There is a paradox in metamathetics that asks the thinker to conceive of “the smallest number not describable in fewer than eleven words.” The paradox arises when the thinker notes that that description itself is ten words long, and thus any number to which it refers is, indeed, describable in fewer than eleven words.
This particular paradox is shown to be a false paradox when it’s pointed out that the number doesn’t necessarily have to exist. That is, any number which it describes would be paradoxical, but if that number doesn’t exist, there is no paradox.
Thinking along these lines, however, engenders thoughts about the whole problem of paradoxes of expressibility. Medieval theologians (such as Cusanus) endorsed the notion of God as a being so great, so Maximum, that complete comprehension of Him is impossible, that we cannot possibly understand God. Similar ideas have been expressed by others (the Taoist notion that “The Tao which can be spoken is not the eternal Tao” comes to mind), but the essential notion is that one is contemplating the uncontemplatable or expressing the inexpressible.
Many observers have pointed out paradox here. This has been taken in a variety of directions, from rationalists using it as justification for a reductio the denies God, to dialetheists using these kinds of problems as justification for a worldview wherein some contradictions hold. But these vectors rely on the notion that this is a true paradox—a notion which does not, I think, hold water.
Let us take the most easily discussable instantiation of this: a concept that is inexpressible. Lao-Tzu wrote that “The Tao which can be spoken is not the eternal Tao.” This line has been translated in ways that are less paradoxical, but this is one of the most famous, and a version with which most Taoists are familiar.
The problem, as has been pointed out, is that this seems to imply that anything said about the Tao is false. If anything said about the Tao is not true, we’ve entered a sort of Liar’s Paradox, where that statement (as well as any of the other text relating to the Tao) is also not true, and therefore of zero epistemic value to us.
The failure, here, is in a confusion of accuracy with entirety. That is, the Tao is, according to Lao-Tzu, something whose fundamental nature, whose entire quality, is beyond articulation or expression. We cannot contain it in words. But that does not mean we cannot contain parts of it in words, or that it does not have some describable qualities.
Such is the case with the Tao. It is impossible to express it directly and entirely. But it is so infinite that it does include certain expressible traits, and thus we are not perjuring ourselves when we claim that the Tao cannot be spoken.
A similar chain of reasoning holds when articulating apparently paradoxical qualities attached to other objects. If we say that God is too great to be conceived of, or more directly if we define God as that which is too great to conceived of, we arise into apparent paradox by pointing out that, in thinking of “an object to be great to be conceived,” we are conceiving of God, and thus are entering paradox. But that misjudges what is happening when we conceive of something beyond conception. When we conceive of an object beyond conception, we’re instantiating a placeholder object and giving, by fiat, the quality of inconceivability. Obviously, this object will be paradoxical, because the only quality it has is its inconceivability, and thus in conceiving of it with that quality, we are conceiving of it in entirety—this stands in violation of that quality, and thus we get paradox. But God, when defined this way, is understood to be MORE than just an unconceivable object; God is Creator or the Absolute Being or whatever else you want to add to God’s totality. But one of God’s qualities is that It is beyond full conception or understanding.
Paradoxes of these sort also resolve when we remember that expression (and even, if you construct it so, conception) is a process of representation. That is, an uttered phrase points to an actual state of affairs or an actual entity; the phrase itself is separate from the actual entity. The words are as a finger, pointing to the moon. We see paradox in these articulations when we confuse the finger for the moon.
In the West, Confucius is often understood to have taught the value of authority and hierarchy. He is often cited (and criticized) for his patriarchy, his insistence on natural social positions, and his demand for imposed order.
But this is not strictly what Confucius meant. Confucius was not trying to impose order on the people, he was trying to build an order―one that could last and was healthy, moral, and effortless. He describes in his works a Way that David Hinton elegantly defines as “the effortless process of human society functioning according to its natural Ritual structure.” So when Confucius gives immediate ethical commands, he is strictly demonstrating those actions which should be exercised to follow the natural Ritual structure of society. This is explicitly demonstrated in chapter two of The Analects, which contains the following passage:
When Lord Meng Yi asked about honoring parents, the Master said: “Never disobey.”
