Philosophy of Mathematics Paper

This is a paper I wrote for my Philosophy of Mathematics class a few years back. I'm including it because it is entirely secular and any non-religious readers I may have might want something of a non-religious nature. The paper discusses the fictionalist view of mathematics against a realist view. Fictionalism claims that mathematics is a fiction. Realism claims that mathematical objects (the number 2, for instance) are real. There are sub-schools of each. I am here defending fictionalism. If you are interested in finding out more about each, you can look at my Works Cited or you can Google/Wikipedia keywords.

Applicability and Mathematical Predictions:
Can Metaphors Save Fictionalism?

Bridges stand and mathematics tells us how. The applicability of mathematics to the physical world is fictionalism’s largest obstacle, and philosophers of mathematics frequently debate whether mathematics’ ability to describe and explain the physical world is enough to defeat the fictionalist agenda. In this paper I will attempt to address this problem briefly; however, I think the problem of applicability covers more than just that bridges do stand, but that bridges will stand. Mathematics is used as more than just a descriptive tool, but as a predictive one as well. That a fiction—fable, metaphor, novel, or heuristic—can describe what happens in the world is one matter and the more easily addressed. Whether it can predict what will happen is another matter and the less easily addressed. If fiction does not have predictive abilities analogous to those of mathematics, then fictionalism cannot be an accurate account of mathematics. I will argue in the course of this paper that what I call a metaphorical-heuristic approach to fictionalism will demonstrate how fiction can predict the world, and how any discrepancy between fiction’s and mathematics’ predictive capabilities are not fatal to the analogy.
Defeating what I might call brute fictionalism is fairly easy because of the descriptive applicability problem. Brute fictionalism would say that mathematical propositions are true in terms of the fiction in which they exist, this fiction being the particular mathematical canon in use. These same propositions are not true in any other contexts, especially the context of the real world, in the same way that the proposition “Wemmick doted on his father” is true only in terms of the fiction Great Expectations. The mathematician does not need to be committed to the existence of mathematical objects outside of the fiction because the truth predicate is simply limited to the terms of the fiction. Mixed contexts create a serious problem for this view. Consider that mathematical analysis of wind pressures, traffic weight, material strength, and mechanical advantage can determine why a particular bridge fell when it did, or that measuring the dimensions of a piano and the doors of a house will determine the best way to move the piano into the living room from the driveway. The problem here is that mathematics, in order to apply to the real world, must be in some sense true outside of the mathematical ‘fiction.’
A more subtle form of fictionalism would allow mathematical propositions to be in some sense true outside of the mathematical fiction, or canon. I call this metaphorical-heuristic fictionalism, which is really just a loose grouping of Yablo’s mathematics-as-metaphor and Balaguer’s mathematics-as-heuristic ideas.[1] I am aware that these are not precisely the same view, but they have enough in common that they are equivalent for my purposes: both take mathematics to be a useful tool that gets to some truth outside of itself in explaining the real world while not being literally true. Yablo describes mathematics as a metaphor, where a metaphor “is an utterance that represents its objects as being like so: the way that they would need to be to make it pretence-worthy—or, more neutrally, sayable—in a game that the utterance itself suggests.”[2] Yablo’s metaphor is a phrase that literally means something to which the speaker is not committed, but also means something non-literal to which the speaker is committed, in such a way that the audience should understand this duality. Mathematics on this view is then not literally true but non-literally true. Balaguer describes mathematics as a heuristic device: “mathematics is relevant not to the operation of the physical world, but to our understanding of the physical world.”[3] Mathematics is not actually true to Balaguer, but simply beneficial to our comprehension of the world. This view is of course different from Yablo’s; Yablo gives some kind of truth to mathematics where Balaguer is more indifferent. However, the point of overlap that I am concerned about is that both focus on mathematics’ ability to help us understand what is going on in the real world without making literal claims of truth. I would tend to side with Yablo on this question, but I do think the idea of a heuristic is relevant to this approach.
The metaphorical-heuristic approach could address the mixed contexts problem because it is not true only within the context of its own fiction; rather, it models a true proposition of the outside world while still remaining fictional. When we analyze why a particular bridge is falling apart, it is useful to look at the forces acting upon the bridge in terms of mathematics; however, the mathematics needs only act as a heuristic device or metaphor for those forces, and not refer to mathematical objects existing in some non-causal way.
