Tag Archives: chora

Lots of questions, part I

There is a button for “publish”. So what?

I started this open notebook  with the goal to disseminate some of my work and ideas.  There are always many subjects to write about, this open notebook has almost 500 posts. Lately the rhythm of about a post every 3 days slowed down to a post a week.  I have not run out of ideas, or opinions. It is only that I don’t get anything in return.

I explain what I mean by getting something in return. I don’t believe that one should expect a compensation for the time and energy converted into a post. There are always a million posts to read.  There is not a lot of time to read them. It is costly in brain time to understand them, and probably, from the point of view of the reader, the result of this investment does not worth the effort.

So it’s completely unreasonable to think that my posts should have any treatment out of the usual.

Then, what can be the motivation to have an open notebook, instead of just a notebook? Besides vanity, there is not much.

But vanity was not my motivation, although it feels very good to have a site like this one. Here is why:   from the hits I can see that people read old posts as frequently as new posts. You have to agree that this is highly unusual for a blog. So, incidentally,  perhaps this is not a blog, doh.

I put vanity aside and I am now closer to the real motivations for maintaining this open notebook.

Say you have a new idea, product, anything which behaves like a happy virus who’s looking for hosts to multiply. This is pretty much the problem of any creator: to find hosts.  OK, what is available for a creator who is not a behemoth selling sugar solutions or other BRILLIANT really simple viruses like phones, political ideas, contents for lazy thinking trolls and stuff like this?

What if I don’t want to sell ideas, but instead I want to find those rare people with similar interests?

I don’t want to entertain anybody, instead that’s a small fishing net in the big sea.

OK, this was the initial idea. That compared to the regular ways, meaning writing academic articles, going to conferences, etc, there might be more chances to talk with interesting people if I go fishing in the high seas, so to say.

These are my expectations. That I might find interesting people to work with, based on common passions, and to avoid the big latency of the academic world, so that we can do really fast really good things now.

I know that it helps a lot to write simple. To dilute the message. To appeal to authority, popularity, etc.

But I expect that there is a small number of you guys who really think as fast as I do. And then reply to me, simultaneously to Marius.Buliga@imar.ro and Marius.Buliga@gmail.com .

Now that my expectations are explained, let’s look at the results. I have to put things in context a bit.

This site was called initially lifeinrio@wordpress.com . I wanted to start a blog about how is it to live in Rio with wife and two small kids. Not a bad subject, but I have not found the time for that side project, because I was just in the middle of an epiphany. I wanted to switch fields, I wanted to move from pure and applied mathematics to somewhere as close as possible to biology and neuroscience. But mind you that I wanted also to bring with me the math. Not to make a show of it, but to use the state of mind of a mathematician in these great emerging fields. So, instead of writing about my everyday life experiences, I started to write to everybody I found on the Net who was not (apparently) averse to mathematics and who was also somebody in neuroscience. You can imagine that my choices were not very well informed, because these fields were so far from what I knew before. Nevertheless I have found out interesting people, telling them about why I want to switch. Yes, why? Because of the following  reasons: (1) I am passionate about making models of reality, (2) I’m really good at finding unexpected points of view, (3) I learn very fast, (4) I understood that pure or applied math needs a challenge beyond the Cold War ones (i.e. theories of everything, rocket science, engineering).  OK, I’ll stop here with the list, but there were about 100 more reasons, among them being to understand what space is from the point of the view of a brain.

I got fast into pretty weird stuff. I started to read philosophy, getting hooked by Plato. Not in the way the usual american thinker does. They believe that they are platonic but they are empiricists, which is exactly the poor (brain) version of platonism. I shall stop giving kicks to empiricists, because they have advanced science in many ways in the last century.  Anyway empiricism looks more and more like black magic these days. Btw, have you read anything by Plato? If you do, then try to go to the source. Look for several sources,  you are not a good reader of ancient Greek.  Take your time, compare versions, spell the originals (so to say), discover the usual phenomenon that more something is appreciated, more shit inside.

Wow, so here is it a mathematician who wants to move to biology, and he uses Plato as a vehicle. That’s perhaps remarkabl…y stupid to do, Marius. What happened, have you ran out of the capacity to do math? Are you out in the field where people go when they can’t stand no more the beauty and hardness of mathematics? Everybody knows, since that guy who wrote with Ramanujan and later, after R was dead, told us that mathematics is for young people. (And probably white wealthy ones.)

