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In This Article Expand or collapse the "in this article" section Time Travel

Introduction, general overviews.

• David Lewis’s Analysis, Its Forerunners and Critics
• Gödel and the Ideality of Time
• Models and Issues from Relativity
• Models and Issues from Quantum Theory
• Causal Loops and Probability
• Time Travel in Many Worlds and the Autonomy Principle
• Travel in Dynamic Time and Multi-Dimensional Time
• General Metaphysical Issues

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• Contemporary Metaphysics
• Foreknowledge
• Laws of Nature
• Persistence
• Philosophy of Cosmology
• Space and Time
• Time and Tense

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Time Travel by Alasdair Richmond LAST REVIEWED: 26 October 2015 LAST MODIFIED: 26 October 2015 DOI: 10.1093/obo/9780195396577-0295

Time travel is a philosophical growth industry, with many issues in metaphysics and elsewhere recently transformed by consideration of time travel possibilities. The debate has gradually shifted from focusing on time travel’s logical possibility (which possibility is now generally although not universally granted) to sundry topics including persistence, causation, personal identity, freedom, composition, and natural laws, to name but a few. Besides metaphysical discussions, some time travel works draw on the philosophies of science, spacetime, and computation. Some interesting forerunners notwithstanding, serious physical interest in time travel begins with Gödel’s 1949a demonstration that general relativity permits space-times that are riddled with closed timelike curves (“CTCs” henceforth). A key philosophical text on time travel is Lewis 1976 and its argument for the logical possibility of certain backward time travel journeys and even for the possibility of casual loops. Lewis concludes that time travel could occur in a possible world, albeit perhaps a strange world that would feature (or seem to feature) strange restrictions on actions. In Lewis’s analysis, a traveler can arrive in the past of the same history they come from provided that the traveler’s actions on arrival are consistent with the history that they come from. So other worlds or multiple temporal dimensions are not necessary to make time travel consistent. Granted, the physics, persistence conditions, agency, and epistemology of agents in such worlds might look weird indeed. Since Lewis, philosophical time travel questions include the following: given that a traveler into the past cannot create any paradoxical outcomes on arrival, what then would stay their hand? Are the constraints on a traveler’s actions admissible within our ordinary understanding of physical law or human agency? Is time travel compatible with dynamic time or even with the existence of time itself? Can backward time travel be physically possible within a single history? If a time traveler meets another stage of him- or herself, is the traveler in two places at once, and what theory of persistence can cope with this puzzling multiplication? Can time-travel spacetimes resolve otherwise intractable computational problems?

Despite several hundred philosophical and scientific articles, book chapters, and Internet resources devoted to philosophical problems posed by time travel, there is currently no full-length monograph or anthology on the subject. The best introduction to the topic in general so far is chapter 8 of Dainton (second edition 2010), Dainton 2010 being the best general philosophical resource available on time and space. The key work is Lewis 1976 , a defense of the logical possibility of backward time travel, from which a large number of subsequent treatments take their cue. A useful overview, albeit largely from a physical science perspective, is Nahin 1999 . Also largely physical in emphasis but comprehensive and thorough is Earman 1995 . Richmond 2003 surveys philosophical work on time travel to date. Arntzenius 2006 details the problems of free action and nomological constraint posed by backward time travel. Arntzenius and Maudlin 2005 is helpful on (especially) problems of physical law. Carroll 2008 is perhaps the best single online resource available on any aspect of time travel. Le Poidevin 2003 is a highly commendable introduction to the philosophy of time in general but especially good on problems of time travel. Bourne 2006 offers some useful arguments and clarifications centered on Gödel’s arguments about time travel and the relations between time travel and the status of times themselves. Earman and Wüthrich 2006 offers scientifically well informed but approachable and philosophically cogent discussions of what physics might, and might not, allow by way of time travel.

Arntzenius, Frank. “Time Travel: Double Your Fun.” Philosophy Compass 6 (2006): 599–616.

DOI: 10.1111/j.1747-9991.2006.00045.x

Entertaining survey of the philosophical terrain around time travel that concentrates particularly on the constraints on action likely to be suffered by travelers in the past. An excellent introduction to the nomological contrivance problem and more. Available online for purchase or by subscription.

Arntzenius, Frank, and Tim Maudlin. “ Time Travel and Modern Physics .” In Stanford Encyclopedia of Philosophy . Edited by Edward N. Zalta. 2005.

Notably acute survey of physical possibilities for time travel, including detailed arguments that backward time travel threatens to create correlations that conflict with standard quantum predictions.

Bourne, C. A Future for Presentism . Oxford: Oxford University Press, 2006.

DOI: 10.1093/acprof:oso/9780199212804.001.0001

Although primarily devoted to defending presentism, chapter 8 offers one of the best treatments of Gödel’s ideality argument around and pp. 132–134 offer some interesting sidelights on the possible compatability of time travel and presentism.

Carroll, John W. A Time Travel Website . 2008–.

Extremely thorough, engagingly-written, well-designed, and continually evolving online resource that offers helpful discussions, well-chosen readings, and helpful animations to boot.

Dainton, Barry. Time and Space . 2d ed. Durham, NC: Acumen, 2010.

Revised and expanded edition of Dainton’s classic 2001 introduction to the philosophy of space and time. Can be highly recommended but notable here for its extensive, essential treatments of time travel, relativity, and Gödel’s “ideality” argument.

Earman, John. “Recent Work on Time Travel.” In Time’s Arrows Today . Edited by Steven F. Savitt, 268–310. Cambridge, UK: Cambridge University Press, 1995.

DOI: 10.1017/CBO9780511622861

Thorough discussion of the then-current state of play in the philosophical and physical literature on time travel. This is still a valuable resource.

Earman, John, and Christian Wüthrich. “ Time Machines .” In Stanford Encyclopedia of Philosophy . Edited by Edward N. Zalta. 2006.

Comprehensive discussion of physical resources for time travel, among other intriguing suggestions, develops the view that physically realistic time machines might be uncontrollable even if they become a possiblility.

Le Poidevin, Robin. Travels in Four Dimensions: The Enigmas of Space and Time . Oxford: Oxford University Press, 2003.

Engaging and clearly written introduction to the philosophy of space and time. Often offers problems and discussions that lend themselves to time travel interpretation. An excellent introductory and pedagogical resource.

Lewis, David. “ The Paradoxes of Time Travel .” American Philosophical Quarterly 13 (1976): 145–152.

The philosophical time travel work. Includes Lewis’s discrepancy definition of time travel: the most useful by far. Invokes the notion of compossibility to disambiguate “Grandfather paradox” arguments and argues that backward time travel and causal loops can occur in (nonbranching) possible worlds. Usefully distinguishes between replacement change and counterfactual change. (This is often cited and sometimes rebutted but never refuted.)

Nahin, Paul. Time Machines: Time Travel in Physics, Metaphysics and Science Fiction . 1st ed. New York: American Institute of Physics, 1999.

DOI: 10.1007/978-1-4757-3088-3

Engaging and comprehensive attempt at surveying all the scientific, philosophical, and fictional literature on time travel. Perhaps slightly more at ease with physics and fiction than with philosophy, but this is a detailed and thorough treatment.

Richmond, Alasdair. “Recent Work: Time Travel.” Philosophical Books 44 (2003): 297–309.

DOI: 10.1111/1468-0149.00308

Survey of the time travel debate from Lewis 1976 onward, sketching links with debates in persistence, philosophy of spacetime and temporal topology. Available online by subscription.

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Time of Logics and Time of Physics

• First Online: 31 May 2017

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• Carlo Proietti 7  

Part of the book series: Boston Studies in the Philosophy and History of Science ((BSPS,volume 326))

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The history of reasoning about time is filled with paradoxes and conundra; contemporary physics is no exception. Logics in general, and more specifically temporal logics, represent a rigorous formal tool in order to solve or clarify problems of this kind. In what follows we will first explain, from the point of view of philosophical logic, what is a paradox and what should count as a solution to it. After that we will illustrate A.N. Prior’s formalization of the traditional paradox of future contingency and determinism . Then we will focus on two modern paradoxes – the twin paradox and the time travel paradox – and show how an adequate temporal logic can help their framing and understanding.

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Knowing that A implies that A is true and knowing that not-A implies that A is false. This is an intuitive property of knowledge that the medievals expressed with the motto nihil scitum nisi verum .

Of course, (a) must hold for the argument to be conclusive and that is where Aristotle and Diodorus diverge in their analysis. Most commentators read the whole chapter IX of Aristotele ( 1941 ) as an attempt to consistently reject (a) and therefore discard the Diodorean argument.

Our object of analysis here is the context-independent (or absolute) notion of time, as opposed to the context-dependent (or phenomenological) one (see Ismael , Chap. 2 of this volume). The latter brings in additional elements of complexity, although, as argued by Ismael , it can be framed in rigorous terms.

The twin paradox dates back to the very early days of relativity theory. Einstein himself stressed as a “peculiar consequence” of the theory the fact that “if at the points A and B of K there are stationary clocks which, viewed in the stationary system, are synchronous; and if the clock at A is moved with the velocity v along the line AB to B, then on its arrival at B the two clocks no longer synchronize, but the clock moved from A to B lags behind the other which has remained at B” (in Einstein 1905 § 4, see also Einstein 1911 ). This scenario was popularized by Langevin ( 1911 ) with the more vivid description of a travel back and forth from Earth in a projectile. Finally, it was Weyl ( 1918 ) who upgraded the example with two twins aging differently, one staying on Earth and one travelling in the projectile. However, neither Einstein nor Langevin nor Weyl called it a paradox, since the situation is perfectly consistent with the theory. The alleged paradoxality originated mostly in philosophical discussion, due to the objection that if motion is relative then the situation should be symmetric – from the local perspective of each twin, the other is the one moving – and therefore there seems to be no reason why the twins should age at a different pace. The reader may consult During ( 2014 ) for a detailed historical account of how the example developed and crystallized into a paradox.

The original source of the paradox, and of its name, is the science fiction novel by Barjavel ( 1944 ) whose main character , an imprudent time traveller, kills his grandfather before the latter meets the time traveller’s grandmother.

This was not because Prior was unaware of special and general relativity, the point is that a Newtonian framework is detailed enough for dealing with traditional paradoxes and conundrums. Indeed, when dealing with a paradox, a good logical analysis consists first and foremost in simplifying the picture in a way that is fine-grained enough for the problem to arise and hopefully to articulate a solution. In the case of determinism and free-will the further complexity induced by relativistic space-time is to a large extent superfluous and therefore negligible.

For a more advanced reading see Belnap ( 1992 ), McCall ( 1994 ) and Strobach ( 2007 ).

See De Interpretatione chap. IX.

See among others Summa theologiae I.14.13 (Aquinas 1964 ) and De Veritate Q. 2  (Aquinas 1949 ).

See Baudry ( 1950 ).

Our approach diverges from Prior’s insofar as his construction was mostly syntactic and employed semantics just as an auxiliary tools. Prior had a specific philosophical motivation for this. However, a semantic approach fits better with our explanatory purpose.

Allowing multiple histories is not merely a logician’s trick: in quantum mechanics a many-world interpretation is actually provided which allows parallel universes as in our scenario.

In what follows we will not fully adhere with Belnap’s presentation. For example, Belnap does not provide truth clauses for temporal operators and does not introduce operators for possible reference frame s . However, to understand some puzzles of space-time relativity, we need to give at least a partial account of how these operators should work. For this purpose we will (freely) borrow from Strobach ( 2007 ).

This is considered, for various reasons, to be the most coherent option by Belnap ( 1992 ).

The truth clause of the operator P * n is defined analogously to that of F * n (see previous subsection), i.e. P * n φ is true at (e,h,f) if and only if φ is true at (e’,h,f) for some e’ such that e’ < < e and d(e,e’) = n w.r.t. the frame f .

See Price and Wharton (Chap. 7 of this volume) and Berkovitz (Chap. 8 ) for a more elaborate discussion of this point (in the framework of retro-causal interpretations of quantum mechanics ).

This solution is pretty much the same as the one provided by Lewis ( 1976 ). In the same paper, Lewis deems as logically impossible to change one’s own past. However, such logical impossibility refers to changing one and the same history by making, e.g., c and ¬c true there. As we have shown, a contradiction may indeed be avoided only by means of two alternative histories. Of course, such an option generates additional problems of trans-world identity and causation which are the object of ongoing philosophical disputes.

Aquinas, T. 1949. De veritate. Quaestiones disputatae . Torino-Roma: Marietti. URL http://books.google.se/booksid=0bRaAAAAYAAJ .

———. 1964. Summa theologiae: Latin text and English translation, introductions, notes, appendices, and glossaries. Summa theologiae . London: Blackfriars. URL http://books.google.se/books?id=sc7WAAAAMAAJ .

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———. 1911. Die Relativitäts Theorie. Naturforschende Gesellschaft, Zürich,Vierteljahresschrift 56: 1–14.

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Acknowledgements

The author is grateful to Ilaria Jemos, Iacopo Carusotto and Jeroen Smid for useful theoretical insights and contributions to clarification, and to Andrew McFarland for a thorough language check of the manuscript. This research was supported by the Swedish Research Council.

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Proietti, C. (2017). Time of Logics and Time of Physics. In: Bouton, C., Huneman, P. (eds) Time of Nature and the Nature of Time. Boston Studies in the Philosophy and History of Science, vol 326. Springer, Cham. https://doi.org/10.1007/978-3-319-53725-2_3

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Søren Kierkegaard’s philosophy sees the human self as a synthesis of contrary and opposing tendencies. This inherent ambiguity of life creates existential anxiety.

T he Concept of Anxiety (1844) and The Sickness Unto Death (1849) are considered part of the psychological work of Danish philosopher Søren Kierkegaard. For him, psychology is related to the science of man as an emerging self-conscious spirit. In these two books, one finds central concepts such as anxiety and despair. Every authentic human being faces anxiety, the weight of freedom, and the possible . One of the ways we find ourselves immersed in despair is by not confronting this anxiety.

Søren Kierkegaard’s Philosophical Figure

Is difficult to grasp Kierkegaard’s ideas without saying something, though tremendously brief, about his own life. Born in 1813 in Copenhagen, Denmark, Kierkegaard was from the start deeply influenced by the religious devotion of his father. At some point, he even completed a pastoral seminary to become a priest and he eventually preached in Copenhagen churches . However, he was never ordained. In fact, his young adult years were taken up with disagreeable hedonistic behaviors. Kierkegaard remains a religious writer and concepts such as anxiety and despair have religious connotations.

Kierkegaard was a melancholic figure. He deemed, as his father claimed, that the family was under the curse of God and that their lives were stained by a “Death Shadow.” Kierkegaard had reasons to believe this curse: five of his brothers died as well as his mother. For this, Kierkegaard judged that he would not live to be thirty-four (Reale & Antiseri, 2008, p. 356). Although economically wealthy, entries in his journal are testimony of his inner isolation and lack of direction, an overwhelming melancholy. One entry reads “I just now come from a party where I was its life and soul (…) everyone laughed and admired me, but I went away (…) and wanted to shoot myself” (Hannay, 1993, p. 4).

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Another key event in his biography is his engagement to Regine Olsen in 1840, who was 18 years old by then. However, the engagement was shortly terminated. According to Kierkegaard, someone who embraces the ideal of the Christian life cannot live as a calm man while being married; moreover, he recognized that he could not be a husband and a father given his melancholy and spectral ways   (Lippitt & Evans, 2023). The years after the failed engagement were very productive for Kierkegaard: Either/Or (1843), Repetition, Fear and Trembling (1843), and The Concept of Anxiety (1844) are some of the books that represent this phase.

Kierkegaard’s Pseudonyms 

Kierkegaard wrote under a variety of pseudonyms in addition to his name. Some of the Pseudonyms he used are Victor Emerita, Johannes de Silentio, Vigilius Haufniensis, and Anti-Climacus among many others. Kierkegaard was also keen in insisting that one should quote the pseudonyms. Therefore, many modern scholars use the pseudonyms when discussing Kierkegaard’s oeuvre. And some argue that certain pseudonyms like Anti-Climacus better depict Kierkegaard’s views. We must ask: why did Kierkegaard use pseudonyms, and give them so much importance? It seems that the pseudonyms represent different life views or voices: the  “aesthetic”, “the ethical” and “the religious” stages of life —without accounting for important subdivisions (Lippitt & Evans, 2023). Generally speaking, the aesthetic sphere relates to hedonism (both physical and intellectual); the ethical is linked to commitment and embracement of social norms; and the religious stage is marked by faith.  Following Lippit & Evans (2023) the spheres could be interpreted as “different views of what gives human life value.”

Instead of engaging in abstract and theoretical descriptions of these spheres of life, Søren Kierkegaard was more interested in embodying them; hence, the pseudonyms. Conceptual understandings are not enough, each sphere of life needs to be embodied in existence—albeit in a literary form. This was also a way of distancing himself from an abstract approach à la Hegel, who was more concerned with the general idea of humans instead of concrete individuals (Reale & Antiseri, 2008, p. 361).

Kierkegaard’s View of Human Existence: Anxiety

In The Concept of Anxiety (1844) and The Sickness Unto Death (1849), Kierkegaard deals with the question of what it means to become a person and how can one fail as a person i.e. how life can be wasted. In the latter work, Kierkegaard— in the voice of Anti-Climacus—starts by writing that the self (Selvet) is a synthesis:

“A human being is a synthesis of the infinite and the finite, of the temporal and the eternal, of freedom and necessity, in short, a synthesis” (Kierkegaard, 1998, p. 38).

There is a similarity to Hegel’s dialectical conception of the human self, but for Kierkegaard, the “synthesis” is continuous, even at the end of life, the self is never complete, and a human being is an unfinished project .  There is a tension between the infinite and the finite, the temporal and the eternal, possibility and necessity, and from such a condition, anxiety arises. For the Danish philosopher and theologian, therefore, “the self is constituted as a synthesis of opposing tendencies (…) but which are ‘held together’ but the Spirit (will)” (Mullen, 1981, p. 47).

The opposition between the finite and the infinite depicts the human capacity to imagine different worlds (infinite) on the one hand and to realize how the world actually is (the finite) on the other. Focusing only on the infinite leads to a world of fantasy and delusion.  The possibility/necessity dyad revolves around the question, “What shall I do with my life?”. It is a personalized version of the infinite and finite tension that focuses on the individual. Here as well, centering oneself only on the possibility is to imagine multiple versions of one’s identity detached from significant determinations. Necessity represents things over which we have no control (or only partly), for example,  our origin, environment, height, etc. (Watkin, 2001, p. 231).

