Archive for the ‘Mathematics’ Category

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Saturday, January 24th, 2009

No prescription tramadol online legally, In his blog series "A programmer's view of the universe", Google engineer Steve Yegge is exploring the ontological insights gained in his years writing software.  His first post told the sad story of a pet fish who died after discovering the nature and limits of its own captivity, its tank, while looking out through the glass walls to a world it could never reach.  His latest post uses the video game Mario Kart to explore the concept of embedded system, where the player controls drivers who race around a track inside a universe completely contained within the computer, but with the illusion of a landscape that stretches off into the horizon.  Here he introduces the concept of the one-way wall, which the players butt up against, for example, when attempting to drive too far off the track.  The rules of the game keep the players safely within prescribed boundaries; the game software itself can monitor and control all activity within the simulated game world, but the game cannot "see into" the containing world outside its confines.

Yegge is probably fully aware of the well-worn philosophical implications of this line of inquiry; he is, after all, a self-proclaimed fan of Douglas Hofstadter, whose almost mystical ruminations on the nature of recursion and embeddedness in Gödel, Escher, Bach are a metaphysical classic in the computer-geek canon.  Seeing oneself as a creature in an embedded system attempting to perceive the workings of the "outer system" - to look through the one-way-wall the other way - has forever been a preoccupation of intelligent and spiritually-inclined seekers.

Flammarion woodcut


I would submit, contra Baudrillard, that while we denizens of the internet may today experience an unprecedented degree of abstraction, humanity has never inhabited a space other than one populated by self-created simulacra, and that indeed, the simulacrum has always been the very nature of the worldview of every individual who has ever lived.  For the human brain is itself an embedded system, fed information through a one-way wall (our sensory input) and developing a model of the world outside to the best of our abilities and according to our internal propensities.

Human communities are also an embedded system, one in which individuals reside, and they share a pool of common assumptions about the world (myths) and the "rules" by which we participate in it, refining and elaborating on the individual's innate sense of right and wrong.  Much like cells organizing themselves in a human body, or ants in a colony, all of human civilization and culture -- indeed all human achievement -- is built up of thousands of individual decisions within a socially prescribed rule system.  Such self-emergent organization -- whereby progressively more complex, richly varied systems construct themselves from a few basic rules -- is very appealing to computer geeks, who can create such embedded systems on their computers with just a few lines of recursive code, like this 1-dimensional cellular automaton:

1d cellular automata


This ties into a conversation I was having the other day with a colleague who had just finished reading another computer-geek classic, Ray Kurzweil's The Singularity is Near.  Kurzweil's work of Utopian fantasy is popular among geeks, since it promises a future world (not too distant, either: according to his projections, within the next 50 years) in which technology will become sufficiently advanced that computer intelligence will outpace human intelligence.

What happens beyond this point is impossible to say, but according to Kurzweil's theory, machine intelligence will recursively create new machines with rapidly accelerating intelligence, Very Cheap Tramadol, which may (among other things) provide capabilities for human immortality, interstellar space travel, or perhaps even, one day, create nanobots that will transform the nature of matter and reality so that all matter will reach a point of "supersaturation" in which the entire universe becomes a gigantic supercomputer.  This event is called the singularity, also known as "nerd rapture".

To which I replied: What if the singularity has already happened?  How would you know, no prescription tramadol online legally.

Singularity speculation (in its modern form) began in 1965 with J. L. Good, but was eagerly adopted in the post-psychedelic era by early transhumanist and Harvard psychologist Dr. Timothy Leary in his eccentric 1979 book The Game of Life, in which he uses the Tarot, the I Ching, Kundalini Yoga, and various other esoteric traditions as metaphors for a projected evolution of life (and accompanying stages in the development of individual consciousness), beginning with single-celled organisms and progressing through fish, lizards, mammals, apes, primitive humans, early civilizations, monarchies, and monotheism, then stepping into increasingly liberated states of self-awareness, reality fabrication, DNA manipulation, and atomic, gravitational, and quantum manipulation, into the final payoff, Tramadol Caps 50mg, fusion into a black hole, which in turn creates a new universe.  (Paul Davies also alludes to this possibility, that new universes could be spawned by sufficiently advanced technology.)

