Infinities in Literature and Mathematics

Infinity art 2-thumb-475x346-243418During this past month, I was having a conversation with a couple of friends who study Latin-American Literature, and I noticed that they were having a hard time understanding how a literary work can have infinite critical interpretations, while at the same time not all its interpretations are critical. Apparently they found this to be contradictory.

I was shocked by their confusion, because to me the idea in question is almost self-evident. But later I came to acknowledge the fact that my friends, who are schooled in the humanities, have little if any notion of the mathematical idea of the infinite. For that reason, I suggest in this essay that the humanities can learn something from the concept of infinities in mathematics.

The problem with Romanticism’s concept of the Infinite

According to Alain Badiou, the history of Western philosophy can be divided into two great periods. First, the era before and including Kant, when mathematical reasoning was considered a singular way of thinking that interrupted the predominance of opinion — or, to put it in philosophical jargon, of Doxa — in philosophical reasoning. And second, the post-Kant era, which gave birth to Romanticism, which was consummated by Hegel, whose philosophical system is powered at its core by the schism between math and philosophy. Following Badiou [1], this schism also lies at the core of 19th century positivism and modern radical empiricism — because arguments put forth by these movements just flip to the other side of the same coin without really solving the problem — and has greatly impacted contemporary thinking, especially in the humanities.

In fact, for Hegel, nature is death Logos, or ideas, and as such it has no real value for human understanding. For him, math was part of nature, and therefore it too has no real meaning for human understanding of the world. He even went as far as to criticize the use of math in physics. For example, he once wrote:

“Whoever approaches this part of physics [Newtonian physics] soon realizes that it is rather a mechanics than a physics of the heavens and that astronomy’s laws derive their origin from another science, from mathematics, rather than actually having been teased from nature or constructed by reason […] All physicists before him [Newton] regarded the relationship between the planets and the sun as a true one, i.e. as a real and physical force. What Newton did was to compare the magnitude of gravity shown by experience for bodies forming part of our earth with the magnitude of celestial motions; he then proceeded to deal with everything else using mathematical reasoning from geometry and calculus. We must be especially wary of this binding of physics with mathematics; we must beware of confusing pure mathematical grounds with physical ones; namely, of blindly taking lines deployed by geometry as helps to construction in proving its theorems for forces or force directions.” [2]

He goes on to describe gravity in terms of planetary needs and desires, as if planets  themselves were capable of some kind of reasoning. Having said that, Hegel declares that the real understanding of the infinity cannot come from math, but has to originate from philosophy (as he understood it).

Following this principle of separating math from philosophy, Hegel develops an idea about infinity that identifies it with the Unity and the Absolute. This idea is widely spread today — it is held by my literary friends, to say, among others — and it has closed the possibility for an entire discipline to grasp the idea of multiple infinities.

For Badiou, the true reason lying behind this schism between math and contemporary humanities is to be found in the idea of historicism. According to historicism, the content of every idea is inescapably attached to the historical context in which it originated. Therefore, all “truths” are such only within a particular historical framework. While this idea may be true for many areas of inquiry, it is definitively not true for mathematical reasoning, precisely because the latter is a type of human reasoning that transcends the bindings of history. In other words, math has shown us that human thinking is not condemned to always be a matter of opinion.

Mathematical Infinities

Thinking about infinity can be a source of really severe headaches. For example, let’s think about integers (…-4, -2, -1, 0, 1, 2, 3, 4… and so on). It is fascinating to note that there are as many integers as there are prime numbers (numbers that can only be divided by 1 and by themselves, as 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97…). Surely this is fascinating, but it is also puzzling! I mean, between 1 and 100 we can find 100 integers and only 25 primes. Then, how is it possible to argue that infinity contains as many integers as it contains primes?

There is an easy way to prove this, and it was proposed by 19th century mathematician Georg Cantor. He first stated that it is possible to count two different sets of things by matching a one to one correspondence between their elements. For example, let’s imagine that I ignore how many fingers I have in my right hand, as well as how many platonic solids there are. Let’s suppose, too, that I never attended school and I don’t how to count. Nevertheless, I’m a stubborn person and still find a way to know how many platonic solids there are in relation to the fingers in my right hand, by the simple means of matching a one on one correspondence between them. One of the possible outcomes of this exercise will be as follows:

fig1

By doing so, now I know that there are at least as many platonic solids as fingers in my right hand.

What Cantor discovered is that it is possible to measure infinities in exactly the same way; as shown in the picture below:

fig2

Thanks to this method, Cantor was able to determine that in fact, there are as many integers, as there are prime numbers. Cantor named these kind of infinities countable infinities — although, as Jame Grime has pointed out, they would be better named enlisting infinities [3]and proved that every countable infinity is as big as the infinity of the integers.

In the next picture, it is shown how it is possible to enlist fractional numbers, too. Therefore, it is possible to assert that there are as many fractional numbers as primes and integers:

fig3

This observation in itself represented a big breakthrough, but what Cantor also found is that there are some types of infinities that simply cannot be enlisted. He called them uncountable infinities. The real numbers are an example of these kind of infinities. To be able to understand this principle, I unsuccessfully tried to enlist all the numbers between 0 and 1. As you can see in the picture:

fig4

By doing a simple exercise, Cantor also proved that in any given list of real numbers, we could always find a number that is not contained in it; and that we can do this by following three simple rules: 1) In writing the first digit of our number we will pay attention to the first digit of the first number of the list; then, in writing the second one, our attention should be on the second digit of the second number of the list… and so on; 2) if the number we are looking at is not a 1, we are going to write down a 1; 3) but if the number we are looking at is a 1, then we are going to write down a 2. The result of this procedure will always be a new number. Don’t believe me? Try it!

fig5

After that, Cantor showed that the uncountable infinities are in fact bigger than the countable ones! So, there are some infinities that are bigger than others! This clearly goes against Hegel’s idea of a Unique and Absolute Infinity.

