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A Kuhnian Approach to the Non-Euclidean Revolution in Geometry Date 2004-7-1 2:45 |
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A Kuhnian Approach to the
_________________________
Non-Euclidean Revolution in Geometry
____________________________________
C. Christopher Smith
csmith@mad.scientist.com
Indiana University/Purdue University Indianapolis
(submitted: November 13, 1997)
"Mathematics offered to the world proof that man can acquire
truths and then destroyed the proof. It was [the] non-Euclidean
[revolution] ... that paved the way for this intellectual disaster."
-- Morris Kline, _Mathematics And The Loss of Certainty_, 99
Introduction
____________
In any academic specialization, certain texts emerge over time as
the classics of that field. In the study of the history and philosophy
of science, one of the essential works is Thomas Kuhn's _The Structure of
Scientific Revolutions_. Published in 1962, this piece has not escaped
its share of controversy, yet its content has influenced a host of
academic areas. In _Structure_, Kuhn provides an archetype for the
understanding of the history of science as intra-disciplinary
cycles of stasis ("normal science") and revolution. _Structure's_
greatest contribution to philosophy was likely Kuhn's breathing new
meaning into the term "paradigm," which he used to refer to a temporary
model for scientific problem-solving which emerges during a
revolutionary era. As Hoyningen-Huene points out in his thorough
treatment of _Structure_, the concept of a scientific revolution
was not a new one. Even before Kuhn, there was an easy association
between such names as Copernicus, Newton, Lavoisier, Darwin, Bohr and
Einstein and their respective scientific revolutions (197). However,
Kuhn broadened the definition of "scientific revolution" and detailed
characteristic traits of revolutionary science. His definition
includes those scientific developments that have minimal or no impact
outside their sector of science, and more cautiously, "unexpected
discoveries of new phenomena or entities" (Hoyningen-Huene 198).
By any definition, the genesis of post-Euclidean geometry was a
revolution, and an ideologically violent one at that, as it laid siege
on geometry's primary text, Euclid's _Elements_ -- which guided that
field for over 2000 years. It also, on its philosophic front, more
subtly attacked popular Kantian views of axioms and of space. Kuhn's
_Structure of Scientific Revolutions_, and its ideological framework,
provides a choice foundation for a study of the geometric revolution and
the vast scope of its impact. One could make a good case that its
influence was comparable to any generated by the largest, and most
widely recognized, of scientific revolutions. The aim of this paper is,
by donning the lens of Kuhn's _Structure_, to explore the geometrical
revolution in its context and to take a rudimentary glance at its impact
on the realms of mathematics, science, and philosophy.
The Context Out of Which the Geometric Revolution Would Emerge
______________________________________________________________
Euclid's _Elements_
_________________
As the eighteenth century waned, geometrical studies were largely
stagnant, as their nucleus was Euclid's _Elements_, a 13-volume treatise
written around 300 B.C. Surviving many translations and cultures in its
2000-year history, _Elements_ comprised essentially the whole of
geometrical knowledge. The work of Euclid was essentially the first
geometric "paradigm" (in the Kuhnian sense), and thus Kuhn aptly
describes it:
In the development of any science, the first received
paradigm is usually to account quite successfully for most
of the observations . . . Further development ordinarily
calls for . . . a refinement of concepts that increasingly
lessens their resemblance to their usual common sense
prototypes. That professionalization leads to . . . a
considerable resistance to paradigm change.
(_Structure_ 64)
The foundation of _Elements_ is Euclid's core of 23 definitions
and 10 assumptions. Euclid separates his assumptions into five "common
notions," general logical statements, and five "postulates," which are
specific to geometry. It is these "postulates" that a historical
treatment must address, as they would provide fodder for Euclid's
critics. Euclid postulated:
1. That one can draw a straight line from any point to any
point.
2. That one can produce a finite straight line
continuously in a straight line.
3. That one can describe a circle with any center and
distance.
4. That all right angles are equal to one another.
5. That, if a straight line falling on two straight lines
make the interior angles on the same side less than two
right angles, the two straight lines, if produced
indefinitely, meet on that side on which are the angles
less than the two right angles.
(Euclid 154-155)
The first four postulates stood solid through centuries of testing, and
even the revolutionary geometers had no doubts about their veracity.
However, if for no other reason than its verbose complexity, the fifth
postulate was the one that astute mathematicians would question. In
1795, John Playfair simplified the terminology of the fifth postulate
and his phrasing has become the most common: "Through a given point can
be drawn only one parallel to a given line" (Wolfe 20).
From this core of definitions and assumptions, Euclid proceeded
to deduce 465 propositions, though not without error. The 2000-year
period of stasis served to correct minor flaws that did surface in
Euclid's _Elements_. The genius of Euclid's work lies not in its
perfection, but in its deduction of such a massive propositional system
from its small core of definitions and assumptions. _Elements_ stood as
the paragon of the axiomatic method, a process which begins with a set
of axioms (postulates) which are accepted without proof as self-evident
truths, and then reasons through a system of logic to arrive at a new
set of propositions (Greenberg 11). Einstein said of _Elements_, "This
admirable triumph of reasoning gave the human intellect the necessary
confidence in itself for its subsequent achievements" (13).
Kant's Philosophy
_________________
A significant figure in the development of the modern mind was the
Prussian philosopher Immanuel Kant (1724-1804). Kant's *piece de
resistance* was his _Critique of Pure Reason _(1781), in which he set
out to approach knowledge from new directions, in the wake of perceived
failures in the rationalist and empiricist thought of the seventeenth
and eighteenth centuries (Martin et al 485-486). Kant makes two
distinctions about the nature of any proposition. First, a proposition
must be either *a priori*, known before experience, or empirical, known
as a result of experience. Secondly, it must also be either analytic
-- if analysis shows that the proposition's predicate is contained
within its subject -- or synthetic if the predicate extends the subject.