Later, when Fan Ch’ih was driving his carriage, the Master said: “Meng asked me about honoring parents, and I said ‘Never disobey.’”
“What did you mean by that?” asked Fan Ch’ih.
“In life, serve them according to Ritual,” replied the Master. “In death, bury them according to Ritual. And then, make offerings to them according to Ritual.”
Were it simply obedience itself which was necessary, there would have been no necessity to explain what was meant by “never disobey.” As it were, the edict to never disobey stems from Confucius’s observation that part of the natural Ritual structure of fifth century BCE China was an acceptance and respect for authority.
We can see here that the demand for obedience does not make Ritual, but it instead comes from that which already is Ritual. Ritual itself is a far subtler and deeper phenomenon. Hinton defines the word Li, herein translated as “Ritual,” to mean “the web of social responsibilities that bind a society together.” The spirituality inherent in the word “ritual” (and the Chinese Li) stems from Confucius’s desire for those social responsibilities to be enacted with a fullness of intentionality often reserved for the religiously sacred.
It is not Confucius’s intent for any of the social responsibilities he describes to be enacted by rote or to simply go through the motions. In fact, such a notion is actually quite repulsive to him:
The Master said: “Governing without generosity, Ritual without reverence, mourning without grief―how could I bear to see such things?”
It’s the lines like these which show that Confucius aimed to fill every social act with a deep-seated awareness of act and intentionality. It was his desire then, first and foremost, to instill in people a sense of awe and profundity in regards to their own activities.
But it was not his intent to teach a doctrine, a series of immediate ethical rules to follow and by which to create a positive, healthy society. He advocates throughout his work a sense of doctrinal flexibility, a way of looking at the world wherein the state of things as they are lead to quick, effective, and healthy decisions.
His immediate ethical teachings advocating authoritative respect and social positioning are a result of his observations about the state of things in ancient China. In a modern Western society, the natural Ritual structure is very different, and therefore the immediate ethics, which stem from the Ritual structure, are likewise altered.
In an open, democratic society the primary virtues of a good citizen are not humility, respect for authority, and knowing one’s place, but rather participation, understanding, and civility. In a society ruled by the laws and the people, it is not correct that one should follow the commands of the authority simply because they have authority. What is correct, what is in accord with the natural Ritual structure of such a society, is to be engaged in the community, critical of the political status quo, and respectful of the rights and opinions of those who are also in the community.
At first blush, this may seem to rebel radically against that which Confucius taught. But Confucius himself was no passive follower. He was engaged at all times with his surroundings, saying that “questioning is itself Ritual.” In his engagement with the absolute dynastic power of ancient Chine, he discovered that hierarchy and social position were important in maintaining a stable society. But in a system where power is diffused and it is widely understood to the people’s responsibility to be critical of the government, a different Ritual structure is present, and that Ritual structure leads to a different genre of immediate ethical social involvement.
But remember that the religiously-loaded word “Ritual” is used to describe this underlying social fabric. It is not enough to simply vote and to go through the motions of democracy. We must engage in that activity with a deep reverence, passion, and deliberation. The Ritual order of a democratic society seeks open participation and involvement from the people, so it is up to the people to execute that charge with zest and a sense of the weight of that activity. In this way can we maintain the strength and vitality of our democratic societies for generations to come. The Master has been influential in Chinese thought for millennia, but his wisdom is not exclusive to Chinese society. Even in the West we can use his words and his wisdom to help strengthen and consolidate our social order, even though the specifics of structure and ideals differ greatly.
We do not have access to the objective world. All of our knowledge, of anything and everything, is filtered through the subjective apparatus of conscious. We cannot speak of the objective world, we cannot know the objective world. As such it does not meaningfully exist as something separate from the subjective world. The subjective, perceived world, then is extended outwards and fills the void of “truth” left by the hole that was the objective space. The metaphysical resolution of the phenomenal/noumenal dichotomy can resolve into a spiritual resolution of the Self/Other dichotomy fairly easily. The phenomenal realm is the world as experienced through the self, the noumenal world is the world of things in themselves. If we strip away the difference, what we are stripping away is the separation of perspectives. Those perspectives that are resolved are “my” perspective and “not-my” perspective.
