Greek Geometers Research Articles

Operationalism: An Interpretation of the Philosophy of Ancient Greek Geometry

I present a systematic interpretation of the foundational purpose of constructions in ancient Greek geometry. I argue that Greek geometers were committed to an operationalist foundational program, according to which all of mathematics—including its entire ontology and epistemology—is based entirely on concrete physical constructions. On this reading, key foundational aspects of Greek geometry are analogous to core tenets of 20th-century operationalist/positivist/constructivist/intuitionist philosophy of science and mathematics. Operationalism provides coherent answers to a range of traditional philosophical problems regarding classical mathematics, such as the epistemic warrant and generality of diagrammatic reasoning, superposition, and the relation between constructivism and proof by contradiction. Alleged logical flaws in Euclid (implicit diagrammatic reasoning, superposition) can be interpreted as sound operationalist reasoning. Operationalism also provides a compelling philosophical motivation for the otherwise inexplicable Greek obsession with cube duplication, angle trisection, and circle quadrature. Operationalism makes coherent sense of numerous specific choices made in this tradition, and suggests new interpretations of several solutions to these problems. In particular, I argue that: Archytas’s cube duplication was originally a single-motion machine; Diocles’s cissoid was originally traced by a linkage device; Greek conic section theory was thoroughly constructive, based on the conic compass; in a few cases, string-based constructions of conic sections were used instead; pointwise constructions of curves were rejected in foundational contexts by Greek mathematicians, with good reason. Operationalism enables us to view the classical geometrical tradition as a more unified and philosophically aware enterprise than has hitherto been recognised.

Greek Geometric Animal Figurines and the Origins of the Ancient Olympic Games

According to the prevailing scholarly opinion, Geometric bronze animal figurines found at Olympia represent cattle and horses which were put under the protection of the divinity in this form. This view is challenged here for various reasons including literary testimony and comparisons with contemporary shrines containing similar dedications (especially Kato Syme on Crete). This paper argues that the bovines depicted were feral, and the figurines were offered by foreign aristocrats visiting the sanctuary especially for the sake of hunting these animals. Similarly, the horse figurines are interpreted as depicting feral equines, which were presumably captured and taken away by the visitors. After examining the cultic regulations related to the Olympic Games (timing, crowns, exclusion of married women and the penteteric periodicity), it is suggested that excessive hunting led to the extinction of some game animals and thus to a radical shift in the cult practice and ultimately resulted in the introduction of athletic events, i.e., in the Olympic Games.

Любые античные вазы из Северного Причерноморья: путешествие к трансцендентальному

The author analyzes the historical background for collecting and trading in the ancient painted pottery (Corinthian, Boeotian, Euboean, Greek geometric, Italiote, Cypriot) which was considered as finds from the South Russia (Crimea, Odessa region and Taman peninsula) during 1890–1910s. At that time these specimens of ancient pottery played an important role as an element bringing more colors to the expected image of “the Classical Hellas in the South Russia”. The pieces are currently available for studies in various Russian museums.

Analysis and demonstration: Wallis and Newton on mathematical presentation

Emulating the Greek geometers, Newton used synthetic demonstration to present the ground-breaking arguments of the Principia . This paper argues that we can better understand Newton's reasons for using geometry by considering John Wallis's interpretation of synthetic demonstration. Wallis condemned demonstration for failing to explain the mathematical truths it presented. He opposed to it a presentation that combined symbolic analysis with a documented account of discovery. In preferring symbols, Wallis was motivated both by the nascent tradition of symbolic analysis and by contemporary interest in artificial languages. Newton maintained Wallis's characterization of Greek demonstration as adapted to common understanding rather than as strictly elucidating, but he inverted the values Wallis associated with synthesis and analysis. In Newton's new account, synthetic demonstration was preferable precisely because it could address inexpert readers without exposing them to the complications of symbols-based analysis. Newton advanced his arguments on behalf of geometry through portraits of ancient mathematicians: Archimedes and Pythagoras.

Method of Analysis: A Paradigm of Mathematical Reasoning?

The ancient Greek method of analysis has a rational reconstruction in the form of the tableau method of logical proof. This reconstruction shows that the format of analysis was largely determined by the requirement that proofs could be formulated by reference to geometrical figures. In problematic analysis, it has to be assumed not only that the theorem to be proved is true, but also that it is known. This means using epistemic logic, where instantiations of variables are typically allowed only with respect to known objects. This requirement explains the preoccupation of Greek geometers with questions as to which geometrical objects are ‘given’, that is, known or ‘data’, as in the title of Euclid's eponymous book. In problematic analysis, constructions had to rely on objects that are known only hypothetically. This seems strange unless one relies on a robust idea of ‘unknown’ objects in the same sense as the unknowns of algebra. The Greeks did not have such a concept, which made their grasp of the analytic method shaky.

Book Review of Corpus Vasorum Antiquorum. Great Britain 25. The British Museum 11: Greek Geometric Pottery, by J. Nicolas Coldstream

Book Review of Corpus Vasorum Antiquorum. Great Britain 25. The British Museum 11: Greek Geometric Pottery, by J. Nicolas Coldstream

(J.N.) Coldstream Greek Geometric Pottery. A Survey of Ten Styles and their Chronology. Updated second edition. Pp. xlii + 502, maps, pls. Exeter: Bristol Phoenix Press, 2008. Cased, £150, US$275. ISBN: 978-1-904675-81-5.

