Mathematical fictions

Transcription

1 Mathematical fictions Jody Azzouni Tufts University Forthcoming in Fiction and Art: Explorations in Contemporary Theory, edited by Ananta Sukla London: Bloomsbury Academic (fall, 2014). Abstract: I describe the philosophically-blinding value of fictional terms in mathematics. It s extremely hard for philosophers to believe that mathematical terms don t refer: the truth of mathematical statements, their indispensability to empirical science, and the apparent necessity of mathematics this has convinced almost every philosopher that mathematical terms must not only refer, but that they refer to strange eternally unchanging objects that we only have access to through some unusual aspect of mathematical practice (e.g., intellectual intuition). Aboutness illusions explain why philosophers (until the twenty-first century) either committed themselves to the position that (1) fictions actually exist, or to the position that (2) sentences that appear to have fictional terms (terms that don t refer) are actually some other kind of sentence all the terms of which do refer. Finally, I explain the unusual properties of mathematical discourse itself as due (in part) to mathematical terms being fictional. 1. Plato and other philosophers on the impossibility of talking about what isn t Plato, like almost every philosopher (even still today), thought it was impossible to talk about what doesn t exist. Because if something doesn t exist, then it isn t anything. And if something isn t anything, then it isn t possible to talk about it. (Because, what could it be that we would be talking about? If we re actually talking about it, then it has to be something.) And so it certainly isn t possible to say something true (or even false) about something that isn t anything. Because for a sentence to be true (or false) is for it to be true (or false) of something. But (again) if something isn t anything (if it doesn t exist) then it isn t something. Here is what Plato once wrote about this: STRANGER: Surely we can see that this expression something is always used of a thing that exists. We cannot use it just by itself in naked 1

2 isolation from everything that exists, can we? THEAETETUS: STRANGER: No. Is your assent due to the reflection that to speak of something is to speak of some one thing? THEAETETUS: STRANGER: Yes. Because you will admit that something stands for one thing, as some things stands for two or more. THEAETETUS: STRANGER: Certainly. So it seems to follow necessarily that to speak of what is not something is to speak of no thing at all. THEAETETUS: STRANGER: Necessarily. Must we not even refuse to allow that in such a case a person is saying something, though he may be speaking of nothing? Must we not assert that he is not even saying anything when he sets about uttering the sounds a thing that is not? (Plato 1963a, 237, D-E; emphasis in translation) We also have a fragment from Parmenides that expresses similar sentiments: It must be that what is there for speaking of and thinking of is; for it is there to be. (Parmenides of Elea, Toronto, Canada: University of Toronto Press, 1984) So the impossibility of talking about nothing is among the oldest of philosophical claims. 2. Talking about nothing and saying true (and false) things about it 2

3 Despite Plato (and nearly every other philosopher who has opined about this), there seems to be lots of things that we say on a regular basis about fictions about things that don t exist. And some of these things are true and some of them are false. Consider the following statements as examples: James Bond is depicted in the first paragraph of the novel Goldfinger as having had two double bourbons and as contemplating life and death. James Bond is portrayed in the fiction of Washington Irving as having died and as having had an interest in early New York. The narrator of the short story Revelation from a Smoky Fire is depicted as presenting himself as believing that his study has a defective chimney. Neither the name James Bond nor the description The narrator of the short story Revelation from a Smoky Fire refer to anything: there is no James Bond, and the narrator of the short story is fictional as well. Nevertheless, the first and third sentences are true; the second is false. As far as meaning (semantics) and grammar are concerned, these previous three sentences look fairly similar to these next three sentences: Page 8 of the July 13 th -19 th 2013 issue of The Economist depicts Jean-Claude Juncker as having resigned as prime minister of Luxembourg. 3