Lastly, the tension created by the temporal and the eternal has the same pattern as the others but with a nuance: given the possibilities of who I can become, the question is now is who should I become? It is an ethical issue. There are many ways in which one can fail at finding the authentic self. The point to remember when looking at these tensions is that one cannot ever know in advance which possibilities of self are real and, therefore, achievable, thus they constitute a source of anxiety (Angest). Life is too ambiguous to establish universal rules to decide . A genuine person (a free person) will have to deal with anxiety and maintain her individuality by the spirit.

Various scholars precisely define anxiety as the consequence of the tension between self-consciousness and biology, or biology and the spiritual life (Watkin, 2001, p. 16). However, necessity encompasses more than just biology; it alludes to everything beyond the control of the individual. In any case, anxiety cannot be resolved; it can only be faced. Amid the uncertainty of the tension between the possible and necessity, a leap towards action is the only way . As Mullen comments: “It all comes down to an act of the will; a leap, if not in the dark, at least at dusk” (1981, p. 50).  Anxiety is also linked to responsibility. One’s freedom implies the responsibility of action, the responsibility of making that leap. In short, anxiety is the possibility of human freedom. Artistically, when discussing existential anxiety, Edward Munch’s painting ‘The Scream’ comes to mind: the silhouette of an alarmed human, aware of their anxiety at the crossroads of the possible and the necessary. Again, existential anxiety is “the possibility of possibility.” Indeed, it is not inaccurate to assume a deep influence from Kierkegaard in Munch’s work  (Stewart, 2016).

Pathologies of the Self: Despair

We said that existential anxiety is a necessary part of a genuine human life. This means that anxiety cannot be relinquished without losing oneself in the process. If an individual does not face anxiety and rather tries to dismiss it or escape from it, she will fall into despair (Fortvivlelse). Hannay writes “Despair is the inherently morbid (…) condition in which the self fails to exploit the unique educative possibilities offered by anxiety” (1993, p. 166). Put differently, despair is the consequence of the lack of willingness and courage to be oneself. Every person, asserts Kierkegaard, is ready to alleviate her anxiety by surrendering some part of her humanity, this is despair. Thus, despair is a form of human corruption in which the individual does not achieve her true self. Despair is not a feeling or an emotion but essentially a condition of the human self (Mullen, 1981, p. 60).

There are types of despair depending on which side of the tension is favored. For instance, the despair of infinitude is characterized by a lack of awareness of real concerns and conditions of realization; is a form of escapism from anxiety into fantasy and self-delusion in which an individual wills the impossible. But there is also despair of finitude depicted by a self that is swallowed by purely temporal values, by the immediacy of earthly pleasures (Watkin, 2001, p. 64). In a similar vein, despair can be understood along the possibility/necessity tension: either the despair of possibility in which I imagine multiple versions of myself all detached from real considerations and actions, or the despair of necessity in which the individual abandons hope and any desire for change and innovation. Mullen explains it poetically: “One can reject himself by living in abstraction fleeing from the world; or one can reject himself by becoming the crowd, fleeing into the world” (1981, p. 63). Both tendencies are pathologies of the self that lead to despair, they are both consequences of being a coward, attempts at escaping anxiety, and genuine selfhood .

Kierkegaard’s Lesson in Our Time

We have only scratched the surface of some key ideas in Kierkegaard’s notions of human existence, anxiety, and despair. However, even at this stage, we can draw relevant insights from Kierkegaard’s philosophy. How often do we harbor the suspicion that life is slipping away, that our true selves remain undiscovered? According to Kierkegaard, individuals sometimes deceive themselves to avoid the pain of committing to the responsibility that anxiety presents. Perhaps this is why we are so attached to our ‘entertainment,’ be it games, consumerism, or more radically, addictions. These distractions allow us to evade the call to confront anxiety and seek our authentic selves.

Before concluding, it’s worth remembering that Søren Kierkegaard is a religious writer, and he eventually links the concept of despair to sin. Sin is a deliberate disease resulting from self-deceptive strategies. A healthy self is one that courageously confronts anxiety, embraces responsibility for one’s life project, and, ultimately, establishes a connection with God.

Hannay, A. (1993). Kierkegaard (Repr). Routledge.

Kierkegaard, S. (1998). The sickness unto death: A Christian psychological exposition for upbuilding and awakening (H. V. Hong, Ed.; Nachdr.). Princeton Univ. Press.

Lippitt, J., & Evans, S. (2023). Søren Kierkegaard. In Z. Edward (Ed.), The Stanford Encyclopedia of Philosophy . <https://plato.stanford.edu/archives/sum2023/entries/kierkegaard/>

Mullen, J. D. (1981). Kierkegaard’s philosophy: Self-deception and cowardice in the present age . A Mentor Book New Americal Library.

Reale, G., & Antiseri, Darío. (2008). Historia de la filosofía. 4, De Spinoza a Kant (1a ed). Universidad Pedagógica Nacional : San Pablo.

Stewart, J. (2016). Kierkegaard’s influence on literature, criticism, and art . Routledge.

Watkin, J. (2001). Historical dictionary of Kierkegaard’s philosophy . Scarecrow Press.

What Is an Existential Crisis?

By Andres Felipe Barrero MA Philosophy, MSc Philosophy, Ph.D. Candidate Andrés has a background in philosophy from Universidad de la Salle in Bogotá, Colombia, where he finished his undergraduate and master`s studies. He completed a second master's at Universität Hamburg, Germany, where he wrote about philosophical theories of Modernity and Secularization. Currently, he is a Ph.D. Candidate at Universität Bremen. His fields of interest include the Philosophy of Language, Philosophy of Religion, Philosophy of Science, Social Theory, Discourse Studies, Corpus Linguistics, and Natural Language Processing.

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Time Travel and Modern Physics

Time travel has been a staple of science fiction. With the advent of general relativity it has been entertained by serious physicists. But, especially in the philosophy literature, there have been arguments that time travel is inherently paradoxical. The most famous paradox is the grandfather paradox: you travel back in time and kill your grandfather, thereby preventing your own existence. To avoid inconsistency some circumstance will have to occur which makes you fail in this attempt to kill your grandfather. Doesn't this require some implausible constraint on otherwise unrelated circumstances? We examine such worries in the context of modern physics.

1. A Botched Suicide

2. why do time travel suicides get botched, 3. topology and constraints, 4. the general possibility of time travel in general relativity, 5. two toy models, 6. remarks and limitations on the toy models, 7. slightly more realistic models of time travel, 8. even if there are constraints, so what, 9. quantum mechanics to the rescue, 10. conclusions, other internet resources, related entries.

You are very depressed. You are suicidally depressed. You have a gun. But you do not quite have the courage to point the gun at yourself and kill yourself in this way. If only someone else would kill you, that would be a good thing. But you can't really ask someone to kill you. That wouldn't be fair. You decide that if you remain this depressed and you find a time machine, you will travel back in time to just about now, and kill your earlier self. That would be good. In that way you even would get rid of the depressing time you will spend between now and when you would get into that time machine. You start to muse about the coherence of this idea, when something amazing happens. Out of nowhere you suddenly see someone coming towards you with a gun pointed at you. In fact he looks very much like you, except that he is bleeding badly from his left eye, and can barely stand up straight. You are at peace. You look straight at him, calmly. He shoots. You feel a searing pain in your left eye. Your mind is in chaos, you stagger around and accidentally enter a strange looking cubicle. You drift off into unconsciousness. After a while, you can not tell how long, you drift back into consciousness and stagger out of the cubicle. You see someone in the distance looking at you calmly and fixedly. You realize that it is your younger self. He looks straight at you. You are in terrible pain. You have to end this, you have to kill him, really kill him once and for all. You shoot him, but your eyesight is so bad that your aim is off. You do not kill him, you merely damage his left eye. He staggers off. You fall to the ground in agony, and decide to study the paradoxes of time travel more seriously.

The standard worry about time travel is that it allows one to go back and kill one's younger self and thereby create paradox. More generally it allows for people or objects to travel back in time and to cause events in the past that are inconsistent with what in fact happened. (See e.g., Gödel 1949, Earman 1972, Malament 1985a&b, Horwich 1987.) A stone-walling response to this worry is that by logic indeed inconsistent events can not both happen. Thus in fact all such schemes to create paradox are logically bound to fail. So what's the worry?

Well, one worry is the question as to why such schemes always fail. Doesn't the necessity of such failures put prima facie unusual and unexpected constraints on the actions of people, or objects, that have traveled in time? Don't we have good reason to believe that there are no such constraints (in our world) and thus that there is no time travel (in our world)? We will later return to the issue of the palatability of such constraints, but first we want to discuss an argument that no constraints are imposed by time travel.

Wheeler and Feynman (1949) were the first to claim that the fact that nature is continuous could be used to argue that causal influences from later events to earlier events, as are made possible by time travel, will not lead to paradox without the need for any constraints. Maudlin (1990) showed how to make their argument precise and more general, and argued that nonetheless it was not completely general.

Imagine the following set-up. We start off having a camera with a black and white film ready to take a picture of whatever comes out of the time machine. An object, in fact a developed film, comes out of the time machine. We photograph it, and develop the film. The developed film is subsequently put in the time machine, and set to come out of the time machine at the time the picture is taken. This surely will create a paradox: the developed film will have the opposite distribution of black, white, and shades of gray, from the object that comes out of the time machine. For developed black and white films (i.e. negatives) have the opposite shades of gray from the objects they are pictures of. But since the object that comes out of the time machine is the developed film itself it we surely have a paradox.

However, it does not take much thought to realize that there is no paradox here. What will happen is that a uniformly gray picture will emerge, which produces a developed film that has exactly the same uniform shade of gray. No matter what the sensitivity of the film is, as long as the dependence of the brightness of the developed film depends in a continuous manner on the brightness of the object being photographed, there will be a shade of gray that, when photographed, will produce exactly the same shade of gray on the developed film. This is the essence of Wheeler and Feynman's idea. Let us first be a bit more precise and then a bit more general.

For simplicity let us suppose that the film is always a uniform shade of gray (i.e. at any time the shade of gray does not vary by location on the film). The possible shades of gray of the film can then be represented by the (real) numbers from 0, representing pure black, to 1, representing pure white.

Let us now distinguish various stages in the chronogical order of the life of the film. In stage S 1 the film is young; it has just been placed in the camera and is ready to be exposed. It is then exposed to the object that comes out of the time machine. (That object in fact is a later stage of the film itself). By the time we come to stage S 2 of the life of the film, it has been developed and is about to enter the time machine. Stage S 3 occurs just after it exits the time machine and just before it is photographed. Stage S 4 occurs after it has been photographed and before it starts fading away. Let us assume that the film starts out in stage S 1 in some uniform shade of gray, and that the only significant change in the shade of gray of the film occurs between stages S 1 and S 2 . During that period it acquires a shade of gray that depends on the shade of gray of the object that was photographed. I.e., the shade of gray that the film acquires at stage S 2 depends on the shade of gray it has at stage S 3 . The influence of the shade of gray of the film at stage S 3 , on the shade of gray of the film at stage S 2 , can be represented as a mapping, or function, from the real numbers between 0 and 1 (inclusive), to the real numbers between 0 and 1 (inclusive). Let us suppose that the process of photography is such that if one imagines varying the shade of gray of an object in a smooth, continuous manner then the shade of gray of the developed picture of that object will also vary in a smooth, continuous manner. This implies that the function in question will be a continuous function. Now any continuous function from the real numbers between 0 and 1 (inclusive) to the real numbers between 0 and 1 (inclusive) must map at least one number to itself. One can quickly convince oneself of this by graphing such functions. For one will quickly see that any continuous function f from [0,1] to [0,1] must intersect the line x = y somewhere, and thus there must be at least one point x such that f ( x )= x . Such points are called fixed points of the function. Now let us think about what such a fixed point represents. It represents a shade of gray such that, when photographed, it will produce a developed film with exactly that same shade of gray. The existence of such a fixed point implies a solution to the apparent paradox.

Let us now be more general and allow color photography. One can represent each possible color of an object (of uniform color) by the proportions of blue, green and red that make up that color. (This is why television screens can produce all possible colors.) Thus one can represent all possible colors of an object by three points on three orthogonal lines x , y and z , that is to say, by a point in a three-dimensional cube. This cube is also known as the ‘Cartesian product’ of the three line segments. Now, one can also show that any continuous map from such a cube to itself must have at least one fixed point. So color photography can not be used to create time travel paradoxes either!

Even more generally, consider some system P which, as in the above example, has the following life. It starts in some state S 1 , it interacts with an object that comes out of a time machine (which happens to be its older self), it travels back in time, it interacts with some object (which happens to be its younger self), and finally it grows old and dies. Let us assume that the set of possible states of P can be represented by a Cartesian product of n closed intervals of the reals, i.e., let us assume that the topology of the state-space of P is isomorphic to a finite Cartesian product of closed intervals of the reals. Let us further assume that the development of P in time, and the dependence of that development on the state of objects that it interacts with, is continuous. Then, by a well-known fixed point theorem in topology (see e.g., Hocking and Young 1961, p 273), no matter what the nature of the interaction is, and no matter what the initial state of the object is, there will be at least one state S 3 of the older system (as it emerges from the time travel machine) that will influence the initial state S 1 of the younger system (when it encounters the older system) so that, as the younger system becomes older, it develops exactly into state S 3 . Thus without imposing any constraints on the initial state S 1 of the system P , we have shown that there will always be perfectly ordinary, non-paradoxical, solutions, in which everything that happens, happens according to the usual laws of development. Of course, there is looped causation, hence presumably also looped explanation, but what do you expect if there is looped time?

Unfortunately, for the fan of time travel, a little reflection suggests that there are systems for which the needed fixed point theorem does not hold. Imagine, for instance, that we have a dial that can only rotate in a plane. We are going to put the dial in the time machine. Indeed we have decided that if we see the later stage of the dial come out of the time machine set at angle x , then we will set the dial to x +90, and throw it into the time machine. Now it seems we have a paradox, since the mapping that consists of a rotation of all points in a circular state-space by 90 degrees does not have a fixed point. And why wouldn't some state-spaces have the topology of a circle?

However, we have so far not used another continuity assumption which is also a reasonable assumption. So far we have only made the following demand: the state the dial is in at stage S 2 must be a continuous function of the state of the dial at stage S 3 . But, the state of the dial at stage S 2 is arrived at by taking the state of the dial at stage S 1 , and rotating it over some angle. It is not merely the case that the effect of the interaction, namely the state of the dial at stage S 2 , should be a continuous function of the cause, namely the state of the dial at stage S 3 . It is additionally the case that path taken to get there, the way the dial is rotated between stages S 1 and S 2 must be a continuous function of the state at stage S 3 . And, rather surprisingly, it turns out that this can not be done. Let us illustrate what the problem is before going to a more general demonstration that there must be a fixed point solution in the dial case.

Forget time travel for the moment. Suppose that you and I each have a watch with a single dial neither of which is running. My watch is set at 12. You are going to announce what your watch is set at. My task is going to be to adjust my watch to yours no matter what announcement you make. And my actions should have a continuous (single valued) dependence on the time that you announce. Surprisingly, this is not possible! For instance, suppose that if you announce “12”, then I achieve that setting on my watch by doing nothing. Now imagine slowly and continuously increasing the announced times, starting at 12. By continuity, I must achieve each of those settings by rotating my dial to the right. If at some point I switch and achieve the announced goal by a rotation of my dial to the left, I will have introduced a discontinuity in my actions, a discontinuity in the actions that I take as a function of the announced angle. So I will be forced, by continuity, to achieve every announcement by rotating the dial to the right. But, this rotation to the right will have to be abruptly discontinued as the announcements grow larger and I eventually approach 12 again, since I achieved 12 by not rotating the dial at all. So, there will be a discontinuity at 12 at the latest. In general, continuity of my actions as a function of announced times can not be maintained throughout if I am to be able to replicate all possible settings. Another way to see the problem is that one can similarly reason that, as one starts with 12, and imagines continuously making the announced times earlier, one will be forced, by continuity, to achieve the announced times by rotating the dial to the left. But the conclusions drawn from the assumption of continuous increases and the assumption of continuous decreases are inconsistent. So we have an inconsistency following from the assumption of continuity and the assumption that I always manage to set my watch to your watch. So, a dial developing according to a continuous dynamics from a given initial state, can not be set up so as to react to a second dial, with which it interacts, in such a way that it is guaranteed to always end up set at the same angle as the second dial. Similarly, it can not be set up so that it is guaranteed to always end up set at 90 degrees to the setting of the second dial. All of this has nothing to do with time travel. However, the impossibility of such set ups is what prevents us from enacting the rotation by 90 degrees that would create paradox in the time travel setting.

Let us now give the positive result that with such dials there will always be fixed point solutions, as long as the dynamics is continuous. Let us call the state of the dial before it interacts with its older self the initial state of the dial. And let us call the state of the dial after it emerges from the time machine the final state of the dial. We can represent the possible initial and final states of the dial by the angles x and y that the dial can point at initially and finally. The set of possible initial plus final states thus forms a torus. (See figure 1.)

Suppose that the dial starts at angle I . The initial angle I that the dial is at before it encounters its older self, and the set of all possible final angles that the dial can have when it emerges from the time machine is represented by the circle I on the torus (see figure 1). Given any possible angle of the emerging dial the dial initially at angle I will develop to some other angle. One can picture this development by rotating each point on I in the horizontal direction by the relevant amount. Since the rotation has to depend continuously on the angle of the emerging dial, ring I during this development will deform into some loop L on the torus. Loop L thus represents the angle x that the dial is at when it is thrown into the time machine, given that it started at angle I and then encountered a dial (its older self) which was at angle y when it emerged from the time machine. We therefore have consistency if x = y for some x and y on loop L . Now, let loop C be the loop which consists of all the points on the torus for which x = y . Ring I intersects C at point < i , i >. Obviously any continuous deformation of I must still intersect C somewhere. So L must intersect C somewhere, say at < j , j >. But that means that no matter how the development of the dial starting at I depends on the angle of the emerging dial, there will be some angle for the emerging dial such that the dial will develop exactly into that angle (by the time it enters the time machine) under the influence of that emerging dial. This is so no matter what angle one starts with, and no matter how the development depends on the angle of the emerging dial. Thus even for a circular state-space there are no constraints needed other than continuity.

Unfortunately there are state-spaces that escape even this argument. Consider for instance a pointer that can be set to all values between 0 and 1, where 0 and 1 are not possible values. That is, suppose that we have a state-space that is isomorphic to an open set of real numbers. Now suppose that we have a machine that sets the pointer to half the value that the pointer is set at when it emerges from the time machine.