But hold your breath. The most exciting part of the story is yet to be told, because Cantor himself was convinced that there should be something between the countable infinities and the uncountable ones. This idea he called the hypothesis of the continuum, and he could never prove it. In 1938 Kurt Gödel proved that it is actually impossible to show that this hypothesis is false; and, even more amazingly, in 1963 Paul Cohen proved that it is also impossible to prove it correct! So we are condemned to live with this ignorance forever!

Now that we have learned some basics of what mathematicians think when they think about the infinity we can finally discuss how this idea can affect the philosophy of literature.

Infinite Interpretations of Literary Works

One of the main things critics of literature do is to interpret literary works. In the past, the notion that a literary work, say a novel, only has one real meaning was widely accepted. But, with the dawn of positivism and the ascension of hermeneutics in the humanities, that notion was left behind. Nowadays almost every critic of literature will argue that a novel has as many meanings as there are interpreters. Most of them will even argue that a novel has, in fact, potentially infinite meanings.

But, if every reader can find different valid meanings in a novel, then, how is it that the work of the literary critic is still relevant? According to French philosopher Paul Recoeur [4], most of the interpretations given to a novel by ordinary people are just quick guesses based in conjectures. For sure, they are valid, but clearly are not as valuable as that of the literary critic, who spends most of her time studying the history of literature and its relation with the social contexts in which it is produced and received.

Also, a critic has to present her interpretation to a whole group of other critics. In doing so, she has to defend her ideas about a novel or a poem with a number of arguments: by relating it to the culture where it was produced, with biographical information about the author, by paying attention to what other people have believed about that specific novel or poem in different moments of history, or even by studying the reasons the publisher had to publish it in the first place.

For all of that, the critic’s interpretation of a literary work tend to be better informed and argued than the ones of casual readers. As Recoeur said, maybe there are no methods to make valid interpretations, but there surely are ways to make those interpretations invalid.

To help clarify this idea, I will call interpretations of casual readers real interpretations, in analogy to the uncountable infinity of the Real numbers; and I will refer to the interpretations given by literary critics asinteger interpretations, in clear analogy to the countable infinity of Integer numbers. Is there any connection between the two types of interpretations? How could we even know!

The main difference between real interpretations and integer interpretations of a literary work is that the first ones are more numerous, easier to produce and, hence, less valuable than the second ones. This is so, because more hard work is needed to produce an integer interpretation than it is to produce a real one (the only ingredient needed for the latter is a mere guess). Georg Cantor paraphrased: the integer numbers are like the bright stars in the sky at night, while real numbers are darkness.

Of course, this is an analogy. Philosophy and math are based on different kinds of reasoning, and I’m pretty confident they will stay this way forever. My general point is that some times, analogies can help us mediate between different kinds of knowledge. And especially, that we should be aware of not confusing infinity with the absolute: in the humanities there may be no “true” interpretations, but clearly some interpretations are better than others.

A Final Thought About the Relation Between Math and Literature

Why should we talk about math in an essay about literature? I have two distinct reasons for this. In the first place, because, in the end, the value of literature and mathematics is that they both enrich our life. Life itself is meaningless but it acquires value to us because we, somehow, have created ways to enjoy our individual and collective existences. And I can’t see any reason to doubt that art, scientific knowledge and thinking itself are noble ways to enrich one’s life.

In contemporary culture, many people value things, including scientific knowledge, according to how practical they are. What we don’t always realize is that while we are wishing for a scientific notion to be useful in a tangible way, what we really want is for it to be relevant to our lives in a way that may contribute to enriching our existence. The most important thing about the Theory of General Relativity is not that it enables us to use GPS systems (which are pretty useful nonetheless, if you ask me); rather, it is important because it is fascinating to grasp it. To understand that, as far as we know, space can be curved and the flow of time can be altered by the effect of mass.

The same thing applies to math. As the mathematician Harold Harley famously stated, “very little of mathematics is useful, and this minority is comparatively dull” [5]. Something like that can be said about art and, in this sense, math and literature can be measured with the same ruler.

Now, I’m not going as far as Alain Badiou when he states that math is ontology itself [6], but I can argue that mathematical thinking can be of great value to metaphysics. This is because, as we have seen, math transcends the limits imposed by doxa and by historicism. Math can also also take the form of a secular truth, at the least according to some recent speculations about the meaning of “nothing” [7].

But, as Badiou — and Plato well before him — has stated, even when math offers true knowledge orepisteme, it alone cannot suffice for wisdom. For mathematical knowledge to turn into wisdom it is necessary to include philosophy, and literature.

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[1] I highly recommend Conditions, by A. Badiou, 2009.

[2] Hegel, De Orbitis Planetarum.

[3] Infinity is bigger than you think.

[4] Interpretation Theory: Discourse and the Surplus of Meaning, by P. Ricoeur, 1976.

[5] A Mathematician’s Apology, by H. Hardy, 2005.

[6] Although some mathematicians do!

[7] The Book of Nothing: Vacuums, Voids, and the Latest Ideas about the Origins of the Universe, by J. Barrow, 2002.

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Jorge Alejandro Laris Pardo is an undergraduate student in the History of México at the University of Yucatán. He is currently working on a thesis about “Science and the Polemics around the Idea of Science in the Yucatan Peninsula in the Times of the Restoration of the Republic (1867-1882),” were he is concerned about studying how social contexts can affect science and the quest for scientific knowledge, without falling into the extremes of postmodernism.

This article previously appeared here, republished under creative commons license.