At this point, Self and Other no longer stand dichotomized.
Thing is, though, that in our phenomenal experience of the world, we (can) still perceive a separation between Self and Other, and the phenomenal world of things-as-they-appear is all there is, which means that the visions of phenomena are “true.” So the Self/Other dichotomy stands.
But we already resolved the Self/Other dichotomy, didn’t we?
At this point, we have two logical options. We can either accept this as a reductio ad absurdum argument that disproves the original thesis (that we do not have access to the objective world) or we can accept that paradox holds.
But logic itself has shown us that logic does not hold all the answers. This is the great epistemological value of Goedel’s Incompletness Theorems. And mathematics has proven contradictions from otherwise accepted axioms (Banach-Tarski Paradox proves, without cheating, that 1=2). From logic’s own reasoning, we must recognize that some truths exist beyond logic’s capacity to explain or understand.
After a two-week hiatus, I present to you the second installment in my series of papers on the relationship between our knowledge and the truth.
Science, providing for us as it does a method with which we can predict and manipulate the world around us, offers a solid and concrete argument for its own possession of the truth. But we must be wary of science, in this regard, for it plays to one of our most basic intellectual drives: rationality. We seek, natively, to make sense of the world. Wittgenstein said that it is impossible to have an illogical thought, and many have contested that envisioning a “colorless green unicorn” is impossible because of the nonsense and paradox inherent in the phrase. It is, however, my contention that the position that the rational inquiry generated by science is somehow superior in its accessibility to the truth stems from an untestable chauvinism and prejudice of the mind.
Science has always been concerned with the testing and honing of its instruments and technology in order to find more accurate results, or to be surer of the accuracy of the results already received. But the sciences—indeed, most fields of inquiry—have been historically lax in testing the source of all these tools and instruments: the thinking apparatus of the scientist (or human in general). The field of study which does concern itself with this apparatus, psychology, is not even yet fully emerged and formed as a science in the Kuhnian sense, and as such science cannot yet claim to be examining its primary tool for inquiry.
This “truth” that comes out of science is marked by rationale, reason, and logical explanation. But if, as some have suggested, our minds are only capable of logical thinking (whether we actively understand the logical path taken or otherwise), it would be necessary for science to reflect that logicality, irrespective of logicality present in that which science tests.
To structure the attack slightly differently, consider this: rationalists and atheists often criticize religious justifications for the existence of God because of the circular reasoning that presents itself in some of the arguments, i.e., “God exists; I know this because the Bible says so; I can trust the Bible because it is the word of God.” However, the assumption that posits the truth of rationality (of the a priori “pure logic” sort and the a posteriori sort of which science is a major part) is equally circular: we know that the world is rational because reason matches up with our perceived perception of the world; reason and rationality stem from the mind; our access to our perception of the world is filtered through our minds; that apparatus which performs the act of reason is the same which filters and understands the world. In both circularly reasoned cases, we have the source of “truth” claiming itself to be true. The case of the human mind is even more devilishly circular, though, because as we turn to examine it for veracity and accuracy, we must examine it through the lens of our own minds, generating a device whose total accuracy and relation the real world is essentially untestable, because we need to use the device in question to test its own accuracy.
What we then find ourselves with is a situation in which we have no reference point. That is, we have no “objective” truth against which to compare our notions of truth to determine how accurate they are. We use notions like “explanation” and “prediction” to attempt to approximate a reference point so as to judge the accuracy of a scientific theory, but what that essentially amounts to is not a measure of the correlation between theory and phenomenon, but rather a measure of the correlation between theory and the prejudiced interpretation of phenomenon.
