A Wonderful Idea: Metagrammatical Numbers

October 17, 2008

Apply Cantor’s diagonal argument to the list of describable numbers, and one arrives at a description of a number which is not describable.  A paradox? Not at all.  Merely proof that higher orders of description exist. That is, some numbers are metagrammatical.  They can be described and computed to arbitrary precision, but necessarily the description is meta to whatever grammar one begins with.  This is delightful!


Details:  We call a number describable if it can be described by a finite expression, with a specified alphabet and a specified grammar, which decides how to compute the number from the description if it is grammatical, and discards expressions which are not grammatical (i.e., we are using ”grammar” to mean both what is admissible as a description, and given an admissible description, what is the protocol to compute the number which is described).  For example, consider the set 


With this set we can describe every rational number, since we can write a rational number as an expression using just these 5 characters.  (If we are careful we can get rid of the parentheses). The grammar throws out nonsensical expressions like 1--/-, and computes the others. Allowing iterative operators (e.g., \Pi, \Sigma), quantifiers (\forall,~\exists), a countable number of variable (v_1,\ldots v_n,\ldots), et cetera, we get descriptions of algebraic numbers and a lot of common transcendental numbers. Still, even with a countable alphabet we can enumerate the number of finite strings of characters, so that the set of describable numbers is countable. This is not too surprising, but if you have never thought about it, it is worth pondering. This set of numbers is a field, for example, and not a trivial example either! Indeed (almost) all numbers that are useful at all are describable. Of course, picking one at random (with equal distribution) is impossible, so probabilists, among others, demand the existence of indescribable numbers.

Let us make a list of all descriptions on the left hand side of our paper, and on the right we will make the corresponding list of numbers, written in binary (which frequently do not terminate, but that’s allowed). Now we’ll construct a number, \chi, that can be computed but is not on our list. Let \chi have as its nth diadic (“diadic” is like “digit” but for base two and not base ten) a 0 if the nth description has a 1 in its nth diadic, and a 1 if the nth description has a 0 in its nth diadic. That is, go down the diagonal of the list of numbers and make the number which has switched each diadic in the diagonal. Have we not described \chi? But it cannot be on our list, since it differs from every number on our list in at least one diadic.

The point is we can never nest a description of a grammar within the grammar itself, although we can nest one grammar inside another. If we could nest a grammar inside itself, we could, in grammar G, write the description

the nth diadic of this number is the opposite (base 2) of the nth diadic of the nth description in G,

and we would have produced the paradoxical situation of having a number which both is and is not on a countable list.

More specifically, let some huge N be the number in an ordering of G, so that the Nth description is the one in the above blockquote, and let it correspond to the real number \chi, which it describes. If such a description can be written, it must be written in a finite number of characters, and so there must be some such number N. Now we ask what is the first diadic in \chi? No problem, our grammar just asks itself what the 1st diadic in the first description is, and adds 1 (mod 2) to it, next it asks what the 2nd diadic in the second description is, and adds 1 (mod 2) to it. Eventually it gets to N and it asks itself about the Nth description, but the Nth description requires it make a list of the first N descriptions to get to the Nth diadic in that number, so the grammar tries to reference itself, and in doing so must reference itself again, in a nested infinite recursive loop. So forget about the fact that the grammar is paradoxically trying to switch one of its own diadics. Forget the paradox that the number is both on and not on the countable list. The first problem is that the grammar goes into an infinite recursion.

In fact, lets go back and do the same thing except instead of switching the nth diadic of the nth describable number, we use the nth diadic of the nth describable number as the nth diadic in our number, say \chi_0. Can you think of a more agreeable number to have on our list? \chi_0 could be any of the countable numbers on our list. Suppose it is the Nth number. What does it have for its Nth diadic? Well, either 0 or 1, since this number must merely be equal to itself. But then \chi_0 is not describable, since its description does not produce a unique value!

This is, essentially, how one approaches the Berry Paradox. In short,

definable in this system

is just never a logical definition (or clause thereof) in that system.

However, once we agree on what a definition or a description is, then these examples tell us that there are deterministically producible, unambiguously definable, finitely expressible numbers and strings which can only be expressed with a grammar meta to the one first chosen.

What a delight!


October 10, 2008



What is Universal?

What is universal?    

Nothing is. (rather, there does not exist something which is universal).

