Rachel Hillman

Hon 213

4/5/01

Forget the years; forget distinctions. Leap into the boundless and make it your home! – Chaung Tzu, 4^th century BCE Taoist Philosopher and Mystic

Metamathematics is the study of the methodology of mathematics. It deals with the underlying structure of mathematical reasoning, and addresses such philosophical topics as ‘proof’, ‘truth’, and ‘meaning’, to name just a few. According to Imre Lakatos, metamathematicians can be divided into two intellectual camps: dogmatists and skeptics (Lakatos 4). Dogmatists are those who argue that truth can be discovered through the power of the human intellect, and skeptics are those who maintain that truth can never be known (unless, perhaps, by mystical experience). In proof-analysis, dogmatists recognize that truth can only be reached if 'true' and 'false' are clearly delineated. When 'true' and 'false' are clearly delineated, the result is mathematical order. On the other hand, when 'true' and 'false' are not clearly delineated, the result is chaos in mathematics. Thus, one of the key problems dogmatists and skeptics grapple with, as evidenced in Lakatos’ Proofs and Refutations, is the conflict between order and chaos. This, as we shall see later, is simultaneously a conflict between binary and non-binary thinking.

The goal of this paper is to correlate the metamathematical debate about chaos and order with other academic and philosophical traditions, for the purpose of demonstrating its universal importance. Mathematical dogmatists and skeptics have counterparts in other intellectual disciplines, the most important of which, for the sake of this discussion, are mysticism and representation theory. Mysticism is a loosely defined term that encompasses a variety of philosophical and religious traditions. Representation theory, which deals with the assigning of meaning through language and other signifying systems, is the combined work of socio-linguists, social critics, communication theorists, philosophers and psychologists. Both branches of study are broad and inclusive, and therefore can only be minimally represented here. Thus, the present discussion of mysticism will draw solely upon the Taoist tradition, represented predominantly by the work of Chuang Tzu, the fourth century BC Chinese philosopher and mystic. Likewise, the present discussion of representation theory will be based almost exclusively on the text Representation: Cultural Representations and Signifying Practices, edited by Stuart Hall.

These three disparate traditions—metamathematics, mysticism, and representation theory—are bound by their common interest in the question of chaos vs. order. Advocates of both perspectives are represented in each of the disciplines. As already mentioned, in metamathematics the proponents of order are the dogmatists, and the proponents of chaos are the skeptics. In mysticism, the proponents of chaos are the Taoists, exemplified by Chaung Tzu, and the proponents of order are members of the opposing philosophical school, the Confucians. While the Taoists reject binary thinking and embrace the chaos of unknowing, the Confucians advocate strict hierarchic social order. In representation theory, there are no opposing groups, only opposing perspectives; they are the reflective approach and the constructionist approach. The reflective approach emphasizes the fixity of meaning, whereas the constructionist approach maintains that meaning is fluid, and forever changing.

Across these three disciplines, proponents of chaos and proponents of order share certain respective beliefs. For instance, all supporters of order believe that truth can be plainly distinguished from falsehood, and can therefore be defined. In order for truth to be defined, a binary distinction must be drawn between 'true' and 'false' and, ultimately, between 'right' and 'wrong', 'good' and bad', and so on. By contrast, proponents of chaos reject binary thinking. They recognize that binary thinking is the basis of conventional social order, which they reject as arbitrary and artificial. Thus, proponents of chaos embrace a worldview that is in constant flux, while proponents of order seek to rigidly define the universe in static terms.

Let us begin by looking more closely at how the conflict between chaos and order gets played out in metamathematics. Recall that mathematical dogmatists look for order in the form of verifiable truth deduced through rigorous intellectual reasoning. By contrast, skeptics believe that truth can never be known, and thus they allow for expanded and reformed mathematical definitions that constantly challenge previously held assumptions. In Lakatos' dialogue, Delta is the quintessential dogmatist. His argument in favor of mathematical order is implicitly based on the idea that ‘true’ is distinct from ‘false’ and that the two cannot co-exist. In other words, a logical statement can either be true or false, but it cannot be both or neither. If a statement is both true and false, or is neither true nor false, it cannot be accepted into the system of mathematical logic. However, in Proofs and Refutations it just so happens, to the Delta's dismay, that such contradictory statements emerge. For example, when Sigma suggests that some mathematical propositions, "although they hinge on true principles, nevertheless admit restrictions or exceptions in certain cases," Delta bemoans, "To agree to a peaceful coexistence of theorems and exceptions means to yield to confusion and chaos in mathematics"(Lakatos 25). Delta also attacks Alpha’s counterexamples (‘monsters’) and says, "I look for order and harmony in mathematics, but you only propagate anarchy and chaos. Our views are irreconcilable" (Lakatos 19). In this instance, Sigma favors a less rigid, more inclusive statement, but Delta rejects it because it does not clearly distinguish between true and false and therefore leads to chaos.

