Useful Fictions 

“I know who I am,” said Don Quixote, “and who I may be, if I choose….”

(Note: Most of this lecture is a condensed and highly dumbed-down version of a lecture given by UI Math professor Robert Ely.)

The first, most obvious and most consistent theme running throughout the novel is that Don Quixote has been driven insane by consuming too much fantasy literature.  Specifically, he’s been reading Medieval Romances; the closest to these that we have read were The Grail and Sir Gawain and the Green Knight, but it’s clear DQ’s novels were closer to Steven King or comic books, or, to make the comparison with more contemporary sources, you might imagine he’s addicted to gaming or watching The Lord of the Rings or Twighlight etc. (Note: I would have used watching Game of Thrones as an example, but that's different; I can quit any time.  No, seriously, I can, or could, if I wanted to.)

Ironically, in some ways this theme harkens back to an argument Plato makes in Chapter X of The Republic:  poetry (or fiction; he’s explicitly discusses Homer) is so beautiful, so seductive, that it ultimately corrupts our understanding of the actual Truth, and, of course, Plato is all about "the Truth"; and so Plato argues that most poetry would be banned in the republic.  I say "ironically" because on the surface Plato and Cervantes have completely opposite ideas about just where or what this thing called "Truth" is (see Don Quixote: Science, Sanity & The First Novel).

But this is only the first -- what we might call the "Modern" -- layer or perspective on this issue.  The novel then asks us is there perhaps still great wisdom in this insanity?  This is perhaps the main question Cervantes' novel poses.   Is Don Quixote just a madman and/or a fool, or is he also a hero?  Or, is he both?  Or, might one need to have an element of insanity to become a hero? Is Cervantes warning us of the dangers of believing in fictional ideals (in accord with Plato), or are we meant to learn from Don Quixote and realize that fictions are useful and even necessary to our own sanity?

We’re interested in how Don Quixote’s view of reality and fantasy – how the two relate to one another – seem to mirror emerging attitudes toward what Liebniz (one of the inventors of the calculus) will later call “useful fictions” in the realm of math.   Understanding these changing mathematical attitudes should help us understand Cervantes’ and Don Quixote’s fantastical view of reality.

How Math and Reality Relate: A Super-Brief History:

Primitive Math: In its simplest and earliest uses, mathematical concepts and formulas represented observable facts about the world.  We all agree that 2+2=4 or 2-2=0 because we can see it or demonstrate it visibly with concrete, "real" examples from the "real" world.  Thus, addition and subtraction merely offer a symbolic language for making statements about the observable world, that is, about things we could all see in the physical, sensory world and readily agree were true (so, primitive math represents Plato's "sensible" knowledge). 

Greek Geometry: Similarly, Classical and Hellenistic Greek mathematicians were generally concerned with making sure the objects they represented mathematically and discussed were real, in the sense that they could be constructed and had real-world referents, like actual objects: circles, triangles, parabolas etc. 

Note, this does not mean that they thought that mathematical objects were tangible.  As we've discussed, for Plato, they are universal forms (Ideal Forms) that real objects participate in.  But, still, intelligible knowledge (math formulas) were Ideal forms of sensible knowledge (again, things we could see/perceive in the physical world).

Example: π  and   are representations of physical objects (things we can observe in the physical world): circles and triangles, or relationships between various elements in those objects. 

Example: Archimedes, to cut a given sphere by a plane so that the volumes of the pieces are in a given ratio. Translated algebraically, this entails solving 3ax² - x³ = 4a²b.  He finds the answer by drawing a particular parabola and hyperbola and finding their intersection.  That’s the solution to a cubic equation for the Greeks. (Note: at this point in the lecture Tom regrets that he a) took geometry over 30 years ago and b) dropped out of the class midway through and, thus, c) has no idea what this paragraph is about.  He did, however, find a picture on the internets, and it looks like this:)
 

Example: Classical era math would not use the variable (the "x" or "y" etc. in a math formula), such as 4y = X, because the variable was too abstract, and so they would represent each leg of the concept 4(1)=4, 4(2)=8, 4(3)=12. 

Important, Key Example: Classic math would not use the concept of 0 or negative numbers because how could a non existent cow exist (0 cows) or how you “have” or “see/perceive” a “negative cow” -1 cow?  So, here's how to think like a Greek mathematician: try to milk -1 cows.  Try to feed a village on 0 cows.  Hint: you can't!  Why?  Because, obviously, 0 and -1 cows are absurd; thus, only a crazy fool would think you could use these concepts in math. (And recall that the concept of 0 will be developed by Arabic mathematicians and introduced to Europe through Spain.)

Early Modern Math:  In 1500-1800 we have a middle period of mixed ontology of mathematical objects—in which some but not all mathematical objects are fictitious.  This is in contrast with Greek mathematics. Of these fictitious entities there are two kinds, “systematic fictions” (my [Robert Ely's] term) and “useful fictions” (Leibniz’ term).

