The Infinity Book by Eric Gorvin

∞ T H E

I N F I N I T Y

B O O K

This book is dedicated to my family as well as the great makers and thinkers who constantly inspire, enrich, and support me through everything that I do. Thank you. Eric Carranza, Andrés Guzmán, Tomás Villaseñor, Patricia Healy McMeans, & Erik Brandt

Da Haunted Attic (Studio). Powderhorn, Minneapolis, MN U.S.A.

S. MPLS, MN

The Infinity Book by Eric Gorvin 2013

1st Edition

U.S.A.

Introduction M o tiv e In my late MCAD education, I took a class called Linear Perspectives with Katharine Kindervater as my professor. Towards the end of the semester, she assigned readings from Brian Rotman's Ad Infinitum and my fixation with infinity began. From then on, I began compiling leagues of readings that I could relate to infinity in its many areas of study. This project is the compilation of a good majority of my research and experiments. As an artist, I find it very important to convert ideas into visual and sonic work to aid in new ways of understanding our world, ourselves, and others. Nothing is worse than perfectly good knowledge that has no vehicle for the world to really understand it. Though I am not a scientist, nor a mathematician, nor an astronomer, I believe that the facts I have found in these fields of study have the potential to elucidate new ideas and new ways of thinking about infinity and our universal relationship to it. So it follows that many shortcomings encountered in our everyday lives have something to do with our inability to see outside of ourselves. We develop a strong relationship with our ego, our singularity(1), and it becomes harder and harder to leave it as the years pass. Itâ&#x20AC;&#x2122;s this relationship that hypnotizes us into feeling that we are the only things in this Universe, or, the only things to which we should devote thought. My argument is quite the contrary. I believe that the inherent truths embedded in the many observables we have at the other end of our senses are a working display of the true nature of our Universe. The only thing preventing us from absorbing these many attributes of our world and then being able to interpret them into usable ontological ideas, is the fact that we are finite and cannot get over it. This overwhelming feeling of being finite causes us to become increasingly self-centered and in turn creates an unwillingness to fully acknowledge our true size, or consider our interpretation of time. Every day each person on earth knows that they are going to die â&#x20AC;&#x201C; that they are finite. It is this fact that drives each and every one of us to get absorbed in the finite and thus forget about the infinite counterpart that makes all of our material world possible. It has been my intention with this book to adjust your focus from the finite world and liberate you into a much more exciting infinite realm. We get easily locked up in ourselves, in finitude, enslaving us to a finite Earth created by man...not one that is the beautiful product of any kind of connected Universe. Learning just a few things about the evidence of infinity will help develop an ability to see both the infinite and the finite in perfect harmony. I hope that you enjoy this book. Sometimes all it takes is being able to see it all at once.

In f inity In Ev e ry Day Drawing relationships between bits of data to create a whole vision is very important in our quest for the infinite. The relationships we make here decide the way we interpret things and understand them. With that being said, it is important to keep an open mind towards everything, allowing for all these connections to become realized ideas. In this way, we can start to see the relationships that are happening around us at all times. This includes observables that are not only in our every day encounters, intents and feelings, but also in nature and the cosmos. It is my intention to have these priorities at the forefront of my discussion. I hope that all data and visual information presented can help in aiding you in your own understanding of infinity.

Expla na tio n This project was born of my own interest in infinity. As you will notice, among other excerpts found in this book, I have included a significant amount of quotations and excerpts from Brian Rotmanâ&#x20AC;&#x2122;s Ad Infinitum. This is the book that initially interested me in this conquest. I have always been perplexed and perhaps somewhat haunted by the obvious and evident balance to things. From the macro to the micro, there are many observations that can be made about the relationships of the different elements of our Universe. This book is not intended to pontificate, nor is it intended that s/he who handles it should take it for a finite set of facts. I have been as accurate as I can in my explorations and experiments, and have gone through all that is possible for me to make the information and facts contained legible and accurate. The amount of information that is available on just one of the many areas covered in this book is vast to say the least. I like to think of this as an overview and an experiment. All parts of the book, from the readings to the illustrations, come from my quest to understand infinity â&#x20AC;&#x201D; my personal exploration hybrid with formal research. The way that this book navigates is through a conduit of language, symbology, and imagery. It is loosely arranged to allow the ability to connect the dots for yourself with the given material. I have done my best in creating a breadth of work to reflect my ever growing interest in the concepts of infinity as well as including the words of extraordinary thinkers from the fringe of recorded history to the present. My only hope is to get you thinking about infinity. With the same credence you give darkness to lightness, hot to cold, I hope that you can apply the same notions of duality to infinite and finite. If we can understand that our own finite quality is infinite in magnitude, we will be able to move toward a new era of positive growth, peace, and enlightenment.

How It Works Bo o k This book includes excerpts from an array of different authors, thinkers, philosophers, mathematicians, etc... All sources are cited using the footnotes I have included on each page.

Text typed in this font is previously published work and should not be taken as my own. Text typed in this font is either myself speaking or cited information. I've covered many different subjects in this book, I understand that some of the information found here is not directly related to infinity to some people. But please try something; let it be related. As I just said on the previous page, these various findings should be thought of in light of infinity to help bring clarity and sense to our views of the Universe. There is also a wide array of imagery. I've authored some of this imagery and tried my best to note and cite an image if it was from an outside source.

Video I performed a number of video "experiments" for this project. Half of their volume consisted of exploring the act of counting in some respect. The other half were related to water, because water was a driving force for me in my meditations on infinity as well as a convenient visual aid for it in our everyday. The playback format of the videos is in a loop structure that echos that of a sine wave. The model of the sine wave (pg. 50) was something that permeated the entire project and generated a lot of my interest in investigating infinity in a sonic realm as well.

Sound Accompanying the video experiments, I explored the audio/music side of this project with my good friend and collaborator, Eric Carranza. The conversations we had at that time (in 2012) heavily informed the sound we created. Having these conversations with Eric provided another human being to convey some of these complex ideas to, and one who would understand and expand with me. Having already spent the better part of our twenties making music together, the process came very naturally to us once I realized that it absolutely must be a component of the project. We ended up completing an eight-song record titled "Through All The Windows, I Only See Infinity". It is meant to be listened to while experiencing the project (this is available on the Internet via Soundcloud or Bandcamp).

On Infinity

On Number

On Water

On Sound

On Physics

On Mathematics

On Nature

On Paradox

111

On Time

129

On Space

FIGURE 1.1 Gorvin, Eric. "Moleskine Entry". The Infinity Project. 2012.

On Infinity

FIGURE 1.2 Gorvin, Eric. "Potential vs. Actual Infinity. The Infinity Project. 2012.

Ad Infinitum1 A B r ie f His t or y & Ov erv ie w Of In f inity A century ago the mathematician Leopold Kronecker

notably about infinitesimal magnitudes, that emerged within

declared that “God made the integers, the rest is the work

these theories. The attempt to eliminate these produced a

of Man.” This much cited remark was intended as a call for

movement toward rigor at the beginning of the nineteenth

constructivist rigor, a polemic against the contemporary

century associated with Carl Friedrich Gauss and Augustin

embrace of infinitism, a Pythagorean reaffirmation of the

Cauchy which repudiated the use or mention of an actual or

originating and privileged status of the whole numbers.

completed infinity in any acceptable mathematical context.

And so it has been interpreted. It was not, one feels, meant

In less than a century the repudiation (despite Kronecker’s

with any theological literalness. And yet . . . is there not

Pythagorean campaign for constructivist renewal) was

in the very idea of their endlessness, their continuation ad

swept aside with the acceptance of Richard Dedekind’s

infinitum, something strange and other about the whole

description of the continuum and Georg Cantor’s theoriza-

numbers, the imprint or trace of some disembodied, tran-

tion of the actual infinite—the set of all integers, all subsets

scendental maker, perhaps?

of integers, all fractions, all points on a line, and so on—as

bona fide mathematical objects.

Potential/Actual The history of mathematics and the history of “the endless”

The third crisis, whose “resolution” forms the horizon

intertwine. Issues of the infinite, of the nonfinishing, that

of present day thinking about the mathematical infinite,

which cannot be limited, bound, or traversed, arise as soon

emerged from the upheaval in the logic and ontology of

as mathematics takes quantity and length as its constituting

number and the paradoxes that occurred in relation to

abstractions and is obliged to make self-reflective sense of

Cantor’s theory of infinite sets. Within this horizon two

the questions “how many numbers?” and “how long, how

opposed accounts of mathematics emerged: the dominat-

divisible a line?”

ing orthodoxy of a Platonism inspired by Gottlob Frege and Joan Brouwer’s anti-Platonic constructivism.

There have been various moments of conflict and dislocation in the history of mathematics when the question of

For most mathematicians (and, one can add, most scien-

the infinite— what it means and how one is to think it—has

tists) mathematics is a Platonic science, the study of time-

pushed itself to the foreground of mathematical discourse.

less entities, pure forms that are somehow or other simply

Each such moment arose in the wake of a speculative mobi-

“out there,” preexistent objects independent of human

lization of infinitary reasoning which had resulted in obscu-

volition or of any conceivable human activity; mathemati-

rity and absurdity intolerable to mathematical thought.

cians discover but never in any sense invent their presence and their properties. Accordingly the integers exist—out

The earliest crisis, associated with Zeno’s infamous paradox-

there—in their entirety as a single set, an unproblematic

es centered on whether one could assume space and time

completed infinity; likewise the set of all subsets of the inte-

were endlessly divisible or whether, on the contrary, they

gers, the set of all subsets of these, and so on, ad infinitum.

were composed of ultimate and indivisible quanta. Zeno,

Mathematics is thus seen to be organized as an infinite hier-

attempting to establish the unreality of motion and plurality

archy of infinite sets within a rigorously axiomatic framework

in the name of Parmenides’ changeless and indivisible One,

by means of which all doubts, circularities, and antinomies

forced each of the possible alternatives into a contradiction.

have been swept away.

The effect of his arguments on classical thought was twofold: an avoidance of any appeal to motion or to the “endless”

Against this there is the constructivist dissent. Brouwer

with in their reasoning on the part of Greek mathematicians;

disputed the coherence of classical, Platonic logic—reject-

and Aristotle’s distinction, arising out of his engagement

ing the law of excluded middle that dictates that for any

with Zeno’s paralogisms and fundamental in all subsequent

mathematical object, 0, either 0 exists or 0 doesn’t exist—

discussions, between a safe and legitimate potential infinite,

by insisting that any mathematical proof of the existence of

an endless coming into being, and a dangerous, paradox-

an object had to be in the form of instructions for arriving

infested completed or actual infinite.

at it—that is, a finitely specifiable procedure that could “in principle” be executed in the mind. The progression of inte-

During the sixteenth and seventeenth centuries Aristotle’s

gers for Brouwer was a potential and not an actual infinity,

interdiction of actual infinity was set aside in the develop-

available in principle as an endlessly producible sequence

ment of the theory of infinite series and the infinitesimal

of purely mental acts, constructions to be performed deep

calculus. The second major problematic of the mathemati-

inside our Kantian-intuition of time.

cal infinite then arose out of inconsistencies and absurdities, 1 Rotman, Brian. “Abstract.” Ad Infinitum - The Ghost in Turing’s Machine. Stanford, CA: Stanford UP, 1993. 3+. Print 2 See chapter “On Number” pg. 25 3 See chapter “On Paradox” pg. 99

∞4 His t or y of t h e ∞ sy mbo lo gy

Infinity Signifiers

John Wallis is credited with introducing the infinity symbol, ∞ (Figure 1.5), in 1655 in his De sectionibus conicis. One conjecture about why he chose this symbol is that he derived it from a Roman numeral for 1000 that was in turn derived from the Etruscan numeral for 1000, and was sometimes used to mean "many." Another conjecture is that he derived it from the Greek letter ω (omega), the last letter in the Greek alphabet. The infinity symbol is also sometimes depicted as a special variation of the ancient ouroboros snake symbol (Figure 1.8). The snake is twisted into the horizontal eight configuration, or circle, while engaged in eating its own tail, a uniquely suitable symbol for endlessness.

FIGURE 1.3

FIGURE 1.4

Self-Reference5 FIGURE 1.5

FIGURE 1.6

De finitio n Self-reference occurs in natural or formal languages when a sentence, idea or formula refers to itself (Figure 1.7). The reference may be expressed either directly—through some intermediate sentence or formula—or by means of some encoding. In philosophy, it also refers to the ability of a subject to speak of or refer to himself, herself, or itself: to have the kind of thought expressed by the first person pronoun, the word “I” in English. Self-reference is studied and has applications in mathematics, philosophy, computer programming, and linguistics. Self-referential statements are sometimes paradoxical.

FIGURE 1.7

4 "Infinity Symbol." Wikipedia. Wikimedia Foundation, 13 Mar. 2012. Web. 20 Apr. 2012. 5 ”Self-reference.” Wikipedia. Wikimedia Foundation, 20 Mar. 2012. Web. 27 Mar. 2012. FIGURE 1.3 – Used in mathematics as well as language (e.g. 3.33... or etc...) FIGURE 1.4 – Found in mathematics, where "x" is any number and the dash over the integer signifies its infinite repeat. FIGURE 1.5 – Universal infinity symbol, see ∞ - History at top of page FIGURE 1.7 — The Treachery Of Images (1928-29) by René Magritte depicts a pipe along with text stating “This is not a pipe.”

Ouroboros6 The Serpent The Ouroboros (or Uroborus) (Figure 1.8) is an ancient symbol depicting a serpent or dragon eating its own tail. The name originates from within Greek language; οὐρά (oura) meaning “tail” and βόρος (boros) meaning “eating”, thus “he who eats the tail”. The Ouroboros represents the perpetual cyclic renewal of life and infinity, the concept of eternity and the eternal return, and represents the cycle of life, death and rebirth, leading to immortality, as in the phoenix.

In Egypt

FIGURE 1.8

The first known appearance of the ouroboros motif is in the Enigmatic Book of the Netherworld, an ancient Egyptian funerary text in the tomb of Tutankhamun, in the 14th century BC. The text concerns the actions of the god Ra and his union with Osiris in the underworld. In an illustration from this text, two serpents, holding their tails in their mouths, coiled around the head and feet of an enormous god, who may represent the unified Ra-Osiris. Both serpents are manifestations of the deity Mehen, who in other funerary texts protects Ra in his underworld journey. The whole divine figure represents the beginning and the end of time.

In Greec e Plato described a self-eating, circular being as the first living thing in the universe—an immortal, mythologically constructed beast7 (pg. 23).

In India Ouroboros symbolism has been used to describe Kundalini energy. According to the 2nd century Yoga Kundalini Upanishad, “The divine power, Kundalini, shines like the stem of a young lotus; like a snake, coiled round upon herself she holds her tail in her mouth and lies resting half asleep as the base of the body”. Another interpretation is that Kundalini equates to the entwined serpents of the Caduceus, the entwined serpents representing commerce in the west or, esoterically, human DNA (Figure 1.9).

FIGURE 2.1 FIGURE 1.9

6 ”Ouroboros.” Wikipedia. Wikimedia Foundation, 28 Mar. 2012. Web. 20 Feb. 2012. 7 Plato, Timaeus FIGURE 1.8, 1.9 Gorvin, Eric. "DNA Strand". The Infinity Project. 2012.

Infinity in Context8 Th e His t or y o f Inf in ity As there is no record of earlier civilizations regarding, conceptualizing, or discussing infinity, we will begin the story of infinity with the ancient Greeks. Originally the word apeiron meant unbounded, infinite, indefinite, or undefined. It was a negative, even pejorative word. For the Greeks, the original chaos out of which the world was formed was apeiron. Aristotle thought being infinite was a privation not perfection. It was the absence of limit. Pythagoreans had no traffic with infinity. Everything in their world was number. Indeed, the Pythagoreans associated good and evil with finite and infinite. Yet, to the Greeks, the concept of infinity was forced upon them from the physical world by three traditional observations:

That time appears to have no end is not too curious. Perhaps, owing to the non-observability of worldending events as in our temporal world of life and death, this seems to be the way the universe is. The second, the apparent conceivability of unending subdivisions of both space and time, introduces the ideas of the infinitesimal and the infinite process. In this spirit, the circle can be viewed as the result of a limit of inscribed regular polygons with increasing numbers of sides. These two have had a lasting impact, requiring the notion of infinity to be clarified. Zeno, of course, formulated his paradoxes by mixing finite reasoning with infinite and limiting processes. The third was possibly not an issue with the Greeks as they believed that the universe was bounded. Curiously, the prospect of time having no beginning did not perplex the Greeks, nor other cultures to this time. With theorems such that the number of primes is without bound and thus the need for numbers of indefinite magnitude, the Greeks were faced with the prospect of infinity. Aristotle avoided the actuality of infinity by defining a minimal infinity, just enough to allow these theorems, while not introducing a whole new number that is, as we will see, fraught with difficulties. This definition of potential, not actual, infinity worked and satis-

fied mathematicians and philosophers for two millennia. So, the integers are potentially infinite because we can always add one to get a larger number, but the infinite set (of numbers) as such does not exist. Aristotle argues that most magnitudes cannot be even potentially infinite because by adding successive magnitudes it is possible to exceed the bounds of the universe. But the universe is potentially infinite in that it can be repeatedly subdivided. Time is potentially infinite in both ways. Reflecting the Greek thinking, Aristotle says the infinite is imperfect, unfinished and unthinkable, and that is about the end of the Greek contributions. In geometry, Aristotle admits that points are on lines but points do not comprise the line and the continuous cannot be made of the discrete. Correspondingly, the definitions in Euclidâ&#x20AC;&#x2122;s The Elements reflect the less than clear image of these basic concepts. In Book I the definitions of point and line are given thusly:

The attempts were consistent with other Greek definitions of primitive concepts, particularly when involving the infinitesimal and the infinite (e.g. the continuum). The Greek inability to assimilate infinity beyond the potential-counting infinity had a deep and limiting impact on their mathematics. Nonetheless, infinity, which is needed in some guise, can be avoided by inventive wording. In Euclidâ&#x20AC;&#x2122;s The Elements, the very definition of a point, a point is that which has no part, invokes ideas of the infinite divisibility of space. In another situation, Euclid avoids the infinite in defining a line by saying it can be extended as far as necessary. The parallel lines axiom requires lines to be extended indefinitely, as well. The proof of the relation between the area of a circle and its diameter is a limiting process in the clock of a finite argument via the method of exhaustion. Archimedes proved other results that today would be better proved using calculus.