Suppose the pointer starts at value I . As before we can represent the combination of this initial position and all possible final positions by the line I . Under the influence of the pointer coming out of the time machine the pointer value will develop to a value that equals half the value of the final value that it encountered. We can represent this development as the continuous deformation of line I into line L , which is indicated by the arrows in Figure 2. This development is fully continuous. Points < x , y > on line I represent the initial position x = I of the (young) pointer, and the position y of the older pointer as it emerges from the time machine. Points < x , y > on line L represent the position x that the younger pointer should develop into, given that it encountered the older pointer emerging from the time machine set at position y . Since the pointer is designed to develop to half the value of the pointer that it encounters, the line L corresponds to x = 1 / 2 y . We have consistency if there is some point such that it develops into that point, if it encounters that point. Thus, we have consistency if there is some point < x , y > on line L such that x = y . However, there is no such point: lines L and C do not intersect. Thus there is no consistent solution, despite the fact that the dynamics is fully continuous.

Of course if 0 were a possible value L and C would intersect at 0. This is surprising and strange: adding one point to the set of possible values of a quantity here makes the difference between paradox and peace. One might be tempted to just add the extra point to the state-space in order to avoid problems. After all, one might say, surely no measurements could ever tell us whether the set of possible values includes that exact point or not. Unfortunately there can be good theoretical reasons for supposing that some quantity has a state-space that is open: the set of all possible speeds of massive objects in special relativity surely is an open set, since it includes all speeds up to, but not including, the speed of light. Quantities that have possible values that are not bounded also lead to counter examples to the presented fixed point argument. And it is not obvious to us why one should exclude such possibilities. So the argument that no constraints are needed is not fully general.

An interesting question of course is: exactly for which state-spaces must there be such fixed points. We do not know the general answer. (But see Kutach 2003 for more on this issue.)

Time travel has recently been discussed quite extensively in the context of general relativity. Time travel can occur in general relativistic models in which one has closed time-like curves (CTC's). A time like curve is simply a space-time trajectory such that the speed of light is never equalled or exceeded along this trajectory. Time-like curves thus represent the possible trajectories of ordinary objects. If there were time-like curves which were closed (formed a loop), then travelling along such a curve one would never exceed the speed of light, and yet after a certain amount of (proper) time one would return to a point in space-time that one previously visited. Or, by staying close to such a CTC, one could come arbitrarily close to a point in space-time that one previously visited. General relativity, in a straightforward sense, allows time travel: there appear to be many space-times compatible with the fundamental equations of General Relativity in which there are CTC's. Space-time, for instance, could have a Minkowski metric everywhere, and yet have CTC's everywhere by having the temporal dimension (topologically) rolled up as a circle. Or, one can have wormhole connections between different parts of space-time which allow one to enter ‘mouth A ’ of such a wormhole connection, travel through the wormhole, exit the wormhole at ‘mouth B ’ and re-enter ‘mouth A ’ again. Or, one can have space-times which topologically are R4, and yet have CTC's due to the ‘tilting’ of light cones (Gödel space-times, Taub-NUT space-times, etc.)

General relativity thus appears to provide ample opportunity for time travel. Note that just because there are CTC's in a space-time, this does not mean that one can get from any point in the space-time to any other point by following some future directed timelike curve. In many space-times in which there are CTC's such CTC's do not occur all over space-time. Some parts of space-time can have CTC's while other parts do not. Let us call the part of a space-time that has CTC's the “time travel region" of that space-time, while calling the rest of that space-time the "normal region". More precisely, the “time travel region" consists of all the space-time points p such that there exists a (non-zero length) timelike curve that starts at p and returns to p . Now let us start examining space-times with CTC's a bit more closely for potential problems.

In order to get a feeling for the sorts of implications that closed timelike curves can have, it may be useful to consider two simple models. In space-times with closed timelike curves the traditional initial value problem cannot be framed in the usual way. For it presupposes the existence of Cauchy surfaces, and if there are CTCs then no Cauchy surface exists. (A Cauchy surface is a spacelike surface such that every inextendible timelike curve crosses it exactly once. One normally specifies initial conditions by giving the conditions on such a surface.) Nonetheless, if the topological complexities of the manifold are appropriately localized, we can come quite close. Let us call an edgeless spacelike surface S a quasi-Cauchy surface if it divides the rest of the manifold into two parts such that a) every point in the manifold can be connected by a timelike curve to S , and b) any timelike curve which connects a point in one region to a point in the other region intersects S exactly once. It is obvious that a quasi-Cauchy surface must entirely inhabit the normal region of the space-time; if any point p of S is in the time travel region, then any timelike curve which intersects p can be extended to a timelike curve which intersects S near p again. In extreme cases of time travel, a model may have no normal region at all (e.g., Minkowski space-time rolled up like a cylinder in a time-like direction), in which case our usual notions of temporal precedence will not apply. But temporal anomalies like wormholes (and time machines) can be sufficiently localized to permit the existence of quasi-Cauchy surfaces.

Given a timelike orientation, a quasi-Cauchy surface unproblematically divides the manifold into its past (i.e., all points that can be reached by past-directed timelike curves from S ) and its future (ditto mutatis mutandis ). If the whole past of S is in the normal region of the manifold, then S is a partial Cauchy surface : every inextendible timelike curve which exists to the past of S intersects S exactly once, but (if there is time travel in the future) not every inextendible timelike curve which exists to the future of S intersects S . Now we can ask a particularly clear question: consider a manifold which contains a time travel region, but also has a partial Cauchy surface S , such that all of the temporal funny business is to the future of S . If all you could see were S and its past, you would not know that the space-time had any time travel at all. The question is: are there any constraints on the sort of data which can be put on S and continued to a global solution of the dynamics which are different from the constraints (if any) on the data which can be put on a Cauchy surface in a simply connected manifold and continued to a global solution? If there is time travel to our future, might we we able to tell this now, because of some implied oddity in the arrangement of present things?

It is not at all surprising that there might be constraints on the data which can be put on a locally space-like surface which passes through the time travel region: after all, we never think we can freely specify what happens on a space-like surface and on another such surface to its future, but in this case the surface at issue lies to its own future. But if there were particular constraints for data on a partial Cauchy surface then we would apparently need to have to rule out some sorts of otherwise acceptable states on S if there is to be time travel to the future of S . We then might be able to establish that there will be no time travel in the future by simple inspection of the present state of the universe. As we will see, there is reason to suspect that such constraints on the partial Cauchy surface are non-generic. But we are getting ahead of ourselves: first let's consider the effect of time travel on a very simple dynamics.

The simplest possible example is the Newtonian theory of perfectly elastic collisions among equally massive particles in one spatial dimension. The space-time is two-dimensional, so we can represent it initially as the Euclidean plane, and the dynamics is completely specified by two conditions. When particles are traveling freely, their world lines are straight lines in the space-time, and when two particles collide, they exchange momenta, so the collision looks like an ‘ X ’ in space-time, with each particle changing its momentum at the impact. [ 1 ] The dynamics is purely local, in that one can check that a set of world-lines constitutes a model of the dynamics by checking that the dynamics is obeyed in every arbitrarily small region. It is also trivial to generate solutions from arbitrary initial data if there are no CTCs: given the initial positions and momenta of a set of particles, one simply draws a straight line from each particle in the appropriate direction and continues it indefinitely. Once all the lines are drawn, the worldline of each particle can be traced from collision to collision. The boundary value problem for this dynamics is obviously well-posed: any set of data at an instant yields a unique global solution, constructed by the method sketched above.

What happens if we change the topology of the space-time by hand to produce CTCs? The simplest way to do this is depicted in figure 3: we cut and paste the space-time so it is no longer simply connected by identifying the line L − with the line L +. Particles “going in” to L + from below “emerge” from L − , and particles “going in” to L − from below “emerge” from L +.

Figure 3: Inserting CTCs by Cut and Paste

How is the boundary-value problem changed by this alteration in the space-time? Before the cut and paste, we can put arbitrary data on the simultaneity slice S and continue it to a unique solution. After the change in topology, S is no longer a Cauchy surface, since a CTC will never intersect it, but it is a partial Cauchy surface. So we can ask two questions. First, can arbitrary data on S always be continued to a global solution? Second, is that solution unique? If the answer to the first question is no , then we have a backward-temporal constraint: the existence of the region with CTCs places constraints on what can happen on S even though that region lies completely to the future of S . If the answer to the second question is no , then we have an odd sort of indeterminism: the complete physical state on S does not determine the physical state in the future, even though the local dynamics is perfectly deterministic and even though there is no other past edge to the space-time region in S 's future (i.e., there is nowhere else for boundary values to come from which could influence the state of the region).

In this case the answer to the first question is yes and to the second is no : there are no constraints on the data which can be put on S , but those data are always consistent with an infinitude of different global solutions. The easy way to see that there always is a solution is to construct the minimal solution in the following way. Start drawing straight lines from S as required by the initial data. If a line hits L − from the bottom, just continue it coming out of the top of L + in the appropriate place, and if a line hits L + from the bottom, continue it emerging from L − at the appropriate place. Figure 4 represents the minimal solution for a single particle which enters the time-travel region from the left:

Figure 4: The Minimal Solution

The particle ‘travels back in time’ three times. It is obvious that this minimal solution is a global solution, since the particle always travels inertially.

But the same initial state on S is also consistent with other global solutions. The new requirement imposed by the topology is just that the data going into L + from the bottom match the data coming out of L − from the top, and the data going into L - from the bottom match the data coming out of L + from the top. So we can add any number of vertical lines connecting L - and L + to a solution and still have a solution. For example, adding a few such lines to the minimal solution yields:

Figure 5: A Non-Minimal Solution

The particle now collides with itself twice: first before it reaches L + for the first time, and again shortly before it exits the CTC region. From the particle's point of view, it is traveling to the right at a constant speed until it hits an older version of itself and comes to rest. It remains at rest until it is hit from the right by a younger version of itself, and then continues moving off, and the same process repeats later. It is clear that this is a global model of the dynamics, and that any number of distinct models could be generating by varying the number and placement of vertical lines.

Knowing the data on S , then, gives us only incomplete information about how things will go for the particle. We know that the particle will enter the CTC region, and will reach L +, we know that it will be the only particle in the universe, we know exactly where and with what speed it will exit the CTC region. But we cannot determine how many collisions the particle will undergo (if any), nor how long (in proper time) it will stay in the CTC region. If the particle were a clock, we could not predict what time it would indicate when exiting the region. Furthermore, the dynamics gives us no handle on what to think of the various possibilities: there are no probabilities assigned to the various distinct possible outcomes.

Changing the topology has changed the mathematics of the situation in two ways, which tend to pull in opposite directions. On the one hand, S is no longer a Cauchy surface, so it is perhaps not surprising that data on S do not suffice to fix a unique global solution. But on the other hand, there is an added constraint: data “coming out” of L − must exactly match data “going in” to L +, even though what comes out of L − helps to determine what goes into L +. This added consistency constraint tends to cut down on solutions, although in this case the additional constraint is more than outweighed by the freedom to consider various sorts of data on L +/ L -.

The fact that the extra freedom outweighs the extra constraint also points up one unexpected way that the supposed paradoxes of time travel may be overcome. Let's try to set up a paradoxical situation using the little closed time loop above. If we send a single particle into the loop from the left and do nothing else, we know exactly where it will exit the right side of the time travel region. Now suppose we station someone at the other side of the region with the following charge: if the particle should come out on the right side, the person is to do something to prevent the particle from going in on the left in the first place. In fact, this is quite easy to do: if we send a particle in from the right, it seems that it can exit on the left and deflect the incoming left-hand particle.

Carrying on our reflection in this way, we further realize that if the particle comes out on the right, we might as well send it back in order to deflect itself from entering in the first place. So all we really need to do is the following: set up a perfectly reflecting particle mirror on the right-hand side of the time travel region, and launch the particle from the left so that— if nothing interferes with it —it will just barely hit L +. Our paradox is now apparently complete. If, on the one hand, nothing interferes with the particle it will enter the time-travel region on the left, exit on the right, be reflected from the mirror, re-enter from the right, and come out on the left to prevent itself from ever entering. So if it enters, it gets deflected and never enters. On the other hand, if it never enters then nothing goes in on the left, so nothing comes out on the right, so nothing is reflected back, and there is nothing to deflect it from entering. So if it doesn't enter, then there is nothing to deflect it and it enters. If it enters, then it is deflected and doesn't enter; if it doesn't enter then there is nothing to deflect it and it enters: paradox complete.

But at least one solution to the supposed paradox is easy to construct: just follow the recipe for constructing the minimal solution, continuing the initial trajectory of the particle (reflecting it the mirror in the obvious way) and then read of the number and trajectories of the particles from the resulting diagram. We get the result of figure 6:

Figure 6: Resolving the “Paradox”

As we can see, the particle approaching from the left never reaches L +: it is deflected first by a particle which emerges from L -. But it is not deflected by itself , as the paradox suggests, it is deflected by another particle. Indeed, there are now four particles in the diagram: the original particle and three particles which are confined to closed time-like curves. It is not the leftmost particle which is reflected by the mirror, nor even the particle which deflects the leftmost particle; it is another particle altogether.

The paradox gets it traction from an incorrect presupposition: if there is only one particle in the world at S then there is only one particle which could participate in an interaction in the time travel region: the single particle would have to interact with its earlier (or later) self. But there is no telling what might come out of L − : the only requirement is that whatever comes out must match what goes in at L +. So if you go to the trouble of constructing a working time machine, you should be prepared for a different kind of disappointment when you attempt to go back and kill yourself: you may be prevented from entering the machine in the first place by some completely unpredictable entity which emerges from it. And once again a peculiar sort of indeterminism appears: if there are many self-consistent things which could prevent you from entering, there is no telling which is even likely to materialize.

So when the freedom to put data on L − outweighs the constraint that the same data go into L +, instead of paradox we get an embarrassment of riches: many solution consistent with the data on S . To see a case where the constraint “outweighs” the freedom, we need to construct a very particular, and frankly artificial, dynamics and topology. Consider the space of all linear dynamics for a scalar field on a lattice. (The lattice can be though of as a simple discrete space-time.) We will depict the space-time lattice as a directed graph. There is to be a scalar field defined at every node of the graph, whose value at a given node depends linearly on the values of the field at nodes which have arrows which lead to it. Each edge of the graph can be assigned a weighting factor which determines how much the field at the input node contributes to the field at the output node. If we name the nodes by the letters a , b , c , etc., and the edges by their endpoints in the obvious way, then we can label the weighting factors by the edges they are associated with in an equally obvious way.

Suppose that the graph of the space-time lattice is acyclic , as in figure 7. (A graph is Acyclic if one can not travel in the direction of the arrows and go in a loop.)

Figure 7: An Acyclic Lattice

It is easy to regard a set of nodes as the analog of a Cauchy surface, e.g., the set { a , b , c }, and it is obvious if arbitrary data are put on those nodes the data will generate a unique solution in the future. [ 2 ] If the value of the field at node a is 3 and at node b is 7, then its value at node d will be 3 W ad and its value at node e will be 3 W ae + 7 W be . By varying the weighting factors we can adjust the dynamics, but in an acyclic graph the future evolution of the field will always be unique.

Let us now again artificially alter the topology of the lattice to admit CTCs, so that the graph now is cyclic. One of the simplest such graphs is depicted in figure 8: there are now paths which lead from z back to itself, e.g., z to y to z .

Figure 8: Time Travel on a Lattice

Can we now put arbitrary data on v and w , and continue that data to a global solution? Will the solution be unique?

In the generic case, there will be a solution and the solution will be unique. The equations for the value of the field at x , y , and z are:

x = v W vx + z W zx y = w W wy + z W zy z = x W xz + y W yz .

Solving these equations for z yields

z = ( v W vx + z W zx ) W xz + ( w W wy + z W zy ) W yz , or z = ( v W vx W xz + w W wy W yz )/ (1 − W zx W xz − W zy W yz ),

which gives a unique value for z in the generic case. But looking at the space of all possible dynamics for this lattice (i.e., the space of all possible weighting factors), we find a singularity in the case where 1−W zx W xz − W zy W yz = 0. If we choose weighting factors in just this way, then arbitrary data at v and w cannot be continued to a global solution. Indeed, if the scalar field is everywhere non-negative, then this particular choice of dynamics puts ironclad constraints on the value of the field at v and w : the field there must be zero (assuming W vx and W wy to be non-zero), and similarly all nodes in their past must have field value zero. If the field can take negative values, then the values at v and w must be so chosen that v W vx W xz = − w W wy W yz . In either case, the field values at v and w are severely constrained by the existence of the CTC region even though these nodes lie completely to the past of that region. It is this sort of constraint which we find to be unlike anything which appears in standard physics.

Our toy models suggest three things. The first is that it may be impossible to prove in complete generality that arbitrary data on a partial Cauchy surface can always be continued to a global solution: our artificial case provides an example where it cannot. The second is that such odd constraints are not likely to be generic: we had to delicately fine-tune the dynamics to get a problem. The third is that the opposite problem, namely data on a partial Cauchy surface being consistent with many different global solutions, is likely to be generic: we did not have to do any fine-tuning to get this result. And this leads to a peculiar sort of indeterminism: the entire state on S does not determine what will happen in the future even though the local dynamics is deterministic and there are no other “edges” to space-time from which data could influence the result. What happens in the time travel region is constrained but not determined by what happens on S , and the dynamics does not even supply any probabilities for the various possibilities. The example of the photographic negative discussed in section 3, then, seems likely to be unusual, for in that case there is a unique fixed point for the dynamics, and the set-up plus the dynamical laws determine the outcome. In the generic case one would rather expect multiple fixed points, with no room for anything to influence, even probabilistically, which would be realized.

It is ironic that time travel should lead generically not to contradictions or to constraints (in the normal region) but to underdetermination of what happens in the time travel region by what happens everywhere else (an underdetermination tied neither to a probabilistic dynamics or to a free edge to space-time). The traditional objection to time travel is that it leads to contradictions: there is no consistent way to complete an arbitrarily constructed story about how the time traveler intends to act. Instead, though, it appears that the problem is underdetermination: the story can be consistently completed in many different ways.

The two toys models presented above have the virtue of being mathematically tractable, but they involve certain simplifications and potential problems that lead to trouble if one tries to make them more complicated. Working through these difficulties will help highlight the conditions we have made use of.