Universality is a partial order (see previous post), which orders according to ability to be translated without significant loss of meaning. The statement “I am hungry” is more universal than the statement “Mayor Madison manufactured a million magnesium missiles” for a variety of reasons. First of all, the context of the second statement demands an understanding of what a mayor is. We’ve had similar town officials in most civilized regions and times, but the meaning changes significantly if we go far enough back, or far enough away. Next, Madison is a name. This is more or less arbitrary; unless of course you know that the Madisons are a particular family with a particular public standing. “Manufactured” cannot exactly translate to “made,” so context requires understanding of post-industrial revolution production. Similarly with “missiles.” “Magnesium” will not translate easily to a society without a periodic table of the elements, and even “million” is inexplicable to a primitive tribe with a number system consisting of the four quantities {1, 2, 3, many.} Et cetera. Furthermore, the sentence strikes any English speaker as having been concocted to alliterate. Therefore, it can be argued that this is part of its meaning, so that translation to a foreign language becomes much more difficult. On the other hand, every society on earth has a word (or hand signal) for “hungry” and in particular some way to declare being in this state (regardless of whether the subjective pronoun is explicitly used). (Think, also, of different ages of individuals, or different levels of accessibility: an uneducated child has a way of saying the first sentence, no way of even understanding the second).

Context, in the above paragraph, is at the scale of societies of humans. We also can speak of the context of certain species. “I am hungry” can be translated to the lexigram language of the bonobo monkeys, whereas we doubt that it’s even meaningful to consider translating this to something a tree would understand (“understand”?!). We can go in the other direction too, of specificity. I can create words I never share with anyone, which refer to complex memories or particular synesthetic experiences such as a smell which reminds me of a shade of brown. Now the problem of translation is between individuals and not societies or species. 

We can arrange academic disciplines according to this partial order. I contend that physics and mathematics are more or less at one end and poetry and film theory are at the other. Not that there couldn’t be personal theories which are significantly less universal than film theory, I believe there are. Nor that there might not be some way of understanding the universe which is more universal than mathematics, but this is much harder to conceive of, indeed an example would most illuminating!

Here I must warn myself that things are not as easy as a partial order, exactly. Advanced mathematics is not very universal in the context of the human race. So here what appear to be most universal are simple statements, referencing basic animal desires. Still if we are to find any common ground with an alien species, especially one that can traverse great distances of intergalactic space, we assume they have a language for expressing “a^N+b^N=c^N, a,b,c,N positive integers, has solutions only for N=1,2”, (which neither the bonobo nor the hobo do (though the hobo could be taught, which is a form of translation)) before we assume they can relate to “hunger.” I.e., on a grand scale, universality is not about accessibility, but about the ability to be translated between contexts which arose more independently. 

(I am willing to admit this may not be the best definition of universal. It seems there are qualities and states such as pain and pleasure, experience, and desire that are arguably more universal than particle physics. There is probably a lot more to say here, but I haven’t thought it out).

I am interested in the common ground which is necessary for translation–those suppositions which some discipline is predicated on–as a measure of that discipline’s universality. Literary theory, for example, requires not only experience with the texts but also the cultural and historical context to parse what has been written. Chemistry, on the other hand, is predicated on the identification of elements and distinguishing elements from the molecules they form. Mathematics is predicated on the notion of a collection of things and on the simplest possible relationships and patterns common to different situations: number, order, position.

How then could I claim mathematics is not universal? the child/monkey/homeless man example is one direction to go. We could even conceive of an alien race who appear to be technological but who have no explicit mathematics, surely. A more interesting direction to substantiate this claim: we can consider mathematics a tool to compensate for weakness. In this way we can argue that mathematics belongs only to the context of beings that need to recognize common patterns in different situations. 

This point is subtle but worth investigating. Nor does it make mathematics very context-dependent. But it does suggest “universal” may not be a quality at all (in that nothing can posses universality), but necessarily only some sort of partial order.

How could mathematics be a tool to compensate for weakness? It can be argued that a great deal of our faculty for analysis originated in the survival mechanism of thinking something out, conceiving of a plan, understanding the nature of animals and plants and ourselves. Also of communicating the results of thinking things out. Of having language for any and all thoughts. If there was never a necessity of survival to communicate we wouldn’t have theories, as we do. Could a creature evolve such that it was able to perform seemingly mathematical or technological feats without having a theory of same? well of course! the birds have no theory of flight, yet to us this is a feat requiring great skill in mathematics and physics. Could the equivalent of flying birds exist, but instead as interplanetary flight? There is no reason why not. Even more so, could it be possible that a creature exists not merely born with an instinct to perform a specific, highly skilled action, but with a subconscious mind that does all the learning and processing and inventing, while a conscious mind observes only desires and suffers a very limited vocabulary, much like a child? This is entirely conceivable. One such creature builds the first space shuttle and upon being asked the question “why?” it says “well tommy built an airplane, it flies, so I wanted one too, but one that goes out into space.” Upon being asked the question “how?” I suppose it says something like “well it has to be able to go fast, and made from really good stuff,” all other details being obvious and trivial, and interpolated with the advanced instinct these creatures have.