For early modern mathematicians, systematic fictions arose through a general algebraic method (credit to Al-Khwarizmi and the other Islamic algebraists), which one applies without concern about the ontological status (whether they are "real").  For instance, when working on solutions of cubic equations of the form x³ = 24x + 3, ((c/3)³ is larger than (d/2)², Rafael Bombelli (1526-1572) finds what we now call "imaginary solutions" (he called, say, 2+3i “2 p di m 3”).  He presents systematic rules for calculating with these, although he notes that “the whole matter seems to rest on sophistry rather than on truth.”  

Other examples include negative numbers and various transcendental numbers (what the heck is e^ipi?). 

Calculus and Useful Fictions:  In response to Zeno’s Dichotomy paradox (you can’t reach the wall, because you must first go halfway, and before that, halfway of the halfway, and so on to infinity; see: http://plato.stanford.edu/entries/paradox-zeno/#Dic), Aristotle says that we can sensibly only talk about potential infinity, not actual infinity.  In other words, a segment can be divided arbitrarily, and never reaching atoms, but we can never talk about the result of this infinite division process. Thus, we cannot posit the existence of infinitely small quantities. We can talk about these heuristically, but our proofs can never rely on them (and the Greeks underwent significant contortions in their proofs to avoid them). This is worth thinking about for a minute: we can use the concept of infinity even though we accept that infinity does not exist. This is part of what Plato was after when arguing there are "ideals" (or true ideas) that do not exist in the physical world in which we live.

All of this new way of thinking about mathematical concepts and their relationship to reality is leading us toward calculus (and, believe it or not, and I know you don't believe it, to Don Quixote).

Why Everyone Wanted to Invent Calculus: The problem with sticking to math that only describes objects is that this math does not describe change, and that's what everyone wanted to chart mathematically, changes like acceleration, velocity, ballistics as an arrow or cannonball flies through the air. All of these require breaking open Zeno's paradox: the ability to measure an infinitesimally small moment in time and space.  That is, if I want to know how fast an accelerating object is going at a given moment, that "given moment" can be reduced, "split in half", over and over again...infinitely. Dammit Zeno!

Solution? Fictional numbers.

Leibniz (1646-1716) describes the infinitesimals that he and many mathematicians in the 1600s used as “useful fictions.” These are mathematical objects that have no real-world referents, but through the use of which we are able to discover new mathematical truths about the real world (useful!). This includes all of calculus, which describes the scientific world with unparalleled effectiveness, but which in the 18th century entirely relied upon infinitesimals (indeed, in Leibniz’ day it was called “the infinitesimal calculus”). 

Calculus measures reality via non-existent concepts:   Real Thing --> Not Real Thing  --> Truth About Real Thing.

These mathematical "useful fictions" are like metaphors:  they are not "real" but they are in many ways "true": or the concept (metaphor, calculus) illuminates reality.

Details about the way in which these are “fictions” for Leibniz:  He justifies them by appeal to imaginary numbers. (A good, short description of "Imaginary Numbers" is here.)  He praises their capacity to achieve universal harmony by the greatest possible systematization, which is a central trait of infinitesimals also. Infinitesimals unify and extend methods for determining quadratures, tangents, etc. to more kinds of functions.

Deleuze: “For him [Leibniz] [infintesimals, indeed differential calculus] is merely a convenient and well-founded fiction. In this respect the question is not that of existing infinity or of the infinitesimal… The question is rather: Is differential calculus adequate for infinitesimal things?...Differential relations intervene only in order to extract a clear perception from minute, obscure perceptions. Thus the calculus is precisely a psychic mechanism, and if it is fictive, it is in the sense that this mechanism belongs to a hallucinatory perception. Calculus surely has a psychological reality, but here it is deprived of physical reality.”

So calculus is a set of symbols mirroring thoughts which can be "true" but do not exist in the physical sense.  Note how this relates us back to Plato's "Intelligible vs. Sensible knowledge: it is essentially a more radical version of Plato's theory, since Plato was interested in how symbols (mathematical formulas) could represent ideal physical, geometric objects (triangles, circles etc.), but here we find those symbols can simply represent an ideal that doesn't exist at all in the sensible/physical realm...but can be used to solve sensible/real world problems!

In other words, you can use this purely abstract, symbolic way of thinking to figure out things like trajectory and acceleration, which are actions, not physical "things"...and this abstract way of thinking helps you understand the behavior of physical things.  But, most importantly for us, in the middle of this process of understanding real things, "the real world", we are forced to used imaginative, "fictional" thinking (in this case: calculus).

Useful indeed—determines the equations for the motions of the planets, the isochrones, tautochrone, etc., the behavior of vibrating strings, optics, fluid dynamics, mechanics and statics (bridges), catenary.

How does this all relate to Don Quixote?  See: Don Quixote Discussion Questions 1