Consider a slight modification of the first simple model proposed to us by Adam Elga. Let the particles have an electric charge , which produces forces according to Coulomb’s law. Then set up a situation like that depicted in figure 9:

Figure 9: Set-up for Elga's Paradox

The dotted line indicates the path the particle will follow if no forces act upon it. The point labeled P is the left edge of the time-travel region; the two labels are a reminder that the point at the bottom and the point at the top are one and the same.

Elga's paradox is as follows: if no force acts on the particle, then it will enter the time-travel region. But if it enters the time travel region, and hence reappears along the bottom edge, then its later self will interact electrically with its earlier self, and the earlier self will be deflected away from the time-travel region. It is easy to set up the case so that the deflection will be enough to keep the particle from ever entering the time-travel region in the first place. (For instance, let the momentum of the incoming particle towards the time travel region be very small. The mere existence of an identically charged particle inside the time travel region will then be sufficient to deflect the incoming particle so that it never reaches L + .) But, of course, if the particle never enters the region at all, then it will not be there to deflect itself….

One might suspect that some complicated collection of charged particles in the time-travel-region can save the day, as it did with our mirror-reflection problem above. But (unless there are infinitely many such particles) this can't work, as conservation of particle number and linear momentum show. Suppose that some finite collection of particles emerges from L - and supplies the repulsive electric force needed to deflect the incoming particle. Then exactly the same collection of particles must be “absorbed” at L + . So at all times after L + , the only particle there is in the world is the incoming particle, which has now been deflected away from its original trajectory.

The deflection, though, means that the linear momentum of the particle has changed from what is was before L - . But that is impossible, by conservation of linear momementum. No matter how the incoming particle interacts with particles in the time-travel region, or how those particle interact with each other, total linear momentum is conserved by the interaction. And whatever net linear momentum the time-travelling particles have when they emerge from L - , that much linear momentum most be absorbed at L + . So the momentum of the incoming particle can't be changed by the interaction: the particle can't have been deflected. (One could imagine trying to create a sort of “S” curve in the trajectory of the incoming particle, first bending to the left and then to the right, which leaves its final momentum equal to its initial momentum, but moving it over in space so it misses L + . However, if the force at issue is repulsive, then the bending back to the right can't be done. In the mirror example above, the path of the incoming particle can be changed without violating the conservation of momentum because at the end of the process momentum has been transferred to the mirror.)

How does Elga's example escape our analysis? Why can't a contintuity principle guarantee the existence of a solution here?

The continuity assumption breaks down because of two features of the example: the concentration of the electric charge on a point particle, and the way we have treated (or, more accurately, failed to treat) the point P , the edge of L + (and L - ). We have assumed that a point particle either hits L + , and then emerges from L - , or else it misses L + and sails on into the region of space-time above it. This means that the charge on the incoming particle only has two possibilities: either it is transported whole back in time or it completely avoids time travel altogether. Let's see how it alters the situation to imagine the charge itself to be continuous divisible.

Suppose that, instead of being concentrated at a point, the incoming object is a little stick, with electric charge distributed even across it (figure 10).

Figure 10: Elga's Paradox with a Charged Bar

Once again, we set things up so that if there are no forces on the bar, it will be completely absorbed at L + . But we now postulate that if the bar should hit the point P , it will fracture: part of it (the part that hits L+ ) will be sent back in time and the rest will continue on above L + . So continuity of a sort is restored: now we have not just the possibility of the whole charge being sent back or nothing, we have the continuum degrees of charge in between.

It is not hard to see that the restoration of continuity restores the existence of a consistent solution. If no charge is sent back through time, then the bar is not deflected and all of it hits L + (and hence is sent back through time). If all the charge is sent back through time, then is incoming bar is deflected to an extent that it misses L + completely, and so no charge is sent back. But if just the right amount of charge is sent back through time, then the bar will be only partially deflected, deflected so that it hits the edge point P , and is split into a bit that goes back and a bit that does not, with the bit that goes back being just the right amount of charge to produce just that deflection (figure 11).

Figure 11: Solution to Elga's Paradox with a Charged Bar

Our problem about conservation of momentum is also solved: piece of the bar that does not time travel has lower momentum to the right at the end than it had initially, but the piece that does time travel has a higher momentum (due to the Coulomb forces), and everything balances out.

Is it cheating to model the charged particle as a bar that can fracture? What if we insist that the particle is truly a point particle, and hence that its time travel is an all-or-nothing affair?

In that case, we now have to worry about a question we have not yet confronted: what happens if our point particle hits exactly at the point P on the diagram? Does it time-travel or not? Confronting this question requires us to face up to a feature of the rather cheap way we implemented time travel in our toy models by cut-and-paste. The way we rejiggered the space-time structure had a rather severe consequence: the resulting space-time is no longer a manifold : the topological structure at the point P is different from the topological structure elsewhere. Mathematical physicists simply don't deal with such structures: the usual procedure is to eliminate the offending point from the space-time and thus restore the manifold structure. In this case, that would leave a bare singularity at point P , an open edge to space-time into which anything could disappear and out of which, for all the physics tells us, anything could emerge.

In particular, if we insist that our particle is a point particle, then if its trajectory should happen to intersect P it will simply disappear. What could cause the extremely fortuitous result that the trajectory strikes precisely at P ? The emergence of some other charged particle, with just the right charge and trajectory, from P (on L - ). And we are no longer bound by any conservation laws: the bare singularity can both swallow and produce whatever mass or change or momentum we like. So if we insist on point particles, then we have to take account of the singularity, and that again saves the day.

Consideration of these (slightly more complicated) toy models does not replace the proving of theorems, of course. But they do serve to illustrate the sorts of consideration that necessarily come into play when trying to spell out the physics of time travel in all detail. Let us now discuss some results regarding some slightly more realistic models that have been discussed in the physics literature.

Echeverria, Klinkhammer and Thorne (1991) considered the case of 3-dimensional single hard spherical ball that can go through a single time travel wormhole so as to collide with its younger self.

The threat of paradox in this case arises in the following form. There are initial trajectories (starting in the non-time travel region of space-time) for the ball such that if such a trajectory is continued (into the time travel region), assuming that the ball does not undergo a collision prior to entering mouth 1 of the wormhole, it will exit mouth 2 so as to collide with its earlier self prior to its entry into mouth 1 in such a way as to prevent its earlier self from entering mouth 1. Thus it seems that the ball will enter mouth 1 if and only if it does not enter mouth 1. Of course, the Wheeler-Feynman strategy is to look for a ‘glancing blow’ solution: a collision which will produce exactly the (small) deviation in trajectory of the earlier ball that produces exactly that collision. Are there always such solutions? [ 3 ]

Echeverria, Klinkhammer & Thorne found a large class of initial trajectories that have consistent ‘glancing blow’ continuations, and found none that do not (but their search was not completely general). They did not produce a rigorous proof that every initial trajectory has a consistent continuation, but suggested that it is very plausible that every initial trajectory has a consistent continuation. That is to say, they have made it very plausible that, in the billiard ball wormhole case, the time travel structure of such a wormhole space-time does not result in constraints on states on spacelike surfaces in the non-time travel region.

In fact, as one might expect from our discussion in the previous section, they found the opposite problem from that of inconsistency: they found underdetermination. For a large class of initial trajectories there are multiple different consistent ‘glancing blow’ continuations of that trajectory (many of which involve multiple wormhole traversals). For example, if one initially has a ball that is traveling on a trajectory aimed straight between the two mouths, then one obvious solution is that the ball passes between the two mouths and never time travels. But another solution is that the younger ball gets knocked into mouth 1 exactly so as to come out of mouth 2 and produce that collision. Echeverria et al. do not note the possibility (which we pointed out in the previous section) of the existence of additional balls in the time travel region. We conjecture (but have no proof) that for every initial trajectory of A there are some, and generically many, multiple ball continuations.

Friedman et al. 1990 examined the case of source free non-self-interacting scalar fields traveling through such a time travel wormhole and found that no constraints on initial conditions in the non-time travel region are imposed by the existence of such time travel wormholes. In general there appear to be no known counter examples to the claim that in ‘somewhat realistic’ time-travel space-times with a partial Cauchy surface there are no constraints imposed on the state on such a partial Cauchy surface by the existence of CTC's. (See e.g., Friedman and Morris 1991, Thorne 1994, and Earman 1995; in the Other Internet Resources, see Earman, Smeenk, and Wüthrich 2003.)

How about the issue of constraints in the time travel region T ? Prima facie , constraints in such a region would not appear to be surprising. But one might still expect that there should be no constraints on states on a spacelike surface, provided one keeps the surface ‘small enough’. In the physics literature the following question has been asked: for any point p in T , and any space-like surface S that includes p is there a neighborhood E of p in S such that any solution on E can be extended to a solution on the whole space-time? With respect to this question, there are some simple models in which one has this kind of extendibility of local solutions to global ones, and some simple models in which one does not have such extendibility, with no clear general pattern. The technical mathematical problems are amplified by the more conceptual problem of what it might mean to say that one could create a situation which forces the creation of closed timelike curves. (See e.g. Yurtsever 1990, Friedman et al. 1990, Novikov 1992, Earman 1995 and Earman, Smeenk and Wüthrich 2009; in the Other Internet Resources, see Earman, Smeenk and Wüthrich 2003). What are we to think of all of this?

Since it is not obvious that one can rid oneself of all constraints in realistic models, let us examine the argument that time travel is implausible, and we should think it unlikely to exist in our world, in so far as it implies such constraints. The argument goes something like the following. In order to satisfy such constraints one needs some pre-established divine harmony between the global (time travel) structure of space-time and the distribution of particles and fields on space-like surfaces in it. But it is not plausible that the actual world, or any world even remotely like ours, is constructed with divine harmony as part of the plan. In fact, one might argue, we have empirical evidence that conditions in any spatial region can vary quite arbitrarily. So we have evidence that such constraints, whatever they are, do not in fact exist in our world. So we have evidence that there are no closed time-like lines in our world or one remotely like it. We will now examine this argument in more detail by presenting four possible responses, with counterresponses, to this argument.

Response 1. There is nothing implausible or new about such constraints. For instance, if the universe is spatially closed, there has to be enough matter to produce the needed curvature, and this puts constraints on the matter distribution on a space-like hypersurface. Thus global space-time structure can quite unproblematically constrain matter distributions on space-like hypersurfaces in it. Moreover we have no realistic idea what these constraints look like, so we hardly can be said to have evidence that they do not obtain.

Counterresponse 1. Of course there are constraining relations between the global structure of space-time and the matter in it. The Einstein equations relate curvature of the manifold to the matter distribution in it. But what is so strange and implausible about the constraints imposed by the existence of closed time-like curves is that these constraints in essence have nothing to do with the Einstein equations. When investigating such constraints one typically treats the particles and/or field in question as test particles and/or fields in a given space-time, i.e., they are assumed not to affect the metric of space-time in any way. In typical space-times without closed time-like curves this means that one has, in essence, complete freedom of matter distribution on a space-like hypersurface. (See response 2 for some more discussion of this issue). The constraints imposed by the possibility of time travel have a quite different origin and are implausible. In the ordinary case there is a causal interaction between matter and space-time that results in relations between global structure of space-time and the matter distribution in it. In the time travel case there is no such causal story to be told: there simply has to be some pre-established harmony between the global space-time structure and the matter distribution on some space-like surfaces. This is implausible.

Response 2. Constraints upon matter distributions are nothing new. For instance, Maxwell's equations constrain electric fields E on an initial surface to be related to the (simultaneous) charge density distribution ρ by the equation ρ = div( E ). (If we assume that the E field is generated solely by the charge distribution, this conditions amounts to requiring that the E field at any point in space simply be the one generated by the charge distribution according to Coulomb's inverse square law of electrostatics.) This is not implausible divine harmony. Such constraints can hold as a matter of physical law. Moreover, if we had inferred from the apparent free variation of conditions on spatial regions that there could be no such constraints we would have mistakenly inferred that ρ = div( E ) could not be a law of nature.

Counterresponse 2. The constraints imposed by the existence of closed time-like lines are of quite a different character from the constraint imposed by ρ = div( E ). The constraints imposed by ρ = div( E ) on the state on a space-like hypersurface are: (i) local constraints (i.e., to check whether the constraint holds in a region you just need to see whether it holds at each point in the region), (ii) quite independent of the global space-time structure, (iii) quite independent of how the space-like surface in question is embedded in a given space-time, and (iv) very simply and generally stateable. On the other hand, the consistency constraints imposed by the existence of closed time-like curves (i) are not local, (ii) are dependent on the global structure of space-time, (iii) depend on the location of the space-like surface in question in a given space-time, and (iv) appear not to be simply stateable other than as the demand that the state on that space-like surface embedded in such and such a way in a given space-time, do not lead to inconsistency. On some views of laws (e.g., David Lewis' view) this plausibly implies that such constraints, even if they hold, could not possibly be laws. But even if one does not accept such a view of laws, one could claim that the bizarre features of such constraints imply that it is implausible that such constraints hold in our world or in any world remotely like ours.

Response 3. It would be strange if there are constraints in the non-time travel region. It is not strange if there are constraints in the time travel region. They should be explained in terms of the strange, self-interactive, character of time travel regions. In this region there are time-like trajectories from points to themselves. Thus the state at such a point, in such a region, will, in a sense, interact with itself. It is a well-known fact that systems that interact with themselves will develop into an equilibrium state, if there is such an equilibrium state, or else will develop towards some singularity. Normally, of course, self-interaction isn't true instantaneous self-interaction, but consists of a feed-back mechanism that takes time. But in time travel regions something like true instantaneous self-interaction occurs. This explains why constraints on states occur in such time travel regions: the states ‘ab initio’ have to be ‘equilibrium states’. Indeed in a way this also provides some picture of why indeterminism occurs in time travel regions: at the onset of self-interaction states can fork into different equi-possible equilibrium states.

Counterresponse 3. This is explanation by woolly analogy. It all goes to show that time travel leads to such bizarre consequences that it is unlikely that it occurs in a world remotely like ours.

Response 4. All of the previous discussion completely misses the point. So far we have been taking the space-time structure as given, and asked the question whether a given time travel space-time structure imposes constraints on states on (parts of) space-like surfaces. However, space-time and matter interact. Suppose that one is in a space-time with closed time-like lines, such that certain counterfactual distributions of matter on some neighborhood of a point p are ruled out if one holds that space-time structure fixed. One might then ask “Why does the actual state near p in fact satisfy these constraints? By what divine luck or plan is this local state compatible with the global space-time structure? What if conditions near p had been slightly different?” And one might take it that the lack of normal answers to these questions indicates that it is very implausible that our world, or any remotely like it, is such a time travel universe. However the proper response to these question is the following. There are no constraints in any significant sense. If they hold they hold as a matter of accidental fact, not of law. There is no more explanation of them possible than there is of any contingent fact. Had conditions in a neighborhood of p been otherwise, the global structure of space-time would have been different. So what? The only question relevant to the issue of constraints is whether an arbitrary state on an arbitrary spatial surface S can always be embedded into a space-time such that that state on S consistently extends to a solution on the entire space-time.

But we know the answer to that question. A well-known theorem in general relativity says the following: any initial data set on a three dimensional manifold S with positive definite metric has a unique embedding into a maximal space-time in which S is a Cauchy surface (see e.g., Geroch and Horowitz 1979, p. 284 for more detail), i.e., there is a unique largest space-time which has S as a Cauchy surface and contains a consistent evolution of the initial value data on S . Now since S is a Cauchy surface this space-time does not have closed time like curves. But it may have extensions (in which S is not a Cauchy surface) which include closed timelike curves, indeed it may be that any maximal extension of it would include closed timelike curves. (This appears to be the case for extensions of states on certain surfaces of Taub-NUT space-times. See Earman, Smeenk, and Wüthrich 2003 in the Other Internet Resources). But these extensions, of course, will be consistent. So properly speaking, there are no constraints on states on space-like surfaces. Nonetheless the space-time in which these are embedded may or may not include closed time-like curves.

Counterresponse 4. This, in essence, is the stonewalling answer which we indicated at the beginning of section 2. However, whether or not you call the constraints imposed by a given space-time on distributions of matter on certain space-like surfaces ‘genuine constraints’, whether or not they can be considered lawlike, and whether or not they need to be explained, the existence of such constraints can still be used to argue that time travel worlds are so bizarre that it is implausible that our world or any world remotely like ours is a time travel world.

Suppose that one is in a time travel world. Suppose that given the global space-time structure of this world, there are constraints imposed upon, say, the state of motion of a ball on some space-like surface when it is treated as a test particle, i.e., when it is assumed that the ball does not affect the metric properties of the space-time it is in. (There is lots of other matter that, via the Einstein equation, corresponds exactly to the curvature that there is everywhere in this time travel worlds.) Now a real ball of course does have some effect on the metric of the space-time it is in. But let us consider a ball that is so small that its effect on the metric is negligible. Presumably it will still be the case that certain states of this ball on that space-like surface are not compatible with the global time travel structure of this universe.

This means that the actual distribution of matter on such a space-like surface can be extended into a space-time with closed time-like lines, but that certain counterfactual distributions of matter on this space-like surface can not be extended into the same space-time. But note that the changes made in the matter distribution (when going from the actual to the counterfactual distribution) do not in any non-negligible way affect the metric properties of the space-time. Thus the reason why the global time travel properties of the counterfactual space-time have to be significantly different from the actual space-time is not that there are problems with metric singularities or alterations in the metric that force significant global changes when we go to the counterfactual matter distribution. The reason that the counterfactual space-time has to be different is that in the counterfactual world the ball's initial state of motion starting on the space-like surface, could not ‘meet up’ in a consistent way with its earlier self (could not be consistently extended) if we were to let the global structure of the counterfactual space-time be the same as that of the actual space-time. Now, it is not bizarre or implausible that there is a counterfactual dependence of manifold structure, even of its topology, on matter distributions on spacelike surfaces. For instance, certain matter distributions may lead to singularities, others may not. We may indeed in some sense have causal power over the topology of the space-time we live in. But this power normally comes via the Einstein equations. But it is bizarre to think that there could be a counterfactual dependence of global space-time structure on the arrangement of certain tiny bits of matter on some space-like surface, where changes in that arrangement by assumption do not affect the metric anywhere in space-time in any significant way . It is implausible that we live in such a world, or that a world even remotely like ours is like that.

