Impossible Objects, Thought Experiments, and the Logic of Fictional Worlds
Simon de Bourcier
Writers of fiction are often credited with creating worlds. Sometimes, in the case of Science Fiction, we literally mean a different planet. Often, though, a fictional world is simply the place where a set of imaginary events occur. A fictional world may differ from the real world by being populated by fictitious characters only in the private sphere, or in the public sphere too, with imaginary politicians, corporations, or countries, or may contain miraculous technologies. In a sense fictional worlds consist in precisely these differences: an author does not have to tell the reader everything about a fictional world (the sun rises in the morning, people start out as babies then get bigger, etc.) because the reader will assume that it resembles the world she knows except when she is told otherwise.
Science does not share fiction’s privilege of being able to presume a common understanding of the world. One of the ways it describes the world is by cataloguing large amounts of data. Edward Mendelson has proposed the term ‘encyclopedic narrative’ to describe a class of text which includes Ulysses, Moby Dick, and Gravity’s Rainbow, implying that these texts imitate the inclusiveness of the eighteenth-century encyclopaedists, or of the type of scientific practice associated with Humboldt and Linnaeus.[i] However, science can also describe the world by formulating laws. The articulation by Newton and Kepler of laws describing the motion of bodies in space is considered among the great achievements of science. In this essay when I talk about a world, I mean a universe defined by a set of natural laws or ‘universal statements’. Might a fictional text create a world by telling the reader, explicitly or implicitly, that the fictional world is governed by different laws than those which govern our own?
We are sometimes told directly about the laws that obtain in the world of a particular fiction. Science Fiction is notorious for using the ‘As you know, captain…’ formula to introduce expositions of made-up scientific theories. But an argument advanced by Karl Popper suggests another way in which the laws governing a fictional world might be implicitly communicated to the reader, and a fictional world, or universe, thereby evoked:
The theories of natural science, especially what we call natural laws, have the logical form of strictly universal statements; thus they can be expressed in the form of negations of strictly existential statements or, as we may say, in the form of non-existence statements (or ‘there is not’ statements).
This is Popper’s elaboration of a more general principle: ‘The negation of a strictly universal statement is always equivalent to a strictly existential statement and vice versa’. He offers an example: ‘the law of the conservation of energy can be expressed in the form: “There is no perpetual motion machine”’.[ii] We can schematize Popper’s theory like this:
‘Energy is neither created nor destroyed’
(the law of the conservation of energy) = ‘There is no perpetual motion machine’
Conversely:
‘There is a perpetual motion machine’ = ‘The law of the conservation of energy is false’
Logically, then, any existential statement – a statement, for example, beginning ‘Once upon a time there was…’ – always implies a world, because it implies a universal statement. It may be that the universal statement we can infer from an existential one is a mere tautology or, in the mathematical sense, trivial. For example, Popper’s logic tells us that the existential statement, ‘Once upon a time there was a girl called Cinderella’ is equivalent to a negation of the universal statement, ‘There are no girls called Cinderella’. Applying Popper’s logic to fiction becomes more interesting when, as with his example, we think about representations of impossible objects. Mark Currie has suggested that ‘the impossible object or world is the very possibility of fiction’.[iii] In fiction, then, there can be a perpetual motion machine.
