Can you think of contexts where a particular class of number is inapplicable?

Think about this for a moment. By “class of number” I mean, for example, negative numbers. Negative numbers, we learn, are great for expressing debt. They apply perfectly well, also, to linear orders where there is no smallest element (temperature isn’t the best choice, since Kelvin has a smallest possible value… year is better, since although we estimate a beginning of the universe, we can’t count days since then, so BC years are appropriately negative). What negative numbers don’t seem to apply to are numbers of family members, times you can iterate a process, dimensions of a space, distances. It is no wonder they too were suspect when first conceived.

How about the class of numbers which are not 1? Possibly it makes no sense to talk of multiple universes (I am of this camp, nevermind the fashionable physics which suggests the idea). For if there is a vantage point from which to observe 2 universes, then causally the two are part of your universe, even if they are entirely independent of each other. We might (should) define universe to be all those things which cause change in what we can perceive (to my mind this is possibly an asymmetric relation: a realm can exist which is broadcasted to us, upon which we have no impact, so that it is part of our universe but we are not part of its). Then to any observer there is, by definition, but one universe.

For number of family members and iterations of a process, it doesn’t generally make sense even to apply positive rational numbers. Particularly if the process is instantaneous. Can you wake up 2 and a third times? Well, not without defining what it means to wake up a non-integer number of times, it certainly is not given. Is it okay that certain numbers have limited scopes to the context to which they apply?

The idea of iterating a process, when put in mathematical terms, gives rise to the old subject of solving functional equations. If I have a function f:R->R from the real line to itself, I can define the square of the function, that is f^2, to be the function composed with itself. Then I can ask if there exists a function g so that g^2=f, i.e., does f have a square root? If f is the function of sending x to -x then normally there exists no function which halves this process (for example, sending x to the average of x and -x certainly doesn’t work). For 1×1 matrices, multiplication and composition are the same thing. So for linear maps from R to R, finding half of the inverting map, [-1], is exactly finding a square root of -1. If you extend the real line to be sitting in a plane, then it becomes clear that you can rotate the plane 90 degrees in either direction to give a process which when iterated twice sends x to -x. This mysterious fake number, i, is nothing more than rotation of a plane by 90 degrees. Do rotations by 90 degrees exist? If anything does, they damn well do. Teachers have been heard saying “i is important because it solves x^2=-1, which you want to be able to do for engineering problems” It is a good teacher that says “i is important because it is a compass point for any object with a quality that takes values in the circle, which certainly includes many physics and engineering problems” So, any phenomenon involving waves involves complex numbers. If waves exist, complex numbers do too.

From above: let us know what it is we are doing with language, so that we may deliberately do much more of it.

Let me clarify.

Whatever the devices used by the individual mathematician, in order for the ideas to become part of a shared theory, some form of written communication is used. This is trivially “linguistic,” and, as I’m using it, includes diagrams and even geometric drawings. But are the choices that go into recording mathematical ideas merely style, more or less independent from content? I’m trying to argue that they are not, that a good deal of the content IS the organization of the (written account of the) collection of ideas.

I’m thinking of developments which dramatically progress
mathematics, not the personal phenomena that come with problem solving. The latter can be written about, but it’s mostly irrelevant to mathematics. Major innovations include the development of the decimal system, certainly this can be seen as both a linguistic and a mathematical innovation. Or Newton’s initial formulation of a differential equation, (just 9 years after calculus had been developed!), or for another example, Eilenberg-Maclane spaces. In both of these examples (and countless not mentioned) you have a heap of theory that goes toward defining objects which become *basic* objects of study. (Included in that theory are theorems, since frequently theorems are necessary to give the context for a definition). Now I can say “suppose I have this K(pi,n) space …” How do I translate that statement into the setting, a few hundred years prior, of Descartes or Newton? How is a differential equation translated into the language of the Greeks? It’s not just that the definitions would be long-winded, but much stronger: that for an adult of moderate intelligence to contemplate articulate questions about 4-dimensional manifolds before Gauss’s time, say, is absurd. In the 21st century it is not. To a great extent this is because us 21st century inhabitants have been given an organized language for asking (and *understanding*) such questions. In summary, language is a lot more than a list of labels, but instead is rich with structure. This structure is crucial to mathematics.