8 Allen, Donald G. "The History of Infinity." Diss. Texas A&M University, 1999. The History of Infinity. Texas A&M University. Web.

These theorems were proved using the method of exhaustion, which in turn is based on the notion of â&#x20AC;&#x153;same ratioâ&#x20AC;?, as formulated by Eudoxus. We say:

This definition requires an infinity of tests to validate the equality of the two ratios, though it is never mentioned explicitly. With this definition it becomes possible to prove the Method of Exhaustion. It is:

The Greeks were reluctant to use the incommensurables to any great degree. One of the last of the great Greek mathematicians, Diophantus, developed a new field of mathematics being that of solving algebraic equations for integer or rational solutions. This attempt could be considered in some way a denial of the true and incommensurable nature of the solutions of such equations.

FIGURE 1.10 Gorvin, Eric. "Untitled Study". The Infinity Project. 2013.

FIGURE 1.11 Gorvin, Eric. "The Universe Without Legs Or Feet". The Infinity Project. 2012.

â&#x20AC;&#x153;The living being had no need of eyes when there was nothing remaining outside him to be seen; nor of ears when there was nothing to be heard; and there was no surrounding atmosphere to be breathed; nor would there have been any use of organs by the help of which he might receive his food or get rid of what he had already digested, since there was nothing which went from him or came into him: for there was nothing beside him. Of design he was created thus, his own waste providing his own food, and all that he did or suffered taking place in and by himself. For the Creator conceived that a being which was self-sufficient would be far more excellent than one which lacked anything; and, as he had no need to take anything or defend himself against any one, the Creator did not think it necessary to bestow upon him hands: nor had he any need of feet, nor of the whole apparatus of walking; but the movement suited to his spherical form was assigned to him, being of all the seven that which is most appropriate to mind and intelligence; and he was made to move in the same manner and on the same spot, within his own limits revolving in a circle. All the other six motions were taken away from him, and he was made not to partake of their deviations. And as this circular movement required no feet, the universe was created without legs and without feet.â&#x20AC;? â&#x20AC;&#x201D;Plato, Timaeus 23

FIGURE 2.1

FIGURE 2.1 Bergamini, David. Mathematics. New York: Time, 1963. Print.

On Number

â&#x20AC;&#x153;The Pythagoreans spoke of two causes in the same way, but added, as an idiosyncratic feature, that the limited and the unlimited and the one were not separate natures, on a par with fire or earth or something, but the unlimited itself and the one itself were taken to be the substance of the things of which they are predicated. This is why they said that number was the substance of everything.â&#x20AC;?â&#x20AC;&#x201D;Aristotle, Metaphysics

Ad Infinitum9 Th e Na t u r al N umbers Now, as far as the purely logical engendering of the hierarchy of infinities is concerned, Kronecker’s Pythagorean point is well put: evidently all infinities, potential as well as actual, whatever differences in psychology, logical coherence, meaning, and ontology constructivism and Platonism might attach to them, are rooted in the integers, the progression of objects that mathematicians, Platonists, and constructivists alike, call the natural numbers. “Natural” because they are given at the outset, taken for granted as a founding, unanalyzable intuition, outside any critique that might demand an account of how they come or came—potentially or actually—to “be.” Indeed, for Platonism the very call for such critique would be senseless, whilst for constructivism the issue is essentially dodged—sold short by constructivism’s immersion in an unexaminedly ideal mentalism. Evidently, if we are to understand the numbers’ relation to God, it is precisely their “endlessness,” as natural and given or as the result of an ideal construction, that needs to be interrogated. But faced with Platonist orthodoxy’s inability to question the integers’ engenderment and the complicity of its constructivist critics, how are we to escape from the sedimented legitimacy and beguiling immediacy of the “natural”? How refuse the claim that numbers are elemental, mathematical or constituents of the actual or potential order of things? How deny that producing or perceiving the endless progression of them is inextricable from the apparatus of rational thought itself? Where do numbers come from? If not from Kant’s transcendental intuition or Kronecker’s God, then where? What seems universally accepted is that numbers are inconceivable — practically, experientially, conceptually, semiotically, historically—in the absence of counting... Counting, whether with fingers, pebbles, notches, tally marks, abacus beads, or notations on chalkboards, paper, and computer screens, is an activity involving signs. And, as an activity, counting works through—it is—significant repetition. How are we–through what discursive apparatus and technology of symbolic persuasion–to imagine a business of repeating the selfsame signifying act without end, of iterating for ever? Or, which will come to the same, what would it mean to deny the possibility of endlessly repeating a signifying act? Is it in fact possible to coherently imagine an activity of iterating that did not—by definition of the very abstracted purity of the repetition that furthers it—go on forever?

9 Rotman, Brian. “Abstract.” Ad Infinitum - the Ghost in Turing’s Machine. Stanford, CA: Stanford UP, 1993. 3+. Print. FIGURE 2.2 Gorvin, Eric. The Infinity Project. 2013.

The Pythagoreans10 A L ife Of N umber At the same time [as Leucippus ami Democritus] and earlier than them were the so-called Pythagoreans, who were interested in mathematics. They were the first to make mathematics prominent, and because this discipline constituted their education they thought that its principles were the principles of all things. Now, in the nature of things, numbers are the primary mathematic principals; they also imagined that they could perceive in numbers many analogues to things that are and that come into being (more analogues than fire and earth and water reveal) -such-and-such an attribute of numbers being justice, such-and-such an attribute being soul and mind, due season another, and so on for pretty well everything else; moreover, they saw that the attributes and ratios of harmonies depend on numbers. Since, then, the whole natural world seemed basically to be an analogue of numbers, and numbers seemed to be the primary facet of the natural world, they concluded that the elements of numbers are the elements of all things, and that the whole universe is harmony and number. They collected together all the properties of numbers and harmonies which were arguably conformable to the attributes and parts of the universe, and to its organization as a whole, and fitted them into place; and the existence of any gaps only made them long for the whole thing to form a connected system. Here is an example of what I mean: ten was, to their way of thinking, a perfect number, and one which encompassed the nature of numbers in general, and they said that there were ten bodies moving through the heavens; but since there are only nine visible heavenly bodies, they came up with a tenth, the counter-earth... They hold that the elements of number are the even and the odd, of which the even is unlimited and the odd limited; one is formed from both even and odd, since it is both even and odd; number is formed from one and, as I have said, numbers constitute the whole universe. Other members of the same school say that there are ten principles, which they arrange in co-ordinate pairs: limit and unlimited; odd and even; unity and multiplicity; right and left; male and female; still and moving; straight and bent; light and darkness; good and bad; square and oblong.11 …

10 11 12 13

Waterfield, Aristotle, Aristotle, Bergamini,

The Pythagoreans spoke of two causes in the same way, but added, as an idiosyncratic feature, that the limited and the unlimited and the one were not separate natures, on a par with fire or earth or something, but the unlimited itself and the one itself were taken to be the substance of the things of which they are predicated. This is why they said that number was the substance of everything.11 … The Pythagoreans, as a result of observing that many properties of numbers exist in perceptible bodies, came up with the idea that existing things are numbers, but not separate numbers: they said that existing things consist of numbers. Why? Because the properties of numbers exist in musical harmony, in the heavens, and in many other cases.11 … The Pythagoreans also claim that there is such a thing as a void. According to them, it enters the universe from the infinite breath because the universe breathes in void as well as breath. What void does, they say, is differentiate things; they think of void as being a kind of separation and distinction when one thing comes after another. This happens first among the numbers, because on their view it is the void that distinguishes one number from another.12

Computing13 Fro m Hu man Fingers to Man-made Brains Counting is an intricate process; of all the earth's creatures only man can do it. Early humans probably formed numbers with their fingers, as some primitive peoples still do. As society became more complicated, man had to make fairly elaborate calculations involving subtraction, multiplication and division, and the devices he used to assist him grew more advanced. By the time of the ancient Greeks, mechanical calculators were in use—and in the 2,000 years since then an array of increasingly sophisticated computing machines has been developed. The culmination was the electronic computer, that wonderful "brain" that can do difficult mathematical problems in a split second and is slowly changing our very civilization. There may indeed be some kernel of truth in the exuberant claim of the great mathematician-philosopher Auguste Comte: "There is no inquiry which is not finally reducible to a question of numbers."

Robin. The First Philosophers: The Presocratics and Sophists. Oxford: Oxford UP, 2000. Print. Metaphysics. Physics. David. Mathematics. New York: Time, 1963. Print.

On Water

The Power In Water14 Th e U n p re d ict a b le Wate r M o le cule If water, the most common substance on earth, suddenly began to behave as its molecular make-up suggests, life would be overwhelmed by a series of unparalleled disasters. Blood would boil in the body, plants and trees would either and die, and the world would be transformed into an arid waste. But water molecules are bound together in ways unlike those of any other compound; for this reason they possess properties that are unique and paradoxical. For example, water is one of the very few substances that are heavier as liquids than as solids. As a liquid, it can creep uphill despite the force of gravity. Water is so benign that immensely diversified forms of life can thrive within it—and so corrosive that, given sufficient time, it will disintegrate the toughest metal. Although it seems to change its form with miraculous ease—sometimes existing simultaneously as a solid, a liquid and a gas around the same river or lake—water actually must yield or absorb prodigious amounts of energy to produce these transformations. In fact, the energy it would take to melt even a small iceberg could drive a large ship across the Atlantic 100 times.

Wa t e r In Three Fo rms Water appears in all three of its physical states as a hot stream of liquid sculpts a jagged hole in a block of ice. Some of the water molecules immediately disperse to form an invisible gas, then quickly cool and condense into tiny water droplets that make up the cloud of mist rising above the ice block . Whenever water takes the form of ice, some liquid and gas are always present.

FIGURE 3.2

14 Leopold, Luna B., and Kenneth Sydney Davis. Water. New York: Time, 1966. Print. FIGURE 3.2, 3.4, 3.5 Leopold, Luna B., and Kenneth Sydney Davis. Water. New York: Time, 1966. Print. FIGURE 3.3 Gorvin, Eric. "Three Forms". The Infinity Project. 2013.

An Iro nclad Molec ular Bond Hydrogen and oxygen have so great an affinity for one another that, given even the slightest nudge, they come together violently, forming water and releasing great quantities of energy. In 1937 the huge dirigible Hindenburg exploded over Lakehurst, New Jersey, when its hydrogen, ignited by a spark, fused with the oxygen in the air; amid the explosive release of energy, water was produced.

FIGURE 3.4

Conversely, it takes a great deal of energy to split water into its components. In fact, in ancient times water was considered a basic, indestructible element of the universe. Not until Henry Cavendish startled the scientific community in 1783 by synthesizing the water molecule did it become clear that the substance is actually a compound made up of one part oxygen and two parts hydrogen. The reason water was long thought to be a single element was that the sturdy water molecule remains intact even when frozen solid or heated to temperatures at which many other compounds disintegrate. For the atoms of the water molecule are laced together by powerful bonds, which can be severed only by the most aggressive agentsâ&#x20AC;&#x201D;such as electrical energy or certain chemicals. One such chemical is potassium; when even a small lump of potassium is dropped into water, it pulls the molecules apart so violently that the container of water may actually explode.

Tensio n at the Rim of a Water Tap The powerful tension that hydrogen bonds, create on a water surface can be seen most clearly at a dripping tap. The horizontal film of water that first appears at the tap's opening acts as if it were a circular piece of very thin transparent rubber. Like an elastic membrane, it slowly bulges as the weight of water it encloses grows greater. But it does not break. Instead, it seems at last to tear itself away from the rim of the tap and to snap around a freely falling drop which, if it were not distorted by air pressure, would be a perfect sphere. Of all possible shapes the sphere is the one having the smallest surface per unit volume. It is the shape in which the falling drop can most tightly, closely pull itself together.

FIGURE 3.5

There, in the homely shape of a falling drop, are demonstrated the molecular forces that give water its peculiar propertiesâ&#x20AC;&#x201D;those rare qualities that make it the one substance most important to the affairs of this planet.

Entry M editatio n s On Wa te r For the whole duration of my studies, the observation of water has played a steady and important part. As our source of life it has undeniable clout as an element. A stream's seamless endlessness was among my first realizations of our very human formed view of infinity â&#x20AC;&#x201D; a truly potential infinity. In fact, it drove me to the depths of thought regarding if anything could be infinite when all things we know of have an end. In observing a waterfall, river, or creek, you accept the endless state of it, hence its serene effect. However, geology and geography have shown that no matter if it is a river, creek, waterfall, or ocean, there was a time when it did not exist the way it does now. Some bodies of water dry up, and some might expand into massive oceans. But what are we to do? We cannot simply observe this river or creek forever. So, according to the rest of our logic, is it safe to say that we accept it as an infinite system for we cannot conceive its end? Does that make anything we cannot conceive the end or limit to infinite? Furthermore, I admire and respect the interesting relationships between water and the rest of the Universe. It's celestial nature makes me think of stars and planets differently, and their relationship to us. It's unusual strength as a chemical bond is oddly in line with it's priority in our lives. It makes us up and some say it to even have the ability to absorb energy from humans, creating crystals that echo the output and quality of energy15. The implications are endless. I think it has a lot to offer us beyond the obvious.

The next four pages are meditations on water out of my Moleskine while I was visiting Two Harbors, MN in the spring of 2011. This is when my ideas as water being a metaphor for, not only a potential but actual, infinity were conceived. I think that when we pull water out of our context with it, and examine what we feel when we see it, and why we are feeling that...it becomes very interesting. Is it because it is a significant part of us (50-60% of our whole body on average, 70% of our brain) that we gain an intense connection to it? Or is it the hypnotizing nature that represents something of an endless object to us? Is it because we need it to survive, so as an organism we have a longing and endless affinity for it? I am interested in the connections able to be developed here.

15 Emoto, Masaru. The Hidden Messages in Water. Hillsboro, Or.: Beyond Words Pub., 2004. Print.

FIGURE 4.2-4.5 Gorvin, Eric. "Moleskine Entries In Two Harbors". The Infinity Project. 2011.

“If the doors of perception were cleansed, everything would appear to man as it is, infinite.”—Robert Irwin, Seeing Is Forgetting The Name Of The Thing One Sees

FIGURE 4.6 Gorvin, Eric. The Infinity Project. 2012.

FIGURE 4.7 NASA Apollo 17. The Blue Marble. December 7, 1972.

On Sound

FIGURE 4.1 Stevens, S. S., and Fred Warshofsky. Sound and Hearing. [Amsterdam]: Time-Life International (Nederland), 1965. Print.

“The eyes are made for astronomy, and by the same token the ears are presumably made for the type of movement that constitutes music.”—Plato, Republic

On Sound The Sine Wa v e It’s important to note, at this point, how integral sound was to my studies and experiments on infinity. I have always found certain attributes of sound to be extra spectacular. It seems to be one of the only things that can consistently bring people of all sorts together for seemingly no tangible reason. And, by that fact alone, I believe it deserves further dissection. One does not need to wander far into the world of sound before being confronted by one of its most basic attributes, the sine wave. The sine wave is a very simple self-reflective wave that can be combined with others to create complex sounds, or chords (read more on opposite page). The sine wave was very inspirational for me in developing the various video experiments I created for Through All The Windows I Only See Infinity, (2012). It seemed to me that this very model — a display of the necessary characteristic of sound to create music — was also a fantastic model of infinity. The process of how a sound even comes into being is a function that necessitates a symbiotic relationship of air compressions that result in the creation of a sound. But in all forms of it, the sound absolutely needs a return sequence in order for it to be audible. This fact within the origin of sound leaves room for questions and ideas to emerge concerning music, it’s attributes, and our universal connection to it.