In this essay, however, I am looking closely at another imaginary object, the time machine. It should be possible to formulate one or more universal statements of which the ‘existential’ statement, ‘There is a time machine’, is the negation, or conversely, which can be expressed in the form: ‘There is no time machine’. One might be an affirmation of presentism: ‘Only the present exists’. John Bigelow gives a historicist explanation of time-travel stories based on the idea that time travel presumes the co-existence of past, present, and future: the ‘utter absence of any time[-]travel stories whatsoever prior to the nineteenth century’ is, he argues, ‘at least partly explained by the utter universality of presentism prior to the nineteenth century’.[iv] Other candidates for a universal statement negated by the statement, ‘There is a time machine’, are, ‘Nothing can be and not be’ – the law of non-contradiction – and, ‘Everything must either be or not be’ – the law of the excluded middle. Stanislaw Lem argues that ‘the presupposition of “journeys in time” […] implies a qualitative difference in the causal structure of the world’. Lem discusses several statements ‘which logic, by virtue of a “disconnected [i.e. excluded] middle” or by virtue of a tautology, asserts are always true’ and sets out to ‘investigate whether there can be worlds in which their veracity ceases’. He concludes: ‘For a real tautology to become a falsehood, the device of travel in time is necessary’.[v]
This implies, conversely, that a narrative involving time travel, since it can give rise to contradiction, must of necessity be a fiction. D. H. Mellor discusses the question of whether time travel is possible by presenting three brief narratives, or thought experiments, involving a time machine which he gives the name TARDIS, borrowed from the television series Doctor Who. Mellor argues that some of the logical objections to the possibility of travelling backwards in time are in fact ‘[s]oluble problems’. One such ‘non-problem’ is presented like this:
Suppose that […] TARDIS is made in Cambridge in 2030 but not used until 2050, and then only to go back to London in 2040, after which it remains in London and is never used again. So while […] there is only one TARDIS before 2040, and after 2050, in between those two years there are two, one in Cambridge and one in London. But how can this be? How can these two different machines be one and the same, as our story implies? Is this not a contradiction, and therefore impossible? If it is, then since backward time travel implies that this is possible, it itself must be impossible.
Not so, for unless some other feature of time travel rules it out, I see nothing impossible in [this scenario], merely a counter-example to the thesis that nothing can be in two or more places at once.[vi]
In the terms of Popper’s logic, the existential statement, ‘There exists a machine which can travel backwards in time’, is logically equivalent to a negation of the universal statement, ‘Nothing can be in two places at once’. Such a machine implies a world in which that universal statement is not true. There may be worlds, Mellor suggests, in which it is not, and therefore the TARDIS is not a logically impossible object.
Mellor next offers this scenario:
Imagine a single TARDIS vanishing in 2050 to arrive in 1950, when it is put in the Science Museum in London. There it remains for a century, until a group of would-be time travellers wondering how to make a time machine remember that there already is one in the Science Museum, complete with instruction manual. So in 2050 they retrieve it from the museum, read the manual and set off for 1950, where…
This implies that the TARDIS ‘was never made, in any way, by anyone’; but this is not necessarily a problem, says Mellor:
It is true that this story, like its predecessor, seems to conflict with what we take to be the laws of nature, such as the laws of the conservation of mass and energy which TARDIS’s appearing in 1950 and disappearing in 2050 would violate. But perhaps these laws […] hold only when and where there is no time travel.[vii]
Mellor closely follows Popper’s rules for the relation of existential statements and universal statements: saying that a certain machine exists amounts to saying that a certain law is not true. The existence of Mellor’s TARDIS is prohibited, like that of Popper’s perpetual motion machine, by the law of the conservation of energy. But though it contradicts what we know, or think we know, about the world, it is not self-contradictory.
Finally, Mellor presents ‘the insoluble problem’ for time travel: ‘if we could travel into the past we could always cause a contradiction, by doing something then that would stop us setting off now. But no one can cause a contradiction; so backward time travel must be impossible’. Now, if Mellor is prepared to dispense with the rule that something cannot be in two places at once, why is he not also prepared to let go of the rule that something can both happen and not happen? For him, the law of contradiction governs not only this world, but every possible world, whereas the laws of physics are contingent. I am going to consider some alternatives to this position arising from the ideas of the early twentieth-century philosopher Hans Vaihinger and those of the physicist Erwin Schrödinger. My purpose is not, ultimately, to refute Mellor’s argument, but to consider some of the implications of how it is made. Formally, it is a reductio ad absurdum: by demonstrating that a certain premise gives rise to a contradiction, Mellor hopes to prove that the premise must be false. He concludes that he has successfully proved that ‘time travel into our past’ is ‘impossible’.[viii] The contradiction he has adduced is a version of what is sometimes termed the ‘grandfather paradox’: if you could travel back in time you could kill your own grandfather, preventing your own birth, so that you cannot go back in time and kill him. The initial hypothesis results in two irreconcilable consequences: you both do and do not kill your grandfather. For Mellor, this contradiction necessitates rejecting the hypothesis.
