KNOWING AND ERRING: THE CONSOLATIONS OF ERROR

Essay On Developmental Epistemology (Excerpt)

by

Robert Kalechofsky

<micah@micahbooks.com>

THE LOSS OF CERTAINTY

The title of this book partially stems from a translation of Ernst Mach's work,
Erkenntnis und Irrtum, written at the end of the nineteenth century. Mach attempted
to base knowing on the physiological processes of perception, perhaps as part
of the effort to make the study of epistemology scientific. Epistemology--the
study of knowledge or how humans come to know--is to modern philosophy as the
study of ethics was to Greek philosophy. Mind is replacing action and soul as
the human center and therefore as the center of philosophy. At the same time,
many today long for certainty in knowledge and find "error" to be
as corrupting as the snake in the garden of Eden. Error, it is felt, poisons
human existence, the admission that error is an inevitable part of life is intolerable.
Our culture's watchword is, "Know the truth and the truth shall make you
free." To question this, many feel, is to render life as a shadow or illusion,
and to question the foundation of existence. At first, religion promised to
slay the dragon of error; then science promised it would do it; then philosophy
tried. The present century opened with the logical optimism of Bertrand Russell's
analytic philosophy, but it is closing otherwise.

A specter haunts the mathematical/logical community since the late nineteenth
century, which has developed into a crisis in the study of epistemology. The
specter stems from the undigested contradictions of Russell et al to the deeper
dissonances of Gödel, which precluded hope of the absoluteness of consistency
and truth. The incontrovertible possibility of ineradicable error has invaded
mathematics/logic. The Russell-like paradoxes had opened Pandora's box, and
even if we manage to resolve one such paradox, the possibility of others is
present. Gödel's theorems--that there are true, unprovable statements and
that there is no absolute proof of the consistency of sufficiently complex mathematical
systems--furthered the inevitable possibility of error and paradox in classical
mathematics. Consonant problems in science arose around the same time, 1900,
particularly in physics, which likewise shook the paradigm of cause-effect relations
among phenomena. This increasingly led to ideas of propensities and statistical
connections linking complex phenomena.

The major response to the dissonance posed by the possibility of inevitable
error has been to encapsulate and effectively to ignore it. To be sure, Formalists
like Hilbert tried to delimit mathematics to more finite methods to avoid the
paradoxes of the infinite and to restrict what is mathematics to symbols, thereby
vitiating the idea of meaning in mathematics. Hilbert's drive to divest mathematics/logic
of error derived from a need for certainty:

"If mathematical thinking is defective where are we to find truth and certitude?"
(Hilbert, "On the Infinite" in Philosophy of Mathematics ed. Benacerraf
and Putnam p.134)

This need for certainty leads to a view of mathematics as an immaculate conception,
unsullied by the earthly dialectical interaction of the cognitive being and
the world. It has led to the avoidance of powerful tools in mathematical reasoning,
such as Aristotle's law of the excluded middle. Intuitionists like Brouwer limited
logic and mathematics to finite processes without recourse to the law of the
excluded middle in infinite domains, and determined that existence be based
on the direct production of the existent.

Both these tendencies to banish error from mathematical/logical thought are
rooted in a need to preserve the faith in the certainty of mathematics as a
pristine heaven walled off from uncertainty and error. A variant of this attitude
may be found in the attempts to separate pure from applied mathematics. Fear
of error-making and uncertainty have played a subterranean and deleterious historical
role in human thought. Mathematics/logic is nonetheless no worse off than the
physical sciences which are, together with their mathematical underpinnings,
fraught with the same potential for error and inconsistency, in addition to
that of their physical models. So, mathematics/logic remains a source of relative
certainty for the more empirical sciences, while it remains in a deductive/empirical
dialectic while probing the nature of things. In amplification of this view
that error, carefully nurtured, can foster the development of important metaphors
and deepened understanding, one can cite the examples of the Dirac delta function
and divergent infinite series, each arising from the powerfully productive but
cognitively ambiguous world of Quantum Theory. The delta function, zero almost
everywhere but infinite at one point was decried by many mathematicians as meaningless,
used by many physicists to understand and apply the theory pretty adequately,
and ultimately understood mathematically as a functional. The "error"
evolved and deepened our understanding both mathematically and physically. Similarly,
the divergent asymptotic series of Quantum Theory were held onto despite their
divergence until they were re-understood (normalized), made convergent and proved
fruitful in understanding the phenomena of the theory.

