Ark - where are you headed?

[SoCurious]

Yeah, I get it. Maybe I'm just a bit confused over how we can understand the inadequacies of Aristotelian-bound syllogistic thought, the limitations of Newtonian-classical physics, the lack of holism between System 1 and System 2 on a personal level, the errors of black and white thinking (binary oppositional patterns), the fascist nature of our "States" (Nations) suppressing the creativity and emotional lives of citizens - "States" that exist by pathologically growing ever more totalitarian day-by-day, yet we can find the idea of a "binary reality" credible...interesting even?

I think I need to back down, be quiet and lurk more. Apologies to you and Ark for interrupting this discussion. Ain't nothing here about me.

Hiya Buddy,

I once worked for a company with a progressive HR department. They instituted a number of interesting courses and had a system of ranking for attendees, the highest ranking being "Curious". I found this intriguing and was actually delighted with the grading system. After all, those who are most curious always seem to be ahead of the curve.

Being curious is a sure-fire way of becoming confused. There's just so much contradictory information, yet our confusions in certain areas are cleared up as we continue to delve into them. A wonderful example is the paleo diet.

I have a special fondness for curious people and relish debating or arguing (as long as it never turns nasty) as this can lead to a growth in knowledge. I would be most upset with myself if I in any way curtailed rather than fostered anyone's curiosity as I'd be responsible for taking away a spark this planet badly needs.

As to confusion, this is a state I've become accustomed to So please, no apologies are necessary or even desired.

SeekinTruth has mirrored me perfectly. I'm extremely interested in how Ark and Laura are working together albeit via different methods. I can understand Laura more easily as I'm better with words than math which is why I'm asking for "Ark for Dummies". It seems we need to understand some underlying concepts before we get to Ark's math, and it seems as if this is where he's leading us.

The way I see it, this thread is all about Ark and his work. I'm just trying to hang on :)

 

[dant]

I had "read" the posted links I have supplied, and I haven't a whit where
I was going with it due to not recalling many of the mathematical rules,
proofs, axioms, corollaries, lemmas, symbols, hypothesis, theories,
methods, techniques, and so on from those who laid the foundations
from the past to the present and from which new ideas may be formed,
based either by deduction or by observable realities.

It is interesting however, that some authors may take the time to
give some examples in real terms, what these abstractions represent,
in (semi-)layman terms. When they do, it is, in a sense, as if the high
energy state of abstraction has dropped to a lower state, thus releasing
a glimpse of light, from which darkness was temporarily subdued, then
returned to its former state? :)

In Conne's papers, there was present, some graphs, diagrams, pictures,
such as the galaxy and the black hole, some spectrographs, and so on.
You might notice the picture of "The Lady in Blue by Gainsborough" which
was present in original form, only to be transformed into checkerboarded
and rearranged form by application of von Neumann algebras, which I still
don't grok. :/

So... it will be interesting if we, the layman, can be taught "ivory tower"
concepts and I am really looking forward to it!

Ark... bring it on! (and thanks!)

 

[ark]

Alain Connes model of Reality

It would be not quite true if I would say that Alain Connes advocated a purely discrete vision of Nature. While he stressed in several places the important part played by discrete structures, in his models of the physical Reality he was joining together both discrete and continuous structures. For in a widely read American popular science journal, Scientific American of August 2006 there is an article entitled Insight that starts with this description:

Acclaimed mathematician Alain Connes takes a geometric approach to learn how spacetime makes particles.

In the article we can read the following brief description of Connes’ model:

… he [Alain Connes] succeeded in creating a noncommutative space that contains all the abstract algebras (known as symmetry groups) that describe the properties of elementary particles in the Standard Model. The picture that emerges from the Standard Model, then, is that of spacetime as a noncommutative space that can be viewed as consisting of two layers of a continuum, like the two sides of a piece of paper. The space between the two sides of the paper is an extra discrete (noncontinuous), noncommutative space.

Of course the term "noncommutative" is a technical jargon, but it simply means that we move to algebra, where AxB is not the same as BxA. We are just departing from numbers, for which 2x3=3x2 and we enter to another world, the world well known today even for engineers - the world of "matrices".

Why should mathematical description of our world be based on simple numbers? Can't it be based on "tables of numbers" instead?

One of the main nonmaterials ingredients of the world seems to be "information". How to code information? In numbers, or in "structures"? So, we need to learn somethong about structures. The simplest ones. What would be the simplest, but nontrivial structure of a monad?

 

[John G]

You might notice the picture of "The Lady in Blue by Gainsborough" which
was present in original form, only to be transformed into checkerboarded
and rearranged form by application of von Neumann algebras, which I still
don't grok.