However, a different view of contradiction is found in Vaihinger’s work. He argues that many of the fundamental conceptual categories of philosophy and science are self-contradictory and should be thought of as fictions. He argues that:
[M]any thought-processes and thought-constructs appear to be consciously false assumptions, which either contradict reality or are even contradictory in themselves, but which are intentionally thus formed in order to overcome difficulties of thought by roundabout ways and by-paths.[ix]
Vaihinger distinguishes ‘real fictions’ from ‘semi-fictions’. Semi-fictions, he says, ‘only contradict reality as given, or deviate from it, but are not in themselves self-contradictory’. He regards all classificatory systems as semi-fictions. Real fictions ‘are not only in contradiction with reality but self-contradictory in themselves’. He gives as an example of the latter ‘the concept of the atom’.[x] He regards the ‘fundamental concepts of mathematics’ – space, time, point, line and surface – as ‘contradictory fictions’, and mathematics as ‘based upon an entirely imaginary foundation, indeed upon contradictions’. He says: ‘Semi-fictions assume the unreal, real fictions the impossible’. A fiction that Vaihinger discusses at some length is differential calculus. Differentials ‘are purely fictional, contradictory constructs by means of which, however, it is possible to subsume the curve under the general concept and the laws of the straight line’. That ‘however’ marks an important dynamic in Vaihinger’s argument: such fictions are contradictory but nonetheless useful. Calculus depends on the idea of infinitesimals. Vaihinger argues that the infinitely great and the infinitely small are both fictions, and that this is demonstrated by the contradictions they produce: the contradictions resulting from the idea of the infinitely small are seen in the Eleatic paradoxes. Calculus nonetheless enables us to perform certain operations very effectively. ‘What is untenable as a hypothesis’, Vaihinger argues, ‘can often render excellent service as a fiction’. Vaihinger argues that the difference between fiction and hypothesis is ‘that the fiction is a mere auxiliary construct, a circuitous approach, a scaffolding afterwards to be demolished, while the hypothesis looks forward to being definitely established’. He goes on to say:
The hypothesis tries to discover, the fiction to invent. The former is therefore often called a découverte; whereas the differential calculus is generally […] called an invention. Thus natural laws are discovered but machines invented. Fictions, as scientific mental instruments without which a higher development of thought is impossible, are invented. […] The atom is not a discovery of natural science but an invention.
Vaihinger’s talk of inventions and machines brings us back to impossible objects like the perpetual motion machine and the time machine. The word invention also recalls the elegantly ambiguous subtitle of H. G. Wells’s The Time Machine, which is An Invention, referring to both the machine and the story. It is, after all, by means of the technologies of fiction that Wells transports us into the future. Stories about machines or inventions are in this respect peculiarly self-referential. A fictional machine may be the means by which a fictional narrative implies or evokes a world, via the logical relation described by Popper, so there is a kind of identity between the story (which depicts a world) and the machine which (logically and semantically) designates that world. For Vaihinger, all fictions are inventions, and they are characterized by contradiction. His supreme example of an ‘exceedingly useful’ fiction is ‘the conceptual world’.[xi]