4 Page 9 of The New York Post portrays Jean-Claude Juncker as hiding in a Moscow airport, and as asking for asylum in Eucador. Susan Carey, in The Origin of Concepts, depicts Quine as committed to representational primitives being limited to a perceptual similarity space. As before, the first and third sentences are true, the second isn t. Truth and falsity don t cease to apply to sentences just because (some of) the terms in those sentences fail to refer. 3. Aboutness intuitions It s tempting to try to avoid the evidence that we talk about nothing but pace Plato and Parmenides we nevertheless say things that are true and false when doing this. One easy way to try to avoid the apparently obvious point that we often say true and false things about what isn t is to argue that these statements (contrary to appearances) really aren t about things that don t exist. They are really about something else that does exist. There are many ways to attempt this, but the idea roughly is that when we are apparently talking about James Bond or the narrator depicted in the short story, Revelation from a Smoky Fire, we are actually talking about the novel Goldfinger or the short story Revelation from a Smoky Fire, itself. Unfortunately, this just isn t true. James Bond doesn t refer to a novel it doesn t refer at all. (There is no James Bond.) Although the phrase the narrator depicted in the short story Revelation from a Smoky Fire, contains the phrase the short story Revelation from a Smoky Fire, it clearly just uses that phrase to enable it to indicate something else something that doesn t exist. In any case, we can (and do) say true (and false) things about nonexistent beings of 4

5 all sorts statements that are clearly about the (nonexistent) characters and not about the (existing) fictions they appear in. Although I hate everything that s ever been written about Sherlock Holmes, I really respect him. Mickey Mouse is more famous than anyone real. That cartoon is sloppily drawn, but nevertheless Sponge Bob is really cute. Another way to try to avoid the obvious fact that we are saying truths and falsities (on a regular basis) using sentences with terms that don t refer to anything, is to instead find something that the terms Mickey Mouse, Sherlock Holmes, and so on, can be taken to refer to. We can say, for example, that (1) Mickey Mouse refers to the idea of Mickey Mouse, or that (2) it refers to the collection of Mickey-Mouse-cartoons, or (if we re desperate enough), that (3) it refers to some sort of (really weird) metaphysical entity that doesn t exist (since Mickey Mouse doesn t exist, is true) but nevertheless has some other sort of being: The weird thing that Mickey Mouse refers to is something that subsists or subexists (or is), but that doesn t exist. Unfortunately, either this doesn t work or it s so crazy that only philosophers would take it seriously. Consider the first two maneuvers. We say, Mickey Mouse doesn t exist. We don t say, Mickey-Mouse-cartoons don t exist since they do; we don t say, the idea of Mickey Mouse doesn t exist since I m thinking of Mickey Mouse right now, and so are you. So these first two moves simply illicitly change the subject matter of the sentences we normally utter: 5

6 from the nonexisting Mickey Mouse to existing Mickey-Mouse ideas or to existing Mickey- Mouse cartoons. What about the third move, Mickey Mouse being some sort of weird metaphysical entity? Perhaps the weird object that the words Mickey Mouse refers to is one that was originally brought into existence by Disney thinking of him. Well, this really is desperate, isn t it? Why invent a lot of weird metaphysics when the alternative is simply to accept the obvious evidence of how we talk and write. Contrary to what Plato and Parmenides and other philosophers have said, we simply do talk about nothing, and say (true and false) things about it. Well, Plato (and company) could reply: You just put your claim in a way that we find impossible to comprehend; you just used the word about. But how can you talk about nothing, and say things about it? That s just incoherent if it s nothing, then there s nothing we re talking about. It s true. When we talk about Mickey Mouse, it doesn t seem to be like talking about Pegasus. Neither exists, but we re nevertheless talking about different things that s how it feels. The first thing that has to be said here is that our use of the word about is fooling us. There is certainly a different subject matter in each case; that is, there are different sentences with different truth values that we use. To talk about Mickey Mouse is to say things that are true (or false) that are different from the things we say when we talk about Pegasus. It s true that Disney invented Mickey Mouse ; it isn t true that Disney invented Pegasus. That s what our aboutness intuitions are actually tracking: they re tracking the fact that sentences with the word Pegasus in them have different truth values than sentences with the word Mickey Mouse in them. But this isn t the same thing as there being two different objects Mickey Mouse and 6

