Does a Postmodernist Philosophy of Mathematics Make Sense

Does a Postmodernist Philosophy of Mathematics Make Sense?

Is “2 + 2 = 5” Correct as Long as One's Personal Situation orPerspective Requires It?

We shall not cease from exploration

And the end of all our exploring

Will be to arrive where we started

And know the place for the first time.

T. S. Eliot

Ilhan M. Izmirli

American University, Washington, D.C.

Abstract

Postmodernism,one of the most influential philosophical and cultural movements of the twentieth century is also one of the mostmisconstrued, partly because of its rejectionto be confined by some well-defined characteristics. In this paper,we will start out by discussing some major principlesof postmodernist philosophy. We will then investigate how theyapply to natural sciences and in particular to mathematics. We will also summarize their manifestations in pedagogy of mathematics. We will conclude the paper by answering the question that was raised by Arthur T. White in his paper Mathematics in the Postmodernist Era:Is 2 + 2 = 5 correct as long as one's personal situation or perspective requires it?

1. Introduction

This paper comprises three sections. In the first section, we will briefly discuss basic tenets of modernism and postmodernism. In the remaining two sections, we will analyze the modernist and postmodernist approaches to natural sciences, to mathematics, and finally to pedagogy of mathematics.

The terms modernism and postmodernism are usually used in reference to cultural movements, especially to arts, architecture, music, and literature, and as such, modernism refers to the period from 1890s to 1940s and postmodernism to the period following the Second World War, especially 1960s onward.

In this studywe will concentrate on the manifestations of modernism and postmodernism in natural sciences and in mathematics, and in particular in their philosophies. This is not as farfetched an exercise as one might tend to think. Indeed, these movements have long been associated with sciences. In fact, it has been argued that modernism began either by Richard Dedekind’s introduction of the idea of cuts to define irrational numbers in 1872, or by Stephan Boltzmann’s introduction of statistical thermodynamics in 1874 (Everdell 1997).

Historically, the passage from medieval to modern thinking exhibited itself as a struggle between the establishment (Church) and emerging sciences (empiricism). Among the pivotal events that determined the outcome of this contention were Nicolaus Copernicus’s (1473 – 1543) positing of the heliocentric system based on observational evidence, Francis Bacon’s (1561 – 1626) argumentsadvocating the use of experimental methods in sciences, Johannes Kepler’s (1571 – 1630) masterful combination of observational evidence with mathematical theory to displace Ptolemy’s model of the universe, Galileo Galilei’s (1564 – 1642) establishment of modern experimental physics, and Isaac Newton’s (1642 – 1727) unification of celestial and terrestrial mechanics through empirical and mathematical methods.

Certainly, each one of the above scholars could be considered a modernist. However, most historians maintain thatEmmanuel Kant (1724 – 1804), who believed that Newton’s Laws could be shown to be true by reason,should be called the “the first real modernist” (Frascina and Harrison 1982, 5). Some argue that August Comte (1798 – 1857),who proposed a multistage development for the human mind[mythical (religious), metaphysical, and the positive stage, where the positive stage was characterized by systematic collection of observable facts], should share the honor (Weston 2001).

Early twentieth century modernism was influenced by the theories of two Austrian scholars, those of Sigmund Freud (1856 – 1939),who argued that human mind had a fundamental structure, and those of Ernst Mach (1838 – 1916), who developed a philosophy of science known as positivism.

Positivism held that scientific laws were summaries of experimental events, constructed for human comprehension of complex data. Thus, for instance, Mach opposed the atomic theory of physics since atoms were too minute to be observed directly. He defended the idea that any physical law was less than the actual fact itself, because it only reproduced that aspect of the fact which was significant for the particular discussion (i.e., abstraction of nature). Thus the goal of physical sciences was to provide the simplest and most economical abstract expression of the pertinent aspects of facts. Mach had a direct influence on the Vienna Circle and the school of logical positivism in general.

Postmodernism originally began as a reaction to modernism, in particular to the pursuit of perfection and harmony of form and functionality of modernist architecture (Bertens 1995). In its most general sense, the term now evokes a cultural, intellectual, and artisticoutlookthat rejects any central hierarchy or organizing principle and embodies complexity, ambiguity, interconnectedness, and contradiction.

In philosophy, the groundwork for postmodernism was laid by the German philosopher Friedrich Nietzche (1844 – 1900), who claimed that will to power was more important than facts or things, and by the Danish philosopher Søren Kierkegaard (1813 – 1855), who argued against objectivity and emphasized skepticism.

The post-colonialist period following the Second World War further contributed to the postmodernist contention of the impossibility of attaining an objectively superior belief system. Postmodernism was further developed by Martin Heidegger (1889 – 1976), Ludwig Wittgenstein (1889 – 1951), and Jacques Derrida (1930 – 2004), who after a careful examination of epistemology, concluded that rationality was not as clearly defined and well-understood as the modernists had asserted (Bertens 1995).

