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the extended edition

Clocks, Orreries, etc.


It will be obvious to many readers that the Millennium Clocks mentioned in Anathem are inspired by The 10,000 Year Clock, a project conceived circa 1995 by Danny Hillis and currently under development by The Long Now Foundation. I first became aware of the idea, and discussed it with Danny and with Stewart Brand, at a Hackers Conference in the mid-1990s. In spring of 1999, as the Long Now Foundation was getting ready to launch a new Clock prototype, Danny asked me and several others to contribute back-of-napkin sketches to the Long Now Foundation's website. These were not intended as serious engineering proposals but as ways of showing how various people were thinking about the Clock. My contribution depicted a system of walls with apertures that would be opened at certain times by the Clock mechanism. A few other elements of the world depicted in Anathem may also be glimpsed in these doodles.


At the end of 1999, when the newspapers were publishing their annual end-of-year news summaries, as well as their decennial end-of-decade summaries and, in some cases, end-of-century and end-of-millennium news roundups as well, I began to think more seriously about the "clock monks" mentioned in these sketches and to wonder whether this might be the germ of a novel. At the time, though, I was just beginning serious work on The Baroque Cycle and so I placed the idea in deep storage and thought about it very little until The System of the World, the last of the Baroque Cycle novels, had been finished. Then I began to see possible connections between it and some other ideas in which I had become interested, and hence dusted it off and began work on the project in earnest.


The design of the Clock, the exploration of possible sites, and the development of prototype components--notably an orrery, which was unveiled in October 2005--have moved steadily forward during the last few years, largely thanks to generous support from (alphabetically by last name) Jeff Bezos, Bill Joy, Mitch Kapor, Nathan Myhrvold, Jay Walker, and an anonymous donor, as well as many others who have lent financial support, ideas, and expertise. I'm indebted to Stewart Brand, Alexander Rose, Doug Carlston, and Tomi Pierce, as well as those mentioned above, for conversations about the Clock project.


Philosophical and Scientific Ideas


If Anathem were nonfiction, the parts of it where philosophy and science are being discussed would be spattered with footnotes in which I would express my indebtedness to the thinkers of Earth who originated the ideas under discussion. Each of these footnotes would be carefully hedged, in roughly the following manner:


"If Person X had never thought up Idea Y and published it in Book Z, then I never could have written this; however, please bear in mind that (a) I have no formal credentials as a philosopher, mathematician, or scientist, and (b) this is a work of fiction, not a peer-reviewed monograph. Accordingly, the manner in which I have used Idea Y here might not stand up to rigorous scrutiny; Person X, if still alive, upon seeing his or her name mentioned in an academic-looking footnote in this context, would, therefore, probably issue a public disclaimer denying all connection with me and the book, and otherwise is rolling over in his or her grave. Dear reader, please know that this footnote serves only to acknowledge an intellectual debt and to give fair credit to Person X; if you really want to understand Idea Y, please buy and read Book Z."


Keeping that general framework in mind, here are the Xs and Ys and Zs.


Julian Barbour's book The Discovery of Dynamics (available in different editions with varying titles, ISBN 978-0195132021) was one of the most important sources of ideas for my earlier Baroque Cycle project. Almost every page of Anathem bears some imprint from his more recent The End of Time (ISBN 978-0965040808).


Barbour's footnotes and bibliographies provide enough citations to keep most readers busy for many years, however one that merits special notice here is John Stewart Bell's Speakable and Unspeakable in Quantum Mechanics (ISBN 978-0521818629), a collection of articles that includes two concise and clear explanations of the topic that, in the terminology of Anathem, passes under the name "worldtracks" or "narratives." They are to be found in the papers entitled The measurement theory of Everett and de Broglie's pilot wave (which, by the way, takes issue with the many-worlds interpretation) and Quantum Mechanics for Cosmologists.


Any material in Anathem that has to do with worldtracks, narratives, and "Hemn space" (which is just the Arbran term for what on Earth is called configuration, phase, or state space) can be traced back directly to the works mentioned above, and their antecedents in the literature. The notion of configuration spaces dates back to Joseph Louis Lagrange's work on generalized coordinates and later refinements  owed to William Rowan Hamilton. The term "phase space," according to Wikipedia, was introduced by Willard Gibbs in 1901; it can be taken as a further generalization of the work of Lagrange and Hamilton.


