fact, the oldest, but … but only if it had a functional peer-review system attached. Then it would be perfect!
It is hard though to come with a model of peer-review for the arXiv. Or for any other green OA publication system, I take the arXiv as example only because I am most fond of. It is hard because there has to be a way to motivate the researchers to do the peer-reviews. For free. This is the main type of psychological argument against having green OA with peer-review. It is a true argument, even if peer-review is made for free in the traditional publishing model. The difference is that the traditional publishing model is working since the 1960′s and it is now ingrained in the people minds, while any new model of peer-review, for the arXiv or any other green OA publication system, has first to win a significant portion of researchers.
Such a new model does not have to be perfect, only better than the traditional one. For me, a peer-review which is technical, open, pre- and post- “publication” would be perfect. PLoS and PeerJ already have (almost) such a peer-review. Meanwhile, us physicists and mathematicians sit on the greatest database of research articles, greener than green and older than the internet and we have still not found a mean to do the damn peer-review, because nobody has found yet a viral enough solution, despite many proposals and despite brilliant minds.
So, why not gamify the peer-review process? Researchers like to play as much as children do, it’s part of the mindframe requested for being able to do research. Researchers are driven also by vanity, because they’re smart and highly competitive humans which value playful ideas more than money.
I am thinking about Google Scholar profiles. I am thinking about vanity surfing. How to add peer-review as a game-like rewarding activity? For building peer communities? Otherwise? Any ideas?
UPDATE: … suppose that instead of earning points for making comments, asking questions, etc, suppose that based on the google scholar record and on the keywords your articles have, you are automatically assigned a part, one or several research areas (or keywords, whatever). Automatically, you “own” those, or a part, like having shares in a company. But in order to continue to own them, you have to do something concerning peer-reviewing other articles in the area (or from other areas if you are an expansionist Bonaparte). Otherwise your shares slowly decay. Of course, if you have a stem article with loads of citations then you own a big domain and probably you are not willing to loose so much time to manage it. Then, you may delegate others to do this. In this way bonds are created, the others may delegate as well, until the peer-review management process is sustainable. Communities may appear. Say also that the domain you own is like a little country and citations you got from other “countries” are like wealth transfer: if the other country (domain) who cites you is more wealthy then the value of the citation increases. As you see, until now, with the exception of “delegation” everything could be done automatically. From time to time, if you want to increase the wealth of your domain, or to gain shares in it, then you have to do a peer-review for an article where you are competent, according to keywords and citations.
Something like this could be tried and it could be even funny.
After the 15 months delay experience had with G&T which was told in the post “Anonymous peer-review after 15 months“, I decided to submit to FoM the article arXiv:0907.1520 “Emergent algebra”, even if this decision seemed to go against the view I hold, namely that gold OA is not the right OA. So I took the risk to disappoint people which have views which I respect, like Orr Shalit with his “Worse than Elsevier, worse than …“. Let me explain why:
- The article Emergent algebra deserves a “stamp of quality”. Providing such stamps is one of the roles of FoM, according to Timothy Gowers. So, I went for such a stamp, because really that’s all this article needs. Moreover,
- I highly respect the mathematicians who initiated FoM and I would be very glad to hear about their opinion on this piece of research which looks like it does not finds it’s place (because it’s revolutionary, I say, but hey, I’m the author, I am allowed to say this).
- I was expecting to get a detailed, useful, fair review from this new journal started by people described at point 2.
- I was curious what will happen, if it will matter that I expressed publicly my dislike for a new gold OA journal, in posts like this ones: Quick reaction on Gowers’ “Why I’ve joined the bad guys” and Second thoughts on Gowers’ “Why I’ve joined the bad guys”. I was NOT expecting to matter, after all math is math and opinions are opinions. But I was still a bit curious.
Today, 30 April 2013, I just received an e-mail from FoM. In a sense, I got my stamp of quality and I express my thanks for it. I reproduce the message:
Dear Dr. Buliga:
I write you in regards to manuscript # Sigma-2013-0027 entitled
“Emergent algebras” which you submitted to the Forum of Mathematics,
Unfortunately, your manuscript has been denied publication in Forum of
Mathematics. Although it is an interesting line of investigation,
based on advice from experts in the area, it was felt that the results
are not compelling enough for publication in Sigma.
Thank you for considering Forum of Mathematics, Sigma for the
publication of your research. I hope the outcome of this specific
submission will not discourage you from the submission of future
Dr. Bruce Kleiner
Forum of Mathematics, Sigma
Editors: Dr. Simon Donaldson, Dr. Bruce Kleiner, Prof. Curtis McMullen
As I said, if people like Donaldson, Kleiner and McMullen say that’s “an interesting line of investigation”, what could I ask more? Ah, maybe a referee report? As I was expecting, see previous point 3? (
As for “experts in the area”, I would like to meet them, because it’s a new area, I invented it.) Or at least, which interpretation is correct, ” although it is an interesting line of investigation, based on advice from experts in the area”, or this one “based on advice from experts in the area, it was felt that …”?
More intrigued I was by the expression, which I never encountered before in a message from a publisher: “your manuscript has been denied publication in Forum of Mathematics”.
It’s a coincidence, it may have no meaning, but I can’t help to notice that in the morning I posted “Research banana republic“, where I take the side of Mike Taylor’s post “Predatory publishers: a real problem“. In that post Mike Taylor criticizes among others the Cambridge University Press, which is the publisher of FoM. In the evening I got the previously written message from FoM concerning “denied” publication of my article. But, but … it’s math, not politics! Nah, it has to be a coincidence.
This is a continuation of Geometric Ruzsa triangle inequalities and metric spaces with dilations . Proposition 1 from that post may be applied to groupoids. Let’s see what we get.
Definition 1. A groupoid is a set , whose elements are called arrows, together with a partially defined composition operation
and a unary “inverse” operation:
which satisfy the following:
- (associativity of arrow composition) if and then and and moreover we have ,
- (inverses and objects) and ; for any we define the origin of the arrow to be and the target of to be ; origins and targets of arrows form the set of objects of the groupoid ,
- (inverses again) if then .
The definition is a bit unnecessary restrictive in the sense that I take groupoids to have sets of arrows and sets of objects. Of course there exist larger groupoids, but for the purpose of this post we don’t need them.
The most familiar examples of groupoids are:
- the trivial groupoid associated to a non-empty set is , with composition and inverse . It is straightforward to notice that and , which is a way to say that the set of objects can be identified with and the origin of the arrow is and the target of is .
- any group is a groupoid, with the arrow operation being the group multiplication and the inverse being the group inverse. Let be the neutral element of the group . Then for any “arrow$ we have , therefore this groupoid has only one object, . The converse is true, namely groupoids with only one object are groups.
- take a group which acts at left on the set , with the action such that and . Then is a groupoid with operation and inverse . We have , which can be identified with , and , which can be identified with . This groupoid has therefore as the set of objects.
For the relations between groupoids and dilation structures see arXiv:1107.2823 . The case of the trivial groupoid, which will be relevant soon, has been discussed in the post The origin of emergent algebras (part III).
The following operation is well defined for any pair of arrows with :
Let be three subsets of a groupoid with the property that there exists an object such that for any arrow we have . We can define the sets , and .
Let us define now the hard functions and with the property: for any we have
(The name “hard functions” comes from the fact that should be seen as an easy operation, while the decomposition (1) of an arrow into a “product” of another two arrows should be seen as hard.)
The following is a corollary of Proposition 1 from the post Geometric Ruzsa triangle inequalities and metric spaces with dilations:
Corollary 1. The function defined by
is injective. In particular, if the sets are finite then
Proof. With the hypothesis that all arrows from the three sets have the same origin, we notice that satisfies the conditions 1, 2 from Proposition 1, that is
- the function is injective.
As a consequence, the proof of Proposition 1 may be applied verbatim. For the convenience of the readers, I rewrite the proof as a recipe about how to recover from . The following figure is useful.
We have and and we want to recover and . We use (1) and property 1 of in order to recover . With comes . From and we recover , via the property 2 of the operation . That’s it.
There are now some interesting things to mention.
Fact 1. The proof of Proposition 2 from the Geometric Ruzsa post is related to this. Indeed, in order to properly understand what is happening, please read again The origin of emergent algebras (part III) . There you’ll see that a metric space with dilations can be seen as a family of defirmations of the trivial groupoid. In the following I took one of the figures from the “origin III” post and modified it a bit.
Under the deformation of arrows given by the operation becomes the red arrow
The operation acting on points (not arrows of the trivial groupoid) which appears in Proposition 2 is , but Proposition 2 does not come straightforward from Corollary 1 from this post. That is because in Proposition 2 we use only targets of arrows, so the information at our disposal is less than the one from Corrolary 1. This is supplemented by the separation hypothesis of Proposition 2. This works like this. If we deform the operation on the trivial groupoid by using dilations, then we mess the first image of this post, because the deformation keeps the origins of arrows but it does not keep the targets. So we could apply the Corollary 1 proof directly to the deformed groupoid, but the information available to us consists only in targets of the relevant arrow and not the origins. That is why we use the separation hypotheses in order to “move” all unknown arrow to others which have the same target, but origin now in . The proof then proceeds as previously.
In this way, we obtain a statement about algebraic operations (like additions, see Fact 2.) from the trivial groupoid operation.
Fact 2. It is not mentioned in the “geometric Ruzsa” post, but the geometric Ruzsa inequality contains the classical inequality, as well as it’s extension to Carnot groups. Indeed, it is enough to apply it for particular dilation structures, like the one of a real vectorspace, or the one of a Carnot group.
Fact 3. Let’s see what Corollary 1 says in the particular case of a trivial groupoid. In this case the operation is trivial
and the “hard functions$ are trivial as well
The conclusion of the Corollary 1 is trivial as well, because (and so on …) therefore the conclusion is
However, by the magic of deformations provided by dilations structures, from this uninteresting “trivial groupoid Ruzsa inequality” we get the more interesting original one!
Think about universities as governments, ruling over researchers and their virtual children, the students. Think about research results as bananas. The “universitary ” governments rule that the only good bananas are those accepted by publishers (mainly private entities, or even intimately associated with universities). In exchange for good bananas the researchers get vanity points, which they exchange for universitary positions or grant funds. They feed their virtual children, the students, some of the good bananas, namely their published books, or published books (validated bananas) from researchers of another, more prestigious university. These books, produced by researchers of one university are bought by another university library from a publisher, by default.
It’s a banana republic:
a banana republic is a country operated as a commercial enterprise for private profit, effected by a collusion between the State and favoured monopolies, in which the profit derived from the private exploitation of public lands is private property, while the debts incurred thereby are a public responsibility.
State = universities
Favoured monopoly = publisher
This post is triggered by Mike Taylor’s post “Predatory publishers: a real problem“.
I have added the page “Series Tracker“, which gives direct access to a part of the posts, namely to series of posts, which I am actively pursuing. The effect is they are continued in parallel, so it might be difficult to notice them in entirety, or to remember they are still active. The page gives direct access to them.
Here is a partial list of series (and stems):
- Geometric Plünnecke graphs part I
- A roadmap to computing with space part I
- Curvature and halfbrackets part I, part II, part III
- Noncommutative Baker-Campbell-Haussdorf formula part I , part II
- Emergent algebras as combinatory logic part I, part II, part III, part IV,
- Towards qubits part I, part II,
- Generalizations of positional number systems for approximate groups part 0, part I,
- Ancient Turing machines (ancient CS) part I, part II, part III, part IV, part V,
- Vision theater part I, part II, part III, part IV, part V, part VI,
- The cartesian disease part I, part II, part III
- ??? (under construction)
See also the page on graphic lambda calculus, which gives access to continued research on this specific subject.
There are other posts marked by “part I”, which means they are stems for other series, which will be added later.
These series concern mainly mathematical research, but for me this is only a part of a larger picture, for an example see the recent series on the cartesian disease, or the older on the theatre of vision.
Not mentioned (yet) are the series of posts on publishing, open access, peer-reviews or the Unlimited Detail technology.
In conclusion, it looks this blog becomes a larger pile of stuff than I previously envisioned and some ordering/structuring becomes a must. Please, if anybody has suggestions about how to better organize, or how to make the valuable parts more visible, or simply what to do with all this stuff, comment here and I’ll appreciate any constructive contribution.
This post is related to “Geometric Ruzsa triangle inequalities and metric spaces with dilations“. This time the effort goes into understanding the article arXiv:1101.2532 Plünnecke’s Inequality by Giorgis Petridis, but this post is only a first step towards a geometric approach to Plünnecke’s inequality in spaces with dilations (it will be eventually applied for Carnot groups). Here I shall define a class of decorated binary trees and a notion of closeness.
I shall use binary decorated trees and the moves R1a and R2a, like in the post “A roadmap to computing with space“:
To these move I add the “linearity moves”
Definition 1. These moves act on the set of binary trees with nodes decorated with two colours (black and white) and leaves decorated with elements of the infinite set of “variable names” . I shall denote by … such trees and by … elements of .
The edges of the trees are oriented upward. By convention we admit to be a subset of , thinking about as an edge pointing upwards which is also a leaf decorated with .
The moves act locally, i.e. they can be used for any portion of a tree from which looks like one of the patterns from the moves, with the understanding that the rest of the respective tree is left unchanged.
Definition 2. The class of finite trees is the smallest subset of with the properties:
- if then , where is the tree
- if then , where is the tree
- if and we can pass from to by one of the moves then .
The class of finite trees is closed also with respect to other operations.
Proposition 1. If then
Let us denote by the tree from the LHS of the first row, by the tree from the middle of the first row and by the tree from the LHS of the second row.
Corollary 1. If then , and .
Proof: If then . By Proposition 1 the trees and can be transformed into , therefore they belong to . Also, by the same proposition the tree can be transformed into , which we proved that it belongs to . Therefore .
I define now when two graphs are close.
Definition 3. Two graphs are close, denoted by , if there is such that can be moved into .
Proposition 3. The closeness relation is an equivalence.
Proof: By the move R1a we have for any . Take now , therefore there is such that can be moved into . We know that by Corollary 1. On the other hand we can prove that can be moved into , therefore .
Finally, let , so there’s a such that can be moved into , and let , so there’s such that can be moved into . Let now given by . Check that can be moved into , therefore .
Question: do you think this proof is more easy to understand than an equivalent proof given by drawings?