The key insight that Leary brings to this fantasy is one of cyclical motion: once a new universe is spawned, the cycle begins anew, for new worlds and new civilizations to repeat this journey.  Indeed, for Leary, each stage of evolution goes through a three-stage cycle of birth, maturation, and decay before leading into the next stage.  This reflects the notion of a cyclical universe in Hindu cosmology, which however sees the course of civilization as one destined toward decay and destruction rather than a state of perfection.  In the Hindu view, it is the individual life that re-enters again and again in new forms to attempt to climb the ladder to liberation. No prescription tramadol online legally, In any event, the key ingredient to this timely yet ancient cosmology -- recursion -- leads us to the insight that no matter the ultimate direction of history or the universe, we exist only in a single moment, a point in motion, brought here by events outside our control and to be swept away by forces outside our control, in death and decay.  We act out our parts as best we can, knowing that we are subsumed in the eternal flux of creation and destruction that define all existence.

This problem is present in all cosmologies (including the modern scientific one), and it's a crucial weakness for the monothestic/Judeo-Christian view, which sees an eternally static Creator God "zapping" the world and time into existence exactly as we see it today, then just as abruptly, destroying it all in a raging fire, whisking his faithful off to eternal, static bliss.  I find this view of the universe pitifully unimaginative and lacking a key ingredient: sacred union, from which we derive imbalance, from which comes change, repetition, recursion, and eternity.  The Judeo-Christian god is deeply flawed for lacking (or, more truthfully, denying) a creative feminine force, which drives this change and evolution.

In fact, to reconcile this problem, Kabbalistic thinkers have developed the idea of tzimtzum: that, in order to make room for creation, the omnipresent God had to withdraw from a part of himself, leaving himself imperfect.  Thus even Yahweh seeks a mystical union with his own creation.  In any event, Cheap Tramadol Fedex Overnight, we know from experience that the nature of reality is the paradox of eternal change, and yet in common Christian thought Yahweh immaturely seeks to deny this, asserting the immutability of his own personality and ego.  (Gary Snyder reports a Mahayana Buddhist commenting, "He needs to do more meditation.")

A frequent counter to the idea of an "unmoved mover" god is the question: well, where did God come from, then?  The answer, "I am that I am", is true yet unsatisfying; this is not the petty, adolescent, egotistical, tribal Yahweh-god but his Mother - the primordial Void, the backdrop in which the creator God appears - talking.  Here's Jesus, emerging from Mother-God's vagina, represented as a Mandorla:

Mandorla


That's more like it!  Every cause must have a cause, and this leads us into a state of "turtles-all-the-way-down", or infinite recursion, which may as well be an eternal cycle, represented as the self-devouring serpent ouroboros:

Ouroborous


The ultimate, all-encompassing religious expression of this concept is contained within this modern image of the Tibetan Buddhist Dharmapala (dharma-protector) Mahakala:

Mahakala


Here, the Great Time-Lord Mahakala advances through time in a blaze of activity, through all dimensions, in perpetual combat with himself as he generates adversaries (his own image) and slaughters and devours them, thereby regenerating himself.  His myriad arms, at once swords and wings, cradle in their clawed hands Bodhisattvas (saints and future Buddhas).  From this eternal recursive creation-destruction, all forms are generated (symbolized in the tiger skin, the leopard prints, the octopus tentacles, the various faces... well, What Is Tramadol Used For, keep looking, you'll see).  The two tracks of infinitely-receding adversary heads are joined in an earring which points to the empty heart of the beast, the Void.  Yet in his hands, amid the fires and the creation and the death and destruction, he protects in a delicate (vaginal/Mandorla-like) clamshell the central Mystery of the sacred union, Yabyum: the eternal, dark Buddha making love to his own ephemeral feminine nature, enraptured in the embrace of the beloved.

In the distance stand other Buddhas, Dharmapalas, and various demons: the Dharmapala replicated, again, across yet other dimensions (other universes, perhaps?).

The lesson that the Dharmapala brings to us is this: wherever you are, whatever you are doing, all things are in flux.  There is always change, there is no Utopian state of perfection, for creation and destruction are intertwined.  Fighting through the one-way wall, stepping out of the matrix, you find yourself in another matrix, and you cannot escape.  Concentrate therefore on the center, this fusion of opposites, from which all things flow, and you will inhabit at once all levels of existence; the nested shells of all matrices outward and inward come into focus, and you see them line up in eternal recursion in all dimensions for all time.  This is Moksha, liberation.