For an exceptionally beautiful account of mathematical physics with particular reference to action principles, read David Oliver's The Shaggy Steed of Physics (ISBN 0-387-40307-8).


The metaphysical thread linking the Baroque Cycle to Anathem begins with Gottfried Wilhelm Leibniz's Monadology, available in various translations, online and otherwise. The idea was submerged for much of the 18th and 19th Centuries but gained currency during the 20th as the inspiration for background-independent formulations of physics, a story that is told, in a form accessible to the general reader, both in Barbour's work and in Lee Smolin's The Trouble with Physics (ISBN 978-0618918683) and Three Roads to Quantum Gravity (ISBN 978-0465078363). Note also that his book The Life of the Cosmos (ISBN 978-0195126648) presents one of the more readable expositions of just how uncannily finely tuned the universe is for supporting life, a topic that comes under discussion in one of the Messals. Lee Smolin's hypothesis as to just how this might have come about is not, however, addressed in Anathem. Moreover, he would take issue with many of the ideas that are batted around in the Messal chapters, particularly Fraa Jad's position that time is an illusion. That being said, I would like to give him thanks and credit for discussing some of these topics with me during the time that I was writing the novel.


David Deutsch is, to my mind, the most eloquent and persuasive supporter of Hugh Everett's Many-Worlds Interpretation of Quantum Mechanics. His (Deutsch's) book The Fabric of Reality (ISBN 978-0140275414) sat unopened by my writing chair for a long time, as I was reluctant to come to grips with the many-worlds idea, but when I finally worked up my nerve to read it, I found that, without it, I could never have moved the Anathem project forward (what Deutsch calls the Multiverse, my characters call the Polycosm). An especially useful and interesting feature of this book is a brief reference to the work of the philosopher David Lewis, about whom more below.


The work of Roger Penrose is relevant to, and has influenced, Anathem in at least five ways:


1. Penrose posits, in The Emperor's New Mind (ISBN 978-0192861986) and Shadows of the Mind (ISBN 978-0195106466), that the human brain takes advantage of quantum effects to do what it does. This has been so controversial that I have found it impossible to have a dispassionate conversation about it with any learned person. The dispute can be broken apart into a number of different sub-controversies, some of which are more interesting than others. The science-fictional premise of Anathem is based on the relatively weak and modest assumption that natural selection has found some way to construct brains that, despite being warm and wet, are capable of exploiting the benefits of quantum computation. Readers who are uncomfortable with the specific mechanism posited by Penrose might also wish to read Henry Stapp's Mind, Matter, and Quantum Mechanics (ISBN 978-3540407614) which posits a quantum brain mechanism that is completely incompatible with Penrose's.

2. In an unpublished, informally circulated manuscript,  and in his 2004 book The Road to Reality (ISBN 0-09-944068-7), Penrose promulgated a novel diagrammatic notational system for writing out mathematical expressions that involve tensors (mathematical entities heavily used by theorists in general relativity and other modern physics). Working from the informal manuscript, Smolin and Rovelli generalized the diagrams into spin networks and spin foams which became the basis of their background-independent formulation of physics.

3. Penrose is an out-and-out Mathematical Platonist who thinks and talks about his Platonism. In The Road to Reality, he even wrote fiction about it (the volume is bookended by a pair of short stories), which, when I discovered it early in the Anathem project, gave me a warm feeling that it wasn't completely insane to try to write a novel about Platonism. I am grateful to him for a brief but (to me) informative discussion of this topic in Seattle in January of 2007.