I don’t know yet what exactly is “computing with space”, but I almost know. Further is a description of the road to this goal, along with an invitation to join.
Before starting this description, maybe is better to write what this explanation is NOT about. I have arrived to the idea of “computing with space” by branching from a beautiful geometry subject: sub-riemannian geometry. The main interest I have in this subject consists in giving an intrinsic (i.e. by using a minimal bag of tools) description of the differential structure of a sub-riemannian space. The fascinating part is that these spaces, although being locally topologically trivial, have a differential calculus which is not amenable to the usual differential calculus on manifolds, in the same way as the hyperbolic space, say, is not a kind of euclidean space, geometrically. I consider very important and yet not well known the discovery of the fact that there are spaces where we can define an intrinsic differential calculus fundamentally different than the usual one (locally, not globally, as it is the case with manifolds which admits different GLOBAL differential structures, although at the local level they are just pieces of an euclidean space). But in this post I shall NOT go to explain this. The road to computing with space branches from this one, however there are paths represented by mathematical hints which are criss-crossing both these roads.
1. In “Dilatation structures I. Fundamentals” I propose, in section 4 “Binary decorated trees and dilatations” a formalism for making easy various calculations with dilation structures (or “dilatation structures”, as I called them at the moment; notice that dilation vs dilatation is a battle won by dilations in math, but by dilatation in other fields, although the correct word historically is dilatations).
This formalism works with moves acting on binary decorated trees, with the leaves decorated with elements of a metric space. It was extremely puzzling that in fact the formalism worked without needing to know which metric space I use. It was also amazing to me that reasoning with moves acting on binary trees gave proofs of generalizations of results involving elaborate calculations with pseudo-differential operators and alike. At a close inspection it looked like somewhere in the background there is an abstract nonsense machine which is just applied to this particular case of metric spaces.
Here is an example of the formalism. The moves are (I use the names from graphic lambda calculus):
Define the following tree (and think about it as being the graphical representation of an operation):
Think that it represents , with respect to the base point . Then we can prove that
which is a kind of associativity relation. The proof by binary trees has nothing to do with sub-riemannian geometry, right? An indirect confirmation is that the same formalism works very well on the ultrametric space given by the boundary of the infinite dyadic tree, see Self-similar dilatation structures and automata.
As a conclusion for this part, it seemed that in order to unravel the abstract nonsense machine from the background, I needed to:
- find a way to get rid of mentioning metric spaces, so in particular to get rid of decorations of the leaves of binary trees by points in in some space, (or maybe use these decorations as a kind of names)
- express this proof based on moves applied to binary trees as a computation, (i.e. as something like a reduction procedure).
Otherwise said, there was a need for a kind of logic, but which one?
… and dilation structures might exist, physically, in some parts of the brain. (See also section 2.4. in “Computing with space …” arXiv:1103.6007 .) I will surely come back to this subject, after learning more, but here are some facts.
The primary source of this post is the article “From A to Z: a potential role for grid cells in spatial navigation” Neural Syst Circuits. 2012; 2: 6. by Caswell Barry and Daniel Bush.
From wikipedia entry for grid cell:
A grid cell is a type of neuron that has been found in the brains of rats, mice, bats, and monkeys; and it is likely to exist in other animals including humans. In a typical experimental study, an electrode capable of recording the activity of an individual neuron is implanted in the cerebral cortex of a rat, in a part called the dorsomedial entorhinal cortex, and recordings are made as the rat moves around freely in an open arena. For a grid cell, if a dot is placed at the location of the rat’s head every time the neuron emits an action potential, then as illustrated in the adjoining figure, these dots build up over time to form a set of small clusters, and the clusters form the vertices of a grid of equilateral triangles. This regular triangle-pattern is what distinguishes grid cells from other types of cells that show spatial firing correlates. By contrast, if a place cell from the rat hippocampus is examined in the same way (i.e., by placing a dot at the location of the rat’s head whenever the cell emits an action potential), then the dots build up to form small clusters, but frequently there is only one cluster (one “place field”) in a given environment, and even when multiple clusters are seen, there is no perceptible regularity in their arrangement.
So, there are place cells and grid cells. Here is what wikipedia says about place cells:
Place cells are neurons in the hippocampus that exhibit a high rate of firing whenever an animal is in a specific location in an environment corresponding to the cell’s “place field”. These neurons are distinct from other neurons with spatial firing properties, such as grid cells, border cells, barrier cells, conjunctive cells, head direction cells, and spatial view cells. In the CA1 and CA3 hippocampal subfields, place cells are believed to be pyramidal cells, while those in the dentate gyrus are believed to be granule cells.
The behaviour of these cells is explained in the Figure 1 from the mentioned article by Barry and Bush:
(Copyright ©2012 Barry and Bush; licensee BioMed Central Ltd.)
The explanation of the Figure 1 reads:
Single unit recordings made from the hippocampal formation. a) CA1 place cell recorded from a rat. The left-hand figure shows the raw data: the black line being the animal’s path as it foraged for rice in a 1m2 arena for 20minutes; superimposed green dots indicating the animal’s location each time the place cell fired an action potential. Right, the same data processed to show firing rate (number of spike divided by dwell time) per spatial bin. Red indicates bins with high firing rate and blue indicates low firing rate, white bins are unvisited, and peak firing rate is shown above the map. b) Raw data and corresponding rate map for a single mEC grid cell showing the multiple firing fields arranged in a hexagonal lattice. c) Three co-recorded grid cells, the center of each firing field indicated by a cross with different colors corresponding to each cell. The firing pattern of each cell is effectively a translation of the other co-recorded cells as shown by superposition of the crosses (right). d) Changes made to the geometry of a familiar environment cause grid cell firing to be distorted (rescale) demonstrating that grid firing is, at least, partially controlled by environmental cues, in this case the location of the arena’s walls. Raw data are shown on the left and the corresponding rate maps on the right. The rat was familiar with the 1 m2 arena (outlined in red). Changing the shape of the familiar arena by sliding the walls past each other produced a commensurate change in the scale of grid firing. For example, shortening the x-axis to 70cm from 100cm (top right) caused grid firing in the x-axis to reduce to 78% of its previous scale, while grid scale in the Y-axis was relatively unaffected. Numbers next to the rate maps indicate the proportional change in grid scale measured along that axis (figure adapted from reference ).
And now, the surprise: scale is indeed a place in the brain. Let’s see Figure 2. from the same article:
(Copyright ©2012 Barry and Bush; licensee BioMed Central Ltd.)
The caption of this figure is:
Grid scale increases along a dorso-ventral gradient in the mEC. Two grid cells recorded from the same animal but at different times are shown, both cells were recorded in a familiar 1m2 arena. Approximate recording locations in the mEC are indicated. The more ventral cell exhibits a considerably larger size of firing fields and distance between firing fields than the dorsal cell.
… and, from the article, (boldfaced by me):
The scale of the grid pattern, measured as the distance between neighboring peaks, increases along the dorso-ventral mEC gradient, mirroring a similar trend in hippocampal place fields [15,25]. The smallest, most dorsal, scale is typically 20 to 25cm in the rat, reaching in excess of several meters in the intermediate region of the gradient [15,26] (Figure (Figure2).2). This may explain how this remarkable pattern was missed by early electrophysiology studies, which targeted ventral mEC and found only broadly tuned spatial firing (for example, ). Interestingly, grid scale increases in discontinuous increments and the increment ratio, at least between the smaller scales, is constant . Grid cells recorded from the same electrode, which are, therefore, proximate in the brain, typically have a common scale and orientation but a random offset relative to each other and the environment . As such, their firing patterns are effectively identical translations of one another and a small number of cells will ‘tile’ the complete environment (Figure (Figure1c).1c). It also appears that grids of different scale recorded ipsilaterally have a common orientation, such that the hexagonal arrangement of their firing fields share the same three axes, albeit with some localized distortions [15,28,29].
That’s just amazing!
Concerning the hypothesis (Hafting, T.; Fyhn, M.; Molden, S.; Moser, M. -B.; Moser, E. I. (2005). “Microstructure of a spatial map in the entorhinal cortex”. Nature 436 (7052): 801–806. Bibcode:2005Natur.436..801H. doi:10.1038/nature03721.) that the grid cells firing fields encode the abstract structure of an euclidean space, I think this is not following from the observations. My argument is that the translation-invariance (of firing patterns in this particular case) emerge by the mechanism of dilation structures and it is, at least up to my actual understanding, an evidence for the existence of these structures in the brain. But of course, there is much to learn and think about.
I continue from “Curvature and halfbrackets, part II“. This post is dedicated to the application of the previously introduced notions to the case of a sub-riemannian Lie group.
1. I start with the definition of a sub-riemannian Lie group. If you look in the literature, the first reference to “sub-riemannian Lie groups” which I am aware about is the series Sub-riemannian geometry and Lie groups arXiv:math/0210189, part II arXiv:math/0307342 , part III arXiv:math/0407099 . However, that work predates the introduction of dilation structures, therefore there is a need to properly define this object within the actual state of matters.
Definition 1. A sub-riemannian Lie group is a locally compact topological group with the following supplementary structure:
- together with the dilation structure coming from it’s one-parameter groups (by the Montgomery-Zippin construction), it has a group norm which induce a tempered dilation structure,
- it has a left-invariant dilation structure (with dilations and group norm denoted by ) which, paired with the tempered dilation structure mentioned previously, it satisfies the hypothesis of “Sub-riemannian geometry from intrinsic viewpoint” Theorem 12.9, arXiv:1206.3093
- there is no assumption on the tempered group norm to come from a Riemannian left-invariant distance on the group. For this reason, some people use the name sub-finsler arXiv:1204.1613 instead of sub-riemannian, but I believe this is not a serious distinction, because the structure of a scalar product which induces the distance is simply not needed for understanding sub-riemannian Lie groups.
- by Theorem 12.9, it follows that the left-invariant field of dilations induces a length dilation structure. I shall use this further. Length dilation structures are maybe a more useful object than simply dilation structures, because they explain how the length functional behaves at different scales, which is a much more detailed information about the microscopic structure of a length metric space than just the information about how the distance behaves at different scales.
This definition looks a bit mysterious, unless you read the course notes cited inside the definition. Probably, when I shall find the interest to pursue it, it would be really useful to just apply, step by step, the constructions from arXiv:1206.3093 to sub-riemannian Lie groups.
2. With the notations from the last post, I want to compute the quantities . We already know that is related to the curvature of with respect to it’s sub-riemannian (sub-finsler if you like it more) distance, as introduced previously via metric profiles. We also know that is controlled by and . But let’s see the expressions of these three quantities for sub-riemannian Lie groups.
I denote by the left invariant sub-riemannian distance, therefore we have .
Now, , where by definition. Notice also that , where is the deformed group operation at scale , i.e. it is defined by the relation:
With all this, it follows that:
A similar computation leads us to the expression for the curvature related quantity
. This last quantity is controlled by a halfbracket, via a norm inequality.
The expressions of make transparent that the curvature-related is the sum of and . In the next post I shall use the length dilation structure of the sub-riemannian Lie group in order to show that is controlled by , which in turn is controlled by a norm of a halfbracket. Then I shall apply all this to , as an example.
This post is a written record of mt thoughts after reading “WTF? The University of California sides with publishers against the public” by Michael Eisen.
I suspected and privately said to reluctant ears that there is something profoundly dishonest, in principle, in the system of fabricating research papers for the love of the number of them, BUT (and the emphasis is here) it works because many researchers love it. Well, maybe not many and maybe not especially the young ones, but many of the researchers with established reputation constructed in the interior of this system.
Is this a naive thought? Surely is for the two categories of people in the academia who sustain it: a big, maybe a majority, maybe not, class of mediocre researchers, formed by those who find a sure and opportunistic path to promotion, tenure, etc, by producing a kind of structured noise which looks like research and, a second class, of managers of the academic realm, having direct interest into the system, mainly, I suspect, because it provides access to power over other people lives. (The second class may be populated by former members of the first, this follows logically from the fact that if the promotion system is based on massive mediocre crap production then the best among the producers tend to be selected by the system.)
My preferred comparison of the fall in the making of the actual academic system is described in “Another parable of academic publishing: the fall of 19th century academic art“. Continuing the comparison, it is true that the production of independent artists surely contained (and still contains now-a-days) a lot of garbage, but it is also true that the academic production of paintings was massively mediocre. What to choose — diversity, from very bad to very good to out of the scale exploratory art — or — uniform mediocrity, with rare dashes of solid, good, surprising or even exceptional academic paintings? In the past, diversity won.
- This post is not directed against UC. At least UC made a statement which is criticised in the post by Michael Eisen. On the contrary, the vast majority of smaller, or less visible academic institutions don’t even make clear their respective positions on this matter. In the background business goes as usual.
- See also the very well written previous post by Eisen, “The Past, Present and Future of Scholarly Publishing “. Just a small quote from the post:
Tonight, I will describe how we got to this ridiculous place. How twenty years of avarice from publishers, conservatism from researchers, fecklessness from universities and funders, and a basic lack of common sense from everyone has made the research community and public miss the manifest opportunities created by the Internet to transform how scholars communicate their ideas and discoveries.
Avizienis introduced a signed-digit number representation which allows fast, carry propagation-free addition (A. Avizienis, Signed-digit number representation for fast parallel arithmetic, IRE Trans. Comput., vol. EC-10, pp. 389-400, 1961, see also B. Parhami, Carry-Free Addition of Recoded Binary Signed-Digit Numbers, IEEE Trans. Computers, Vol. 37, No. 11, pp. 1470-1476, November 1988.)
Here, just for fun, I shall use a kind of approximate groups to define a carry-free addition-like operation.
1. The hypothesis is: you have three finite sets , , , all sitting in a group . I shall suppose that the group is commutative, although it seems that’s not really needed further if one takes a bit of care. In order to reflect this, I shall use for the group operation on , but I shall write for the set of all elements of the form with and .
We know the following about the three (non-empty, of course) sets :
- , , , , ,
- (where is the cardinal of the finite set ),
- (that’s what qualifies as a kind of an approximate group).
Let now choose a bijective function and two functions and such that
(1) for any and any we have .
2. Let be the family of functions defined on with values in , with compact support, i.e. if belongs to , then only a finite number of have the property that . The element is defined by for any . If with then is the smallest index with .
3. The structure introduced at 1. allows the definition of an operation on . The definition of the operation is inspired from a carry-free addition algorithm.