In the meantime, humanity will continue to attempt to construct a utopia out of machines.  It's what we do, worker bees that we are. Futile though it is, it's in our nature.  It is clear that while we may construct bigger and more fantastic fishtanks for ourselves, Ultram Overdose, we cannot escape fishtank existence.  But perhaps we don't really want to.

No matter.  Onward.  Upward.  Inward.  Outward.  Always.

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Sunday, January 18th, 2009

No prescription tramadol online, If a religion is defined as a system of thought which requires belief in unprovable truths, then mathematics is the only religion that can prove it is a religion!

via Paul Davies, The Mind of God, Penguin Books, 1992.

If mathematics describes the universe, it must describe a universe which contains mathematics, The Drug Tramadol, and therefore, Non Prescription Tramadol, itself. Self-reference in mathematics is not merely esoteric, it's fundamental, Ultram Medicine, and nowadays quite commonly understood. Purchase tramadol online cheap, In fact, self-reference of formal systems is ubiquitous in computer science: one can write a program in a language X, that can compile and execute other programs in language X, Companies That Make Tramadol, including itself.  Recursion on all kinds of scales appears everywhere in computer science; languages such as LISP are built around it.

Look again at the Penrose diagram, it seems a reasonable approach to the problem of fundamentalism is to apply a scientific (evolutionary) viewpoint to its existence.  Instead of addressing our species' reluctance to evolve with outrage, let's understand and accept it as a part of the same natural system that brought us here in the first place.  Let's turn the system on itself.

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Sunday, November 30th, 2008

Gustav Doré’s “Paradisio Canto 31″

[Dante's] Paradisio Canto 31 On sale tramadol without prescription, (a view of God, the source of Light) by Gustav Doré.

E8

A 2-dimensional projection of an 8-dimensional representation of E8, an 248-dimensional mathematical structure of "extreme symmetry" proposed by physicist Garrett Lisi's Exceptionally Simple Theory of Everything as describing all fundamental interactions in physics.  See Lisi's TED talk.

A fascinating discussion is occurring in the theoretical physics realm in the pursuit of a "theory of everything".  The main rival of Lisi's theory, Getting High On Tramadol, string theory, is also very much in theoretical stages, International Online Tramadol, and is considerably more baroque in its explanations, theoretical implications, and mathematics.  Among the heated partisan debates within the physics community, I admire Lisi's stance: "I think scientists should be able to weigh the pros and cons of different theoretical models for themselves and follow what interests them, Medication Tramadol, without pressure in one direction or another... I welcome criticism and skepticism, Tramadol Stay In System, and I encourage people to look through the mathematics for themselves."  He seems to embrace the possibility that he may be proven wrong, which is the healthy way to practice science.  And even if he is wrong, we're still left with the beautiful E8 Lie group, which existed before Lisi's theory and exhibits some extraordinary properties in its own right, Avesto Online Tramadol.

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Sunday, April 6th, 2008

While New Atheist Ordering tramadol no prescription, views (espoused by Dawkins, Daniel Dennett, Sam Harris, et al.) tend to focus on the field of evolutionary biology, the sphere of physicists and mathematicians seems to have a much more nuanced view of the relationship between science and metaphysics. In particular, mathematics stands distinct from science per se in that it does not make specific verifiable claims of empirical fact (much of mathematics does not directly relate to the physical world at all); rather, it provides a formal structure, a framework, by which we can communicate and generalize abstract notions. In some sense, these decontextualized abstract notions themselves can be said to have an existence independent of physical manifestation and our ability to perceive them. These three ideas - abstract (Platonic/mathematical) ideal, imperfect manifestation in physical reality, and the human mind's perception of pattern in the physical world and transference of physical phenomena into abstract ideals - lie at the foundation not only of mathematics, but true religion as well.