4. He did pioneering work in aperiodic tiling problems. The Decagon problem that figures in Anathem is of this class. Please note, however, that the problem described in my novel, while inspired by Peter J. Lu and Paul J. Steinhardt's study of geometric tilings in Central Asian mosques, is, in its details, altogether fictional. A mathematician could probably prove that my version of it is not merely fictional but ridiculous; but the same mathematician would, however, readily agree that aperiodic tilings are an important and fascinating branch of mathematics. There is no really apt place to wedge this fact into these acknowledgments, but it's worth mentioning that Hao Wang, who is about to be talked about in the section on Gödel, also worked on related problems, inventing a set of tiles or dominoes that can, properly used, can function as Turing machines, and that later turned out to tile the plane aperiodically.

5. In The Emperor's New Mind, Penrose has written a particularly lucid and readable exposition of the concept of phase space, alluded to above.


Kurt Gödel is the one thinker who knit together all of the other influences mentioned in these Acknowledgments. I need to mention a few things about Gödel now, gleaned from several different sources, and it seems most expedient to list all of the sources first. So, here they are:


Palle Yourgrau. Gödel meets Einstein: Time Travel in the Gödel Universe (1999) ISBN 978-0812694086


Palle Yourgrau. A World Without Time: The Forgotten Legacy Of Godel And Einstein (2004) ISBN 978-0465092949


Rebecca Goldstein. Incompleteness: The Proof and Paradox of Kurt Godel (2005) ISBN 978-0393051698


Hao Wang. A Logical Journey: from Gödel to Philosophy (1997) ISBN 978-0262231893


Kurt Gödel. Collected Works. Edited by Solomon Feferman, John W. Dawson Jr. et al. (1990) ISBN 0195147219


Freeman Dyson. Private communication, 2007


Verena Huber-Dyson. Private communication, 2006


The picture that is clear from the above sources is that Gödel (like Penrose) was an out-and-out Mathematical Platonist who devoted a great deal of time and care to thinking about his Platonism and striving to place it on a sound metaphysical footing. During the second half of his life--the entire time that he spent at the Institute for Advanced Studies, from 1940 onwards--he devoted the preponderance of his working hours to this project. As outlined by Hao Wang in A Logical Journey and by Howard Stein's Note to 1949a in Volume 2 of Gödel's Collected Works, the general plan was to build up a rigorous metaphysics founded on Leibniz's monadology. Since Kant had lodged objections to Leibniz's metaphysics, and since Gödel respected, but did not agree with, Kant, Gödel needed, early in his program, to address and overcome Kant's objections. During the late 1940's Gödel famously developed solutions to Einstein's field equations proving that, in a universe that was rotating, it would be physically possible to travel backwards in time. He seems to have done so not only because it was interesting physics, but also to refute Kant's views on time and space, almost as if the rotating universe solution was a mere lemma that had to be got out of the way before he could proceed with the main project. Freeman Dyson has stated that, later in his life, Gödel would call him on the telephone from time to time and ask him "Have they found it yet?" meaning "Have astronomers found evidence that the universe rotates?"  Freeman Dyson, of course, was one of the prime movers behind Project Orion (see George Dyson's Project Orion: the True Story of the Atomic Spaceship ISBN 0805072845). If the universe does in fact rotate, one could exploit Gödel's solution to travel backwards in time by constructing a sufficiently powerful spaceship. The only technologically plausible way of building such a ship in our time would be through the use of Orion-like technology. It is extraordinary that Kurt Gödel, the man who calculated that such a journey was physically possible provided that the universe rotated, worked in the same Institute with Freeman Dyson, the man who helped conceived the ship that would be needed in order to make the journey, and that he occasionally checked in with Dyson to find out whether the universe did in fact rotate. This combination of circumstances is one of the kernels around which Anathem developed.


As explained by Hao Wang, the general model that Gödel used to think about mathematical Platonism ran something like this:


1. Entities that are the subject matter of mathematics exist independently of human perceptions, definitions and constructions.

2. The human mind is capable of perceiving such entities.


Item 1 above seems uncontroversial to many and is believed, at least to some extent, by nearly all mathematicians as well as many who adopt a "common sense" approach to such questions; as an example, anyone who believes that 3 was a prime number a billion years ago, agrees at least to some extent with Item 1.