These distinctions create four sectors into which to classify all
propositions: analytic empirical, analytic *a priori*, synthetic empirical
and synthetic *a priori*. However, Kant points out that analytic
empirical statements are non-existent "because in forming such a
judgment, I need not go out of the sphere of my conceptions, and
therefore recourse to the testimony of experience is quite unnecessary"
(_Critique of Pure Reason_ 7).
Kant based his understanding of mathematics primarily upon
Euclidean geometry. As Hintikka says, "all the different features of
Kant's philosophy of mathematics... are strongly and immediately
suggested by Euclid's procedure" (203). Kant demonstrates that
Euclid's theorems are synthetic *a priori* statements, i.e., that their
knowledge comes independent of experience and that their predicates
extend their subjects. He makes the claim that Euclidean propositions
are all *a priori*, and that Euclid's postulates are synthetic. Since
consequences of synthetic statements are synthetic, and since Euclid
developed his theorems from his postulates, the Euclidean theorems are
synthetic and thus synthetic *a priori* (Trudeau 115-116). Due to the
synthetic *a priori* axioms of Euclidean geometry, Kant understood space
as necessarily Euclidean. However, some thinkers (most notably,
Bertrand Russell) maintain that Kant was merely using Euclidean geometry
as an example to demonstrate that any geometry must contain a set of
synthetic *a priori* statements. However, to do so would be to wrench
Kant from his context, where, as Henderson says, "'geometry' meant
Euclidean geometry, the only geometry known for two thousand years,
... [and] 'space' was Euclidean space" (12).
Euclidean Geometry as "Normal Science" circa 1800
_________________________________________________
Thus, at the dawn of the nineteenth century, geometry was clearly
in a state of what Kuhn calls "normal science." Euclid's _Elements_ was
recognized as "the scientific achievement . . . supplying the
foundation for further practice" (Kuhn 10). In accord with Kuhn's two
criteria for the definition of a paradigm of "normal science,"
Euclidean geometry had a vast and enduring school of devotees and it was
open enough to accommodate a set of unresolved problems -- i.e., those
questions surrounding the fifth postulate (10). Geometrical research
also flowed only from esoteric circles of academic practitioners who
all accepted Euclid's paradigm, and there was an obvious lack of any
new texts in the field (Kuhn 20). Kuhn himself superbly describes the
state of geometry circa 1800, "Perhaps the most striking feature of the
normal research problems . . . is how little they aim to produce major
novelties, conceptual or phenomenal" (35).
One can also understand Kant's philosophy of geometry as a normal
paradigm, although, for obvious reasons, it cannot adhere strictly to
Kuhn's scientific definition. Kant's _Critique of Pure Reason_ was the
foundational text, which, in its role as a "critique," subverted the
waning paradigms of rationalism and empiricism. Kant attracted his
share of followers, yet many matters, particularly ones of terminology,
were still open to resolution. Thus, the Euclidean and Kantian
paradigms together comprised a massive ideological fortress that
dominated the modern academic arenas of mathematics and philosophy,
as well as many scientific disciplines including physics and optics.
Crisis: The Battleground of Euclid's Fifth Postulate
_____________________________________________________
Kuhn's notion of "Crisis"
_________________________
Kuhn devotes the seventh chapter of _Structure_ to defining the
concept of paradigmatic crisis. The Kuhnian crisis is born out of the
"profound awareness" of an anomaly. This recognition of anomaly then
leads to "proliferation of versions of a theory, . . . a very usual
symptom of crisis" (71). Kuhn builds solid support for his definitions
through various examples chosen from the history of science. Although
Kuhn never mentions geometry, the early nineteenth century developments
in this field conform quite well to his definition of crisis.
A Long History of Questions
___________________________
As a probable result of its wordy complexity, Euclid's fifth
postulate never attained a status of good repute. Kuhn says, "Failure
of existing rules is the prelude to a search for new ones" (Structure
68), and while the parallel postulate never clearly failed, it was
dubiously successful. Thus mathematicians, in light of this anomaly,
began the search for alternatives. Ptolemy, a contemporary of Euclid's,
wrote a book on the fifth postulate, which included a fallacious attempt
at a proof. In the fifth century A.D. Proclus -- in his _Commentary on
the First Book of Euclid_ -- demonstrated the fallacy of Ptolemy's
proof. He then produced his own "proof" which C. S. Claudio would
eventually show to beg the question (Bonola 13). Several attempts at a
proof arose from the Arabic world near the beginning of the second
millennium A.D. One "proof" proffered by the Egyptian Abu Ali Ibn
al-Haytham, also fell short of its goal, primarily because of its
reliance on diagrams, which may mislead, as they are only
representations of the forms described by Euclid. Another psuedo-proof
came from the Persian Nasiraddin (1201-1274), who also fell victim to
circular definitions (Wolfe 28). Aristotle addressed the problem of
these early attempts to reconcile this anomaly in his _Prior Analytics_,
"This is what those persons do who suppose that they are constructing
parallel straight lines: for they fail to see that they are assuming
facts which it is impossible to demonstrate unless the parallels exist"
(II.16: 4-7).
Bonola records the progress of the study of the fifth postulate
during the sixteenth and seventeenth centuries (12-13, 17). The most
notable of these attempts was that of the Italian Giordano Vitale who
reduced "the question of equidistant straight lines to the proof of the
existence of one point H upon DC, whose distance from AB is equal to
the segments AD and BC." John Wallis (1616-1703) revived the work of
Nasiraddin, and in 1663 developed his own proof, but yet again, as both
Wolfe (30) and Bonola (16-17) note, his result was a mere re-statement
of Euclid's fifth postulate. The above mathematicians collectively made
many contributions to geometry and other mathematical arenas. However,
despite their success in other mathematical pursuits, they were
unsuccessful in their attempts to prove the anomalous fifth postulate.