Let us illustrate this argument in a different way by assuming that wormhole time travel imposes constraints upon the states of people prior to such time travel, where the people have so little mass/energy that they have negligible effect, via the Einstein equation, on the local metric properties of space-time. Do you think it more plausible that we live in a world where wormhole time travel occurs but it only occurs when people's states are such that these local states happen to combine with time travel in such a way that nobody ever succeeds in killing their younger self, or do you think it more plausible that we are not in a wormhole time travel world? [ 4 ]

There has been a particularly clear treatment of time travel in the context of quantum mechanics by David Deutsch (see Deutsch 1991, and Deutsch and Lockwood 1994) in which it is claimed that quantum mechanical considerations show that time travel never imposes any constraints on the pre-time travel state of systems. The essence of this account is as follows.

A quantum system starts in state S 1, interacts with its older self, after the interaction is in state S 2 , time travels while developing into state S 3 , then interacts with its younger self, and ends in state S 4 (see figure 13).

1 3 develops into 2 4 .

Similarly, suppose that:

1 3 develops into 2 4 , 1 3 develops into 2 4 , and 1 3 develops into 2 4 .

This clarification of why Deutsch needs his mixtures does however indicate a serious worry about the simplifications that are part of Deutsch's account. After the interaction the old and young system will (typically) be in an entangled state. Although for purposes of a measurement on one of the two systems one can say that this system is in a mixed state, one can not represent the full state of the two systems by specifying the mixed state of each separate part, as there are correlations between observables of the two systems that are not represented by these two mixed states, but are represented in the joint entangled state. But if there really is an entangled state of the old and young systems directly after the interaction, how is one to represent the subsequent development of this entangled state? Will the state of the younger system remain entangled with the state of the older system as the younger system time travels and the older system moves on into the future? On what space-like surfaces are we to imagine this total entangled state to be? At this point it becomes clear that there is no obvious and simple way to extend elementary non-relativistic quantum mechanics to space-times with closed time-like curves. There have been more sophisticated approaches than Deutsch's to time travel, using technical machinery from quantum field theory and differentiable manifolds (see e.g., Friedman et al 1991, Earman, Smeenk, and Wüthrich 2003 in the Other Internet Resources, and references therein). But out of such approaches no results anywhere near as clear and interesting as Deutsch's have been forthcoming.

How does Deutsch avoid these complications? Deutsch assumes a mixed state S 3 of the older system prior to the interaction with the younger system. He lets it interact with an arbitrary pure state S 1 younger system. After this interaction there is an entangled state S ′ of the two systems. Deutsch computes the mixed state S 2 of the younger system which is implied by this entangled state S ′. His demand for consistency then is just that this mixed state S 2 develops into the mixed state S 3 . Now it is not at all clear that this is a legitimate way to simplify the problem of time travel in quantum mechanics. But even if we grant him this simplification there is a problem: how are we to understand these mixtures?

Now whatever one thinks of the merits of many worlds interpretations, and of this understanding of it applied to mixtures, in the end one does not obtain genuine time travel in Deutsch's account. The systems in question travel from one time in one world to another time in another world, but no system travels to an earlier time in the same world. (This is so at least in the normal sense of the word ‘world,’ the sense that one means when, for instance, one says “there was, and will be, only one Elvis Presley in this world.”) Thus, even if it were a reasonable view, it is not quite as interesting as it may have initially seemed.

What remains of the killing-your-earlier-self paradox in general relativistic time travel worlds is the fact that in some cases the states on edgeless spacelike surfaces are ‘overconstrained’, so that one has less than the usual freedom in specifying conditions on such a surface, given the time-travel structure, and in some cases such states are ‘underconstrained’, so that states on edgeless space-like surfaces do not determine what happens elsewhere in the way that they usually do, given the time travel structure. There can also be mixtures of those two types of cases. The extent to which states are overconstrained and/or underconstrained in realistic models is as yet unclear, though it would be very surprising if neither obtained. The extant literature has primarily focused on the problem of overconstraint, since that, often, either is regarded as a metaphysical obstacle to the possibility time travel, or as an epistemological obstacle to the plausibility of time travel in our world. While it is true that our world would be quite different from the way we normally think it is if states were overconstrained, underconstraint seems at least as bizarre as overconstraint. Nonetheless, neither directly rules out the possibility of time travel.

If time travel entailed contradictions then the issue would be settled. And indeed, most of the stories employing time travel in popular culture are logically incoherent: one cannot “change” the past to be different from what it was, since the past (like the present and the future) only occurs once. But if the only requirement demanded is logical coherence, then it seems all too easy. A clever author can devise a coherent time-travel scenario in which everything happens just once and in a consistent way. This is just too cheap: logical coherence is a very weak condition, and many things we take to be metaphysically impossible are logically coherent. For example, it involves no logical contradiction to suppose that water is not molecular, but if both chemistry and Kripke are right it is a metaphysical impossibility. We have been interested not in logical possibility but in physical possibility. But even so, our conditions have been relatively weak: we have asked only whether time-travel is consistent with the universal validity of certain fundamental physical laws and with the notion that the physical state on a surface prior to the time travel region be unconstrained. It is perfectly possible that the physical laws obey this condition, but still that time travel is not metaphysically possible because of the nature of time itself. Consider an analogy. Aristotle believed that water is homoiomerous and infinitely divisible: any bit of water could be subdivided, in principle, into smaller bits of water. Aristotle's view contains no logical contradiction. It was certainly consistent with Aristotle's conception of water that it be homoiomerous, so this was, for him, a conceptual possibility. But if chemistry is right, Aristotle was wrong both about what water is like and what is possible for it. It can't be infinitely divided, even though no logical or conceptual analysis would reveal that.

Similarly, even if all of our consistency conditions can be met, it does not follow that time travel is physically possible, only that some specific physical considerations cannot rule it out. The only serious proof of the possibility of time travel would be a demonstration of its actuality. For if we agree that there is no actual time travel in our universe, the supposition that there might have been involves postulating a substantial difference from actuality, a difference unlike in kind from anything we could know if firsthand. It is unclear to us exactly what the content of possible would be if one were to either maintain or deny the possibility of time travel in these circumstances, unless one merely meant that the possibility is not ruled out by some delineated set of constraints. As the example of Aristotle's theory of water shows, conceptual and logical “possibility” do not entail possibility in a full-blooded sense. What exactly such a full-blooded sense would be in case of time travel, and whether one could have reason to believe it to obtain, remain to us obscure.

• Deutsch, D. 1991. “Quantum mechanics near closed timelike curves,” Physical Review D , 44: 3197-3217.
• Deutsch, D. and Lockwood, M. 1994. “The quantum physics of time travel,” Scientific American , 270 (3): 68-74.
• Earman, J. 1972. “Implications of causal propagation outsider the null cone,” in Foundations of Space-Time Theory , Minnesota Studies in the Philosophy of Science , Vol VII, Earman, J., Glymour, C., and Stachel, J. (eds), pp. 94-108. Minneapolis: University of Minnesota Press.
• Earman, J. 1995. Bangs, Crunches, Whimpers and Shrieks: Singularities and Acausalities in Relativistic Spacetimes , New York: Oxford University Press.
• Earman, J., Smeenk, C., and Wüthrich, C. 2009.“Do the laws of physics forbid the operation of a time machine?,” Synthese , 169 (1): 91-124.
• Echeverria, F., Klinkhammer, G., and Thorne, K. 1991. “Billiard ball in wormhole spacetimes with closed timelike curves: classical theory,” Physical Review D , 44 (4): 1077-1099.
• Friedman, J. et al. 1990. “Cauchy problem in spacetimes with closed timelike lines,” Physical Review D , 42: 1915-1930.
• Friedman, J. and Morris, M. 1991. “The Cauchy problem for the scalar wave equation is well defined on a class of spacetimes with closed timelike curves,” Physical Review letters , 66: 401-404.
• Geroch, R. and Horowitz, G. 1979. “Global structures of spacetimes,” in General Relativity, an Einstein Centenary Survey , S. Hawking and W. Israel (eds.), Cambridge: Cambridge University Press.
• Gödel, K. 1949. “A remark about the relationship between relativity theory and idealistic philosophy,” in Albert Einstein: Philosopher-Scientist , P. Schilpp (ed.), La Salle: Open Court, pp. 557-562.
• Hocking, J., and Young, G. 1961. Topology , New York: Dover Publications.
• Horwich, P. 1987. “Time travel,” in Asymmetries in time , Cambridge, MA: MIT Press.
• Kutach, D. 2003. “Time travel and consistency constraint”, Philosophy of Science , 70: 1098-1113.
• Malament, D. 1985a. “’Time travel’ in the Gödel universe,” PSA 1984, 2: 91-100, P. Asquith and P. Kitcher (eds.), East Lansing, MI: Philosophy of Science Association.
• Malament, D. 1985b. “Minimal acceleration requirements for ‘time travel’ in Gödel spacetime,” Journal of Mathematical Physics , 26: 774-777.
• Maudlin, T. 1990. “Time Travel and topology,” PSA 1990, 1: 303-315, East Lansing, MI: Philosophy of Science Association.
• Novikov, I. 1992. “Time machine and self-consistent evolution in problems with self-interaction,” Physical Review D , 45: 1989-1994.
• Thorne, K. 1994. Black Holes and Time Warps, Einstein's Outrageous Legacy , London and New York: W.W. Norton.
• Wheeler, J. and Feynman, R. 1949. “Classical electrodynamics in terms of direct interparticle action,” Reviews of Modern Physics , 21: 425-434.
• Yurtsever, U. 1990. “Test fields on compact space-times,” Journal of Mathematical Physics , 31: 3064-3078.

How to cite this entry . Preview the PDF version of this entry at the Friends of the SEP Society . Look up this entry topic at the Internet Philosophy Ontology Project (InPhO). Enhanced bibliography for this entry at PhilPapers , with links to its database.

• Earman, J., Smeenk, C. and Wüthrich, C., 2003, “ Take a ride on a time machine ”, manuscript available at the PhilSci Archive, University of Pittsburgh.
• Time Travel in Flatland (Cal Tech Particle Theory Group)

determinism: causal | -->Gödel, Kurt: contributions to relativity theory --> | time machines | time travel

Acknowledgments

Thanks to Edward N. Zalta, who spotted that we incorrectly stated one of the consequences of Maxwell's equations as E = div(ρ) rather than as ρ = div( E ).

Copyright © 2009 by Frank Arntzenius Tim Maudlin < twm3 @ nyu . edu >

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Stanford Humanities Today

Arcade: a digital salon, a sketch map of a lost continent: the republic of letters.

This essay offers a historical traveler’s report on a strange imaginary land, one that had few of the distinctive marks by which we usually identify a state. It did have a distinctive name: Respublica literarum , the Republic of Letters. Its citizens agreed that they owed it loyalty, and almost all of them spoke its two languages—Latin, which remained the language of all scholars from 1500 to 1650 or so, and still played a prominent role thereafter, and French, which gradually replaced it in most periodicals and almost all salons. But it had no borders, no government, no capital. In a world of sharp and well-defined social hierarchies—a world in which men and women wore formal costumes that graphically revealed their rank and occupation—its citizens insisted that all of them were equal, and that any special fame that one of them might enjoy had been earned by his or her own efforts. As one observer put it in 1699, “The Republic of Letters is of very ancient origin . . . It embraces the whole world and is composed of all nationalities, all social classes, all ages and both sexes . . . All languages, ancient as well as modern, are spoken. The arts are joined to letters, and artisans also have their place in it . . . Praise and honor are awarded by popular acclaim.” [1] The Republic of Letters imagined itself as Europe’s first egalitarian society, even if it did not always enact these high ideals in the grubby reality of its intellectual and professional practices.

The citizens of the Republic were the last Europeans who could plausibly claim that they were masters of their entire civilization. We live in a world of specialists. From engineering and mathematics to philosophy and criticism, success means something specific: defining a problem precisely and solving it in a formal, definitive way. Only other specialists, we believe, can or should tell us if such problems have been solved. We find ourselves baffled and worried when, as has recently happened in the super-specialized realms of mathematics and physics, the specialists don’t seem to agree about who proved the Poincaré conjecture or whether string theory will ever reveal something about the real world.

Specialists and professionals are, for the most part, recent creations: the denizens of modernity, a world in which every highly educated man or woman has a particular function, and has obtained a formal license to practice it. [2] In pre-modern Europe, specialists existed: but even those who proudly described themselves as “mathematicians” or “critics” practiced their arts in a broad context. For the whole system of formal education was geared to produce generalists. Every learned person became a classicist at school. Specialists in the ancient higher faculties—medicine, law, and theology—imported their humanistic training into these fields, and changed the humanities by bringing medical, legal and theological perspectives to bear on them. [3] And specialists in a more modern sense often did the same. Even the most gifted mathematicians studied Greek and Latin and history at school and logic and philosophy at college, before they turned to numbers. Nowadays we remember Leibniz and Newton as scientists, the great men who created the calculus and modern physics, and Leibniz as a philosopher as well. Though the two men took justified pride in their extraordinary achievements in what now seem these central fields, they also pursued their interests into many neighboring ones. Leibniz was a productive, critical historian and a penetrating student of the origin and development of human languages. Newton spent years of his life performing alchemical experiments, reworking the history of the ancient world, reconstructing the Temple of Solomon and trying to interpret the prophecies of Daniel and the Book of Revelation. Thousands of pages of tightly written notes record his efforts in these multiple fields—each of which he apparently took as seriously as the rest. Both men wrote Latin as easily as the modern languages, and often chose to use it—or, in Leibniz’s case, French—when addressing issues of importance to a wide audience.

One way to imagine the Republic, then, is as a sort of Pedantic Park: a world of wonders, many of them man-made, inhabited by scholarly dinosaurs. The Republicans haunted the massive, classical libraries that patrons preferred where busts of the heroes of letters stared down from the laden shelves; stared politely at the rhinoceros horns, skis and Etruscan weapons artfully heaped on the walls and shelves of cabinets of curiosity, and savored the circular spectacle of the anatomy theater—at least in the winter, when the corpses didn’t stink. [4] Each of these preferred habitats reflected their eclectic tastes: every one was an encyclopedia—designed to teach lessons, some of them material or visual, about man and nature, science and history—and a laboratory, in which new forms of knowledge took shape. Leiden University’s celebrated anatomy theater, for example, bristled with human and animal skeletons, neatly arrayed to teach comparative anatomy. Men and horses—or at least their articulated bones—held up banners with Latin devices, designed to impress on visitors the moral lessons of mortality as well as the technical ones of zoology. Skeletal figures of Adam and Eve drove home the theological moral. The university’s equally celebrated library offered visitors books and manuscripts, city views, globes, and atlases—and learned conversation, during its few weekly hours of opening. [5]

The dinosaurs themselves came in many forms. Most of them were mild plant-eaters. But the gentle savants were flanked by vast lumbering monsters of erudition, like Athanasius Kircher, who lived a life of adventure, physical and intellectual, that Indiana Jones might have envied. Kircher climbed into the crater of Vesuvius in pursuit of an understanding of volcanoes, helped Bernini design the Fountain of the Four Rivers, amused himself working out the magic tricks of the conjurers who performed next to the Fountain in the Piazza Navona, played football against the Dominicans—and speculated, in ways that frightened and angered the religious authorities, about heliocentric astronomy and the pre-diluvial history of Egypt and China. He preserved his complex thoughts on all of these subjects in a vast array of massive, heavily illustrated Latin folios—more than any modern scholar could possibly read, much less write. While his books reportedly found few buyers, they won generous support from patrons and widespread interest—as well as considerable ridicule—from readers. [6]

The Republicans of Letters were not uniformly distinguished for integrity and generosity. Noel Malcolm has compellingly argued that Kircher’s pursuit of secrecy—not to mention his unfounded claims to mysterious knowledge about ancient Egypt and much else—put him in conflict with the Republic’s principles of openness, transparency, and full citation of evidence. But Kircher’s aggressive pursuit of knowledge, patronage and fame was not unusual in this world. Under the feet of the giants, swift, vicious little raptors fought and tore their way to prominence with equal energy: for example, Justus Lipsius, the brilliant textual critic who offered to recite the text of Tacitus, with a knife held to his throat, to be plunged in if he made a mistake. Lipsius’s moral writings made him the widely respected prophet of high Stoicism, the preferred moral code for scholars in an age of absolutism and religious war. In recent decades, scholars have emphasized his sophisticated pursuit of the lessons of Tacitus and Polybius and his ability to codify the results of sixteenth-century antiquarianism for a new readership, which he reconfigured so successfully that they proved indispensable for the practical purposes of military reform. Some have assumed that Lipsius—and his Stoical philosophy—can stand as a kind of moral counter for the early Republic of Letters as a whole. But he ensured his reputation as a textual critic by snitching other scholars’ clever emendations, which he recorded in a working copy of Tacitus now held in Leiden. Sadly, his efforts to erase the evidence did not succeed. [7]

Yet Kircher and Lipsius, for all their idiosyncrasies, offer what seem still to be powerful models for the conduct of intellectuals. They devised ways to conduct research with rigor, even when its results were uncomfortable; to publicize their results, without fear or favor; and, again and again, to rise above their prejudices without losing their convictions—as both men did when they maintained friendships that spanned ideological and political borders. And they also both collaborated, to spectacular effect, with artists, who gave their books a radically new visual form and, in Kircher’s case, realized his vision of ancient Egypt in the piazzas of modern Rome. They still, I now believe, have much to teach us about the forgotten premodern intellectual worlds that they inhabited and explored, and also, perhaps, about how modern intellectuals could and should serve the public good in our own poisoned public sphere.

We must remember, first of all, that the period in which the Republic of Letters flourished most was no golden age for Old Europe. The phrase Respublica literarum appeared in the fifteenth century, as a euphonious way to describe “the literary enterprise” or “the cause of letters.” The Republic itself, however, really began to take shape in the consciousness of scholars around 1500, as Erasmus became the leader of a self-conscious avant garde of scholars bent on reforming the Church and the universities. [8] And this process unrolled only a few years before Martin Luther and his Reformation split the Catholic church, which had been unified for more than a thousand years, forever.

The Republic barely survived the years from 1550 onwards, when militant Catholics and Calvinists in France and the Low Countries created what amounted to the first national revolutionary parties, organized by cells and inspired by absolutist ideologies, and fought civil wars of terrifying brutality. These years were marked by such terrible events as the Massacre of Saint Bartholomew and the assassinations of Kings Henry III and IV of France and William of Orange—assassinations made easier, as Lisa Jardine has shown, by the rise of the handgun, a curse then as now. [9] Yet the Republic staggered onwards, and even flourished, during the first half of the seventeenth century, even as almost all the European powers found themselves drawn into the Thirty Years’ War that turned Germany, then known as the Holy Roman Empire, into an impoverished and backwards set of principalities.