FIGURE 4.2 Gorvin, Eric. The Infinity Project. 2012.

Sound Experience S ou n d & Pre ssu re 16 The experience we call sound results from our detection of pressure changes in a medium such as air. In a diagram of a pressure wave, peaks represent moments of relatively high pressure or compression of the air molecules. Valleys represent low pressure or partial vacuum, which is called rarefaction. A vibrating string in a piano produces alternating zones of compression and rarefaction in the air, resulting in sound. The distance from one peak to the next is a single cycle. A single cycle per second is called one Hertz (abbreviated Hz) after the German physicist Heinrich Hertz.

A S u m Of Sine Wa v es 17 The role of sine waves as the building blocks of complex sounds was first made clear in 1801 by a brilliant French mathematician, Jean Baptiste Fourier, who was not even studying sound at the time. Fourier was investigating the way heat flows through an object. This led him to a powerful mathematical technique—Fourier analysis—which reduces any series of waves, no matter how complex, to a series of simple sine waves. The sum of the sine waves equals the original complex wave.

Fourier’s resolution of sounds into their constituent sine waves moved a giant step farther. The sine waves that make up a musical note turn out to bear a most simple relationship to one another. Each is an overtone, or harmonic, of the fundamental note—i.e., the lowest note—and the frequency of each harmonic is a multiple of the fundamental frequency. Bowing the A string of a violin, for example, generates not only the fundamental A note of 440 vibrations per second—but also the second harmonic, one octave higher, of 880 vibrations, the third harmonic of 1,320 vibrations, and so on. The connection between harmonic frequencies and the length of a vibrating string can be seen by shaking a rope tied to a tree. Shake the rope slowly and a single wave runs along the rope. That single wave represents the fundamental frequency of vibration—the first harmonic. Then shake the rope twice as rapidly and two short waves will travel to the tree. With the frequency of vibration now twice as great, the length of each wave is halved. Shaking the rope three times as fast, at triple the fundamental frequency, will generate the third harmonic three distinct waves, each one-third the length of the fundamental (Figure 4.3).

Fourier’s great achievement lay in relating the complexity of wave motion-any wave, whether of radio, heat, light, sound or water—to a mathematical idea as old as ancient Greece. More than 2,000 years before Fourier’s time, the Greek mathematician Pythagoras had discovered a simple numerical relationship in the sounds of music. He pointed out that the lengths of the plucked strings whose vibrations gave the notes of the scale can be expressed as ratios of whole numbers. Thus if one string sounds the note C, another string 16/15 as long will sound the next lower note, B, one 18/15 as long will sound the A below that, one 20/15 as long will sound G, and so on down the scale. A string twice as long as the original will sound C again, but one octave lower.

FIGURE 4.3

FIGURE 4.4

16 Dewey, Russ. “Sound and the Auditory Waveform | in Chapter 04: Senses | from Psychology: An Introduction by Russ Dewey.” Table of Contents for Psychology: An Introduction by Russ Dewey. 2007. Web. 21 Apr. 2012. 17 Stevens, S. S., and Fred Warshofsky. Sound and Hearing. [Amsterdam]: Time-Life International (Nederland), 1965. Print. FIGURE 4.3-4.4 Stevens, S. S., and Fred Warshofsky. Sound and Hearing. [Amsterdam]: Time-Life International (Nederland), 1965. Print.

Waveform as Color18 Th e E le ct ro Magn e tic S pe ctrum The electromagnetic spectrum is the totality of light emitted from the sun (Figure 4.5). It consists of different wavelengths of electromagnetic radiation, including light, radio waves, and X-rays. We name regions of the spectrum rather arbitrarily, but the names give us a general sense of the energy of the radiation; for example, ultraviolet light has shorter wavelengths than radio light. However, the only region in the entire electromagnetic spectrum that our eyes are sensitive to is the visible region. Gamma rays have the shortest wavelengths, < 0.01 nanometres (about the size of an atomic nucleus). This is the highest frequency and most energetic region of the electromagnetic spectrum. Gamma rays can result from nuclear reactions and from processes taking place in objects such as pulsars, quasars, and black holes. X-rays range in wavelength from 0.01 – 10 nm (about the size of an atom). They are generated, for example, by super-heated gas from exploding stars and quasars, where temperatures are near one million to ten million degrees Fahrenheit. Ultraviolet radiation has wavelengths of 10 – 310 nm (about the size of a virus). Young, hot stars produce a lot of ultraviolet light and bathe interstellar space with this energetic light. Visible light covers the range of wavelengths from 400 – 700 nm (from the size of a molecule to a protozoan). Our sun emits the most of its radiation in the visible range, which our eyes perceive as the colors of the rainbow. Our eyes are sensitive only to this small portion of the electromagnetic spectrum.

FIGURE 4.5

Infrared wavelengths span from 710 nm – 1 millimeter (from the width of a pinpoint to the size of small plant seeds). At a temperature of 37 degrees Celsius, our bodies give off infrared wavelengths with a peak intensity near 900 nm. Radio waves are longer than 1 mm. Since these are the longest waves, they have the lowest energy and are associated with the lowest temperatures. Radio wavelengths are found everywhere: in the background radiation of the universe, in interstellar clouds, and in the cool remnants of supernovae explosions, to name a few. Radio stations use radio wavelengths of electromagnetic radiation to send signals that our radios then translate into sound. Radio stations transmit electromagnetic radiation, not sound. The radio station encodes a pattern on the electromagnetic radiation it transmits, and then our radios receive the electromagnetic radiation, decode the pattern and translate the pattern into sound.

18 "The Electromagnetic Spectrum." HubbleSite. N.p., n.d. Web. 03 Feb. 2013. FIGURE 4.5 Gorvin, Eric. "Electromagnetic Spectrum / The Spectrum of Light". The Infinity Project. 2013.

The Spectrum of Light

10-6 nm

Gamma Rays

1 nm

X Rays

Violet Ultraviolet Radiation

1 µm

Blue Green Yellow

Infrared Radiation

400 nm

Visible Light (Entire Color Spectrum)

Orange Red

700 nm

1 mm 1 cm

Microwaves

UNITS

Radio Waves

nm = µm = mm = cm = m= km =

Nanometer Micrometer Millimeter Centimeter Meter Kilometer

1 km

100 km

FIGURE 4.6 Gorvin, Eric. The Infinity Project. 2012.

Entry This day was very productive. It is a Friday so the most of my day was spent doing research and exploration for the infinity project. I had a very successful day. I showed Erik what I had been making with my infinitely divisible circle and the pencil being whittled video and he was very excited about the initial power of the pencil video. I hope to produce more work like this. After the success with the pencil experiment (this concept <><><>...) I went to my music library to find my recent acquisitions: BBC sound libraries. I recently ripped in a handful of them just to have on my computer, and among them are sounds from Africa as well as a whole mess of nice field recordings of thunderstorms and rain. So I grabbed a few samples and dropped them into my computer to edit. I was wondering what the effects would be on performing the same experiment with sound. I started with the thunderstorm tracks and was instantly enthused. The thunder, rain, and all other natural features of the recording sounded almost identical backwards as they did forwards. I had a moment of thinking about nature being independent of time (whoa!) and got very excited about what else in the world was similar in this way. I found through more experiments that even the sounds of animals share this characteristic, and I theorize that with no concept of language, we might feel the same about our voices. After creating this shape (<>) with the field sound (the front half forward and the back half reversed to meet back up with the forward part ad infinitum) I began trying this concept on an entire musical composition, or rather half of a composition, playing it forward and then looping it back on itself. A completely symmetrical song19. I am excited thinking about the further implications of these ideas. I am also excited that my good friend and long-time collaborator, Eric Carranza, is helping me in an effort to conquer the sonic realm of this infinite odyssey. There is still much to explore.

19 Experiments In Finite. Through All The Windows I Only See Infinity. Tracks "Lament" and "Tents". 2012.

On Physics

FIGURE 5.2 Gorvin, Eric. The Infinity Project. 2012.

â&#x20AC;&#x153;For there are two ways in which distance and time (and, in general, any continuum) are described as infinite: they can be infinitely divisible or infinite in extent. So although it is impossible to make contact in a finite time with things that are infinite in quantity, it is possible to do so with things that are infinitely divisible, since the time itself is also infinite in this way. And so the upshot is that it takes an infinite rather than a finite time to traverse an infinite distance, and it takes infinitely many rather than finitely many nows to make contact with infinitely many things.â&#x20AC;? â&#x20AC;&#x201D;Aristotle, Physics

The Physical World20 Classical VS Quantum Mechanics In current scientific debate, scientists and

the motion of these dynamic variables in the

bystanders alike are trying to answer the

system to then tell us what happens to a given

loaded question, 'what are our physical laws

system when something is done to it. In other

and how do they work?' For the last 100 years,

words, this is a deterministic world which

the two sides of the argument stood as Newto-

answers

nian,

our

questions

with

formulaic

and

Quantum

most often predictable answers. In 1927, every-

Mechanics. The differences in the two sides

thing changed for this classical vision of

science. The scientific published works of Nieles

this

Classical debate

Mechanics,

lie

within

and

the

notions

observables and unobservables, entering into

Bohr

amplify

preliminary

conversations with realism and anti-realism

and make the problem of Quantum Mechanics a

as well an established discourse with chance

real debate.

and the question of determinism in our world.

This debate can be found at the heart of much

Where classical mechanics describe our world

scientific discourse that occurs. There is even

on a very macro scale, it is restricted to

a movement to create a universal concept that

those things which are directly observable

encompasses both classical and quantum mechan-

objects. It does not, however, delve into the

ics, called quantum field theory. The problem

wild world of the quanta. When the laws of

that most scientists find with it, however, is

classical mechanics are applied in our new

that the two players have characteristics that

atomic and sub-atomic world, problems occur

undoubtedly negate the other. They disagree

almost

upon investigation into their dependent attri-

cal system, we can see millions of particles,

butes making them whole, concise theories. The

summed together by objects that create the

following paragraphs will explain classical

view we have of our world and how it functions.

mechanics, quantum mechanics, the exchange and

The bizarre part is that when you zoom in

somewhat cosmic relativity that they have with

to the quantum level, all the particles that

one another, as well as how we are supposed to

make up these so-called solid bodies of mass

understand all of this.

are completely non-deterministic... without a

instantly.

Upon

work

observing

Einstein

classi-

set position or state! The discovery of these Classical Mechanics

anomalies and their relation (or lack there of)

Classical mechanics was introduced by Isaac

to the hard cold laws of classical mechanics

Newton in 1687 in what is one of the most

spurred the birth of quantum theory that began

significant

really addressing this problem.

scientific

books

ever

written,

Philosophiæ Naturalis Principia Mathematica. Newton wrote of our world, or universe as being

Quantum Mechanics

a system with static properties. These static

To start, in quantum mechanics, the state of a

properties are things that do not change with

system on the microscopic atomic level is not

time, such as the mass of an object or the

characterized by a set of dynamic variables.

density of something. It is important that we

On the contrary, it is defined with a state

are clear that this is a known system with

function. The state function has a functional

unchanging properties.

dependence, however, on the “possible position” of the particle. Where state function is

Continuing onward, we have dynamic variables

denoted with z and possible position is denoted

which are things that can change, but are

with x, our equation would read z(x). How the

set with initial conditions. Then there are

state of this particle then changes with time

actions taken on these dynamic variables that

(denoted with t) would read z(x, t). The product

then determine how the system changes with

of this equation is where we witness the occur-

time. To articulate this further, we need an

rence of a “wave function” which hits the nerve

equation

time-

of our debate on indeterminism and we begin to

dependence of the system. Mathematically, this

understand the nature of this particle. Is it

is solved with the equation that determines

a particle or a wave?

motion

that

governs

the

20 Eric Gorvin, The Physical World, 2011. Print. *All sources listed below. Tang, C. L. Fundamentals of Quantum Mechanics: For Solid State Electronics and Optics. Cambridge, UK: Cambridge UP, 2005. Print. “Combining Relativity and Quantum Theory.” Physics. N.p., n.d. Web. 20 Apr. 2011. “Quantum Mechanics vs. Relativity? - Yahoo! Answers.” Yahoo! Answers. N.p., n.d. Web. 23 Apr. 2011. “Quantum gravity - Wikipedia, the free encyclopedia.” Wikipedia. N.p., n.d. Web. 20 Apr. 2011. “Quantum indeterminacy - Wikipedia, the free encyclopedia.” Wikipedia. N.p., n.d. Web. 20 Apr. 2011. “Quantum mechanics - Wikipedia, the free encyclopedia.” Wikipedia. N.p., n.d. Web. 20 Apr. 2011.

The “famous double slit experiment” (Figure

photon is in an indeterministic state. When it

5.3)

performed originally by Thomas Young in a

is being observed, it, as all other tests on

paper entitled, “Experiments and Calculations

electrons, etc. chooses to behave like a deter-

Relative to Physical Optics,” published in 1803

ministic particle, baffling scientists and the

was one of the first understandings of photons

world alike.

as having wave-like properties as well as what

led to the building blocks of quantum theory. In

Conclusions

a classical double-slit experiment, particles

Conclusively, there is no conclusion and there

(in

our

case

photons-elementary

particles,

every

conclusion.

Einstein

went

his

the quanta of light) are propelled toward a

deathbed trying to crack the quantum code. He

wall with two slits in it, ultimately ending

had debates with Bohr that lasted for years,

up on a different wall behind it, quantifying

neither ever being able to accept one anoth-

the results.

er’s respective theories. Erwin Schrödinger went to great lengths to even illustrate how

Upon initial observation, the photons created

ridiculous the idea was by devising his famous

wall,

thought experiment involving a cat sealed in

showing that photons behave like waves. A good

a box with a radioactive compound that had

way of conceptualizing this is to think about

a 50/50 chance of decaying and mixing with

dropping two rocks, side by side, in a pond

cyanide to kill the cat. So probabilistically

and observing the waves. Two concentric waves

the cat has a 50/50 chance of living, but on

interfere and create a number of concentra-

the quantum level, the cat should be consid-

tion points on the wall where the crests of

ered both alive, dead, and neither. Without our

the two circles interfere with each other. In

observation, the cat and every particle depen-

this experiment, the wall displayed a pattern

dent on it is in an indeterministic state.

much like if you were to push an actual wave

We can see the absurdity in this, obviously,

through those 2 slits.

because upon opening the door, we will expe-

interference

pattern

the

back

rience an alive cat or a dead cat, never an Scientists

were

baffled

the

notion

in-between cat.

particle behaving like a wave, but Young as

well as hundreds of scientists preceding him

So, how are we to think of this today? As

weren’t satisfied with such odd results for some-

the

thing that was assumed to have properties of a

conversations have not stopped happening, the

particle just like everything else. In the next

discourse is still very much alive. I believe

experiment, they devised a way to shoot one

that there has to, and must be, a solution that

photon at a time in intervals, hypothesizing

involves both classic and quantum mechanics.

that was no way for a wave pattern to be made

It seems to me that the cosmic, vague evasive-

without a mass of particles to create a wave-like

ness of our sub-atomic world and the explain-

impact. They were wrong. After an hour or so,

able, macro nature of our observable world

it was evident that the particles were still

must be in a conversation with each other. Is

making an interference pattern.

it too crazy to think that the same apparent

quantum

field

theory

builds

steam,

the

“free-thinking” particle could be fabricating Unsatisfied, scientists put a detection device

the laws of classical mechanics? Is it too

at the mouth of the slits to observe how a

wild to think that all we know — gravity, laws

particle could possibly interfere with itself

of motion, and the behavior of light — could

to create such a pattern. Upon trying to observe

be determined by particles reacting to our

these photons, they went back to behaving like

observation? And from our observations, do we

regular particles, leaving two concentration

ascribe them value? Perhaps these two sides of

points on the back wall just as you would get

physics have more in common than they think.

from firing tennis balls through two slits in a

It may be that both the chaotic nature of the

wall. So the action of observing the particle

quanta and the determinism of our day-to-day

made it behave differently. This, obviously

physical experience are intimately connected,

being an unexplainable phenomenon, has been

allowing for everything in the universe to be

tested and re-tested to know conclusive avail.

as it is.