Gerolamo Saccheri
_________________
The next milestone in the history of the study of the fifth
postulate was the work of Gerolamo Saccheri (born 1667) as expounded in
a book published just before his death in 1733 entitled _Euclides ab omni
naevo vindicatus_ (Euclid Freed from Every Flaw). Greenberg notes that,
despite its bold claims, Saccheri's work ironically remained virtually
unknown outside of esoteric academic circles until it was revived by
Beltrami over 150 years later (154). Saccheri took a *reductio ad
absurdum* approach to proving the fifth postulate. He thereby developed
a geometry using Euclid's first four postulates and the negation of the
fifth postulate in the hopes of uncovering a contradiction in such a
geometry. Saccheri's geometry was based on the quadrilateral ABCD
where AD equals CB and angles A and B are right angles.
There are two possible negations to the equivalent of Euclid's fifth
postulate, which says that angles C and D are both right angles:
1. that angles C and D are both obtuse
2. that angles C and D are both acute
Bonola elaborates how Saccheri found a contradiction in the case of the
obtuse angles (24-26). However, despite his great effort in
demonstrating the first twenty-six of Euclid's propositions with the
"acute hypothesis" (Bonola 22), Saccheri was unable to produce a
contradiction in the case of the acute angles.
Despite the fact that he had unwittingly developed what was, in
essence, the first non-Euclidean geometry, Saccheri could not accept the
veracity of his work, and is reputed to have said, "The hypothesis of
the acute angle is absolutely false, because [it is] repugnant to the
nature of the straight line" (Greenberg 155). Though unsuccessful in
resolving the geometric anomaly, Saccheri deserves twofold recognition
for his contributions to the geometric crisis. First, his understanding
of Euclid's fifth postulate was open-ended and did not exclude the
possibility of its falsity. Secondly, it would be through Saccheri's
work that further non-Euclidean "versions" of geometrical theory would
arise.
Lambert, Klugel and Legendre
____________________________
Several other mathematicians made noteworthy geometric
contributions to the pre-revolutionary atmosphere. The first of these is
Johann Lambert (1728-1777), whose posthumously published _Theorie der
Parallellinien_ came to essentially the same conclusions as Saccheri.
This work of Lambert's quotes a dissertation by G.S. Klugel which
critically analyzes the work of Saccheri and demonstrates the failure of
the "most celebrated attempts to demonstrate the fifth postulate"
(Bonola 51). Klugel also makes the prophetic suggestion that non-
intersecting straight lines diverge and his reasoning for such is a
notable landmark in the philosophy of geometry. He says that the
paradox of such a notion is "not the result of a rigorous proof, nor a
consequence of the definitions of straight lines, but rather something
derived from experience and the judgment of our senses" (Bonola 51).
There is one other notable precursor to the development of
non-Euclidean geometry, Adrien Marie Legendre (1752-1833). Goldberg
notes that Legendre most likely worked independently of a knowledge of
the work of Saccheri (157). While previous attacks on the fifth
postulate focused on elucidating problems, Legendre's version of anomaly
rectification consisted of attempting to demonstrate that the fifth
postulate should indeed be a theorem instead of a postulate (Bonola 53).
While his efforts toward this end were ultimately unsuccessful, his
writings -- such as the collected works in _Reflections on Different
Manners of Demonstrating the Theory of Parallels or the Theorem on the
Sum of the Three Angles of a Triangle_ -- displayed extraordinary clarity
and attracted many readers. Though his contributions to the technical
assault on Euclid's fifth postulate were minimal, he succeeded in
arousing interest in the questions that surround this postulate (Wolfe
34). Although 2000 years of study failed to rectify the anomalous fifth
postulate, it did serve to lay a solid foundation on which a non-
Euclidean geometry could be built in the first half of the nineteenth
century.
The Development of a non-Euclidean Geometry
___________________________________________
Kuhn says that a scientific revolution is "seldom completed by a
single man" (_Structure_ 7). History typically credits three essentially
independent mathematicians with creating the first non-Euclidean
version geometrical theory: the German Carl Friedrich Gauss (1777-1855),
the Russian Nikolai Lobachevsky (1793-1856) and the Hungarian Janos
Bolyai (1802-1860). Their non-Euclidean geometry proffered a new
understanding of Euclid's fifth postulate: that, regardless of its
veracity, it might not be necessary. Gauss's work was the earliest
of the three, but it remained unpublished because Gauss feared the
turmoil that such a system would wreak upon the contemporary thought
dominated by Kantian philosophy. However, he did admit details of his
work through correspondence with a circle of his most trusted friends.
Lobachevsky has the distinction of being the first to publish his work,
as early as 1829. However, due to the cultural and technological
barriers of that era, when Bolyai published his work in 1832, he thought
he had earned that distinction.
Gauss, a Hesitant Revolutionary
_______________________________
Building on the work of the non-Euclidean predecessors,
particularly Klugel, Carl Friedrich Gauss was the first to develop a
non-Euclidean geometry. Through an analysis of his correspondence,
Bonola pinpoints the origin of Gauss's efforts to be approximately
1792 (65). Gauss began in the same manner as those before him -- by
attempting to prove the veracity of the fifth postulate. True to the
Kuhnian model of a scientific revolution (_Structure_ 7), Gauss's
realization of the logical possibility and consistency of a
non-Euclidean geometry was not a spontaneous revelation, but rather a
gradual and pained triumph over "the inherited prejudice against it"
(Bonola 66). The first record of Gauss's mentioning his development of
a non-Euclidean geometry was in a letter to Taurinus in 1824, but
therein he also begs for the secrecy of this pronouncement. Wolfe
presents a translation of the letter in its entirety (46-47) and it is
clear that Gauss's motivation for desiring privacy stemmed from the
fear of denouncing the Kantian ideas about the nature of space. He
indeed had a great reputation to uphold, as the extent of his
mathematical work was far-reaching. Halsted, in his 1891 appendix to
his translation of Lobachevsky's _Theory of Parallels_, proclaims that at
his apex Gauss was "the most powerful mathematician in the world" (47).
The inopportune announcement of his covert geometric work might have
quickly merited the scorn of less astute geometers and philosophers.
