I've just started reading The Language Instinct by Steven Pinker. It's a book directed at laymen to the field of linguistics. I had been enjoying how he shows that we are genetically-specified to acquire language. I concur with this notion. Then I came to the chapter, entitled How Language Works and here I had a problem. My problems led me to begin thinking about what at first sight appears to be a simple question.
Are there an infinite number of ways we can combine words of any language (in our case English) to form meaningful sentences? Simple it seems huh? Not so fast. Pinker repeatedly tells us yes, we can make an infinite number of sentences in this or any other language. He uses algebraic combinatorics to show this. But, to me this doesn't seem so. The implication of my dissent is that we have an upper supremum on what meaningful statements we can make in any language.
Take English, with 26 letters we can form a large finite set of words. I've read, from lexical sources estimates that English has approximately 500,000 to as high as 800,000 words. The combinations of those words arithmetically are seemingly infinite. But, they are not. If we specify that the sentences formed must be meaningful in this set it is limited. What do I mean by meaningful? We will discover that shortly. Pinker gives an example like this in the above-named chapter. The happy dog eats candy. He then goes on to show we could form another sentence by simply adding another happy to the given one. And based on this any given sentence could be infinitely extended by adding more words. So, using this example let's say we took the sentence The happy dog eats candy and added every single one of the 800,000 English words to the end of it, one at a time. This would amount to taking The happy dog eats candy + 1, The happy dog eats candy + 2....The happy dog eats candy + N, where N is every English word up to 800,000. So, we now have The happy dog eats candy + 800,000. We can also take 800,000 sentences times 800,000 sentences with one word changed and we would then have 800,000 x 800,000 or 6.4 *10^11 sentences. A very large number no doubt. Still a finite number by the way. Some of those sentences would be meaningless. We could go on to take other combinations, by taking 6.4 *10^11 one word at a time so 6.4*10C sub(1), meaning combination of 6.4*10 each word taken one at a time. And this process could be repeated infinitely. Would all of these constructions make sense? Pinker points out that integer sets become infinite by simply appending 1 to the end of any given number. Yet, constructions in spoken or written language must make sense to the listener or reader. Because combinations are infinite, doesn't make any combinatorial construction valid in a language. Why doesn't he see this? What he forgets to note is that the integers are mathematically called an infinite ring. Not being a mathematician, I can forgive him for this glaring mistake in his book. Anyway, we can take the numbers 0 to 9 and keep augmenting them by using a rule of repetition, not so with human languages. For instance, it is easy to imagine an integer that goes on forever. In fact, we all know a non-integer that does this, e.g. π. Can you imagine an endless word that would be meaningful in any language? Of course not! Take the word good, if we repeated like such and considered it ONE word:
good good good good good good good good good good good good good good good good good good good good good good
Going on forever.
Would it then mean that
this word is the greatest good? Yes,
the pun is intended here. No, it wouldn't mean any such thing. It would become
meaningless, idiotic, the babble of the insane, and physically impossible to do
for one thing. An infinite word is only
a conceptual possibility not a reality.
And with this comment, so are infinite sentences: just conceptual possibilities, that we can well understand, yet not real world constructions we can experience
and know perceptually. Underlining my point is the plain fact that the concept of a word necessarily means limitation. We conceive of a word as a finite unit of sound and meaning, which in linguistic jargon is a phoneme and a morpheme. So, too is a sentence. This is true because we only perceive a limited set of things and our conceptions are built from our perceptual experiences. No, it doesn't mean we can't conceive an infinity. It simply means, we can't experience such a state. And thus, nor can our languages allow us to experience such a state. Oh yes, there will be those that will go on to construct clever sentences that appear to go on infinitely. These sentences will be conceptually infinite no doubt. Most of these constructions use a subject that refers to itself and names an object. For instance: The man that saw a man, and that man saw a man, and that man saw a man. This sentence starts with a subject, e.g. man, then in each succeeding reference to this noun subject, it is referring to the previous noun subject man. It is a reflexive structure. The initial man subject is the base from which all others are generated. Sentences of this type have the verisimilitude of infinite construction. Yet, they don't. Sentences of this type run smack into some form of the Burali-Forti Paradox . This paradox occurs in the theory of transfinite ordinal numbers. It applies to numbers and just as easily can apply to words in a sentence structured in ordinal sequence.