Piaget's developmental cognitive/epistemological models of knowing suggest that
error making, awareness of dissonance, and the development of cognitive structures
are important bases of creatively understanding problems of consistency and
paradoxes. Mathematics is rooted in the human cognitive structure, and therefore
is and should be treated as a developmental discipline with an empirical component
intertwined with its essentially theoretical and speculative bases. Consonant
with the views of Lakatos and Popper, mathematics should function somewhat parallel
to the other sciences: that is, proofs should, wherever possible, adduce models
of validation, counter examples should be actively sought, and understanding
of error-making in mathematical/logical processes should be promoted as ways
of deepening our knowing processes through acknowledging the role error plays
in our cognitive processes.

The paradoxes and errors resulting from infinite processes and the law of the
excluded middle should not be the occasion for ridding ourselves of the culprits.
Let us use them carefully and creatively, conceiving of mathematics/logic as
rooted in an empirical mode in our minds and the outer universe, but as always
based in a developmental psychological/cultural framework.

TOWARD A METAPHYSICS OF KNOWING

All knowledge is filtered through metaphors, and all knowledge is metaphoric.
The remainder of this book will explore the ramifications of this view, its
validation, and what appear to be its consequences. The metaphors of our knowing
processes are not like free floating atoms: they form structures, adhere to
our thought processes and interact in ways that characterize what we say we
know. These structures, in our deliberations about the world, help account for
our knowledge about it and for the errors we make in understanding the world.
They form connecting links between the minds of, say, children, and those of
sophisticates such as philosophers and scientists. One of the inevitable problems
resulting from this is to characterize the nature of these developmental structures
and the errors which they may lead us to commit. It is also necessary and interesting
to understand how the errors become known as errors to other people or to the
erring mind itself: How does cognitive dissonance occur and how can it be resolved?

The other side of the coin of erring is the structural network involved in our
knowing processes. The taxonomy we form in science, the language structures
we use in understanding are developmental in nature; since they emanate from
humans they are, at their roots, both psychological and dynamically connected
to the world about us. Our task is to elicit these structures of knowing, both
in their psychic manifestations, in the cognitive structures which develop in
our interaction with our world, and as these structures map out the nature of
our world.

A paradigm which illustrates and deepens our metaphysics of knowing may be found
in the following which concerns erring and knowing in a mathematical and physical
context:

A root of the aphorism attributed to the ancient Greeks, that nature abhors
a vacuum may be found in Aristotle's physics, where he analyzes the motion of
a falling body. Aristotle said, in effect, that an object falls slowly in a
highly viscous medium, more quickly in thinner oil, and even more quickly in
air. He concluded that the speed of an object is inversely proportional to the
viscosity of the medium in which it falls. Ergo an object would fall with infinite
speed in a vacuum (which has zero viscosity), which is impossible, therefore
there can be no vacuum. (His famous dictum was "Nature abhors a vacuum.")
This error occurs in scientific literature up to the 17th century and is found
in Kepler's writings before his elliptical orbit theory appears. Kepler, while
investigating the motion of Mars with respect to the Sun, noted that its speed
at its furthest point from the Sun is slower than its speed at its closest point.
He concluded that the speed of Mars was inversely proportional to its distance
from the Sun. Students in today's schools (including colleges) when presented
with Aristotle's and Kepler's analyses, without being forewarned that they are
both errors, generally agree with their analyses and conclusions. In other words,
both historically and psychologically, assumptions about numerical connections
among things tend to be linear or inversely linear. This suggests a psychological
preference for certain modes of cognition that span the centuries. Furthermore,
among sophisticated scientists this kind of error does not occur after the 17th
century. The key idea needed to clarify these phenomena was that of function.
As the analytical notion of function developed in the 17th century and thereafter,
and the multiple possibilities of the ways in which ideas like y=f(x) could
manifest themselves, then the immediate assumption of linearity or inverse proportionality
was no longer tenable when investigating relations between two variables. When
the notion of function broadened to that of transformation, fixed points and
invariants as foci of stability gained in physical, mathematical, and psychological
importance. In the last hundred years, ideas of invariants grew, which led to
Felix Klein's geometric theory based on the invariants of various geometric
transformations and to Einstein's theory of relativity, which could be re-named
the theory of absolutes, because it focuses on invariant laws and properties
of the universe. Similarly the stability and the invariance of the atom can
be understood as predicted by quantum theory based on discrete transfer of energy
rather than on continuous transfer. Such ideas of invariance are importantly
involved in Piaget's genetic epistemology, where knowing structures are understood
by focussing on the invariants of knowing processes. In spite of important cognitive
developments, the idea that "nature abhors a vacuum," persists.