We can only understand this as follows: When the photon behaves like a wave, it is actually going through both slits and neither. The

FIGURE 5.3

“When the province of physical theory was extended to encompass microscopic phenomena through the creation of quantum mechanics, the concept of consciousness came to the fore again. It was not possible to formulate the laws of quantum mechanics in a fully consistent way without reference to the consciousness.”—Eugene Wigner, Remarks on the Mind-Body Question

String Theory21 Th e C om p on e n t s o f S trin g Th e o ry Within the framework of string theory, the familiar fundamental particles such as electrons and quarks are not really particles at all - they are actually tiny vibrating strings of about the Planck length (about 10-33 cm). As such, they are undetectable to our particle accelerators, which would have to utilize energies about a million billion times greater than those accessible at present in order to determine that what seems to be a point particle is in reality a string. Strings may represent the “bottom layer” of the universe’s fundamental constituents. That is to say, although strings have extent in space, they are not made up of anything else—they are the last level of the sub-structure of the universe. However, there are subtle theoretical suggestions that strings may in fact have a substructure in their own right, thereby simultaneously destroying their chance at being the “fundamental” components of the universe and opening fascinating new doors of intricate theoretical physics. Either way, string theory seems capable of either providing itself as a final theory or leading us to a more complete theory that is a TOE (Theory Of Everything). It should be noted here that string theory provides for many different types of strings. There are both open and closed strings, with closed strings forming a loop and open strings, logically, having both ends free—a cut loop. For reasons that will later become apparent, the universe seems to incorporate closed strings, although open strings may also be present. Recent research has also revealed that “strings” may actually have many different dimensions, from the one-dimensional strings originally postulated to a two-dimensional membrane to many analogous structures in higher dimensions. Physicists have taken to calling strings “branes,” and defining each’s dimensional extent with a number, i.e. a one-brane is a one-dimensional string. These will also be explored later in the text.

S trin g Th e o ry ’s Power of Unification String theory postulates the existence of tiny vibrating strings that correspond to the observed elementary particles. Strings can undergo an infinite number of different vibrational patterns, called resonances, whose evenly-spaced peaks and troughs fit exactly along its spatial extent. By analogy, the strings of a guitar can similarly undergo an infinite number of vibrational patterns that meet the same requirement, though we only come in contact with a few of them. These recognizable vibrations are perceived by human ears as different musical notes. Similarly, the vibrations which strings undergo not only correspond to, but actually create, the different masses and charges observed in the various elementary particles. In other words, an elementary particle’s precise properties are caused by the vibrations of its string. This connection is best illustrated for the mass of a particle. A vibrational pattern’s energy is related to its amplitude, or the maximum height of a wave peak (or depth of a trough) and the wavelength, or the distance between one peak and the next. Greater amplitude and greater wavelength correlate with greater energy - that is, the more frenetic the vibrations of the string, the greater energy it has. Since energy is related to mass by Einstein’s famous equation E=mc2, high vibrational energies correspond to high-mass particles.

Entry I feel like it is generally difficult to think about something like this [infinity] because of its non-relevance to the daily comings and goings of humans. And I understand this. I understand it well from studying quantum and classic mechanics. The quantum level requires a bit of imagination to understand. To think about such a chaotic system that is in fact indeterminate but it is somehow orchestrating a system that is determinate, means that all of the things happening from day to day that seem so sure, operate by chance, not because it is law. So in the same light, thinking of this infinitely erratic system being the "solid" world you live in is a challenging nugget of knowledge to ruminate on.

21 "Introduction to String Theory." Thinkquest. Oracle Foundation. Web. 21 Apr. 2012.

FIGURE 5.4 Gorvin, Eric. The Infinity Project. 2012.

FIGURE 6.1 Gorvin, Eric. "Moleskine Entry on Pi". The Infinity Project. 2012.

On Mathematics

Ad Infinitum22 P rob le m s In Ma th e matics

Across the span of Western thought, infinity has been a notoriously troublesome idea, difficult to pin down, full of paradox, and seemingly connected in some way or other with the divine. But whatever its philosophicotheological obscurities and contradictions, infinity in mathematics, as a phenomenon and an effect, is neither difficult to pin down nor hard to come by. One meets it immediately in elementary situations when, for example, one tries to divide a number by zero or compute the tangent of 90 degrees or express the fraction 1/3 as a decimal, or fundamentally, when one writes the ideogram, ” . . . “ of mathematical continuation to signal that the progression of whole numbers 1, 2, 3, . . . be continued without end. And one meets it just as immediately in non-elementary mathematics. It is at work in the very idea of the geometrical continuum of points on a line and their integer-based real number descriptions—two linked abstractions which ground all post-Renaissance mathematics. And it is the founding signified, the crucial ontological term, in contemporary mathematics’ description of itself as an infinite hierarchy of infinite sets.

century mathematical scene. A century ago, as part of the initiating phase of this axiomatics, Richard Dedekind asked how numbers were to be defined. And he did so by means of a double question—”Was sind und was sollen die Zahlen?”—what are [sind] the numbers and what might/should [sollen] they be? Dedekind’s answer—that numbers were a certain sort of sets—is now part of the very picture of a ‘’naturalized” infinity of numbers that will be put into question here. Nevertheless, we might adapt his question, replacing his modality of being by a modality of being thought, and ask, “What is the infinite and how are we to think it?”

How, one might ask, has mathematics so successfully tamed and incorporated the infinite? Incorporated it moreover at such a basic, elemental level, in so allpervasive a way? How does infinity get to be an exact, rigorously specified mathematical object—an object about which mathematics delivers “true” and “objective” knowledge? Mathematics starts from the integers. Its entire formalism opens out from the sequence 1, 2, 3, ... that mathematicians call the “natural” numbers. The question can therefore be particularized: What does it mean to say of these numbers that they are infinite, that they form a progression which is endless? In what sense are they natural, that is to say, before, independent, and outside of us? Precisely this question frames the present essay.

at the same time a play of imagination and a discourse of written symbols. Should we not then pose the question of the mathematical infinite as a question of language, as part of an overall study of the nature and practice of mathematical signs—as part, that is, of a semiotics of mathematics? The answer is that we should, but that— obviously enough—we need to develop a semiotics of mathematics in order to do so.

Of course, part of any answer is historical. The presentday interpretation of the mathematical infinite has a long story behind it; from Anaximander and Zeno, through Euclid, Aristotle, Carl Friedrich Gauss, culminating in Gottlob Frege’s logic and Georg Cantor’s infinities articulated within an axiomatics that dominates the twentieth

To pursue zero further would be to have to say more about its role at the origin of this formalism. But such a project would require a critique of mathematical logic. In particular, one would need to unravel the assumptions behind the claim that is made for Boolean logic, with its referential appa-

But, as we shall see, being thought in mathematics always comes woven into and inseparable from being written. We are never presented with the pure idea of infinity as such. How could we be? Instead, we meet only with certain mathematical inscriptions that in turn are connected to other inscriptions through a complex and openended spread of sense and interpretation. Thinking in mathematics is always through, by means of, in relation to the manipulation of inscriptions. Mathematics is

In this sense this book [Ad Infinitum] starts where a previous one on mathematical signs—Signifying Nothing— felt itself able to sign off. In that text, having chased zero and zero-like signs across different cultural codes, I called a halt at its presence within the binary formalism of Boolean logic governing the contemporary computer:

Rotman, Brian. “Preface.” Ad Infinitum -- the Ghost in Turing’s Machine. Stanford, CA: Stanford UP, 1993. 3+. Print

ratus of truth and falsity, to be the grammar of all mathematical, hence all scientific-technical, hence all supposedly neutral, culturally invariant, objective, true/false assertions about some prior “real” world. To do this would require a semiotics that went beyond zero to the whole field of mathematical discourse. A semiotics which, in order to begin at all, would have to demolish the widely held metaphysical belief that mathematical signs point to, refer to, or invoke some world, some supposedly objective eternal domain, other than that of their own human—time bound, changeable, subjective and finite—making. What I present here is not a further stalking of zero but a pursuit of infinity. However, the intimacy—both historical and conceptual—of the coupling between the two, between nothing/zero and all/infinity, manifest for example in the immediate production of infinity as the reciprocal of zero, makes any explication of the mathematical infinite a supplement to that of zero. Neither is what follows a philosophical critique of the metaphysical system, the rampant Platonism, that threads its way through the contemporary interpretation of mathematics—though, naturally, I cannot avoid engaging with Platonism, given its near universal acceptance. Rather it is a critique, in the form of an interrogation from first principles, of what seems to me an altogether more subtle metaphysical principle that permeates the entire subject. This is the principle of ad infinitum continuation that is inseparable from the mathematical community’s wholesale-acceptance of the view that the numbers are “natural,” and its failure to ask the question of where these numbers could possibly come from. With this failure comes its inability to perceive either the need or the point of asking what the infinite is and how we are to think of it.

FIGURE 6.2 Gorvin, Eric. The Infinity Project. 2012.

FIGURE 6.3 Gorvin, Eric. The Infinity Project. 2012.

â&#x20AC;&#x153;Mathematics are the result of mysterious powers which no one understands, and which the unconscious recognition of beauty must play an important part. Out of an infinity of designs a mathematician chooses one pattern for beautyâ&#x20AC;&#x2122;s sake and pulls it down to earth.â&#x20AC;?-Marston Morse

History in Math23 A n In t e r vie w W ith Rev iel N etz How do mathematicians define infinity? Something which is equal to some of its parts. That's really the technical definition. Is there a difference between the mathematical concept of infinity and infinity in terms, say, of space or time— the philosophical concept? That's the curious thing. Infinity became a really clear and well-defined quantity mathematically in the late 19th century, which it wasn't before and which makes it rather different from what we ordinarily talk about when we talk about infinity, namely, about something very, very big. In mathematics nowadays, when we think about infinity, we think about a set whose properties are different from those of ordinary sets. Can you explain? Well, the defining property of infinity today is that a set's cardinality [the number of elements in a given mathematical set] is equal to the cardinality of some real subset of that set. The thread that links the technical notion of infinity and the more popular/philosophical notion of infinity is the sense that infinity's hugeness being beyond reach endows it with certain paradoxical properties. A standard example would be what was taken to be something rather paradoxical, the fact that you can take, say, the number 1 and correlate it with the number 2, take the number 2 and correlate it with the number 4, take the number 3 and correlate it with the number 6. In this way you can gradually, step by step, correlate all integer numbers (1, 2, 3, 4, etc.) with all even numbers (2, 4, 6, 8, etc.). And that's funny, because this would imply at first glance that the number of integers is equal to the number of even integers. That's the paradoxical property of infinity.

This discomfort with infinity lasted until the invention of the calculus. How did that change things? What happens with the calculus is that you find ways to calculate infinitely long series. I've mentioned that in order to get from Point A to Point B, you first have to cross half the way, then half of the half, which is a quarter, then half of the half of the half, which is an eighth. This seems paradoxical, because it looks like we can define it this way: Let's have a series, and the series is 1/2 plus 1/4 plus 1/8, etc., going on to infinity. It would appear that the series that has infinitely many terms should be infinitely large. What you find with the calculus is that there are ways of dealing with infinitely large objects in some ways that are still finite in other ways. There are ways of calculating with infinitely large objects. To start with, this was done on a rather intuitive basis. It was just a series of observations that things could be done and seem to work properly. It was only in the 19th century that precise techniques for dealing with infinitely large magnitudes emerged. They found that once you allow yourself to do those things in practice, then you can do as a matter of calculation—real calculation with numbers—things that before you'd done purely by operating with geometrical configurations. You can think not in the qualitative terms of geometrical configurations, but in the quantitative terms of dealing with numbers and series of numbers. That's very powerful, because numbers are very powerful. They are precise and manageable in ways that geometrical configurations are not. Essentially, then, you now had a tool that is much more useful for science. So what appears to be a quaint, paradoxical realm—the realm of infinite magnitude—is actually very practical. It's something that allows you to extend operations with numbers to any domain whatsoever. That's what happened from the 17th century onward. How did Georg Cantor's set theory refine mathematicians' thinking about infinity? Well, the essence of the calculus is that you deal with infinitely large objects. But you never had to define infinity itself, and you never had to worry about the nature of infinity, primarily because you always dealt with the very same kind of infinity—roughly speaking, the infinity of points making up a line, the infinity of all the real numbers between, let's say, 0 and 1. That's the type of thing they were worried about in the calculus from the 17th century to the 19th century.

23 Netz, Reviel. "Working With Infinity." Interview. PBS/NOVA. Public Broadcast System, 30 Sept. 2003. Web. Edited by Peter Tyson FIGURE 6.4 Gorvin, Eric. The Infinity Project. 2012.

But they didn't think about what infinity is, because for one thing they didn't think about what a set is. What is a set? And then what would be the difference between a finite and an infinite set? This is something that Cantor did in the late 19th century. Cantor developed the notion of a set, and the notion of an infinite set, a set that has infinitely many objects.

In finity To da y So is infinity an active field of study for mathematicians today? Oh, infinity is what mathematics is about. Infinity always was what mathematics is about. It's just that there are many different ways of dealing with it. In the Greek context, you dealt with situations that give rise to infinity by transforming them into geometrical representations. So, for example, you effectively dealt with how many lines it takes to fill a certain rectangle by looking at various geometrical configurations. You never called infinity by its name.

FIGURE 6.5

Cardinal Number24 D efinition In mathematics, cardinal numbers, or cardinals for short, are

In the Scientific Revolution, you did those things with the recognition that you're dealing with infinity, and from the 19th century onwards you began dealing with actual infinite setsâ&#x20AC;&#x201D;a mathematics of infinity. The kinds of infinity that are allowed into the game define the kind of mathematics you're doing even today. That's the fundamental division between different kinds of mathematics. Do you allow in infinities beyond the first, most simple infinity or not? There are different kinds of mathematics dependent on that. Primarily you're dealing with various situations arising from various different kinds of infinity.

a generalization of the natural numbers used to measure the cardinality (size) of sets. The cardinality of a finite set is a natural numberâ&#x20AC;&#x201D;the number of elements in the set. The transfinite cardinal numbers describe the sizes of infinite sets. Cardinality is defined in terms of bijective functions. Two sets have the same cardinal number if and only if there is a bijection between them. In the case of finite sets, this agrees with the intuitive notion of size. In the case of infinite sets, the behavior is more complex. A fundamental theorem due to Georg Cantor shows that it is possible for infinite sets to have different cardinalities, and in particular the set of real numbers

Finite thingsâ&#x20AC;&#x201D;yes, of course, there are important fields of mathematics that deal with finite situations, and these are fascinating fields in their own right. But the fundamental thrust of mathematics is dealing with infinite situations.

and the set of natural numbers do not have the same cardinal number. It is also possible for a proper subset of an infinite set to have the same cardinality as the original set, something that cannot happen with proper subsets of finite sets.

24 "Cardinal Number." Wikipedia. Wikimedia Foundation, 04 Mar. 2013. Web. 12 Jan. 2013. FIGURE 6.5 is the first and smallest infinite cardinal. 0

Mathematical Induction De fin itio n 25 Mathematical induction can be informally illustrated by reference to the sequential effect of falling dominoes. Mathematical induction is a method of mathematical proof typically used to establish that a given statement is true for all natural numbers (positive integers). It is done by proving that the first statement in the infinite sequence of statements is true, and then proving that if any one statement in the infinite sequence of statements is true, then so is the next one.

E n t ry By using mathematical induction, could one assume something that is within the same system (i.e. molecules, cosmos, and humans) will have the same properties as the next thing, no matter how big or how small?

In fin it e De sce n t 26 Another variant of mathematical induction—the method of infinite descent—was one of Pierre de Fermat’s favorites. This method of proof works in reverse, and can assume several slightly different forms. For example, it might begin by showing that if a statement is true for a natural number n it must also be true for some smaller natural number m (m < n). Using mathematical induction (implicitly) with the inductive hypothesis being that the statement is false for all natural numbers less than or equal to m, we can conclude that the statement cannot be true for any natural number n.

25 "Mathematical Induction." Wikipedia. Wikimedia Foundation, 10 Mar. 2012. Web. 20 Apr. 2012. 26 “Infinite Descent.” Wikipedia. Wikimedia Foundation, 20 Mar. 2012. Web. FIGURE 6.6-6.7 Gorvin, Eric. The Infinity Project. 2012.

d Approximating Pi27 A rc h im e de s's M eth o d Around 250 B.C., the Greek mathematician Archimedes

diameter

calculated the ratio of a circle's circumference to its diameter. A precise determination of pi, as we know this ratio today, had long been of interest to the ancient Greeks, who strove for precise mathematical proportions in their architecture, music, and other art forms. Close approximations of pi had been known for over 1,000 years. Archimedes’ value, however, was not only more accurate, it was the first theoretical, rather than measured, calculation of pi.

Circumference 1

2 2 1

3 2 3

π=

C d

1 M illion D igits of π

FIGURE 6.8

π28 C alcu la t in g π Co n tempo rarily When a circle's diameter is 1 unit, its circumference is π units. In Euclidean plane geometry, π is defined as the ratio of a circle's circumference C to its diameter d. The ratio C/d is constant, regardless of a circle's size. For example, if a circle has twice the diameter d of another circle it will also have twice the circumference C, preserving the ratio C/d. This definition depends on results of Euclidean geometry, such as the fact that all circles are similar, which can be a problem when π occurs in areas of mathematics that otherwise do not involve geometry. For this reason, mathematicians often prefer to define π without reference to geometry, instead selecting one of its analytic properties as a definition. A common choice is to define π as twice the smallest positive x for which the trigonometric function cos(x) equals zero.