7 Pegasus that our talk is targeting, and that our aboutness intuitions are marking as different objects. There is something else (something psychological) that s driving our aboutness intuitions. This is that when we are fantasizing, or making up stories, or dreaming, or otherwise thinking up imaginary beings, our psychological methods of thinking about real objects are the only mental tools we have to manage this. (We subconsciously borrow our methods of thinking about real things when we re instead making up stories or fantasizing or dreaming.) We have a capacity for object-directed thinking that we can detach from the real things that we normally think about, and that we can continue to operate with even though there are no objects involved any longer. And we also borrow our ordinary ways of talking about objects when fantasizing (for example). That s why we speak of thinking about hobbits or elves or dwarves, even though there are no hobbits or elves or dwarves; that s why we experience our talking about (or thinking about) hobbits as different from our thinking about elves even though there is nothing we are thinking about (or talking about) in both cases. Because our imaginative faculties operate by borrowing mental tools that we use to think about real things there is a strange cognitive cost to the process: our minds create aboutness illusions. And it s our succumbing to these aboutness illusions (when doing philosophy) that in turn generates strange claims and metaphysical positions ones that date back to Plato and before. Aboutness illusions misfocus our minds: instead of our feeling the difference between talking about Pegasus and talking about Hercules as due to the differences in the truth values of sentences in which the words Pegasus and Hercules appear, we instead feel the difference as one about different objects we are talking about while simultaneously we are aware at the same time that there are no such objects as Pegasus and Hercules they are fictions. 7

8 Closely related to this is that we have to be very careful about the word about : it s treacherous. Sometimes it s being used with respect to objects that we are thinking and talking about; and sometimes it s being used even though no objects are involved in our thinking or talking at all. In this second case, thinking about Mickey Mouse is compatible with our saying that there is no Mickey Mouse. Unfortunately, telling ourselves (or others) all this stuff about aboutness illusions won t eliminate our experience of these illusions that arises whenever we transact with the nonexistent. These illusions are just like optical illusions: No matter how much we stare and stare at an optical illusion, we can t make it go away just by saying to ourselves (for example): I know those lines are the same length even though they appear not to be. Here too, we ll always have the overwhelming cognitive impulse to experience our thinking about Pegasus and our thinking about Hercules as kinds of thinking about objects and objects that are different. Even when we know it isn t true. There is no way to escape these aboutness illusions. Not for us. Not for us humans. We can imagine other creatures that are different from us in this respect. These creatures also talk about the nonexistent, but they don t do it psychologically by subconsciously borrowing the psychological mechanisms that they use to talk about real objects. Instead, they focus entirely on discourse, on words. They say the same things we do, but when they think about Pegasus and Hercules, they only notice that sentences in which these words appear can differ in truth values ( Pegasus is depicted as a flying horse, Hercules is depicted as a flying horse. ) I imagine that reading fiction isn t as enjoyable for these creatures as it for us because we re able to engage our emotional reactions to fictional creatures precisely because those same emotional mechanisms get engaged when we talk and think about real objects. And maybe there 8

9 would be other differences between these creatures and us too. The important point is that these creatures, unlike us, wouldn t experience aboutness illusions. One of our collective delusions about ourselves is that our intellects when we don t interfere with those intellects by being emotional or biased or whatever are cool and dispassionate and objective. Not so. Intellectual cognition is full of bizarre and irrational dispositions and drives. Our brains (and so our minds) just like our bodies are the uneven products of evolution. As a result we have all sorts of cognitive appetites and lusts. Many of these appetites and lusts speaking frankly are metaphysical. (I know how weird this sounds but it s true, it really is.) In particular, what I ll call the aboutness appetite is a desire (an obsession, really) to believe that if we say something true then we are talking about something. And this aboutness appetite is impossible to shake. We are a strange animal in many ways. This is one of those unexpected ways that we re strange. We re strange because we have these amazing minds with amazing capacities one of which is the capacity to entertain thoughts about what isn t there. But amazing mental abilities of any sort always come with intricate (cognitive) mechanisms that (in some way or other) don t quite work right. Because, after all, nothing works perfectly. Really. 4. Some of the ways that mathematical entities are strange Suppose someone says: Okay, I ll accept all this. I ll agree with you that (strangely) this is how our minds work and how our language works when we make up stories or when we mythologize. We talk about nothing and somehow we get away with it. But surely we can isolate all that sort of talk and thought from our serious talking and thinking, the kind of serious talking 9