In the first chapter of his excellent The Road to Reality, Sir Roger Penrose introduces a diagram illustrating this threefold relationship:

The Penrose Trinity

Plato made it clear that the mathematical propositions -- the things that could be regarded as unassailably true -- referred not to actual physical objects (like the approximate squares, triangles, circles, spheres, Tramadol Line, and cubes that might be constructed from markers in the sand, or from wood or stone) but to certain idealized entities. He envisaged that these ideal entities inhabited a different world, distinct from the physical world, ordering tramadol no prescription. Today, we might refer to this world as the Platonic world of mathematical forms....

This was an extraordinary idea for its time, and it has turned out to be a very powerful one.... It tells us to be careful to distinguish the precise mathematical entities from the approximations that we see around us in the world of physical things. Moreover, it provides us with the blueprint according to which modern science has proceeded ever since. Ordering tramadol no prescription, Scientists will put forward models of the world -- or, rather, of certain aspects of the world -- and these models may be tested against previous observation and against the results of carefully designed experiment....

If the model itself is to be assigned any kind of "existence", then this existence is located within the Platonic world of mathematical forms. Of course, one might take a contrary viewpoint: namely that the model is itself to have existence only within our various minds, rather than to take Plato's world to be in any sense absolute and "real". Yet, there is something important to be gained in regarding mathematical structures as having a reality of their own. For our individual minds are notoriously imprecise, unreliable, Tramadol 120, and inconsistent in their judgments....

Thus, mathematical existence is different not only from physical existence but also from an existence that is assigned by our mental perceptions. Yet there is a deep and mysterious connection with each of those other two forms of existence: the physical and the mental, ordering tramadol no prescription. [In the figure above] I have schematically indicated all of these three forms of existence -- the physical, the mental, and the Platonic mathematical -- as entities belonging to three separate "worlds", drawn schematically as spheres. The mysterious connections between the worlds are also indicated, where in drawing the diagram I have imposed upon the reader some of my beliefs, or prejudices, concerning these mysteries.

It may be noted, with regard to the first of these mysteries -- relating the Platonic mathematical world to the physical world -- that I am allowing that only a small part of the world of mathematics need have relevance to the workings of the physical world.... Likewise, in relation to the second mystery, whereby mentality comes about in association with certain physical structures (most specifically, healthy, Online tramadol without prescription, wakeful human brains), I am not insisting that the majority of physical structures need induce mentality.... Ordering tramadol no prescription, Finally, for the third mystery, I regard it as self-evident that only a small fraction of our mental activity need be concerned with absolute mathematical truth. ... These three facts are represented in the smallness of the base of the connection of each world with the next, the world being taken in a clockwise sense in the diagram. However, it is in the encompassing of each entire world within the scope of its connection with the world preceding it that I am revealing my prejudices....

On this view, everything in the physical universe is indeed governed in completely precise detail by mathematical principles....


In William Blake's Newton, we see this very same threefold mystery: Here, the man emerges from the earth and produces a representation of the geometry (γεωμετρία; geo = earth, metria = measure) of which the earth is made.


Blake’s Newton


Newton (1795) by William Blake


Likewise, we can easily see the Platonic significance of the Christian trinity as well.

Medieval trinity illumination (origin unknown)


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Here, the Holy Trinity of Christianity is presented as a triangle: Father (Pater, which might stand for "earth"), Son (Filius, Smoke Tramadol, "man"), and Holy Spirit (Spiritus Sanctus, "platonic ideal") in an triangular relationship, three equal and essential parts of God. The connections between them ("non est") assert the independence of the parts, while their connections to the center ("est") assert their importance in the integral whole. We might say also that the human conception of mathematical truth is not the truth itself; that manifestation in the physical world is not the Platonic form; and that the physical world consists of much more than human minds. But the relationship remains the same: timeless, illustrated through myriad imperfect forms, this knowledge has been transmitted from ancient (pre-scientific) times to us today, and we continue to find new manifestations of the Trinity. There's a lot more to be said here; this touches upon the significance of mythology, sacred geometry, and mysticism: these have always been vehicles by which humans could codify and transmit eternal truths. We may use science to perceive the details, but we use religious/mythological and mathematical abstraction to incorporate, Paypal tramadol online, generalize, and interpret this knowledge.


In this way, true religious/spiritual knowledge does not properly occupy the space of empirical scientific knowledge, but rather serves as a counterpart to abstract mathematical knowledge, existing the the Platonic (meta) realm, rather than making specific claims of empirical truth in the physical realm.

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