In “Mathematics and Bleak House,” John P. Burgess objects to a metaphorical approach to fictionalism on grounds that I think are generally ill-founded.[4] Burgess’ objection to metaphor runs as follows: “Metaphor is of course not a genre of fiction but a figure of speech, and I think that to speak as Yablo does of a metaphor running on for volumes and volumes and volumes is to stretch the concept of ‘metaphor’ well beyond the breaking point.”[5] Whether or not Yablo’s metaphor is past the breaking point aside, the approach to fictionalism that I am suggesting is best does not require that the metaphor “runs on” for so many volumes because mathematics need not be one massive metaphor; it is a series of metaphors. Each mathematical object and each relationship is a separate metaphor.[6] This would satisfy all of Burgess’ requirements of fiction: some metaphors are attributable to authors and some are traditional; some metaphors are recycled and altered over time; some entities reappear from metaphor to metaphor and are “beings of a different order.”[7] While metaphor may not be a genre of fiction, it is an element of fiction and perhaps fiction is a metaphor: William Golding was not committed to the literal truth of The Lord of the Flies, but he was committed to the society it was supposed to represent.[8] I generally find Burgess’ objection to metaphorical fictionalism faulty, and will continue to use it.
So metaphorical-heuristic fictionalism can explain how a fictional mathematics can describe and explain physical events and objects. It is less obvious that fictional mathematics can predict events in the physical world. An argument that such mathematics cannot might follow two general routes. The first is that the analogy between fiction and mathematics is not accurate. We can and often have safely assumed that mathematics can predict physical phenomena. When civil engineers build bridges, they can fairly accurately predict how long the bridges will stand; when NASA technicians launch spacecraft, they can fairly accurately predict how long it will take for the shuttle to exit the atmosphere; when artillery-man fired their canons, they could generally determine where the shells would land. Using mathematics to determine any course of action almost always requires that the mathematics has some predictive power over not just its own propositions, but events in the real world. Where mathematics fails to predict something accurately, it is often not the fault of the mathematics itself but a poorly measured or unnoticed variable in the equation. Fiction, on the other hand, seems upon first examination to lack this predictive power. The Mouse that Roared, anti-fictionalists might argue, has never given political scientists the tools to predict European politics and The Lord of the Flies has never given child psychiatrists an accurate predictive model for playground society. Since fiction has never predicted the future, anti-fictionalists might argue, it is more likely that the fault lies not with some mistake in the variables, but the fiction itself. Because prediction is crucial to the question of the applicability of mathematics, and because fiction cannot accurately model the predictive capabilities of mathematics, fictionalism of even a metaphorical-heuristic variety cannot work.
The second argument against fictionalism on grounds on predictability explains why there is this disconnect in the analogy. The problem may rest in the idea of construction. Fiction is constructed. Specifically, metaphors and other heuristic fictions are constructed to describe something which already exists. This means that they are reactive, and, intuitively, a reactive system is not designed to be proactive or predictive. For instance, a common metaphor like “a growing number of these leaks can be traced to Starr’s office”[9] is coined simply to describe some event in the real world that is easier to say and understand than the more literal alternative of “leaks can be traced to Starr’s office with increasing frequency.”[10] It has no predictive power. If mathematics is only designed to be a heuristic in understanding mathematics, then it can only react to knowledge we already have. No new knowledge can necessarily be modeled by mathematics, and therefore mathematics should not be capable of prediction. Since mathematics clearly is capable of prediction, it must be more than a reactive construct, and so it cannot be a fiction of any stripe, brute or metaphorical-heuristic.
Neither of these approaches is fatal to fictionalism, and I will demonstrate this with examples of accurate predictions that assorted fictions have made. To use one of Balaguer’s examples,[11] Animal Farm accurately models the history of the Russian Revolution. However, it also models the history of many other communist or popular revolutions that occurred after its publication, such as those in Cuba, the Congo, and Iran. The specifics are not the same on a literal level, but they are metaphorically fairly accurate. Another of Orwell’s books, Nineteen Eighty-Four, concerns the development of Newspeak in Oceania, which seems prescient of the rhetoric of modern politicians and propagandists, and the telescreens are remarkably similar to the camera-surveillance common in modern urban culture. Mary W Shelley wrote Frankenstein long before life’s creation by artificial means was possible, but now that it is, many of the philosophical debates in the text concerning responsibility, ownership, and parentage are relevant, and its more metaphorical meanings are mapping the course of discussion on pollution, nuclear energy and weapons, GE, and AI. More personally, Sir Philip Sidney’s Astrophil and Stella foreshadows the psychology of many unrequited lovers since his text’s publication in the late 1500s, and Lord Byron’s Don Juan modeled the melancholy individual that the author would become only after writing the epic poem. A brief survey of English literature demonstrates that fictions have made accurate predictions, contrary to the argument outlined above.