10 and thinking we soberly engage in when we re doing real science. 1 When we re doing science (one might say) we re only talking about real things because made-up things don t matter to science: we re trying to find out about the world when we do science, we re not interested in the fantasies we dream up when we re trying to entertain ourselves (or soothe small children into falling asleep in the evening). And so, serious talk about what is is surely all the talk we need for science; and serious talk about what is must include mathematics because mathematics is needed for science. Unfortunately, this is where things get weird (again). I m going to explain this in the next few sections. I ll tell you now, though, what the upshot will be: our mathematics is full of fictions of the same sort that show up in our storytelling. More strongly: no mathematical terms refer. But it s going to take a little while to establish this, and the argument is a bit intricate. Here s the first point. Mathematics, right from its established beginning among the ancient Greeks as a deductive science, involved reference to all sorts of strange entities. There were supposed to be points, for example: and these were supposed to have no dimensions at all. And there were lines too: these were supposed to have only one dimension, although lines were also supposed to be composed only of points. (And so you could worry many people did and do: how can things with no dimension be packed together tightly enough so that the result is that they can be made into something with one dimension?) And triangles, they were strange too: their angles sum to exactly (no more and no less) than 180 degrees. When the ancient Egyptians 1 I don t think this isolating-discourse move works. Not because it shows a certain disrespect for literature studies although it does but because implicit in the strategy is that we can isolate our talk about literary and mythological fictions from our scientific discourse. But that s false. An important part of certain sciences psychology and sociology, for example is the phenomena of our thinking about, writing and speaking about, and even seeing, things that don t exist. I won t pursue this thread any further now because the specific topic of this paper is mathematical fictions and so I ll grant the isolation thesis for the sake of argument. But the interested reader can look at my Talking about nothing for details about why it s wrong. 10

11 measured triangles empirically drawing them in sand (or whatever), they never got exactly 180 degrees. They always got less or more. Try it. You will too. As mathematics has developed, the list of strange entities it traffics in has only grown larger and stranger: complex numbers, higher-dimensional figures (including various infinitedimensional ones), sets, levels of infinity, Turing machines,. This is an extremely short list of only some of the many odd entities that mathematical terms refer to. These entities are strange in part because of the properties attributed to them: properties nothing on Earth seems to have. They are strange also because we don t seem to have discovered them the way we discover other objects (by noticing them hanging around in our environment). Many people presume mathematical objects furthermore are eternal, that they re not located in space and time, and that they never change in any of their properties. Plato, in particular, was so mesmerized by these entities that he built his metaphysical views around their ontological centrality. 5. Why Plato (and many other philosophers) thought mathematical objects have to exist Euclidean geometry, once it emerged as a deductive science as a proof-driven science revealed itself to have a number of extremely striking properties unusual properties shared more generally with other branches of mathematics but with nothing else. These properties were already clear to the ancient Greeks, and had a profound influence on Greek philosophers. First, and perhaps most dramatically, we are able to reason from what we already know about Euclidean objects to new facts about those objects that are unexpected and even surprising. I ve already mentioned the angle result. A major joy of mathematical practice is precisely this: showing a result that no one (including yourself) sees coming. 11

12 But second, one can see by means of this reasoning by proofs that the results have to be true. A symptom of this is an amazing confluence in viewpoint among people who are sophisticated enough to follow the proofs in question (which, of course, not everyone is). Everyone who can follow these proofs agrees on the results. It s important to see why these two properties of mathematical proof should be called amazing. The reason is that the ancient Greeks saw that these properties were so manifestly absent from discourse in nearly every other area of human life. (This is still true today.) Plato, in particular, developed a lifelong obsession with extending this aspect of mathematical proof to an important area where he saw enormous and destructive disagreement ethics. Of course, he didn t succeed, and no one else after him has succeeded either. Third, there is that this geometrical knowledge is valuable to other areas of life. It isn t like games chess, for example where facility in a game has pretty much no value anywhere else (except, say, in entertainment). Euclidean geometry, for example, was coupled by the ancient Greeks with empirical claims about space, about the movement of celestial bodies, about various properties of physical objects, and it yielded empirical results that were as unexpected as the pure mathematical results were. Notice how these considerations make compelling the view that we haven t made up mathematical objects the way that we ve made up fantasy objects that, despite how strange mathematical objects are, they have to exist. First, there is the mistaken view I opened this paper with, exemplified by quotations from Plato and Parmenides: mathematical statements are true (or false: the interior angles of Euclidean trianges sum to 360 degrees ). A statement that s true or false has to be about something that exists. 12