All this is to say nothing of such horrors as the witch trials of the same period, that deprived thousands, most of them women, of life itself on the grounds that they had had intercourse with devils, called up storms to destroy crops, and stolen men’s penises, which they hid in birds’ nests; or of the imposition of censorship in much of the Catholic world; or of the systematic oppression of Jews. It was a harsh world, as one might expect, since the men who ruled it generally cherished absolute convictions about life, the universe, and everything. As Brad Gregory has recently shown in a stunning book, hundreds of men and women died for their religious beliefs, willingly and bravely, after the Reformation began. Not a single one of the officers who imposed these punishments and watched them carried out, whether Catholic or Protestant, Lutheran or Calvinist, seems to have felt any qualms about inflicting martyrdom—to say nothing of converting to the martyrs’ faith, or to have repented after watching the martyrs suffer. [10]

Yet across this ocean of darkness—so historians have been learning over the last half-century and more—small bands navigated in fragile craft: little communities of scholars, whose members did their best to maintain a different kind of society, with its own rules and its own values. Many of the Republicans of Letters held official positions in universities, courts, or academies, and some used their skills to express imperial and national visions. [11] Others managed to hold official positions and still work for the Republic’s own vague but vital ideals of peace and tolerance. [12] But some were barred from most such posts by their conviction that they could not serve the state or the institutional church without being swallowed up by it and forced to violate the teachings of their consciences.

Whether the Republicans belong to an establishment or were hunted by one, they lived lives characterized by movement and distance. Protestants and Catholics alike crossed nations, borders, sometimes whole worlds. The distinguished older literature on the Republic of Letters—the books and articles of Annie Barnes, Erich Haase, Paul Dibon—concentrated on the Huguenot Refuge. They treated it, as Hugh Trevor-Roper summed up in a brilliant, over-generalized article, as the refuge not only of Protestant intellectuals and artisans, but also of an Erasmian ideal of tolerance. [13] More recently, students of the Catholic world have placed the Jesuits, always in purposeful motion, on the same imaginary map, and shown that they and the Huguenots—even as they denounced one another—cultivated the same fields of study, from natural science to the art of reading history, and sometimes even pursued the same ideals of civility and collaboration. The network of correspondents that linked a José de Acosta, writing on the natural and moral history of the New World in Peru, to fellow Jesuits from Rome to China—and the knowledge transmission belts that, as Paula Findlen has shown, brought the results of Kircher’s Egyptology from Rome to the cloister of Sor Juana in New Spain—were as global as those that brought the Samaritans in Palestine and the Jacobite Patriarch Ignatius Na’amatallah in Rome into productive contact with the Huguenot Joseph Scaliger in Leiden. [14] The Republic of Letters existed, first and foremost, as a palimpsest of people, books, and objects in motion.

Motion, of course, was always difficult and sometimes dangerous in the premodern world. But it also had much to offer. Travel—as we have learned from Justin Stagl, Joan-Pau Rubiés, and Paola Molino—became an art in the sixteenth and seventeenth centuries. [15] The Republicans of Letters drew up for each city they visited a questionnaire with spaces for geographical setting and urban form, natural resources and crafts, literary and religious life. They used this as a hermeneutic, which enabled them to read urban spaces as if they were texts (just as the antiquaries knew how to read texts so as to set them back into the three-dimensional cities where they had originally been produced). As they traveled, they learned about the diverse characters of nations. But at the same time, they followed the thinnest capillaries of the Republic of Letters, seeking out their fellow Republicans in their local habitats. In city after city they performed the homosocial rituals of their kind, offering the respects of friends elsewhere and entering signatures, epigrams and rebuses into one-another’s alba amicorum . [16] By doing so, they made deposits in a bank of social and cultural capital that would serve them throughout their lives.

Distance and motion had other functions as well. Above all, distance lent prestige—especially to such apparently glittering centers of new intellectual life as Louvain in the early sixteenth century, Leiden and Prague a century later, and Amsterdam and London later still—not to mention Paris and Rome, the eternal beacons of erudition and antiquity. By contacting a dominant figure in one of these glittering galaxies of talent and receiving a testimonial of warm approval, one could win credit in one’s own local, competitive environment. That is why, as Mario Biagioli has recently taught us, Galileo set such store by Johannes Kepler’s approval of the findings that he announced in the Sidereal Messenger . The mathematician beleaguered by local critics and competitors in northern and central Italy was bathed in a glowing nimbus of support by the detailed letter sent by the Imperial Mathematician from far-off Prague. [17]

The stars that glittered most brightly across distance were usually cities. The Republicans often had to spend time and provide services at courts. But they liked cities that enjoyed a certain measure of autonomy, and whose governors did not share the general belief that torture and execution were the appropriate tools for reducing religious and intellectual dissidents to order and submission. The citizens of the Republic also needed to perch near certain urban institutions: libraries, for example, and the printing presses that gave men and women of letters their only power, that of publicity. Their favored places, the capitals of their imagined state, included Strasbourg, a border town, cosmopolitan and tolerant; Leiden and Amsterdam, the Dutch trading centers, in which Catholics and Calvinists, Anabaptists and Jews rubbed elbows in mutual tolerance—and all of them joined to reject what they called “the Genevan Inquisition,” when doctrinaire preachers tried to carry out an ideological cleansing; and, of course, Basel, where Erasmus and other irenic souls found a spiritual home—a city ever hospitable to refugees from oppression in their native countries. London and Berlin also figured on the imaginary maps of the Republic, since both cities harbored many of the refugee French Protestants who made up a major share of the republic’s population.

Cities, after all, offered unique intellectual resources. In such forcing-houses of talent and research as the Lime Street community in Elizabethan London, reconstructed with great skill by Deborah Harkness in The Jewel House , clusters of artisans and apothecaries, Paracelsians and natural historians made their shops and gardens into a vast collective laboratory—something like an embodiment of Bacon’s supposedly Utopian New Atlantis, in which intellectual workers of very different kinds coordinated their efforts to force nature to reveal her secrets. Yet this local world had foreign as well as British inhabitants and was closely connected, by contacts made in travel and correspondence networks that passed through the Low Countries to the rest of Europe, with a vast range of impressive foreign contacts, who appreciated its lively, fertile culture. [18] The active, engaged, and sometimes quarrelsome form of collaboration between artisans and scholars that Harkness has turned up in London had become a standard feature of cultural life in Renaissance Italy, and would characterize the Republic throughout its history.

Cities also were the habitats of most of the learned women—beneficiaries, as Sarah Ross and April Shelford have taught us, of surprisingly ardent support from the Latin-speaking patriarchy—who created such salons as that of the Dames des Roches in Poitiers, and who, in the later seventeenth and eighteenth centuries, entered such once male preserves as the realm of classical philology. [19] Yet country houses, with their alluring Mannerist gardens, also offered islands of civility, many of them superbly stocked with books and antiquities, and historians of the Republic of Letters in the German and Austrian lands have emphasized the role of these aristocratic enclaves. [20]

Wherever they found jobs or refuge, the Republicans continued to respond to changes in the world outside the scholarly enclaves. Gradually they ditched backward-looking Latin for the up-to-date language of civilization, French, and took their campaigns against persecution and oppression to a wider public, even as the wars of Louis XIV turned much of northern Europe into a wasteland, and systematic oppression and abuse almost destroyed France’s own Protestant communities. [21]

The citizens of the Republic carried no passports, but they could recognize one another by certain marks. Not wealth, of course; then as now, scholar did not rhyme with dollar. But they looked for learning, for humanity, for generosity, and they rewarded those who possessed these qualities. Any young man, and more than a few young women, could pay the price of admission. Just master Latin—and, ideally, Greek, Hebrew, and Arabic; become proficient at what now seem the unconnected skills of mathematics and astronomy, history and geography, physics and music; turn up at the door of any recognized scholar from John Locke in London to Giambattista Vico in Naples, bearing a letter from a senior scholar, and greet your host in acceptable Latin or French—and you were assured of everything a learned man or woman could want: a warm and civilized welcome, a cup of chocolate (or, later, coffee); and an hour or two of ceremonious conversation on the latest editions of the classics and the most recent sightings of the rings of Saturn. [22]

If this state had no maps, no administrative officials, and no borders, how do we know it existed at all? And how can we define it more precisely? We know the Republic, in the first instance, from what its citizens tell us about it. The documents in which they discussed it form the primary archive from which we can draw both descriptions and evaluation. Not all contacts were informal. Traditional historiography has emphasized the scientific societies that took shape over the course of the seventeenth century, and whose officers and members did their best to establish new criteria and methods for the proper study of nature: the Accademia del Cimento, the Lincei, the Académie des Sciences and the Academia Naturae Curiosorum. [23] More recent studies have taught us to see these as one particularly vital species within a larger genre. The Republicans of Letters created many local communities of savants, dedicated to the search for religious or secular truths, or both at once. In some cases, as in the sixteenth-century Neapolitan academies that dedicated themselves to hunting out the secrets of nature or the seventeenth-century utopian brotherhoods that took shape in Tübingen or elsewhere, individuals or groups drew up formal rules for membership and elaborate protocols for the proper pursuit of intellectual life. [24] In others, like the Cabinet of the brothers Dupuy or Théophraste Renaudot’s Bureau d’Addresse, membership and activities formed more spontaneously. [25] In any number of cases, only a textual record remains, raising questions that remain difficult to resolve about whether a given society functioned in the material and social world. [26] Nonetheless, these organizations clearly played vital roles. They made clear that intellectual life needed a social foundation—and needed it all the more as Europe’s age of religious war progressed. And they helped to create the epistolary networks that gave the Republic its true circulatory system.

It is above all in the thousands of surviving letters—letters that combined the official and professional with the personal in a way that in the pre-modern world seemed entirely natural—that the outlines, highways and capitals of the Republic can be glimpsed most vividly. Tucked into letters were the reports on barometric experiments and the movements of falling bodies, the specimens of Egyptian mummy and New World flora, the drawings of rhinoceros horn and Roman feet, the descriptions of newly discovered manuscripts of ancient texts, the historical and political information that enabled men and women to know what was happening in the great world outside their little town, and to compile the great syntheses of political, historical, philosophical and scientific information that we still read: the work of Grotius on natural law, Galileo on natural philosophy, Locke on the nature of property. To a world that has largely abandoned letters except when asking for money in a good cause, these epistles—with their formal Latin salutations and intimate details of urinalysis and kidney stones, astrological predictions and monstrous births—may seem quaint. In their day, however, they constituted the fragile but vital canals that connected and animated intellectual commerce in the far-flung parts of the republic. The strands of long-term correspondence formed a capillary system along which information could travel from papal Rome to Calvinist strongholds in the north, and vice versa—so long as both had inhabitants, as they did, who wished to communicate. [27]

The constant writing and sending of letters was more than a system for collecting and exchanging information. The citizens of the Republic saw it as a moral duty: at once the only way to show their sympathy and affection for those from whom political and religious borders separated them and the only way to enter into a regular relationship with the greats who glittered far away. Consider just one instance: Erasmus, the great teacher and letter-writer, whose textbooks dominated the schools and universities of northern Europe until the middle of the seventeenth century, and whose own correspondence fills twelve volumes in the great modern edition published by the Clarendon Press. Erasmus treated the letter as a literary genre in its own right, and set down rules for the composition of effective, eloquent letters. In one of his textbooks—the aptly named On Copiousness in Words and Ideas —he went further, listing hundreds of ways to say “As long as I live, I shall remember you” and “Thank you for the letter” in elegant, correct Latin. The effort seems disproportionate to the task, until we realize that, as Kathy Eden has made clear, Erasmus deeply believed both in the community of intellectual and literary property (“all the property of friends is held in common,” he liked to say, quoting the ancient Greek thinker Pythagoras) and in the connection between the language one used and the state of one’s mind and soul.

The scholar, for Erasmus, must school himself or herself to write, over and over again, to critics as well as supporters, enemies as well as friends, professing friendship and concern. By doing so, one would knit up the raveled sleeves of particular relationships. But one would also become a true friend, one genuinely devoted to and concerned for others. The vast series of letters that fill dozens of volumes in every great European library are the relics of a great effort, inspired by Erasmus and many others after him, to create a new kind of virtual community: one sustained not by immediate, direct contact and conversation so much as by a decades-long effort of writing and rewriting. [28]

These exchanges, as Anne Goldgar and Brian Ogilvie have taught us, followed, or were supposed to follow, a strict code. Write to another scholar and you engaged yourself to reply to future letters in reasonable time, to give credit to your correspondent for information received and suggestions accepted, and to call him or her a friend—a term that had a strong formal meaning. [29] Yet for all the coded formality of their Latin and French, for all their authors’ desperate efforts to create personae on paper, and for all the breaches of epistolary etiquette that fused the circuits of this vast mechanism of exchange, many of them remain very moving. [30]

As this example suggests, some of the Republic’s qualities give it a genuine contemporary relevance. Like us, its citizens made conscious efforts to create communities, both of people and of information, that crossed political, linguistic and religious borders. Like us, they did their best to manage the vast amount of information to which they had access. The early modern period, as Ann Blair, Richard Yeo, Daniel Rosenberg and Noel Malcolm have shown, witnessed multiple efforts to capture, organize and make available to all the vast amounts of information flooding into Europe from travelers, compiled by scientific observers, and excavated by historians—a flood not only reproduced, but magnified, by the printing press. The tools they forged included not only scholarly correspondence of a personal sort, but also both technical and literary models for stockpiling information and making it available: the bibliography, the filing cabinet, the compilation of “historia litteraria” and the journal. [31] From the 1660s onwards, a swarm of new printed publications, in both Latin and the modern languages, compiled new information, reviewed new books, and made it possible, for the first time, for intellectuals across Europe to have reliable, regular information on the doings of scholars—and kings—in every other corner of the European world. [32]

Trade had become global again in the fifteenth century. Now information also joined the global flow, as Huguenots in exile in Berlin and Potsdam informed the European world about recent science and scholarship in French. Kircher, admired and envied in Rome, drew information from fellow Jesuits around the world as he charted the underground movements of rivers and lava flows and the ancient migrations of peoples. Vico, isolated but well-informed in Catholic, southern Naples, used Dutch journals published in Latin as his primary sources for the new theories of Spinoza and Locke. Like the blogs that have accelerated the movement of facts and ideas in recent years, the new journals and publishing houses had a profoundly unsettling effect on political and social authorities. The Republic of Letters stood, in the first instance, for a kind of intellectual market—one in which values depended, in theory at least, not on a writer’s rank but on the quality of his or her work.

The Republic, moreover, was more than a sprawling series of social and intellectual networks, loosely linked by curiosity about nature and history. It would be wrong to suggest that it had a single ideology or an official set of beliefs—even that of the Radical Enlightenment recently reconstructed with such brio by Jonathan Israel. [33] Its citizens, after all, included Catholics of different sorts, Protestants of every flavor, a few Sephardic and an even smaller number of Ashkenazi Jews—in addition, as time went on, to Unitarians and others who abandoned all the established churches. Patriotic Dutch scholars presumably felt a shiver a pride—and patriotic British ones just a shiver—when a Dutch fleet sailed up the Medway and burned much of the English navy.

Moreover, and more important, many of the most erudite scholars, Catholic, Calvinist, and Lutheran, pursued their research largely, or even primarily, for partisan reasons: in order to ensure the triumph of a religion—or, in other cases, a ruling house. Jesuits in China and elsewhere showed a deep interest in alien systems of belief and practice: yet their primary goal was the conversion of the world to Catholicism, and even a Matteo Ricci found it easy to draw the line between Chinese beliefs that he saw as compatible with Christianity and those that were not. [34] The large-scale research enterprises mentioned in my recent book—the teams of scholars assembled to study the history of the Church—found patrons because they promised to supply weapons to be used in confessional strife. [35] Even those who consciously tried to see the merits in others’ programs and practices were still driven, much of the time, by theological convictions. Isaac Casaubon’s demonstration that the philosophical Hermetica were late rested on his mastery of the language and technical philology. But it was motivated by an absolute conviction that neither Cardinal Baronio, whose work he dissected at enormous length, nor a pagan like Hermes who supposedly anticipated Christian truths, could possibly have written in good faith. [36]

And yet certain views, shared in greater and lesser degree by the Republicans, ran counter to the confessionalism of the time. Some of them, for a start, believed that it was simply wrong—morally wrong and intellectually wrong—to break off communications with those who didn’t share their religious beliefs or their political views. Knowledge and sociability, after all, mattered most: and restrictions could only hamper the flow of information and ideas. That helps to explain why a long series of Vatican librarians, in the heart of papal Rome, admitted Protestant scholars as freely as Catholic ones.

A fair number went further. In an age of brutal persecution, when torture was the standard legal method on the Continent for extracting information and confessions, scholarly citizens of the Republic of Letters pointed out, forcibly and clearly, that torture could make people confess not only crimes they had not committed, but crimes that no one could commit—a thesis that is anything but quaint or antiquated today. They also became the first to argue in detail that the vast tottering structure of dogma that underpinned the persecution of witches was far too rickety to bear so great a weight. An early citizen of the Republic, Johannes Reuchlin, dared the disapproval of influential men and women across Germany when he wrote a powerful legal defense of the right of Jews to retain their own books, which influential Catholics wanted to burn. [37] Another citizen of the Republic, Sebastian Castellio, first elaborated an even more radical idea, one that flew in the face of religious authority from Saint Augustine on. Once a great admirer of John Calvin’s, Castellio was horrified by Calvin’s part in the arrest and execution of the heretic, Michael Servetus, in 1553. He took action—the sort of action that the citizens of the Republic took. Working with local eminences like Bonifacius Amerbach and Thomas Platter as well as émigrés from Holland and France, all of whom shared his loathing of coercion and violence, Castellio compiled excerpts from early Christian and contemporary works to show that the state had no right to persecute those who did not accept the beliefs of its established church. The Basel printer Joannes Oporinus, whose list included authors of every conceivable ideology and religion, published this work, as subversive in import as it was mild in form, under a false imprint. Each piece of the mosaic Castellio assembled added a new color to his polemical palette. Writing under the pseudonym Martin Bellius, Castellio argued that persecution was literally un-Christian. Those who executed heretics seemed to think that Jesus had been a “Moloch or a god of the sort,” who commanded human sacrifice. Writing as Basilius Montfort, he mounted a more tightly defined political argument. Secular rulers had no right to punish anyone on grounds of belief: “He who suffers persecution on account of his faith stands either in the truth or in error. If he is right he must not be harmed. If he is wrong he must be forgiven. Christ asked God to forgive those who crucified him, for they knew not what they were doing. Would this not apply more greatly to those who allow themselves to be crucified for him?” [38] Castellio’s arguments were hardly rigorous. In the end, he argued, one should judge people by their conduct—a theologically naïve view that could not be reconciled with any Protestant understanding of grace and salvation. His convictions stemmed partly from his wide reading in the early, more radical writings of Martin Luther and the great polemics against persecution by Sebastian Franck, and in part from the lived experience of Basel, where Castellio had seen that men and women of different religions could manage and negotiate their differences—as, some years later, the Hebraist Johannes Buxtorf would manage to publish important editions of Hebrew texts, even though the Jewish printers refused to work on Saturdays, and the Christians on Sunday.