We have, then, two symmetrical versions of what contradiction means, but with quite different emphases. For Mellor, it is the final nail in a logical coffin, but he nevertheless manifests a certain enjoyment in spinning the yarns which constitute his thought experiments. Conversely, Vaihinger accepts that a contradictory fiction can never be true, but emphasizes its explanatory power. A contradiction may mean the end of a hypothesis, but it can be the beginning of a powerful fiction. Does this difference correspond to the difference between scientific models of the world and fictional worlds? Popper makes the falsification of hypotheses the defining activity of empirical science,[xii] but fictional worlds can embrace narratives that not only contradict the given world, but are themselves contradictory or paradoxical. However, Vaihinger’s claim that our fundamental conceptual structures are all fictions tends to problematize the argumentative use of contradiction. Moreover, to show that a hypothesis gives rise to a contradiction or paradox may not simply necessitate abandoning the hypothesis, but may demand a re-examination of some fundamental ideas about reality. A narrative which tracks the consequences of a hypothesis without having to physically conduct an experiment is known as a ‘thought experiment’. I earlier applied this term to Mellor’s brief time-travel tales, but not all thought-experiments share their simple rhetorical structure. Wells’s Scientific Romances have been characterized by Brian Stableford as ‘literary thought-experiments’, and Ursula Le Guin suggests that many Science Fiction stories can be read as thought experiments.[xiii] Conversely, some scientific thought experiments can resemble more sophisticated fictions in the way they deal with contradiction. One of the most famous in the history of science, set out by Erwin Schrödinger in a paper first published in 1935, illustrates the way that such a narrative can transcend the simple reductio structure in which contradiction necessitates the abandonment of a hypothesis. It can instead challenge the reader to undertake the imaginative work of re-casting the contradiction as a paradox, and the hypothesis as a useful fiction.
‘Schrödinger’, according to John Gribbin, ‘as upset as Einstein by the implications of quantum theory, tried to show the absurdity of those implications’.[xiv] James T. Cushing says that Schrödinger offers ‘an argument (but not, of course, a logical impossibility proof) against the completeness of quantum mechanics (that is, against the Copenhagen interpretation of quantum mechanics)’.[xv] The Copenhagen interpretation of quantum mechanics, set out by Niels Bohr in 1927, holds that quantum mechanics, which describes subatomic phenomena in terms of statistical probabilities, offers the most complete possible description of those phenomena. Schrödinger’s argument specifically concerns the quantum-mechanical concept of the wave-function or ψ-function. Any measurement of a physical system on the atomic level involves an interaction between that system and a measuring apparatus, between a microsystem and a macrosystem. Ideally, the state of the macrosystem will tell us reliably about the state of the microsystem. When we perform a measurement, we want the interaction to affect the macrosystem but not the microsystem. Now, Quantum mechanics tells us that we cannot know everything about certain microsystems. We can only describe their states, mathematically, in terms of a probabilistic wave-function, or ψ-function.[xvi] Peter Byrne explains the wave-function like this:
Physicists use mathematical entities called wave functions to represent quantum states. A wave function can be thought of as a list of all the possible configurations of a superposed quantum system, along with numbers that give the probability of each configuration’s being the one, seemingly selected at random, that we will detect if we measure the system.[xvii]
In such a mathematical description, certain variables are ‘blurred’ – they do not have a definite value, only a probability of having one of a range of values. The combination of microsystem and macrosystem can also only be described in terms of a wave-function, which is initially a product of the wave-functions of the two systems, and then evolves over time in a way that is described by what is known as the Schrödinger equation.