13 But second, mathematics is empirically valuable. How can something that we make up be empirically valuable? If you hire someone to build a bridge for you, and you look at his notes, you won t be surprised or puzzled (or disturbed) if equations, numbers, drawings of geometrical objects, and so on, show up. You will be (very) surprised and puzzled (and disturbed) if copious (or any) references to unicorns, hobbits, elves, and so on, show up. Mathematics is incredibly valuable for science contemporary physics simply wouldn t exist without it. How can something full of terms that don t refer possibly be of value in this way? And third, the following explanation for the necessity of deductive reasoning seems available. Deductive reasoning (recall) seems to yield not just truths, but necessary truths: The interior angles of a triangle have to sum to 180 degrees; adding 2 to 2 has to yield 4. Well (a philosopher might say and many have): here s a reason for this. Mathematical objects themselves are necessarily unchanging and eternal. It s that the mathematician (by intuition/perception of these mathematical objects, say) is trafficking with these eternal unchanging objects that makes the reasoning itself necessary. Let s admit it. There is something just a bit mystical about this line of thought but mysticism, after all, can be attractive. (To some people, anyway; other people find it repulsive.) Anyway, reasons like these convinced ancient philosophers that mathematical terms have to refer, and have to refer to strange objects. Reasons like these convince many contemporary philosophers as well. 6. Why the belief in mathematical objects existing is based on a mistake The evidence despite all the considerations I mentioned in section 5 that mathematical objects (of all sorts) don t exist is actually pretty good. Here s the general kind of strategy that 13

14 can be used to establish this. If we can tell a story about how the belief in something arose by a kind of error for example by thinking that a dream one had wasn t a dream or by confusing a play of shadows with the movements of a very quick magical being of some sort then that s evidence that the names and descriptions introduced to talk about those beings that one dreamed up (or thought one saw when one didn t) actually don t refer to anything at all. A story like this can be told about the emergence of the metaphysical belief in mathematical entities, although the errors involved are pretty sophisticated. Here s how it goes. Start with the original claim by Plato that it s impossible to say something true or false about what doesn t exist. This is a mistake; but it s an understandable mistake (based, as it is, on aboutness illusions). Given this mistake, however, it follows immediately that if mathematical statements are true, then they have to be about something. But do mathematical statements have to be true? Yes, because it s their being true that makes them valuable in empirical science. We use mathematical statements together with truths from empirical science to derive new results about an empirical subject matter that are true as well. We measure the sides of a rug; the length and width of that rug are empirical facts. We then use a bit of mathematics (that the area of a square is its length times its width) and deduce the area of the rug. That is, we reason with mathematical statements, and when we reason, if what we start with is true, then our conclusions have to be true as well. But we can t apply this model of reasoning if mathematical statements aren t true. (If it s not true that the area of a square is the product of its length and its width, then measuring the sides of a rug won t help us to deduce what its area is.) So the initial error here is that if the sentences of a discourse (like mathematics) are true, then the terms in those sentences have to refer. So now consider terms in Euclidean geometry 14

15 like point and line. These terms can t refer to the actual lines and points that people draw because those objects haven t the same properties that Euclidean lines and points have: the lines people draw have irregular widths, the points they make on paper take up varying amounts of space. (The sides of a rug aren t Euclidean lines.) The mistaken conclusion then is that the Euclidean terms, point and line must refer to something else (something else that the points and lines we actually draw remind us of, but they must be things which don t actually occur anywhere on Earth). Of course, Plato s initial error and it s an error that everyone made for a very long time rules out the possibility that the terms point and line, as used in geometry don t refer to anything at all. 7. It s because mathematical terms don t refer that mathematics is deductively tractable There is still this to explain. How can a discourse with terms that refer to nothing at all nevertheless be valuable to empirical science and in ordinary life the way mathematics is? I m going to explain this and more; I m going to explain that it s the fact that the terms in mathematics don t refer that makes mathematics valuable. The key to understanding this is to isolate the two ways that geometry is valuable, separate them, and explain them in different ways. First, recall a point I made in section 5 about how unusual mathematics is. You can start with a few definitions about points and lines and triangles (and so on), and a few axioms about the properties of these things (five in all), and then (if you re good enough or if you ve studied enough) you can go on to prove all sorts of other unexpected things about these objects. I described this as rather unique it s a phenomenon that doesn t really occur outside mathematics. 15