So perhaps it is not as simple as a partial order, after all. My theory of consciousness as a partial order may have fallen apart, as well, though I have an idea to resurrect both, in new and more robust form! That’s for next post, but now I’ll conclude with God’s lack of mathematics.

It seems reasonable to talk about God as some sort of limit (in the mathematical sense) of the partial order of consciousness. Little omega, a limit ordinal. Still the singularity of God’s mental capacity puts God in a completely solitary context. I don’t buy the model of God and humans as actors with free will, capable of interacting and learning from each other. It makes no sense to me. An omniscient being has no need for communication, does not answer questions. I don’t know, this area is difficult because the concept of omniscience isn’t even comprehensible, may not even make sense at all. Still, to the extent it does make sense, recognizing patterns is not a necessity of such a creature. That is, if you know everything, you have no need to identify the commonalities between different atoms and classify them all according to a model atom and call them “atoms.” Instead you have every possible nuanced difference of matter available to your imagination, and you observe the universe without any reason to compare atoms, since no two are alike anyway. Interestingly, I imagine a sequence of creatures which converge to this God, and each has more and more theories and levels of mathematics and abilities to identify patterns, yet in the limit all these dissolve. Though, as I’ve suggested here, there are probably other sequences of creature which converge to the same God, without language or theories. At some point you have to know all the theories of mankind, possibly all the possible theories, since this is arguable anything in the universe which can be known.

Consciousness is a Partial Order

October 10, 2008


Consciousness is a Partial Order

I will now state the obvious about consciousness in animals: 

Consciousness is not exclusive to human beings.

This seems absolutely obvious to me, though some disagree. The argument for consciousness in other animals goes something like this:

Begin with Descartes’ skepticism: we know only that we are conscious; we cannot be certain that other humans think as we do. Next, we grant the possibility that other minds do think as we do AS we notice a remarkable fact: if other minds do in fact think, then they must perceive us similarly to how we perceive them. That is, although others are but images, sounds and textures to us, we would be similar images, sounds and textures to the minds within others, were these minds to think as ours does. From this perspective it is entirely reasonable to believe that other humans are conscious. (A super genius is more justified to be a solipsist, perhaps, but most of us are not).

To extend this to some non-human we still require some behavior on its part that we relate to: a behavior which, in us, is correlated to some thought, feeling or idea.

An example used in the philosophy of mind is the example of a dog chasing a squirrel. The squirrel jumps behind a tree just as the dog’s view is obscured by a bush. Then the dog runs to the tree and starts jumping against the trunk, barking up into the branches. In this case we say the dog thinks the squirrel is in the tree. It is less a statement about knowing what is going on in the mind of the dog and more a statement about recognizing and relating to motivation and intention, in noting to some extent the dog behaves as we do. Not entirely, but so much more than the tree or the rocks in ground, of which we can only figuratively ascribe intentions to. It is not figurative to say “the dog wants to catch the squirrel.”

This example can also substantiate the claim that some linguistic capability exists in other animals (without mentioning signing chimps or talking African gray parrots). With some mental conception of the world, e.g., having a concept of a squirrel or a concept of tree, we can argue that the dog has a linguistic representation of the world, although the language hasn’t matured to the point of being used for communication.

That consciousness exists in nature in more or less a continuum, which extends from flies to philosophers, is obvious to me. There exists a postmodern trend to challenge linear, hierarchical, simplified models, as well as anthropocentrism. In the case of consciousness I think the anthropocentric view is warranted. Also, a partial order exists, whereby we rightfully say “the dog is more conscious than the fly,” “the philosopher is more conscious than the dog,” (and ergo the philosopher is more conscious than the fly, but we knew this). Still we find that the dog and the cat have different ways of thinking, and frequently it is hard to compare the two with a binary, so instead we will say consciousness exists as a partial ordering, both on the set of species and on the set of individuals.

Some have argued that animals operate according to instinct alone, whereas humans have a second level of thinking called the conscious state. These people are misguided. The distinction between instinct and consciousness is difficult to pinpoint in humans, instead these terms are useful only in recognizing that certain thoughts and actions of humans seem much more automatic (e.g., catching a dropped egg), or cannot be fully explained (e.g., not trusting a new acquaintance), or are explained in terms of a primitive desire becoming dominant (e.g., as one might explain an act of adultery, “…I couldn’t help myself”). The distinction in humans between instinct and not-instinct is not clear. Nor do we believe that a human engaging in an act of instinct is temporarily unfeeling (the adulterer is capable of much feeling during the “helpless to raw desire” episode).