Definition 1. If both are equal to then . Otherwise, let be the smallest index such that one of or is different than . The element is defined by the following algorithm:
- for any ,
- let , . Repeat:
4. We may choose the functions such that the operation is starting to become interesting. Before doing this, let’s remark that:
- the operation is commutative as a consequence of the fact that is commutative,
- the operation is conical, i.e. it admits the shift , as automorphism ( a property encountered before for dilation structures on ultrametric spaces, see arXiv:0709.2224 )
Proposition 1. If the functions satisfy the following:
- for any
- , for any , for any ,
then is a conical quasigroup.
For reasons partly explained some time ago in this blog, as well as for more recent ones, I want to explore if there’s future life for me outside the academia. I became very skeptical about the ways of this small world (in particular about the pressure for traditionally published articles versus arXiv and other true OA variants which still lack for mathematicians), as well as about the value of this accumulation of salami-sliced research results which are the fashion now-days. Originality and strength to pursue high-risk but high-reward research are just not what is needed in academic (mathematical) research. I am more of a theory and models builder type, again apparently not needed in this narrow problem-solving world. Interdisciplinary knowledge and nose for the next big ideas are not on the demands lists in academic (mathematical, at least) circles, according to my experience. For all these reasons, but mainly because of the slow starvation of academic research I am witnessing around me, I look for other options. Preferably privately funded, preferably high-risk high-reward research. Or maybe concerning critical assessment of new ideas, or even communicating in nontrivial ways, out of the manual. Or something completely unexpected. I don’t know, therefore I have to ask. Please direct any comment to my address Marius.Buliga@gmail.com.
The article “Minimal generating sets of Reidemeister moves“ by Michael Polyak gives some minimal generating sets of all oriented Reidemeister moves. The question I am asking in this post is this: how big can be non-generating sets of moves?
The oriented Reidemeister moves are the following (I shall use the same names as Polyak, but with the letter replaced by the letter ):
- four oriented Reidemeister moves of type 1:
- four oriented Reidemeister moves of type 2:
- eight oriented Reidemeister moves of type 3:
Among these moves, some are more powerful than others, as witnessed by the following
Theorem. The set of twelve Reidemeister moves formed by:
- four moves of type 1
- two moves of type 2, namely R2a and R2b
- six moves of type 3, namely R3b, R3c, R3d, R3e, R3f, R3g
does not generate the remaining moves R2c, R2d, R3a and R3h.
I postpone the proof for another time, here I just want to emphasize that the moves R2c, R2d, R3a and R3h are more special than the rest of them. The proof which I have is via the graphic lambda calculus, but probably other proofs exists. Please tell me if you know a proof of this theorem.
UPDATE: see a proof at mathoverflow.
Is there any difference between writing a research article and writing a blog post on a research subject? This is what I would like to understand.
In my mind research articles should become a subset of … how should I call them, maybe “research posts”. There are obvious advantages on the side of the research posts, as well as some disadvantages. I think it is revealing that the advantages have an objective flavour, while the disadvantages have more of a subjective one, mostly being related to the bad image the blog posts have among the “serious” researchers.
I have already experimented with this idea:
- the Tutorial on graphic lambda calculus has served as the template for arXiv:1302.0778 On graphic lambda calculus and the dual of the graphic beta move,
- the post Geometric Ruzsa triangle inequalities and metric spaces with dilations became arXiv:1304.3358 Geometric Ruzsa triangle inequality in metric spaces with dilations, with very few modifications,
- I just submitted on arXiv the article arXiv:1304.3694 Origin of emergent algebras, based on the posts The origin of emergent algebras, part II and part III, as well as parts of Emergent algebra as combinatory logic (part I),
- now I am struggling with writing a shorter (and somehow dumber) version of arXiv:1207.0332 Local and global moves on locally planar trivalent graphs, lambda calculus and -Scale, because I was too hurried and spoiled by the freedom this blog gives me to do research, so in the first version of the article I just write too much, without enough motivation. Therefore I decided (based on a peer-review which I appreciated) to concentrate first only on what the graphic lambda calculus can do with the gates corresponding to the application, the abstraction and the fan-out: the lambda calculus part of the graphic lambda calculus, along with the braids formalism part. The emergent algebra part is for later.
The format of the articles from this list is as much as possible similar with the one of the research posts. As an example I mention the use of links inside the text, including direct links to the (preferably OA) versions of the cited articles. See the post Idiot things that we we do in our papers out of sheer habit by Mike Taylor for more examples of the same “habits” which I already renounced in some of my papers.
The new habit of giving exactly the link to the article, instead of a numbered citation in the bibliography, as well as giving the link to ANY source which is used in the research article (as for example a wikipedia page for a first time encounter of a term, along with the invitation for further study from another sources), is clearly one of the advantages a blog post has over a traditionally written article.
However, it is difficult to find the good balance between the extreme freedom of a blog post and the more constraint one of a research article (although the blog of Terence Tao is a very good example – maybe the best I know about – that such a balance may be attainable).
My guess is that at some point open peer-review and this change of habits concerning writing research articles will meet.________________
UPDATE: See the post Blogging as post-publication peer review: reasonable or unfair? by Dorothy Bishop, as well as the comment by Phillip Lord which I reproduce here:
Why stop there? If Author self-publishing can provide rapid feedback on “properly” published science, then they can also provide dissemination of that science in the first place.
Scientific publishing has too long been about credit and promotion. It’s time it returned to what it really should be and what it originally was: communication.
I continue from Packing and unpacking arrows in graphic lambda calculus and I want now to describe another sector of the graphic lambda calculus, called the strand networks sector.
Strand networks are a close relative to spin networks (in fact they may be used as primitives, together with some algebraic tricks in order to define spin networks). See the beautiful article by Roger Penrose, Angular momentum: an approach to combinatorial space-time, in Quantum Theory and Beyond, ed. T. Bastin, Cambridge University Press, Cambridge, 1971. where strand networks appear in and around figure 17.
However the strand networks which I shall write about here are oriented. Moreover, a more apt name for those would be “states of strand networks”. The main thing to retain is that the algebraic glue can be put over them (for example, as concerns free algebras generated by those, counting states, grouping them into equivalence classes which are the real strand networks and so on).
In graphic lambda calculus we may consider graphs in which are obtained by the following procedure. We start with a finite collection of loops and we clip together, as we please, strands from these loops. How can we do this? Is simple, we choose our strands we want to clip together and the we cross them with an arrow. Here is an example for three strands:
We want to make this clipping permanent, so to say, therefore we apply as many graphic beta moves as are needed, three in this example:
The beautiful thing about this procedure is that in the middle of the graph from the right hand side we have now three arrows with the same orientation. So, let’s pack them into one arrow. The packing procedure for three arrows, with the same orientation, is the following:
So, we pack the arrows, we connect them and we unpack afterwards.
With the previous trick, we may define now decorated arrows of strand networks, but mind that we have bundles, packs of oriented strands, so the decoration has to be more precise than just indicating the number of strands.
That is about all, only we have to be careful to discern among clipped strands which become decorated arrows and clipped strands which become nodes of the strand network. But this is rather obvious, if you think about.
Conclusion. Let’s think about the purpose of the algebraic glue which is needed in order to define a spin network and then an evolving spin network. As you can see all comes down to another way of deciding which is the sequence of applications of (graphic) beta moves, which is this time probabilistic. It is another type of computation than the one from the lambda calculus sector, where we use the combinators and zippers (as well as some strategy of evaluation) in order to make a classical computation. So at the level of graphic lambda calculus, it becomes transparent that both are computations, but with slightly different strategies. (This is a kind of a theorem, but to put it in a precise form one has to make ornate notations and unnecessary orderings, so it would be just a manifestation of the cartesian disease, which we don’t need.) The last point to mention is that, compared to the lambda calculus strategy (based on combinators plus choice of evaluation), this computation which comes from physics, is more cartesian disease free.
This post is for a new discussion on UD, because the older ones “Discussion about how an UD algorithm might work” and “Unlimited detail challenge: the most easy formulation” have lots of comments.
UPDATE: It looks like Oliver Kreylos (go and read his blog as well) extremely interesting work on LiDAR Visualization is very close to Bruce Dell’s Geoverse. Thanks to Dave H. (see comments further) for letting me know about this. It is more and more puzzling why UD was welcomed by so much rejection and so many hate messages when, it turns out, slowly but surely, that’s one of the future hot topics.
Please post here any new comment and I shall update this post with relevant information. We are close to a solution (maybe different than the original), thanks to several contributors, this project begins to look like a real collaborative one.
Note: please don’t get worried if your comment does not appear immediately, this might be due to the fact that it contains links (which is good!) or other quirks of the wordpress filtering of comments, which makes that sometimes I have to approve comments by people who already have previous approved comments. (Also, mind that the first comment you make have to be approved, the following ones are free of approval, unless the previous remarks apply.)
I am very intrigued by the following idea, which is not new (see further), but I have not seen it discussed in the small world of mathematics. Peer-reviews have two goals which could be separated, for the benefit of a better communication of research among scholars:
- to filter submitted articles as a function of the soundness of the research work,
- to assess the level of interest of the research work.
The first goal is a must, the second one opens the gate to abuse and subjective bias.
I learned about the idea of separating these two goals from the presentation by Maria Kowalczuk which I shared in this post. Afterwards, I looked on the net to find more. It seems this idea was pioneered by PLoS ONE, with PeerJ being the latest adopter. The following citation is taken from “Open and Shut?:UK politicians puzzle over peer review in an open access environment” (2011):
* On splitting traditional peer review into two separate processes: a) assessing a paper’s technical soundness and b) assessing its significance — a model pioneered by open-access publisher PLoS ONE, and now increasingly being adopted by traditional publishers …
Q162 Chair: We have heard that pre-publication peer review in most journals can be split, broadly, into a technical assessment and an impact assessment. Is it important to have both?
Dr Torkar: … It is fairly straightforward to think about scientific soundness because it should be the fundamental goal of the peer review process that we ensure all the publications are well controlled, that the conclusions are supported and that the study design is appropriate. That is fairly straightforward as a very important aspect which should be addressed as part of the peer review process.
The question of the importance of impact is more difficult. When we think about high impact papers we think about those studies which describe findings that are far reaching and could influence a wide range of scientific communities and inform their next-stage experiments. Therefore, it is quite important to have journals that are selective and reach out to a broad readership, but the assessment of what is important can be quite subjective. That is why it is important, also, to give space to smaller studies that present incremental advances. Collectively, they can actually move fields forward in the long term.
Dr Patterson: … [B]oth these tasks add something to the research communication process. Traditionally, technical assessment and impact assessment are wrapped up in a single process that happens before publication. We think there is an opportunity and, potentially, a lot to be gained from decoupling these two processes into processes best carried out before publication and those better left until after publication.
One way to look at this is as follows. About 1.5 million articles are published every year. Before any of them are published, they are sorted into 25,000 different journals. So the journals are like a massive filtering and sorting process that goes on before publication. The question we have been thinking about is whether that is the right way to organise research. There are benefits to focusing on just the technical assessment before publication and the impact assessment after publication … Online we have the opportunity to rethink, completely, how that works. Both are important, but we think that, potentially, they can be decoupled …
Dr Lawrence: … [I]t is not known immediately how important something is. In fact, it takes quite a while to understand its impact. Also, what is important to some people may not be to others. A small piece of research may be very important if you are working in that key area. Therefore, the impact side of it is very subjective.
Dr Read: … Separating the two is important because of the time scale over which you get your answer. The impact is much longer. I guess the technical peer review is a shorter-term issue.
What intrigues me the most is that, even if the idea comes from the publication of medical research, it rather looks easy to implement in a model of math publication. Indeed, is it not the first purpose of peer-review of a math article to decide the soundness of mathematical results from within?
In mathematics we have the proof, which is highly optimized for independent check. True or false, sound or flawed, right? In principle at least, the peer-review in mathematics should serve mainly as a filter for sound results. The reality is different, I think experiences like the ones described in this post (browse through the provided links too, maybe), are by no means exceptional, but rather common.
As for the interest level, it is well known that in mathematics one never ever knows the long term effect of a mathematical result. It is common in mathematics that results have a big latency, that articles may become suddenly relevant decades and even centuries after their publication. It should be common-sense in mathematics that one cannot rely on the peer-review assessment of level of interest. But is not and, more often than not, under the umbrella of the relevance for the journal publication lurk darker things, like conflict of interests, over-protection of one field of work against stranger researchers intrusion, or exclusion based on club membership. In few words: the second role of peer-review is used for masking power games.
All that being said, let’s contemplate if the idea of keeping only the first role of peer-reviews is feasible. It certainly is, if it works already for PeerJ and PLoS ONE, why would not work for new models of mathematical publication, where it would be easier to apply? As usual, the most difficult part is to start using it.
Questions: how can be implemented an open, perpetual peer-review with the main goal of assessment of soundness of mathematical research? Would a system based on comments and manifold contributions from blogs, as the Retraction Watch for example, be part of the the soundness decision, or a part of the level of interest decision? Would, for example, the wikipedia model be better for the soundness part of peer-reviews and “comments in blogs” dreaded by some mathematicians would be better for assessing the local (in time and space) level of interest?
Your informative or critical comments would be great!
Via the post “Peer pressure: the changing role of peer review” at BioMed Central blog, which I highly recommend as reading for those interested in the problem of peer review. I embed further the presentation “Future of peer review” by Maria Kowalczuk, because I think it exactly applies to publishing in mathematics. There’s a lot to learn form, or to discuss about.
UPDATE: Here is an almost similar presentation, from dec. 2011, by Iain Hrynaszkiewicz: (pdf)
I continue from “Curvature and halfbrackets, part I“, with the same notations and background.
In a metric space with dilations , there are three quantities which will play a role further.
1. The first quantity is related to the “norm” function defined as
Notice that this is not a distance function, instead it is more like a norm of with respect to the basepoint , at scale . Together with the field of dilations, this “norm” function contains all the information about the local and infinitesimal behaviour of the distance . We can see this from the fact that we can recover the re-scaled distance from this “norm”, with the help of the approximate difference (for this notion see on this blog the definition of approximate difference in terms of emergent algebras here, or go to point 3. from the post The origin of emergent algebras (part III)):
(proof left to the interested reader) This identity shows that the uniform convergence of to , as goes to , is a consequence of the following pair of uniform convergences:
- that of the function which converges to
- that of the pair (dilation, approximate difference) to , see how this pair appears from the normed groupoid formalism, for example by reading the post from the post The origin of emergent algebras (part III).