The problem with the anti-fictionalist argument here is that a reactive construction can, in theory, make predictions. So long as the system that the metaphor or heuristic describes is organized and at least moderately understood, the metaphor or heuristic should be able to make those predictions. Consider code-breaking. If a series of information encrypted into a certain code is used to create a code-translating procedure, then that procedure should be able to break another series of information encrypted in the same code and to accurately encrypt information into the code. The procedure should be able to predict how a certain piece of information would appear when encrypted, even though it is a reactive construction. There is no reason to believe that a reactive construction cannot make predictions about an organized system.
The anti-fictionalist might then misguidedly object that some fiction can make somewhat accurate predictions some of the time, but that this is still not sufficient to posit an analogy between fiction and mathematics because most fiction makes mostly inaccurate predictions most of the time, and the some fictions and the some of the time are mostly not somewhat accurate, but entirely inaccurate. I agree that most fiction makes inaccurate predictions, but I do not agree that it is fatal. Most fiction is admittedly imperfect, even metaphorical ones. Creating a mathematics that is as near to perfection as is theoretically possible has been the occupation of millions of skilled thinkers for thousands of years. No metaphor and no piece of literature have had such a devoted parentage. We cannot expect the sorts of fictions that we are dealing with in metaphor or literature to even approach the degree of perfection that mathematics has reached. Instead, we must imagine a theoretically ideal fiction that has had hundreds of wise and gifted authors tinker with it over thousands of years so that it has reached a state of true genius, capable of many predictions of both the vastly political and intimately personal. At this point we may say that this fiction is comparable to mathematics. Of course, such a fiction does not exist, but, very importantly, it could exist. On the other side, just as imperfect fiction fails to predict the world accurately, imperfect mathematics also fails to predict the world accurately. Imperfect mathematics is hard to find, but it has existed; it must have, since mathematics is occasionally refined when certain methods of proof are discredited. In these cases, the imperfect mathematics would occasionally make erroneous predictions.
Even this theoretically near-perfect fiction could conceivably make false predictions, the anti-fictionalist might say, and the anti-fictionalist would be correct if the fiction had as its subject matter human activity. This is because authors often choose as subject matter problems that are not even theoretically predictable. Presumably, the majority of applications of mathematics are theoretically predictable. There are perhaps exceptions in quantum physics and social sciences, but generally people use mathematics to predict events that have only one physically possible outcome. The goal is to figure out what that outcome happens to be. The subject matter of most fictions, however, has more than one physically possible outcome. I am assuming that humans, and possibly though not necessarily animals, have free will. This means that until the point a decision is made, that decision is undetermined and therefore unpredictable. This does not mean that likely decisions can not be predicted, but the actual decision cannot be predicted with perfect accuracy. Since each fiction will contain at least dozens and potentially millions of decisions, the final result is much harder to predict with perfect accuracy. If a fiction were written that contained no free will decisions at all—say a fiction concerning the life of a particular particle in a particular environment—then that fiction can be expected to make accurate predictions about the actions of any particle that exists or will exist in the real world that has the same properties and environment. That no one has written such a fiction does not mean that it could not theoretically exist. It just means that no one has been motivated to write that fiction because it is boring or unfit for sale or, likely, both.[12] The upshot of all of this is that fiction would be analogous to mathematics if it had the same devoted parentage and the same subject matter. The fact that it has neither does not mean that it theoretically could not have either. An ideal fiction about a theoretically predictable subject could then be a working analogy for mathematics.
A crucial objection that fictionalism must face is applicability problems, and while mathematics’ descriptive and explanatory abilities are incorporated into metaphorical-heuristic fictionalism where they are not in brute fictionalism, an anti-fictionalist might object that fiction does not share mathematics’ predictive abilities. I believe that fiction does make accurate predictions some of the time, and those times it does not are not fatal to the analogy because fiction is further from the ideal than mathematics is and because the chosen subject matter of fiction can be predicted with complete accuracy even theoretically. Fictionalism, even of a metaphorical-heuristic variety, has other tough objections to face, such as Burgess’ accusations that the difference between fictionalism’s account and realism’s account of mathematics is essentially meaningless,[13] and these objections may well be fatal. However, it is my understanding that metaphorical-heuristic fictionalism can adequately face the applicability problems concerning description, explanation, and prediction.