Burali-Forti Paradox (BFP) can be applied to the simple infinite sentence cited above. First, we must have an idea of what this paradox means.
BFP creates a paradox from 3 existence conditions:
I'm not even sure if
all human languages are potentially infinite. We know new words are invented
all the time. That too is questionable because in order for word creation
to make an infinite set of meaningful sentences we'd have to do it in infinite
time. Moreover, the word creation couldn't be infinite, unless we started making
repetitive words using the augmentation rules like arithmetic. I believe we'd
start creating meaningless words. Example: Say we want a new word. Take
the word fell and add another l to get felll, but that
would not be a meaningful English word. Since the set of letters we have now is
limited (26 in English) the corresponding combinations (words) are too. Unless
we allow infinite repetition of letters, like we do with integer arithmetic,
infinite creation of new words would lead to a meaningless arrays of symbols.
There is a philosophic implication here: Within infinite sets, are finite sets.
If we allow infinite members to be in a set (by repetition of letters or
other devices) we begin to lose meaning. It's worst if we have infinite symbols
(I'm referring to the letter symbols). Say we wanted the 26 letter English
alphabet increased infinitely. This could in fact be done. Add another curve to
B, or an additional line to T, etc. We could increase the
primitives that compose words infinitely. If we have an infinite variety of
symbols, we could never make one symbol unique. If an infinitude of symbols
could stand for one of our most basic pronouns I, how could we ever
uniquely specify ourselves? By analogy, if we had an infinity of words, how
could we ever specify say tree? This is similar to the Axiom of Choice
in set theory mathematics. This axiom says in non-technical language that there
must be a way to specify members of a set. For example, in a set (a,b,c) we
must able to indicate that a is not b and b is not c, or c is not a, etc. The
paradoxical consequence of an infinite alphabet is that every member WOULD BE
UNIQUE, and thus no member could be used uniquely. Setting aside whether
there are infinite words, languages as they stand now, certainly don't allow an
infinite set of sentences if those sentences are required to be meaningful in
said language. Here is where I know the objection can be raised: just what do
you mean by meaningful? In fact, Pinker again is himself helpful on this
point. He gives an example from Noam Chomsky of a grammatical correct yet
meaningless sentence. Colorless green ideas sleep furiously. This is an
example of a sentence I'd say could be formed, that is not meaningful and thus
not a valid sentence. Now what do I mean by meaningful? I am referring to sentences the we can experience in our perceptual world. This perception applies to imagined sentences too, an imagined state of being is still a perceptual state of being. It is meaningful in the sense that it is a state of being that we understand through our experiences in the physical world, no matter how counter-real it is. Think of a dream in which you have encounters that are counter-factual, for instance you fly in the air, you see beings that don't exist in the concrete physical world. I would still say, this dream state experience is rooted in your conscious experiences, it is determined by it and thus subject to it. You could create ideas, images, even fantastical tales of things you never physically experienced, yet you could never do this, without reference to the perceptual experiences you have already had, in the language you speak and read. The very nature of your state of being traps you in this respect. In simple terms, to have infinite linguistic license, we must be infinite beings, with experiences of infinite states of being. This we do not have. An infinite being would have no need to use the 1st person personal pronoun I to name itself, it could have.. well... an infinite number of names for its self-reference. We mere humans can't. This is the point of meaningful.Of course there is no infinite being by the way. The profound conclusion of
what I'm suggesting is this: We can only say so much in a given language. There
is only so much we can actually write and say. It's huge in its finitude,
it is not endless. To get an idea of the enormity of our language look at Robert Mannell's estimate, a linguist from Australia, whom also believes in the finitude of language Infinite Number of Sentences. I love this site and the arithmetic estimate approach it takes. It makes us realize how great our human capacity for language is, while noting that it's not the God-like infinity we believe. Any mathematician knows this for sure. Dr. Mannell gives give you its arithmetical scope. Some would make the leap and declare: well
that means we can only have a finite number of thoughts too. This would be a fallacious deduction. Pinker again is instructive here. He shows that words and their supersets, sentences are not at heart what thought is. Thought is more a brain
activity that recognizes manipulation of representations of reality. For
instance, you can think left and right without words for those geometric
directions. Or you can feel thirsty without having a word to express this desire for water. What it does imply is not every thought can be articulated. I can
see this as being true. Intuitively, I
had thoughts or to use the colloquial term, feelings that I can not
express in spoken or written language. Even more interesting is the question: do we need to name things to have a concept of multitude? Like direction in space, we perceive separate objects with our eyes. Can we understand the concept of numeration, without a system of naming those multitudinous things? I believe we can. Though, it would be difficult to recognize 10 things without 10 names for them, but it could be done. Now I'm moving off point so let's get back to the real issue. I'm actually not sure if we are limited in
our thought process. We could be. If the idea that our brains are discrete
state machines (sometimes called finite state machines), which can process
units of input to our brains, is true, then there is an upper limit on what we
can think too.
