Book project: Metric spaces with dilations

On the mathematics front, here is a link to a book project

Metric spaces with dilations

(working version! chech for updates)

As it is now, it’s not much more than a merger of previous research papers, without the nice figures from “Computing with space…” , but from seeing all in one place you may get a sense of what is this all about.

This first version has been prepared as a basis for the minicourse

“Carnot-Caratheodory spaces as metric spaces with dilations”

held at a January 2011 Summer School in Rio de Janeiro.

In the same place I started being interested in exploring this frontier between neuroscience and mathematics…

Koenderink and Changizi

Jan Koenderink is a leading researcher in vision. He proposed the concept of
“scale-space representation” in relation to the understanding of how the front-end visual system works.

His paper “The brain a geometry engine” starts with:

According to Kant, spacetime is a form of the mind. If so, the brain must be a geometry engine. This idea is taken seriously, and consequently the implementation of space and time in terms of machines is considered. This enables one to conceive of spacetime as really ldquoembodied.rdquo

Later he writes:

There may be a point in holding that many of the better-known brain processes are most easily understood in terms of differential geometrical calculations running on massively parallel processor arrays whose nodes can be understood quite directly in terms of multilinear operators (vectors, tensors, etc).
In this view brain processes in fact are space.

This is a very interesting idea! As far as I understand, Koenderink is saying that somehow brain processes involved in vision and (external) space are similar!

In my opinion this is something to explore. However, my take is that this superb idea is clouded by his reliance on linear algebra and differential calculus of the exterior euclidean space (see “vectors, tensors, etc” as well as his derivation of the gaussian filter from invariance with respect to the same euclidean structure). If said brain processes are space and if those brain processes are a kind of computation (in a sense to be explained later) then space should appear as the result of a computation in the front-end visual system. No euclidean a priori!

Are those brain processes a kind of computation? The answer depends on what computation means. Anyway, nobody doubts that logical boolean computations are orthodox computations.

See then the following paper by Mark Changizi “Harnessing vision for computation” or check this Wired post

Scientists Build Visual Circuits to Harness your Brain’s GPU”

The abstract of the paper is:

Might it be possible to harness the visual system to carry out artificial computations, somewhat akin to how DNA has been harnessed to carry out computation? I provide the beginnings of a research programme attempting to do this. In particular, new techniques are described for building `visual circuits’ (or `visual software’) using wire, NOT, OR, and AND gates in a visual modality such that our visual system acts as `visual hardware’ computing the circuit, and generating a resultant perception which is the output

My conclusion: this is experimental proof that at least some brain processes related to vision can do something which simulates logical computation.

Computing with space

This is the first in a series of postings concerning computing with space. I shall try to give a gentle introduction to – and later discussion around – the ideas from the paper

Computing with space: a tangle formalism for chora and difference

We shall talk about:

– mathematics of metric spaces with dilations

Bateson viewpoint that the map is the territory, as opposed to Korzybski dictum “the map is not the territory”.

– Plato’ Timaeus 48e-53c where he introduces the concept of “chora”, which means “space” or “place”

– research in the neuroscience of vision, like Jan Koenderink paper “Brain a geometry engine”

and many other.

Older papers of mine on this subject: arXiv:1009.5028 “What is a space? Computations in emergent algebras and the front end visual system” and the arXiv:1007.2362 “Introduction to metric spaces with dilations”.

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