AN EPISTEMOLOGICAL REBELLION

Cognition should be studied in its relation to both epistemology and psychology.
The separation of the two disciplines of psychology and philosophy/epistemology
early in the nineteenth century allowed the infant psychology to develop and
gain independent maturity, but it should now return to a resumed dialectical
interaction. This would be a fruitful interdisciplinary linkage, fostered by
Piaget's not-always successful, but provocative and illuminating theory of developmental
structures, by some of the non-behaviorist studies in artificial intelligence,
and by the historico-philosophical approach of Thomas Kuhn.

The basic ideas which sustain this investigation into knowing as a developmental
process are structure and error. All knowing, from that of children to that
of adults, from early primitive societies to philosophically sophisticated and
scientifically based cultures, takes place within a structural framework and
is subject to erring as an essential aspect of the cognitive process. The structures
are both cultural (sociological) and individual (psychological) and may be characterizable
mathematically (as in some of Piaget and Kalechofsky) or verbally. Erring is
part of the developmental knowing structures. It is the other side of the coin
of knowing.

That knowing, whether philosophical, mathematical, scientific or whatever, is
a human activity, necessitates a psychological component. However, this common-sense
statement, is anathematical to the classical view of science as "objective"
and is fiercely denied by many "gatekeepers." C. G. Hempel's response
to a question at a symposium on Kuhn's contribution to the History and Philosophy
of Science (M. I. T., May, 1990) concerning a psychological underpinning for
Kuhn's view of paradigms as used in science, was that he could not accept psychology
as a basis for understanding the history and philosophy of science, exemplifies
the opposition to the idea of knowing as a human activity.

A variation of Hempel's attitude is found in Wittgenstein. One would have thought
that the experience of teaching children (in Austria), and experiencing the
involvement with the daily problems of teaching-learning would have informed
him of the psychological, developmental underpinnings of epistemology, because
the epistemological, developmental enterprise is operational at all times during
an individual's life. However, the idea of the epistemological style which proceeds
as though one is an adult disembodied head was a sophisticated fiction foisted
by aspects of Wittgenstein, the Vienna Circle, the Positivist tradition and
their more modern forms.

In his later writings, Wittgenstein did not refer explicitly to the effect of
his teaching experience on his understanding of the knowing process, but he
did refer to psychological-like models of picturing knowledge. However, as with
most philosophers, the psychological-developmental process of knowing was not
considered by him and when, for example, he referred to the principle of identity
(A=A) he denied it any psychological basis. From a Piagetian and developmental
viewpoint, the principle of identity can be linked to the child's initial structuring
of the invariance of the object and is an important cognitive underpinning in
the development of ideas of invariance and knowing processes in the child and
the adult.

© Robert Kalechofsky 1997

*The above is the first 15 pages of the book, Knowing and Erring, The Consolations
of Error: Essay On Developmental Epistemology, Marblehead, Massachusetts: Micah
publications, ISBN 0-916288-45-5 1. (Micah Publications, 255 Humphrey Street,
Marblehead, MA 01945. http://www.micahbooks.com or micah@micahbooks.com)*

**'Knowing and Erring' it is being carried by <ionsystems.com> for online
distribution
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