Download 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 or
Perspective 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.
[email protected]
Abstract
Postmodernism, one of the most influential philosophical and cultural movements of the twentieth
century is also one of the most misconstrued, partly because of its rejection to be confined by some welldefined characteristics. In this paper, we will start out by discussing some major principles of
postmodernist philosophy. We will then investigate how they apply 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 study we 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) arguments
advocating 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 that Emmanuel 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 artistic outlook that 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)
2. Applications to Natural Sciences
Modernism advocated 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’s perception 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 abnormalities transpire. 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:
A. 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.
B. Natural sciences are embedded in and hence are limited by culture and language.
C. There is no boundary to scientific progress. Science is a process.
D. Suppose S(t) is a scientific theory (time dependent) such that S(t0) is the scientific explanation
for a natural phenomenon N at t = t0. We expect that as t →∞, S(t) → N.
3. 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 to A. 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 180 0 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.
References
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