Archive for the ‘Space and Time’ Category

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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Matter, Energy, Space, and Time

February 17, 2014

sq_MESTHere’s a nice little fourfold that I’ve overlooked until now, kind of like a four-leaf clover. If you search for images of matter, energy, space, and time, several websites show a diagram quite similar to this, and some have interesting things to say.

One shows free energy instead of energy in general. Free energy is energy that is able to do work, and that is an important distinction, especially if you are thinking about the Four Causes. Then the four can be considered fundamental resources.

There are some other things to consider from these references. One is that there is the familiar equivalence between matter and energy. Another is the equivalence principle between gravity and acceleration.

Also, there are obvious analogies between this fourfold and the Four Elements and the Four States of Matter. sq_Noether's TheoremI have also used the pairs space-time and matter-energy as two parts of the fourfold Noether’s Theorem, the other two parts being symmetry and conservation. There is also the fourfold Spacetime that has some similarities to this one. And let’s not forget the atomism of On the Nature of Things by Lucretius.

Be sure and watch the upcoming television series Cosmos: a Spacetime Odyssey, hosted by Neil deGrasse Tyson. I’m sure there will be much talk of Matter, Energy, Space, and Time.

Notes:

There is even a term coined for Matter-Energy-Space-Time: MEST. Of course, MEST also stands for Math, Engineering, Science, and Technology! No, sorry, that’s STEM!

http://lornaalkana.com/portfolio/graphic-design/ticket-space-time-energy-matter/

Links:

http://abyss.uoregon.edu/~js/ast123/lectures/lec09.html

http://www.zamandayolculuk.com/cetinbal/htmldosya1/ModernScience.htm

http://billmoyers.com/2014/01/10/the-new-cosmos-a-spacetime-odyssey/

[*7.90, *8.32]

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Human Limitations

July 10, 2013

birth_death_nature_nurture Everything existing in the universe is the fruit of chance and necessity.

Democritus

What are humans limited by? We are limited by time and by space.

We are limited in time because we are mortal beings, bound to the past by our birth and to the future by our death, delimiting the short interval of our lifetimes.

We are limited in space because we are finite beings. At each point of time we are constrained by what we are and what we might become. We are bound both by our actuality and our possibility. Our nature defines our actuality and our nurture sets our possibility.

Certainly, we can transcend our limitations, but this just means they weren’t really the true limits after all. We can’t say for certain what our real limits are, so that’s a good thing.

This fourfold also reminds me of the following modal square of opposition. At birth, much is possible. At death, all becomes impossible. Nurture is contingent and Nature is necessary.

modal_squareReferences:

http://plato.stanford.edu/entries/logic-deontic/

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

[*7.174]

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The Duality of Time and Information, V2

May 2, 2013

The states of a computing system bear information and change time, while its events bear time and change information.

from The Duality of Time and Information by Vaughn Pratt

The most promising transformational logic seems to us to be Girard’s linear logic.

— from Rational Mechanics and Natural Mathematics by Vaughn Pratt

Here we have three duals: information – time, state – event, and bear – change.

References:

Vaughan Pratt / The Duality of Time and Information http://boole.stanford.edu/pub/dti.pdf

Vaughan Pratt / Time and Information in Sequential and Concurrent Computation http://boole.stanford.edu/pub/tppp.pdf

Vaughan Pratt / Rational Mechanics and Natural Mathematics http://chu.stanford.edu/guide.html#ratmech

[*5.170]

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The Space-Time Tetrahedron

March 29, 2013

spacetime_tetrahedronIn honor of April Fool’s Day, I present the Space-Time Tetrahedron.

The Internet is full of crackpot and nutty websites, and one of the most famous concerns the so-called Time Cube.

Since the ideas expressed on this website are pretty much ignored, it is hoped that it could at least find some recognition as being slightly crackpot or somewhat nutty.

Let’s examine the similarities between Time Cube theory and my theory.

The number four is very important in each. In fact, the Time Cube is a fourfold, although quite a puzzling one.

The notions of space and time are crucial to Time Cube; many of my fourfolds involve some aspect of space and time or spacetime.

How about the differences? I don’t think one can find any common claims between the two theories. At least I hope not!

References:

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

http://knowyourmeme.com/memes/the-time-cube

[*7.150]

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Equivalent Exchange

February 8, 2013

structure-functionThere is an old riddle that asks, what is the value of a human body? It goes on to enumerate the quantities of the various chemicals that constitute an average body, and then price those chemicals to come up with a total material value. This might come as a shock to the one who tried to answer the riddle, because the value is so low.

Of course that analysis ignores the real value of the body because it ignores the spatial arrangement of those chemicals in the various tissues and organs of the body. It ignores the form of the body as an hierarchical arrangement of structured parts. These parts are very valuable for someone needing an organ donation, for example.

This static analysis is lacking as well because it neglects the dynamic actions that a body can perform. Taken as a set of discrete individual acts, this is like considering the individual chemicals that constitute a body. One can enumerate the acts too, as in counting the number of breaths or the number of heart beats over a lifetime.

But we are still not finished, because this enumeration of actions also overlooks the functions that a body can perform, as arrangements of actions in time. These functions are the most important value of a body because they include but are not limited to being alive, thinking, and feeling.

So it seems to me that there is a heirarchy here, that goes from parts to structure to actions to function, with parts being at the lowest level and function being at the highest. Each level of the hierarchy is dependent on the level below, but the nature of a level is fundamentally different from the level below it, as well as the level above it. Please see my previous post Structure-Function.