Anyone who espouses (1), however, must supply an account of how it is that the human mind is capable of obtaining knowledge about mathematical entities, which, according to (1), are non-spatiotemporal and do not stand in a normal causal relationship to the entities that make up the physical universe. Various arguments have been put forward to explain this seeming paradox; for a useful summary, see Mark Balaguer's entry on Platonism in Metaphysics in the Stanford Encyclopedia of Philosophy and for a more thoroughgoing treatment read his Platonism and Anti-Platonism in Mathematics (ISBN 978-0195143980)


Gödel's approach to (2) is as follows:


2a. "Something besides the [physical] sensations actually is immediately given." Gödel refers to these givens as "data of the second kind."

2b. "It by no means follows, however, that the data of this second kind, because they cannot be associated with actions of certain things upon our sense organisms, are something purely subjective. Rather they, too, may represent an aspect of objective reality, but, as opposed to the sensations, their presence in us may be due to another kind of relationship between ourselves and reality."

2c. "I conjecture that some physical organ is necessary to make the handling of abstract impressions (as opposed to sense impressions) possible...Such a sensory organ must be closely related to the neural center for language."


The first two quotes above are from Gödel's 1964 paper What is Cantor's continuum problem? and the third is from Hao Wang's A Logical Journey. These three postulates will hereinafter be denoted as the second kind of data; the other kind of relationship; and Gödel's Organ.


The word "organ" in English usually connotes an identifiable lump of flesh, such as a pancreas, and so the notion of Gödel's Organ may be startling to some readers who might suppose that Gödel was imagining a specific piece of the brain (Descartes' infamous pineal gland comes to mind), which, unlike the rest of the brain, possessed the other kind of relationship to the Platonic Mathematical World and so was capable of picking up the data of the second kind. It is worth keeping in mind that English was not Gödel's first language and that he expressed reservations about his fluency as late as the early 1960s (see for example Gregory H. Moore's Note to 1947 and 1964 in Gödel's Collected Works). Suspecting some infelicity in translation, I have taken this matter up with Verena Huber-Dyson, a native German-speaking logician who crossed paths with Gödel in Princeton (though she did not discuss this particular topic with him), and she has confirmed my suspicion that the word Organ in German can be read as meaning something like a faculty or capability. The correct reading of (2c), therefore, is probably that the brain as a whole, and not some specific part of it, possesses the faculty or capability in question.


Gödel appears not to have supplied a detailed account of how the said Organ might work, and so for a metaphysician who is attempting to follow the route he was blazing, the trail seems to dead-end here. Items (2a) and (2b) both appear in Gödel's 1964 paper What is Cantor's Continuum Problem? which is a revised and expanded version of a 1947 paper of the same title, and so it is safe to infer that Gödel was thinking about these matters at least as early as the mid-1940s and that they were still very much on his mind in the late 1950s and early 1960s. Hao Wang states that in the period 1953-1959 Gödel spent a great deal of effort on his Carnap paper "in which he tries to prove that mathematics is not syntax of language and argues in favor of some form of Platonism." and that 1959 marked the beginning of his studies of the work of Edmund Husserl (1859-1938) which continued to occupy him to the end of his life. The relevance of Husserl to Gödel's metaphysical program is summarized by Hao Wang: "To carry out his program [to find an exact theory of metaphysics, structured after Leibniz's monadology--NS] Gödel had to take into account Kant's criticisms of Leibniz. He saw Husserl's method as promising a way to meet Kant's objections." See also Palle Yourgrau's A World Without Time, pp. 104-107, for a much better explanation of this than I can provide here.


Husserl is famously difficult to read (for what it is worth, his books are the most difficult material I have ever read in my entire life) but plowing through his works can be made somewhat more palatable if one conceives of it as a sort of detective story, the goal of which is to figure out what it was that Gödel saw in these pages that he thought might solve the metaphysical problem he had set for himself. The scope of this quest can, fortunately, be narrowed by attending to section 5.3 (page 164 in my edition) of Hao Wang's A Logical Journey, in which he quotes Gödel as saying that Husserl's most important works were Ideas [pertaining to a pure phenomenology and to a phenomenological philosophy] (ISBN 9024728525 and ISBN 0792307135) and Cartesian Meditations (ISBN 902470068X).