In an 1831 letter to Schumacher, he also coined the term "non-Euclidean"
in reference to this new sector of geometry. Gauss's recognition as the
silent father of non-Euclidean geometry is vastly due to the work of
Engel and Staeckel who, in the later half of the nineteenth century,
compiled the corps of Gauss's correspondence and personal writing into
their _Die Theorie der Parallellinien von Euklid bis auf Gauss_.
Nikolai Lobachevsky
___________________
The Russian Lobachevsky studied and later became a professor at
the University of Kazan. It was there, on February 26, 1826, that
Lobachevsky gave a monumental presentation to the mathematics and
physics division. In this lecture, he suggested that a new version of
geometry could exist where more than one line could be drawn through a
point parallel to a second line (Meschkowski 34). However, no
transcription of that talk has ever surfaced (Wolfe 54). A full
description of his thoughts on parallel lines first appeared in an
article entitled _On the Principles of Geometry_ which appeared in
1829-1830 editions of the Kazan Bulletin. Lobachevsky wrote these
pieces in Russian, and thus they failed to reach a large audience.
However, Rosenfeld notes that even in the academic community of Kazan,
his work "met with incomprehension" (208). Only later in his life when
he transcribed his work into German and French, with the express hopes
of reaching a larger audience, did he receive recognition for his
contributions to non-Euclidean geometry. In 1846, Gauss obtained a
copy of the German transcription, which he richly lauded in a letter to
Schumacher (Wolfe 55-56). Bolyai was to learn of the work of
Lobachevsky through Gauss. Wolfe states that there is no evidence that
Lobachevsky knew of Bolyai's work (56). Despite his massive
contribution to geometry, Lobachevsky lived in essential obscurity, and
even found himself fired from his professorship at Kazan in 1846.
Janos Bolyai
____________
Janos Bolyai had an innate fascination with Euclid's parallel
postulate. His father and mathematics instructor, Farkas Bolyai, was a
colleague of Gauss's who shared with Gauss the common interest of
critically analyzing the fifth postulate. The elder Bolyai even
registered his own attempt at proving this postulate in 1804, though
Gauss easily demonstrated its error. Farkas, in all his fatherly
wisdom, advises Janos:
You must not attempt this approach to parallels. I know
this way to its very end. I have traversed this
bottomless night, which extinguished all light and joy of
my life. I entreat you, leave the science of parallels
alone. . . . I thought I would sacrifice myself for the
sake of the truth. I was ready to become a martyr who
would remove the flaw from geometry and return it
purified to mankind. . . . I turned back when I saw that
no man can reach the bottom of this night. I turned back
unconsoled, pitying myself and all mankind.
(qtd. Meschkowski 31)
However, Janos, in his child-like persistence, was not so easily
dissuaded. In an 1826 correspondence with his father, he speaks of his
progress, "out of nothing I have created a strange new universe" (Wolfe
51). Farkas evidently underwent a change of mind, as he subsequently
encouraged the younger Bolyai to publish his findings expediently as
their time had come. The elder Bolyai even accommodated his son by
allowing Janos to include his work as an appendix to a book detailing
the attempts to prove the parallel postulate. This text, the _Tentamen_,
appeared in 1831, soon after Lobachevsky's work appeared unheralded in
Kazan.
Bolyai's greatest contribution was the idea that Euclidean
geometry was possibly a mere subset of a larger, more general geometry
of space. Thus, he laid a foundation for projective geometry and also
for the eventual resolution of the geometric crisis. He also, as Wolfe
notes, had faith in the impossibility of proving the fifth postulate,
though he was never able to confirm this intuition (53). Though Gauss
spoke approvingly of his work, Janos Bolyai -- a man of "fiery
temperament" (Greenberg 179) -- brashly derided Gauss's decision not to
reveal his non-Euclidean work. In 1851, he wrote, "In my opinion, and as
I am persuaded, . . . all the reasons brought up by Gauss to explain why
he would not publish anything in his life are powerless and void; for in
science, as in common life, it is necessary to clarify things of public
interest which are still vague, and to awaken, to strengthen and to
promote the lacking or dormant sense for the true and right" (Greenberg
179). Regardless, Gauss and Bolyai both helped to spark the
non-Euclidean revolution, as did Lobachevsky.
A Summary of the Kuhnian Approach to the Geometric Crisis
_________________________________________________________
The preceding historical outline has served to show the
conforming of the geometric paradigm crisis to Kuhn's definition. In
the era after the recognition of Euclid's fifth postulate as an anomaly,
several theories arose about the nature of this parallel postulate.
These versions manifest themselves in the varying approaches that
geometers took in their attempts to eliminate the anomaly. As history
witnessed the impending birth of a post-Euclidean paradigm of geometry,
the creation of new versions of possible meanings of the parallel
postulate -- like a birthing mother's contractions -- became more
frequent and more intense.
In Chapter VI of _Structure_, Kuhn uses the discovery of oxygen to
demonstrate how an emergent paradigm requires not only modified theory,
but also a modification of the understanding of the world (53ff). There
are many parallels between Kuhn's example and the development of
non-Euclidean geometry. First, the credit for both of these discoveries
goes to a triad of scientists working independently, who share the
honors because of the obscurity of publications and other complications.