With this definition of the “norm” function, I can now introduce the first quantity of interest, which measures the difference between the “norm” function at scale and the “norm” function at scale :
The interpretation of this quantity is easy in the particular case of a riemannian space with dilations defined by the geodesic exponentials. In this particular case
because the “norm” function equals the distance between (due to the definition of dilations with respect to the geodesic exponential).
In more general situations, for example in the case of a regular sub-riemannian space, we can’t define dilations in terms of geodesic exponentials (even if we may have at disposal geodesic exponentials). The reason has to do with the fact that the geodesic exponential in the case of a regular sub-riemannian manifold, is not intrinsically defined as a function from the tangent of the geodesic at it’s starting point. That is because geodesics in regular sub-riemannian manifolds (at least those which are classically, i.e. with respect to the differential manifold structure, smooth , are bound to have tangents only in the horizontal directions.
As another example, think about a sub-riemannian Lie group. Here, we may define a left-invariant dilation structure with the help of the Lie group exponential. In this case the quantity is certainly not equal to , excepting very particular cases, as a riemannian compact Lie group, with bi-invariant distance, where the geodesic and Lie group exponentials coincide.
2. The second quantity is the one which is most interesting for defining (sectional like) curvature, let’s call it
3. Finally, the third quantity of interest is a kind of a measure of the convergence of to , but measured with the norms from the tangent spaces. Now, a bit of notations:
for any three points ,
for any three points and
for any two points .
With these notations I introduce the third quantity:
The relation between these three quantities is the following:
Suppose that we know the following estimates:
higher order terms, with and ,
higher order terms, with and ,
higher order terms, with and ,
Lemma. Let us sort in increasing order the list of the values and denote the sorted list by . Then .
The proof is easy. The equality from the Proposition tells us that the modules of , and can be taken as the edges of a triangle. Suppose then that , use the estimates from the hypothesis and divide by in one of the three triangle inequalities, then go with to in order to arrive at a contradiction .
The moral of the lemma is that there are at most two different coefficients in the list . The coefficient is called “curvdimension”. In the next post I shall explain why, in the case of a sub-riemannian Lie group, the coefficient is related to the halfbracket. Moreover, we shall see that in the case of sub-riemannian Lie groups all three coefficient are equal, therefore the infinitesimal behaviour of the halfbracket determines the curvdimension.
In my opinion, the best parts of the cartesian method are:
- doubt as a tool for advancing understanding, i.e this part of the rule 1: “never to accept anything for true which I did not clearly know to be such [...] to comprise [...] so clearly and distinctly as to exclude all ground of doubt”,
- and this part of rule 3, taken out of context, seen as a belief, a state of mind of the researcher: “by commencing with objects the simplest and easiest to know, I might ascend [...] to the knowledge of the more complex”
This is, in a nutshell, that part of the scientific method which apply even to mathematicians (who don’t do experiments). If there is any need to tell, I strongly believe in the scientific method, although I recently understood that I have problems with that parts of the cartesian method which I now think will become obsolete, due to better and faster communication among scientists and due to the confrontation with big data (which will require techniques adapted to the recognition of the fact that reality might be complex enough so that a model of it does not fit wholly into one human brain).
The following are just doubts and questions which might be naive or based on ignorance, but nevertheless I write them here, in the hope of informed comments.
Remember again I am a mathematician, so maybe I disappoint some readers (I hope not my readers) by saying that I have no problem with Cantor diagonal argument for the proof of the fact that the set of reals is uncountable. What makes me feel less comfortable is the impression, which might be false, that famous theorems in logic, like Godel, or Turing, show in fact that there are limits to the enumeration part of the cartesian method. Am I right? To make a comparison, suppose I’am a physicist with infinite powers and I say: I made the experiment of counting all them Turing machines and I get results which are contradictory with older experiments. I deduce then, by the scientific method, that the counting procedure itself, any I might think about, is flawed, because all the other parts have been checked independently. It means I am not allowed to use the part of the cartesian method which comprises enumerations if I want to understand the reality. Reality is like this, point. If I count then I am led astray, like, as another comparison, if I suppose that a quantum object has a arbitrarily well localized position and moment, then I am led astray. (Not that I think there is any connection between logic theorems based on Cantor diagonal argument and quantum mechanics.)
So, what is left of logic if counting arguments are eliminated? Anybody knows? I hope so.
UPDATE: Gromov has now a sequel to his ergobrain paper, which I commented before, see Gromov’s Ergobrain and How not to get bored, by reading Gromov and Tao. The sequel is called “Ergostuctures, Ergologic and the Universal Learning Problem: Chapters 1, 2.“. My interest is stirred by this new term “ergologic”, which may be of some interest to the readers of some posts from this blog, namely those regarding the cartesian theater and/or the graphic lambda calculus.
Analysis is the process of breaking a complex topic or substance into smaller parts to gain a better understanding of it. The technique has been applied in the study of mathematics and logic since before Aristotle (384–322 B.C.), though analysis as a formal concept is a relatively recent development.
As a formal concept, the method has variously been ascribed to Alhazen, René Descartes (Discourse on the Method), and Galileo Galilei. It has also been ascribed to Isaac Newton, in the form of a practical method of physical discovery (which he did not name or formally describe).
The four rules of the cartesian methods are:
The first was never to accept anything for true which I did not clearly know to be such; that is to say, carefully to avoid precipitancy and prejudice, and to comprise nothing more in my judgement than what was presented to my mind so clearly and distinctly as to exclude all ground of doubt.
The second, to divide each of the difficulties under examination into as many parts as possible, and as might be necessary for its adequate solution.
The third, to conduct my thoughts in such order that, by commencing with objects the simplest and easiest to know, I might ascend by little and little, and, as it were, step by step, to the knowledge of the more complex; assigning in thought a certain order even to those objects which in their own nature do not stand in a relation of antecedence and sequence.
And the last, in every case to make enumerations so complete, and reviews so general, that I might be assured that nothing was omitted.
In order to justify my claim that the cartesian method is an analysis (or compression) technique, I shall comment these rules one by one.
1. this rule is made by several parts:
- “never to accept anything for true which I did not clearly know to be such”
- “to comprise nothing more in my judgement than what was presented to my mind”
- “so clearly and distinctly as to exclude all ground of doubt”
The first part looks like a thinking hygiene: be sure about your hypotheses.
The second part has to do with the limitations of our brains capacity to process a complex topic. As such, these limitation have nothing to do with the topic under study. Of course we can’t advance our understanding of a subject if we can’t wholly grasp it in our minds. However, is important to remember that when we splice it in smaller, more understandable parts, we introduce an element which has nothing to do with the subject of study, but with our capacity of understanding (and our prejudices, indeed, as witnessed by the fact that the same research subject is spliced differently in different epochs or places, according to cultural prejudices and not biological “computing power” reasons).
The third part has entirely to do with our limitations. In order to understand the topic, we have to use techniques which “exclude all ground of doubt”. The great importance of doubt as a tool for understanding is one of the most viral parts of the cartesian method. It is one of the main ingredients of the scientific method.
2. two parts here as well:
- “to divide each of the difficulties under examination into as many parts as possible”
- “and as might be necessary for its adequate solution”
While the first part is clearly an analysis technique, the second part tell that the purpose of understanding is to find a solution for a sequence of problems. Each small part, each difficulty has to be solved. To make a comparison, say that we have a huge cake to eat, so we chip at it with our small mouths, claiming that our goal is to well chew each bite.
This is a compression technique: we divide the cake into bites, then, as we chew each bite, we forget about the others. The bad part is that the cake is not just the sum of the bites.
3. This is the most problematic rule, because here is given total priority to the understanding over the subject of understanding. The analysis and compression technique from the rule 2 is taken to extreme: first is suggested something like an eager evaluation
- “to conduct my thoughts in such order that”
- “by commencing with objects the simplest and easiest to know, I might ascend [...] to the knowledge of the more complex”
- “little and little, and, as it were, step by step”
… then, in order to be sure that the eager evaluation works,
- “assigning in thought a certain order even to those objects which in their own nature do not stand in a relation of antecedence and sequence”
otherwise, who cares about the subject of understanding as long as I can produce an working algorithm? Then, we study the algorithm and we forget what was all this about. This looks like the most harmful part of the cartesian method and the main source of the cartesian disease…
4. … until we read the last rule:
- “in every case to make enumerations so complete, and reviews so general, that I might be assured that nothing was omitted”
But if we already renounced at the subject of study and we already (recursively) replaced it with an artificial division, enumeration and analysis technique, this rule is only a proclamation of the superiority of understanding of reality over the reality itself: if the understanding of reality is internally coherent, then it is as good as the reality itself.
Conclusion. The cartesian method is designed as a technique for understanding performed by one mind in isolation, severely handicapped by the bad capacity of communication with other isolated minds. It was a very efficient technique, which is now challenged by two effects of its material outcomes:
- better communication channels provided by the www,
- mechanical, or should I say digital, applications of the method which largely surpass the capacity of understanding of one human mind, as witnessed for example by the first computer aided mathematical proofs, or for another example by the fact that we can numerically model physical phenomena, without understanding rigorously why the method works.
UPDATE: if you read this, then you might be interested to read “Descartes, updated” at The “Putnam Program” blog.
Here is a little story about curvature and Lie brackets in the wider context of dilation structures and sub-riemannian Lie groups. The background of this post is provided by the following links:
- Sub-riemannian geometry from intrinsic viewpoint, course notes, arXiv:1206.3093 [math.MG], especially section 2.5 “Curvdimension and curvature” and 12.3 “Coherent projections induce length dilation structures”,
- Curvdimension and curvature of a metric profile part I, part II, part III,
- Noncommutative Baker-Campbell-Haussdorf formula, the problem and a suggested solution.
We are in the frame of a metric space with dilations (aka “dilation structure” or “dilatation structure”). I introduced these spaces under the name “dilatation structures” in the article Dilatation structures I. Fundamentals, arXiv:math/0608536 [math.MG], but see the course notes for the most advanced formulation. In particular regular sub-riemannian manifolds, riemannian manifolds and Lie groups with a left invariant distance induced by a completely non-integrable (i.e. generating) distribution are examples of such spaces.
There are two problems, both without a clear solution yet, concerning this class of spaces:
- Problem 1: how to define a good notion of curvature? In particular, how to define an intrinsic notion of curvature for a regular sub-riemannian manifold, such that, for example, the curvature of a Carnot group is null (i.e. Carnot groups are flat)? On one side, I have a definition, the one about the curvature of a metric profile, which I think is good, but I have troubles computing it for particular examples. On the other side, all other notions of curvature, like for example those coming from the Wasserstein distance, fail spectacularly for sub-riemannian spaces.
- Problem 2: for sub-riemannian Lie groups, or more general for groups with dilations (i.e. left-invariant dilation structures on a topological group), how to define a good generalization of a Lie bracket? As I explained in the posts about the noncommutative BCH formula, this problem comes from the two ways we may interpret the Lie bracket. The first way, classical, says that the Lie bracket is an object which measures the noncommutativity of the group operation. This is in fact a statement which applies not to the Lie bracket, but to the commutator. The Lie bracket itself is related to the second order variation of the commutator (i.e. the commutator of two elements of the group, which are -small, equals times the Lie bracket of those elements plus terms of higher order in . The trouble with this definition of the Lie bracket is that if we replace the “usual” dilations associated to a Lie group, those coming from one-parameter subgroups, by more general dilations then all the reasoning crushes in at least two places. The first place is that in the tangent space at the identity of the group we have a noncommutative (but nilpotent) addition operation instead of the commutative plus operation, i.e. a Carnot group instead of a vector space structure, which makes hard to understand what sense the BCH formula makes. The second place is that instead of the term, there is nothing clear about the “expansion” of the commutator with respect to in this more general case. Another interpretation of the Lie bracket comes from the fact that a half of the Lie bracket measures the speed of the deformation of the group operation by dilations, at . So, in this interpretation, the half of the Lie bracket appears as the first order variation of the deformed group operation. This can be generalized to the more general case of a group with dilations and it leads to the notion of a halfbracket.
My purpose is to explain a link between the curvature (defined as the curvature of a metric profile) and the halfbracket.
With the notations from dilation structures, we know that for any sufficiently close and any sufficiently small we have the uniform estimate
Here is the distance between and , seen as elements of the tangent space at . The quantity
is the deformation of the distance by dilations , centered at , of coefficient .
Let’s take, as an example, the case of a riemannian manifold with geodesic exponential and dilations defined by:
In this case
where is related to the sectional curvature at (we suppose that are not collinear). This gives a curvdimension equal to and also a notion of sectional curvature.
But in general all we can hope is an estimate of the form
where is the curvdimension, and the trick is to have an estimate for the curvdimension. In the next post I shall use halfbrackets in order to estimate the curvdimension, for sub-riemannian Lie groups.
The wording “cartesian disease”, already used in this post (see also my comments there), means an abuse or misuse of the cartesian method. I shall use in the argumentation the citation available at the last given link, because there is a concentrate of the method, formulated in such a precise, equilibrated and astonishingly actual words.
The abuse of some part of the method consists in the excess of use of one ingredient, in parallel with a lack of use of another ingredient provided by Descartes’ method.
In the following are examples of cartesian disease, listed according to the place in the citation which is relevant for the abuse characteristic in the respective example. (Please read, however, the whole text in order to identify the countermeasures which the respective abuses ignore.)
- architectural constructs where many, if not most, of people live. The relevant citation from Descartes is: “Thus it is observable that the buildings which a single architect has planned and executed, are generally more elegant and commodious than those which several have attempted to improve, by making old walls serve for purposes for which they were not originally built. Thus also, those ancient cities which, from being at first only villages, have become, in course of time, large towns, are usually but ill laid out compared with the regularity constructed towns which a professional architect has freely planned on an open plain; so that although the several buildings of the former may often equal or surpass in beauty those of the latter, yet when one observes their indiscriminate juxtaposition, there a large one and here a small, and the consequent crookedness and irregularity of the streets, one is disposed to allege that chance rather than any human will guided by reason must have led to such an arrangement.” Horror, boredom, social problems appeared from living in such functionally designed, but culturally void places, despite the good will of the creators.