Even within that narrowed scope, attempting to summarize such a body of work in this context is a mug's game. It is noteworthy, however, that in the fifth and final Cartesian Meditation Husserl suddenly begins talking about monads and invoking Leibniz. Exactly how he gets there and what he means cannot be understood without steeping oneself in Husserl-thought. But Occam's Razor suggests that this is the most promising lead in the intellectual detective story mentioned above.


Husserl finished his Cartesian Meditations circa 1933 and Gödel published his last work in 1974, four years before his death. One might then wonder what, if anything, has been going on in this field during the succeeding thirty years? Since my objective has been to write an interesting novel, and not to conduct serious philosophical research, I can't claim to have carried out a rigorous and systematic search of the literature. But the works of Edward N. Zalta of Stanford University, and various of his collaborators mentioned below, are a trove of serious, peer-reviewed scholarly metaphysical work that addresses many of the topics mentioned above. As practicing philosophers, Zalta and his collaborators are able to advance and improve on ideas that I can only read about as historical artifacts. The papers that Zalta has authored and co-authored talk specifically about Plato, Leibniz's theory of concepts, Husserl's phenomenology, and Gödel's account of Mathematical Platonism, and so when Steven Horst drew Zalta to my attention, I confess to experiencing a certain silent-upon-a-peak-in-Darien moment. This is a large body of work, and rather than trying to summarize it here, I will draw two features to the reader's attention.


Computational Metaphysics. Zalta is a strong believer in formalizing philosophy, i.e., translating philosophical assertions from prose into expressions written out using the symbols of formal logic. Once this has been done, it becomes possible to compare different philosophers' ideas in the same way as a physicist might compare two different theories by writing them out as equations, then using the rules of mathematics to determine whether they contradict each other, or reduce to the same thing. It is true of both mathematics and formal logic that the rules are few and simple, but that applying them to the kinds of equations or assertions that arise in practice can become astonishingly complicated--in some cases, too complicated for humans to manage without error. Zalta has been using a computer code--an "automated reasoning system" called PROVER9--to resolve such conundrums. This strongly recalls Leibniz's scheme for a characteristica universalis, and indeed Zalta and his co-author Branden Fitelson quote Leibniz in the opening paragraph of their 2007 paper Steps Toward a Computational Metaphysics. Later in the same paper they allude to work in progress on using the system to prove theorems in Zalta's 2000 paper A (Leibnizian) Theory of Concepts, in which he reduces statements from Leibniz's metaphysics into a series of theorems expressed in formal logic. In other papers, Zalta and his collaborators Francis Jeffry Pelletier and Bernard Linsky tackle the work of Plato, Husserl, Gödel, and David Lewis, of whom more below. So this answers the question posed above ("what has been going on in the last thirty years?"). As a side note, computational metaphysics also provided the idea for some of the music that is alluded to in Anathem. Specifically, the tree-dwelling, loincloth-wearing fraas mentioned during the Aut of Inbrase at Tredegarh are carrying out--albeit very slowly--a computation along the lines of what PROVER9 does. They are trying to solve a deep problem in metaphysics, and it is taking them a long time because they don't have access to computers.


David Lewis and the Plurality of Worlds. Zalta has done a great deal of philosophizing on modes of predication, which brings him into contact with the work of the late philosopher David Lewis. Zalta and Linsky addressed Lewis's ontology in their 1991 paper Is Lewis a Meinongian? I think it's correct to say that they take Lewis seriously but they don't agree with him. Lewis wrote a book entitled On the Plurality of Worlds (ISBN 978-0631224266) which might have a familiar ring to those who have read Anathem, since the messal attended by Fraa Erasmas bears the same name. In it, Lewis lays out a metaphysics called modal realism, which (to sum it up very crudely) asserts that possible worlds really exist, and are as real as the world we live in. The relevance of modal realism to Anathem is obvious. As mentioned earlier, David Deutsch mentions Lewis's work in his writings on the many-worlds interpretation of quantum mechanics.