Secondly, in accord with Kuhn, the definition of what exactly
constitutes a "discovery" became somewhat obscured. Was it contingent
with the first to publish his work? Was it instead the first publication
to garner recognition? Or was it the first to develop such a system,
even it remained forever unpublished? In the same manner which Kuhn
describes the discovery of oxygen, the discovery of non-Euclidean
geometry occurred after 1820, but certainly before 1850 (55). Kuhn
states, "discovering a new sort of phenomenon is necessarily a complex
event, one which involves recognizing both that something is and what
it is" (55). Gauss, Lobachevsky, and Bolyai certainly demonstrated the
theoretical existence of non-Euclidean geometry. However, despite their
mutual vague recognition of the threat posed to Kantian ideas of space
and despite Bolyai's undeveloped intuitions about a larger inclusive
geometry, none of these geometers had a very strong grasp on the
implications of their work. Thus, the crisis was to emerge into the
limelight and escalate into a full-scale war of paradigms
Paradigm Clash: Euclid and Kant versus Gauss, Lobachevsky, Bolyai and
_____________________________________________________________________
Hemholtz
________
Kuhn on Paradigm Conflict
_________________________
In Chapter Eight of _Structure_, Kuhn describes the probable
reaction of a scientific discipline to crisis. Therein, he describes
how paradigm shifts generally occur through "ad hoc modifications" (78)
of the old paradigm, rather than complete rejection of the old paradigm
in favor of the new. The anomalies and counter-examples (e.g., the
Gauss/Lobachevsky/Bolyai non-Euclidean geometry) serve to focus the
efforts of that particular scientific field on "reconstruction of the
field from new fundamentals, a reconstruction that changes some of the
field's most elemental theoretical generalizations as well as many of
its paradigm methods and applications" (85). Kuhn continues developing
these thoughts in Chapter Nine, where he notes that the period of
response to the crisis is the true "scientific revolution" (92). The
following sections will describe the ideological clashes of the
geometric revolution, and then detail the mathematical, scientific
and philosophical resolution that shaped a new paradigm.
Why Was the Struggle So Intense?
________________________________
The geometrical "paradigm shift" was difficult primarily because
the Euclidean paradigm had built up a great deal of inertia during its
era of more than 2000 years. The work of Euclid was adequate in its
depiction of the contemporary understanding of space, and few people
gave due consideration to the new Gauss-Lobachevsky-Bolyai geometry
because they saw it as a theoretical novelty. However, the largest
contributor to this inertia was the adherence of the contemporary
thought to Kantian ideals. Two aspects of Kant's thought were
particularly resistant to the new geometrical paradigm: his
understanding of axioms and his theory about the nature of space.
In his _Critique of Pure Reason_, Kant classifies mathematical
axioms as "*a priori* synthetical principles" (411). He also understands
axioms as necessarily true (Broad 61), thus "self-evident" (Kant 412).
If non-Euclidean geometry was true, it would boldly challenge all of
these claims. First, the Gauss-Lobachevsky-Bolyai version of postulate
five, which says that "through a given point, not on a given line, more
than one line can be drawn not intersecting the given line" (Wolfe 66)
is not readily self-evident. Secondly, the truth of non-Euclidean
geometry would undermine the necessity of the truth of Euclid's fifth
postulate. Hence, one aspect of the non-Euclidean revolution was the
struggle to define what a mathematical axiom is. Broad captures the
essence of this struggle:
Most of us in fact find it impossible to imagine and very
difficult to conceive an external world in which Euclid's
axioms did not hold. This is what makes Kant's assertion
that they are synthetic *a priori* propositions, founded
on some kind of intuitive cognition of particular
existents, so very plausible. . . (64)
Kant, in his _Critique_, says, "Geometry is a science which
determines the properties of space synthetically and yet *a priori*" (25).
The development of non-Euclidean geometry would attack this Kantian
conception because if two opposite yet valid geometries give different
descriptions of space, then neither could be determinative of space.
Broad provides several other criticisms of the Kantian understanding of
space, in light of the present post-revolutionary geometrical paradigm.
Among the most noteworthy of these criticisms, is his demonstration that
Kantian spatial theory could neither adequately "account for" nor
"justify" the belief that "all objects of sense-perception will obey the
laws of Euclidean geometry" (48). Thus, the non-Euclidean revolution
would adamantly challenge Kant's theory of space.
Early Victories in the War of Paradigms
_______________________________________
In 1854, Bernhard Riemann, a student of Gauss, proposed a second
sort of non-Euclidean geometry. The basis of Riemannian geometry is the
definition of lines and surfaces in the context of an elliptical surface.
One should note that this new geometry followed from the case of the
obtuse angles on Saccheri's quadrilateral, but Riemann evaded the
contradiction met by his predecessors by redefining the nature of a
line. Saccheri, et al had assumed a line was of infinite length and
thus without end. Since elliptical surfaces are the basis for Riemann's
geometry, a line will indeed have a finite length (its circumference)
but will be without end (Golos 168). The genesis of Riemannian
geometry was important not only because it provided yet another
non-Euclidean system for Euclid's advocates to refute, but also because
it highlighted the vital role of definition and context in the
development of geometrical systems. As Kline says in _Mathematics
and the Loss of Certainty_, "Every axiom system contains undefined terms
whose properties are specified only by the axioms. The meaning of these
terms is not fixed, even though intuitively we have numbers or lines in
mind" (192).
The publication of Eugenio Beltrami's _Saggio di Interpretazione
Della Gemetria Non Euclidea_ in 1868 was another landmark in the war of
paradigms. In this historic piece, Beltrami demonstrated the
impossibility of proving Euclid's fifth postulate. He did so by using a
pseudospherical model and differential geometry (Greenberg 226).
Bonola notes that, though monumental, Beltrami's proof was an implicit
one (234). In 1871, Felix Klein would offer a more direct proof for
this conjecture based on projective geometry (Greenberg 226). These
developments would deal a wounding blow to the Kantian philosophy of
geometry, which requires the necessity of all Euclid's five postulates.
The _MIND_ Battle
_______________
Kuhn notes that "the paradigm period in particular is regularly
marked by frequent and deep debates over legitimate methods, problems
and standards of solution" (47-48). The British journal of psychology
and philosophy, _MIND_, featured one such crucial battle of the
non-Euclidean revolution in its pages over the years 1876-1878. Hermann
von Hemholtz argued as a proponent of the non-Euclidean paradigm, and
J.P.N. Land countered with a defense of the Euclidean paradigm. Thssion
served to arouse public interest in non-Euclidean thought, as
was Hemholtz's intention (I 21).