- the cohort of dictators, along with their respective ideologies, specific to the 20th century. The relevant citation from Descartes is: “In the same way I fancied that those nations which, starting from a semi-barbarous state and advancing to civilization by slow degrees, have had their laws successively determined, and, as it were, forced upon them simply by experience of the hurtfulness of particular crimes and disputes, would by this process come to be possessed of less perfect institutions than those which, from the commencement of their association as communities, have followed the appointments of some wise legislator.”
- giving value to uninformed, ignorant common sense, encouraging simple reasoning, hence less energy consuming, more viral, over more sophisticated reasoning. Devaluing knowledge. The relevant citation from Descartes is: “In the same way I thought that the sciences contained in books (such of them at least as are made up of probable reasonings, without demonstrations), composed as they are of the opinions of many different individuals massed together, are farther removed from truth than the simple inferences which a man of good sense using his natural and unprejudiced judgment draws respecting the matters of his experience.”
- the almost eradication of geometrical thinking in mathematical education and research, performed mainly in the 20th century, with great success. The relevant citation from Descartes is: “Then as to the analysis of the ancients and the algebra of the moderns, besides that they embrace only matters highly abstract, and, to appearance, of no use, the former is so exclusively restricted to the consideration of figures, that it can exercise the understanding only on condition of greatly fatiguing the imagination; and, in the latter, there is so complete a subjection to certain rules and formulas, that there results an art full of confusion and obscurity calculated to embarrass, instead of a science fitted to cultivate the mind.”
- running for the ultimate explanation, for the grand theory of everything, ultimately confusing the compression technique which is the cartesian method, designed for being able to fit into our small brains this huge reality at once, with the reality itself. The relevant citation from Descartes is: “The first was never to accept anything for true which I did not clearly know to be such; that is to say, carefully to avoid precipitancy and prejudice, and to comprise nothing more in my judgement than what was presented to my mind so clearly and distinctly as to exclude all ground of doubt.”
- abuse of division, creation of a myriad of problems, which is a tremendously efficient technique for advancing little by little our understanding of something, but it has the disadvantage of giving the impression that said problems are the goal and not just a mean for understanding. Excessive division of research interests. A very good resource for nowadays publishers, as well as for thousands and thousands of researchers specialized into solving problems for the sake of it. The relevant passage from Descartes is: ” The second, to divide each of the difficulties under examination into as many parts as possible, and as might be necessary for its adequate solution.”
- using, for compression needs, of an unnecessary and unnatural one dimensional formulation of understanding and then thinking exclusively in such terms, forgetting that this streaming is a technique for easier understanding and not a part of the subject of the study. This abuse is present everywhere in CS and probably is the main barrier in front of a better understanding of how the brain, or more generally how the living world works. Neurons have “tasks”, edges “are detected” and so on. The relevant passage from Descartes is:”… assigning in thought a certain order even to those objects which in their own nature do not stand in a relation of antecedence and sequence. And the last, in every case to make enumerations so complete, and reviews so general, that I might be assured that nothing was omitted.”
I invite you to discover in the exceptional text of Descartes the countermeasures.
From Discours de la méthode pour bien conduire sa raison, et chercher la vérité dans les sciences, by René Descartes, beginning of Part II.
The citation is taken from here.
“I was then in Germany, attracted thither by the wars in that country, which have not yet been brought to a termination; and as I was returning to the army from the coronation of the emperor, the setting in of winter arrested me in a locality where, as I found no society to interest me, and was besides fortunately undisturbed by any cares or passions, I remained the whole day in seclusion, with full opportunity to occupy my attention with my own thoughts. Of these one of the very first that occurred to me was, that there is seldom so much perfection in works composed of many separate parts, upon which different hands had been employed, as in those completed by a single master. Thus it is observable that the buildings which a single architect has planned and executed, are generally more elegant and commodious than those which several have attempted to improve, by making old walls serve for purposes for which they were not originally built. Thus also, those ancient cities which, from being at first only villages, have become, in course of time, large towns, are usually but ill laid out compared with the regularity constructed towns which a professional architect has freely planned on an open plain; so that although the several buildings of the former may often equal or surpass in beauty those of the latter, yet when one observes their indiscriminate juxtaposition, there a large one and here a small, and the consequent crookedness and irregularity of the streets, one is disposed to allege that chance rather than any human will guided by reason must have led to such an arrangement. And if we consider that nevertheless there have been at all times certain officers whose duty it was to see that private buildings contributed to public ornament, the difficulty of reaching high perfection with but the materials of others to operate on, will be readily acknowledged. In the same way I fancied that those nations which, starting from a semi-barbarous state and advancing to civilization by slow degrees, have had their laws successively determined, and, as it were, forced upon them simply by experience of the hurtfulness of particular crimes and disputes, would by this process come to be possessed of less perfect institutions than those which, from the commencement of their association as communities, have followed the appointments of some wise legislator. It is thus quite certain that the constitution of the true religion, the ordinances of which are derived from God, must be incomparably superior to that of every other. And, to speak of human affairs, I believe that the pre-eminence of Sparta was due not to the goodness of each of its laws in particular, for many of these were very strange, and even opposed to good morals, but to the circumstance that, originated by a single individual, they all tended to a single end. In the same way I thought that the sciences contained in books (such of them at least as are made up of probable reasonings, without demonstrations), composed as they are of the opinions of many different individuals massed together, are farther removed from truth than the simple inferences which a man of good sense using his natural and unprejudiced judgment draws respecting the matters of his experience. And because we have all to pass through a state of infancy to manhood, and have been of necessity, for a length of time, governed by our desires and preceptors (whose dictates were frequently conflicting, while neither perhaps always counseled us for the best), I farther concluded that it is almost impossible that our judgments can be so correct or solid as they would have been, had our reason been mature from the moment of our birth, and had we always been guided by it alone.
It is true, however, that it is not customary to pull down all the houses of a town with the single design of rebuilding them differently, and thereby rendering the streets more handsome; but it often happens that a private individual takes down his own with the view of erecting it anew, and that people are even sometimes constrained to this when their houses are in danger of falling from age, or when the foundations are insecure. With this before me by way of example, I was persuaded that it would indeed be preposterous for a private individual to think of reforming a state by fundamentally changing it throughout, and overturning it in order to set it up amended; and the same I thought was true of any similar project for reforming the body of the sciences, or the order of teaching them established in the schools: but as for the opinions which up to that time I had embraced, I thought that I could not do better than resolve at once to sweep them wholly away, that I might afterwards be in a position to admit either others more correct, or even perhaps the same when they had undergone the scrutiny of reason. I firmly believed that in this way I should much better succeed in the conduct of my life, than if I built only upon old foundations, and leaned upon principles which, in my youth, I had taken upon trust. For although I recognized various difficulties in this undertaking, these were not, however, without remedy, nor once to be compared with such as attend the slightest reformation in public affairs. Large bodies, if once overthrown, are with great difficulty set up again, or even kept erect when once seriously shaken, and the fall of such is always disastrous. Then if there are any imperfections in the constitutions of states (and that many such exist the diversity of constitutions is alone sufficient to assure us), custom has without doubt materially smoothed their inconveniences, and has even managed to steer altogether clear of, or insensibly corrected a number which sagacity could not have provided against with equal effect; and, in fine, the defects are almost always more tolerable than the change necessary for their removal; in the same manner that highways which wind among mountains, by being much frequented, become gradually so smooth and commodious, that it is much better to follow them than to seek a straighter path by climbing over the tops of rocks and descending to the bottoms of precipices.
Hence it is that I cannot in any degree approve of those restless and busy meddlers who, called neither by birth nor fortune to take part in the management of public affairs, are yet always projecting reforms; and if I thought that this tract contained aught which might justify the suspicion that I was a victim of such folly, I would by no means permit its publication. I have never contemplated anything higher than the reformation of my own opinions, and basing them on a foundation wholly my own. And although my own satisfaction with my work has led me to present here a draft of it, I do not by any means therefore recommend to every one else to make a similar attempt. Those whom God has endowed with a larger measure of genius will entertain, perhaps, designs still more exalted; but for the many I am much afraid lest even the present undertaking be more than they can safely venture to imitate. The single design to strip one’s self of all past beliefs is one that ought not to be taken by every one. The majority of men is composed of two classes, for neither of which would this be at all a befitting resolution: in the first place, of those who with more than a due confidence in their own powers, are precipitate in their judgments and want the patience requisite for orderly and circumspect thinking; whence it happens, that if men of this class once take the liberty to doubt of their accustomed opinions, and quit the beaten highway, they will never be able to thread the byway that would lead them by a shorter course, and will lose themselves and continue to wander for life; in the second place, of those who, possessed of sufficient sense or modesty to determine that there are others who excel them in the power of discriminating between truth and error, and by whom they may be instructed, ought rather to content themselves with the opinions of such than trust for more correct to their own reason.
For my own part, I should doubtless have belonged to the latter class, had I received instruction from but one master, or had I never known the diversities of opinion that from time immemorial have prevailed among men of the greatest learning. But I had become aware, even so early as during my college life, that no opinion, however absurd and incredible, can be imagined, which has not been maintained by some on of the philosophers; and afterwards in the course of my travels I remarked that all those whose opinions are decidedly repugnant to ours are not in that account barbarians and savages, but on the contrary that many of these nations make an equally good, if not better, use of their reason than we do. I took into account also the very different character which a person brought up from infancy in France or Germany exhibits, from that which, with the same mind originally, this individual would have possessed had he lived always among the Chinese or with savages, and the circumstance that in dress itself the fashion which pleased us ten years ago, and which may again, perhaps, be received into favor before ten years have gone, appears to us at this moment extravagant and ridiculous. I was thus led to infer that the ground of our opinions is far more custom and example than any certain knowledge. And, finally, although such be the ground of our opinions, I remarked that a plurality of suffrages is no guarantee of truth where it is at all of difficult discovery, as in such cases it is much more likely that it will be found by one than by many. I could, however, select from the crowd no one whose opinions seemed worthy of preference, and thus I found myself constrained, as it were, to use my own reason in the conduct of my life.
But like one walking alone and in the dark, I resolved to proceed so slowly and with such circumspection, that if I did not advance far, I would at least guard against falling. I did not even choose to dismiss summarily any of the opinions that had crept into my belief without having been introduced by reason, but first of all took sufficient time carefully to satisfy myself of the general nature of the task I was setting myself, and ascertain the true method by which to arrive at the knowledge of whatever lay within the compass of my powers.
Among the branches of philosophy, I had, at an earlier period, given some attention to logic, and among those of the mathematics to geometrical analysis and algebra,–three arts or sciences which ought, as I conceived, to contribute something to my design. But, on examination, I found that, as for logic, its syllogisms and the majority of its other precepts are of avail–rather in the communication of what we already know, or even as the art of Lully, in speaking without judgment of things of which we are ignorant, than in the investigation of the unknown; and although this science contains indeed a number of correct and very excellent precepts, there are, nevertheless, so many others, and these either injurious or superfluous, mingled with the former, that it is almost quite as difficult to effect a severance of the true from the false as it is to extract a Diana or a Minerva from a rough block of marble. Then as to the analysis of the ancients and the algebra of the moderns, besides that they embrace only matters highly abstract, and, to appearance, of no use, the former is so exclusively restricted to the consideration of figures, that it can exercise the understanding only on condition of greatly fatiguing the imagination; and, in the latter, there is so complete a subjection to certain rules and formulas, that there results an art full of confusion and obscurity calculated to embarrass, instead of a science fitted to cultivate the mind. By these considerations I was induced to seek some other method which would comprise the advantages of the three and be exempt from their defects. And as a multitude of laws often only hampers justice, so that a state is best governed when, with few laws, these are rigidly administered; in like manner, instead of the great number of precepts of which logic is composed, I believed that the four following would prove perfectly sufficient for me, provided I took the firm and unwavering resolution never in a single instance to fail in observing them.
The first was never to accept anything for true which I did not clearly know to be such; that is to say, carefully to avoid precipitancy and prejudice, and to comprise nothing more in my judgement than what was presented to my mind so clearly and distinctly as to exclude all ground of doubt.
The second, to divide each of the difficulties under examination into as many parts as possible, and as might be necessary for its adequate solution.
The third, to conduct my thoughts in such order that, by commencing with objects the simplest and easiest to know, I might ascend by little and little, and, as it were, step by step, to the knowledge of the more complex; assigning in thought a certain order even to those objects which in their own nature do not stand in a relation of antecedence and sequence.
And the last, in every case to make enumerations so complete, and reviews so general, that I might be assured that nothing was omitted.
The long chains of simple and easy reasonings by means of which geometers are accustomed to reach the conclusions of their most difficult demonstrations, had led me to imagine that all things, to the knowledge of which man is competent, are mutually connected in the same way, and that there is nothing so far removed from us as to be beyond our reach, or so hidden that we cannot discover it, provided only we abstain from accepting the false for the true, and always preserve in our thoughts the order necessary for the deduction of one truth from another. And I had little difficulty in determining the objects with which it was necessary to commence, for I was already persuaded that it must be with the simplest and easiest to know, and, considering that of all those who have hitherto sought truth in the sciences, the mathematicians alone have been able to find any demonstrations, that is, any certain and evident reasons, I did not doubt but that such must have been the rule of their investigations. I resolved to commence, therefore, with the examination of the simplest objects, not anticipating, however, from this any other advantage than that to be found in accustoming my mind to the love and nourishment of truth, and to a distaste for all such reasonings as were unsound. But I had no intention on that account of attempting to master all the particular sciences commonly denominated mathematics: but observing that, however different their objects, they all agree in considering only the various relations or proportions subsisting among those objects, I thought it best for my purpose to consider these proportions in the most general form possible, without referring them to any objects in particular, except such as would most facilitate the knowledge of them, and without by any means restricting them to these, that afterwards I might thus be the better able to apply them to every other class of objects to which they are legitimately applicable. Perceiving further, that in order to understand these relations I should sometimes have to consider them one by one and sometimes only to bear them in mind, or embrace them in the aggregate, I thought that, in order the better to consider them individually, I should view them as subsisting between straight lines, than which I could find no objects more simple, or capable of being more distinctly represented to my imagination and senses; and on the other hand, that in order to retain them in the memory or embrace an aggregate of many, I should express them by certain characters the briefest possible. In this way I believed that I could borrow all that was best both in geometrical analysis and in algebra, and correct all the defects of the one by help of the other.”