The term “equivalent exchange” comes from an alchemy-centric anime, where through a process called “human transmutation” it is attempted to recreate a body starting from a pile of the basic chemicals that constitute it. Even with some high-powered magic, the attempt fails due to the fact that much more is required to fashion a body or even a living, breathing person.

http://chemistry.about.com/b/2011/02/06/how-much-are-the-elements-in-your-body-worth.htm

http://www.datagenetics.com/blog/april12011/index.html

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Structure-Function

July 3, 2012

Aristotle’s Four Causes is an important fourfold that seems to be the basis for many of the fourfolds, both original and not, presented in this blog. Two of the causes, efficient and material, are acceptable to modern scientific inquiry because they can be thought of as motion and matter, respectively, but the other two causes, formal and final, are not. Why is that?

The formal cause is problematic because the formal is usually considered to be an abstract concept, a construction of universals that may only exist in the human mind. The final cause is also problematic because it is associated with the concept of telos or purpose. There, too, only human or cognitive agents are allowed to have goals or ends. So for two causes, efficient and material, all things may participate in them, but for the two remaining, formal and final, only agents with minds may.

These problems may be due to the pervasive influence of what the recent philosophical movement of Object Oriented Philosophy calls correlationism: ontology or the existence of things is limited to human knowledge of them, or epistemology. The Four Causes as usually described becomes restricted to the human creation and purpose of things. Heidegger’s Tool Analysis or Fourfold, which also appears to have been derived from the Four Causes, is usually explained in terms of the human use of human made things: bridges, hammers, pitchers. Even scientific knowledge is claimed to be just human knowledge, because only humans participate in the making of this knowledge as well as its usage.

Graham Harman, one of the founders of Speculative Realism of which his Object Oriented Ontology is a result, has transformed Heidegger’s Fourfold so that it operates for all things, and so the correlationism that restricts ontology to human knowledge becomes a relationism that informs the ontology for all things. Instead of this limiting our knowledge even more, it is surprising what can be said about the relations between all things when every thing’s access is as limited as human access. However, this transformation is into the realm of the phenomenological, which is not easily accessible to rational inquiry.

I wish to update the Four Causes, and claim that they can be recast into a completely naturalistic fourfold operating for all things. This new version was inspired by the Four Operators of Linear Logic. Structure and function are commonplace terms in scientific discourse, and I wish to replace formal and final causes with them. It may be argued that what is obtained can no longer be properly called the Four Causes, and that may indeed be correct.

First, let us rename the efficient cause to be action, but not simply a motion that something can perform. I’m not concerned at the moment with whether the action is intentional or random, but it must not be wholly deterministic. Thus there are at least two alternatives to an action. I’m also not determining whether one alternative is better than the other, so there is no normative judgement. An action is such that something could have done something differently in the same situation. This is usually called external choice in Linear Logic (although it makes more sense to me to call it internal choice: please see silly link below).

Second, let us call the material cause part, but not simply a piece of something. Instead of the material or substance that something is composed of, let us first consider the parts that constitute it. However, a part is not merely a piece that can be removed. A part is such that something different could be substituted for it in the same structure, but not by one’s choice. Like an action, I am not concerned whether one of the alternatives is better than the other, but only that the thing is still the thing regardless of the alternative. This is usually called internal choice.

Next, we will relabel the formal cause to be structure, but not simply the structure of the thing under consideration. Ordinarily structure is not a mere list of parts, or a set of parts, or even a sum or integral of parts, but an ordered assembly of parts that shapes a form. Ideally structure is an arrangement of parts in space. However, in this conceptualization, structure will be only an unordered list of parts with duplications allowed.

Last, instead of final cause we will say function, but not simply the function of the thing as determined by humans. Ordinarily function is not a mere list of actions, or a set of actions, or a sum or integral of actions, but an ordered aggregate of actions that enables a functionality. Ideally function is an arrangement of actions in time. However, like structure, function will be only an unordered list of actions with duplications allowed.

As we transform the Four Causes from made things to all things, both natural and human-made, we will later examine how that changes them.

References:

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

http://en.wikipedia.org/wiki/Object-oriented_ontology

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

http://wiki.cmukgb.org/index.php/Internal_and_External_Choice

[*6.144, *7.32, *7.97]

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Noether’s Theorem

May 4, 2012

Nature is thrifty in all its actions.

    — Pierre Louis Maupertuis

From Wikipedia:

Noether’s (first) theorem states that any differentiable symmetry of the action of a physical system has a corresponding conservation law. The theorem was proved by German mathematician Emmy Noether in 1915 and published in 1918. The action of a physical system is the integral over time of a Lagrangian function (which may or may not be an integral over space of a Lagrangian density function), from which the system’s behavior can be determined by the principle of least action.

Noether’s theorem can be stated informally:

If a system has a continuous symmetry property, then there are corresponding quantities whose values are conserved in time.

Note:

Symmetries are transformations or exchanges in space or time that leave systems structurally or functionally equivalent to what they were before. The equivalence may or may not be an identity, but only the same in appearance or behavior.

Conservation laws are equivalences for quantitative properties of systems. A given property of matter or energy is quantitatively the same before and after, or continuously through space or time. The functional measure of this property remains constant.

So consider an analogy between Noether’s Theorem and the concept of Equivalent Exchange: for (symmetrical, differentiable) exchanges, there are properties that are equivalent (conserved)!

http://en.wikipedia.org/wiki/Noether’s_theorem

http://en.wikipedia.org/wiki/Action_%28physics%29

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

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

http://math.ucr.edu/home/baez/noether.html

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