On a quick aside, it should be noted that it is, despite the loud claims of rationalists, entirely possible to entice the mind to escape rational limitations. Many Eastern religious and pseudo-religious practices aim towards exactly such a goal, and there is ample anecdotal evidence to suggest that many entheogenic practices lead relatively quickly to a temporary escape from the rational. People who have experienced such states oftentimes report intellectual and sensory abandonment of such basic presuppositions as causality and dichotomy. While this is not the place for the discussion of such matters, the fact that such experiences are possible and even relatively common among certain sectors of society is important to note when we discuss the relationship between the mind and the world and science’s role in that relationship.
Returning to the role science plays in attempting to illuminate for us some degree of truth, it has been argued that science portrays the truth of phenomena, and that its relationship to the truth of a noumenal realm is unknowable and, in the end, irrelevant. However, phenomena being understood as events as they appear to an observer, we have a very real problem once we have established the inability of an individual to judge the accuracy of their own mind: the observer can’t even be sure if the phenomenon in question is appearing a certain way. While this seems a bit ridiculous at first, we must recognize that scientific analysis of phenomena happens, by necessity, after the phenomena in question have occurred, with the scientist operating by memory and recordings. Even non-skeptics will agree that human memory is extremely fallible, and often flawed. Memory and recordings both are subject to that undetectable super-imposition of a rational prejudice discussed earlier. This moves the data used for scientific analysis into the same realm as our entire “truth” notion: those data now must exist in a truth-space without reference point. It is not that the data are not accurate, but rather that they are of unknown and unknowable accuracy.
At this point, clearly, all knowledge, not just science, is suspect. With our inability to accurately gauge the accuracy of our own minds, we cannot with any degree of precision ever know the accuracy of any data, analysis, or thought we encounter. So where does that leave science, the monolith of rational pursuits of truth? My claim: in the same world as other untestable truth pursuits.
If we roughly separate modes of thought into the rational, the intuitive, and the epiphanic, we can also roughly attach certain pursuits of truth to each category. Science would be a rational pursuit of truth, “common sense” could be considered an intuitive pursuit of truth, and religion could be considered an epiphanic pursuit of truth. In all three domains we see the exact same limitations that were considered in the discussion of science: the mechanism that we use to pursue truth has itself unknowable accuracy, leaving the pursuit in the domain of the unknowably accurate. This means that the truths originating from these various sources of knowledge have qualitatively identical claims on our acceptance of those truths as, indeed, true. That qualitatively identical value is, roughly, “indeterminable.”
Now, it should be clearly understood that I am not (nor could I possibly) argue that science or any other form of truth-seeking does not give us truth. What I claim is that we can’t know if it gives us truth. That is, the modes we use of logic and reasoning may be in perfect accordance with the modes of operation of the objective universe, but our lack of access to the objective universe—and our lack of access to an objective understanding of our own subjective mechanisms—means that we cannot possibly ever know with any degree of certainty how accurate our perceptions and thoughts are. From this examination we can state not that science does not give us truth, but rather that science gives us only unknown truth.
This is the first in a series of essays on the nature of truth and our relationship to it. Enjoy, dear readers.
A definition of science being something often discussed and hotly argued, I shall not attempt one here. Instead, a quick list of the functions of science shall be given. First, it seems clear that science purposes to approach some sort of understanding of the world. A weaker version of this thesis is that science gives us a framework within which we can operate and that we use as a filter for our interfacing with the world. Second, science allows us to make predictions about a future state based on initial conditions and the rules which we generate through the pursuit of science. Third, science allows us to manipulate the world around us and shape it as we so choose.
There is a fourth function of science, as well, that deserves mention. Science can also be a purpose unto itself. There are those for whom the scientific method of inquiry is simply a joy to behold in and of itself, and for its own sake. There are those for whom scientific inquiry is of itself a beautiful process. Indeed, I think it can be safely said that most people devoted to science are at least partially enraptured with the pursuit for that very reason. This reason, however, is more concerned with science as an aesthetic exercise than it is with science as a knowledge-oriented exercise, and it is knowledge with which I wish to deal.
It is clear how the third pursuit of science—manipulation—stems from the second pursuit—prediction. It is our ability to predict natural phenomena based on rules and initial conditions that allows us to generate those initial conditions necessary to create a desired effect.