Can we travel back in time?
Below is an article I wrote about time travel based on the Lorentz Contraction in Special Relativity. Yes, we can travel back in time, oddly the result of time travel is unexpected.
English is a Germanic language and thus still compounds its words. Is it possible to take any English compound word reverse its parts and form a meaningful sentence with the new word? I look at 190 bi-compound words in this brief study below and find some not to be expected results. I plan to augment it later with some projections as to how much compounding we can do with English. Again it's not infinite. Not every English word can marry another and make sense. It gets very subjective what we can consider a meaningful compound. For instance take swallowdead, which I just made up, is that meaningful in a poetic sense.
Brief Analysis of English Compound Words
The Lorentz Contraction and Reverse Time Travel
Return to Portal Philosophies, Science, Mathematics, and Music
1. Every well-ordered set has a unique ordinal number.
Well-ordered means the set is arranged from least number to greatest. The unique order number is the number that defines the place of every number before it. Thus a 5th ordinal number applies to a set (1,2,3,4)
2. Every segment of ordinals has an ordinal number, which is greater than any ordinal in that segment.
This means a subset of numbers arranged in least to greatest has an ordinal that is not a part of it and must be greater than its ordinal in the subset. The example above applies again. So, the subset (1,2,3,4) of natural numbers has an 5th ordinal, which is greater than any of the ordinals in the set. E.g. the number that names all the numbers in order before itself.
3. There is a set B of all ordinals in natural order (least to greatest) that is well-ordered (has a smallest member).
Here is where a contradiction arises. If such a number say x in B exists, it is greater than itself, by condition 2. This contradiction arises precisely because set B is an infinite ordinal set. Being such a set, an ordinal for any subset of it will always have a successor ordinal that is larger than the ordinals in this subset, and not a part of this subset.
We can apply this to the sentential string example. To see this, we will number the objects of these sentences as such:
The man that saw a man (1), and that man saw a man (2), and that man saw a man (3), and that man saw an Nth man (where Nth represents every succeeding sighting of a man to infinity).
Generating this set infinitely, we would have the natural numbers, well-ordered like such: {1,2,3,4,5,6,7,8,10,11,12,13,14,15,...∞}. It should be noted here 0 is not in the set. Since there is no 0th man that saw anything. The Nth ordinal of this set would have to be not in the infinite natural set by the Burali-Forti Paradox. This would mean that there exists a man such that, that man DID NOT see the previous man in the order specified. Which, in turn would mean that even conceptually this infinite sentence would break down in its meaningfulness. Or to state it differently, it would not be true and thus not an infinite sentence. For those that follow Cantor's proof of transfinite numbers, I must point out the following. Cantor proved that there are infinite sets that transcend infinity, he was never able to determine the order of these sets. Thus BFP is not subject to this proof. Even if there is an Nth man beyond the infinite that saw a man, we can not determine his ordinality.
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