3.14 1592 6535 8... diameter

Circumference

Continued on the next 6 pages.

27 Groleau, Rick. "Approximating Pi." PBS. PBS, 01 Sept. 2003. Web. 02 Jan. 2013. 28 "Pi." Wikipedia. Wikimedia Foundation, 04 Jan. 2012. Web. 18 Feb. 2012. FIGURE 6.8 Gorvin, Eric. "Illustration of Archimedes's method." The Infinity Project. 2013.

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The Golden Section29 Fib on a c ci Sequence The golden section (Figure 7.10) is a symmetrical relation built from asymmetrical parts. Two numbers, shapes or elements embody the golden section when the smaller is to the larger as the larger is to the sum. That is, a : b = b : (a + b). In the language of algebra, this ratio is 1 : φ = 1 : ( 1 + √5 ) / 2, and in the language of trigonometry, it is 1 : (2 sin 54°). Its approximate value in decimal terms is 1: 1.61803. The second term of this ratio, φ (the Greek letter phi), is a number with several unusual properties. If you add one to cp, you get its square ( φ x φ ). If you subtract one from φ, you get its reciprocal (1 / φ ). And if you multiply φ endlessly by itself, you get an infinite series embodying a single proportion. That proportion is 1 : φ. If we rewrite these facts in the typographic form mathematicians like to use, they look like this:

Here each term after the first two is the sum of the two preceding. And the farther we proceed along this series, the closer we come to an accurate approximation of the number φ. Thus 5 : 8 = 1 : 1.6; 8 : 13 = 1:1.625 ; 13 : 21 = 1 : 1.625; 21 : 34 = 1: 1.619, and so on. In the world of pure mathematics, this spiral of increase, the Fibonacci series, proceeds without end. In the world of mortal living things, of course, the spiral soon breaks off. It is repeatedly interrupted by death and other practical considerations – but it is visible nevertheless in the short term. Abbreviated versions of the Fibonacci series, and the proportion 1 : φ, can be seen in the structure of pineapples, pinecones, sunflowers, sea urchins, snails, the chambered nautilus, and in the proportions of the human body as well.

φ + 1 = φ2 φ–1=1/φ φ-1 : 1 = 1 : φ = φ : φ2 = φ2 : φ3 = φ3 : φ4 = φ4 : φ5 ... If we look for a numerical approximation to this ratio, 1 : φ, we will find it in something called the Fibonacci series, named for the thirteenth-century mathematician Leonardo Fibonacci. Though he died two centuries before Gutenberg, Fibonacci is important in the history of European typography as well as mathematics. He was born in Pisa but studied in North Africa. On his return, he introduced Arabic numerals to the North Italian scribes. As a mathematician, Fibonacci took an interest in many problems, including the problem of unchecked propagation. What happens, he asked, if everything breeds and nothing dies? The answer is a logarithmic spiral of increase. Expressed as a series of integers, such a spiral takes the following form: 0 • 1 • 1 • 2 • 3 • 5 • 8 • 13 • 21 • 34 • 55 • 89 • 144 • 233 • 377 • 610 • 987 • 1597 • 2584 • 4181 • 6765 • 10,946 • 17,711 • 28,657 ...

FIGURE 6.11

29 Bringhurst, Robert. The Elements of Typographic Style. Point Roberts, WA: Hartley & Marks, 2004. Print. FIGURE 6.11 Bringhurst, Robert. The Elements of Typographic Style. Point Roberts, WA: Hartley & Marks, 2004. Print.

Mysterious Mathematics30 A S p iralled Shell

N a ture's Spirals

The cut-away of a chambered nautilus shell shows its compartments. Only the outermost is the animal's home at any given time. Collectively, these chambers form an equiangular spiral: the black spiral intersects all the white radii at exactly the same angle, so that the angles A, B, C and so on around the shell are always identical to one another (Figure 6.12).

Nature never has been content with simple shapes, but has created all kinds of intricate mathematical designs, including a variety of spirals. For example, the shell of the chambered nautilus is an equiangular, or logarithmic, spiral (Figure 6.12): the curve of the spiral always intersects the outreaching radii at a fixed angle. Logarithmic spirals also occur in the curve of elephants' tusks, the horns of wild sheep and even canaries' claws. Similar, though less precise, spirals are formed by the tiny florets in the core of daisy blossoms. The eye sees these spirals as two distinct sets, radiating clockwise and counter-clockwise, with each set always made up of a predetermined number of spirals. Most daisies have 21 and 34. Similar arrangements of opposing spirals are found in pine cone scales (5 one way, 8 the other), the bumps on pineapples (8 and 13) and the leaves of many trees. This phenomenon is made all the more mysterious by its relationship with a certain mathematical sequence known by the nickname of its medieval discoverer, Leonardo ("Fibonacci") daPisa. The Fibonacci series is produced by starting with 1 and adding the last two numbers to arrive at the next: 1, 1, 2, 3, 5, 8, 13, 21, 34, etc. The daisy's spiral ratio of 21 : 34 corresponds to two adjacent Fibonacci numbers, as do the pine cone's 5:8 and the pineapple's 8: 13 and the same is true of many other plants with a spiral leaf-growth pattern.

FIGURE 6.12

30 Bergamini, David. Mathematics. New York: Time, 1963. Print. FIGURE 6.12 Bergamini, David. Mathematics. New York: Time, 1963. Print. FIGURE 6.13 Hurlbut, Cornelius S. The Planet We Live On. New York: H. N. Abrams, 1976. Print.

On Nature

Growth & Continuity31 Life 's U n ive r s a l S ca ling L a ws Biological systems have evolved branching networks that transport a variety of resources. We argue that common properties of those networks allow for a quantitative theory of the structure, organization, and dynamics of living systems. Nearly 100 years ago, the eminent biologist D'Arcy Thompson began his wonderful book On Growth and Form by quoting Immanuel Kant. The philosopher had observed that "chemistry... was a science but not Science...for that the criterion of true Science lay in its relation to mathematics." Thompson then declared that, since a "mathematical chemistry" now existed, chemistry was thereby elevated to Science; whereas biology had remained qualitative, without mathematical foundations or principles, and so it was not yet Science. Although few today would articulate Thompson's position so provocatively, the spirit of his characterization remains to a large extent valid, despite the extraordinary progress during the intervening century. The basic question implicit in his discussion remains unanswered: Do biological phenomena obey underlying universal laws of life that can be mathematized so that biology can be formulated as a predictive, quantitative science? Most would regard it as unlikely that scientists will ever discover "Newton's laws of biology" that could lead to precise calculations of detailed biological phenomena. Indeed, one could convincingly argue that the extraordinary complexity of most biological systems precludes such a possibility. Nevertheless, it is reasonable to conjecture that the coarse-grained behavior of living systems might obey quantifiable universal laws that capture the systems' essential features. This more modest view presumes that, at every organizational level, one can construct idealized biological systems whose average properties are calculable. Such ideal constructs would provide a zeroth order point of departure for quantitatively understanding real biological systems, which can be viewed as manifesting "higher order corrections" due to local environmental conditions or historical evolutionary divergence. The search for universal quantitative laws of biology that supplement or complement the Mendelian laws of inheritance and the principle of natural selection might seem to

be a daunting task. After all, life is the most complex and diverse physical system in the universe, and a systematic science of complexity has yet to be developed. The life process covers more than 27 orders of magnitude in massâ&#x20AC;&#x201D;from molecules of the genetic code and metabolic machinery to whales and sequoiasâ&#x20AC;&#x201D;and the metabolic power required to support life across that range spans over 21 orders of magnitude. Throughout those immense ranges, life uses basically the same chemical constituents and reactions to create an amazing variety of forms, processes, and dynamical behaviors. All life functions by transforming energy from physical or chemical sources into organic molecules that are metabolized to build, maintain, and reproduce complex, highly organized systems. Understanding the origins, structures, and dynamics of living systems from molecules to the biosphere is one of the grand challenges of modern science. Finding the universal principles that govern life's enormous diversity is central to understanding the nature of life and to managing biological systems in such diverse contexts as medicine, agriculture, and the environment.

Entry I find it very intriguing that we can start understanding the biology of our world in the act of talking about relationships. I like thinking about the fact that we share the same growth rate and a lot of the same biological characteristics across all the various forms of life (Figure 7.2). I also believe that this is a unique insight into our own connection with nature, exemplifying the beauty in its infinite properties. If we can conclude that all things in nature grow and evolve proportionately, I feel we can start applying that same logic when considering relationships of stars and nebulae to atoms and molecules, helping us further understand our own place in the levels and ratios of existence.

31 West, Geoffrey B., and James H. Brown. "Life's Universal Scaling Laws." Physics Today (2004): 36. University of New Mexico Biology Department. Web.

FIGURE 7.2

FIGURE 7.2 Gorvin, Eric (Illustration) 2013. Information from: West, Geoffrey. "Geoffrey West: The Surprising Math of Cities and Corporations." Lecture. TEDGlobal. Ted.com. Web.

The Rhizome32 Fo rm a l D e fi n i ti o n (B ota ny) rhizome (Figure 7.3): a continuously growing horizontal underground stem that puts out lateral shoots and adventitious roots at intervals. Buds that form at the joints produce new shoots. Thus, if a rhizome is cut by a cultivating tool it does not die, as would a root, but becomes several plants instead of one.

In Nature A rhizome as subterranean stem is absolutely different from roots and radicles. Bulbs and tubers are rhizomes. Plants with roots or radicles may be rhizomorphic in other respects altogether: the question is whether plant life in its specificity is not entirely rhizomatic. Even some animals are, in their pack form. Rats are rhizomes. Burrows are too, in all of their functions of shelter, supply, movement, evasion, and breakout. The rhizome itself assumes very diverse forms, from ramified surface extension in all directions to concretion into bulbs and tubers. When rats swarm over each other. The rhizome includes the best and the worst: potato and couchgrass, or the weed. Animal and plant, couchgrass is crabgrass.

D e l e u ze & G u a t ta ri Let us summarize the principal characteristics of a rhizome: unlike trees or their roots, the rhizome connects any point to any other point, and its traits are not necessarily linked to traits of the same nature; it brings into play very different regimes of signs, and even nonsign states. The rhizome is reducible to neither the One or the multiple. It is not the One that becomes Two or even directly three, four, five etc. It is not a multiple derived from the one, or to which one is added (n+1). It is comprised not of units but of dimensions, or rather directions in motion. It has neither beginning nor end, but always a middle (milieu) from which it grows and which it overspills. It constitutes linear multiplicities with n dimensions having neither subject nor object, which can be laid out on a plane of coinsistency, and from which the one is always subtracted (n-1). When a multiplicity of this kind changes dimension, it necessarily changes in nature as well, undergoes a metamorphosis. Unlike a structure, which is defined by a set of points and positions, the rhizome is made only of lines; lines of segmentarity and stratification as its dimensions, and the line of flight or deterritorialization as the maximum dimension after which the multiplicity undergoes metamorphosis, changes in nature. These lines, or ligaments, should not be confused with lineages of the arborescent type, which are merely localizable linkages between points and positions...Unlike the graphic arts, drawing or photography, unlike tracings, the rhizome pertains to a map that must be produced, constructed, a map that is always detachable, connectable, reversible, modifiable, and has multiple entranceways and exits and its own lines of flight.

FIGURE 7.3

As H uman A rhizome ceaselessly establishes connections between semiotic chains, organizations of power, and circumstances relative to the arts, sciences, and social struggles. A semiotic chain is like a tuber agglomerating very diverse acts, not only linguistic, but also perceptive, mimetic, gestural, and cognitive: there is no language in itself, nor are there any linguistic universals, only a throng of dialects, patois, slangs, and specialized languages. There is no ideal speaker-listener, any more than there is a homogeneous linguistic community. Language is, in Weinreich's words, "an essentially heterogeneous reality.'' There is no mother tongue, only a power takeover by a dominant language within a political multiplicity. Language stabilizes around a parish, a bishopric, a capital. It forms a bulb. It evolves by subterranean stems and flows, along river valleys or train tracks; it spreads like a patch of oil. It is always possible to break a language down into internal structural elements, an undertaking not fundamentally different from a search for roots. There is always something genealogical about a tree. It is not a method for the people. A method of the rhizome type, on the contrary, can analyze language only by decentering it onto other dimensions and other registers. A language is never closed upon itself, except as a function of impotence.

32 Deleuze, Gilles, and FĂŠlix Guattari. "Introduction: Rhizome." A Thousand Plateaus: Capitalism and Schizophrenia. Minneapolis: University of Minnesota, 1987. N. pag. Print. FIGURE 7.3 J.U. & C.G. Lloyd. Drugs and Medicines of North America. Cincinnati: J.U. & C.G. Lloyd, 1884. Print.

Ma p s The rhizome is altogether different, a map and not a tracing. Make a map, not a tracing. The orchid does not reproduce the tracing of the wasp; it forms a map with the wasp, in a rhizome. What distinguishes the map from the tracing is that it is entirely oriented toward an experimentation in contact with the real. The map does not reproduce an unconscious closed in upon itself; it constructs the unconscious. It fosters connections between fields, the removal of blockages on bodies without organs, the maximum opening of bodies without organs onto a plane of consistency. It is itself a part of the rhizome. The map is open and connectable in all of its dimensions; it is detachable, reversible, susceptible to constant modification. It can be torn, reversed, adapted to any kind of mounting, reworked by an individual, group, or social formation. It can be drawn on a wall, conceived of as a work of art, constructed as a political action or as a meditation. Perhaps one of the most important characteristics of the rhizome is that it always has multiple entryways; in this sense, the burrow is an animal rhizome, and sometimes maintains a clear distinction between the line of flight as passageway and storage or living strata (cf. the muskrat). A map has multiple entryways, as opposed to the tracing, which always comes back "to the same." The map has to do with performance, whereas the tracing always involves an alleged "competence."

The tree and root inspire a sad image of thought that is forever imitating the multiple on the basis of a centered or segmented higher unity. If we consider the set, branches-roots, the trunk plays the role of opposed segment for one of the subsets running from bottom to top: this kind of segment is a "link dipole," in contrast to the "unit dipoles" formed by spokes radiating from a single center. Even if the links themselves proliferate, as in the radicle system, one can never get beyond the One-Two, and fake multiplicities. Regenerations, reproductions, returns, hydras, and medusas do not get us any further. Arborescent systems are hierarchical systems with centers of significance and subjectification, central automata like organized memories. In the corresponding models, an element only receives information from a higher unit, and only receives a subjective affection along preestablished paths. This is evident in current problems in information science and computer science, which still cling to the oldest modes of thought in that they grant all power to a memory or central organ.

â&#x20AC;&#x153;Principles of connection and heterogeneity: any point of a rhizome can be connected to anything other, and must be. This is very different from the tree or root, which plots a point, fixes an order.â&#x20AC;?Â 93

Make Rhizomes!

Th e Mid d le Those things which occur to me, occur to me not from the root up but rather only from somewhere about their middle. Let someone then attempt to seize them, let someone attempt to seize a blade of grass and hold fast to it when it begins to grow only from the middle."Why is this so difficult? The question is directly one of perceptual semiotics. It's not easy to see things in the middle, rather than looking down on them from above or up at them from below, or from left to right or right to left: try it, you'll see that everything changes. It's not easy to see the grass in things and in words (similarly, Nietzsche said that an aphorism had to be "ruminated"; never is a plateau separable from the cows that populate it, which are also the clouds in the sky).

Make rhizomes, not roots, never plant! Don't sow, grow offshoots! Don't be one or multiple, be multiplicities! Run lines, never plot a point! Speed turns the point into a line! Be quick, even when standing still! Line of chance, line of hips, line of flight. Don't bring out the General in you! Don't have just ideas, just have an idea (Godard). Have short-term ideas. Make maps, not photos or drawings. Be the Pink Panther and your loves will be like the wasp and the orchid, the cat and the baboon.

Picking Up Speed

â&#x20AC;&#x153;Many people have a tree growing in their heads, but the brain itself is much more a grass than a tree.â&#x20AC;?

A rhizome has no beginning or end; it is always in the middle, between things, interbeing, intermezzo. The tree is filiation, but the rhizome is alliance, uniquely alliance. The tree imposes the. verb "to be," but the fabric of the rhizome is the conjunction, "and ... and ... and . . . "This conjunction carries enough force to shake and uproot the verb "to be ... Where are you going? Where are you coming from? What are you heading for? These are totally useless questions. Making a clean slate, starting or beginning again from ground zero, seeking a beginning or a foundationâ&#x20AC;&#x201D;all imply a false conception of voyage and movement (a conception that is methodical, pedagogical, initiatory, symbolic ... ). But Kleist, Lenz, and Buchner have another way of traveling and moving: proceeding from the middle, through the middle, coming and going rather than starting and finishing. American literature, and already English literature, manifest this rhizomatic direction to an even greater extent; they know how to move between things, establish a logic of the AND, overthrow ontology, do away with foundations, nullify endings and beginnings. They know how to practice pragmatics. The middle is by no means an average; on the contrary, it is where things pick up speed. Between things does not designate a localizable relation going from one thing to the other and back again, but a perpendicular direction, a transversal movement that sweeps one and the other away, a stream without beginning or end that undermines its banks and picks up speed in the middle.