No, what happened was that the air of Rio gave me the guts I have lost during the education process. Plato’s Timaeus spoke to me in nontrivial ways, in particular. I have understood that I am really on the side of geometers, not on the side of language people. And that there is more chance to understand brains if we try to model what the language people assume it works by itself, the low level, non rational processes of the brain. Those who need no names, no language, those highly parallel ones. For those, I discovered, there was no math to apply.  You may say that for example vision is one of the most studied subjects and that really there is a lot of maths already used for that. But if you say so then you are wrong.  There is no model of vision up to now, which explains how biological vision works without falling into the internal or external homunculus fallacies. If you look to computer vision, you know, you can do anything with computers, provided you have enough of them and enough time. There is a huge gap between computer vision and biological vision, a fundamental one.

OK, when I returned home to Bucharest I thought what if I reuse the lifeinrio.wordpress.com and transform it into chorasimilarity.worpress.com? This word chorasimilarity is made of “chora”, which is the feminine version of “choros”, which means place or space. Plato invented the “chora” as a term he used in his writings. “Similarity” was because of my background in math: I was playing with “emergent algebras”, which I invented previously of going on the biology tangent. In fact these emergent algebras made me think first that it is needed a new math, and that maybe they are relevant for biological vision.

I stop a bit to point to the post Scale is a place in the brain, which is about research on grid cells and place cells (research which just got a Nobel in medicine in 2014).

Emergent algebras are about similarity. They make visible that behind is hidden an abstract graph rewrite system. Which in turn can be made concrete by transforming it into chemistry. An artificial chemistry.  But also, perhaps, a real one. Or, the brain is most of it chemistry. Do you see how everything gets in place?  Chora is just chemistry in the brain. Being universal, it is not surprising that we distilled, us humans, a notion of space from that.

There is a lot of infrastructure to build in order to link all these in a coherent way.

Penrose’ combinatorial space time as chora

Roger Penrose, among other extraordinary things he did, proposed an approach to combinatorial space-time by way of spin-networks. Here is a link to his amazing paper

Roger Penrose, Angular momentum: an approach to combinatorial space-time, in Quantum Theory and Beyond, ed. T. Bastin, Cambridge University Press, Cambridge, 1971.

taken from this page of John Baez.

With the new knowledge gradually constructed lately, I returned to read this classic and it just downed on me that a strand-network (Penrose paper page 19 in the pdf linked above) can be given a structure of a collection of choroi, see the post

Entering chora, the infinitesimal place

or better go to the paper “Computing with space“.

This is food for thought for me, I just felt the need to communicate this. It is not my intention to chase for theories of everything, better is to understand, as a mathematician, what is worthy to learn and import from this field into what interests me.

Entering “chora”, the infinitesimal place

There is a whole discussion around the key phrases “The map is not the territory” and “The map is the territory”. From the wiki entry on the map-territory relation, we learn that Korzybski‘s dictum “the map is not the territory” means that:

A) A map may have a structure similar or dissimilar to the structure of the territory,

B) A map is not the territory.

Bateson, in “Form, Substance and Difference” has a different take on this: he starts by explaining the pattern-substance dichotomy

Let us go back to the original statement for which Korzybski is most famous—the statement that the map is not the territory. This statement came out of a very wide range of philosophic thinking, going back to Greece, and wriggling through the history of European thought over the last 2000 years. In this history, there has been a sort of rough dichotomy and often deep controversy. There has been violent enmity and bloodshed. It all starts, I suppose, with the Pythagoreans versus their predecessors, and the argument took the shape of “Do you ask what it’s made of—earth, fire, water, etc.?” Or do you ask, “What is its pattern?” Pythagoras stood for inquiry into pattern rather than inquiry into substance.1 That controversy has gone through the ages, and the Pythagorean half of it has, until recently, been on the whole the submerged half.

Then he states his point of view:

We say the map is different from the territory. But what is the territory? […] What is on the paper map is a representation of what was in the retinal representation of the man who made the map–and as you push the question back, what you find is an infinite regress, an infinite series of maps. The territory never gets in at all.

Always the process of representation will filter it out so that the mental world is only maps of maps of maps, ad infinitum.

At this point Bateson puts a very interesting footnote:

Or we may spell the matter out and say that at every step, as a difference is transformed and propagated along its pathways, the embodiment of the difference before the step is a “territory” of which the embodiment after the step is a “map.” The map-territory relation obtains at every step.