(J.N.) Coldstream Greek Geometric Pottery. A Survey of Ten Styles and their Chronology. Updated second edition. Pp. xlii + 502, maps, pls. Exeter: Bristol Phoenix Press, 2008. Cased, £150, US$275. ISBN: 978-1-904675-81-5. - Volume 60 Issue 1

The Euclidean Mousetrap

In his doctoral dissertation On the Principle of Sufficient Reason, Arthur Schopenhauer there outlines a critique of Euclidean geometry on the basis of the changing nature of mathematics, and hence of demonstration, as a result of Kantian idealism. According to Schopenhauer, Euclid treats geometry synthetically, proceeding from the simple to the complex, from the known to the unknown, synthesizing later proofs on the basis of earlier ones. Such a method, although proving the case logically, nevertheless fails to attain the raison d'etre of the entity. In order to obtain this, a separate method is required, which Schopenhauer refers to as analysis, thus echoing a method already in practice among the early Greek geometers, with however some significant differences. In this essay, I here discuss Schopenhauer's criticism of synthesis in Euclid's Elements, and the nature and relevance of his own method of analysis.

Collaborative interviewing in mathematical analogy technique (Part I): socialization in cognitive beavioural therapy

CLIMATE is an acronym for Collaborative Interviewing in Mathematical Analogy Technique. CLIMATE is a procedure based on the theories of ancient Greek scholars: i. Socrates, who introduced the dialectic method, ii. Plato, Socrates' best disciple who postulated the method of analysis and synthesis, iii. Aristotle, Plato's most known student and iv. Pappus of Alexandria, a great Greek geometer, who postulated the method of analysis and synthesis in mathematics. According to Plato, Socrates in order to prove to Menno that all I know is that I know nothing used a mathematical paradigm to an illiterate servant boy. When socializing a patient into CBT using CLIMATE the therapist follows a similar process.

On the use of the term diastēma in ancient Greek constructions

I examine the use of the term diastēma by Greek geometers in both plane and spherical constructions. I show that while diastēma may be translated as radius in plane constructions, this will not work on the sphere. These investigations have some implications for how we think of construction in Greek mathematics in general.

Low Chronology and Greek Protogeometric and Geometric Pottery in the Southern Levant

The article deals with the absolute chronology of the Iron Age in the Southern Levant and the Aegean world. It first discusses the chronological gap between the conventional dating of Protogeometric pottery in Greece, and its appearance in the Southern Levantine assemblages. The question then asked is whether the Low Chronology recently proposed by Finkelstein will free us from the difficulties raised by the prevailing dating. A brief examination shows that one cannot use the accepted dating of Greek Protogeometric and Geometric pottery to support the ‘Finkelstein correction’, but rather his Low Chronology provides, for the first time, a basis for the absolute chronology of the Dark Age in Greece. At the end the article proposes a common denominator to reconcile the differences of opinion between major chronological approaches. These may be explained by the differences in the length of existence of the relevant strata.

Significant Others: The Male-Female Pair in Greek Geometric Art

After a century of scholarship, Greek Geometric figural art continues to pose fundamental problems of identification and interpretation. The intractable nature of basic iconographic questions is in part the result of the historiographic circumstances that formed the present canon of major and minor Geometric works. This study focuses on the specific iconographic theme of the male-female couple and traces its disappearance from histories of Geometric art as a result of academic and cultural biases operating in the early 20th century. Two examples, a pair of terracotta heads from Amyklai and a group of misidentified female figurines from Geometric bronze tripods, illustrate the progressive marginalization of the theme from the art historical mainstream. An inquiry into the origins of scholarship on Geometric sculpture demonstrates how crucial contextual information could be lost and what its restoration can offer in terms of a richer corpus of Geometric iconography with which to approach the challenges of interpretation. Emphasizing the iconographic flexibility of Geometric style, this study suggests that the image of a warrior with female companion was newly introduced in the mid-eighth century by elite patronage and found widespread social utility in the changing world of early Iron Age Greece.

Aristoteliantoposand Greek geometry∗

In this article it is argued that Aristotle borrowed his rhetorical concept of topos from Greek geometry. Passages in Proclus’ Commentary on the First Book of Euclid show that topos as the term for ‘geometrical locus’ was used as early as in the fourth century by Greek geometers in or associated with the Academy. It is further argued that the adjective aτoπoς had its origin in Greek science, in the technical sense ‘contrary to aτoπoς’, i.e. ’impossible’, ‘illogical’.