> 2. Isn’t the structural atomism Andrew mentions, “. . . statements and
> derivations in some atomic, axiomatic system of symbols” distinct
> from the structural aspects of the objects of mathematical theory? I
> guess we could take a nominalist position that there are no mathemat-
> ical objects, only the language that constitutes mathematical theories.
> Would a strict nominalist tell us that we are mistaken when we claim
> to be thinking in non-linguistic geometric imagery prior to enunciating
> mathematical statements about those images?

I can’t really address this, as I don’t really understand the
nominalist position. Am I confusing the structural aspects of the
“theory of mathematics” with “structural aspects of the objects of
mathematical theory.” Maybe I am. We can steer the ship back over to that topic.

> 3. Andrew kicks the hornets’ nest (or would have twenty years ago) in
> making points about “A Transformative Hermaneutics of Quantum
> Gravity”, i.e., the Sokol project. A question I would like to study is
> how the unintelligibility of Po Mo writing and the unintelligibility of
> mathematical writing compare. For example, three years ago or so,
> Gayatri Spivak wrote an article in “Art Forum” on Badiou. I couldn’t
> understand much of that article. Compare this with the fact that
> any of us wouldn’t understand much of Categories for the Working
> Mathematician without a great deal of work. Are these two cases
> essentially the same, or is there a significant difference lurking here?

I didn’t mean to make a point about that work, but the title serves as a great example of far-reaching abstraction (in natural language). If we know a priori that some work is mathematically sound, albeit requiring, say, a billion pages of difficult math to build up to, then perhaps, a priori, we can claim that the work is fundamentally different than a social philosophy text that requires a billion pages of background, although it might be more
interesting to note the similarities. The distinction between the
math and the philosophy is founded on the idea that math can be
formally correct, i.e., that it can be checked by automation.
Philosophy cannot, both for practical reasons and for the fact that social philosophy (any philosophy, really) requires a point of view (in our case human, of a certain culture, etc.). In math we can claim a conclusion is true even if we don’t understand the proof, provided, say, we understand the algorithm that computed the proof, and we understand the terms in the conclusion. The obvious example is the four colored theorem, or suppose the
Goldbach conjecture had been solved by a program. By contrast, what would it mean to say that a philosophical phrase (e.g., “God is Dead!”) is sound because some billion generated lines of code concluded it. Nonesense! Espousing such a slogan in philosophy is a way of referencing one’s own understanding of the justification, and is not the assertion of a FACT.

Of course, conclusions in mathematics can also be in terms which
require billions of pages of code to parse. So instead of something as simple as the Goldbach conjecture, the program
generates a line relating objects which it has defined, and which are incapable of being understood by a human, yet the human can check the code and conclude that, barring bugs, the conclusion
must be correct. This computer generated conclusion is much more akin to the philosophical assertion which requires billions of pages of justification, since in both cases the conclusion is essentially gibberish.

I won’t say anything more concrete about MacLane vs. Badiou, but would be interested in hearing another’s take on the comparison.

> 4. Meter, millimeter, micrometer, nanometer, . . . . This kind of termi-
> nology is similar to the would-be lengthette, isn’t it? The suffixes
> ’eme’ and ’etic’ seem to be used by philosophers of linguistics. Split
> infinitives are evolving to acceptable status, ’lite’ is understood as a
> variant of ’light’ albeit with a special meaning akin to that of ’je-
> june’. Dennis notes jargon such as texting abbreviations (in an email
> that you might not have received) that arises too quickly for me. I
> have heard that German grammar was reconstructed by grammari-
> ans (18th century?) after Latin grammar and that this constituted a
> major structural change. Ray, you can probably help us here.