FIGURE 7.4 Hurlbut, Cornelius S. The Planet We Live On. New York: H. N. Abrams, 1976. Print.

FIGURE 7.4 Infinity Magazine. 1966. Print.

“To be rhizomorphous is to produce stems and filaments that seem to be roots, or better yet connect with them by penetrating the trunk, but put them to strange new uses. We're tired of trees. We should stop believing in trees, roots, and radicles. They've made us suffer too much. All of arborescent culture is founded on them, from biology to linguistics. Nothing is beautiful or loving or political aside from underground stems and aerial roots, adventitious growths and rhizomes.” —Deleuze & Guattari, A Thousand Plateaus

FIGURE 8.1 Gorvin, Eric. The Infinity Project. 2012.

On Paradox

Zeno's Paradoxes33 Ze n o O f Elea The paradoxes of Zeno of Elea are objects of beauty and charm, and sources of intense intellectual excitement. Using everyday occurrences, such as a footrace or the flight of an arrow, Zeno shows that simple considerations lead to profound difficulties. In his attempt to demonstrate the impossibility of plurality, motion, and change, he points to problems lying at the very heart of our concepts of space, time, motion, continuity, and infinity. Since these concepts play fundamental roles in philosophy, mathematics, and physics, the implications of the paradoxes are far-reaching indeed. It is perhaps amusing to be confronted by a simple argument which purports to demonstrate the unreality of something as obviously real as motion; it is deeply intriguing to find that the resolution of the paradox requires the subtlety of modern physics, mathematics, and philosophy. It is difficult to think of any other problem in science or philosophy which can be stated so simply and whose resolution carries one so far or so deep. Bertrand Russell was hardly exaggerating when he said, 'Zeno's arguments, in some form, have afforded grounds for almost all theories of space and time and infinity which have been constructed from his time to our own.'

Th e His t orical Zeno Precious little is known about Zeno of Elea. His fame derives mainly from four paradoxes of motion attributed to him by Aristotle. None of Zeno's writings have survived, but a few passages in other authors are purported to be direct quotations. At best, we have less than two hundred of Zeno's own words, and the paradoxes of motion are not included in this corpus. It is known that Zeno lived in the fifth century B.C., and that he was a devoted disciple of Parmenides. Parmenides maintained that reality is one, immutable, and unchanging; all plurality, change, and motion are mere illusions of the senses. Zeno, according to Plato's testimony, propounded a series of arguments designed to show the absurdity of the views of those who made fun of Parmenides. Zeno was no mere sophist whose sole aim was to confound by verbal trickery, nor was he a skeptic who denied the possibility of all knowledge. He seriously accepted the Parmenidean view, and posed his paradoxes as real difficulties for those who held a different metaphysic. Whether his arguments were directed specifically against the Pythagoreans, against some other particular philosophical school, or more generally against any view that affirmed plurality is still the subject of historical debate.

100

In addition to his paradoxes of motion, several other arguments of Zeno have come down to us, the most important being a paradox of plurality. Since the denial of plurality is the central thesis of Parmenides, it is likely that this paradox plays an even more fundamental role for Zeno than the more famous paradoxes of motion. Moreover, regardless of Zeno's estimate of the relative importance of the two kinds of paradoxes, we shall see that the paradox of plurality is logically more basic than the paradoxes of motion. Aristotle credits Zeno with the invention of dialectic, a method frequently exemplified in Greek philosophy (but related only indirectly to dialectic as conceived by Hegel and Marx). The Greek dialectical method involves a dialogue between two speakers, one of whom propounds and defends a thesis while the other attempts to reduce it to absurdity by deriving a contradiction, a method familiar from the Platonic dialogues. Dialectic involves extensive use of the argument by reductio ad absurdum. Although this form of argument had probably been discovered by mathematicians somewhat earlier than Zeno, he quite possibly imported the technique into philosophy and gave it a central place in philosophical method. This accomplishment secures for Zeno an important position in the early history of logic. Zeno's paradoxes have been the object of extensive historical research especially in the last hundred years. Due to the scantiness of material, various interpretations and reconstructions are possible. On some reconstructions Zeno is guilty of elementary logical and mathematical errors, while on others he displays extraordinary logical and mathematical acumen. Vlastos attributes to Zeno "crudities and blunders," and Booth claims that 'Zeno's arguments . .. involved elementary fallacies; they were not uttered with that full and marvelous understanding which some scholars have attributed to him.' Russell, on the other hand, says that Zeno 'invented four arguments, all immeasurably subtle and profound. . . .' While there may be serious doubt about the subtlety and profundity of the arguments Zeno actually propounded, there can be no doubt that subtle and profound problems have arisen from the consideration of his paradoxes.

33 Salmon, Wesley C. Zeno's Paradoxes. Indianapolis: Bobbs-Merrill, 1970. Print.

Th e Argu me n ts

#1 Achilles and the Tortoise

The orientation of [this book] is systematic rather than historical. Zeno's paradoxes have interested philosophers of all periods, but until the middle of the nineteenth century the paradoxes were almost always regarded as mere sophisms which could be removed with little trouble. In the last hundred years, however, they have been taken very seriously, and in the twentieth century have become the subject of vigorous philosophical discussion. This controversy, still continuing in the professional journals, testifies to the fact that Zeno has raised issues still very much alive. The articles selected for inclusion in [this book] represent attempts to deal with these issues; they are concerned only incidentally with the historicity of the arguments. There is this much historical justice in the approach: if Zeno is not the father of these problems, he certainly is their grandfather. We shall be concerned with five of Zeno's arguments, namely, the four famous paradoxes of motion and a "paradox of plurality" (which we shall construe as a geometrical paradox). Since we do not have any text which路 even pretends to quote Zeno directly on the four paradoxes of motion, we shall have to be content with paraphrases. Our primary source is Aristotle, who makes it quite clear that he is not quoting Zeno. Since Russell presents (below) a standard version of the paradoxes of motion, I shall simply attempt to make as clear as possible the gist of each.

Imagine that Achilles, the fleetest of Greek warriors, is to run a footrace against a tortoise. It is only fair to give the tortoise a head start. Under these Circumstances, Zeno argues, Achilles can never catch up with the tortoise, no matter how fast he runs. In order to overtake the tortoise, Achilles must run from his starting point A to the tortoise's original starting point T0 (see Figure 8.2). While he is doing that, the tortoise will have moved ahead to T1 Now Achilles must reach the point T1 While Achilles is covering this new distance, the tortoise moves still farther to T2. Again, Achilles must reach this new position of the tortoise. And so it continues; whenever Achilles arrives at a point where the tortoise was, the tortoise has already moved a bit ahead. Achilles can narrow the gap between him and the tortoise, but he can never actually catch up with him. This is the most famous of all of Zeno's paradoxes. It is sometimes known simply as "The Achilles."

FIGURE 8.2

FIGURE 8.2 Gorvin, Eric. "Zeno's Paradoxes". The Infinity Project. 2012.

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# 2 Th e Dic h o to my

#3 The Arrow

This paradox comes in two forms. According to the first, Achilles cannot get to the end of any racecourse, tortoise or no tortoise; indeed, he cannot finish covering any finite distance. Thus he cannot even reach the original starting point To of the tortoise in the previous paradox. Zeno argues as follows. Before the runner can cover the whole distance he must cover the first half of it.

In this paradox, Zeno argues that an arrow in flight is always at rest. At any given instant, he claims, the arrow is where it is, occupying a portion of space equal to itself. During the instant it cannot move, for that would require the instant to have parts, and an instant is by definition a minimal and indivisible element of time. If the arrow did move during the instant it would have to be in one place at one part of the instant, and in a different place at another part of the instant. Moreover, for the arrow to move during the instant would require that during the instant it must occupy a space larger than itself, for otherwise it has no room to move. As Russell says," It is never moving, but in some miraculous way the change of position has to occur between the instants, that is to say, not at any time whatever". This paradox is more difficult to understand than "Achilles and the Tortoise" or either form of "The Dichotomy," but another remark by Russell is apt: "The more the difficulty is meditated, the more real it becomes"

Then he must cover the first half of the remaining distance, and so on. In other words, he must first run one-half, then an additional one-fourth, then an additional one-eighth, etc., always remaining somewhat short of his goal. Hence, Zeno concludes, he can never reach it. (This form of the paradox has very nearly the same force as "Achilles and the Tortoise," the only difference being that in "The Dichotomy" the goal is stationary, while in "Achilles and the Tortoise" it moves, but at a speed much less than that of Achilles.) The second form of "The Dichotomy" attempts to show, worse yet, that the runner cannot even get started (Figure 8.3). Before he can complete the full distance, he must run half of it . But before he can complete the first half, he must run half of that, namely, the first quarter. Before he can complete the first quarter, he must run the first eighth. And so on. In order to cover any distance no matter how short, Zeno concludes, the runner must already have completed an infinite number of runs. Since the sequence of runs he must already have completed has the form of a regression (...1/16, 1/8. 1/4. 1/2,), it has no first member, and hence, the runner cannot even get started.

FIGURE 8.3

102

FIGURE 7.2-7.6 Gorvin, Eric. The Infinity Project. 2012.

FIGURE 8.4

# 4 The Stadium

En di ng Comments

Consider three rows of objects, A, B, and C, arranged as indicated in the first position of Figure 8.5. Then, while row A remains at rest, rows B and C move in opposite directions until all three rows are lined up as shown in the second position (Figure 8.6). In the process, C passes twice as many B's as A's; it lines up with the first A to its left, but with the second B to its left. According to Aristotle, Zeno concluded that "double the time is equal to half."

It has been suggested that Zeno's arguments fit into an overall pattern. "Achilles and the Tortoise" and "The Dichotomy" are designed to refute the doctrine that space and time are continuous, while "The Arrow" and "The Stadium" are intended to refute the view that space and time have an atomic structure. Thus, it has been argued, Zeno tries to cut off all possible avenues of escape from the conclusion that space, time, and motion are not real but illusory.

Some such conclusion would be warranted if we assume that the time it takes for C to pass to the next B is the same as the time it takes to pass to the next A, but this assumption seems patently false. It appears that Zeno had no appreciation of relative speed, assuming that the speed of relative to B is the same as the speed of C relative to A. If that were the only foundation for the paradox we would have no reason to be interested in it, except perhaps as a historical curiosity. It turns out, however, as both Russell and Owen show, that there is an interpretation of this paradox which gives it serious import.

How, one might ask, has mathematics so successfully tamed and incorporated the infinite? Incorporated it moreover at such a basic, elemental level, in so allpervasive a way? How does infinity get to be an exact, rigorously specified mathematical object—an object about which mathematics delivers “true” and “objective” knowledge? Mathematics starts from the integers. Its entire formalism opens out from the sequence 1, 2, 3,... that mathematicians call the “natural” numbers. The question can therefore be particularized: What does it mean to say of these numbers that they are infinite, that they form a progression which is endless? In what sense are they natural, that is to say, before, independent, and outside of us?

Suppose, as people occasionally do, that space and time are atomistic in character, being composed of spaceatoms and time-atoms of nonzero size, rather than being composed of points and instants whose size is zero. Under these circumstances, motion would consist in taking up different discrete locations at different discrete instants. Now, if we suppose that the A's are not moving, but the B's move to the right at the rate of one place per instant while the C's move to the left at the same speed, some of the C's get past some of the B's without ever passing them. C begins at the right of B and it ends up at the left of B, but there is no instant at which it lines up with B; consequently, there is no time at which they pass each other—it never happens.

FIGURE 8.5

FIGURE 8.6

103

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Change & Identity34 Th e S h ip o f These u s The "problem of change and identity" is generally explained with the story of the Ship of Theseus: In ancient times, there was a ship, called the "Theseus" after its famous former owner. As the years wore on, the Theseus started getting weak and creaky. The old boards were removed, put into a warehouse, and replaced with new ones. Then, the masts started tottering, and soon they, too, were warehoused and replaced. And in this way, after fifty years, this ship now has all new boards, masts, and everything. The question then arises: Is the ship in the harbor, now called S2, the same ship as the ship that was in the harbor, fifty years ago (called S1, for convenience)? In other words, is S2 really the "Theseus"? There is one answer which is a little too easy and quick. One might say: "No, of course not. The Theseus has changed a lot, so it's not the same ship. At the end of your life, you're not going to be the same person as you were, when you were a teenager. You're going to change a lot in the meantime." However, this is not quite answering the intended question. What is intended by the question is the sense of the word, "same", in which an old woman is the same person at the end of her life as she is at the beginning of her life. Certainly, the word, "same", has such a sense. After all, one implicitly depends on it when one says, for example, "She has changed a lot". In order for someone to change a lot, there has to be one person who underwent the change. (One could perhaps reject that sense, saying that objects do not change over time.) Going back to the definition of "change", an object changes with respect to a property if the object has that property at one time, and at a later time, the object does not have the property. What changes is the fact that the object has a particular property. The only way that that fact can change is if the object remains in existence. One can therefore think of a continuing object as the ground of change, or the arena where change occurs, as it were. To get back to the Theseus, the question is: Has the Theseus merely changed a lot, or is the Theseus gone, being replaced by a new ship?

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One may say, "Sure, it's just a refurbished Theseus, greatly changed to be sure, but still the Theseus". If one thinks in this manner, then consider what happens when the story is extended further. Suppose someone buys all the planks, masts and whatever that is stored in the warehouse, and out of all of those materials, and absolutely no others, he builds a ship according to the same plans that were used to build the ship, christened "the Theseus". And this ship, called S3, is launched and sits on the other side of the harbor where S2 sits. Is S3 the same as S1? In other words, is this recently-constructed ship the same ship as the ship originally called the "Theseus", considering that S3 was built out of the same materials, and according to the same plans as S1. One could take this concept even further by considering not only the properties of the ship, but also the subject matter of the "ship". What if instead the warehoused planks, masts, and other materials were used to build something completely different from a ship, like a house. (A concept explored by the artist Simon Starling, who turned a shed into a working boat and then back into a shed, winning him the 2005 Turner Prize.) The same materials and supplies are being used; yet they have taken on a new form. This relates to the concept of recreation vs. destruction. Inevitably, the problem arises: How can one ever say that both S2 and S3 are the same ship as S1, the original Theseus? This is because if they were both the same as S1, then they would have to be the same as each other. This follows from transitivity, which states that if x = y and x = z, then y = z. With S2 and S3 being clearly different ships, sitting on opposite sides of the harbor, three choices present themselves: 1) S2 is the same ship as S1; 2) S3 is the same ship as S1; or 3) neither is the same ship as S1, and S1 has ceased to exist.

How does one then decide which is the correct answer in this case? It is difficult to tell. Whenever one makes an

34 "Identity and Change." Wikipedia. Wikimedia Foundation, 5 Mar. 2012. Web. 9 Apr. 2012.

Entry identity claim (i.e. a claim which states that two things are the same), one almost always uses two different descriptions. Sometimes, one may say, "x = x", like "I am I", but such claims are not particularly interesting or informative. The interesting identity claims are claims where two different descriptions are used for one and the same thing. As an example, take these two descriptions: "the Morning Star", and "the Evening Star". Sometimes, one can look in the sky just before dawn, and see a very bright point of light — that has been called "the Morning Star". And then also, one can look in the sky just after sunset, and see a very similar point — that has been called "the Evening Star". The Morning Star is, in fact, identical to the Evening Star — both are the planet Venus. As such, they are "two" things, only in description, but in actuality, are one and the same thing under two different descriptions. It is a similar case with S1, S2, and S3, those being three different abbreviations, standing for the following descriptions: "S1", referring to the ship which sat in the harbor fifty years ago, newly christened "the Theseus"; "S2", referring to the ship which sits in the harbor now, with the new planks; and "S3", referring to the ship which sits in the harbor, recently constructed out of the old planks.

When one, therefore, asks a question like, "Is S2 the same as S1?", one can be understood to mean this: "Is the ship which sits in the harbor now, with the new planks, the same ship as the ship which sat in the harbor fifty years ago, newly christened 'the Theseus'?" Do those two descriptions refer to the same thing, or do they not?

The problem of identity & change over time seems like it may be better fit, if anything, in the chapter "On Time". But, I think it suits us better to talk about it here. To me, the Ship of Theseus talks about this problem of time and change, but more broadly it's talking about identity. I think this is the really important part. We are always naming things, identifying with things, defining our lives by things, and everything becomes about things that we can name. In fact, this very problem in semantics has become the root study of many famed philosophers. The problem with identifying something is that you are not just identifying it in space, but also in time. This creates a paradoxical situation where the thing is being defined by the thing and it doesn't work anymore—hence, "the problem of change and identity". This problem permeates much of infinity and should certainly considered when thinking about its relationship with paradox and the self-reflective process that identity constantly undergoes.