It is further clear that science’s ability to make accurate predictions stems directly from the fact that it provides a framework within which to understand the world around us. That is, without a framework of understanding we could not possibly have a reference point from which we could being making informed guesses about the future. There can be no “informed” without “information.”
So it seems that the primary pursuit of science is to provide us with some sort of articulable understanding of phenomena that can then be used for other, secondary aims. At this point, we must ask ourselves about the nature of the phenomena that science understands and the nature of that understanding.
The phenomena that science examines are exactly that—phenomena. They are events in the world of perception, including that portion of the world whose perception is only open to us due to the advancements of science (phenomena at tiny scales, great distances, or in domains such as extremely low-frequency electromagnetic radiation serve as examples). To fall back to the Kantian dualism, it is impossible for science to investigate the “noumenal” realm, as it were. Despite the fact that science allows us to observe events that would be otherwise outside of our domain of experience, we only have phenomenal access to those events, and we cannot through science access any sort of “deeper” truth, penetrating past phenomena into noumena. Indeed, because science deals precisely with the world as we interact with it, science lacks the possibility of even determining if there exists a deeper, noumenal world.
An immediate reaction to this is to question the value of any understanding that comes from science. We must ask ourselves whether or not, given this inability of science to discern “true” nature (or even to discern whether or not there is a true nature to discern), science is inherently impotent. Stated lofty goals of “objective” understanding and a quest for “truth about the natural world” are left by the wayside, bringing the scientific pursuit back down to a trifling subject with limited and restricted power of understanding, just like any other human endeavor.
But this need not be the limiting amputation it at first appears to be. The phenomenal world is defined as the world to which we have access. Putting aside unshareable and untestable (to an outside observer) mystical and religious experiences, it is impossible for us to gain experiential access to any world beyond the phenomenal. As such, from our subjective position, it is not incorrect to act as though the phenomenal world is the only world that exists. While a “noumenal” world may in fact underlie the world which we experience, since we cannot access or interact with it, it is safe to operate as though it did not exist, much the same way the existence or non-existence of Mars is not something which most of us must take into consideration in our day-to-day activities.
As such, science can give us understanding about our world, the world in which we live. It cannot tell us about the foreign worlds to which we do not have access, but that capability is not demanded from it. Nor is it relevant to our operation inside the phenomenal world. Since we do not have to operate with consideration of the noumenal world, science can seek to give us a picture of the entire world with which we interact, and thus of our entire world.
I have already stated that science gives a framework with which we may understand the phenomena that present themselves to us. But perhaps more intriguingly, science allows us to perceive new phenomena and even to create ones that could not have existed beforehand. It is intuitively understood that science comes from the world, or rather that the world generates science. That is, the phenomena present themselves as facts and we coalesce those facts into scientific theories. But science also goes to the world, changing and manipulating existing phenomena and creating entirely new ones—television is a phenomenon impossible without scientific knowledge. In this respect we see that science as a field is not simply the extraction of information and principles from the phenomenal world, but also the injection of phenomena into the information and principles at hand.
In a very real sense, science as an exploratory field is in dialectic repartee with the phenomenal world around us. As the world shapes science, so science shapes the world.
Science itself understands the capacity for such intermingling of causation. The Heisenberg Uncertainty Principle rests on the observation that as we observe and understand certain phenomena, we must affect the objects of those phenomena and thus create change and new truths in our accessible world.
This state of dialogue between science and phenomena leads us to a question of finality. If the reader will pardon the trite expression, where does it all end? Will it ever be possible for science to fully understand all phenomena with which we can possibly come into contact? It would have to seem as though the answer would be “no.” Even if science gets a complete hold over all phenomena that could have occurred before the development of science (although we’re blurring the line between the phenomenal and noumenal here), science itself can continue to create new phenomena that puzzle the researchers involved. If the phenomenal universe operates on a finite set of principles, it is entirely possible to accept the idea that we could master all the fundamental principles, but when we observe a new phenomenon, we would still have to tease out which principles were involved in the phenomenon and how exactly they manifested.