16 Here s a way of seeing why that is. Consider the actual points and lines that people draw, and imagine trying to define them, and deduce properties about them. It s extremely hard. One of the reasons it s so hard is that these actual lines and points are unruly: they aren t particularly straight (even if we re careful) and they range fairly widely in shape (even if we sharpen our pencils really really well). The effect of this is that not much easily follows about their properties. These objects aren t deductively tractable. A subject matter is deductively tractable if it s possible to deduce interesting consequences from a chosen set of statements. Most subject matters aren t deductively tractable; most of the concepts in most subject matters are unruly the way the concepts of actual lines and points are. In fact, the real discovery about Euclidean geometry was that by inventing certain odd concepts points that have no dimension, lines that are perfectly straight and have no width, and so on a small number of statements about them could be written down, and numerous interesting and unexpected results could be proven on that basis. It s very hard to discover a set of concepts like that. Creative (pure) mathematics is not unreasonably described as the invention of sets of concepts that are deductively tractable sets of concepts (and principles) on the basis of which interesting and unexpected results can be proven. Notice the point. Actual points and lines aren t deductively tractable: writing down principles about them isn t going to yield something that we can prove stuff from. It s precisely the fact that the points and lines of geometry are given properties that nothing real has that enables them to be part of the basis of a deductively-tractable mathematical science. I said before that most subject matters aren t deductively tractable. That isn t exactly right. Deductive tractability, actually, is a property of mathematics that s shared with certain interesting games. Some games come with rules by which moves in them can be executed. (In 16

17 this way, games are similar to drawing diagrams in Euclidean geometry: certain ways of drawing lines and curves are allowed and others aren t allowed.) What makes a game interesting is how unexpected the moves and countermoves can be; what kinds of intricate and unexpected scenarios the rules allow. So, for example, Tic Tac Toe is (all things considered) pretty boring. In fact, you can hit on an easy strategy that will prevent you ever losing a game. Checkers is a bit more interesting; and chess is quite complex. I m clearly thinking here of games the rules of which allow strategies and counterstrategies, and in which luck is miminized as a factor. These games (with pieces and rules) are interesting only if two conditions are fulfilled. First, the rules have to be ones that can be manipulated by people successfully. They can t be too hard to implement. (This is the role of tractability in the phrase deductive tractability.) But second, the rules have to result in something unexpected. Otherwise the games are boring. A deductive-tractability condition on mathematics is the same constraint as the one that s operative in games. 8. Why is mathematics empirically valuable? Still left to be explained is this: why should mathematics be empirically useful? After all, games pretty much aren t. We invent various properties that chess pieces have: knights can only move in an L-shape, kings can only move to adjacent squares. These are idealized properties we re attributing to chess pieces, of course. Real wooden pieces can t move at all, or people can push them in any direction whatsoever. The result is a game fun and interesting, but not something that s particularly relevant to knowledge about anything else. On the other hand, when we idealize or abstract the notions of points and lines in Euclidean geometry treat them as items that have no dimensions or no widths, etc., the result isn t just deductive 17

18 tractability (a fun process of drawing diagrams to prove mathematical theorems). It results in a subject matter with lots of empirical applications? But why? Why should the resulting discourse be of any use empirically? Well it isn t always. A lot of people don t know this, but an enormous amount of pure mathematics especially mathematics invented in the twentieth and twenty-first centuries has no empirical value whatsoever. It s striking but something of an historical accident that the earliest mathematics had empirical value. 2 What makes a mathematical discourse empirically valuable? Well, there s no simple answer to this question it very much depends on the particular mathematics and on the particular empirical science that the mathematics is applied to. But in the case of Euclidean geometry, it s pretty easy to see what it is about the mathematics that makes it valuable. The key is that the particular mathematical posits of Euclidean geometry: Euclidean points, Euclidean lines, and (generally) Euclidean figures, can be treated as items that actual points, lines and figures in sand, in drawings, etc., approximate. What approximation means is this. Take a triangle that you ve drawn on a piece of paper. What it means to say that the triangle you ve drawn approximates a Euclidean triangle can t be that it resembles the Euclidean triangle. (Something that s real can t resemble something that doesn t exist. What could that even mean? Things that don t exist don t resemble anything because they aren t anything.) What it means can be illustrated instead by this example: The narrower and straighter you draw actual lines, and the more carefully you measure the interior angles of the actual lines you draw, the closer the 2 Well, maybe not. There were games way back when, and some of these games were purely mathematical. Chess is purely mathematical its instantiation in wooden pieces is as significant to its status as a kind of mathematics as drawing diagrams on paper is to Euclidean geometry. But games of this sort weren t even recognized as mathematics precisely because the examples of mathematics that everyone was focused on (including philosophers) were empirically-valuable mathematics. 18