According to the Copenhagen interpretation the microsystem is not actually in either one state or another before we measure it, the probabilistic element of its wave-function simply an expression of our ignorance about it. Rather, the wave-function is a complete description of its state. The system only jumps into one state or another when it is measured – this is sometimes referred to as the ‘collapse’ of the wave-function. Since the combined microsystem and macrosystem can only be described in terms of a wave-function, the Copenhagen interpretation further implies that the state of the macrosystem is also completely described by a probabilistic equation: ‘we are left with no definite state for the observing apparatus’.[xviii] Until a measurement is made, the macrosystem, that is, a measuring device on the scale of everyday objects, is not in either one state or another, but in a blurred or uncertain condition. This is the aspect of the theory which Schrödinger seizes upon. It is possible to conceive events on a very small scale exhibiting a certain blurriness, but much harder to accept such a state of affairs with regard to the objects of the everyday world. It seems ‘simply wrong’:
One can even set up quite ridiculous cases. A cat is penned up in a steel chamber, along with the following device (which must be secured against direct interference by the cat): in a Geiger counter there is a tiny bit of radioactive substance, so small, that perhaps in the course of the hour one of the atoms decays, but also, with equal probability, perhaps none; if it happens, the counter tube discharges and through a relay releases a hammer which shatters a small flask of hydrocyanic acid. If one has left this entire system to itself for an hour, one would say that the cat still lives if meanwhile no atom has decayed. The ψ-function of the entire system would express this by having in it the living and dead cat (pardon the expression) mixed or smeared out in equal parts.[xix]
In other words, as Cushing explains, when the box is opened ‘a macrosystem, the cat, would be thrown into one state or another by our mere act of observation provided the wave function gives a complete, objective description of reality’.[xx] This does not mean that we kill the cat when we open the box. A forensic feline pathologist could tell us how long the cat has been dead for. Our observation causes the cat to have died that long ago. Schrödinger himself explains the implications of the cat experiment as follows:
It is typical of these cases that an indeterminacy originally restricted to the atomic domain becomes transformed into macroscopic indeterminacy, which can then be resolved by direct observation. That prevents us from so naively accepting as valid a ‘blurred model’ for representing reality. In itself it would not embody anything unclear or contradictory. There is a difference between a shaky or out-of-focus photograph and a snapshot of clouds and fog banks.
For Schrödinger the cat presents a problem for the ‘blurred model’, in other words for Bohr’s contention that quantum mechanics offers a complete description of reality rather than a measure of our ignorance about reality. It seems to imply that the quantum mechanical wave-function (or ψ-function) cannot describe a reality at the atomic level that is itself blurred or indeterminate. The obvious alternative is that the blurring or indeterminacy is epistemological rather than ontological, that ‘the indeterminacy is not even an actual blurring, for there are always cases where an easily executed observation provides the missing knowledge’. However, Schrödinger has already established that ‘it is not possible […] to ascribe, to the momentarily unknown or not exactly known variables, nonetheless determinate values, that we simply don’t know’. It is in the face of this ‘dilemma’, he says, that the Copenhagen interpretation ‘rescues itself or us by having recourse to epistemology’; but this epistemological turn is a more radical one than merely making the wave-function an index of our ignorance:
We are told that no distinction is to be made between the state of a natural object and what I know about it, or perhaps better, what I can know about it if I go to some trouble. Actually – so they say – there is intrinsically only awareness, observation, measurement. If through them I have procured at a given moment the best knowledge of the state of the physical object that is possibly attainable in accord with natural laws, then I can turn aside as meaningless any further questioning about the ‘actual state’, inasmuch as I am convinced that no further observation can extend my knowledge of it – at least, not without an equivalent diminution in some other respect[.]
Schrödinger further explains that, in this interpretation, ‘a variable has no definite value before I measure it; […] measuring it does not mean ascertaining the value that it has’. Accepting the Copenhagen interpretation means that we must accept that the cat is neither alive nor dead (or alive and dead) until we open the box. The Copenhagen interpretation, as Schrödinger represents it, avers that there is no contradiction involved in the alive-and-dead cat because it has no ‘actual state’: there is only ‘awareness, observation, measurement’.[xxi]
By presenting the cat scenario as ‘ridiculous’, Schrödinger is arguably signalling that it is to be understood as an example of reductio ad absurdum. By showing that to adopt the ‘blurred’ or statistical model of events at the atomic level results in having to accept that a cat can be alive and dead at the same time, he demonstrates that such a model is untenable. This is to assume, with Mellor, that for a cat or a grandfather to both die and not die constitutes a contradiction. Cushing’s commentary on the cat paradox makes the opposite assumption: it constitutes ‘an argument (but not, of course, a logical impossibility proof)’. His view is that ‘Schrödinger seems to bank on our visceral (negative) reaction to the suggestion that our mere act of observing the cat actually itself produces a live cat or a dead cat’.[xxii] We are faced with a choice between rejecting the hypothesis that leads to a contradiction, or revising our view of what constitutes a contradiction.