Castellio’s book won few adherents at first. Yet his ideas percolated into the minds of writers like Montaigne and even, so it seems, a few rulers—notably William of Orange and Elizabeth of England. [39] Other citizens of the Republic carried his enterprise onwards, using the literary tools at their disposal since they lacked political ones. The battle against religious prejudice and persecution did not end, of course, in this period, any more than it has ended since. Pierre Bayle—a later citizen of the Republic, a brilliant, bitter critic of absolutism in State and Church who lived in Holland and tormented the authorities with his dazzling pamphlets—shocked many readers when he argued that a society of atheists could live together at peace. And the great philosophes of the eighteenth century—men like Voltaire, who famously left his refuge near Geneva in order to confront the forces of darkness over the Calas case—argued cases like Castellio’s, casting them in a far more radical key. Such characteristic Enlightenment attitudes grew from the speculations of learned men, forced into exile for their beliefs and instructed in the bitter school of political and religious experience that compulsion should never play a role in matters of belief.

Belief in such challenging principles as the free communication of ideas, tolerance in principle if not always in practice, open contact with those of other faiths, and publication of results even when they raised theological difficulties manifested themselves not only in such famous and controversial cases as that of Servetus, but in the everyday life and work of scholars, many of them less courageous—and stiff-necked—than Castellio. And here I pass from cartography to chorography. Like most of those who study the Republic, I have examined only one corner of this vast realm at close range, and in the course of this more limited and detailed scrutiny I have come to see just how the Republicans of Letters used their general canons of conduct to regulate particular, technical forms of inquiry. Starting out in the 1970s, I wanted to understand how men and women could master the whole range of period disciplines and texts, from astronomy to philology; to learn what it felt like to be as skillful at interpreting ancient history as at reading the movements of the planets. So I set out to reconstruct a single discipline that has nowadays largely been forgotten: technical chronology, the formal study of the dates at which events happened in ancient and medieval history. Even in the early modern period, the field was known to be obscure—Johannes Kepler, who knew and loved the subject, noted that books with “Chronology” in their titles didn’t sell. [40]

Still, chronology was a hot field in its day. [41] Leiden University, the most innovative one in Europe, paid the French scholar Joseph Scaliger a higher salary than the law professors and allowed him to forgo lecturing and devote himself to research because of the world-wide reputation he had made as the leading expert on chronology. And it was hard. Chronology demanded, and demands, extraordinary skills: to practice it you had to be able to interpret ancient texts, decipher ancient inscriptions, and even to plot the dates of the eclipses and other astronomical events mentioned in ancient texts, which provide the only absolute dates we have. Chronology posed problems that remain extremely difficult, and to some extent unsolved: for example, how to reconcile the dynasty lists and dates of the Old and New Testaments with one another and with those preserved in secular texts. Its practitioners, accordingly, had to walk fine lines. They could not, in theory, force or falsify any of the evidence. But as soon as they chose one biblical datum to rely on, they laid themselves open to the charge of neglecting others that contradicted it.

A great many early modern scholars wielded this rather scary palette of technical skills with ease and dexterity. Scaliger and his Jesuit critic Denis Petau were probably the best known experts in the field. But chronology also fascinated great astronomers like Copernicus and Kepler, the important composer and musical theorist Seth Calvisius, the erudite Anglican churchman James Ussher, and the most original historical thinker of the whole pre-modern period, Giambattista Vico. These men did extraordinary, wrenchingly difficult work, with meticulous care. By 1700 they had crafted the basic armature of dates on which modern scholars still hang the flesh and blood of ancient and medieval history. Yet they had also begun to study data from other cultures, such as the dynasty lists of the Chinese, that called their basic assumptions into question, and in the end they could not save their beautiful theories from the impact of these obdurate facts. [42]

Scaliger and Kepler, Calvisius and Petau turned out to be as phenomenally learned and analytical, as wide-ranging in their interests and as precise and prophetic in their results as I had believed before I ventured into the labyrinths of their books and manuscripts. But these toiling giants had many human failings. They misreported one another’s ideas; they failed to give credit where credit was clearly due; they ripped one another’s work apart with a zeal that would have been far better spent on other objects.

Academic gossip described chronology as rife with extravagant and willful hypotheses: chronologers, like clocks, never agreed. And some chronologers lived up to these clichés. The erudite Jesuit numismatist and textual critic Jean Hardouin, for example, decided after decades of chronological and philological study that pretty much all the Greek and Latin classics—except for Pliny’s Natural History , which he had edited, and a few other texts—had been forged in the thirteenth century by an “atheistic sect” led by Frederick II of Hohenstaufen. He drew these conclusions partly from an exhaustive study of ancient coins, partly from the texts, which he subjected to the kind of endlessly skeptical scrutiny that adherents of the Baconian theory inflict on the text of Shakespeare. Protestants and Catholics alike were shocked by his radicalism, which provoked bitter debates and unsettled all practitioners of chronology, and all too many of them responded by making the Jesuit order as a whole collectively guilty for Hardouin’s individual fancies. [43] More important—and more generally—it became clear that chronologers, like so many other Republicans of letters, were hypersensitive to slights. Any letter that showed insufficient respect, any publication of a fact or utterance of a word that might reflect badly on them, and they turned on discussion partners and rent them—even when, as Kepler haplessly tried to convince Seth Calvisius, the offended party had simply misunderstood the offender’s use of German.

Yet even as I realized that my chronologers were not such consistent models of scholarly and human virtue as I had hoped, I also found them working hard and effectively to raise bridges across the most profound ideological and theological gaps. Scaliger—a fierce Calvinist who believed, as many of his co-religionists did, that the Pope was the Antichrist—told his students to view the great Catholic church history by Cardinal Baronio with respect. “Every history is good,” he explained: all information mattered, and you could learn far more from a great scholar whose opinions you didn’t share than from a charlatan with whom you went to church. [44]

Chronology could be brutally polemical, but it could also provide an ideal public stage for demonstrations of tolerance, since the chronologer was constantly required to negotiate agreement between sources of radically different origin and nature—a delicate operation at best, and sometimes impossible. To study the Christian past you had to understand the Jewish calendar: not just the sequence of years and months, moreover, but also the nature of religious holidays and observances. Scaliger and his close friend, Isaac Casaubon, realized in the last decades of the sixteenth century that they could not reconstruct the sequence of events or understand the meaning of individual episodes in the Gospels themselves without mastering Jewish scholarship. The Last Supper—as Scaliger pointed out in his first chronology, to the astonishment of erudite theologians—was an adaptation of the Jewish Passover Seder. To understand this primal Christian event, one must read a Passover Haggadah. [45]

But the Haggadah did not clear up all the problems—for example, that of how Jesus had apparently been condemned and executed on days when Jews were prohibited from appearing in court. How were Christians to gain this esoteric knowledge? Scaliger and Casaubon were masters of language, steeped in the Bible. Learning to read Hebrew, in the first instance, had only required them to work out which words were which in the Hebrew version of Genesis, since they already knew it by heart in Latin and French. But the Bible offered nowhere near all the information they needed. To understand exactly the world in which Jesus preached, they had to explore the entire enchanted castle of Jewish learning—chronicles, rabbinical commentaries, even the Talmud. And for guidance through these labyrinths they turned to Jews. Scaliger worked for six months with a very learned convert, Philippus Ferdinandus, who helped him to see that many of Jesus’ precepts, in the Gospels, did not contradict, but actually reflected, Jewish moral teachings. Casaubon invited a learned Jew from Italy, Jacob Barnet, to stay with him for a month at his lodgings at Drury Lane in London. At every meal—one would love to know what they ate—the two men eagerly discussed Jewish texts—including, evidently, the legal ones that Casaubon could not read on his own. Barnet showed Casaubon discussions of Jewish burial practice—which made clear, to the brilliant Calvinist, that Jesus had been buried in a normal, Jewish fashion, rather than, as Cardinal Baronio maintained, in a new way that became the basis of Catholic burial practice. [46]

Neither Scaliger nor Casaubon was especially philo-Semitic in everyday life. But the ethics of scholarship as they understood it brought them into intimate contact with Europe’s local Others. And the contact had a tremendous effect on both of them. Scaliger, the most arrogant scholar in an age when scholarly arrogance flourished, admitted after Ferdinandus died that no Christian could hope to understand the Talmud and other Jewish texts as his friend had. He wept, in a very human way, for his human and intellectual loss. Jacob Barnet, whom the Oxford authorities had destined for public conversion in Saint Mary’s Church, rebelled, ran away, and wound up confined, in miserable conditions, in the university jail, Bocardo. Casaubon—a mild man, bent with age and unremitting study—intervened. He denounced Barnet’s treatment as a violation of Christian ethics; in fact, he went so far as to appeal to King James I—himself an erudite man—on Barnet’s behalf, and his pleas succeeded. A king’s man brought a warrant to Oxford, removed Barnet from prison and put him on the next ship to France. He soon turned up again, man of parts that he was, as an expert on Jewish matters at the French royal court, where he and Giulio Cesare Vanini enjoyed passing gossip about the stinginess of British patrons. The openness that men like Scaliger and Casaubon showed to others whose faith and culture they definitely did not share offered them no practical advantages and could have caused them endless difficulties. It is a tribute to the regulative principles of the Republic—and a sign of their historic impact—that these men behaved as they did.

Both Casaubon and Scaliger, in the course of their work, took on board ideas and ways of doing things that would have shocked—that did shock—some of their contemporaries. Neither went so far as those seventeenth-century Amsterdam Jews who appalled their co-religionists by adopting a form of Karaism—one based not on contact with actual Karaites, but on the descriptions given by Christian Hebraists. [47] Still, Casaubon, after his years of biblical study and his intensive work with Barnet, sometimes even prayed in Hebrew. Scaliger, after even longer years of historical study, made a chronological discovery so profound that even his brilliance could not cope with it: the Egyptian dynasty list of Manetho, according to which history had begun not only before the Flood, but before the Creation. Both men made clear that Christianity represented, in many ways, less a break with the Judaism of the time of Jesus than a new development within it. Historical research was supposed to rear and trim neat, tidy structures, which showed the hand of providence, working to bring Christianity into being. Instead, Scaliger’s and Casaubon’s dangerous ideas and destructive practices undermined the authorities they were expected to support. Yet both men published what they learned—and by doing so disturbed and irritated more orthodox thinkers across the whole European world.

Finally, chronology has a chronology of its own—one that helps us to understand the larger chronology of the Republic of Letters. In the late 1650s, Isaac Vossius—whose father Gerardus had brought him up within the formidable learned tradition of Dutch humanism—shocked the world of learning. He helped to arrange the reprinting in Amsterdam of the Jesuit Martino Martini’s history of China, which set the beginning of the Middle Kingdom so early that only the Greek Old Testament, with its longer chronology, could fit it in after the Flood. And then he published a pamphlet, first in Latin and then in Dutch, in which he insisted that the longer chronology of the Greek Bible, rather than the shorter one of the Hebrew, deserved credence. In doing this Vossius—as should by now be clear—did not forge a new thesis. Ever since Scaliger, chronologers had weighed the difficulties of early history and the virtues of the different biblical versions, and admitted the impossibility of arriving at firm results—in the privacy of their letters and conversations or in the relative privacy of vast Latin treatises. But Vossius—to borrow a term from my late friend and colleague Gerald Geison—turned what had been private science into public science—so public that it provoked a series of pamphlet-sized refutations, which did nothing to soothe the scholarly waters. [48]

A few years later he would literally make the private public by printing Joseph Scaliger’s table talk—which showed that in his chimney corner, talking to his students in a pithy mixture of Latin and French, Scaliger had entertained similarly bold ideas about the duration of history and the incompleteness of the Hebrew Bible. [49]

In these middle years of the seventeenth century, fissures and cracks opened up in many of the fields that mattered to the Republicans. Science and scholarship underwent dramatic transformations in key and tone, as great treatises in Latin gave way to pamphlets in the vernacular, and detailed arguments in dark libraries to lively debates in public venues—and as what had been the practices of erudition revealed what could be a powerful potential to call ancient authorities into question. Traditional histories of the Enlightenment, often centered on France, have tended to treat the eighteenth century in almost Hegelian terms—as the time when the world spirit turned from erudition to philosophy, and scholars became marginal figures in the world of learning. [50] In chronology—as in other fields—older traditions of exegesis and newer methods of historical scholarship, both designed to clarify and confirm traditional structures, ended by destroying them. [51]

The citizens of the early modern Republic of Letters created a virtual community not of those who shared beliefs, but of those who differed. They made up rules for civility: rules that could be used to judge the conduct of all those who offered their intellectual wares for sale in the new, largely free market. They developed new tolerances, for thinkers who disagreed with them on fundamental matters and for facts that challenged their most basic verities. What unified these efforts was a shared, if inchoate and incomplete, respect for truth, for civility, and for the integrity of the human being—a respect not founded, perhaps, on deep philosophical or theological arguments, and often violated in practice, but solid enough to make them bold when they confronted what they saw as superstitions. One of the most prominent citizens of the Republic, Jean Le Clerc—a Swiss, born in Geneva, who moved to Amsterdam to enjoy intellectual freedom—put it well: “If a thing is bad in itself, the example of the ancients does not make it better. Nothing should stop us from improving on them. The Republic of Letters has finally become a land of reason and light, and not of authority and blind faith, as it was for too long. Nowadays numbers prove nothing, and there are no more cabals.” [52] True, in this case Le Clerc was defending not freedom of speech or religion but the use of footnotes in historical texts: but he and his contemporaries wrote with equal clarity and power about religious and political oppression.

Naturally, practices were always more complex—and sometimes much darker and more oppressive—than precepts. Citizens of the Republic of Letters who violated its rules, like Vossius, could change them—but only at the cost of suffering abuse and exclusion. The same fate awaited those who parodied its customs too radically—for example, those who devised erotic readings of the story of the Fall, and took the eating of the apple as an allegory for sexual intercourse. These rash young man found themselves, as Kristine Haugen and Martin Mulsow have shown, deprived of their jobs, hunted from their homes, and forced into poverty and obscurity, by the very fellow scholars who would have defended them if they had not breached the Republic’s codes of decorum. [53] Senior academics, then as now, often showed less tolerance for their junior colleagues than for almost anyone else.

Still, the Republic of Letters provided a stage where free exchange of opinions could sometimes be proclaimed and performed in a new way. Though its story has often been treated as coextensive with that of the eighteenth-century Enlightenment, such accounts foreshorten the traditions of scholarship, debate and sociability that connect the humanist sodalities of Renaissance Florence and Rome to the academies, public libraries, Masonic lodges and salons of the seventeenth and eighteenth centuries. This complex and inspiring history remains to be written.

[1] Noelle d’Argonne, Mélanges d’histoire et le littérature, recuillis par M. de Vigneul-Marville , 2 vols. (Paris, 1699-1700), quoted in Paul Dibon, “Communication in the Respublica Litteraria in the 17th Century,” Res publica litterarum 1 (1978): 43; for an elegant brief survey of the Republic and its institutions, see Didier Masseau, L’invention de l’intellectuel dans l’Europe du xviii e siècle (Paris: Presses universitaires de France, 1994). Collections of essays that help to set the scene include Herbert Jaumann, ed., Die europäische Gelehrtenrepublik im Zeitalter des Konfessionalismus = The European Republic of Letters in the Age of Confessionalism (Wiesbaden: Harrassowitz, 2001), and Rudolf Keck, Erhard Wiersding, and Klaus Wittstadt, eds., Literaten—Kleriker—Gelehrte. Zur Geschichte der Gebildeten im vormodernen Europa (Cologne: Böhlau, 1996).

[2] Though the Republic, like the rest of European society, was certainly patriarchal and hierarchical, women played vital public roles in the culture of erudition. See most recently April Shelford, Transforming the Republic of Letters: Pierre-Daniel Huet and European Intellectual Life, 1650-1720 (Rochester, NY: University of Rochester Press, 2007), and more generally the pioneering study by Dena Goodman, The Republic of Letters: A Cultural History of the French Enlightenment (Ithaca, NY: Cornell University Press, 1994).

[3] For case studies in these processes of interdisciplinary scholarship, see Gianna Pomata and Nancy Siraisi, eds., Historia: Empiricism and Erudition in Early Modern Europe (Cambridge, MA: MIT Press, 2005). One of the richest full-scale studies is Siraisi, History, Medicine, and the Traditions of Renaissance Learning (Ann Arbor: University of Michigan Press, 2007).

[4] Paula Findlen, Possessing Nature: Museums, Collecting, and Scientific Culture in Early Modern Italy (Berkeley: University of California Press, 1994); Horst Bredekamp, The Lure of Antiquity and the Cult of the Machine: The Kunstkammer and the Evolution of Nature, Art and Technology , trans. Allison Brown (Princeton, NJ: Markus Wiener, 1995); Bredekamp, Die Fenster der Monade: Gottfried Wilhelm Leibniz’ Theater der Natur und Kunst (Berlin: Akademie, 2004). For excellent recent surveys of the habitats of early modern savants, see The Cambridge History of Science, III, Early Modern Science , ed. Lorraine Daston and Katharine Park (Cambridge: Cambridge University Press, 2006).

[5] Anthony Grafton, Bring Out Your Dead: The Past as Revelation (Cambridge, MA: Harvard University Press, 2001), chap. 6.

[6] Daniel Stolzenberg, ed., The Great Art of Knowing: The Baroque Encyclopedia of Athanasius Kircher (Stanford: Stanford University Libraries, 2001); Paula Findlen, ed., Athanasius Kircher: The Last Man Who Knew Everything (New York: Routledge, 2004) .