The ‘ette’ suffix was an example of what we might call a structural innovation in our language. Indeed it does not exist as I have defined it (again, if I want accuracy but am unable to be precise, I should not use ‘a few nanometers’ to mean a relatively small length). In category theory we have the ‘co’ prefix which is extremely general. The introduction of such modifiers is clearly more substantial than the labeling of some very specific instance of some thing.

I was making a point about how language is only infrequently modified deliberately, to be more structurally robust. It’s mostly aside to the topic here.

> 5. Finally a point that is tangentially related to Andrew’s note. Struc-
> turalism was fashionable in several parts of the liberal arts in the 1950’s
> to 1980’s, but went out of fashion, declared by many to be irrelevant.
> Here is what Peter Caws says (Structuralism, 2000, p.105):
>
> In fact the stress on “formal models” in this statement of
> Chomsky’s points up a general problem for structuralism.
> It is perfectly true that outside some domains of linguistics
> very little progress has been made in developing interesting
> or fruitful formalisms: as we shall see in the next chapter,
> L´vy-Strauss’s attempts, which count among the most am-
> e
> bitious, seem often at once simplistic and contrived. But
> even within linguistics the results have not been much hap-
> pier, and my own view is that the whole attempt to conduct
> the structuralist enterprise in terms of rigorous mathemati-
> cal models is a case of what I call “spurious formalism.” A
> nonspurious formalism has to meet two conditions: what it
> deals with must be precisely specifiable in formal language
> (in the ideal case, moreover one frequently encountered in
> the physical sciences, it will be quantifiable), and this specifi-
> cation must make possible formulations and operations that
> would not be possible in ordinary language. This last con-
> dition is hardly ever met in structuralist studies; the main
> points can nearly always be conveyed discursively, and the
> introduction of formulas and technical expressions is often
>
> 2
>
> merely ornamental and plays no real role in the argument.
>
> In relation to Andrew’s (2), mathematics and mathematical physics
> could in principle be written in natural language, just as they could
> be written in some first-order formal language. It’s not that it is
> impossible, just that it is impractical in the extreme. Caws’s point is
> that the opposite is true of most structuralist efforts in the arts and
> human sciences, that is, natural languages serve better than various
> formalisms of a mathematical kind.

I am, at present, fascinated by what I will call the role of language
in mathematics. In using this phrase I wish to call to my own
attention the similarities between [natural language, and how it is
used to imbue theory with structure] and [language in mathematics]. I
want to make the point that mathematical innovations are linguistic
innovations, and that this observation is important, though frequently
overlooked. Let us draw an analogy between the material reductionist
and the mathematical formalist. In the former, all utterances are
understood to be rough shorthand for statements in terms of atomic
particles or some finer physical stratum. In the latter, all
mathematical theorems and their proofs are understood to be precise
shorthand for statements and derivations in some atomic, axiomatic
system of symbols. While it is sufficient (and often necessary) to
offer a definition of a mathematical term by way of previously defined
terms, (i.e., on a tree of definitions rooted in the formal base language),
one immediate question following such a definition is why this
definition at all? We are lead to believe that “mathematics”
encompasses not only the naming of natural and relevant formal
constructions, but that assigning arbitrary definitions, and remarking
with arbitrary observations, is also mathematics, provided the
definitions are formal and the observations are proved. That is, the
essential in mathematics is its absolute precision, and not in the
particular choices of structure in the theory. And why not? after all,
who can account for particular choices? We call these *tastes*, their
justification resides in a wholly different realm than the
justification for a line in a proof. They depend on the whims of
complex and erudite connoisseurs, not on reason. But when a student
asks why the property of a subgroup H in G, that gHg^{-1}=H, is given
a name (let alone why the particular name), what can they be answered
with? The usual answer would go along the lines of “normal subgroups
are extremely important, ubiquitous, characterize kernels of
homomorphisms, provide for series which classify groups, (etc.)”. That
is, the student is given a structuralist justification, and not “it is
a matter of taste to consider this property important and not, rather,
some other arbitrary property,” which would constitute a very weak answer. I
argue the reason normal subgroups have a name, (versus, say, subgroups
which have order any of {7, 48, 184}) a) is because there are multiple
contexts where normalcy is referenced, and b) has as much to do with
the essence of mathematics as does the precision of the definition of
normal. Indeed, it is the appreciation for concepts such as “normal”
that leads people to mathematics, and not merely the precision, as is
typically perceived.