[1] Stephen Yablo, “Apriority and Existence,” in New Essays on the A Priori, ed. Paul Boghassian and Christopher Peacoche (New York: Oxford U. P., 2000), 198-228; Mark Balaguer, “A Fictionalist Account of the Indispensable Applications of Mathematics,” Philosophical Studies 83, (1996): 291-314.
[2] Yablo, “Apriority,” 213.
[3] Balaguer, “Fictionalist Account,” 298.
[4] John P. Burgess, “Mathematics and Bleak House,” Philosophica Mathematica 12, no. 3, (2004): 18-36.
[5] Burgess, “Bleak House,” 21.
[6] We are already aware that mathematical notation is symbolic. ‘2’ in no way visually suggests duality, and ‘=’ does so only metaphorically in the representation of two equal line segments.
[7] Burgess, “Bleak House,” 22. Burgess argues that fables, having these attributes, are better comparisons to mathematics than metaphors or novels, but I think that metaphors fulfill these requirements. Various famous writers coined their own metaphors which have since been reused, while other metaphors—“the leg of a table”—are traditional. Metaphors are adopted and adapted frequently, like fables and mathematical theorems, and they contain recurrent and alien entities: reservoirs of abstract objects—patience, courage, and self-control, for instance—appear very frequently and are certainly “beings of a different order.”
[8] Burgess also objects to the term heuristic, but that rests primarily in his definition of it, which differs radically from my own. To Burgess, a heuristic fictionalist says that mathematicians have always meant mathematics to be understood non-literally. Most mathematicians do not intend this, and Burgess argues on these grounds that revolutionary fictionalism, which seeks to mend mathematics, if preferable to heuristic fictionalism. Since this is not my definition of heuristic fictionalism, this objection hardy applies to my version. Anyway, as Yablo demonstrates that people use metaphors frequently without knowing that they are using them non-literally, Burgess’ objection does not seem relevant even in terms of his own definition. Since the purpose of my paper is not to criticize Burgess’ understanding of fictionalism, however, I will not dwell on this point.
[9] Yablo, “Apriority,” 214.
[10] Of course, ‘growing number’ is not the only metaphor in that sentence: ‘leaks’ and ‘traced’ are also metaphors, and ‘Starr’s office’ is a metonym.
[11] Balaguer, “Fictionalist Account,” 306.
[12] Even if I am wrong in assuming that free will exists, human decision-making may well be beyond human understanding, and thus beyond human ability to construct, if only because of the number of variables that are unnoticed or improperly measured. Not only does this seem likely to me, I also think that human behaviour, if not free, would still only be predictable with a perfected mind-reading device, be it neurological or psychic. And if this does occur, then good fiction will be able to make accurate predictions and the analogy is strengthened. Whether human activity is unpredictable because of mental privacy and therefore a theoretically unpredictable subject, or human activity is predictable via mind-reading and therefore a theoretically predictable subject, my argument remains in the same position as it would if humans had free will.
[13] Burgess, “Bleak House,” 35.

Works Cited

Balaguer, Mark. “A Fictionalist Account of the Indispensable Applications of Mathematics.” Philosophical Studies 83, (1996): 291-314.

Burgess, John P. “Mathematics and Bleak House.” Philosophica Mathematica 12, no. 3 (2004): 18-36.

Yablo, Stephen. “Apriority and Existence.” In New Essays on the A Priori, edited by Paul Boghossian and Christopher Peacoche, 197-228. New York: Oxford U. P., 2000.