[7] See esp. Wilhelm Kühlmann, Gelehrtenrepublik und Fürstenstaat: Entwicklung und Kritik des deutschen Späthumanismus in der Literatur des Barockzeitalters (Tübingen: Max Niemeyer, 1982); Mark Morford, Stoics and Neostoics: Rubens and the Circle of Lipsius (Princeton, NJ: Princeton University Press, 1991); Jacob Soll, “Amelot de La Houssaye (1634-1706) Annotates Tacitus,” Journal of the History of Ideas 61, no. 2 (2000): 167-87; Soll, Publishing The Prince : History, Reading, and the Birth of Political Criticism (Ann Arbor: University of Michigan Press, 2005); Grafton, Bring Out Your Dead , chap. 12.

[8] See most recently Constance Furey, Erasmus, Contarini, and the Religious Republic of Letters (Cambridge: Cambridge University Press, 2006).

[9] Lisa Jardine, The Awful End of Prince William the Silent: The First Assassination of a Head of State with a Handgun (London: HarperCollins, 2005).

[10] Brad Gregory, Salvation at Stake: Christian Martyrdom in Early Modern Europe (Cambridge, MA: Harvard University Press, 1999).

[11] See e.g. R. J. W. Evans, The Making of the Habsburg Monarchy, 1550-1700: An Interpretation (Oxford: Clarendon Press, 1979).

[12] For case studies, see e.g. R. J. W. Evans, Rudolf II and His World: A Study in Intellectual History, 1576-1612 (Oxford: Clarendon Press, 1972; corrected ed., London: Thames and Hudson, 1997); Howard Louthan, Johannis Crato and the Austrian Habsburgs: Reforming a Counter-Reform Court (Princeton, NJ: Princeton Theological Seminary, 1994); Louthan, The Quest for Compromise: Peacemakers in Counter-Reformation Vienna (Cambridge: Cambridge University Press, 1997).

[13] Annie Barnes, J ean Le Clerc (1657-1736) et la république des lettres (Paris: Droz, 1938); Erich Haase, Einführung in die Literatur des Refuge: der Beitrag der französischen Protestanten zur Entwicklung analytischer Denkformen am Ende des 17. Jahrhunderts (Berlin: Duncker & Humblot, 1959); Paul Dibon, Regards sur la Hollande du siècle d’or (Naples: Vivarium 1990); Hugh Trevor-Roper, “The Religious Origins of the Enlightenment,” in Religion, the Reformation, and Social Change (London: Macmillan, 1967).

[14] See e.g. Mordechai Feingold, ed., Jesuit Science and the Republic of Letters (Cambridge, MA: MIT Press, 2003).

[15] Justin Stagl, Apodemiken: eine räsonnierte Bibliographie der reisetheoretischen Literatur des 16., 17. und 18. Jahrhunderts (Paderborn: Schöningh, 1983); Stagl, A History of Curiosity: The Theory of Travel, 1550-1800 (Chur: Harwood, 1995); Joan-Pau Rubiés, “Instructions for Travellers: Teaching the Eye to See,” History and Anthropology 9 (1996): 139-90; Rubiés, Travel and Ethnology in the Renaissance: South India Through European Eyes, 1250-1625 (Cambridge: Cambridge University Press, 2000); Rubiés, “Travel Writing as a Genre: Facts, Fictions and the Invention of a Scientific Discourse in Early Modern Europe,” Journeys: The International Journal of Travel and Travel Writing 1 (2000): 5-33; Paola Molino, “Alle origini della Methodus Apodemica di Theodor Zwinger: la collaborazione di Hugo Blotius, fra empirismo ed universalismo,” in Codices Manuscripti (forthcoming).

[16] For a superb example and study of the album amicorum, see Chris Heesakkers, ed., Een netwerk aan de basis van de Leidse universiteit. Het Album amicorum van Janus Dousa: facsimile-uitgave van hs. Leiden UB, BPL 1406 (Leiden: Leiden University Library, 2000).

[17] Mario Biagioli, Galileo’s Instruments of Credit: Telescopes, Images, Secrecy (Chicago: University of Chicago Press, 2006).

[18] Deborah Harkness, The Jewel House: Elizabethan London and the Scientific Revolution (New Haven, CT: Yale University Press, 2007).

[19] Sarah Ross, The Birth of Feminism: Woman as Intellect in Renaissance Italy and England (Cambridge, MA: Harvard University Press, 2009); Shelford, Transforming the Republic of Letters ; see also David Norbrook, “ Women, the Republic of Letters, and the Public Sphere in the Mid-Seventeenth Century ,” Criticism 46 (2004): 223-40, and Carol Pal, “Republic of Women: Rethinking the Republic of Letters, 1630-1680” (PhD diss., Stanford University, 2006).

[20] See e.g. Otto Brunner, Adeliges Landleben und europäischer Geist. Leben und Werk Wolf Helmhards von Hobburg , 1612-1688 (Salzburg: Müller, 1949); Manfred Fleischer, Späthumanismus in Schlesien. Ausgewählte Aufsätze (Munich: Delp, 1984); Morford, Stoics and Neostoics.

[21] On the further development of the Republic, see inter alia Goodman, Republic of Letters , and Lawrence Brockliss, Calvet’s Web: Enlightenment and the Republic of Letters in Eighteenth-Century France (Oxford: Oxford University Press, 2002).

[22] For a fascinating comparative study of the similarities and differences between two provinces of the Republic, see John Robertson, The Case for the Enlightenment: Scotland and Naples, 1680-1760 (Cambridge: Cambridge University Press, 2005).

[23] The vast literature on these institutions goes back to such classics as Harcourt Brown, Scientific Organizations in Seventeenth Century France (1620-1680) (Baltimore: Williams and Wilkins, 1934). More recent studies include Roger Hahn, The Anatomy of a Scientific Institution: The Paris Academy of Sciences, 1666-1803 (Berkeley: University of California Press, 1971); Stephen Shapin and Simon Schaffer, Leviathan and the Air-Pump: Hobbes, Boyle, and the Experimental Life (Princeton, NJ: Princeton University Press, 1985); and David Freedberg, The Eye of the Lynx: Galileo, His Friends, and the Beginnings of Modern Natural History (Chicago: University of Chicago Press, 2002). For a rapid recent review, emphasizing the connections between academies and courts, see Bruce Moran, “Courts and Academies,” in Cambridge History of Science, III , 251-71.

[24] See William Eamon, Science and the Secrets of Nature: Books of Secrets in Early Modern Culture (Princeton, NJ: Princeton University Press, 1994); Donald Dickson, The Tessera of Antilia: Utopian Brotherhoods and Secret Societies in the Early Seventeenth Century (Leiden: Brill, 1998).

[25] See respectively Klaus Garber, “Paris, die Hauptstadt des europäischen Späthumanismus. Jacques Auguste de Thou und das Cabinet Dupuy,” in Res publica litteraria: Die Institutionen der Gelehrsamkeit in der frühen Neuzeit , ed. Sebastian Neumeister and Conrad Wiedemann (Wiesbaden: Harrassowitz, 1987), 1:71-92; Antoine Coron, “‘Ut prosint aliis’: Jacques Auguste de Thou et sa bibliothèque,” in Histoire des bibliothèque françaises, II: Les bibliothèques sous l’Ancien Régime , ed. Claude Jolly (Paris: Promodis, 1988), 101-25; Howard Solomon, Public Welfare, Science, and Propaganda in Seventeenth-Century France: The Innovations of Théophraste Renaudot (Princeton, NJ: Princeton University Press, 1972); Simone Mazauric, Savoirs et philosophie à Paris dans la première moitié du XVIIe siècle: les conférences du Bureau d’adresse de Théophraste Renaudot, 1633-1642 (Paris: Publications de la Sorbonne, 1997); and Kathleen Anne Wellman, Making Science Social: The Conferences of Théophraste Renaudot, 1633-1642 (Norman: University of Oklahoma Press, 2003).

[26] For a superb case study, see Didier Kahn, “The Rosicrucian Hoax in France (1623-24),” in Secrets of Nature: Astrology and Alchemy in Early Modern Europe , ed. Anthony Grafton and William Newman (Cambridge, MA: MIT Press, 2001), 235-344.

[27] On these and other technical aspects of communication in the Republic of Letters—as well as on much more—see the fine synthesis of Hans Bots and Françoise Waquet, La République des Lettres (Paris: Belin, 1997), as well as Bots and Waquet, eds., Commercium litterarium, 1600-1750 (Amsterdam: APA-Holland University Press, 1994).

[28] Kathy Eden, Friends Hold All Things in Common: Tradition, Intellectual Property, and the Adages of Erasmus (New Haven, CT: Yale University Press, 2001).

[29] Anne Goldgar, Impolite Learning: Conduct and Community in the Republic of Letters, 1680-1750 (New Haven, CT: Yale University Press, 1995); Brian Ogilvie, The Science of Describing: Natural H istory in Renaissance Europe (Chicago: University of Chicago Press, 2006).

[30] On the practices of correspondents in the Republic, see Toon Van Houdt, Jan Papy, Gilbert Tournoy, and Constant Matheeussen, eds., Self-Presentation and Social Identification: The Rhetoric and Pragmatics of Letter-Writing in Early Modern Times (Leuven: Leuven University Press, 2002), and the forthcoming proceedings of the conference on “Observation in Early Modern Letters, 1500-1650,” held at the Warburg Institute on June 29-30, 2007, and organized by Charles Burnett and Dirk van Miert.

[31] See Ann Blair, The Theater of Nature: Jean Bodin and Renaissance Science (Princeton, NJ: Princeton University Press, 1997); Blair, “Reading Strategies for Coping with Information Overload, ca. 1550-1700,” Journal of the History of Ideas 64 (2003): 11-28; Richard Yeo, “Ephraim Chambers’s Cyclopaedia (1728) and the Tradition of Commonplaces,” Journal of the History of Ideas 57 (1996): 157-75; Yeo, “A Solution to the Multitude of Books: Ephraim Chambers’s Cyclopaedia (1728) as ‘The Best Book in the Universe,’” ibid., 64 (2003): 61-72. See in general Daniel Rosenberg, “Early Modern Information Overload,” ibid., 64 (2003): 1-9, and the recent, revisionist case study by Noel Malcolm, “William Harrison and His ‘Ark of Studies’: An Episode in the History of the Organization of Knowledge,” The Seventeenth Century 19 (2004): 196-232.

[32] See Goldgar, Impolite Learning , and Martin Gierl, Pietismus und Aufklärung: theologische Polemik und die Kommunikationsreform der Wissenschaft am Ende des 17. Jahrhunderts (Göttingen: Vandenhoeck & Rupprecht, 1997). For earlier efforts to impose order, see also Wilhelm Schmidt-Biggemann, Topica universalis: eine Modellgeschichte humanistischer und barocker Wissenschaft (Hamburg: Meiner, 1983), and Helmut Zedelmaier, Bibliotheca universalis und bibliotheca selecta: das Problem der Ordnung des gelehrten Wissens in der frühen Neuzeit (Cologne: Böhlau, 1992); for a broader range of similar enterprises, see Frank Büttner, Markus Friedrich, and Helmut Zedelmaier, eds., Sammeln, Ordnen, Veranschaulichen: zur Wissenskompilatorik in der Frühen Neuzeit (Münster: LIT, 2003).

[33] See Jonathan Israel’s influential Radical Enlightenment: Philosophy and the Making of Modernity, 1650-1750 (Oxford: Oxford University Press, 2001), now somewhat revised and considerably extended in his own Enlightenment Contested: Philosophy, Modernity, and the Emancipation of Man, 1670-1752 (Oxford: Oxford University Press, 2006).

[34] Howard Goodman and Anthony Grafton, “ Ricci, the Chinese, and the Toolkits of Textualists,” Asia Major 3, no. 2 (1990): 95-148.

[35] Anthony Grafton, Worlds Made by Words: Scholarship and Community in the Modern West (Cambridge, MA: Harvard University Press, 2009).

[36] Anthony Grafton, Defenders of the Text: The Traditions of Scholarship in an Age of Science, 1450-1800 (Cambridge, MA: Harvard University Press, 1991), chap. 5.

[37] Johannes Reuchlin, Recommendation Whether to Confiscate, Destroy, and Burn all Jewish Books , ed. and trans. Peter Wortsman, critical introduction by Elisheva Carlebach (New York: Paulist Press, 2000); Erika Rummel, The Case against Johann Reuchlin: Religious and Social Controversy in Sixteenth-Century Germany (Toronto: University of Toronto Press, 2002).

[38] Hans Guggisberg, Sebastian Castellio, 1515-1563: Humanist and Defender of Religious Toleration in a Confessional Age , ed. and trans. Bruce Gordon (Aldershot: Ashgate, 2003), 73-152, esp. 88-90.

[39] On the complex history of Castellio’s reputation, see Hans Guggisberg, Sebastian Castellio im Urteil seiner Nachwelt vom Späthumanismus bis zur Aufklärung (Basel: Helbing & Lichtenhahn, 1956). For another study in late humanists’ views on toleration, see Gerhard Güldner, Das Toleranz-Problem in den Niederlanden im Ausgang des 16. Jahrhunderts (Lübeck: Matthiesen, 1968).

[40] See Grafton, Worlds Made by Words , chap. 6.

[41] Unexpectedly, it has become one again in Russia, where the polemical chronologies of the distinguished mathematician Anatoly Fomenko, in which he argues that all of world history has been falsified to make Muscovite society and culture seem younger than those of ancient Greece and Rome, are best-sellers.

[42] For the development of the field in early modern Europe, see in general Anthony Grafton, Joseph Scaliger , 2 vols. (Oxford: Clarendon Press, 1983-1993); Grafton, Defenders of the Text ; Paolo Rossi, The Dark Abyss of Time: The History of the Earth and the History of Nations from Hooke to Vico , trans. Lydia G. Cochrane (Chicago: University of Chicago Press, 1984); Sicco Lehmann-Brauns, Weisheit in der Weltgeschichte: Philosophiegeschichte zwischen Barock und Aufklärung (Tübingen: Niemeyer, 2004); Helmut Zedelmaier, Der Anfang der Geschichte: Studien zur Ursprungsdebatte im 18. Jahrhundert (Hamburg: Meiner, 2003). On the wider history of scholarship in this period, see e.g. Françoise Waquet, Le modèle français et l’Italie savante: conscience de soi et perception de l’autre dans la République des lettres (1660-1750) (Rome: Ecole Française de Rome; Paris: Boccard, 1989); Helmut Zedelmaier and Martin Mulsow, eds., Die Praktiken der Gelehrsamkeit in der frühen Neuzeit (Tübingen: Niemeyer, 2001); C. R. Ligota and J.-L. Quantin, eds., History of Scholarship (Oxford: Clarendon Press, 2006).

[43] Grafton, Bring Out Your Dead , chap. 10.

[44] Grafton, Scaliger , 2:699, 703. For a wide-ranging study that sets another branch of late Renaissance erudition, antiquarianism, into the context of the Republic of Letters, see Peter Miller, Peiresc’s Europe: Learning and Virtue in the Seventeenth Century (New Haven, CT: Yale University Press, 2000).

[45] Grafton, Scaliger , 2:316-22; Debora Shuger, The Renaissance Bible: Scholarship, Sacrifice, and Subjectivity (Berkeley: University of California Press,1994), chap. 1.

[46] For a full treatment, see Anthony Grafton and Joanna Weinberg, Rabbi Isaac Casaubon: A Renaissance Hellenist Meets the Jews (Cambridge, MA: Harvard University Press, forthcoming).

[47] Marina Rustow, “Karaites, Real and Imagined: Three Cases of Jewish Heresy,” Past and Present 197 (2007): 35-74.

[48] Gerald Geison, The Private Science of Louis Pasteur (Princeton, NJ: Princeton University Press, 1995).

[49] For an erudite and elegant recent treatment of the Vossius controversy and its context, see Eric Jorink, Het “boeck der natuere": nederlandse geleerden en de wonderen van Gods Schepping, 1575-1715 (Leiden: Primavera Pers, 2006).

[50] See Claudine Poulouin, “Les érudits de l’Académie des Inscriptions et Belles-Lettres, marginaux des Lumières?,” in Les marges des Lumières françaises (1750-1789) , ed. Didier Masseau (Paris: Droz, 2004), 199-204. Cf. Mélanie Traversier, “De l’érudition à l’expertise: Saverio Mattei (1742-1795), ‘Socrate imaginaire’ dans la Naples de Lumières,” Revue historique 309 (2007): 91-136; and for a radically different presentation on the grandest of scales, see J. G. A. Pocock, Barbarism and Religion , 4 vols. to date (Cambridge: Cambridge University Press, 1999-2005).

[51] Cf. Noel Malcolm, Aspects of Hobbes (Oxford: Clarendon Press, 2002); Jonathan Sheehan, “Sacred and Profane: Idolatry, Antiquarianism and the Polemics of Distinction in the Seventeenth Century,” Past and Present 192 (2006): 35-66. Cf. more generally Martin Mulsow, Die drei Ringe: Toleranz und clandestine Gelehrsamkeit bei Mathurin Veyssière La Croze (1661-1739) (Tübingen: Niemeyer, 2001); Mulsow, “Practices of Unmasking: Polyhistors, Correspondence, and the Birth of Dictionaries of Pseudonymity in Seventeenth-Century Germany,” Journal of the History of Ideas 67 (2006): 219-50.

[52] Jean Le Clerc, Parrhasiana (Amsterdam: chez les héritiers d’Antoine Schelte , 1699-1701), 1:144.

[53] Kristine Haugen, “Imagined Universities: Public Insult and the Terrae Filius in Early Modern Oxford,” History of Universities 16 (2000): 1-31; Martin Mulsow, “Unanständigkeit. Zur Missachtung und Verteidigung des Decorum in der Gelehrtenrepublik der Frühen Neuzeit,” Historische Anthropologie 8 (2000): 98-118; Mulsow, Die unanständige Gelehrtenrepublik: Wissen, Libertinage und Kommunikation in der Frühen Neuzeit (Tübingen: Niemeyer, 2007; English translation forthcoming; Ann Arbor: University of Michigan Press).

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17 Lying in the Grey Zone

• Published: April 2024
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This chapter examines the role and legal status of State lying in grey zone conflicts, focusing on States lying to other States in their inter-State communications. The chapter begins by examining the utility of State lying in grey zone conflicts, looking at the example of Russia’s 2014 annexation of Crimea. While there is presently no international law prohibition on State lying, the chapter argues that we should extend the existing laws on insincere State utterances to include lying by States. Because the existing laws are explained by the principle of good faith, the proposed prohibition on State lying would only apply when States are coordinating plans of action, creating problems for its application in grey zone conflicts. The chapter looks to resolve some of these difficulties, before concluding with a summary of the argument.

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