In this post, which continues Currying by using zippers and an allusion to the Cartesian Theater, I want to explain how we may pack and unpack two arrows into one, in the realm of graphic lambda calculus. An algebrization of graphic lambda calculus may be deduced from this, but one step of it, namely enumerating arbitrarily the nodes of a graph in , suffers from the same cartesian disease which was exposed in the previously mentioned post. But nevermind, it is at least funny to show that the usual ways of CS thinking may be used to transform this apparently more general frame of graphic lambda calculus into a 1D string submitted to local algebraic manipulations.
We start from the following sequence of three graphic beta moves.
With words, this figure means: we pack the 1, 2, entries into a list, we pass it trough one arrow then we unpack the list into the outputs 3, 4. This packing-unpacking trick may be used of course for more than a pair of arrows, in obvious ways, therefore it is not a restriction of generality to write only about two arrows.
We may apply the trick to a pair of graphs in , call them and , which are connected by a pair of arrows, like in the following figure.
With the added packing and unpacking triples of gates, the graphs , are interacting only by the intermediary of one arrow.
In particular, we may use this trick for the elementary gates of abstraction and application, transforming them into graphs with one input and one output, like this:
Let’s look now at the graphic beta move:
If we use the elementary gates transformed into graphs with one input and one output, the move becomes this almost algebraic, 1D rule:
Finally, the packing-unpacking trick described in the first figure becomes this:
An interesting discussion started at Retraction Watch, in the comments of the post Brian Deer’s modest proposal for post-publication peer review. Let me repeat the part which I find interesting: post-publication peer review.
The previous post “Peer-reviews don’t protect against plagiarism and articles retraction. Why?“ starts with the following question:
After reading one more post from the excellent blog Retraction Watch, this question dawned on me: if the classical peer-review is such a good thing, then why is it rather inefficient when it comes to detecting flaws or plagiarism cases which later are exposed by the net?
and then I claimed that retractions of articles which already passed the traditional peer-review process are the best argument for an open, perpetual peer-review.
Which brings me to the subject of this post, namely what is peer-review for?
Context. Peer-review is one of the pillars of the actual publication of research practice. Or, the whole machine of traditional publication is going to suffer major modifications, most of them triggered by its perceived inadequacy with respect to the needs of researchers in this era of massive, cheap, abundant means of communication and organization. In particular, peer-review is going to suffer transformations of the same magnitude.
We are living interesting times, we are all aware that internet is changing our lives at least as much as the invention of the printing press changed the world in the past. With a difference: only much faster. We have an unique chance to be part of this change for the better, in particular concerning the practices of communication of research. In front of such a fast evolution of behaviours, a traditionalistic attitude is natural to appear, based on the argument that slower we react, a better solution we may find. This is however, in my opinion at least, an attitude better to be left to institutions, to big, inadequate organizations, than to individuals. Big institutions need big reaction times because the information flows slowly through them, due to their principle of pyramidal organization, which is based on the creation of bottlenecks for information/decision, acting as filters. Individuals are different in the sense that for them, for us, the massive, open, not hierarchically organized access to communication is a plus.
The bottleneck hypothesis. Peer-review is one of those bottlenecks, traditionally. It’s purpose is to separate the professional from the unprofessional. The hypothesis that peer-review is a bottleneck explains several facts:
- peer-review gives a stamp of authority to published research. Indeed, those articles which pass the bottleneck are professional, therefore more suitable for using them without questioning their content, or even without reading them in detail,
- the unpublished research is assumed to be unprofessional, because it has not yet passed the peer-review bottleneck,
- peer-reviewed publications give a professional status to authors of those. Obviously, if you are the author of a publication which passed the peer-review bottleneck then you are a professional. More professional publications you have, more of a professional you are,
- it is the fault of the author of the article if it does not pass the peer-review bottleneck. As in many other fields of life, recipes for success and lore appear, concerning means to write a professional article, how to enhance your chances to be accepted in the small community of professionals, as well as feelings of guilt caused by rejection,
- the peer-review is anonymous by default, as a superior instance which extends gifts of authority or punishments of guilt upon the challengers,
- once an article passes the bottleneck, it becomes much harder to contest it’s value. In the past it was almost impossible because any professional communication had to pass through the filter. In the past, the infallibility of the bottleneck was a kind of self-fulfilling prophecy, with very few counterexamples, themselves known only to a small community of enlightened professionals.
This hypothesis explains as well the fact that lately peer-review is subjected to critical scrutiny by professionals. Indeed, in particular, the wave of detected plagiarisms in the class of peer-reviewed articles lead to the questioning of the infallibility of the process. This is shattering the trust into the stamp of authority which is traditionally associated with it. It makes us suppose that the steep rise of retractions is a manifestation of an old problem which is now revealed by the increased visibility of the articles.
From a cooler point of view, if we see the peer-review as designed to be a bottleneck in a traditionally pyramidal organization, is therefore questionable if the peer-review as a bottleneck will survive.
Social role of peer-review. There are two other uses of peer-review, which are going to survive and moreover, they are going to be the main reasons for it’s existence:
- as a binder for communities of peers,
- as a time-saver for the researchers.
I shall take them one-by-one. What is strange about the traditional peer-review is that although any professional is a peer, there is no community of peers. Each researcher does peer-reviewing, but the process is organized in such a manner that we are all alone. To see this, think about the way things work: you receive a demand to review an article, from an editor, based on your publication history, usually, which qualifies you as a peer. You do your job, anonymously, which has the advantage of letting you be openly critical with the work of your peer, the author. All communication flows through the editor, therefore the process is designed to be unfriendly with communications between peers. Hence, no community of peers.
However, most of the researchers who ever lived on Earth are alive today. The main barrier for the spread of ideas is a poor mean of communication. If the peer-review becomes open, it could foster then the appearance of dynamical communities of peers, dedicated to the same research subject. As it is today, the traditional peer-review favours the contrary, namely the fragmentation of the community of researchers which are interested in the same subject into small clubs, which compete on scarce resources, instead of collaborating. (As an example, think about a very specialized research subject which is taken hostage by one, or few, such clubs which peer-reviews favourably only the members of the same club.)
As for the time-saver role of peer-review, it is obvious. From the sea of old and new articles, I cannot read all of them. I have to filter them somehow in order to narrow the quantity of data which I am going to process for doing my research. The traditional way was to rely on the peer-review bottleneck, which is a kind of pre-defined, one size for all solution. With the advent of communities of peers dedicated to narrow subjects, I can choose the filter which serves best my research interests. That is why, again, an open peer-review has obvious advantages. Moreover, such a peer-review should be perpetual, in the sense that, for example, reasons for questioning an article should be made public, even after the “publication” (whatever such a word will mean in the future). Say, another researcher finds that an older article, which passed once the peer-review, is flawed for reasons the researcher presents. I could benefit from this information and use it as a filter, a custom, continually upgrading filter of my own, as a member of one of the communities of peers I am a member of.
I am continuing from the post Towards qubits: graphic lambda calculus over conical groups and the barycentric move. My goal here is to give a set of axioms for a “projective conical space”. Let me recall the following facts:
- affine conical spaces are the non-commutative equivalent of affine spaces. An affine conical space is constructed over a conical group as an affine space is constructed over a vector space. Conical groups are generalizations of Carnot groups, in the sense that in the realm of Lie groups the basic example of a conical group is a Carnot group. A conical Lie group is a contractive Lie group and therefore, by a theorem of Siebert, if it is simply connected then it is a nilpotent Lie group with a one-parameter family of contractive automorphisms. Carnot groups (think about examples as the Heisenberg group) are conical Lie groups with a supplementary hypothesis concerning the fact that the first level in the decomposition of the Lie algebra is generating the whole algebra.
- an affine conical space is an usual affine space if and only if it satisfies the barycentric move. In this case and only in this case the underlying structure of the conical group is commutative.See arXiv:0804.0135 [math.MG] for the introduction of “non-commutative affine geometry”, called here “affine conical geometry”, which generalizes results from W. Bertram Generalized projective geometries: From linear algebra via affine algebra to projective algebra, Linear Algebra and its Applications 378 (2004), 109 – 134.
- afine conical spaces are defined in terms of a one-parameter family of quandle operations (called dilations). More specifically an affine conical space is generated by a one-parameter family of quandles which satisfy also some topological sugar axioms (which I’ll pass). More precisely, affine conical spaces are self-distributive uniform idempotent right quasigroups. Uniform idempotent right quasigroups were introduced and studied under the shorter name “emergent algebras” in arXiv:0907.1520 [math.RA], see also arXiv:1005.5031 [math.GR] for the context of studying them as algebraic-topologic generalizations of dilation structures (introduced in arXiv:math/0608536 [math.MG]), as well as for the description of symmetric spaces as emergent algebras.
- in affine conical geometry there is no notion of incidence or co-linearity, because of non-commutativity lurking beneath. However, there is a notion of a collinear triple of points, as well as a ratio associated to such points, but such collinear triples correspond to triples of dilations (see further what “dilation” means) which, composed, give the identity. Such triples give the invariant of affine conical geometry which corresponds to the ration of three collinear points in the usual affine geometry.
In the post Towards qubits I I explained (or linked to explanations) this in the language of graphic lambda calculus. Here I shall not use it fully, instead I shall use a graphical notation with variable names. But I think the correspondences between these two notations are rather clear. In particular I shall interpret identities as moves in trivalent graphs.
1. Algebraic axioms for affine conical spaces. (Topological sugar not included). We have a non-empty set and a commutative group of parameters with operation denoted multiplicatively and neutral element . Think about as being or even where is a field.
On is defined a function (Bertram uses the letter instead, I am using ). This function is to be interpreted as a -parametrized family of operations. Namely we denote:
This family of operations, called dilations, satisfies a number of algebraic axioms (as well as topological axioms which I pass), making them in particular into a family of quandle operations. I shall give these axioms in a graphical form, by using the transparent, I hope, notation:
Combinations (i.e. compositions) of dilations appear therefore as oriented trees with trivalent planar nodes decorated by the elements of , with leaves (but not the root) decorated with elements from .
The algebraic axioms of affine conical spaces are stating identities between certain compositions of dilations. Graphically these identities will be representes, as I wrote, as moves applied to such oriented trees.
Here are these axioms in graphical form:
(1) this is equivalent with the move ext2 from graphical lambda calculus: (i.e. extensionality move 2)
(3) this is equivalent with the move R2 from the graphical lambda calculus (i.e. Reidemeister move 2, all Reidemeister moves 2 are equivalent in this formalism)
(4) this is the self-distributivity axiom, which could be called move R3b with the notations of Polyak
2. Algebraic axioms for projective conical spaces. The intention is to propose a generalization of the same type, this time for projective spaces, of the one from W. Bertram Generalized projective geometries: General theory and equivalence with Jordan structures, Advances in Geometry 3 (2002), 329-369.
This time we have a pair of spaces . Think about the elements as being “points” and about the elements as being “lines”, although, as in the case of affine conical geometry, there is no proper notion of incidence (except, of course, for the “commutative” particular case).
A pair geometry is a triple where is the set of pairs (say point-line) in general position. Compared to the more familiar case of incidence systems, the interpretation of is “the point is not incident with the line “. The triple satisfies some conditions which I shall write after introducing some notations.
For any and any we denote:
Let also denote
(Pair geometry 1) for any and for any the sets and are non-empty,
(Pair geometry 2) for any pair of different points there exists and it’s unique a line such that and are not in ; dually, for any pair of different lines there exists and it’s unique a point such that and are not in .
Remark. This is the definition of a pair geometry given by Bertram. I shall keep further only (Pair geometry 1) because I feel that (Pair geometry 2) has too much “incidence content” which might be not non-commutative enough. So, for the moment, (Pair geometry 2) is in quarantine. As a first suggestion coming into mind, it might well turn out that it can be replaced by a more lax version saying that there is a number such that is covered by the reunion of sets (and a similar dual formulation for . As it is, (Pair geometry 2) corresponds to such a formulation for .
We want the following:
- for any point the space is an affine conical space,
- for any line the space is an affine conical space,
- these structures are glued together by some axioms.
Let’s pass through these three points of the list.
1. that means we shall put a structure of dilation operations on every . It is natural then to mark the dilation operations not only by elements of the group , but also by . More concretely that means we introduce for any a function
which, for any it takes a pair of lines , with and returns .
We ask that for any the dilations satisfy axioms (1), (2), (3) of affine conical spaces.
2. in the same way, we want that every to have a structure of dilation operations. We have therefore, for any another function (but I shall use the same letter )
which, for any it takes a pair of points , with and returns .
We ask that for any the dilations satisfy axioms (1), (2), (3) of affine conical spaces.
3. the gluing axioms are generalizations of axioms (PG1), (PG2) of Bertram. In the mentioned article, Bertram explains that these two axioms lead to eight identities. From those eight, six of them are different. From those six, Bertram is using the barycentric axiom to eliminate two of them, which leaves him with four identities. I shall not use the barycentric axiom, because otherwise I shall fall on the commutative case, but I shall eliminate as well these two axioms> Therefore I shall have four moves which will replace the Reidemeister move 3 axiom , i.e. the self-distributivity move (4) from affine conical spaces.
Remark. Bertram adds some sugar over (PG1) and (PG2) which serves to be able to construct tangent structures further. I renounce at those in favor of my topological sugar which I pass, for the moment.
Remark. As we saw that the axioms of affine conical spaces are practically corresponding to the Reidemeister moves, it is natural to expect that the four axioms correspond to either: the Roseman moves, or to some 2-quandle definition. I need help and suggestions here!
I shall write further the four axioms which replace the axiom (4), that is why I shall name them (4.1) … (4.4). As previously I shall use a graphical notation, which my visual brain finds more easy to understand than the notation using multiple compositions of functions with 4 arguments (however, see Bertram’s notations involving adjoint pairs). Also, there are limits to my capacity to write latex formulae which are well parsed in this blog.
So, here is the notation for dilations which I shall use for writing those four axioms:
Let’s look at the first line. For any we have an associated dilation operation taking as input a pair of points . Graphically this is represented by a node with two inputs and an output, together with a planar embedding (i.e. the local planar embedding tells us which is the left input and which is the right output), and with a supplementary input which points to the center of the circle (node), serving to identify the node as the dilation in the space . Similar comments could be made about the second line of the figure.