Another roadblock to total scientific knowledge will be human activity. Even if science progresses so that it understands the basic, fundamental principles driving all phenomena, the collective ambiguity that is human nature will serve as a monstrous obstacle to total understanding of everything. The placement of the individual scientist or a collection of scientists within the fabric of society will limit the objectivity of the inquiry into human nature, and the understanding of people by people will always have very real epistemological concerns, centered around Wittgensteinian concerns about the limitations of an inherently subjective perspective.
Despite these blocks in the understanding that science can deliver, science is still providing us with a very real understanding of the world within which we operate. As such, it is fair to state that science provides us with some form of “truth” about the world. It is a truth that does not expand to all possible domains, but that does not change the fact that it is true. A triangle’s internal angles do not have to sum to one hundred and eighty degrees in any possible space, but that does not render false the assertion that a triangle in Euclidean space does have that property. In the same vein, while science may tell us nothing about any noumenal world, it does give us truth about the phenomenal world, which is by definition the domain with which science is concerned.
The Luddites held that the changes brought on by the Industrial Revolution would lead to a moral degradation of society. Many modern Luddites often argue the same thing—that technological progress opens up the door for deeper and deeper immoralities and creates a darker and darker world.
This is not an incorrect assessment. Nuclear technology has given mankind the capacity to blow itself into oblivion, multiple times over. The internet has facilitated a whole new genre of ways to screw people over, from identity theft to bank fraud and internet-distributed child pornography. The printing press has been used to teach people closemindedness, destructiveness, and hate.
But that’s not all that any of these technologies have done.
Nuclear technology has also given us nuclear power, a clean and efficient energy source that could effectively power the world. The internet has given us nearly infinite founts of data and has facilitated the spread of messages of peace and goodness and charity like no technology before it. The printing press has been used to teach people to think and to provide basic literacy throughout most of the world.
The Luddites, see, saw a half-empty glass. But progressivists who see only the good of technology are like bright-eyed optimists. Oftentimes more fun to be around, yes, but also oftentimes blatantly ignoring the risks associated with various ventures. See, technology progress is neither a great evil nor a great good. It, like most everything else, simply is.
Every new technological advancement brings with it a potential for good and for evil. Recently, biologists have figured out how to manufacture brand new species using entirely fabricated DNA. The potential use of such technology for good is extraordinary: we could theoretically build a bacteria that fought along the immune system to ward off unwanted infections; we could create colonies that could break down oil into non-harmful components and then, when their job was done, simply deactivate and die off; we could use this technology to scrub the atmosphere and help fix much of the damage we’ve done. At the same time, the entropic, destructive uses of such technology are equally rife: an infection could be engineered to only target people showing certain genetic characteristics unique to a particular race; rampantly reproduced cancer-like colonies could be designed to choke out entire ecosystems; species could be built to produce toxic gases to render certain vast stretches of the planet inhospitable.
This technology, then, is neither good nor evil. It simply IS. It increases the potential action pool of the human species. Gives us more room within which to operate. Yes, the potential power is terrifying. But yes, also, the potential power is glorious.
This is not just true of technology. Ideas on the whole often exhibit this characteristic of an amorality, of a simple being. Religion is often touted by secularists to be a great evil, the cause of much of the world’s great terror and suffering and strife. But religion has also caused great good. Religion has led many who were suffering into a place of peace. Religious groups activate and mobilize for disaster relief faster than many governments, and often in greater numbers and with greater aid. Yes, religion has been used to preach hate, intolerance, exclusion, and dogmatic narrow-mindedness. But religion has also been used to preach love, tolerance, acceptance, and inquiry. Religion simply is.
It is up to us, as individuals, to wield these weapons properly and for good. The problem, of course, is that we have so many different ideas of that which is “right” and that which is “just.” But we must conquer this in ourselves before we can conquer it in others. We can only begin to use our vast power rightly when we understand, personally, what that idea means. I have my notion, yes, and perhaps you have yours.
But I have faith in the species. I have faith that we’ll do rightly. I have faith in your ability to choose correctly. And even if you or anyone else chooses incorrectly, I have faith in the power of the species to fight back and fix the errors committed by its brethren.
