This is a pedagogical introduction covering maps of metric spaces, Gromov-Hausdorff distance and its “physical” meaning, and dilation structures as a convenient simplification of an exhaustive database of maps of a metric space into another:
I am intrigued by this part of the post from NEW
“The public talk by Cumrun Vafa puts out the classic message that strings have come to the rescue of physics, unifying QM and gravity, and that:
Smooth geometry of strings seems to explain all known interactions (at least in principle)”
Why “smooth”? Probably only because this is in the comfort zone of many.
However, there are two new fields of mathematics which deserve to be taken into consideration by physicists (or not, not my problem in fact):
- Nonsmooth calculus, see for an intro this excellent review by Juha Heinonen .
- Additive combinatorics, see this book by Terence Tao, Van Vu.
That’s the future!
UPDATE: (24.03.2012) Congratulations to Endre Szemeredi, the Abel Prize Laureate 2012, “for his fundamental contributions to discrete mathematics and theoretical computer science, and in recognition of the profound and lasting impact of these contributions on additive number theory and ergodic theory.”
… and modern physics, maybe in 50 years.
In the middle of the 19th century, in France, just before the impressionist revolution, painting was boring. Under the standards imposed by the Academie des Beaux Arts, paintings were done in a uniform technique, concerning a very restrictive list of subjects. I cite from wikipedia:
”Colour was somber and conservative, and the traces of brush strokes were suppressed, concealing the artist’s personality, emotions, and working techniques.”
This seems very much similar with the situation in today’s mathematical research.
Read the rest here: “Boring mathematics, artistes pompiers and impressionists”.
PS: anyone who understands from this post that I think mathematics is boring has the attention span of a gnat.
Further I reproduce, with small modifications, a comment to the post
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.
Looks to me there is something wrong with the Cartesian Theater term.
Short presentation of the Cartesian Theater, according to wikipedia (see previous link):
The Cartesian theater is a derisive term coined by philosopher Daniel Dennett to pointedly refer to a defining aspect of what he calls Cartesian materialism, which he considers to be the often unacknowledged remnants of Cartesian dualism in modern materialistic theories of the mind.
Descartes originally claimed that consciousness requires an immaterial soul, which interacts with the body via the pineal gland of the brain. Dennett says that, when the dualism is removed, what remains of Descartes’ original model amounts to imagining a tiny theater in the brain where a homunculus (small person), now physical, performs the task of observing all the sensory data projected on a screen at a particular instant, making the decisions and sending out commands.
Needless to say, any theory of mind which can be reduced to the Cartesian Theater is wrong because it leads to the homunculus fallacy: the homunculus has a smaller homunculus inside which is observing the sensory data, which has a smaller homunculus inside which …
This homunculus problem is very important in vision. More about this in a later post.
According to Dennett, the problem with the Cartesian theater point of view is that it introduces an artificial boundary (from Consciousness Explained (1991), p. 107)
“…there is a crucial finish line or boundary somewhere in the brain, marking a place where the order of arrival equals the order of “presentation” in experience because what happens there is what you are conscious of.”
As far as I understand, this boundary creates a duality: on one side is the homunculus, on the other side is the stage where the sensory data are presented. In particular this boundary acts as a distinction, like in the calculus of indications of Spencer-Brown’ Laws of Form.
This distinction creates the homunculus, hence the homunculus fallacy. Neat!
Why I think there is something wrong with this line of thought? Because of the “theater” term. Let me explain.
The following is based on the article of Kenneth R Olwig
but keep in mind that what is written further represents my interpretation of some parts of the article, according to my understanding, and not the author point of view.
There has been a revolution in theater, started by
“…the early-17th-century court masques (a predecessor of opera) produced by the author Ben Jonson (the leading author of the day after Shakespeare) together with the pioneering scenographer and architect Inigo Jones.
The first of these masques, the 1605 Masque of Blackness (henceforth Blackness ), has a preface by Jonson containing an early use of landscape to mean scenery and a very early identification of landscape with nature (Olwig, 2002, page 80), and Jones’s scenography is thought to represent the first theatrical use of linear perspective in Britain (Kernodle, 1944, page 212; Orgel, 1975).” (p. 521)Ben Johnson,
“From the time of the ancient Greeks, theater had largely taken place outside in plazas and market places, where people could circle around, or, as with the ancient Greco-Roman theater or Shakespeare’s Globe, in an open roofed arena. Jones’s masques, by contrast, were largely performed inside a fully enclosed rectangular space, giving him control over both the linear-focused geometrical perspectival organization of the performance space and the aerial perspective engendered by the lighting (Gurr, 1992; Orrell, 1985).” (p. 522, my emphasis)
“Jonson’s landscape image is both enframed by, and expressive of, the force of the lines of perspective that shoot forth from “the eye” – notably the eye of the head of state who was positioned centrally for the best perspectival gaze.” (p. 523, my emphasis)
“Whereas theater from the time of the ancient Greeks to Shakespeare’s Globe was performed in settings where the actor’s shadow could be cast by the light of the sun, Jones’s theater created an interiorized landscape in which the use of light and the structuring of space created an illusion of three dimensional space that shot from the black hole of the individual’s pupil penetrating through to a point ending ultimately in ethereal cosmic infinity. It was this space that, as has been seen, and to use Eddington’s words, has the effect of “something like a turning inside out of our familiar picture of the world” (Eddington, 1935, page 40). It was this form of theater that went on to become the traditional `theater in a box’ viewed as a separate imagined world through a proscenium arch.” (p. 526, my emphasis)
I am coming to the last part of my argument: Dennett’ Cartesian Theater is a “theater in a box”. In this type of theater there is a boundary,
a distinction, as in Dennett argument. We may also identify the homunculus side of the distinction with the head of state.
But this is not all.
Compared with the ancient Greeks theater, the “theater in a box” takes into account the role of the spectator as the one which perceives what is played on stage.
Secondly, the scenic space is not “what happens there”, as Dennett writes, but a construction already, a controlled space, a map of the territory and not the territory itself.
Conclusion: in my view (contradict me please!) the existence of the distinction (limen) in the “Cartesian theater”, which creates the homunculus problem, is superficial. More important is the fact that “Cartesian theater”, as “theater in a box”, is already a representation of perception, having on one side of the limen a homunculus and on the other side a scenic space which is not the “real space” (as for example the collection of electric sparks sent by the sensory organs to the brain) but instead is as artificial as the homunculus, being a space created and controlled by the scenographer.
Litmus test: repeat the reasoning of Dennett after replacing the “theater in a box” preconception of the “theater” by the older theater from the time of ancient Greeks. Can you do it?
On the beautiful idea of “aerography”, later.
On the mathematics front, here is a link to a book project
(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…
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 embodied.
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.
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.