A comparison of mathematical dogmatism, as exemplified by Delta, and mystical philosophy, as exemplified by Chuang Tzu, reveals two antithetical belief systems. This is primarily due to the fact that Chuang Tzu totally rejects conventional binary notions of right/wrong, good/bad, beautiful/ugly, and true/false. As evidence of this rejection, consider the following quotation:

As the above passage indicates, Chuang Tzu’s teachings are based on the principle that everyone and everything are part of an "inexpressible absolute" (Watson 20), and that it is artificial to distinguish between them. The illusion of separateness, according to Chuang Tzu, is a dream from which we one day awake, although we do not know that we are dreaming (Watson 43). Moreover, Chuang Tzu extends this concept of unconscious dreaming to mean that it is impossible to ever really know anything with certainty. He says, "Suppose I try saying something. What way do I have of knowing that if I say I know something I don’t really not know it? Or what way do I have of knowing that if I say I don’t know something I don’t really know it?" (Watson 41). This sort of argument is exactly in line with metamathematical skeptics who maintain that it is impossible to ever really know anything, and thus it is wholly counter to mathematical dogmatism (Lakatos 5). Chuang Tzu would consider the principles of mathematical dogmatism to be illusory and artificial, while dogmatists would likely think of Chuang Tzu’s teachings as chaotic and confusing. This comparison is significant because it is evidence not only of a disparity within and between academic disciplines, but also of two opposing worldviews fighting a conceptual battle on multiple intellectual fronts.

Scholars regard Chuang Tzu as one of the most profound philosophical minds in human history. As Burton Watson, professor of East Asian Languages and Cultures at Columbia University, put it, "[the work of Chuang Tzu] constitutes one of the fiercest and most dazzling assaults ever made upon man's conventional system of values, [and] upon his conventional concepts of time, space, reality and causation as well" (Watson 5). Nevertheless, Chuang Tzu's work was challenged during his lifetime by an alternate philosophical movement, Confucianism, which emphasized the importance of hierarchic social order and proper behavior. Unlike Taoism, Confucianism stressed the importance of moral guidelines for ethical behavior (Watson 10). Obviously, this entailed making clear distinctions between ‘good’ and ‘bad’ conduct. To a Confucian, good conduct meant obeying strict principles of social respect and obligation. For instance, according to Mencius, one of the foremost Confucian thinkers, the greatest act of goodness was giving one’s parents a proper burial. Mencius also emphasized the extension of good conduct into the public sphere. His idea of social order involved a strict hierarchic arrangement--a benevolent, but nevertheless authoritarian, ruler had control over his subjects, whose duty it was to respect and obey him as they would their parents. Thus, for Mencius, moral conduct had significant political applications. By contrast, Taoism was basically apolitical (Watson 10). Chuang Tzu was far more concerned with the individual than he was with social organization. This is undoubtedly due to his rejection of binary thinking and conventional moral values. In his writing, Chuang Tzu satirizes the Confucians and, through his reasoning, demonstrates the absurdity of their strict moral teaching. He says, "What does the way rely upon, that we have true and false? What do words rely upon, that we have right and wrong? . . What one calls right the other calls wrong; what one calls wrong the other calls right" (Watson 34). The illusion of binary thinking is a key component of Chuang Tzu’s mysticism, and it is the principal rationale behind his disinterest in social order.

Constructionist representation theorists describe this relationship between binary thinking and order at length. For example, theorists such as Stuart Hall argue that meaning is predicated upon the establishment of difference, and that the most common method of marking difference is by means of binary opposition. In other words, we know that ‘RED’ is a color that is different from ‘GREEN’ and that under certain conditions these two colors have opposite meanings (in the language of traffic lights, red means ‘stop’ and green means ‘go’). We also know that ‘father’ means ‘not mother’, ‘black’ means ‘not white’, ‘good’ means ‘not bad’, and so on (Hall 17). Without marking difference, Hall argues, human beings could not make meaning, nor could they communicate, nor could they form societies. In turn, human societies follow the patterns of marking difference established through language, which explains why culture is so often divided along lines of class, race, and gender difference. Thus, representation theorists, like their mystical counterparts, maintain that social order is directly linked to binary thinking.

Furthermore, a constructionist would argue that the relationship between a sign and the concept it represents is arbitrary and imposed, and that, therefore, meaning can never really be fixed, but is constantly changing. In this sense, constructionists are not unlike those students in Lakatos’ dialogue that constantly redefine the term ‘polyhedra’ and propose altered and expanded versions of the original conjecture. These students reflect an awareness of the fact, as a constructionist would put it, that "things don’t mean; we construct meaning" (Hall 25). In general, the constructionist perspective affords less weight to the notion of inviolable, absolute truth, and is more in line with the skeptics than with the dogmatists. This is illustrated as well by the similarity between constructionist ideas and Chuang Tzu’s teachings. For instance, Chuang Tzu writes:

Here, Chuang Tzu reflects upon the arbitrary nature of words and the changeability of meaning. This is an aspect of his skepticism, and is part of his grand philosophical scheme that questions all known conventional wisdom.

