The Cartesian Method

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.”

Packing and unpacking arrows in graphic lambda calculus

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 GRAPH, 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.

reg_1

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 GRAPH, call them A and B, which are connected by a pair of arrows, like in the following figure.

reg_2

With the added packing and unpacking triples of gates, the graphs A, B 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:

reg_3

Let’s look now at the graphic beta move:

beta_move_1

If we use the elementary gates transformed into graphs with one input and one output, the move becomes this almost algebraic, 1D rule:

reg_4

Finally, the packing-unpacking trick described in the first figure becomes this:

reg_5

Peer-review, what is it for?

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.

Axioms for projective conical spaces (towards qubits II)

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 X  and a commutative group of parameters (\Gamma, \cdot, 1) with operation denoted multiplicatively \cdot(\varepsilon, \mu) = \varepsilon \mu and neutral element 1. Think about \Gamma as being (0,+\infty) or even K^{*} where K is a field.

On X  is defined a function \delta: \Gamma \times X \times X \rightarrow X (Bertram uses the letter \mu instead, I am using \delta). This function is to be interpreted as a \Gamma-parametrized family of operations. Namely we denote:

\delta(\varepsilon, x, y) = \delta^{x}_{\varepsilon} y = x \circ_{\varepsilon} y

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:

dilat_1

Combinations (i.e. compositions) of dilations appear therefore as oriented trees with trivalent planar nodes decorated by the elements of \Gamma, with leaves (but not the root) decorated with elements from X.

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)

dilat_2

(2) this is equivalent with the move R1a from graphical lambda calculus (i.e. Reidemeister move R1a, following the notation from Michael Polyak “Minimal generating sets of Reidemeister moves“)

dilat_3

(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)

dilat_4

(4) this is the self-distributivity axiom, which could be called move R3b with the notations of Polyak

dilat_5

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 (X,X'). Think about the elements  x \in X as being “points” and about the elements  a \in X' 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 (X,X',M) where M \subset X \times X' is the set of pairs (say point-line) in general position. Compared to the more familiar case of incidence systems, the interpretation of (x,a) \in M is “the point x is not incident with the line a“.  The triple satisfies some conditions which I shall write after introducing some notations.

For any x \in X and any a \in A we denote:

V_{x} = \left\{ b \in X' \mid (x,b) \in M \right\}  and    V_{a} = \left\{ y \in X \mid (y,a) \in M \right\}.

Let also denote

D = \left\{ (x,a,y) \in X \times X' \times X \mid (x,a), (y,a) \in M \right\} and D' = \left\{ (a,x,b) \in X' \times X \times X' \mid (x,a), (x,b) \in M \right\}.

We ask:

(Pair geometry 1) for any x \in X and for any a \in X' the sets V_{x} and V_{a} are non-empty,

(Pair geometry 2) for any pair of different points x,y \in X there exists and it’s unique a line a \in X' such that (x,a) and (y,a) are not in M; dually, for any pair of different lines a,b \in X' there exists and it’s unique a point x \in X such that (x,a) and (x,b) are not in M.

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 N such that X is covered by the reunion of N  sets V_{x} (and a similar dual formulation for X'. As it is, (Pair geometry 2) corresponds to such a formulation for N = 3.

We want the following:

  1. for any point x \in X the space V_{x} is an affine conical space,
  2. for any line a \in X' the space V_{a} is an affine conical space,
  3. 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 V_{x}. It is natural then to mark the dilation operations not only by elements of the group \Gamma, but also by x. More concretely that means we introduce for any \varepsilon \in \Gamma  a function

\delta_{\varepsilon}: D' \rightarrow X'

which, for any x \in X it takes a pair of lines (a,b), with a,b \in V_{x} and returns \delta_{\varepsilon}(a,x,b) \in V_{x}.

We ask that for any x \in X the dilations \varepsilon \mapsto \delta_{\varepsilon}(\cdot, x, \cdot) satisfy axioms (1), (2), (3) of affine conical spaces.

2. in the same way, we want that every V_{a} to have a structure of dilation operations. We have therefore, for any \varepsilon \in \Gamma another function (but I shall use the same letter \delta)

\delta_{\varepsilon}: D \rightarrow X

which, for any a \in X' it takes a pair of points (x,y), with x,y \in V_{a} and returns \delta_{\varepsilon}(x,a,y) \in V_{a}.

We ask that for any a \in X' the dilations \varepsilon \mapsto \delta_{\varepsilon}(\cdot, a, \cdot) 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:

proj_1

Let’s look at the first line. For any a \in X' we have an associated dilation operation taking as input a pair of points x,y \in V_{a}. 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 V_{a}. 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 \Gamma.

(4.1)     first part of (PG1) proj_2

______________________

(4.2) second part of (PG1)

proj_3

______________________

(4.3)  first  part of (PG2)

proj_5

______________________

(4.4) second part of (PG.2)

proj_6

______________________

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.

 

Geometric Ruzsa triangle inequalities and metric spaces with dilations

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 X be a non-empty set and \Delta: X \times X \rightarrow X be an operation on X which has the following two properties:

  1. for any a, b, c \in X we have \Delta(\Delta(a,b), \Delta(a,c)) = \Delta(b,c),
  2. for any a \in X   the function z \mapsto \Delta(z,a) is injective.

We may use weaker hypotheses for \Delta, namely:

  1. (weaker) there is a function F: X \times X \rightarrow X such that F(\Delta(a,b), \Delta(a,c)) = \Delta(b,c) for any a, b, c \in X,
  2. (weaker) there is a function G: X \times X \rightarrow X such that a \mapsto G(\Delta(a,b), b) is an injective function for any b \in X.