The alive-and-dead cat is for Schrödinger a contradiction that presents a problem for the Copenhagen interpretation, but his paper is also a challenge to the reader to re-imagine the world so that it can accommodate the paradox. The relation of the cat to the world can of course be thought of in terms of Popper’s logic: ‘there is an alive-and-dead cat’ is equivalent to the negation of the law ‘nothing is both alive and dead’. An alive-and-dead cat implies a world in which this law is not true. Although for Mellor for something to both be and not be is unacceptable, the possibility of both being and not being is, to borrow Currie’s phrase, ‘the possibility of fiction’. Fictional worlds possess the paradoxical status of existing without being real (or the other way round). Fictionality thus presents itself as the excluded middle between being and non-being. Recall that for Vaihinger our entire ‘conceptual world’ partakes of such fictionality.[xxiii] Schrödinger’s paradox implies that, if the Copenhagen interpretation is correct, Vaihinger is right.
We can further illuminate the difference between the defeat of a hypothesis by a contradiction, and a thought experiment or fiction which entails a paradox, by asking what exactly the difference is between paradox and contradiction. Laurence R. Horn explains that the existence of paradoxes is problematic for the law of non-contradiction, but, historically, has not completely unseated it:
In the Western tradition, the countenancing of true contradictions is typically – although not exclusively – motivated on the basis of such classic logical paradoxes as ‘This sentence is not true’ and its analogues (the Liar, the Barber, Russell’s paradox), each of which is true if and only if it is not true.[xxiv]
Horn represents Schrödinger’s cat paradox, similarly, as a challenge to the law of non-contradiction.[xxv] Where to draw the line between contradiction and paradox is a decision facing both physicists and mathematicians: according to David Foster Wallace, before Georg Cantor’s ‘invention’ of transfinite mathematics, paradoxes of infinity were dealt with
by first fudging the distinction between a paradox and a contradiction, and then by applying a kind of metaphysical reduction: if allowing infinite quantities like the number of points on a line or the set of all integers led to paradoxical conclusions, there must be something inherently wrong or nonsensical about infinite quantities, and thus ∞-related entities couldn’t really “exist” in a mathematical sense.[xxvi]
If a premiss leads to a paradoxical conclusion, one can treat the paradox as a contradiction, and therefore as grounds for dismissing the original premiss: ‘there exists a set of all sets which do not contain themselves’, ‘a time machine exists which can travel back in time’, and ‘quantum mechanics gives a complete description of subatomic phenomena’ are rejected as false or incoherent. On the other hand, if we recognize that paradox is not the same as contradiction, we can think in more sophisticated ways about these existential statements and the worlds they imply.
R.M. Sainsbury argues that what differentiates a paradox from a contradiction is that in the case of paradoxes we cannot simply dismiss the premisses which entail contradictory consequences as incoherent. Russell’s paradox, as Sainsbury explains, involves ‘the class of all classes that are not members of themselves’, which Sainsbury terms R. R ‘is a member of itself if and only if it is not a member of itself’. This is clearly a contradiction. But:
To have a contradiction is not necessarily to have a paradox. Recall the Barber paradox […]. The barber shaves all and only those who do not shave themselves. Who shaves the barber? By reasoning similar to that used to derive Russell’s paradox, we find that the barber shaves himself if and only if he does not.
We respond to the barber paradox simply by saying there is no such barber. Why should we not respond to Russell’s paradox simply by saying that there is no such class as R? The difference is this: nothing leads us to suppose that there is such a barber; but we seem to be committed, by our understanding of what it is to be a class, to the existence of R.[xxvii]
The decision, then, to consider an apparently contradictory state of affairs as a contradiction or as a paradox amounts to a decision about whether we are committed to the original set of premisses from which it derives. Mellor is not committed to the premiss that TARDIS can travel back in time, so his contradiction functions as a reductio; but physicists who are committed to the validity of quantum mechanics must regard the alive-and-dead cat as a paradox.