Of course the term "noncommutative" is a technical jargon, but it simply means that we move to algebra, where AxB is not the same as BxA. We are just departing from numbers, for which 2x3=3x2 and we enter to another world, the world well known today even for engineers - the world of "matrices".

Why should mathematical description of our world be based on simple numbers? Can't it be based on "tables of numbers" instead?

One of the main nonmaterials ingredients of the world seems to be "information". How to code information? In numbers, or in "structures"? So, we need to learn somethong about structures. The simplest ones. What would be the simplest, but nontrivial structure of a monad?

I know Tony Smith uses Clifford Algebra and its ability to make big algebras via lots of Cl(8)s as his von Neumann algebra "monade" and I know Ark uses Clifford Algebra and is on an editorial board for a Clifford Algebra journal. Ark's group theory papers appear in that journal but I get lost a lot going from the very binary Clifford Algebra to the not so binary looking groups. Clifford Algebra handles matrices too.

 

[Buddy]

@Richard: I understand and thank you for your patience.

@Bluelamp: I feel I left something unclarified. I don't experience that "Being and Non-Being" as a description of a binary reality because there is so much more to that quote and that is why I like it so much. I can't explain why exactly, but if you look at the TAO or ying/yang symbol you may see what I mean. In that symbol, I see the same duality everyone else in humanity sees but I also see the "S" superposed with the "O". To me, this is symbolic of quantum reality AND a duality. Water, flow, the feminine creative principle, eternal (e)motion, eternal flux, boundless potential, etc., etc.

So, in short, I 'see' 'the two' and 'sense or feel the third' and I therefore have some way of noticing classicist tomfoolery even though I can hardly avoid it linguistically myself. Hope this makes some sense as I would prefer to remain uncommitted to the nut ward.

Again, please excuse the interruption. Back to the topic...

[quote author=Ark]
So, we need to learn somethong about structures. The simplest ones. What would be the simplest, but nontrivial structure of a monad?[/quote]

I don't know if the above monad is a strict reference to a Leibniz monad or not. The simplest structures I consider myself aware of deal with information and are found in living cells. In fact, they're called the cell's computer.

According to Hammeroff, "tubulin" comprise the nervous system of a cell. Cells use structures of micro-tubules for sex and their nervous system. These structures form a kind of cytoskeletal architecture - a bone-like support and are thought of as an on-board computer. So, the cytoskeleton is a dynamic scaffolding and a micro-tubule is a major component.

But we might not wish to speak of biology here, so I pass to others...

 

[thevenusian]

What would be the simplest, but nontrivial structure of a monad?

Maybe a circle?

It is discrete and binary (a single line and the space defined by it- or one thing and no-thing- 1 and 0).

Or in 3 dimensions, a sphere.

 

[ark]

From an interview with Alain Connes:

Concerning heuristics : you have written several times that geometry is on the side of intui-tion. On the other hand, formulas seem to play a leading role in the way you work.

Ah, yes, absolutely. I can think much better about a formula than about a geometrical object because I never trust that a geometric picture, a drawing, is suffciently generic. I don't really have a geometrical mind. When there is some geometry problem, and I succeed in translating it into algebra, then it's fine. There are several steps, ¯rst the translation, then the purely algebraic thinking. I always try to distinguish between the intuitive side, the geometrical one, and the linguistic one, the algebraic one, in which one manipulates formulas, and I think much better on that side. For me, algebra unfolds in time : I can see a formula live and turn and exist in time, whereas geometry has something instantaneous about it and I have much more di±culty with it. As far as I go, formulas create mental pictures.

...

What is more important for you in your mathematical work : unity or evolution ?

It's diffcult to decide. Every mathematician has a kind of Ariadne's thread which he follows from his starting point, and that he should absolutely try not to break. So, there is a unity, a kind of trajectory, which makes you start from a place, and because you have started there, in a slightly bizarre and special place, you have a certain originality, a certain perspective, different from that of the others. And this is essential, otherwise, you put everybody in the same mould, everybody would have the same reactions to the same questions. This is not what we want, we want different people who have their own approaches, their own methods. So there is a unity in the trajectory, which is not at all the unity of mathematics. The unity of mathematics, you discover bit by bit, when you realize that extremely different trajectories, of extremely different people, get closer to the same vibrant heart of mathematics. But what I have felt above all is the unity, the fdelity to a trajectory.

 

[SoCurious]

The unity of mathematics, you discover bit by bit, when you realize that extremely different trajectories, of extremely different people, get closer to the same vibrant heart of mathematics.

I was wondering if this could be applied to life in general. I guess it can, as long as there is a common goal.