Philosophers are not interested in the "Ship of Theseus" problem per se, but to a more basic problem which is this: How does one decide that X is the same as Y, where X describes something at one time, and Y describes another thing at a later time? This is called the "problem of identity over time", or alternatively, the "problem of change".

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FIGURE 8.8 Bergamini, David. Mathematics. New York: Time, 1963. Print.

“It is clear that nothing can be in itself as its primary place. Zeno's puzzle—that if places exist then they will be in something—is not difficult to resolve. For nothing prevents the primary place of a thing from being in something else— but not in it as in a place.”—Early Greek Philosophy, pg. 157

“If there are many, they must be just as many as they are and neither more nor less than that. But if they are as many as they are, they would be limited. If there are many, things that are are unlimited. For there are always others between the things that are, and again others between those, and so the things that are are unlimited.”—Aristotle, Physics

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FIGURE 9.1 Christopher Dela Pole, "Distant Lovers". 2009.

On Time

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“But now the sight of day and night, and the months and the revolutions of the years, have created number, and have given us a conception of time, and the power of enquiring about the nature of the universe; and from this source we have derived philosophy, than which no greater good ever was or will be given by the gods to mortal man.”—Plato, Timaeus

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Formal Time35 De finitio n Time is a dimension in which events can be ordered from the past through the present into the future, and also the measure of durations of events and the intervals between them. Time has long been a major subject of study in religion, philosophy, and science, but defining it in a manner applicable to all fields without circularity has consistently eluded scholars. Nevertheless, diverse fields such as business, industry, sports, the sciences, music, dance, and the live theater all incorporate some notion of time into their respective measuring systems. Some simple, relatively uncontroversial definitions of time include "time is what clocks measure" and "time is what keeps everything from happening at once".

In Ph ilo so phy Two contrasting viewpoints on time divide many prominent philosophers. One view is that time is part of the fundamental structure of the universeâ&#x20AC;&#x201D;a dimension independent of events, in which events occur in sequence. Sir Isaac Newton subscribed to this realist view, and hence it is sometimes referred to as Newtonian time. The opposing view is that time does not refer to any kind of "container" that events and objects "move through", nor to any entity that "flows", but that it is instead part of a fundamental intellectual structure (together with space and number) within which humans sequence and compare events. This second view, in the tradition of Gottfried Leibniz and Immanuel Kant, holds that time is neither an event nor a thing, and thus is not itself measurable nor can it be travelled.

continuum called spacetime bring questions about space into questions about time, questions that have their roots in the works of early students of natural philosophy. Furthermore, it may be that there is a subjective component to time, but whether or not time itself is "felt", as a sensation or an experience, has never been settled. Temporal measurement has occupied scientists and technologists, and was a prime motivation in navigation and astronomy. Periodic events and periodic motion have long served as standards for units of time. Examples include the apparent motion of the sun across the sky, the phases of the moon, the swing of a pendulum, and the beat of a heart. Currently, the international unit of time, the second, is defined in terms of radiation emitted by caesium atoms. Time is also of significant social importance, having economic value ("time is money") as well as personal value, due to an awareness of the limited time in each day and in human life spans.

In Ph y sics Time is one of the seven fundamental physical quantities in the International System of Units (Figure 9.2). Time is used to define other quantities â&#x20AC;&#x201C; such as velocity â&#x20AC;&#x201D; so defining time in terms of such quantities would result in circularity of definition. An operational definition of time, wherein one says that observing a certain number of repetitions of one or another standard cyclical event (such as the passage of a free-swinging pendulum) constitutes one standard unit such as the second, is highly useful in the conduct of both advanced experiments and everyday affairs of life. The operational definition leaves aside the question whether there is something called time, apart from the counting activity just mentioned, that flows and that can be measured. Investigations of a single

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35 "Time." Wikipedia. Wikimedia Foundation, 15 Mar. 2013. Web. FIGURE 9.2 Gorvin, Eric. "International System of Units". The Infinity Project. 2013.

FIGURE 9.2

FIGURE 9.3 Williams, John E., and Charles Elwood Dull. "Matter and Energy." Modern Physics. New York: Holt, Rinehart and Winston, 1968. 25. Print.

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“This time of the present is explicated as a sequence Constantly rolling through the now, a sequence Whose directional sense is said to be singular and irreversible. Everything that occurs rolls out of an infinite future into an irretrievable past...” —Martin Heidegger, The Concept of Time

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Infancy & History36 Th e C ontin u u m Since the human mind has the experience of time but not its representation, it necessarily pictures time by means of spatial images. The Graeco-Roman concept of time is basically circular and continuous. Puech writes: "Dominated by a notion of intelligibility which assimilates the full, authentic being to what is in him and corresponds to him, to the eternal and the immutable, the Greek regards movement and becoming as inferior degrees of reality, where correspondence is at best only understood as permanence and perpetuity, in other words as return. Circular movement, which guarantees the unchanged preservation of things through their repetition and continual return, is the most direct and most perfect expression (and therefore the closest to the divine) of the zenith of the hierarchy: absolute immobility." In Plato's Timaeus time is measured by the cyclical revolution of the celestial spheres and defined as a moving image of eternity. 'The creator of the world constructed a moving image of eternity, and, in ordering the heavens, from eternity one and unshifting he made this image which ever moves according to the laws of number and which we call time.' Aristotle confirms the circular nature of time in these terms: "...and so time is regarded as the rotation of the sphere, inasmuch as all other orders of motion are measured by it, and time itself is standardized by reference to it. And this is the reason of our habitual way of speaking; for we say that human affairs and those of all other things that have natural movement...seem to be in a way circular, because all these things come to pass in time and have their beginning and end as it were 'periodically'; for time itself is conceived as coming round; and this again because time and such a standard rotation mutually determine each other. Hence, to call the happenings of a thing a circle is saying that there is a sort of circle of time…" The first outcome of this conception is that time, being essentially circular, has no direction. Strictly speaking, it has no beginning, no middle and no end—or rather, it has them only in so far as its circular motion returns unceasingly back on itself. A singular passage in Aristotle's Problemata explains that from this point of view it is impossible to say whether we are before or after the Trojan War:

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"Do those who lived at the time of the Trojan War come before us, and before them those who lived in an even more ancient time, and so on to infinity, those men most remote in the past coming always before the rest? Or else, if it is true that the universe has a beginning, a middle and an end; that what in aging reaches its end to find itself therefore back at the beginning; if this is true, on the other hand, that the things that are closest to the beginning come before, what then prevents us from being closer to the beginning than those who lived at the time of the Trojan War?… If the sequence of events forms a circle, since the circle has indeed neither beginning nor end, we cannot, by being closer to the beginning, come before them any more than they can be said to come before us." But the fundamental character of the Greek experience of time—which, through Aristotle's Physics, has for two millennia determined the Western representation of time—is its being a precise, infinite, quantified continuum. Aristotle thus defines time ..as 'quantity of movement according to the before and the after and its continuity is assured by its division into discrete instants [the now], analogous to the geometric point. The instant in itself is nothing more than the continuity of time, a pure limit which both joins and divides past and future. As such, it is always elusive, and Aristotle expresses its paradoxically nullified character in the statement that in dividing time infinitely, the now is always 'other'; yet in uniting past and future and ensuring continuity, it is always the same; and in this is the basis of the radical 'otherness' of time, and of its 'destructive' character: "And besides, since the 'now' is the end and the beginning of time, but not of the same time, but the end of time past and the beginning of time to come, it must present a relation analogous to the kind of identity between the convexity and the concavity of the same circumference, which necessitates a difference between that with respect to which it bears the other."

36 Agamben, Giorgio. Infancy and History: The Destruction of Experience. London: Verso, 1993.

3 Conceptions of Time Outlo o ks 37 Contemporary science represents a progressive view of the world that relies on the incessant accumulation of data to improve knowledge and reach through it new stages of enlightenment. By virtue of such accumulation, Science knows today more than yesterday and less than tomorrow (the technological applications serving as measure of her knowledge). As a result we sense a forward movement in time which expands the physical powers of humanity. This movement conveys the idea that there is more enlightenment and power farther on the same road, science itself being the vehicle taking us to our destination. From this progressive perspective, the present appears illuminated if compared to the past, but dark when compared to the future. Accordingly, the present looks like a "twilight zone" from which we perceive the light ahead and the darkness behind. We notice the present's intrinsical weakness only as it plunges into the past, being turned into "Darkness"--a metaphor for ignorance, superstition, and inability to develop complex technologies (Figure 9.6). On the other hand, the study of antiquity, and particularly of mythology, invites an opposite posture. "Recent sources" are seldom as respectable as the most ancient, while "later additions" often awake suspicion. It is as if a light were shining in a remote past. The closer to that light an author is, the more "authentic" his testimony will be. For it is assumed that later authors "invented", "filled gaps", "committed errors", or even "lied", thus obscuring the real meaning of their tradition. A typical traditionalist cherishes the past, persuaded that a legacy of wisdom was condensed in it for all times to come. He believes that as Time gradually removes us from the original source of everything, we forget our nature, our identity, and our purpose. One hundred and eighty degrees separate the traditionalist view from the progressive (Figure 9.7). Yet a third conception may be distinguished, generally embraced by religions contemplating salvation. For the salvationist, revelation is the light of the past, and redemption that of the future. In this manner both preceding views are combined: the past enlightens man, and the future redeems him. The Progressive and Salvationist views tend to be linear in their appreciation of Time whereas the Traditionalist conception often implies that light and darkness recur in a circle or cycle (Figure 9.8).

Progressive View Past

Present

Future

FIGURE 9.6

Today is worse than tomorrow but better than yesterday. The past is the darkest or â&#x20AC;&#x153;worstâ&#x20AC;? section of time.

Traditionalist View Past

Present

Future

FIGURE 9.7

Today is worse than yesterday but better than tomorrow. The future is the darkest or "worst" section of time.

Salvationist View Past

Present

Future

FIGURE 9.8

Revelation is in the past and redemption is in the future. The present is the darkest or "worst" section of time.

Linea r & Cyclical Time 38 Ancient cultures such as Incan, Mayan, Hopi, and other Native American Tribes, plus the Babylonians, Ancient Greeks, Hinduism, Buddhism, Jainism, and others have a concept of a wheel of time, that regards time as cyclical and quantic consisting of repeating ages that happen to every being of the Universe between birth and extinction. In general, the Judaeo-Christian concept, based on the Bible, is that time is linear, beginning with the act of creation by God. The general Christian view is that time will end with the end of the world.

37 Parada, Carlos, and Maicar FĂśrlag. "Chronos." Greek Mythology Link. N.p., 1997. Web. 38 "Time." Wikipedia. Wikimedia Foundation, 15 Mar. 2013. Web.

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The Arrow Of Time A n In t roductio n 39 The words past and future do not appear in the laws of physics. There is no difference between one direction of time and the other in the ultimate elementary laws of physics. What we do is we assign the words past and future to the direction of lower entropy and higher entropy. Entropy is a way of measuring the disorderliness of any system—whether it is the universe or a cup of coffee or anything like that. So if things are precisely organized, in particular if they’re segregated so that your cream is over here and your coffee is over here, that’s low entropy, that’s a high amount of organization. Then you mix them together and the entropy goes up. And it’s a law of nature if you leave things by themselves entropy always goes up. Things go from being orderly, very delicately organized, to being messier, to being more disorganized. And so we tend therefore to associate time with not just change, but a certain directed kind of change. The direction of time in which the entropy was lower we call the past. The direction of time in which entropy is increasing we call the future. That’s what defines the arrow of time. And all of that is part of this underlying dynamic caused by the fact that the universe started very organized and is becoming ever more disorganized as it expands and cools and things happen.

In t e r vie w W it h Sea n Carro ll 40 What is the arrow of time? The past is different from the future. One of the most obvious features of the macroscopic world is irreversibility: heat doesn't flow spontaneously from cold objects to hot ones, we can turn eggs into omelets but not omelets into eggs, ice cubes melt in warm water but glasses of water don't spontaneously give rise to ice cubes. We remember the past, but not the future; we can take actions that affect the future, but not the past (we can't unto our mistakes). We are all born, then age, then die; never the other way around. The distinction between past and future seems to be consistent throughout the observable universe. The arrow of time is simply that distinction, pointing from past to future. Why is there such an arrow? Irreversible processes are summarized by the Second Law of Thermodynamics: the entropy of a closed system will (practically) never decrease into the future. It's a bedrock foundation of modern physics.

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What's "entropy"? Entropy is a measure of the disorder of a system. A nice organized system, like an unbroken egg or a neatlyarranged pile of papers, has a low entropy; a disorganized system, like a broken egg or a scattered mess of papers, has a high entropy. Left to its own devices, entropy goes up as time passes. But entropy decreases all the time; we can freeze water to make ice cubes, after all. And life evolved. Not all systems are closed. The Second Law doesn't forbid decreases in entropy in open systems -- by putting in the work, you are able to tidy up your room, decreasing its entropy but still increasing the entropy of the whole universe (you make noise, burn calories, etc.). Nor is it in any way incompatible with evolution or complexity or any such thing. Should we be surprised? The first mystery of the arrow of time is that it's nowhere to be found in the fundamental laws of physics. Those laws work perfectly well if we run processes backwards in time. (More rigorously, for every allowed process there exists a time-reversed process that is also allowed, obtained by switching parity and exchanging particles for antiparticle—the CPT Theorem.) Nevertheless, the macroscopic world we observe is full of irreversible processes. The puzzle is to reconcile microscopic reversibility with macroscopic irreversibility. And how do we reconcile them? The observed macroscopic irreversibility is not a consequence of the fundamental laws of physics, it's a consequence of the particular configuration in which the universe finds itself. In particular, the unusual lowentropy conditions in the very early universe, near the Big Bang. Understanding the arrow of time is a matter of understanding the origin of the universe. Wasn't this all figured out over a century ago? Not exactly. In the late 19th century, Boltzmann and Gibbs figured out what entropy really is: it's a measure of the number of individual microscopic states that are macroscopically indistinguishable. An omelet is higher entropy than an egg because there are more ways to re-arrange its atoms while keeping it indisputably an omelet, than there are for the egg. That provides half of the explanation for the Second Law: entropy tends to increase because there are more ways to be high entropy than low entropy. The other half of the question still remains: why was the entropy ever low in the first place?

39 Levin, David. "Sean Carroll on Time." Nova Beta. PBS, 08 Nov. 2011. Web. 08 June 2012 40 "Arrow of Time FAQ : Cosmic Variance." Cosmic Variance. Discovery Magazine, 3 Dec. 2007. Web. 08 June 2012.

A New Rationality41 Is the origin of the Second Law really cosmological? We never talked about the early universe back when I took thermodynamics. Trust me, it is (or trust Richard Feynman, if you don't trust me). Of course you don't need to appeal to cosmology to use the Second Law, or even to "derive" it under some reasonable-sounding assumptions. However, those reasonable-sounding assumptions are typically not true of the real world. Using only time-symmetric laws of physics, you can't derive time-asymmetric macroscopic behavior (as pointed out in the "reversibility objections" of Lohschmidt and Zermelo back in the time of Boltzmann and Gibbs); every trajectory is precisely as likely as its timereverse, so there can't be any overall preference for one direction of time over the other. The usual "derivations" of the second law, if taken at face value, could equally well be used to predict that the entropy must be higher in the past -- an inevitable answer, if one has recourse only to reversible dynamics. But the entropy was lower in the past, and to understand that empirical feature of the universe we have to think about cosmology.

Earlier this century in The Open Universe: An Argument for Indeterminism, Karl Popper wrote, “Common sense inclines, on the one hand, to assert that every event is caused by some preceding events, so that every event can be explained or predicted .... On the other hand,... commonsense attributes to mature and sane human persons . . . the ability to choose freely between alternative possibilities of acting.” This “dilemma of determinism,” as William James called it, is closely related to the meaning of time. Is the future given, or is it under perpetual construction? A profound dilemma for all of mankind, as time is the fundamental dimension of our existence. It was the incorporation of time into the conceptual scheme of Galilean physics that marked the origins of modern science. This triumph of human thought is also at the root of the main problem addressed by this book: the denial of what has been called the arrow of time. As is well known, Albert Einstein often asserted, “Time is an illusion.” Indeed time, as described by the basic laws of physics, from classical Newtonian dynamics to relativity and quantum physics, does not include any distinction between past and future. Even today, for many physicists it is a matter of faith that as far as the fundamental description of nature is concerned, there is no arrow of time. (Continued on pg. 124)

41 Prigogine, I., and Isabelle Stengers. The End of Certainty: Time, Chaos, and the New Laws of Nature. New York: Free, 1997. Print

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FIGURE 9.6 Gorvin, Eric. "Moleskine Entry". The Infinity Project. 2013.