While the constructionist approach focuses on the changeability of meaning, the reflective approach does not. The reflective approach is based on the assumption that language (which can refer to any method of communication) "functions like a mirror to reflect true meaning as it already exists in the world" (Hall 24). This approach assumes that meaning is fixed, and does not allow for the variability of interpretation. I equate the reflective approach to mathematical dogmatism because, in a sense, it relies on the preconception that truth is absolute and can be known objectively. As mentioned earlier, in order for truth to be known, a distinction must be made between true and false. Because this is a binary distinction, we can say that the reflective approach endorses binary thinking, which, subsequently, is tantamount to social order.

Because Lakatos’ mathematicians are not concerned with placing value judgments on their theorems, lemmas, and conjectures.,we might say that they are not binary thinkers. However, the most basic form of binary thinking is distinguishing between true and false, which is precisely what mathematicians and, in particular dogmatists, seek to do. Clearly, the ‘order’ and ‘harmony’ that Delta seeks is not unlike the socio-linguistic order described in representation theory. Delta embodies not only the mathematical dogmatist perspective that truth exists and can be determined through the human intellect, but also the binary worldview that insists that ‘true’ and ‘false’ be clearly delineated. This matter becomes increasingly complicated as the other students not only redefine ‘polyhedra’, but also quibble over the meaning of numerous other words. What becomes apparent is that these mathematicians must assign meaning to terms in order to progress with their proof, and that the assigning of meaning is, to some extent, arbitrary. A restricted definition is selected from a range of possibilities. The mechanics of this process are similar to those described by the constructionists of representation theory, a fact which raises problems for those who look for clearly delineated truth and order in mathematics. How can truth be absolute if meaning is not fixed, but fluid?

Unfortunately, this conflict between mathematical dogmatists and skeptics, order and chaos, has no resolution at the present. Lakatos’ dialogue hovers undecidedly between skepticism and dogmatism, and, in the end, is left unresolved, leaving the reader uncertain as to which perspective will prevail. Perhaps neither perspective will ever prevail. I would venture to say that advocates of chaos and advocates of order are represented in all disciplines in philosophical movements. The great divide that separates these two groups is the battle between fixity and fluidity. On one hand, there is an orderly static universe, and on the other there is a forever-changing universe. Perhaps the greatest question that philosophy has yet to resolve is whether or not chaos and order, fixity and fluidity, dogmatism and skepticism, must be mutually exclusive. Perhaps the two worldviews can co-exist, but that remains to be seen.

Scholars regard Chuang Tzu as one of the most profound philosophical minds in human history. He is known for his mythical, somewhat equivocal parables, and for his esoteric teaching--that conventional values and ideas are essentially meaningless, and that 'freeing oneself from the world' is the answer to all suffering (Watson 5). As Burton Watson, professor of East Asian Languages and Cultures at Columbia University, put it, "[the work of Chuang Tzu] constitutes one of the fiercest and most dazzling assaults ever made upon man's conventional system of values, [and] upon his conventional concepts of time, space, reality and causation as well" (Watson 5). As this statement suggests, Chuang Tzu rejects the 'commonsense' belief in quantifiable reality and heartily embraces the chaos of unknowing. In this sense, his teachings are very much aligned with skepticism, but stand in marked contrast to mathematical dogmatism. This comparison is significant because it is evidence not only of a disparity within and between academic disciplines, but also of two opposing worldviews fighting a conceptual battle on multiple intellectual fronts. In fact, Chuang Tzu's work was challenged during his lifetime by an opposing philosophical movement, Confucianism, which emphasized the importance of hierarchic social order and proper behavior. In the discussion that follows, I will demonstrate how these two opposing philosophies echo the debate between the mathematical dogmatists and the skeptics.

Representation theory deals with the construction of meaning through language and other signifying systems. There are opposing perspectives in representation theory, known as the reflective approach and the constructionist approach. In the reflective approach, meaning in language is thought to abide in the object that is described. According to this perspective, the meaning of 'rose' comes from the rose itself (Hall 24). The constructionist approach rejects this view, and argues instead that meaning is constructed by an observing entity. Implicit in the reflective approach is the idea that meaning is fixed, and therefore verifiable, whereas the constructionist approach emphasizes the ever-changing, fluid nature of meaning. The constructionist approach also provides a framework with which to assess the relationship between order and binary thinking that is essential to the discussion that follows.

Representation: Cultural Representations and Signifying Practices. Hall, Stuart, ed. Sage Publications. 1997.

Lakatos, Imre. Proofs and Refutations: The Logic of Mathematical Discovery.Cambridge University Press. 1976.

Chuang Tzu: Basic Writings. Watson, Burton, trans. Columbia University Press. 1964.