Archive for the ‘Space and Time’ Category

Noether-Pauli-Jung, V2

April 10, 2019

What happens when the fourfold of Noether’s Theorem is spliced together with the fourfold of Pauli-Jung? Both have Space-Time and Matter-Energy. The former has Conservation and Symmetry, and the latter has Causality and Synchronicity. And if Space-Time and Matter-Energy are both divided into Space and Time and Matter and Energy, one obtains this eight-fold.

Causality means that some action or cause in time (say a process) of things in space can have an effect (another process, say) on different things in space, and Synchronicity means that different events (say processes) separated in space can have non-causal relationships between them.  Conservation means the consistency of a quantity of matter or energy or matter-energy through time, and Symmetry means the consistency of a measure of a structure or form through space.

I am reminded of my fourfold Four Bindings, consisting of Chains, Grids, Blocks, and Cycles. Causality and Synchronicity are Chains (or non-chains for the latter) Space and Time are Grids (or flexible meshes), Matter and Energy are Blocks (or chunks of stuff), and Symmetry or Conservation are Cycles (of the group-theoretic kind or the equivalence class kind or just loops).

Further Reading:

https://en.wikipedia.org/wiki/Causality

https://en.wikipedia.org/wiki/Synchronicity

https://en.wikipedia.org/wiki/Symmetry

https://equivalentexchange.wordpress.com/2012/05/04/noethers-theorem/

https://equivalentexchange.wordpress.com/2018/01/23/atom-and-archetype/

https://equivalentexchange.blog/2016/04/06/four-bindings/

This is a reworking of a previous six-fold diagram that I believe is served better as an eight-fold.

https://equivalentexchange.blog/2018/03/29/noether-pauli-jung/

[*10.68, *10.155, *11.55]

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Four Forms Make a Universe

March 12, 2019

How could I not love a paper with this title? I’ve struggled with it for a bit, and I’ve only managed a couple of diagrams relating the author’s LICO (Linear Iconic) alphabet made up of 16 letters. However, I see that there are a few other papers by Schmeikal available on ResearchGate that look easier to understand. But also however, the first one says to read the “Four Forms” paper first!

At any rate, I present a sixteen-fold of the LICO alphabet, and another of the binary Boolean operators that are in a one-to-one mapping with LICO. There is much to understand from these papers, including much syncretism between various mathematical sixteen-folds, so please forgive me if I don’t explain it all with immediate ease. However, I believe it is well worth the effort to understand.

(Please note that the characters of the LICO alphabet are oriented so that the bottoms of the letters are downward, but the Boolean operators are oriented so that the bottoms of the equations are towards the right angles of the triangles.)

The title comes from the result that four elements of LICO can reproduce the other twelve via linear combinations. These four forms are 1) Boolean True (A or ~A), 2) A, 3) B, and 4) A=B. These are within the interior right-hand triangles in the LICO diagram. Of course, it is well known from Computer Science that the NAND operator (~A or ~B) can also generate all other fifteen operators, but this is by multiple nested operations instead of simple Boolean arithmetic. There are several other “universal” binary gates that can do this as well.

Two other representations that have four elements that can generate the other twelve via linear combinations come from CL(3,1), the Minkowski algebra. These representations are called “Idempotents” and  “Colorspace vectors”. Because of this algebra’s association with space and time in relativity, Schmeikal claims that LICO has ramifications in many far-ranging conceptualizations.

Further Reading:

Bernd Schmeikal / Four Forms Make a Universe, in Advances in Applied Clifford Algebras (2015), Springer Basel (DOI 10.1007/s00006-015-0551-z)

https://link.springer.com/article/10.1007/s00006-015-0551-z

At http://www.researchgate.net:

Bernd Schmeikal / Free Linear Iconic Calculus – AlgLog Part 1: Adjunction, Disconfirmation and Multiplication Tables

Bernd Schmeikal / LICO a Reflexive Domain in Space-time (AlgLog Part 3)

https://en.wikipedia.org/wiki/Spacetime_algebra

https://www.allaboutcircuits.com/technical-articles/universal-logic-gates/

[*9.145, *11.50]

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An Eightfold Metaphysics

February 1, 2019

If one juxtaposes the fourfold Space-Time-Matter-Energy with the fourfold Structure-Function-Part-Action, one sees that there are several relations between them. These are simple common-sense relations, not modern-physics types of relations. Below they are listed by Space-type relations and Time-type relations.

Space is required for the extension of Structures.
Matter constitutes Structures.
Parts in an arrangement make up Structures.

Structures extend and are organized in Space.
Matter is located in Space.
Parts occupy Space.

Time allows the expression of Functions.
Energy is required for the operation of Functions.
Actions in a sequence constitute Functions.

Functions have duration and reoccur in Time.
Energy requires and is dependent on Time.
Actions take place within Time.

Several analogies are also evident in this diagram.