Hemholtz's initial discourse on "The Origin and Meaning of
Geometrical Axioms" appeared in the July 1876 issue of _MIND_. He attacks
the problems with the Kantian understanding of axioms as outlined above,
saying "The main difficulty in these inquiries [into geometrical axioms]
is and always has been the readiness with which results of everyday
experience become mixed up as apparent necessities of thought" (I 302).
In support of his position, he employs the analogy of how beings in a
two-dimensional world would experience a third dimension. This anecdote
is a hauntingly similar precursor to Edwin A. Abbott's classic tale
_Flatland_, which would appear in 1884. Hemholtz held that if the human
mind can imagine space other than Euclidean space, then the Euclidean
postulates are no longer necessary as Kant had declared (I 314). He
proceeds to proffer an imaginative depiction of Beltrami's
pseudospherical space as support for his claim. At the end of his
description Hemholtz concludes that "pseudosphericaace would not
seem to us very strange, comparatively speaking; we should only at first
be subject to illusions in measuring by eye the size and distance of
remote objects" (I 317). In a manner reminiscent of the Riemannian
emphasis on the relative aspects of definition and context, Hemholtz
closes his arguments:
I would again urge that the axioms of geometry are not
propositions pertaining only to the pure doctrine of
space. As I said before, they are concerned with quantity. We
can speak of quantities only when we know of some way
by which we can compare, divide and measure them. (I 319)
In 1877, J.P.N. Land responded -- via _MIND_ -- to Hemholtz's
contentions. Land begins his retort by recognizing that Hemholtz has
indeed presented some of the "chief difficulties" of Kantian spatial
theory (38). He proceeds to make a claim that there is a bold division
between the espistemological approaches of science and philosophy,
stating that "Science has no suspicion of a distinction between
'objectivity' and 'reality'" (38-39). Such a distinction is necessary,
Land says, lest the questions these epistemologies ask become obscured
(40). Such a proposition leads him to claim that there indeed may be
separate geometries of mind, as those of non-Euclidean geometry, and of
the empirical (44). Land draws the Kantian conclusion that the axioms
of "geometry proper" are derived from the experience of an objective
reality, and thus that the development of a consistent alternative
geometry is a mere scientific novelty (45).
A year later, _MIND_ featured a counter-response from Hemholtz.
The highlight of this piece is Hemholtz's deconstruction of Land's idea of
a transcendental geometry. He notes that the only way that such a
geometry could arise, even with the assumption of transcendental axioms,
is through the flawed empirical tools of perception (II 219-220). He
declares, "If we really have an innate and indestructible form of
space-intuition involving the axioms with it, their objective scientific
application to the phenomenal world would be justified only in so far as
observation and experiment made it manifest that physical geometry,
grounded in experience, could establish universal propositions agreeing
with the axioms" (II 221). Thus, Hemholtz concludes his remarks with
the bold stance that the Kantian concept of transcendental axioms is
"unproved," "unnecessary" and "wholly irrelevant" (II 225).
These discussions in _MIND_, and their parallel debates in the
French journals _Revue Philosophique_ and _Revue de Metaphysique et de
Morale_ (Sommerville 84ff), embody the tension that emanated from the
warring geometrical paradigms. As these transcribed debates waned and
the nineteenth century faded, the works of Hemholtz and others had
served their purpose. There was a more widespread recognition of the
inadequacy of the Euclidean paradigm outside the mathematical arena.
Kuhn and the "Resolution of Revolutions"
________________________________________
Kuhn entitled chapter twelve of _Structure_ "The Resolution of
Revolutions." Herein, he states that resolution comes "only after
the sense of crisis has evoked an alternate candidate for a paradigm"
(145). The competing paradigms then enter a period of verification to
see which "fits the facts better" (147). One of the primary points of
this chapter is that "the proponents of competing paradigms practice
their trades in different worlds" (150). This concept is vital because
it successfully depicts the intersection of the academic realms of
science and philosophy. Science, the pursuit of knowledge about the
natural world, presupposes a philosophy of the nature of the world. As
Kuhn notes, paradigm conflict often stems more from differing
philosophical assumptions about the world than from erroneous scientific
practice. The intent of the following sections is to follow the
resolution of the geometric revolution, and thereby to glimpse at the
pre- and post-conflict worlds.
Mathematical Solidification of a Post-Euclidean Geometrical Paradigm
____________________________________________________________________
Poincare Re-defines the Role of Axioms
______________________________________
If, as Hemholtz postulated, the Kantian understanding of geometric
axioms as transcendental was inadequate, then questions should arise
regarding what the true role of axioms was. One of the catalysts in the
French discussion in _Revue Metaphysique_, Henri Poincare, explored the
realm of such inquiries. He believed, like the English Philosopher John
Stuart Mill had as early as 1843 (556), that human adherence to the
Euclidean paradigm was a psychological one which stemmed from our
educational processes (374). Thus, Poincare took a middle ground
rejecting both the hypothesis that axioms are *a priori* intuitions and
the one which states that these axioms are experiential in nature. To
him, geometrical axioms were mere conventions. Kolakowski summarizes
Poincare's perspective well, saying "We have chosen the Euclidean system
not because of our store of experiences obliges us to do so, but because
it is the most convenient in our everyday contacts with solid bodies"
(141). Poincare's understanding of axioms is the generally accepted one
today, although as Greenberg notes, a school of opponents does exist
(293).
The Development of a More General Projective Geometry
_____________________________________________________
The geometrical war of paradigms left many questions about whether
Euclidean or non-Euclidean geometry was the better descriptor of space.
While the popular philosophic debates over geometrical paradigms raged,
geometers were developing a more general geometrical system that would
incorporate the Euclidean geometry as well as the non-Euclidean
geometries of Gauss-Lobachevsky-Bolyai and Riemann. This projective
geometry is so called because it involves the projection of the
Euclidean or non-Euclidean planes onto a circle. Cayley was the first
to describe projective geometry in 1859, but his work focused only on
the projection of Euclidean surfaces. Klein expanded Cayley's geometry
to include non-Euclidean surfaces in 1871 (Bonola 163-164). Projective
geometry today, with a few revisions to its Euclidean sector submitted
by Hilbert in the early twentieth century, represents our best
understanding of geometry.