Therefore, this time we are working with trees made by 4-valent nodes, each node having three inputs and one output and moreover with a triple of two inputs and the output with an orientation given. The leaves, but not the root of such a tree are decorated by points or lines. There should be other constraints on this family of trees, coming from the fact that if the input which points to the center of the circle correspond to a point then the other inputs should correspond to lines, and so on. For the moment I pass over this, probably a solution would be to colour the edges, by using two colors, one for points, the other for lines, then express the constraints in terms of those colors.
As previously, the nodes are decorated by elements of the commutative group .
(4.2) second part of (PG1)
(4.3) first part of (PG2)
(4.4) second part of (PG.2)
In a future post I shall give:
- a theorem of characterization of projective conical spaces, of the same type as the theorem of characterization of affine conical spaces
- examples of non-commutative projective conical spaces, in particular answering to the question: what is the natural notion of a projective space of a conical group (more particularly, if we think about Carnot groups as being non-commutative vector spaces, then who are their associated non-commutative projective spaces?).
UPDATE: The axioms (4.1) … (4.4) take a much more simple form if we use choroi and differences, but that’s also for a future post.
In arXiv:1212.5056 [math.CO] “On growth in an abstract plane” by Nick Gill, H. A. Helfgott, Misha Rudnev , in lemma 4.1 is given a proof of the Ruzsa triangle inequality which intrigued me. Later on, at the end of the article the authors give a geometric Ruzsa inequality in a Desarguesian projective plane, based on similar ideas as the ones used in the proof of the Ruzsa triangle inequality.
All this made me write the following.
Let be a non-empty set and be an operation on which has the following two properties:
- for any we have ,
- for any the function is injective.
We may use weaker hypotheses for , namely:
- (weaker) there is a function such that for any ,
- (weaker) there is a function such that is an injective function for any .
Prop. 1. Let be a non empty set endowed with an operation which satisfies 1. and 2. (or the weaker version of those). Then for any non empty sets there is an injection
where we denote by
In particular, if are finite sets, we have the Rusza triangle inequality
where denotes the cardinality of the finite set .
I shall give the proof for hypotheses 1., 2., because the proof is the same for the weaker hypotheses. Also, this is basically the same proof as the one of the mentioned lemma 4.1. The proof of the Ruzsa inequality corresponds to the choice , where is a group (no need to be abelian). The proof of the geometric Ruzsa inequality corresponds to the choice , with the notations from the article, with the observation that this function satisfies weaker 1. and 2.
Proof. We can choose functions and such that for any we have . With the help of these functions let
We want to prove that is injective. Let . Then, by 1. we have . This gives an unique . Now we know that . By 2. we get that qed.
In a metric space with dilations we have the function approximate difference based at and applied to a pair of closed points . This function has the property that converges uniformly to as goes to . Moreover, there is a local group operation with as neutral element such that , therefore the function satisfies 1. and 2.
As concerns the function , it satisfies the following approximate version of 1.:
- (approximate) for any which are sufficiently close and for any we have, with the notation , the relation
We say that a set is separated if for any , the inequality implies . Further I am going to write about sets which are closed to a fixed, but arbitrary otherwise point .
Prop2. In a metric space with dilations, let and let be finite sets of points included in a compact neighbourhood of , which are closed to , such that for any the sets and are separated. Then for any there is an injective function
Proof. As previously, we choose the functions and . Notice that these functions depend on but this will not matter further. I shall use the notation liberally, for example means . Let’s define the function by the same formula as previously:
Let and be pairs such that . From 1. (approximate) and from the uniform convergence mentioned previously we get that
There is a function such that implies (the last from the previous relation) . For such a , by the separation of we get .
Let . From the hypothesis we have . This implies, via the structure of the function and via the uniform convergence, that (by compactness, this last does not depend on ). By the same reasoning as previously, we may choose such that if . This implies qed.
Finding the following in a CS research article:
… understanding the brain’s computing paradigm has the potential to produce a paradigm shift in current models of computing.
almost surely would qualify the respective article as crackpot, right? Wrong, for historical and contemporary reasons, which I shall mention further.
The brain differs from modern computing systems in many ways. The first striking difference is its use of heterogeneous components: unlike the components of a modern computer, the components of the brain (ion channels, receptors, synapses, neurons, circuits) are always highly diverse – a property recently shown to confer robustness to the system . Second, again unlike the components of a computer, they all behave stochastically – it is never possible to predict the precise output they will produce in response to a given input; they are never “bit-precise”. Third, they can switch dynamically between communicating synchronously and asynchronously. Fourth, the way they transmit information across the brain is almost certainly very different from the way data is transmitted within a computer: each recipient neuron appears to give its own unique interpretation to the information it receives from other neurons. Finally, the brain’s hierarchically organised, massively recurrent connectively, with its small-world topology, is completely different from the interconnect architecture of any modern computer. For all these reasons, understanding the brain’s computing paradigm has the potential to produce a paradigm shift in current models of computing.
Part of the efforts made by HBP are towards neuromorphic computing. See the presentation Real-time neuromorphic circuits for neuro-inspired computing systems by Giacomo Indiveri, in order to learn more about the history and the present of the subject.
2. As you can see from the presentation, neuromorphic computing is rooted in the article “A logical calculus of the ideas immanent in nervous activity” by Warren Mcculloch and Walter Pitts,1943, Bulletin of Mathematical Biophysics 5:115-133. This brings me to the “history” part. I shall use the very informative article by Gualtiero Piccinini “The First Computational Theory of Mind and Brain: A Close Look at McCulloch and Pitts’s ‘Logical Calculus of Ideas Immanent in Nervous Activity’”, Synthese 141: 175–215, 2004. From the article:
[p. 175] Warren S. McCulloch and Walter H. Pitt’s 1943 paper, ‘‘A Logical Calculus of the Ideas Immanent in Nervous Activity,’’ is often cited as the starting point in neural network research. As a matter of fact, in 1943 there already existed a lively community of biophysicists doing mathematical work on neural networks. What was novel in McCulloch and Pitts’s paper was a theory that employed logic and the mathematical notion of computation – introduced by Alan Turing (1936–37) in terms of what came to be known as Turing Machines – to explain how neural mechanisms might realize mental functions.
About Turing and McCulloch and Pitts:
[p. 176] The modern computational theory of mind and brain is often credited to Turing himself (e.g., by Fodor 1998). Indeed, Turing talked about the brain first as a ‘‘digital computing machine,’’ and later as a sort of analog computer. But Turing made these statements in passing, without attempting to justify them, and he never developed a computational theory of thinking. More importantly, Turing made these statements well after the publication of McCulloch and Pitts’s theory, which Turing knew about. Before McCulloch and Pitts, neither Turing nor anyone else had used the mathematical notion of computation as an ingredient in a theory of mind and brain.
[p. 181] In 1936, Alan Turing published his famous paper on computability (Turing 1936–37), in which he introduced Turing Machines and used them to draw a clear and rigorous connection between computing, logic, and machinery. In particular, Turing argued that any effectively calculable function can be computed by some Turing Machine – a thesis now known as the Church–Turing thesis (CT) – and proved that some special Turing Machines, which he called ‘‘universal,’’ can compute any function computable by Turing Machines. By the early 1940s, McCulloch had read Turing’s paper. In 1948, in a public discussion during the Hixon Symposium, McCulloch declared that in formulating his theory of mind in terms of neural mechanisms, reading Turing’s paper led him in the ‘‘right direction.’’
On McCulloch and “the logic of the nervous system”:
[p. 179] In 1929, McCulloch had a new insight. It occurred to him that the all-or-none electric impulses transmitted by each neuron to its neighbors might correspond to the mental atoms of his psychological theory, where the relations of excitation and inhibition between neurons would perform logical operations upon electrical signals corresponding to inferences of his propositional calculus of psychons. His psychological theory of mental atoms turned into a theory of ‘‘information flowing through ranks of neurons.’’ This was McCulloch’s first attempt ‘‘to apply Boolean algebra to the behavior of nervous nets.’’ The brain would embody a logical calculus like that of Whitehead and Russell’s Principia Mathematica, which would account for how humans could perceive objects on the basis of sensory signals and how humans could do mathematicsand abstract thinking. This was the beginning of McCulloch’s search for the ‘‘logic of the nervous system,’’ on which he kept working until his death.
On Pitts, McCulloch and logic:
[p. 185-186] In the papers that Pitts wrote independently of McCulloch, Pitts did not suggest that the brain is a logic machine. Before McCulloch entered the picture, neither Pitts nor any other member of Rashevsky’s biophysics group employed logical or computational language to describe the functions performed by networks of neurons. The use of logic and computation theory to model the brain and understand its function appeared for the first time in McCulloch and Pitts’s 1943 paper; this is likely to be a contribution made by McCulloch to his joint project with Pitts. [...]
Soon after McCulloch met Pitts, around the end of 1941, they started collaborating on a joint mathematical theory that employed logic to model nervous activity, and they worked on it during the following two years. They worked so closely that Pitts (as well as Lettvin) moved in with McCulloch and his family for about a year in Chicago. McCulloch and Pitts became intimate friends and they remained so until their death in 1969. According to McCulloch, they worked largely on how to treat closed loops of activity mathematically, and the solution was worked out mostly by Pitts using techniques that McCulloch didn’t understand. To build up their formal theory, they adapted Carnap’s rigorous (but cumbersome) formalism, which Pitts knew from having studied with Carnap. Thus, according to McCulloch, Pitts did all the difficult technical work.
A citation from McCullogh and Pitts paper [p. 17 from the linked pdf]
It is easily shown: first, that every net, if furnished with a tape, scanners connected to afferents, and suitable efferents to perform the necessary motor-operations, can compute only such numbers as can a Turing machine; second, that each of the latter numbers can be computed by such a net; and that nets with circles can be computed by such a net; and that nets with circles can compute, without scanners and a tape, some of the numbers the machine can, but no others, and not all of them. This is of interest as affording a psychological justification of the Turing definition of computability and its equivalents, Church’s -definability and Kleene’s primitive recursiveness: If any number can be computed by an organism, it is computable by these definitions, and conversely.
Comment by Piccinini on this:
[p. 198] in discussing computation in their paper, McCulloch and Pitts did not prove any results about the computation power of their nets; they only stated that there were results to prove. And their conjecture was not that their nets can compute anything that can be computed by Turing Machines. Rather, they claimed that if their nets were provided with a tape, scanners, and ‘‘efferents,’’ then they would compute what Turing Machines could compute; without a tape, McCulloch and Pitts expected even nets with circles to compute a smaller class of functions than the class computable by Turing Machines.
I have boldfaced the previous paragraph because I find it especially illuminating, resembling the same kind of comment as the one on currying I gave in the post “Currying by using zippers and an allusion to the Cartesian Theater“.
[p. 198-199] McCulloch and Pitts did not explain what they meant by saying that nets compute. As far as the first part of the passage is concerned, the sense in which nets compute seems to be a matter of describing the behavior of nets by the vocabulary and formalisms of computability theory. Describing McCulloch–Pitts nets in this way turned them into a useful tool for designing circuits for computing mechanisms. This is how von Neumann would later use them (von Neumann 1945).
I am continuing the post Applications of UD by two comments, one concerning Google, the other Kinect.
Google: There are many discussions (on G+ in particular) around A second spring of cleaning at Google, mainly about their decision concerning Google Reader. But have you notice they are closing Google Building Maker? The reason is this:
Compare with Aerometrex, which uses UD:
So, are we going to see application 2 from the last post (Google Earth with UD) really soon?
Kinect: (I moved the update from the previous post here and slightly modified) Take a look at the video from Kinect + Brain Scan = Augmented Reality for Neurosurgeons
They propose the following strategy:
- first use the data collected by the scanner in order to transform the scan of the patient’s brain into a 3D representation of the brain
- then use Kinect to lay this representation over the real-world reconstruction of the patient’s head (done in real time by Kinect), so that the neurosurgeon has an augmented reality representation of the head which allows him/her to see inside the head and decide accordingly what to do.
This is very much compatible with the UD way (see application point 3.) Suppose you have a detailed brain scan, much more detailed than Kinect alone can handle. Why not using UD for the first step, then use Kinect for the second step. First put the scan data into the UD format, then use the UD machine to stream only the necessary data to the Kinect system. This way you have best of both worlds. The neurosurgeon could really see microscopic detail, if needed, correctly mapped inside patient’s brain. What about microscopic level reconstruction of the brain, which is the real level of detail needed by the neurosurgeon?
I shall tell you the story of this article, from its inception to its publication. I hope it is interesting and funny. It is an old story, not like this one, but nevertheless it might serve to justify my opinion that open peer-review (anonymous or not, this doesn’t matter) is much better than the actual peer-review, in that by being open (i.e. peer-reviews publicly visible and evolving through contributions by the community of peers), it discourages abusive behaviours which are now hidden under the secrecy, motivated by a multitude of reasons, like conflict of interests, protection of it’s own little group against stranger researchers, racism, and so on .
Here is the story.
In 2001, at EPFL I had the chance to have on my desk two items: a recent article by Bernard Dacorogna and Chiara Tanteri concerning quasiconvex hulls of sets of matrices and the book A.W. Marshall, I. Olkin, Inequalities: Theory of Majorisation and it’s Applications, Mathematics in science and engineering, 143, Academic Press, (1979). The book was recommended to me by Tudor Ratiu, who was saying that it should be read as a book of conjectures in symplectic geometry. (Without his suggestion, I would have never decided to read this excellent book.)
At the moment I was interested in variational quasiconvexity (I invented multiplicative quasiconvexity, or quasiconvexity with respect to a group), which is still a fascinating and open subject, one which could benefit (but it does not) from a fresh eye by geometers. On the other hand, geometers which are competent in analysis are a rare species. Bernard Dacorogna, a specialist in analysis with an outstanding and rather visionary good mathematical sense, was onto this subject from some time, for good reasons, see his article with J. Moser, On a partial differential equation involving the Jacobian determinant, Annales de l’Institut Henri Poincaré. Analyse non linéaire 1990, vol. 7, no. 1, pp. 1-26, which is a perfect example of the mixture between differential geometry and analysis.
Therefore, by chance I could notice the formal similarity between one of Dacorogna’s results and a pair (Horn, Thompson) of theorems in linear algebra, expressed with the help of majorization relation. I quickly wrote the article “Majorization with applications to the calculus of variations“, where I show that by using majorization techniques, older than the quasiconvexity subject (therefore a priori available to the specialists in quasiconvexity), several results in analysis have almost trivial proofs, as well as giving several new results.
