Tag Archives: emergent algebra

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 🙂

Rigidity of algebraic structure: principle of common cause

I follow with a lot of interest the stream of posts by Terence Tao on the Hilbert’s fifth problem and I am waiting impatiently to see how it connects with the field of approximate groups.

In his latest post Tao writes that

… Hilbert’s fifth problem is a manifestation of the “rigidity” of algebraic structure (in this case, group structure), which turns weak regularity (continuity) into strong regularity (smoothness).

This is something amazing and worthy of exploration!
I propose the following “explanation” of this phenomenon, taking the form of the:

Principle of common cause: an uniformly continuous algebraic structure has a smooth structure because both structures can be constructed from an underlying emergent algebra (introduced here).

Here are more explanations (adapted from the first paper on emergent algebras):

A differentiable algebra, is an algebra (set of operations A) over a manifold X with the property that all the operations of the algebra are differentiable with respect to the manifold structure of X. Let us denote by D the differential structure of the manifold X.
From a more computational viewpoint, we may think about the calculus which can be
done in a differentiable algebra as being generated by the elements of a toolbox with two compartments “A” and “D”:

– “A” contains the algebraic information, that is the operations of the algebra, as
well as algebraic relations (like for example ”the operation ∗ is associative”, or ”the operation ∗ is commutative”, and so on),
– “D” contains the differential structure informations, that is the information needed in order to formulate the statement ”the function f is differentiable”.
The compartments “A” and “D” are compatible, in the sense that any operation from “A” is differentiable according to “D”.

I propose a generalization of differentiable algebras, where the underlying differential structure is replaced by a uniform idempotent right quasigroup (irq).

Algebraically, irqs are related with racks and quandles, which appear in knot theory (the axioms of a irq correspond to the first two Reidemeister moves). An uniform  irq is a family of irqs indexed by elements of a commutative group (with an absolute), such that  the third Reidemeister move is related to a statement in terms of uniform limits of composites of operations of the family of irqs.

An emergent algebra is an algebra A over the uniform irq X such that all operations and algebraic relations from A can be constructed or deduced from combinations of operations in the uniform irq, possibly by taking limits which are uniform with respect to a set of parameters. In this approach, the usual compatibility condition between algebraic information and differential information, expressed as the differentiability of algebraic operations with respect to the differential structure, is replaced by the “emergence” of algebraic operations and relations from the minimal structure of a uniform irq.

Thus, for example, algebraic operations and the differentiation operation (taking   the triple (x,y,f) to Df(x)y , where “x, y” are  points and “f” is a function) are expressed as uniform limits of composites of more elementary operations. The algebraic operations appear to be differentiable because of algebraic abstract nonsense (obtained by exploitation of the Reidemeister moves) and because of the uniformity assumptions which allow us to freely permute limits with respect to parameters in the commutative group (as they tend to the absolute), due to the uniformity assumptions.

Hilbert fifth’s problem without one parameter subgroups

Further I reproduce, with small modifications, a comment   to the post

Locally compact groups with faithful finite-dimensional representations

by Terence Tao.

My motivation lies in the  project   described first time in public here.  In fact, one of the reasons to start this blog is to have a place where I can leisurely explain stuff.

Background:    The answer to the  Hilbert fifth’s problem  is: a connected locally compact group without small subgroups is a Lie group.

The key idea of the proof is to study the space of one parameter subgroups of the topological group. This space turns out to be a good model of the tangent space at the neutral element of the group (eventually) and the effort goes towards turning upside-down this fact, namely to prove that this space is a locally compact topological vector space and the “exponential map”  gives a chart  of  (a neighbourhood of the neutral element of ) the group into this space.

Because I am a fan of differential structures   (well, I think they are only the commutative, boring side of dilation structures  or here or emergent algebras)   I know a situation when one can prove the fact that a topological group is a Lie group without using the one parameter subgroups!

Here starts the original comment, slightly modified:

Contractive automorphisms may be as relevant as one-parameter subgroups for building a Lie group structure (or even more), as shown by the following result from E. Siebert, Contractive Automorphisms on Locally Compact Groups, Math. Z. 191, 73-90 (1986)

5.4. Proposition. For a locally compact group G the following assertions are equivalent:
(i) G admits a contractive automorphism group;
(ii) G is a simply connected Lie group whose Lie algebra g admits a positive graduation.

The corresponding result for local groups is proved in L. van den Dries, I. Goldbring, Locally Compact Contractive Local Groups, arXiv:0909.4565v2.

I used Siebert result for proving the Lie algebraic structure of the metric tangent space to a sub-riemannian manifold in M. Buliga, Infinitesimal affine geometry of metric spaces endowed with a dilatation structure, Houston Journal of Mathematics, 36, 1 (2010), 91-136. arXiv:0804.0135v2

(added here: see  in Corollary 6.3 from “Infinitesimal affine …” paper, as well as Proposition 5.9 and Remark 5.10 from the paper  A characterization of sub-riemannian spaces as length dilation structures constructed via coherent projections, Commun. Math. Anal. 11 (2011), No. 2, pp. 70-111 , arXiv:0810.5042v4 )

When saying that contractive automorphisms, or approximately contractive automorphisms [i.e. dilation structures], may be more relevant than one-parameter subgroups, I am thinking about sub-riemannian geometry again, where a one-parameter subgroup of a group, endowed with a left-invariant distribution and a corresponding Carnot-Caratheodory distance, is “smooth” (with respect to Pansu-type derivative) if and only if the generator is in the distribution. Metrically speaking, if the generator  is not in the distribution then any trajectory of the one-parameter group has Hausdorff dimension greater than one. That means lots of problems with the definition of the exponential and any reasoning based on differentiating flows.