The new system of post-Euclidean geometry fits the two criteria
that Kuhn set for the scientific acceptance of a new paradigm in
_Structure_. First, the "new candidate must seem to resolve some
outstanding and generally recognized problem that can be met in no other
way" (169). The post-Euclidean geometry provided solutions for two
primary and related problems. Projective geometry provided a sufficient
response to the questions that surrounded Euclid's fifth postulate, and
its inclusiveness quelled the crisis between Euclidean and non-Euclidean
systems of geometry. Kuhn's second criteria was that emerging paradigm
should "promise to preserve a relatively large part of the concrete
problem-solving ability that has accrued to science through its
predecessors" (169). Although it provided solutions for questions
generated by the Euclidean paradigm, the post-Euclidean geometry
retained a great deal of Euclid's system. In fact, Euclidean geometry
-- with Hilbert's corrections -- was wholly contained within projective
geometry, and the non-Euclidean sectors of the new paradigm retained
most of the Euclidean structure and logic, despite their differing
postulates. In accord with Kuhn's definitions, the mathematical
community accepted the projective geometry. Thus, it arose as the new
geometrical paradigm.
The Scientific Acceptance of the Post-Euclidean Paradigm
________________________________________________________
The Role of Riemannian Geometry in Einsteinian Physics
______________________________________________________
Despite the fact that Beltrami and Klein had demonstrated
non-Euclidean geometry's validity by proving the impossibility of
proving the fifth postulate, it retained its status as a theoretical
novelty through the end of the nineteenth century. The work of Einstein
would change that view by applying non-Euclidean geometry to his new
paradigm of physics. Specifically, Riemann's elliptical geometry played
a vital role in Einstein's general theory of relativity. Einstein
claimed that Riemann's geometry provided an apt description of the
curvature of space caused by the gravitational pull of matter,
specifically large masses like stars (Gamow 137). The first
demonstration of this theory was his application of it to the recognized
anomaly in the orbit of Mercury. The results of this experiment showed
that the general theory of relativity yielded an exact prediction of the
anomaly (Clark 257-259). Further testing confirmed the universality of
Einstein's general theory and it was for this work that he received the
Nobel Prize for Physics in 1923.
In _Structure_, Kuhn proffers the Einsteinian revolution in
physics as an example in his chapter on the "Nature and Necessity of
Scientific Revolutions" (92 ff). This section is pertinent to a
discussion of the non-Euclidean revolution for two reasons. First, the
Einsteinian revolution whose basis was a post-Euclidean geometry subverted
the Newtonian paradigm for which Euclidean geometry was foundational.
Secondly, and more importantly, the theoretical progression of the
twentieth century revolution in physics and the non-Euclidean revolution
are extraordinarily similar. Kuhn points out that Newton's model of
physics was only a special instance of Einstein's paradigm (99).
Similarly, Euclid's geometry was an instance of the more general
projective geometry. Though speaking of the revolution in physics,
Kuhn captures the essence of the tension between the geometric paradigms:
But the objection continues, no theory can possibly
conflict with one of its special cases. If
Einsteinian science seems to make Newtonian dynamics
wrong, that is only because some Newtonians were so
incautious as to claim that Newtonian theory yielded
entirely precise results . . . Since the could not have
had any evidence for such claims, they betrayed the
standards of science when they made them. In so far
as Newtonian theory was ever a truly scientific
theory supported by valid evidence, it still is. Only
extravagant claims for the theory -- claims that were
never properly parts of science -- can have been shown by
Einstein to be wrong. Purged of these merely human
extravagances, Newtonian theory has never been
challenged and cannot be. (99)
If one were to substitute post-Euclidean for Einsteinian and Euclidean
for Newtonian, this passage would be no less true. In this case, the
bold non-scientific claims had their origins in the work of Kant, who
had no pretense of being scientific. However, his work would evolve
over time to dominate the popular understanding of geometry, and to form
the philosophical basis of the pre-revolutionary world. The purging of
the Kantian ideals amalgamated into the science of geometry was a
primary achievement of the non-Euclidean revolution. In the words of
Bertrand Russell, "As the mathematical results shook themselves free
from philosophical controversies, they assumed gradually a stable form,
from which further development, we may reasonably hope, will take the
form of growth rather than transformation" (Essay 50).
A Glimpse at the Philosophical Impact of the non-Euclidean Revolution
_____________________________________________________________________
Kant's Philosophy of Geometry Falls
___________________________________
The debates in _MIND_ and the other works of Hemholtz served to
call
into question the world defined by Kant's understanding of the role of
geometry. However, these arguments and the evidence offered by the
development of projective geometry served merely to alter the Kantian
philosophy of geometry. In his epic work of 1897, _An Essay on the
Foundations of Geometry_, Bertrand Russell details the history of the
non-Euclidean revolution and expounds a neo-Kantian philosophy of
geometry. Given his post-Euclidean context, Russell realizes that
neither Euclidean nor non-Euclidean geometry is the necessary foundation
of space. However, he posits that the more general projective geometry
is the *a priori form* of an external space.
Russell determined a set of axioms that he believed would cement
the apriority of projective geometry:
I. We can distinguish different parts of space, but all
parts are qualitatively similar, and are
distinguished only by the immediate fact that they
lie outside one another.
II. Space is continuous and infinitely divisible; the
result of infinite division, the zero of extension,
is called a point.
III. Any two points determine a unique figure, called a
straight line, any three in general determine a
unique figure, the plane. Any four determine a
corresponding figure of three dimensions, and for
aught that appears to the contrary, the same may be
true of any number of points. But this process comes
to an end, sooner or later, with some number of
points which determine the whole of space.
(Essay 132)
He proceeded to demonstrate that projective geometry, as determined by
his axioms, was "wholly *a priori*" as it followed as a logical extension
of his axioms and thus, was independent of experience (146).