The basic methodology of postmodernism is deconstruction. The term was introduced by Jacques Derrida in the 1960s. Deconstruction involves the close reading of texts to demonstrate that rather than being a unified whole, any given text has irreconcilably contradictory meanings. Since language is arbitrary,the theses of a text are undermined by its own contradictions. In other words, meaning is indeterminate.

Basic Tenets of Modernism
There exists an external, ultimate reality that can be discovered.
Scientific truths are certain, absolute, and objective.
Reason is the ultimate judge of truth, goodness, and beauty.
Rationality is the only way of organizing knowledge.
Sciences can produce general laws, i.e., they are universal. Scientific progress is the only way to social progress.
Language is transparent and represents the perceivable world.
The world is ordered.
Natural sciences replace tradition, supernatural is to be rejected.
Logic replaces religious authority.
Basic Tenets of Postmodernism
There is no ultimate reality. The Cartesian view of an objectively existing external world is to be rejected.
There is no absolute truth. There is only interpretation. There is no a priori dogma. Science changes over time (Lyotard 1984)
Knowledge is neither eternal nor universal. Reality is a cultural construct that changes over time and place. Knowledge embodies the values of those who are powerful enough to create and disseminate it (Foucault 1988). Knowledge is constructed by the human mind, not discovered.
There is no standard, objective, or universal moral law. Morality is a human construct.
Knowledge has an essentially pluralistic character. One should give equal status and eminence to divergent, contradictory, and ill-fitting interpretations of a phenomenon and speak of multiple truths.
Ambiguity and disorder are to be tolerated.
Language is arbitrary.
Intellect is replaced by will.
Morality is replaced by relativism.
Reason is replaced by emotion.

Postmodernism, obviously, has been the target of severe scholarly criticism. To the critics of postmodernism the French philosopher Michel Foucault (1926 – 1984) responded

It is understandable that some people should weep over the present void and hanker instead, in the world of ideas, after a little monarchy. But those who for once in their lives have found a new tone ... I believe will never feel the need to lament that the world is error, that history is filled with people of no consequence... (Foucault 1988, 330)

1. Applications to Natural Sciences

Modernismadvocated and supported the cause of empiricism and inductive methodology in natural sciences. However, in mid-twentieth century, philosophers of science began to question this assumption.

One of the best known of these philosophers was Karl Popper (1902 – 1994). Popper was critical of inductive methods used in science, and argued that inductive evidence was limited, for it was impossible to observe the entire known universe at all times. There would always be the possibility of a future observation to refute a theory based solely on inductive evidence. Moreover, all observations would reflect a point of view and hence would be shaped by the observer’sperception of the particular phenomenon.

Popper instead proposed an alternative method called falsification. Sciences progress when a theory is shown to be wrong and a new theory that better explains the phenomena is introduced. Thus, scientists should always try to disprove their theories rather than constantly try to prove it.

Thomas Kuhn (1922 -) developed a similar concept called a paradigm shift. Scientists have a worldview (paradigm). A paradigm is an interpretation rather than an objective explanation. Scientists accept the dominant paradigm until some incongruities or abnormalitiestranspire. They then begin to question the basis of the paradigm itself; new theories emerge and challenge the dominant paradigm, and eventually one of these theories become the new paradigm, creating a revolution in the scientific outlook.

In general, the postmodernist interpretation of natural sciences does not intend to provide a new understanding of a theory or of a specific aspect of that science. All it does is to point out that natural sciences might not be as objective or as assumption-free as we have come to believe, implying that the fundamentals of the scientific explanation of a natural phenomenon are not a static, immutable set of laws, but rather simple paradigms that are merely human interpretations of that natural phenomenon, and as such are dependent on the time and place. The replacement of Newtonian paradigm by the Einsteinian one is a clear example. From a postmodernist point of view, scientific explanation is neither objective nor neutral.

Our perception of the postmodernist interpretation of natural sciences can be summarized as follows:

1. Since there can be no neutral observers and since all experiments are theory-laden, no natural science can ever be an exact reflection of Nature. No real phenomenon can be as simple as a scientific theory that tries to explain it.
2. Natural sciences are embedded in and hence are limited by culture and language.
3. There is no boundary to scientific progress. Science is a process.
4. Suppose S(t) is a scientific theory (time dependent) such that S(t0) is the scientific explanation for a natural phenomenon Nat t = t0. We expect that as t→∞, S(t) →N.

1. Applications to Mathematics

There are relatively few papers that investigate whether a postmodernist interpretation of mathematics makes sense. Two papers by Moslehian (2003, 2008) claim it does whereas the paper by White (2009) argues otherwise. There is a more or less neutral paper by Beck (2008) on postmodernist pedagogy of mathematics.

3.1 Modernist point of view of mathematics(MM)

MM is promoted by the absolutist schools of philosophy of mathematics, namely, logicism, formalism, intuitionism, and mathematical realism (Platonism). According to these schools, mathematics is static, infallible, abstract, eternal, perfect, certain, precise, and absolute.