Entry I keep thinking of time more and more relatively. It's easy to get caught up in the scientific side of things. Once I learned of the arrow of time I got really caught up in the lack of possibility for any kind of infinite recursion. But when we talk about the arrow of time, we're really talking about entropy. And entropy is really talking about the movement of order to disordered and its relationship to the development of the universe. So, in effect, the arrow of time is really talking about the genesis of the universe, the origin of matter. Since I'm talking about infinite, imaginary time (not just what has occurred in the last 14 billion years â&#x20AC;&#x201D; though, this is also important to know), I'm looking for the what's next, and what's after that. If we can manage to zoom out even further from our 14 billion year old universe and look at what is at the start and the finish line of its trajectory, it becomes more and more obvious to me that everything appears to be an arbitrary node traveling through space. There may be many universes everywhere doing what we are doing - traveling from a state of low entropy to high entropy - and there may even be universes that travel backwards in time thus reversing the arrow of time all together (many prefer to think that this might be the eventuality of our universe!). If it is theoretically possible for the arrow of time to be reversed, I have to conclude that time (the 4th dimension) is just as much a malleable and changeable of a dimension as the other 3 dimensions. I like to think that mastering the fourth dimension is at the frontier of human evolution. I believe that if we can successfully harness time as it's own changeable medium, we will begin to understand some of the truths associated with it much more thoroughly, getting us much closer to understanding why we exist.

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Yet everywhere in chemistry, geology, cosmology, biology, and the human sciences—past and future play different roles. How can the arrow of time emerge from what physics describes as a time-symmetrical world? This is the time paradox. The time paradox was identified only in the second half of the nineteenth century after the Viennese physicist Ludwig Boltzmann tried to emulate what Charles Darwin had done in biology in an effort to formulate an evolutionary approach to physics. The laws of Newtonian physics had long since been accepted as expressing the ideal of objective knowledge. As they implied the equivalence between past and future, any attempt to confer a fundamental meaning on the arrow of time was resisted as a threat to this ideal. Isaac Newton’s laws were considered final in their domain of application, somewhat the way quantum mechanics is now considered to be final by many physicists. How then can we introduce unidirectional time without destroying these amazing achievements of the human mind? Since Boltzmann, the arrow of time has been relegated to the realm of phenomenology. We, as imperfect human observers, are responsible for the difference between past and future through the approximations we introduce in our description of nature. This is still the prevailing scientific wisdom. Certain experts lament that we stand before an unsolvable mystery for which science can provide no answer. We believe that this is no longer the case because of two recent developments: the spectacular growth of nonequilibrium physics and the dynamics of unstable systems, beginning with the idea of chaos. Over the past several decades, a new science has been born, the physics of nonequilibrium processes, and has led to concepts such as self-organization and dissipative structures, which are widely used today in a large spectrum of disciplines, including cosmology, chemistry, and biology, as well as ecology and the social sciences. The physics of nonequilibrium processes describes the effects of unidirectional time and gives fresh meaning to the term irreversibility. In the past, the arrow of time appeared in physics only through simple processes such as diffusion or viscosity, which could be understood without any extension of the usual time-reversible dynamics. This is no longer the case. We now know that irreversibility leads to a host of novel phenomena, such as vortex formation, chemical oscillations, and laser light,

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all illustrating the essential constructive role of the arrow of time. Irreversibility can no longer be identified with a mere appearance that would disappear if we had perfect knowledge. Instead, it leads to coherence, to effects that encompass billions and billions of particles. Figuratively speaking, matter at equilibrium, with no arrow of time, is “blind,” but with the arrow of time, it begins to “see.” Without this new coherence due to irreversible, nonequilibrium processes, life on earth would be impossible to envision. The claim that the arrow of time is “only phenomenological,” or subjective, is therefore absurd. We are actually the children of the arrow of time, of evolution, not its progenitors. The second crucial development in revising the concept of time was the formulation of the physics of unstable systems. Classical science emphasized order and stability; now, in contrast, we see fluctuations, instability, multiple choices, and limited predictability at all levels of observation. Ideas such as chaos have become quite popular, influencing our thinking in practically all fields of science, from cosmology to economics. As we shall demonstrate, we can now extend classical and quantum physics to include instability and chaos. We are then able to obtain a formulation of the laws of nature appropriate for the description of our evolving universe, a description that contains the arrow of time, since past and future no longer play symmetrical roles. In the classical view-and here we include quantum mechanics and relativity-laws of nature express certitudes. When appropriate initial conditions are given, we can predict with certainty the future, or “retrodict” the past. Once instability is included, this is no longer the case, and the meaning of the laws of nature changes radically, for they now express possibilities or probabilities. Here we go against one of the basic traditions of Western thought, the belief in certainty. As stated by Gerd Gigerenzer et al. in The Empire of Chance, “Despite the upheavals in science in the over two millennia separating Aristotle from the Paris of Claude Bernard, they shared at least one attitude of faith: Science was about causes, not chance. Kant even promoted universal causal determinism to the status of a necessary condition of all scientific knowledge.” There were, however, dissenting voices. The great physicist James Clerk Maxwell spoke of a “new kind of knowledge” that would overcome the prejudice of determinism. But, on the whole, the prevailing opinion was that probabilities were states of mind rather than states of the world. This is so even today in spite of the fact that

41 Prigogine, I., and Isabelle Stengers. The End of Certainty: Time, Chaos, and the New Laws of Nature. New York: Free, 1997. Print

quantum mechanics has included statistical concepts in the core of physics. But the fundamental object of quantum mechanics, the wave function, satisfies a deterministic, time-reversible equation. To introduce probability and irreversibility, the orthodox formulation of quantum mechanics requires an observer. Through his measurements, the observer would bring irreversibility to a time-symmetric universe. Again, as in the time paradox, we would be responsible in some sense for the evolutionary patterns of the universe. This role of the observer, which gave quantum mechanics its subjective flavor, was the main reason that prevented Einstein from endorsing quantum mechanics, and it has since led to unending controversies. The role of the observer was a necessary concept in the introduction of irreversibility, or the flow of time, into quantum theory. But once it is shown that instability breaks time symmetry, the observer is no longer essential. In solving the time paradox, we also solve the quantum paradox, and obtain a new, realistic formulation of quantum theory. This does not mean a return to classical deterministic orthodoxy; on the contrary, we go beyond the certitudes associated with the traditional laws of quantum theory and emphasize the fundamental role of probabilities. In both classical and quantum physics, the basic laws now express possibilities. We need not only laws, but also events that bring an element of radical novelty to the description of nature. This novelty leads us to the “new kind of knowledge” anticipated by Maxwell. For Abraham De Moivre, one of the founders of the classical theory of probabilities, chance can neither be defined nor understood. As we shall illustrate, we are now able to include probabilities in the formulation of the basic laws of physics. Once this is done, Newtonian determinism fails; the future is no longer determined by the present, and the symmetry between past and future is broken. This confronts us with the most difficult questions of all: What are the roots of time? Did time start with the “big bang”? Or does time preexist our universe? These questions place us at the very frontiers of space and time. A detailed explanation of the cosmological implications of our position would require a special monograph. Briefly stated, however, we believe that the big bang was an event associated with an instability within the medium that produced our universe. It marked the start of our universe but not the start of time. Although our universe has an age, the medium that produced our universe has none. Time has no beginning, and probably no end.

But here we enter the world of speculation. The main purpose of this book is to present the formulation of the laws of nature within the range of low energies. This is the domain of macroscopic physics, chemistry, and biology. It is the domain in which human existence actually takes place. The problems of time and determinism have remained at the core of Western thought since the pre-Socratics. How can we conceive of human creativity or ethics in a deterministic world? This question reflects a profound contradiction in Western humanistic tradition, which emphasizes the importance of knowledge and objectivity, as well as individual responsibility and freedom of choice as implied by the ideal of democracy. Popper and many other philosophers have pointed out that we are faced with an unsolvable problem as long as nature is described solely by a deterministic science. Considering ourselves as distinct from the natural world would imply a dualism that is difficult for the modern mind to accept. Our aim in this work is to show that we can now overcome this obstacle. If 'the passion of the western world is to reunite with the ground of its being,' as Richard Tarnas has written, perhaps it is not too bold to say that we are closing in on the object of our passion. Mankind is at a turning point, the beginning of a new rationality in which science is no longer identified with certitude and probability with ignorance. We agree completely with Yvor Leclerc when he writes, 'In the present century we are suffering from the separation of science and philosophy which followed upon the triumph of Newtonian physics in the eighteenth century. Jacob Bronowski beautifully expressed the same thought in this way: 'The understanding of human nature and of the human condition within nature is one of the central themes of science.' At the end of this century, it is often asked what the future of science may be. For some, such as Stephen W. Hawking in his Brief History of Time, we are close to the end, the moment when we shall be able to read the “mind of God.” In contrast, we believe that we are actually at the beginning of a new scientific era. We are observing the birth of a science that is no longer limited to idealized and simplified situations but reflects the complexity of the real world, a science that views us and our creativity as part of a fundamental trend present at all levels of nature.

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FIGURE 9.7 Harold E. Edgerton courtesy E.G.&G. Inc., Boston.

FIGURE 9.8 Differential Equation: Differential equations are prominent in engineering, physics, economics, mathematics and many more disciplines. The Second Law of Thermodynamics (The Arrow of Time) is one of many famous differential equations.

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FIGURE 10.1 Guzmán, Andrés. The Infinity Project. 2012.

On Space

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The Universe42 P yt h a g ore a n s, On S pace In the first book of his work on Pythagorean philosophy Aristotle writes that the universe is one, and that time and breath and the void, which differentiates the places of all individual things, are drawn into the universe from the unlimited. ... All the things that exist must be either limiting or unlimited, or both limiting and unlimited. But they cannot be only unlimited. So since they evidently arise neither from things that are all limiters nor from things that are all unlimited, it clearly follows that the universe and its components were harmonized out of both things which limit and things which are unlimited. And the facts of things also make this clear, since some things arise from limiters and are limiters, while others arise from both limiters and unlimiteds and both limit and fail to impose limit, and others arise from unlimiteds and are plainly unlimited. ... On the subject of nature and harmony, this is how things stand: the being of things, qua eternal, and nature, itself are accessible only to divine and not human knowledge except that it is impossible for any of the things that exist and are known by us to have arisen without the prior existence of the being of the things out of which the universe is composed, namely limiters and unlimiteds. Now, since these sources existed in all their dissimilarity and incompatibility, it would have been impossible for them to have been made into an orderly universe unless harmony had been present in some form or other. Things that were similar and compatible had no need of harmony, but things that were dissimilar and incompatible and incommensurate had to be connected by this kind of harmony, if they are to persist in an ordered universe. ... The first thing to be harmonized, the one, in the centre of the sphere, is called the hearth. ... The universe is single. It originally arose from the centre, and from the centre upwards and downwards in the same way. For what is above the centre is the opposite in disposition to what is below, in the sense that to lower things the lowest part is like the highest part, and the same goes for the upper things too. For the relation to the centre is the same in either case, except that their positions are reversed.

FIGURE 10.2

FIGURE 10.3

FIGURE 10.4

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42 Waterfield, Robin. The First Philosophers: The Presocratics and Sophists. Oxford: Oxford UP, 2000. Print. FIGURE 10.2-10.3 Chaisson, Eric. Cosmic Dawn: The Origins of Matter and Life. Boston: Little, Brown, 1981. Print. FIGURE 10.4 Crab Nebula. FIGURE 10.5 (opposite page) Andromeda Galaxy. The closest galaxy to our own.

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Powers Of 1043 Out to Spa ce f ro m Earth

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43 Powers of Ten, 1978. Pyramid Films, 1978. Film.

In to Sk in/ Qu antu m

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â&#x20AC;&#x153;The Pythagoreans also claim that there is such a thing as a void. According to them, it enters the universe from the infinite breath because the universe breathes in void as well as breath. What void does, they say, is differentiate things; they think of void as being a kind of separation and distinction when one thing comes after another. This happens first among the numbers, because on their view it is the void that distinguishes one number from another.â&#x20AC;?â&#x20AC;&#x201D;Aristotle

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On Relationships Dark M atter Distribu tio n in Our Univ erse

FIGURE 10.6

Pho to gra ph o f Bra in N euro n s

FIGURE 10.7

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FIGURE 10.6 Bizony, Piers. Science: The Definitive Guide. London: Quercus, 2010. Print.

Flight Pattern s in th e USA

FIGURE 10.8

M ap Of Th e In tern e t

FIGURE 10.9

FIGURE 10.8 "Flight Patterns." UCLA Design|Media Arts User Pages. Web. 01 May 2012. FIGURE 10.9 Courtesy of The Opte Project

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FIGURE 10.10 Gorvin, Eric. "Studio Relationships". The Infinity Project. 2012.

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FIGURE 10.11 (Previous page) Gorvin, Eric. "Spiderwebs". The Infinity Project. 2013. FIGURE 10.12 Gorvin, Eric. "Ellipse Study". The Infinity Project. 2012.

â&#x20AC;&#x153;The universe is single. It originally arose from the centre, and from the centre upwards and downwards in the same way. For what is above the centre is the opposite in disposition to what is below, in the sense that to lower things the lowest part is like the highest part, and the same goes for the upper things too. For the relation to the centre is the same in either case, except that their positions are reversed.â&#x20AC;?â&#x20AC;&#x201D;Philolaus

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FIGURE 10.13 Gorvin, Eric. "Mars & The Moon". The Infinity Project. 2012.

Entry In & Ou t The relativity of the relationships astonishing. Just help immensely in

the Universe is something I've been thinking a lot about. All of that exist from the micro to macro view of our Universe/earth are being aware of our size in relation to the rest of the Universe can thinking about infinity.

In my first year of school at MCAD, I took the Introduction to Graphic Design class with Kindra Murphy. She assigned some kind of collage project as our first in-class project, and I don't really remember what the prompt was. But I do remember that I chose to make a hypothetical advertisement about going to space as a vacation spot or recreational destination. "See the world!" was the headline. As primitive as the collage was, my supporting argument was that I thought that if I everyone in the world went into orbit for even just one hour, that it would be enough to change anyone's perspective permanently. There are some things that you can't unsee, good and bad. I would argue that even the simplest of people would be changed forever. I would imagine that all of our differences with eachother would subside, and that the human race might just flourish and exist peacefully. A lot of people said that wasn't possible... But I dream of a day (that I believe to be rather soon) where going to space is just as expensive as international travel. I think this could be the first step in the humbling and unification of man-kind.

On Drea ms I like to dream up the fast-forward time lapse of the history of our earth, and watch as mankind makes its brief cameo. I like to think of all the other sped up time lapses of the rest of the planets in the Universe as well. Through all of the billions of years, and billions of planets, orbiting billions of stars, I like to think about all of the countless times civilizations shined as bright as ours. I like to think of all the permutations of different kinds of creatures and civilizations that might exist. I like to think of pure forms, I like to think of polluted forms. I like to dream that there are creatures like jellyfish that are the size of stars, or the size of whole star clusters, moving at a pace relative to their own size and unfathomable to ours. Observing the miracles of our Earth, I like to think about how they can be observed in the cosmos. I like to think about death. I like to dream of cyclical systems where void and matter are one. I like to conjure new galaxies where trees float above the land of their planets in a warm mist. I like to think about monsters. I like to think about dinosaurs, and how they are our closest idea to what is possible in alien life elsewhere. I like to hope that we will make contact with outside life. I like to think we will. I like to think about the Universe. I like understanding that it's you and me.

FIGURE 10.12 (next page) The Orion Nebula.

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“It was as if everything she thought, felt, remembered, had an aura; behind the briefest eye-blink, least flicker of touch, a shadow, a kind of ghost presence. This aura wasn't part of her, nor was it in any sense that she could fathom produced by her, nor did it seem answerable to her, even though—of this she was certain—only she was aware of it. Neither did it seem to precede her and her business: it wasn't there before— couldn't be anywhere—before she felt what she was able to feel, before she perceived what she—her body—decided she could perceive. And yet there it was, ghost of things present, faint pulsation of the real; at times like the glow on the surface of the universe, at others the dark outline of a world dazzled by there being nothing in it but its own presence. Often it was neither light, nor dark, nor anything visible, but just a presence—simply there—clinging to the motion of her being in space–like the field of a magnet, or radiation from the earth's rocks. Of late it had occurred to her: perhaps it was the aura that was real, felt things, had a body, sat and moved through space and perceived the countless pulsations of light and energy in the universe, and that she was the shadow clinging to it, following it around, copying its business before it had time to look around and be aware of who she was and how her very presence was no more than a confirmation of the aura's desperate need for something— anything—to keep it company.”—Brian Rotman, Aura 149

∞

The Infinity Project was created by

Eric Gorvin in Minneapolis, MN USA. 2011-2013 .

CONTACT ericgorvin@gmail.com ericgorvin.com @ericgorvin facebook.com/ericgorvin ericgorvin.tumblr.com soundcloud.com/theinfinityproject theinfinityproject.bandcamp.com

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