Prop. 1. Let X be a non empty set endowed with an operation \Delta which satisfies 1. and 2. (or the weaker version of those). Then for any non empty sets A, B, C \subset X there is an injection

i: \Delta(C,A) \times B \rightarrow \Delta(B,C) \times \Delta(B,A),

where we denote by \Delta(A,B) = \left\{ \Delta(a,b) \mid a \in A, b \in B \right\}

In particular, if A, B, C are finite sets, we have the Rusza triangle inequality

\mid \Delta(C,A) \mid \mid B \mid \leq \mid \Delta(B,C) \mid \mid \Delta(B,A) \mid,

where \mid A \mid denotes the cardinality of the finite set A.

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 \Delta(a,b) = -a + b, where (X,+) is a group (no need to be abelian). The proof  of the geometric Ruzsa inequality corresponds to the choice \Delta(a,b) = [b,a], with the notations from the article, with the observation that this function \Delta satisfies weaker 1. and 2.

Proof.  We can choose functions f: \Delta(C,A) \rightarrow C and g: \Delta(C,A) \rightarrow A such that for any x \in \Delta(C,A) we have x = \Delta(f(x),g(x)). With the help of these functions let

i(x,b) = (\Delta(b,f(x)), \Delta(b, g(x))).

We want to prove that i is injective. Let (c,d) = i(x,b) = i(x',b'). Then, by 1. we have x = x' = \Delta(c,d).  This gives an unique e = f(x) = f(x'). Now we know that \Delta(b, e) = \Delta(b,f(x)) = c = \Delta(b', f(x')) = \Delta(b', e). By 2. we get that b = b'     qed.

____________

In a metric space with dilations  (X, d, \delta)  we have the function  approximate difference \Delta^{e}_{\varepsilon} (a,b) based at e \in X and applied to a pair of closed points a, b \in X. This function has the property that (e,a,b) \mapsto \Delta^{e}_{\varepsilon}(a,b) converges uniformly to \Delta^{e}(a,b) as \varepsilon goes to 0. Moreover, there is a local group operation with e as neutral element such that \Delta^{e}(a,b) = -a+b, therefore the function \Delta^{e} satisfies 1. and 2.

As concerns the function \Delta^{e}_{\varepsilon}, it satisfies the following approximate version of 1.:

  1. (approximate) for any e, a, b, c \in X which are sufficiently close and for any \varepsilon \in (0,1) we have, with the notation a(\varepsilon) = \delta^{e}_{\varepsilon} a,  the relation

\Delta^{a(\varepsilon)}_{\varepsilon}(\Delta^{e}_{\varepsilon}(a,b), \Delta^{e}_{\varepsilon}(a,c)) = \Delta^{e}_{\varepsilon}(b,c).

We say that a set A \subset X is \varepsilon separated if for any x, y \in A,  the inequality  d(x,y) < \varepsilon  implies  x = y.  Further I am going to write about sets which are closed to a fixed, but arbitrary otherwise point e \in X.

Prop2.  In a metric space with dilations, let p >0 and  let A, B, C be finite sets of points included in a compact neighbourhood of e, which are closed to e \in X, such that for any  \varepsilon \in (0,p)  the sets B and \Delta^{e}_{\varepsilon}(C,A) are \mu separated. Then for any \varepsilon \leq C(\mu) there is an injective function

i: \Delta^{e}_{\varepsilon}(C,A) \times B \rightarrow \Delta^{e}_{\varepsilon}(B,C) \times \Delta^{e}_{\varepsilon}(B,A).

Proof. As previously, we choose the functions f and g. Notice that these functions depend on \varepsilon but this will not matter further.  I shall use the O(\varepsilon) notation liberally, for example y = x + O(\varepsilon) means d(x,y) \leq O(\varepsilon).  Let’s define the function i by the same formula as previously:

i(x,b) = (\Delta^{e}_{\varepsilon}(b,f(x)), \Delta^{e}_{\varepsilon}(b, g(x))).

Let (x,b) and (x',b') be pairs such that i(x,b) = i(x',b') = (c_{\varepsilon}, d_{\varepsilon}). From 1. (approximate) and from the uniform convergence mentioned previously we get that

x = x' + O(\varepsilon) = \Delta^{e}(c_{\varepsilon}, d_{\varepsilon}) + O(\varepsilon).

There is a function C(\mu)  such that \varepsilon \leq C(\mu) implies (the last from the previous relation) O (\varepsilon) < \mu.  For such a \varepsilon, by the separation  of \Delta^{e}_{\varepsilon}(C,A) we get x = x'.

Let z = f(x). From the hypothesis we have \Delta^{e}_{\varepsilon}(b, z) = \Delta^{e}_{\varepsilon}(b',z). This implies, via the structure of the function \Delta^{e} and via the uniform convergence, that b' = b + O(\varepsilon)  (by compactness, this last O(\varepsilon) does not depend on A, B, C). By the same reasoning as previously, we may choose C(\mu) such that d(b,b') < \mu if \varepsilon \leq C(\mu). This implies b = b'   qed.

Neuroscience and computation: hand-in-hand

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.

1. The cited formulation comes from the site of the Human Brain Project, one of the most amazing collaborations ever. More specifically, it is taken from here, let me cite more:

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 [1]. 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 computationintroduced 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  \lambda-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).

Applications of UD (part II)

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?