Another typical perception is that mathematicians invent words and
generate new questions along the path of solving a given problem.
Hence the endless pursuit that is mathematics. Indeed, this perception
is accurate and many a mathematician corroborates it to the outside
world. Related is the myth that generalization=improvement. What is
less often discussed, in the same discussion, is that *any*
philosophical inquiry leads both to new words and questions and to a
generalized view. This is because the vocabulary introduced today
becomes naturalized in tomorrow’s theory. And generalization, when it
happens naturally, is a sign of sophistication. You might work with a
group for some time, because it is the symmetries of some aspect of
this problem you are working on. At some point you identify some
interesting properties of this group. A collection of these properties
fit together nicely, and so you christen the collection of properties.
Now you cannot help but wonder a few things about ****** groups, can
you? The construct of being a ****** group not only has a formal
definition, but it has semantic value to you now. It has an identity,
one that can and should be explored. When I think of generalization I
think of the fact that the Library of Babel has volumes on various
theories (albeit mostly inconsistent and fictional). For every
coherent theory or novel or case study book, there are books which
reference these (although referencing one book from another presents
its own problem, doesn’t it?) so that the theory can be built up to
unimaginable structures (let’s just allow ourselves to adhere labels
to a few of the bindings). Already in the humanities you have a
running gag about names of papers such as the famous “Transgressing
Boundaries, Toward a Transformative Hermaneutics of Quantum Gravity”
(or make up your own). Yet, with enough time, one could very well be
interested in various hermaneutics of quantum gravity, transformative
and otherwise. Why not? But the important thing is not whether some
particular esoteric hodgepodge of theory *has* meaning, but whether it
has interest to you, at your current level of sophistication. As the
levels of sophistication are unbounded, at some point there are
volumes of theory which will be utterly inaccessible to any human. We
have no choice but to call such a text “meaningless.” On a similar
note, one does not begin group theory by studying a subsuming theory,
such as the theory of magmas, not only because it is difficult to
handle such generalization from the beginning, but because groups are
interesting on their own.

Finally, on a somewhat related note, I wish to point out that computer
scientists are frequently fluent in many programming languages and are
frequently at work designing new ones, to improve structural issues of
those existing. I wonder why we do not do this more actively with our
natural language. We evolve our natural language like the brain has
evolved, by packing on new to old, and by tiny discrete changes here
and there. Could our language be more expressive or better organized?
Here is an example: suppose I want to connote a relatively small
amount of some unit, given some context, and I want to be accurate but
not precise (meaning I want to say exactly what I mean, while
remaining vague). It would be ideal if I could attach the same
diminutive suffix to any dimension, so for example, lengthette, timette,
massette, forcette. Not only does no such suffix exist, but if I begin
an essay by “let ette be the suffix which connotes relative smallness
of the linear dimension to which it is appended,” well, let’s just
say this would be unorthodox. I imagine if enough people did this
language would organize itself a little more according to deliberate
design, as is arguably the case in mathematics.