The implicit bargain between a fictional text and the reader is that the reader will regard any contradiction as a paradox. Just as the definition of R is compelling and coherent enough that we do not sacrifice it when a paradox arises, fictional worlds do not fall apart at the first appearance of contradiction. The barber who shaves all the men in the village who do not shave themselves (and no-one else) may not exist in any real village, but we can imagine him setting up shop in a fictional one. Alain Badiou argues that Russell’s paradox implies that ‘it is not true that to a well-defined concept there necessarily corresponds the set of objects which fall under this concept’. For Badiou this is an ‘obstacle to the sovereignty of language’,[xxviii] but I am suggesting that it implies that language’s domain is not confined to the realm of possible objects and worlds, but extends over impossible ones too. Ruth Ronen suggests that possible worlds philosophy allows us to think of fiction as ‘one among other categories of cultural products that present non-actual states of affairs through language (the same is true of conditionals, descriptions of worlds of desire, belief, and anticipation, and mythical versions of the world)’.[xxix] So, language can say, ‘If there were a barber who shaved all the a men in the village who didn’t shave themselves, and no-one else…’ Or, ‘Oh how I wish there were a barber who…’ Or, most importantly, ‘Once upon a time there was a barber who…’ Logic tells us there is no such barber, but language is able to describe him, and fiction can invite us to take on the task of imagining the world in which he exists.
Ronen claims that possible worlds ‘provide for the first time a philosophical explanatory framework that pertains to the problem of fiction’.[xxx] However, the applicability of possible worlds philosophy to fiction is limited, according to Peter Stockwell, by the fact that in fiction ‘the basis of traditional possible worlds theory – logic – is as amenable to alternativity as any other system’.[xxxi] Stockwell insists – with Mellor – that it is legitimate to conceive as ‘possible’ other worlds in which different laws of nature apply – a TARDIS can be in two places at once, or exist without ever having been made – but not a world where the fundamentals of logic do not apply, where cats and grandfathers both die and do not die. Nevertheless, I have been arguing that what constitutes a contradiction is not always self-evident. Schrödinger’s thought experiment, like Russell’s set of all sets which do not contain themselves, provides an example of an apparent contradiction which there are good reasons for viewing as a paradox. It invites us to consider the possibility that our world is like the fictional barber’s village, and can accommodate contradictions.Matt Hills acknowledges the objection raised by Stockwell, but suggests that by adopting a ‘less rigid view of possible worlds philosophy’ one can enable a fruitful dialogue between fiction and possible worlds theory.[xxxii] Specifically, one must overcome the rigid differentiation of possible worlds which conform to classical logic and impossible worlds which do not. In his elaboration of an ontology founded on Cantor’s mathematics of transfinite sets, Badiou seizes on the way in which Cantor responded to the paradoxes of infinity: ‘As often happens, the invention consisted in turning a paradox into a concept’.[xxxiii] Badiou’s use of the word ‘invention’ resonates with the connection I am affirming between impossible machines, like Wells’s time machine, and the fictional worlds contained in paradoxes. It is by choosing to see paradox where some see only contradiction that writers and readers of fiction allow impossible objects to imply worlds.
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Notes
[i] Edward Mendelson, ‘Gravity’s Encyclopedia’, in Mindful Pleasures: Essays on Thomas Pynchon, ed. by George Levine and David Leverenz (Boston: Little, Brown, 1976), p.161.
[ii] Karl Popper, The Logic of Scientific Discovery (London: Routledge, 1992), pp. 68-9.
[iii] Mark Currie, About Time (Edinburgh: Edinburgh University Press, 2007), p. 85.
[iv] John Bigelow, ‘Presentism and Properties’, Philosophical Perspectives, 10 (1996), pp. 35-6.