 

[ana]

When I think of the simplest structure of a monad I think of minerals, rocks and plants ie first density beings who have a more limited awareness.

First density beings manifest properties that are distinguishable from our point of view because we are endowed with more properties and so we can actually differentiate and perceive them as we do. If we were unable to distinguish the properties between these levels of monads we would be part of their realm, entering in their composition/structure.

Isn’t it an example of information found in structures?

 

[StrangeCaptain]

To be honest, I have no right to say much on these subjects as I simply am not ready to think meaningfully. However, I would like to remain in the flow of this thread, so I will just say that when asking what the simplest, non-trivial structure of a monad could be, I first decide just to think of a monad as a set of 2 thingies that we will give the symbols 0 and 1. I some some other definitions of a monad after a quick internet search, but they seemed a bit exotic for our current discussion.

In my mind, the structure would come from what actions we want take on the 2 thingies. In other words, how do we choose one of the two thingies? The most accessible, non-trivial, and natural actions would be the binary arithmetic operations. Is there some novel ways of choosing one of the thingies that we want to consider other than these operations?

And is there merit to thinking of the 2 thingies as actions of some kind themselves rather than static elements? In other words, they would not be our fundamental, discrete building blocks but the the 2 fundamental actions we take on whatever our building blocks would be?

 

[Thomas Alan]

I've been following this thread with fascination. Much of it way over my head. (I struggled, with little success, with high school algebra and geometry. I could not see why it was important to learn them.) Still, it is challenging and fun to try to grasp the concepts.

Time does seem to be an illusion to me. Things change, we experience time as movement. Yet there is something in me that has not moved at all, that seems to be the unmoving observer of the movie.

Difficult to explain what I see, even the explanation is in time and space. :)

Mac

 

[Mountain Crown]

The simplest ones. What would be the simplest, but nontrivial structure of a monad?

It seems to me that the concept of space comes to us through objects or abstract geometric figures that outline (define) a demarcation, i.e., a figure showing different spaces – that which is occupied by the shape vs. that which not of the shape.

If we‘re to determine the most fundamental unit of space (monad) then it follows that it would be the simplest form which if reduced or divided would no longer define any space.

At first it seems that this reduction would lead us to consider a point, but isn’t a point dimension-less thereby not satisfying the condition? Unless location is enough to allow demarcation.

Maybe a monad would be the least amount of points (locations) that would allow a defined space?

Just some thoughts.

 

[Approaching Infinity]

If we‘re to determine the most fundamental unit of space (monad) then it follows that it would be the simplest form which if reduced or divided would no longer define any space.

At first it seems that this reduction would lead us to consider a point, but isn’t a point dimension-less thereby not satisfying the condition? Unless location is enough to allow demarcation.

Maybe a monad would be the least amount of points (locations) that would allow a defined space?

That was my first thought, but then again, I am totally out of my league when it comes to these concepts. 0 and 1 did bring to mind matter and anti-matter, however, or particles and anti-particles. Can particles be said to code for information? And if so, is there any 'simpler' structure that can be said to do so?

 

[John G]

So, in short, I 'see' 'the two' and 'sense or feel the third' and I therefore have some way of noticing classicist tomfoolery even though I can hardly avoid it linguistically myself.

Classical isn't bad. Einstein is a great starting point. The idea is to merge classical and quantum. Venusian mentioned the circle/sphere and you can add hypersphere to that. This relates to imaginary numbers and losing properties like commutativity and Feynman-path-like shortcuts through different types of spacetime. Just like Laura's use of binary is different than the ordinary use, so is Ark's.

 

[John G]

If we‘re to determine the most fundamental unit of space (monad) then it follows that it would be the simplest form which if reduced or divided would no longer define any space.

At first it seems that this reduction would lead us to consider a point, but isn’t a point dimension-less thereby not satisfying the condition? Unless location is enough to allow demarcation.

Maybe a monad would be the least amount of points (locations) that would allow a defined space?

That was my first thought, but then again, I am totally out of my league when it comes to these concepts. 0 and 1 did bring to mind matter and anti-matter, however, or particles and anti-particles. Can particles be said to code for information? And if so, is there any 'simpler' structure that can be said to do so?

There are certainly models like Feynman checkerboards with points spaced say the Planck length apart and these vertex points would have Spinor information that specifies matter/antimatter and which quark/electron/neutrino. The spinor information is a matrix and is kind of the square of what you'd think the minimum information would be; the extra information is kind of for quantum purposes I think. Links coming out from a vertex point would have Adjoint information specifying types of force particles (photon,weak boson,gluon, graviton).