I am no physicist, and I don’t read physics journals. But through the swamps of academic literature, vague abstracts and summaries often reach the common man. In recent days, I’ve noticed an upswing in the amount of discussion going on in regards to models of gravity that differ from the currently accepted Einstein-Hilbert action of general relativity. From different actions that are still relativistic to completely different models of gravity that break away entirely from Einstein’s famous curved-space model.
The interest, here, lies in the fact that general relativity has been having some problems of late. Relativity and quantum mechanics both explain their respective domains extremely well—relativity usually handling the very, very large in space and quantum usually handling the very, very small. But for some time it’s been known that when the two theories intersect, problems arise. The classic example of a problem domain is the deep mechanics of black holes, where physicists must deal with systems that are very, very massive and very, very tiny.
General relativity has also led to other problems, from the Universe beginning as a singularity (which is mathematically undefinable) to the whole Dark Energy/Dark Matter problem.
In short, while general relativity brought us deeper insight into the mechanisms of gravity than any other model before it, it opened questions that have stood open for decades, defying any attempt to solve within the relativistic model.
Thomas Kuhn, the man who wrote The Structure of Scientific Revolutions, the seminal work on the growth and development of the sciences, said that when a scientific model begins defying attempts to solve certain problems, the field moves into a state of extra-ordinary science, wherein scientists begin to get more creative and daring when proposing solutions to existing problems. In science, it is usually held that theories should be minimally modified to solve new problems. That is, if a model must be modified in order to solve a new problem, the modification which least changes the model is ideal. During a period of extra-ordinary science, however, scientists will escape that rule and begin seeing what happens when drastic, radical changes are made.
If the unsolved problems are then solved by one of these drastic, radical changes to the model, we enter a period of paradigm shift, where over roughly the span of a generation (usually), the new model is adopted and then fleshed out, changing the space of the scientific discussion inside the relevant field permanently.
It appears as though we may have begun entering into gravity’s newest period of extra-ordinary science. With radical new theories of the gravitational mechanism sprouting up, the academic community as a whole will begin to be more accepting of radical changes to the model. As the academic community becomes more accepting, more scientists will be daring enough to consider radical changes. And, eventually, one model may click.
We grew up with Einstein’s general relativity as the model of that basic force which keeps us attached to the earth. In most cases, so did our parents and grandparents. But our children might not. Our children may know that gravity is the result of some emergent property of entropy or they might know that the Big Bang is part of an infinite gravitational cycle. They might know this, because science might know this. But we know that general relativity is right, don’t we? Science knows that general relativity is right, doesn’t it? It makes one question what it means to “know” something, doesn’t it?
There is a debate that takes many forms, but one of those forms boils down to the question of cooperation versus competition. In a macro sense, is it more effective to bring interested parties together to cooperate towards a goal, or drive them apart and incite them to compete?
Competition has this tendency to create friction, to separate people, to create artificial boundaries between operating agents. It has a tendency to make us prejudicial and hateful. It generates, often, irreconcilable differences between those competing as enemies. With good reason—in a competitive framework, where a win for one side implies a loss for the other, it’s difficult to cause others to lose if one likes those others.
Perhaps even more damaging is the truth that, in competition, competitors often lose seek of the original aims of the competition and the competition becomes the goal unto itself. We see this in sports when the goal of a player moves from the young, idyllic, “play to have fun” to the older, aggressive, “play to win.” We see this in business when the goal of a corporation moves from providing a valuable good or service into maximizing profit.
But empirically, we see great good come from this competitive mindset. Fantastic technological leaps forward have resulted from a necessity to innovate in order to keep up with the rest of the field in essentially every industry known to man. Professional level, competitive sports have resulted in athletes showing almost super-human talent and progressing to positively remarkable levels of skill in the pursuit of the win. Competitive admissions to schools and universities have resulted in those schools teaching and forming the cream-of-the-crop.