  1. Space : Structures :: Time : Functions
  2. Space : Parts :: Time : Actions
  3. Space : Matter :: Time : Energy
  4. Structures : Parts :: Functions : Actions
  5. Structures : Matter :: Functions : Energy
  6. Parts : Matter :: Actions : Energy

Notice that analogies 1.-3. are “contains” relations, and 4.-6. are “part of” relations. One obtains the nesting of entities:

Space > Structures > Parts > Matter
Time > Functions > Actions > Energy

I suppose that any analogy could be shown in a figure like the one on the right (or even written as A / B // C / D) or with the two squares side by side, and that sets of four or six analogies that overlap could be nicely shown as above. If so, I wonder what can be gained by such representations? Probably individually not so much but with overlaps I think it would be interesting.

Further Reading:

The study of parts and wholes relations is called Mereology.

https://en.wikipedia.org/wiki/Mereology

https://plato.stanford.edu/entries/mereology/

[*10.157, *11.26, *11.34]

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The Seasons and the Zodiac

August 22, 2018

This isn’t a bad little diagram of the four seasons along with the twelve zodiac names and symbols. However, it might be oriented wrong by convention or going clockwise instead of counter-clockwise.  Interestingly, old horoscope charts that show what was in the sky (the positions of the zodiac stars and the eight or nine planets in regards to the twelve “astrological houses”) at the time of a person’s birth were shown using the outside ring of twelve triangles instead of the more familiar circle that is used today. The inner square might be for notes or some nice drawing.

Further Reading:

https://en.wikipedia.org/wiki/Zodiac

https://www.completehoroscope.org/envelope-diagram-horoscope.htm

https://equivalentexchange.blog/2014/10/31/the-four-seasons/

[*10.145]

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Fourfold Physicalism

February 17, 2018

It is not enough for a wise man to study nature and truth, he should dare state truth for the benefit of the few who are willing and able to think. As for the rest, who are voluntarily slaves of prejudice, they can no more attain truth, than frogs can fly.

— From Man a Machine, by Julien Offray de La Mettrie

Further Reading:

https://equivalentexchange.wordpress.com/2012/07/03/structure-function/

https://equivalentexchange.wordpress.com/2013/10/09/things-happen/

https://equivalentexchange.wordpress.com/2014/02/01/relations-all-the-way-down/

https://equivalentexchange.wordpress.com/2015/06/09/four-primary-relations/

https://en.wikipedia.org/wiki/Man_a_Machine

Notes:

Structures are built from parts.
Parts are reductions of structures.
Functions are assembled from actions.
Actions are the constituents of functions.

[*8.132, *9.104, *10.10]

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Atom and Archetype

January 23, 2018

A few weeks ago I ran across this nice review of the book Atom and Archetype: the Pauli-Jung letters 1932-1958. This is a collection of letters exchanged between psychiatrist Carl Jung and physicist Wolfgang Pauli over a course of years. Evidently, Pauli was quite the metaphysician and Jung was intrigued by Einstein’s physics of relative space and time. Together in dialectic they argued and struggled to join together the disparate notions of mind and matter.

What mainly caught my eye was a diagram that I’ve slightly altered and shown above. I’ve mainly just replaced energy with matter-energy for two reasons: first because matter and energy are inter-convertible and second because matter conditions space. This results in similarity to the fourfold diagram for Lucretius that I’ve shown before, consisting of Particles, the Void, Falling, and Swerving.

Further Reading:

https://www.brainpickings.org/2017/03/09/atom-and-archetype-pauli-jung/

Carl Jung and Wolfgang Pauli / Atom and Archetype: the Pauli-Jung letters 1932-1958, Princeton University Press; Updated edition (July 21, 2014)

https://www.amazon.com/Atom-Archetype-Pauli-Letters-1932-1958/dp/069116147X/ref=sr_1_1?s=books&ie=UTF8&qid=1516537804&sr=1-1&keywords=atom+and+archetype

https://equivalentexchange.wordpress.com/2011/10/10/lucretius-on-the-nature-of-things/

https://equivalentexchange.wordpress.com/2014/02/17/matter-energy-space-and-time/

https://equivalentexchange.wordpress.com/2015/01/16/wave-particle-duality/

https://equivalentexchange.wordpress.com/2012/05/04/noethers-theorem/

[*10.60]

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A Digital Universe, V2

August 9, 2017

A digital universe – whether 5 kilobytes or the entire Internet – consists of two species of bits: differences in space, and differences in time. Digital computers translate between these two forms of information – structure and sequence – according to definite rules. Bits that are embodied as structure (varying in space, invariant across time) we perceive as memory, and bits that are embodied as sequence (varying in time, invariant across space) we perceive as code. Gates are the intersections where bits span both worlds at the moments of transition from one instant to the next.

— George Dyson, from Turing’s Cathedral

Further Reading:

George Dyson / Turing’s Cathedral: the origins of the digital universe

Notes:

Embodied as Structure, Perceived as Memory

Invariant across Time: ¬ΔT
Varying in Space: ΔS

Embodied as Sequence, Perceived as Code

Varying in Time: ΔT
Invariant across Space: ¬ΔS

https://equivalentexchange.wordpress.com/2012/04/12/a-digital-universe/

[*7.82, *7.83, *7.153, *10.14]

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The Paradoxes of Zeno

April 21, 2016

sq_zeno_paradoxes5“Suppose,” said Zeno, “that Achilles and a tortoise are planning to race”.