19 result will be to 180 degrees. This is true about everything you can prove in Euclidean geometry about Euclidean figures: there are ways of drawing real figures so that the results will more and more closely approximate results from Eucidean geometry. Notice that this explanation for why Euclidean geometry is empirically valuable avoids treating Euclidean objects as ones that exist (in some weird way) in particular, this explanation avoids talking about nonexistent entities resembling real things. Plato, by contrast, hypothesized that circles on earth resembled perfect circles the ones studied in Euclidean geometry. This would also explain why Euclidean geometry is empirically valuable. If things in domain B resembles things in domain A, then studying the things in domain A will shed light about the things in domain B. But an explanation like that requires the objects in domain A to exist. I ve given a different explanation: it doesn t focus on the objects of Euclidean geometry at all which is a good thing since those objects don t exist. It focuses instead on the truths of Euclidean geometry: it explains why the truths about the objects we draw on (relatively flat) pieces of paper or in sand, or whatever, are going to approximate the truths of Euclidean geometry under certain circumstances. 9. What does it mean to say that mathematical truths are necessary? There is one more thing left to explain. Recall that I mentioned that proofs in Euclidean geometry had a strange effect: They struck the people who followed them as necessarily true. This was once explained, I mentioned, by the idea that mathematical objects are themselves necessary. So on views like this the impression of necessity is due to some sort of perception of the objects that the deductions are about. 19

20 This is a deeply appealing suggestion. One of the things that makes it so appealing is its simplicity. Unfortunately, if we are convinced (on the grounds I ve suggested) that there are no Euclidean objects that there are no mathematical objects then this isn t going to work as an explanation. We need another one. And we have one. It has taken, literally, a few thousand years to get clear about what s going on here, with deduction in the Euclidean case, and generally but we ve done it. I m going to conclude this paper by sketching out the solution. The first step is to realize that the sense of necessity isn t one about the truths of Euclidean geometry that they must be true. Rather, it s a sense of necessity that: if the premises of Euclidean geometry are true, then the deduced results must be true. One thing that contributed to this insight was the discovery of non- Euclidean geometries in the late eighteenth/early nineteenth centuries. The premises of Euclidean geometry, that is, could be false. And the results therefore could be false too. (If you draw a triangle on a sphere, its angles always sum to greater than 180 degrees.) So the sensation of necessity was therefore recognized to lie in the impression of logical implication: that the premises of a deduction necessarily impel the conclusions one proves to follow from those premises. It isn t that mathematical truths are necessary; it s that if the premises are true, the conclusions have to be true. And this is because of a logical relationship between the mathematical premises and the mathematical conclusion: that the conclusion logically follows from the premises. And by the end of the nineteenth century/beginning of the twentieth century, we had done even more than this, we d apparently codified the logical principles by which mathematical reasoning (and arguably, all deductive reasoning) operates. Up until this point the logical rules 20