Inspired by Bateson, I want to explore from the mathematical side the point of view that there is no difference between the map and the territory, but instead the transformation of one into another can be understood by using tangle diagrams.

Let us imagine that the exploration of the territory provides us with an atlas, a collection of maps, mathematically understood as a family of two operations (an “emergent algebra”). We want to organize this spatial information in a graphical form which complies with Bateson’s footnote: map and territory have only local meaning in the graphical representation, being only the left-hand-side (and r-h-s respectively) of the “making map” relation.

Look at the following figure:

In the figure from the left, the “v” which decorates an arc, represents a point in the “territory”, that is the l-h-s of the relation, the “u” represents a “pixel in the map”, that is the r-h-s of a relation. The relation itself is represented by a crossing decorated by an epsilon, the “scale” of the map.

The opposite crossing, see figure from the right, is the inverse relation.

Imagine now a complex diagram, with lots of crossings, decorated by various
scale parameters, and segments decorated with points from a space X which
is seen both as territory (to explore) and map (of it).

In such a diagram the convention map-territory can be only local, around each crossing.

There is though a diagram which could unambiguously serve as a symbol for
“the place (near) the point x, at scale epsilon” :

In this diagram, all crossings which are not decorated have “epsilon” as a decoration, but this decoration can be unambiguously placed near the decoration “x” of the closed arc. Such a diagram will bear the name “infinitesimal place (or chora) x at scale epsilon”.

A difference which makes four differences, in two ways

Gregory Bateson , speaking about the map-territory relation

“What is in the territory that gets onto the map? […] What gets onto the map, in fact, is difference.

A difference is a very peculiar and obscure concept. It is certainly not a thing or an event. This piece of paper is different from the wood of this lectern. There are many differences between them, […] but if we start to ask about the localization of those differences, we get into trouble. Obviously the difference between the paper and the wood is not in the paper; it is obviously not in the wood; it is obviously not in the space between them .

A difference, then, is an abstract matter.

Difference travels from the wood and paper into my retina. It then gets picked up and worked on by this fancy piece of computing machinery in my head.

… what we mean by information — the elementary unit of information — is a difference which makes a difference.

(from “Form, Substance and Difference”, Nineteenth Annual Korzybski Memorial
Lecture delivered by Bateson on January 9, 1970, under the auspices of the Institute of General Semantics, re-printed from the General Semantics Bulletin, no.
37, 1970, in Steps to an Ecology of Mind (1972))

This “difference which makes a difference” statement is quite famous, although sometimes considered only a figure of speach.

I think it is not, let me show you why!

For me a difference can be interpreted as an operator which relates images of the same thing (from the territory) viewed in two different maps, like in the following picture:

This figure is taken from “Computing with space…” , see section 1 “The map is the territory” for drawing conventions.

Forget now about maps and territories and concentrate on this diagram viewed as a decorated tangle. The rules of decorations are the following: arcs are decorated with “x,y,…”, points from a space, and the crossings are decorated with epsilons, elements of a commutative group (secretly we use an emergent algebra, or an uniform idempotent right quasigroup, to decorate arcs AND crossings of a tangle diagram).

What we see is a tangle which appears in the Reidemeister move 3 from knot theory. When epsilons are fixed, this diagram defines a function called (approximate) difference.

Is this a difference which makes a difference?

Yes, in two ways:

1. We could add to this diagram an elementary unknot passing under all arcs, thus obtaining the diagram

Now we see four differences in this equivalent tangle: the initial one is made by three others.
The fact that a difference is selfsimilar is equivalent with the associativity of the INVERSE of the approximate difference operation, called approximate sum.

2. Let us add an elementary unknot over the arcs of the tangle diagram, like in the following figure

called “difference inside a chora” (you have to read the paper to see why). According to the rules of tangle diagrams, adding unknots does not change the tangle topologically (although this is not quite true in the realm of emergent algebras, where the Reidemeister move 3 is an acceptable move only in the limit, when passing with the crossing decorations to “zero”).

By using only Reidemeister moves 1 and 2, we can turn this diagram into the celtic looking figure

which shows again four differences: the initial one in the center and three others around.

This time we got a statement saying that a difference is preserved under “infinitesimal parallel transport”.

So, indeed, a difference makes four differences, in at least two ways, for a mathematician.

If you want to understand more from this crazy post, read the paper 🙂