Such is the beginning of a famous thought experiment by an ancient philosopher. Since athletic Achilles was much faster than the slow tortoise, he let the tortoise start first. But alas, he could never catch up to it, since every time Achilles made it to where the tortoise had been, the tortoise had moved just a little further ahead. Of course Achilles was faster so he had to pass the tortoise quickly unless it had started near the finish line. So, paradox!

Most of the paradoxes of Zeno were about fractions and entireties of time and space. Can an infinite series of fractions of space add up to a finite entirety of space in a finite entirety of time? Some might say that integral calculus solves these basically mathematical problems, yet others think they point to metaphysical issues as regards to the discreteness and the continuity of time and space.

This fourfold reminds me of my previous fourfold Spacetime which dealt with succession (as parts of time), location (as parts of space), extension (as wholes of space), and duration (as wholes of time). It must have been in the back of my mind.

References:

https://en.wikipedia.org/wiki/Zeno%27s_paradoxes

https://equivalentexchange.wordpress.com/2011/09/23/spacetime/

[*9.122]

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Relative Time

November 14, 2014

sq_relative_time For what more terrifying revelation can there be than that it is the present moment? That we survive the shock at all is only possible because the past shelters us on one side and the future on another.

— From Orlando, by Virgina Woolf

As we can see on the previous four-folds of space and time, all have a degree of conditioning to the location and orientation of an observer. In other words, there are no absolute frameworks of space or time. That does not mean that they are not useful conventional and conceptual tools.

What would a four-fold of Relative Time be like? Because time seems to be linear instead of two dimensional, relative time would be very different than relative directions. What if we contrast our understanding of what happened in time with what actually occurred? What if we compare our thoughts of an imagined future with what becomes realized?

One could contrast an individual’s notions of past and future with a group or society’s notions of past and future. Or one could contrast an individual’s or society’s recalled past and imagined future with the actual past and the realized future. Some might argue that there is no actual past, but only the past we think or recall that it is. Similarly, those or others might argue that there is no realized future, because once the future becomes the present it has already slipped into the past that we can now only recall.

As the future becomes realized, the imagined future is discarded or blended into it to become our recalled past. As we understand more about the real past, our recalled past may be discarded or blended into it to become our new recalled past. Or one can refuse that knowledge and believe whatever suits them.

[*8.99, *9.60]

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Four Dimensional Space-time

September 1, 2014

sq_4d_spacetime

Here’s a simple fourfold I’ve been ignoring just because it’s so trivial, but that triviality can be deceiving. Space-time as formulated in special relativity has four dimensions: three of space and one of time. Our everyday experience shows us the three dimensions of space: length, width (or breadth), and depth (or height), but time is a different kind of thing because we cannot see or move forward and backward through time with our eyes or body, like we can along the axes of space.

Personally, only our memory and imagination can let us range through time. Of course, after the invention of language and more recent technologies, the spoken word, writings, photographs, audio recordings, and videos can also be used. But it’s not the same as shifting one’s gaze along the length of something or moving one’s body across a width.

So, we can move semi-freely through the three spatial dimensions but our movement in time seems to be fixed into a relentless forward motion that we have no control over. And because gravity pulls us down onto the surface of the world, one of the spatial dimensions (depth or height) is more limiting than the other two.

sq_ll2Thus another interesting comparison to this fourfold is to that of linear logic. One observation is that length and width can be considered reversible but depth and time can be considered somewhat irreversible. That’s not true of course, but because of gravity it is easier to descend than to ascend, and it’s far easier to move into the future than into the past. But we can see into the distant past, just not our own, as we turn our telescopes to the heavens.

Space without time could have four or even higher dimensions, but we have no empirical evidence that it is so. Mathematically, however, we can easily construct multidimensional spaces. One representation of four dimensional space is by using quaternions, which have four dimensions to the complex numbers’ two. Tuples of real numbers or even vector spaces can also be used. However, the geometry of space-time is not Euclidean; it is described by the Minkowski metric.

Novels about characters living in different numbers of spatial dimensions are an interesting way to learn and think about them. The very first was Flatland by Edwin Abbott Abbott, about a being limited to two dimensions that learns about a third outside his experience when a three dimensional being comes to visit. Just recently I’ve finished reading Spaceland by Rudy Rucker, about an ordinary human person limited to the three dimensions of space that learns about the fourth dimension by similar reasons.

Links:

http://en.wikipedia.org/wiki/Special_relativity

http://en.wikipedia.org/wiki/Minkowski_space

http://en.wikipedia.org/wiki/Four-dimensional_space

http://en.wikipedia.org/wiki/Flatland

http://en.wikipedia.org/wiki/Flatland_%282007_film%29

http://en.wikipedia.org/wiki/Spaceland_%28novel%29

[*8.72]

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