What happens in choice, when alternate outcomes are considered, one is preferred, and then acted out? The universe has encoded itself within itself, in a simplified model which informs events yet to occur. Equipped only with the knowledge of Newtonian mechanics some observer concludes that a car, headed south along highway 81 en route to Asheville, NC, will collide with a tree in approximately 12.03 seconds. Another observer skilled in the art of brain-state reading concludes that the tree will be avoided as the curve in the highway is followed, and that in approximately 6 hours the driver of the car will meet his son in a restaurant, eat and pay the bill. How mysterious is it that the universe encodes past and future in simplified models contained in mind?

i. Objectness

Long before we need worry about free will and consciousness, we need to worry about what objects are. These will be the nouns of any truth claim we make. Before there is *choice*, there is *person*, and how to deal with the distinction of being a *person* object is not as easy as it may seem. Even in a low resolution cellular automaton (i.e., one with few cells) the problem remains. Is a glider an object? Is every configuration of cells an object? We might suppose that every configuration of cells theoretically has a name, or could be given a name, and even that names in our language must be shorthand for collections of cells (that meaning in language must essentially be built from these building blocks). But then something so simple as ‘glider’ is necessarily shorthand for a list of trillions of configurations. This seems like a faulty way of looking at things. If glider includes not only the same two 5-cell configurations of cells, up to translation, but also larger things which exhibit a gliding property, the problem is harder. Certainly if an object appeared to maintain its shape more or less, as it translated itself through space, perhaps even fizzling out at some time, we’d be tempted to call it a glider. This is especially true if our instruments of detection are unable to detect individual cells, so we cannot discern a glider’s states at that finest level. No one argues that natural language is not fuzzy, as it unarguably is, but then how do we interpret a fuzzy truth claim, in reductionist terms?

ii. Diagonalization

We’ve built in our imaginations a cellular world with trillions of cells, and in this world a creature has been formed. That creature is constantly bombarded with gliders of various sizes and from these collisions, (and internal happenings), the creature processes thought and outputs gliders, as statements. This creature I’m thinking of is essentially a human, or a near approximation. Now, let’s say we agree on an interpretation of its language (i.e., the waves of gliders it sends out of its mouth, each wave differing in shape enough so that a discrete language can be understood, as English is). What can it say in this language? One thing it can say is “the universe is a cellular automaton with the following rule of evolution…” What can it not say? It cannot say “I will now give names to the 10^(10^10) cellular configurations possible in this universe, beginning with ‘aardvark,’…” Indeed, since each utterance takes up space (for the gliders to carry the waves of speech) the utterances are quite limited in the amount of information they can carry (necessarily less than the total number of configurations possible in the same tiny amount of space, let alone the universe). Now, theoretically we can offer the utterer all the time he wants to longwindedly describe each fine detail (indeed each cell) of some object, and terminate after finite time. But who, or what, is his audience that can reassemble the information into a model that contains as much information as the original object? This is one reason computers cannot calculate the evolution of the universe, because you don’t even get to specify the initial condition without generating an infinite descending loop! It almost seems absurd to expect more than fuzziness from meaning in language, but of course our language is not fuzzy, and perhaps this is where some of the confusion lies. Language is ridiculously precise. Unlike facial expressions or performed music, it is exact and codifiable. Yet still we discern subtlety and nuance in our favorite authors, after reading hundreds of thousands of their words (or even a good cadence in a paragraph or sentence).

iii. Emergence

I want to draw attention to the fact that fuzzy terms (as in the referent is fuzzy, such as with natural language) aren’t just ‘fuzzy,’ as opposed to being precise, as a sort of deficit. But instead, that there is meaning in a fuzzy term that is essentially lost with the attempt to make it precise. I am thinking of the cellular human, and imagining her, let’s call her Frida, holding a ball and commenting “it is round.” This roundness property, which seems so elementary, is in reality a reflection of the ball’s resemblance to other objects previously perceived by Frida. So the process goes: an object is in front of Frida, some waves of gliders emanate from it (or rather “bounce” off it), carrying information about the object into Frida’s sensory apparati, then Frida’s brain momentarily gets a hold of that information. Within a few seconds most of the information is gone, but some faint ghost remains, a ghost which somehow holds information about the object which is general, and connects to yet other things Frida has seen. This object would not be classified under the blanket abstraction ’round’ were it not for the ‘intrinsic’ cellular make up of the ball, but the abstraction cannot be said to be an intrinsic quality of the ball, supported only by the physical state of that ball. I think some would find this distinction too subtle, but it’s an absolutely crucial difference to understanding how meaning works and how the reductionist is wrong.