[v] Stanislaw Lem, ‘The Time-Travel Story and Related Matters of SF Structuring’, trans. by Thomas H. Hoisington and Darko Suvin, in Science Fiction: A Collection of Critical Essays, ed. by Mark Rose (Englewood Cliffs, NJ: Prentice-Hall, 1976), pp. 72-88 (pp. 81).
[vi] D. H. Mellor, ‘Time Travel’, in Time, ed. by Katinka Ridderbos (Cambridge: Cambridge University Press, 2002), pp. 46-64 (pp. 57-61).
[vii] Mellor, pp. 61-2.
[viii] Mellor, pp. 61-4.
[ix] Hans Vaihinger, The Philosophy of ‘As If’. A System of the Theoretical, Practical, and Religious Fictions of Mankind, trans. by C. K.Ogden (London: Routledge, 2000), pp. xlvi-xlvii.
[x] Vaihinger, 16-18.
[xi] Vaihinger, pp. 51-88.
[xii] Popper, pp. 30-42.
[xiii] Brian Stableford, Scientific Romance in Britain 1890-1950 (London: Fourth Estate, 1985), p. 29. Ursula K. Le Guin, The Language of the Night: Essays on Fantasy and Science Fiction (London: The Women’s Press, 1989), p. 31.
[xiv] John Gribbin, In Search of Schrödinger’s Cat: Quantum Physics and Reality (London: Black Swan, 1991), p. 2.
[xv] James T. Cushing, Philosophical Concepts in Physics: The Historical Relation Between Philosophy and Scientific Theories (Cambridge: Cambridge University Press, 1998), p. 313.
[xvi] Cushing, pp. 311-313.
[xvii] Peter Byrne, ‘The Many Worlds of Hugh Everett’, Scientific American, December 2007 <http://www.sciam.com/article.cfm?id=hugh-everett-biography> [accessed 13th May 2010] (para. 8 of 44).
[xviii] Cushing, p. 311.
[xix] Erwin Schrödinger, ‘The Present Situation in Quantum Mechanics’, in Quantum Theory and Measurement, ed. by John Archibald Wheeler and Wojciech Hubert Zurek (Princeton, NJ: Princeton University Press, 1983), pp. 152-67 (p. 157).
[xx] Cushing, p. 312.
[xxi] Schrödinger, pp. 157-8.
[xxii] Cushing, p. 313.
[xxiii] Vaihinger, p. 63.
[xxiv]Laurence R. Horn, ‘Contradiction’, in The Stanford Encyclopedia of Philosophy, ed. by Edward N. Zalta (Spring 2009 Edition). <http://plato.stanford.edu/archives/spr2009/entries/contradiction/> [accessed 29th June 2010] (§4, para. 4 of 10)
[xxv] Horn, para. 9.
[xxvi] David Foster Wallace, 2005. Everything and More: A Compact History of ∞ (London: Phoenix, 2005), p. 39.
[xxvii] R. M. Sainsbury, Paradoxes, 2nd edn (Cambridge: Cambridge University Press, 1995), pp. 107-8.
[xxviii] Alain Badiou, Logics of Worlds: Being and Event, 2, trans. by Alberto Toscano (London: Continuum, 2009), p.153.
[xxix] Ruth Ronen, Possible Worlds in Literary Theory (Cambridge: Cambridge University Press, 1994), p. 21.
[xxx] Ronen, pp. 6-7.
[xxxi] Peter Stockwell, The Poetics of Science Fiction (Harlow: Longman, 2000), p. 145.
[xxxii] Matt Hills, ‘Time, Possible Worlds, and Counterfactuals’, in The Routledge Companion to Science Fiction, ed. by Mark Bould, Andrew M. Butler, Adam Roberts and Sherryl Vint (London: Routledge, 2009), pp. 433-41 (p. 434).
[xxxiii] Alain Badiou, Being and Event, trans. by Oliver Feltham (London: Continuum, 2006), p. 267.