I submitted the article to many journals, without success. I don’t recall the whole list of journals, among them were Journal of Elasticity, Proceedings of the Royal Society of Edimburgh, Discrete and Continuous Dynamical Systems B.
The reports were basically along the same vein: there is nothing new in the paper, even if eventually I changed the name of the paper to “Four applications of majorization to convexity in the calculus of variations”. Here is an excerpt from such a report:
“Usually, a referee report begins with a description of the goal of the paper. It is not easy here, since Buliga’s article does not have a clear target, as its title suggests. More or less, the text examines and exploits the relationships between symmetry and convexity through the so-called majorization of vectors in Rn , and also with rank-one convexity. It also comes back to works of Ball, Freede and Thompson, Dacorogna & al., Le Dret, giving a few alternate proofs of old results.
This lack of unity is complemented by a lack of accuracy in the notations and the statements. [...] All in all, the referee did not feel convinced by this paper. It does not contain a striking statement that could attract the attention. Thus the mathematical interest does not balance the weak form of the article. I do not see a good argument in favor of the publication by DCDS-B.”
At some point I renounced to submit it.
After a while I made one more try and submit it to a journal which was not in the same class as the previous ones, (namely applied mathematics and calculus of variations). So I submitted the article to Linear Algebra and its Applications and it has been accepted. Here is the published version Linear Algebra and its Applications 429, 2008, 1528-1545, and here is an excerpt from the first referee report (from LAA)
“This paper starts with an overview of majorization theory (Sections 1-4), with emphasis on Schur convexity and inequalities for eigenvalues and singular values. Then some new results are established, e.g. characterizations of rank one convexity of functions, and one considers applications in several areas as nonlinear elasticity and the calculus of variation. [...] The paper is well motivated. It presents new proofs of known results and some new theorems showing how majorization theory plays a role in nonlinear elasticity and the calculus of variation, e.g. based on the the notion of rank one convexity.
A main result, given in Theorem 5.6, is a new characterization of rank one convexity (a kind of elliptic condition) [...] This result involves Schur convexity.
Some modiﬁcations are needed to improve readability and make the paper more self-contained. [...] Provided that these changes are done this paper can be recommended for publication.”
PS. The article which, from my experience, took the most time from first submission to publication is this one: first version submitted in 1997, which was submitted as well to many journals and it was eventually published in 2011, after receiving finally an attentive, unbiased peer-review (the final version can be browsed here)The moral of the story is therefore: be optimistic, do what you like most in the best of ways and be patient.
PS2. See also the very interesting post by Mike Taylor “The only winning move is not to play“.
I reproduce further the message sent by a journal, 15 months after the submission. The message contains comments made by anonymous referees.
Let me stress that:
- I trust the journal, otherwise why submitting to it?
- this is the result of the anonymous peer-review after they sit on the paper for 15 months,
- I suppose that the extracts from the reports, provided to me by the managing editor, are the most significant ones, otherwise the message makes no sense,
- the message is reproduced as it is, with the exceptions of links, added by me, the numbering of the referees’ comments and very few comments of my own, [between brackets], where I really could not stop myself.
I removed the names of the managing editor and journal, in order to present the comments in a more objective light.UPDATE: in fact, there are no reasons to protect maybe the most sloppy peer-review report obtained in the most ridiculous amount of time. The journal is Geometry and Topology. The referees were anonymous, so they could pretend they read the article (but see referee’s comments (3), (6) and (7) which are evidence that at least one of them did not bother to read it in FIFTEEN MONTHS). As for the reason of rejection, because what is written in the reports cannot be one, rationally, I can only speculate.
The achieved effect is to keep this paper out of publishing for a year and a half. This article was written in 2009 and it has been previously submitted to:
- Commentationes Mathematicae Universitatis Carolinae (july 2009), retired from CMUC (april 2010), after no referee report in sight; the reply of the editor was: “I am sorry for the long time you had to wait, and I understand your decision. I informed already our referee (who appologizes as well, but points out that your paper is very difficult to read and check)”
- Groups, Geometry and Dynamics, (april 2010), received the following answer (august 2010): “I am sorry for such a late answer, but after consulting with referees and other editors, we arrived to the conclusion that your paper does not fit the scope of our journal.”
- Electronic journal of Combinatorics (january 2011), answered (january 2011): “I have looked at your paper in the arXiv. It’s really outside the scope of E-JC. You need to send it to one of the journals mentioned in your reference list or some other geometric/algebraic journal.”
- Constructive Approximation (nov. 2011), received answer (nov. 2011): “I had an editor quickly look over your manuscript. He suggested that the article is more appropriate for a journal that publishes papers in algebra and geometry. Here is the response we received from the editor: —————- The paper is interesting. It studies the relationship between algebraic and differential structure on manifolds with sub-Riemannian geometry, in particular, Carnot-Caratheodori metric. Such questions are natural in the framework of rigidity theory. I do not know why he submitted the paper to CA. It should be sent to a journal that publishes papers in algebra and geometry. It can be a general journal (like Journal of LMS) or a more specialized journal like “Geometry and Topology” or IJAC (“International Journal of Algebra and Computation”). —————-”
- Geometry and Topology (2 dec. 2011),answered (11 march 2013), see further.
The article just not fits in a place.
Here is the message from the managing editor of Geometry and Topology:
“Dear Prof. Buliga,
I regret to inform you that your paper
has been rejected by G&T . I attach an extract from the referee’s report. Your paper was sent to two referees, and their conclusion was that the though the results are interesting he paper cannot be published in the current form. The referees’ reports are addressed to the editors, so I only give you at the end of this message some extracts from both reports. The editors would be willing to consider your paper again if it is appropriately rewritten.
On behalf of the editorial board I would like to thank you for giving us the opportunity to consider this paper for G&T.
Yasha Eliashberg , Managing Editor of Geometry & Topology
Selected comments from the referees:
(1) I find the paper a bit confusing. the comments give the impression that the theory encompasses manifolds (and sub-Riemannian manifolds), but, if I am not misled, the results only concern groups. I find the quandle characterization of Carnot groups striking. but the paper needs rewriting.
(2) The second paragraph of the abstract should say “… are related to racks and quandles…” Quotes in TeX should be of the form “this phrase is in quotations.” Many people make that mistake — it is an annoying feature of TeX, but I find it off-putting when and author does not fix this. The second item on page 2 the word should be: information line -14 page 2 “obtain” should be “obtained.”
(3) The paper starts Introduction, Outline, Motivation and I still don’t know what the author wants to achieve. I am not an expert here, but I still want some ideas. What, for example is a Carnot group?
[So, there are 3 sections of the article full of explanations of the ideas, but not enough. Moreover, Carnot groups are defined in Definition 4.6 in the article. Even without reading the article, as this referee, one can still find what a Carnot group is, for example by using google, or a trip to the library]
(4) newtheorem environments default to italics. 3.1,3.2 etc should be set in roman font.
(5) After definition 3.3, example 4.1 should be given. Also, delta should be exemplified. Here there is something very interesting from the point of view of quandles: delta is NOT necessarily an automorphism. The reader needs examples to continue.
(6) On page 6 above Remark 3.4, I don’t see the definitions for the original circle operations.
[They are in Definition 3.3, exactly where the referee does not see them.]
I can’t think of a scenario in which )this( is a meaningful grouping. There are several grouping operations provided in TeX; please use one.
At (a), (b), and (c) I got lost in the notation.
… It took me some time to see that the advantage is the statement of Prop.3.5 and the remarks that follow. [Thank you!]
(7) I think that the point of the paper is the paragraph on page 10 above Def. 4.6. I would like to see some more exposition about this early.
[Section 2 "Motivation" is dedicated to this.]
8) What is in the definition 4.6? In 5.3, do X and Y have the same first operation? Why not an empty diamond or filled one for Y?
(9) My overall impression is that the author has a very nice big idea: geometry and algebraic structures emerge togther in a manifold situation. He is arguing that irqs and/or symmetric spaces give rise to tangent-like structures. But I would really like a careful exposition with detailed examples written for a less selected audience.”
UPDATE 2: I submitted the article to another journal. I hope it is clear that what I am after is fair, rock-solid peer review, which I could learn from and which could improve the article. It is obvious that I think the emergent algebra idea is gorgeous, as the author of it I naturally support it, but not to the point of not accepting critics. But they have to be real critics, with substance and then I welcome them and follow them. That is why I submitted the same file, the one available from arXiv; that is already version 3 and previous versions do not differ much. If I change it then I have to produce a new version, but the substance of the article is well enough (according to my powers) communicated, but not really understood by any of the referees until now. It can be improved, even enormously, and I shall do it eventually, but for the moment, this is really not worthy if the referees are put off by the use of ” or by “obtain -obtained” matters. Maybe I am wrong, surely I am wrong in this respect, but also my past experience tells me that it does not matter how well I present a new idea if the referees don’t really read the paper. This is not meaning I encourage sloppy articles, if you read it then you will see that the paper is densely and carefully written, maybe even too optimized, as the referee’s comment (6) shows. But hey, we are in the internet age, there is no reason anymore to present things as if they are designed to be read in front of a (bored) audience. We have google, we are multitasking, we use hypertext.
In this post, following “Emergent algebras and combinatory logic (part III)“, I define the class of “computable” graphs in and then I show you how differentiability (a generalization of Pansu notion of differentiability on Carnot groups) can be seen as related to this “computability”.
I am using the notations from the last post. See also the post Combinators and zippers for the other notions which I shall use here. The tutorial on graphic lambda calculus may be consulted, if necessary.
1. The combinatory logic sector. This is the collection of graphs in denoted by which is generated by:
- untyped lambda calculus zippers,
- arrows and loops,
- the diamond,
- the termination gate,
with the operation of linking outputs to inputs,under the constraint that we get graphs with a single output, which has to be either:
- the upper side output of the diamond,
- or the upper side output of a zipper
and with the moves:
- this sector contains the untyped lambda calculus sector,
- conjecture: any graph is can be transformed into a finite collection of graphs in the untyped lambda calculus sector.
2. Emergent computable graphs. I define the sector of as the collection of graphs in which are generated by the reunion of and , with the collection of moves which belong to one of the sectors, that is all moves excepting the dual of the graphic beta move and the ext1 move.
This sector can be described as the collection of graphs in generated by:
- graphs in ,
- gate and the gates for any ,
- the approximate sum gate and the approximate difference gate , for any ,
with the operations of linking output to input arrows and with the following list of moves:
3. Computably differentiable graphs. A graph which has at least one input and one output is computably differentiable if the following graph is in
I shall explain in a future post the meaning of this definition, after writing a bit about differentiability in emergent algebras and about the Pansu differentiability.
Where is this fascination about UD from? Think about it a bit from a visual point of view (and mind that I am not writing about the exact, whatever it turns out to be, algorithm, but about principles). The information coming to the brain by the visual path is ridiculously small compared to the complexity of the world. Yes, when we look at something, the world takes care of the intricacies of ray-tracing for us. But what about the visual world reconstructed in our brains? There is no ray-tracing, there are no octrees, nor coordinates. Again, I repeat that despite all the very useful knowledge people have about robotic vision, this knowledge fails spectacularly to explain how we, humans, or simple creatures like flies, see. This cartezian point of view, based on coordinatizing the exterior world and treating it like a given geographical space, is not, neuroscience tells us, how we manipulate space in the brain. Again , I repeat that algorithms, which are devices invented to solve problems, are not the right mean for trying to understand this problem, despite the amazing ideas that CS might give concerning the understanding of the world as some big system which we act upon and which it acts upon us. That is because our little grey cells (but let us not forget about those of the humble fly) don’t work by proposing themselves to solve problems. This is just another disease inflicted by the cartezian viewpoint. (I hope that at this stage you can still make the difference between mine and the average crackpot’s talking.)
If we look at the other side, the one of neuroscience, what we find? Sloppy data, compared to physics, due to the complexity of the systems studied, lack and even despise of mathematical knowledge, again compared with physics, due to the fact that these new sciences are in their infancy.
But somehow, as it has always been, somewhere there is an armchairian (word invented by Scott Aaronson), an ancient greek philosopher kind, which could get rid of the cartezian disease and see clearly a system through the huge pile of data. My bet goes to a mathematician, but I may be biased.
The least interesting application of UD is in the games industry. Here are some other, more appealing, at least to me:
- Google street view with UD. Pro: they (google) only need to change the format of their pictures database (and for example deduce some 3D data from different pictures of the same place made from different POVS, a lot of work, but they surely have the means). Con: there cannot be a too precise such tool, for obvious reasons, from security to privacy. Anyway, imagine you walk on the street (in street view) and you see instead of a sequence of photos a continuous 3D world.
- Google earth with UD, you can pass continuously from a scale to another, like flying in a super space ship. Better yet, at really small detail you pass continuously from google earth to street view.
- Not only geographical databases are interesting to see, but also very complex objects, like in the medical realm. I remember now that in the post on Digital Materialization and Euclideon, I noticed a similarity of goals between the DM project and the UD project. In the video of the interview provided on this blog by Dave H., at a certain point Bruce Dell mentions the difficulty of creating the 3D real objects (like a dragon or what ever head) and the easiness of looking at the virtual version created by scanning and then visualizing the scan with the UD machine. This is quite similar to one of the applications of DM, which consists in preserving cultural objects by using DM. Or by using UD, why not?
- If we talk about games, then the world of a MMORPG could be put in UD form, probably.
In Sub-riemannian geometry and Lie groups , Part I, section 4.2, I introduced the strange notion of “uniform group”. Instead of saying that an uniform group is just a topological group (which has an unique uniformity associated to it), I proposed the following construction.
1. Double of a group. To any group we associate its double group with the group operation
I introduce also the following three functions:
, (where is the neutral element of the group ),
Remark that all these functions are group morphisms. Notice especially the morphism , which is nothing but the group operation of , seen as a group morphism from to .
2. Uniform group. For my purposes I introduced the notion of an uniform group: an uniform group is a group , together with two uniformities, one on , the other on , such that the three morphisms from the point 1. are uniformly continuous.
So, instead of one uniformity, now I use two. Why? Well, we may eliminate one of the uniformities, the one on . Indeed, suppose that is an uniform group. Then take on the smallest uniformity which makes the three morphisms uniformly continuous. Now we have two uniformities, one on the double of , the other on the double of the double of . We may repeat the procedure indefinitely, pushing to infinity the pair of uniformities.
(TO BE CONTINUED)