But what do they mean, “all animals act according to instinct alone”? This is to say the behavior of animals indicates that both slugs and chimpanzees act according to the same natural mechanical drive, whereas humans are radically different. This is clearly false. If a plague had wiped out all animal species but homo sapiens before the advent of language we might be in a place today to assert “consciousness is unique to humans; in fact humans are radically different from every other form of life we know of.” But this is not the case! We see facets of our own consciousness in the behavior of plenty of species of animals, be it language, community, desire, pain, capability to learn, etc.

The other problem with having such a blunt definition of instinct is that it cheats animals, each species, of having nuanced levels of consciousness. Indeed, a dog knocking a table out of the way to move through a doorway is less instinctual than is the decision to pee in 10 places instead of just one. Dogs have thoughts and actions which can be described as more instinctual or less instinctual, and this is a finer, more useful definition of instinct, than is “all animals act according to instinct alone.”

Of course, if consciousness is a partial order on the set of species then humans are the maximal element, and as such it is easy to distinguish this species from others, by defining instinct to be the level of consciousness strictly less than that of humans. This arbitrary distinction does not change the fact, argued above, that animals exhibit plenty of evidence for thinking in ways similar to how we think.

One thing that is particularly interesting in acknowledging the partial ordering of consciousness is the possibility for creatures which are greater than human. As I see it there is no finite limit to this ordering either. We can always conceive of a creature whose language is richer than ours, whose memory is fundamentally keener, with emotional states that make Mozart’s passion for music look like a cow’s desire for grass. 

This leads me to the next assertion, slightly less obvious or well formed: intelligence is to the fine tuners at the base of the violin, as consciousness is to the tuning pegs. They are really the same thing, but on different scales. A topic for another post…

Extra Reality

October 10, 2008

FRIDAY, APRIL 25, 2008


Extra Reality

A being will be called extra-reality if it is ultimately imperceptible to humans.  

Prototype: Pacman lives on screen, pacman is sufficiently advanced to discover fundamental laws of universe:

Power pellet grants invulnerability for time t, inversely proportional to level of advancement.

I travel at a constant speed, can stop only head on into a wall, etc.

Now, this fictitious pacman must be an early success of A.I, if it can analyze its own universe and deduce natural laws. However, I ask, is it possible for the pac creature to further deduce how it came into being, what programming language it is written in, which country the hardware is housed in, or even what a country is? If it is capable of growing in intellect and knowledge should it someday be able to accurately describe the motivations that drove some human programmer to give it life?

I’ll depart momentarily to discuss the problem with induction in the foundations of science, as I see it.

Science is a game of bouncing back and forth from deduction to induction and back. 

Observe. Abstract. Deduce. Predict. Observe.

The laws that are abstracted and function to predict and explain are discovered, and then reinforced, by observation. If a theory fails, it is modified or replaced. Then science never makes the statement “prediction P MUST be true,” but rather “if P is not true, our theory must be altered or replaced.” So questions of “why” can never fully be answered. Answers to “why” follow from deduction: a woman sitting in a cardboard box, left with nothing but canned food, water, and a few axioms of set theory, emerges after some months and surprisingly has theorems in mind that coincide with ours on the outside world. She follows deductive reasoning, concludes *[your favorite theorem]* MUST be true. In the process she has answered the “why,” as well.

But science cannot be practiced in this way. Each “why” demands another level of mystery, and until things become fairly reliable, really anything can happen. We need to observe in order to induce. But then anything that is beyond our ability to observe is beyond our induction, even if the phenomenon is causally relevant to that which we can observe.

For example? Pacman! It may very well be the case that the reason pacman exists is that a programmer was obsessed with mazes, early video games, and artificial intelligence and therefore created the pondering pacman. Pacman’s story of his own origins might include this fact, except for the fact that pacman was made in an entirely formal universe, which might have had a million different origins, each ridiculously distinct. So really the best pacman can do is say “space is black, the pellets are white, there are never more than 150 small pellets on a board,…,and there may be many things outside this universe, but we will never know.” As useless as that last statement is it is certainly more profoundly true than “…, and that is the entire grand unified theory of the universe and everything.”

At this point I may have a reader who is concerned I am working up to an argument for Intelligent Design. This would be unjustified guilt by association. Instead the reader may safely infer I think staunch atheists are fools, and flaunting a serious lack of imagination is something I find irritating if not just boring. I do insist that any reasonable philosopher can entertain ideas of creation without falling into religion’s old clichés , but that is not my point to make today.

[to be continued…]