## 1.2 The Land of Counterexamples

We will now take Borges’ Library of Babel as the discrete analog of what might be called the land of counterexamples. Specifically, suppose we agree that space is continuous, and matter exists in this continuous space in some array of forms at each point. I mean by this that there is a set $X$, possibly a space in its own right, and that each point in space can be occupied by vacuum or by a value from $X$. Then if $I_k$ is the cube of side lengths $k$, we denote by $A_k=X^{I_k}$ the set all possible configurations of matter which fit into the cube $I_k$. Then we sit back and dream about $A=A_\infty$ which is the direct limit of the $A_k$. For those to whom this description is too mathematical the translation is: $A$ is the collection of all possible (bounded, meaning not stretching off to infinity) configurations of matter. This we might call the Continuous Library of Babel (CLB), since it is essentially just the continuous analog of said library. Now, if nature wants for half-rhino-half-chickadees, CLB does not. Think of it as a sort of library of congress, where every physical creation sits, on cubby shelves of increasing size which eventually get arbitrarily large. A philosophers’ Costco.

Exercises: what might be wrong with this parameterization? (hint: there are quite a few things possibly wrong). What are some interesting shelf-lifes to objects in $A$?

Now, think about what the library of Babel has to say, on its shelves, about the CLB. Other than gibberish and lies, one finds guides to understanding the contents of CLB. Like a book that tells you about different birds, so too the library of Babel has guides to help you understand the beasts of the CLB. One problem becomes immediate. The entire library does not contain enough information to specify any object but possibly an infinitesimally few special objects, such as the empty object, or an object that consists of 2 points of matter, exactly 3 meters apart. The majority of everyday objects seem to be indescribable. For example, if one insists that common nouns should reference specific configurations of matter, thereby allowing English sentences about matter to fall into the two categories TRUE and FALSE, then what is a “book”? If one takes a book–just a common book mind you, one whose “bookness” would not be in question– and alters it physically by removing a point of matter, then it should still be a “book,” since no human would ever even detect the difference, indeed vast more differences are occurring molecularly within any given book within any given second. But according to our parameterization there are an infinite number of such configurations, all of which easily fall into what is indisputably understood to be a book. Then it is hopeless to think we can even describe what it means to be a book, let alone have any sort of theory of anything physical at all.

This brings to my mind Zeno’s arrow paradox, which essentially asks the question “is motion inherent in a physical state?” I.e., said in this way, are there configurations of matter that would spring forth as a flying arrow, upon their creation, since matter in motion is different than matter at rest? or is the standard model of matter and motion accurate, that they are independent. Certainly some kinds of potential energy are purely physical, such as a cocked spring, or the head of a match. In theory we could build, atom by atom, pool balls which are pressed into the banks of tables, ready to instantly spring. Can we not then design a flying arrow? I don’t know, and I don’t know if anyone does. What does the uncertainty principle say here? If the universe were a cellular automaton, then motion would be physical, as demonstrated in the gliders, which we know their motion by inspecting them at instances.

Let’s back up and give ourselves a simpler world to explore. After all, the human mind with all the tools at its disposal does not have and never will have infinite precision. What happens if some particle is essentially indivisible, and essentially repeats itself in identical copies, and let’s even go so far as to say only a finite number of locations exist for it to occupy (although it should be a big number; our world becomes rather unrealistic if it is significantly less than 10^100). Here we only need worry about a finite number of configurations. In such a simplified world one can argue that we can have a precise theory after all, i.e., a set of TRUE statements about matter. It seems our library of Babel might have something to say about this land. However, even here, it will take volumes of data to specify one type of object, and we’ll have to leave many many configurations unlabeled. And here’s another problem, even supposing we define some common object by volumes of data which describe it perfectly, suppose now we want to form a true sentence out of 2 or 3 such terms. Won’t the sentence almost necessarily be wrong? Maybe every description of a book contains descriptions of “pages,” so the statement “every book has at least one page” is TRUE. It seems to me that building much theory about objects which we expect to be precisely defined will eventually break down, but perhaps we’re going too far. No one has ever described what is and what is not a book, molecule by molecule, so clearly that’s an artificial requirement of a theory. Could one argue that it is at least possible, in theory? Well, in one sense “no,” since “books” have never been defined in this way. Either

(1) the new definition would precisely reinforce what we already know, namely that THIS is a book and THAT is not.

or

(2) the new definition would tell us that what we thought was a book was in fact not a book, or the other way around.

When we dropped Pluto’s status as a planet we experienced something like (2) here. We decided that the term “planet” should refer to something of a particular size orbiting a star, for organizational purposes, and so had to either refer to 10 such things around our sun or 8. This kind of house cleaning is fine in science, but it’s of course ridiculous to imagine that someday we’ll have a clean definition for “book,” that is amenable to precise physical theories (e.g., books burn at F451=TRUE).

Enter the parade of horribles. For every type of physical object there are countless things–real or imaginary–which debatably belong or not to the category. For example, there are books made of plastic, greeting cards with pages, electronic and audio books, pamphlets, etc., and all of these things are real. There are also books the size of the sun, books which moan when their pages are turned, books which kill all who read their vile words, etc. These are imaginary. Many of the imaginary objects exist in the CLB. For example, with enough care we might design something book-like which moaned when its pages were turned. Is it a book? Hard to tell. In fact, the best way to view this question is not as a TRUE/FALSE question, but more as a hypothetical in which more information is needed. What other objects are around? Do the moaning books moan from some chemical in their pages or are they discovered to be sentient? Are there many many creatures of different types which all resemble books? The world in which they exist, what that world has and what it doesn’t have, will help us to decide how to classify them. This is partly why it’s so hard to discuss things in the CLB and why it is so easy to discuss things in our world. Because all the intermediaries are cut out of the picture from the beginning and throughout time, we are at liberty to discuss, for example, what different kinds of animals there are, and mostly be in agreement about what we mean. We may draw examples again from CLB, but for now we turn to a similar world, for the sake both of reiterating this point and also for making a slightly different point about emergence.