3.2 Postmodernist point of view of mathematics (PMM)

PMM deconstructs absolutism and deems certainty to be an unattainable idea. In other words,

• Mathematics is a fallible and corrigible discipline that is subject to constant change
• A mathematical truth is never absolute but is to be interpreted relative to a background
• Like all other scientific entities, mathematical objects arise from the needs of human societies

• Mathematical proofs depend on a set of axioms assumed to be self-evident and true by human beings, and hence are subjective and time dependent (Lakatos 1976)

• Mathematical knowledge is a representation which is no more or no less true than any other representation

• Mathematical concepts, theories, and methods are socially constructed
• Mathematics is a dynamic endeavor (Ernest 1991).

3.3 Postmodernist Pedagogy of Mathematics (PMPM)

A postmodernist approach to pedagogy of mathematics emphasizes experimental mathematics, in congruence with the fallible and quasi-empirical nature of mathematics. Topics that are not introduced at an early stage in a modernist pedagogy such as examples involving nonlinear systems, examples that lead to fractals and chaos theory, and examples that are relevant to naturally occurring discontinuous phenomena, should be made a part of the curriculum at an early stage.

Since postmodern epistemology measures knowledge on its utility and functionality, the use of computers in discovery and proof of mathematical ideas should be encouraged. It may be true that computational proofs imply a probability but not the certainty of a mathematical result, but based on the above criterion, these are just as valid as the classical axiom-definition-conjecture-proof technique. Teachers of mathematics should also emphasize intuitive explanations and alternate solution methods. To show the dynamic character of mathematics, topics such as non-Euclidean geometries should be standard parts of the mathematical discourse.

One of the most significant contributions of postmodernist approach to pedagogy of mathematics would be the rejection of the Aristotelian Law of The Excluded Middle that something is either true or false, and to replace it by fuzzy logic based on “degrees of truth.”

Fuzzy sets have been introduced by Lotfi A. Zadeh in 1965. If A is a (classical) set, then any x either belongs or does not belong toA. We can describe this by a characteristic function ΧA(x) defined as

ΧA(x) = 1 if x belongs to A

and

ΧA(x) = 0 otherwise

By contrast, fuzzy set theory permits the gradual assessment of the membership of elements in a set. If F is a fuzzy set, this is described with the aid of a membership function ΧF(x) with values in the unit interval [0,1]. Thus, the characteristic functions of classical sets are special cases of the membership functions of fuzzy sets.

Fuzzy logic, introduced by Wilkinson (1963), is a form of multi-valued logic where the degree of truth of a statement can range between 0 and 1 and is not constrained to the two truth values, say, true = 1 and false = 0 as in classic logic.

3.4 White’s Argument

In his paper Mathematics in the Postmodernist Era, White writes

A mathematician, I believe, is quite likely to be motivated by the Platonic view that mathematics is external to human mind, that mathematical truth is discovered and – within a given system of axiomatic assumptions that it has the desirable quality of being absolute (White 2009, 2).

Note that there is almost nothing objective in this argument. In fact, defining the quality of absoluteness in mathematics by the term “desirable” is nothing short of replacing reason by emotion! We go on:

This traditional view today is being deconstructed by some mathematicians and many mathematics educators. The notion of mathematics as objective and eternal is being replaced, among mathematics educators, by the postmodernist notion of “social constructivism”(White 2009, 2)

In other words, traditional values should not be deconstructed. But then, by the same token, shouldn’t we be defending Aristotelian concept of motion? Or Ptolemaic view of the universe? Or should mathematics be considered as somehow differently than physics or astronomy? If so, what would be an objective, logical reason for this distinction? When postmodernist claim that mathematical concepts, theories and methods are socially constructed, they simply mean that the times and societies define “mathematical rigor.” Certainly some of Euler’s proofs of certain (mathematically correct) results, which were acceptable in 18th century Europe, would fail to satisfy the level of rigor required in a standard modern analysis class. This neither diminishes the genius of Euler nor the beauty and utility of his theorems – it just implies that mathematics is just as dynamic as other natural sciences.

What would go wrong if mathematics was subjected to the same natural progress other disciplines are allowed to enjoy?

Absolutism is deliberately replaced by cultural relativism, as if 2 + 2 = 5 were correct as long as one’s personal situation or perspective required it to be correct (White 2009, 2).

First of all, cultural relativism is out of context in this setting. When postmodernists claim that a mathematical truth is never absolute, they mean it is to be interpreted relative to a background. Certainly 2 x 5 = 1 is true in mod (3) arithmetic. No sane mathematician or educator would go around redefining addition or any other mathematical construct because his or her “personal situation” requires it to be correct. The Platonic fact that the sum of the interior angles of a triangle being exactly 1800 was challenged neither because the personal situation of Lobachevski nor because the personal perspective of Riemann warranted it, but because the resulting geometries turned out to be no more or no less correct that the Euclidean one.

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