Could such things have resulted from totally cooperative social systems? I don’t think so. We see, obviously, much progress coming from cooperation. The International Space Station is a cooperative effort between nations all over the world. The internet’s odd, new open-source community, wherein thousands of people cooperate and volunteer their time for a common goal. Cooperation is, clearly, at least sometimes effective, and it rarely presents the collateral costs that are engendered by competition.
The end result is that, clearly, competition has its downfalls. But for sheer getting-shit-done capacity, it’s really not easy to beat. The declaration that competition is universally hurtful, universally cost prohibitive, is patently incorrect. But so, to, is the declaration that competition is always the best response. The most effective path, then, would be to keep competition and cooperation mixed and balanced against each other in harmony. What we, in building our lives, and our civil servants, in building the social structure around us, need to do is carefully and intelligently decide where to build competitive environments and where to construct cooperative frameworks. How we decide where to build which framework is a question that must be left to the intelligence and sense of the designer. But with practice, we can develop the clear thought and open eyes that are necessary to correctly see the situations and build the solutions necessary to move forward.
I begin with mathematics.
The study of the vastness of numbers has led to powerfully interesting results. If we begin by examining the Naturals—the counting numbers 1, 2, 3, etc—we can quickly and easily see that there exist an infinite number of these objects. If we next examine the Reals—those numbers existing on a continuum, containing all natural numbers, all numbers expressible as a fraction, and all numbers NOT expressible as a fraction, such as pi—we also see, relatively intuitively, that there are an infinite number of THESE objects. But a famous result shows that there are more Reals than there are Naturals, with the magnitude of the Naturals referred to as a “countable” infinity and the magnitude of the Reals called an “uncountable” infinity.
This, at first, seems strange. How can one infinity be larger than another? This result was first proven by Georg Cantor. A famous demonstration known today as the Cantor Diagonalization Argument proves this in so elegant a fashion as to be essentially uncontestable. For today, let us simply accept the truth of this separation between values of infinity. I shall allow the curious reader to investigate this independently.
We next move on to language.
Language is built around sentences expressing ideas. Those sentences are built of words. At any given moment, a language has a finite number of words. Language morphs and changes and evolves over time, adding a degree of complexity to the expressive power of a language, but I am concerned right now with language as it stands in any given instant.
When we build sentences, we combine these finite words into sentences. A given sentence is finite in length, but there is no theoretical limit to the length of an individual sentence. I’ve read sentences in books that span pages, and with embedded clauses it is trivial to demonstrate that a sentence can be of any length.
But it is clear that, however many words are in a sentence, that number must be a Natural number. That is to say, to claim a sentence has 7.6 words doesn’t mean anything. For a sentence to follow the structural rules of grammar, that sentence cannot have a fractional number of words. More to the point—and even less arguably—a sentence cannot feasibly have an irrational number (a number only expressible as a non-repeating, non-terminating decimal) of words.
What this tells us, in essence, is that there are a countable number of sentences. Not that the sentences can actually be counted by a human being, but that the infinity of sentence variation is the same size as the number of Natural numbers—a countable infinity.
A book or a paper or an argument, something built of sentences and used to convey ideas, is also necessarily finite in length. Once again, there is no theoretical limit to the number of sentences in a book, but whatever number of sentences are placed in a book, that number will be finite. So the number of feasible books or papers or arguments is countably infinite.
The point of this, then, is this: at any given moment, the number of ideas expressible in a language is countably infinite.
I find this to be quite interesting. If the number of ideas expressible through language is countable, what if the number of ideas which exist is uncountable? It seems to me as though this uncountability of the idea-space is likely true, since it is fair to call all possible numbers an idea, and so long as we accept the Reals we’ve immediately granted a number of ideas minimally equal to the number of Reals.
This means, in short, that there are ideas that language cannot possible express. As a matter of fact, there are an infinitely large number of ideas that cannot possibly be expressed in a language at a given moment.
So where are those other ideas? Well, perhaps some are expressible in non-linguistic media. Music, images, movements, and the like. Perhaps some are simply inexpressible. This is something to ponder, something to consider. It is important, I think, because it suggests the untruth of the often-believed idea that we can only think that which we can linguistically express.
I’d be interested to know where this takes the reader.