21 that we seemed to reason with had been implicit; no one had written them down explicitly. Now this had happened. But almost immediately, another discovery about logic intruded early in the twentieth century. This is that alternative logics are possible. Once the rules of logic are written down, it s clear that other (different) rules can be written down instead. It was thus realized (by some philosophers, anyway) that the impression of necessity that we experience when contemplating the logical principles we re committed to, and that we don t experience when contemplating some other set of logical principles (that we re not committed to) is a cognitive illusion too. There is no metaphysical necessity to logic any more than to mathematics. The deductive rules that we use may be valuable up to a point, they may be the ones we re always cognitively inclined to believe are the right deductive rules. But that doesn t mean that they are. The story I ve just told about where our impression of necessity is coming from is a complex story containing both elements from cognitive science, as well as normative considerations about what forms of reasoning are optimal for the sciences and for our ordinary lives. And, it s simply not possible to get into all the details of the story here. My only point in bringing it up at all is to show that a certain kind of explanation isn t called for. The explanation I m thinking of (that isn t needed) is the one I ve already mentioned: our impression of the necessity of mathematical proof is due to the necessity of the objects talked about in those proofs, the necessity of mathematical objects. Thinking that an explanation like this is needed can also drive people to believe that mathematical terms must refer that they can t be fictions. I ve just shown that we don t need this explanation; contemporary cognitive science (and philosophical insights about the nature of logic) enable us to tell a quite different story. But it s worth mentioning that the explanation I m rejecting was one philosophers believed (one or 21

22 another version of) for a long time. And it s also worth mentioning that it generated puzzles that philosophers worried about for centuries, that seemed insolvable, and that philosophers still worry about. (See Benacerraf 1965.) A version of a puzzle that Plato grappled with is this: how do we manage to experience mathematical objects when they re not in space and time? (Plato s answer, nearly enough, is recollection of experiences from an earlier life (reincarnation): We ve all had previous lives where in some way we were in a position to actually perceive mathematical objects. See Plato 1963b.) The modern story I want to tell is better. This is one that takes its inspiration from a description of our psychological capacities to tell truths about things that don t exist, truths that are nevertheless valuable. 10. Fictions in mathematics compared to fictions in fiction. The most obvious cases of fictions the most famous cases of fictions Sherlock Holmes, Mickey Mouse, Hercules, and so on are also in some ways the most misleading cases of fiction. For they really are cases where the fictions pretty much play only an entertainment role in our lives. Focusing on cases like these, therefore, gives the impression that fictions are only involved in entertainment. I ve tried to show two things in this paper. First, I ve tried to show how valuable (invaluable) fictions are outside of their role in entertainment. And second, I ve tried to convey how philosophically puzzling fictions are. Furthermore, the philosophical puzzles that fictions raise are deep puzzles. They are deep not only in the sense that it has taken over two thousand years to finally understand them; but they are also deep in the sense that understanding them has called for the development of a lot of intellectual capital. It wasn t a matter of solving a logical conundrum by detecting a false step in our reasoning (in Plato s reasoning, for example, as quoted in the opening pages of this paper). Rather, it was a matter of 22

23 getting clear about how reasoning actually works, getting clear about aspects of how our cognitive faculties enable us to engage in imaginative thinking (and talking), and getting clear about aspects of how our language works too. But these are the sorts of things it takes centuries to work out. Further reading Azzouni, Jody Applying mathematics: An attempt to design a philosophical problem. The Monist 83:2, Azzouni, Jody Proof and ontology in Euclidean mathematics. In New trends in the history and philosophy of mathematics, ed. Tinne Hoff Kjeldsen, Stig Andur Pedersen, Lise Mariane Sonne-Hansen, Denmark: University of Southern Denmark. Azzouni, Jody The compulsion to believe: Logical inference and normativity. In (Gerhard Preyer, Georg Peter, eds. Philosophy of mathematics: Set theory, measuring theories, and nominalism, Frankfurt: Ontos Verlag. Azzouni, Jody Why do informal proofs conform to formal norms? Foundations of Science 14: Azzouni, Jody Talking about nothing: Numbers, hallucinations, and fictions. Oxford: Oxford University Press. Benacerraf, Paul Mathematical truth. Journal of Philosophy 70: Plato. 1963a. Sophist. In Plato: The collected dialogues, ed. Edith Hamilton and Huntington Cairns. Princeton, N.J.: Princeton University Press, Plato. 1963b. Phaedo. In Plato: The collected dialogues, ed. Edith Hamilton and Huntington Cairns. Princeton, N.J.: Princeton University Press, Plato. 1963c. Meno. In Plato: The collected dialogues, ed. Edith Hamilton and Huntington Cairns. Princeton, N.J.: Princeton University Press,