In his foreword to Russell's Essay, Morris Kline demonstrates the
inadequacy of Russell's neo-Kantian philosophy of geometry. First, he
notes the development of topology which, in its descriptions of space,
generalizes upon projective geometry, which itself is a generalization
of Euclidean geometry (iv). Secondly, and of more import, Kline states
that the biggest opposition to Russell's understanding of space has come
from Einstein's general theory of relativity. The primary point of
failure in Russell's defense is the question of the homogeneity of space
(v). Russell implicitly assumed the homogeneity of space in the
"qualitatively similar" nature of points in the first of his *a priori*
postulates. Einstein's theory elaborated upon Riemann's hypothesis that
the existence of matter and gravity cause a non-constant curvature of
space, and hence the non-homogeneity of space. Experimentation confirmed
Einstein's theory, and thus a fatal blow shattered Russell's claims for
the apriority of projective geometry. The world defined by Euclidean
geometry and Kantian spatial ideals finally yielded to a post-Euclidean
Einsteinian one.
The Influence of the non-Euclidean Revolution Upon Contemporary
_______________________________________________________________
Philosophy
__________
In one respect the non-Euclidean revolution served to purge
philosophy from the understanding of geometry. However, in another, it
served to foster the growth of the philosophical mindset of
anti-foundationalism which would come to influence the twentieth century
philosophical understanding of the world and, in particular,
contextualist thought. John Dewey, oft-recognized as "one of the
philosophic grandfathers . . . of contextual approaches" (Dervin 5),
demonstrates one way that the non-Euclidean revolution was
anti-foundational:
The logical looseness of the Euclidean postulate regarding
parallels suggested operations previously unthought of,
and opened up new fields--those of the hyper-geometries.
Moreover, the possibility of combining various existing
branches of geometry as special cases of more
comprehensive operations . . . led to the creation of
mathematics of a higher order of generality. . . .
Once the idea of possible operations, indicated by
symbols, is discovered, the road is opened to operations
of ever increasing definiteness and comprehensiveness. Any
group of symbolic operations suggests further operations
that may be performed (157-158).
Due to the realization of such systems, and the progression of
mathematical logic to the apex of Godel's incompleteness theorem,
mathematicians and philosophers began to grasp that, as Riemann found,
axioms and definitions must emerge from context. The words of one of
the primary contextualists, Gregory Bateson, describe this progression,
"Without context there is no meaning" (13). Kuhn seconds this notion in
_Structure_ with his description of an analogous case to that of Riemann's
line: "It follows that scientific concepts like that of the element can
scarcely be invented independent of context" (142).
Georgoudi and Rosnow define contextualism: "[It] emphasizes the
fact that human activity does not develop in a social vacuum, but is
rigorously situated within a sociohistorical context of meanings and
relationships. Thus, in order to understand what an act is or what it
involves . . . , one necessarily has to examine its context, which is to
say the surrounding sociopolitical and historical condition in which the
act unfolds" (82).
Under this definition, Kuhn's approach to scientific revolutions
was more contextual than previous ones, because it sought to define
revolutions out of the context of individual scientific disciplines and
their communities of practitioners. King says, in his excellent summary
of the contextual nature of Kuhn's approach entitled _Reason, Tradition
and the Progressiveness of Science_, that "what Kuhn has done is to show
how such a theory [of scientific change] might be developed by attacking
the problem of how concrete ways of doing science, or more specifically
the authority structures that uphold them, are modified, disrupted and
perhaps overthrown in the face of changes in scientific thought and
technique" (115). Though some critics have questioned
the appropriateness of such a contextual approach to the history of
science (e.g., Popper), others (e.g., Wolin and Hollinger) eagerly apply
a Kuhnian contextual approach to various other fields of study. Kuhn's
_Structure_ explicitly delineated a contextual approach to the history of
science, but it did so only with a wave of controversy. Thus, the non-
Euclidean revolution indirectly was a contributor to the ideological
foundation on which Kuhn would build his contextual understanding of
science.
Summary: Kuhn and the Non-Euclidean Revolution
_______________________________________________
The previous sections have attempted to narrate the history of the
geometric revolution via the framework laid by Thomas Kuhn in his epic
work _On the Structure of Scientific Revolutions_. The pre-revolutionary
Euclidean paradigm, and its practice in a world shaped by Kantian ideals,
had settled to a state of "normal science" by the late eighteenth
century. However, the recognition of Euclid's fifth postulate as
anomalous long preceded this stasis. A crescendo of crises
unsuccessfully attempting to right this anomaly ensued; the result of
which was a variety of understandings of the nature of the parallel
postulate. The peak crisis was the genesis of the
Gauss-Lobachevsky-Bolyai non-Euclidean geometry. The period following
the emergence of non-Euclidean geometry was one of paradigm conflict,
marked by many discussions of the definitions that shaped geometry and,
philosophically, the world in which geometry was done. Little did Janos
Bolyai realize the extent to which his exclamation at having "created a
strange new universe" (Wolfe 51) would ring true. This new world, based
on contextual definitions and Einsteinian relativity instead of the
absolute *a priori* world of Kant and Euclid, emerged as a result of the
geometric paradigm war. The new geometric paradigm took shape from the
resolution of the mathematical, scientific and philosophical conflicts,
and in accord with Kuhn, it "[implied] a new and more rigid definition
of the field" (19).
Although Kuhn, Hoyningen-Huene and others do not recognize the
geometric revolution as a major scientific revolution, it did register a
profound impact on human thought. The geometric paradigm shift led to
contextual approaches to definitions and the Einsteinian revolution in
physics. Thus it heralded the coming demise of the modern approach to
knowledge. George Bruce Halsted, summarized its impact well in an 1899
article entitled "Report on Progress in Non-Euclidean Geometry":
"Hereafter no one may neglect it [the genesis of the post-Euclidean
geometrical paradigm] who attempts to treat the fundamentals in geometry
or philosophy" (557).
========================================================
Works Cited
___________
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