It is worth pointing out that I came to the work of Deutsch and of Zalta along completely different lines of inquiry. My interest in physics brought me to the former, my interest in the Plato-Leibniz-Husserl-Gödel lineage to the latter. When both of them ended up speaking of the same philosopher, David Lewis, I got a feeling--perhaps nothing more than self-delusion--that a circle had magically closed.


There are more circles yet, though. In their 1995 paper Naturalized Platonism vs. Platonized Naturalism, Zalta and LInsky set out a theory of "Principled Platonism," as opposed to "Piecemeal Platonism," the latter being their term for the garden-variety form of Mathematical Platonism espoused by many mathematicians. This is not the place to explain Principled Platonism, but I believe it is reasonable to say that it has certain resonances with Lewis's modal realism in that it posits a plenitude of abstract mathematical objects, as opposed to the sparse (hence "Piecemeal") model that tends to be presumed by other Mathematical Platonists.


The MIT cosmologist Max Tegmark, working from a different set of premises, has been saying some things that align nicely with the work of Zalta and LInsky  While most of Tegmark's work is in the field of cosmology per se, he has written several papers on what might fairly be called metaphysics, most notably and recently The Mathematical Universe in which he posits that the physical world is an abstract mathematical structure. In the section on what he terms the Level IV multiverse, he too invokes David Lewis and describes a hypothesis that is strikingly similar to Zalta's Principled Platonism.


Other sources of inspiration and ideas


The electrodynamic tether used by our heroes for propulsion in space is a legitimate invention, pioneered by the late Dr. Robert Forward and currently under development by Dr. Rob Hoyt and his colleagues at Tethers Unlimited. I am indebted to Hoyt for an exchange of email in which he explained to me what was and was not achievable by such a device, however any technical errors in the pages of the novel are chargeable to some combination of artistic license and failure to understand on my part, and should not be blamed on Hoyt or Forward.


Jaron Lanier and I had several conversations that, in one way or another, helped make Anathem better.


In the same way, I am indebted to several friends whom I am tempted to refer to as "the usual suspects," since it has become something of a tradition for them to read my books in manuscript form, and offer useful commentary: Jeremy Bornstein, Steven Horst, Alvy Ray Smith, and Jaime Taaffe.


Jeremy Bornstein has also taken the lead in drawing up grammatical rules and vocabulary for Orth. Steve Wiggins invented a cuneiform-style alphabet in which to write it, and David Stutz has used it in the Anathem music project. All of these endeavors are explained in better detail on the linked pages.


The Theory Group at Microsoft Research, inaugurated in 1997 by Nathan Myhrvold and headed up, until recently, by Jennifer Chayes and Christian Borgs, has brought many mathematicians to Seattle, some for permanent employment and some for visits ranging from hours to years. I have no formal connection to this organization, and it had nothing definite to do with the development of Anathem, and yet there is an intangible connection that is important enough to be worth mentioning. Much as bread rises better when it's warm, the mere presence of so many gifted mathematicians has changed my social environment in such a way as to make it easier to conceive and write a book such as Anathem.


Anathem could easily be construed as being consistently, even stridently anti-religion. I propose a more ambiguous interpretation. The book is written entirely from the point of view of the avout, who in general take a dim view of religious-minded people, largely because the ones who tend to draw their attention are the ones whose behavior excites strong emotions of fear or contempt: the conspicuous frauds, clowns, and charlatans who are as prevalent on Arbre as they are on Earth. In the pages of the novel I have tried to hint at the existence of other religious-minded people who are little noticed and rarely remarked upon because of the comparative dignity and restraint of their beliefs and their practices. This too is meant to reflect the way things are on Earth. If I'd meant Anathem to be an anti-religion screed, I would not have dedicated it to my parents, who are lifelong attendees of mainstream Protestant college-town churches in which essentially all of the parishioners believe in evolution and would not dream of interpreting scripture literally.


Miscellaneous topics


Anyone interested in "The Conjecture of Saunt Mandarast" might wish to read Peter Ward and Donald Brownlee's Rare Earth: Why Complex Life Is Uncommon in the Universe (ISBN 978-0387952895).


The concept known on Arbre as "Gardan's Steelyard" is roughly equivalent to Occam's Razor.