It is only too easy to imagine ’roundness’ is a concrete quality either enjoyed intrinsically by an object or not. However, terms which are overtly contextual as opposed to physical are readily available and make up the majority of the words we use. Take ‘majority,’ for example, and define it in terms of cells in such a way that nearly all usages found in English literature can be said to reference it. It can’t be done. We can say what a majority of cells being on in a given region means, more or less, but that is not directly referenced by my usage above, nor when we stumble upon it in literature, say in the phrase “tyranny of the majority.”

It is observed, also, that terms which don’t lend themselves easily to reduction, are frequently not necessarily more complex, given a context. This is exactly the point, for if it was necessary to give a description of X with complexity proportional to the extent X resists reduction, then we are in a reductionist framework, and X is just really complex. But the human mind doesn’t work this way. It is frequently possible to communicate great generalities to children, who would have no way of understanding the reduction to finer physical parts. How can one insist “being on one’s best behavior in a restaurant” is really a property or action of physical particles, or even a deep sociological action, when neither of these can be comprehended by the child, while the statement itself is easily understood? Part of the answer is that in building a vast framework of complexity, certain terms become contextually simple, while being intractably complex from a ground-up perspective. You don’t build “one’s best behavior” from scratch. The other part of the answer is that a child begins with such a framework. Knowledge does not stick to an empty slate (not even a blackboard!), but children have a robust way of making sense of generalities from the beginning. My main point here is that meaning is emergent, held together by a framework harder to imagine than a strict partial order. There are more lateral connections in the network of meaning.

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 ?

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.

Andrew– “hello book!”

Book– “hello Andrew!, thank you for reading me. #”

A– “how did you know my name?”

B– “I was written with you in mind!#”

A–”whose mind?”

B–”Isn’t it a bit early for those types of questions? We have 409 more pages to go. #”

A–”I don’t like your attitude, where can I find another talking book? One that is more accommodating?”

B–”surprisingly, there is one right behind you, middle of the 4th shelf up! #”

A–”Are you lying?”

B–”Yes, of course!#”

There are even books that look like this, where you can read the pages out of order, in an attempt to screw up the book, but which account for that, and still read in a linear dialog.

Of course this assumes that your conversation is essentially deterministic, and there should be ways to make this dialog fail. For example, you might ask a lot of questions about neighboring books, and even if your book maintains an English dialog, it will be forced to lie if all the possible ways to tell the truth are located in other parts of the library (where they very well might be lying). Also, it may be that whenever you encounter such a book you don’t have the ‘correct’ conversation, so that in an infinity of time one finds no meaningful conversation.

Other interesting implications: There are dense and eloquent mathematical proofs in the library which require tens of thousands of volumes to prove. Similarly there are books of cultural or psychological theory that build on tens of thousands of other volumes, these of case studies and theory development. Indeed, entire sets of volumes exist to pinpoint the meaning of individual words. It is easy to conceive of some great intelligence that can comprehend some of these books, books which must be all but gibberish to any human. What truths might they hold? Is there any limit to the level of sophistication of theories in that library? One might think so, as the library holds a finite amount of information. But as some theories take many volumes to present, we might as well consider our alphabet to consist of volumes in the library, and strings to be sequences of volumes, so by this language, any finite amount of information can be learned in an afternoon of particularly fast speed reading (say, a few trillion volumes a second, for a layman’s guide to everything). (What is called Quine’s reductio goes in the oppose direction, reminding us that the library might consist of just 2 books: all possible ‘books’ of length one character, on a binary alphabet: 0 and 1. Then any literature can be found as a specific tour through this library, stopping at each of the two books many many times).

Exercise: write a few interesting ‘implications’ about the library. Write about possible limitations, if you see any.