Unthinking Thinking: Jorge Luis Borges, Mathematics, and the New Physics (review)

Reviewed by: Unthinking Thinking: Jorge Luis Borges, Mathematics, and the New Physics John Mitchell Floyd Merrell . Unthinking Thinking: Jorge Luis Borges, Mathematics, and the New Physics. West Lafayette: Purdue UP, 1991. 300 pp. $29.50. In his Preface Merrell comments on the book's title and its relation to both Borges and science: " 'unthinking thinking' reveals the present inquiry's underlying strain. The title, I believe, is appropriately ambiguous. It implies either the project of unthinking traditional Western thought, or, paradoxically, thinking without there being any accompanying process of thought (an inevitably abortive attempt by sheer intellection to approach a mystical insight). Both interpretations are germane [End Page 502] to the nature of Borges's work. . . ." Merrell's approach is a deductive one, a methodology that he traces to Borges's own writing and to certain scientific traditions beginning with the Greek geometers. The writings of Borges mark a point for the folding of disciplines and texts, a referent for "intertextuality" between science and the humanities. And, for Merrell, they provide as well a "springboard" to a multiplicity of issues surrounding the postmodern. One of the great strengths of the book is Merrell's careful placement of both Borges and the "new physics" within a larger historical context which we might designate as the history of science and the history of ideas. Thus, Chapter One examines the age-old debate between nominalism and realism and the extent to which Borges participates in this controversy in his metaphysical writings, for example, "The Zahir" and "The Aleph." Chapter Two on paradox and Three on models of infinity and the problematics of totalization open up the postmodern conflicts full throttle. Merrell works both with Borges's delight in shattering "ideological truths" and with the scientific modalities of chaos theory, taxonomy, and systems theory. Chapter Three is particularly useful in its expositions on mathematics, the end of certainty, and the infinite. Merrell follows these initial metaphysical chapters with an Interlude which summarizes the emergence of the "new physics" and which also draws effective parallels between the fiction-making processes of scientists and writers. The brief history of science (handled in a self-reflexive postmodern manner) Interlude makes smooth the way for the following chapter on the "universe of Einsteinian physics and quantum mechanics." Chapter Four, "The Universe as Library," maps the relation between Einstein's "thought experiments" which resulted in his conceptualization of a space-time continuum, and Borges's own meditations on the universe in "The Library of Babel." Space merges with Time (and Timelessness) in Chapter Five in which Merrell links "The Aleph" with the new physics phenomena of black holes and singularities. Chapter Six extends beyond Space and Time into the mysteries of quantum mechanics where "reality" dissolves difference and Chance, represented by Borges in "The Lottery of Babylon" and by "broken symmetries" in subnuclear particles, disrupts the Newtonian universe. Merrell's project reaches its conclusion in the final chapter on Language. It is here within the bounds of Derrida's "text" that science and art, thought experiment and writing blend together in a "totality of intertextuality." This gives us the multiple realities, the thousand plateaus of the postmodern modality and, as Merrell demonstrates, it also creates in Borges's work the "profound conflict" between this postmodern recognition and the nostalgia for a Modernist (or premodernist?) grounding. Of course, this conflict is one of the keys in making Borges's Quixotic struggles a map for our postmodern labyrinths. John Mitchell Purdue University Copyright © 1992 Purdue Research Foundation

GREEK GEOMETRIC NARRATIVE ART AND ORALITY

GREEK GEOMETRIC NARRATIVE ART AND ORALITY

Some Challenging Constructions

In his book A Survey of Geometry, Howard Eves (1966, 183) states, “The Greek geometers of antiquity devised a game— which judged on … [challenge, variety and simplicity] must surely stand at the very top of any list of games to be played alone.”

A Study on the Chronological Parallels between the Greek Geometric Style and Sialk B Painted Pottery

A Study on the Chronological Parallels between the Greek Geometric Style and Sialk B Painted Pottery

Greek Geometric Pottery at Hama and its Implications for Near Eastern Chronology

(1985). Greek Geometric Pottery at Hama and its Implications for Near Eastern Chronology. Levant: Vol. 17, No. 1, pp. 131-138.

E. Gjerstad et al.: Greek Geometric and Archaic Pottery found in Cyprus (Swedish Institute in Athens, Acta, Series in 4°, XXVI). Pp. 86; 113 plates. Stockholm: Swedish Institute in Athens, 1977. Paper.

E. Gjerstad <i>et al.</i>: Greek Geometric and Archaic Pottery found in Cyprus (Swedish Institute in Athens, Acta, Series in 4°, XXVI). Pp. 86; 113 plates. Stockholm: Swedish Institute in Athens, 1977. Paper.

Evolution

EVOLUTION SAMUEL STEARNS, M.D.* Evolving man moves faster, but error brings disaster. Each species meets its cul-de-sac; life moves one way, no turning back. We hope for orthogenesis, but fossils teach us as it is. Herr Spengler saw Man's history as cyclical and set, while Toynbee thought that war would end the brainiest species yet. Ecologists warn of human fate when all has been consumed; if Engels and Karl Marx were right, our society is doomed. But measuring by 5000 years implies myopia; * 1 1 1 Perkins Street, Jamaica Plain, Massachusetts 02130. 58 Samuel Steams ¦ Evolution Nature's riches spill sans end from her cornucopia. I hope that de Chardin saw well the vector of creation; though I dispute his Omega point, I like his ideation. Our parts achieve a special goal because of their anlagen; how each grows different from the whole leaves open manyfragen. I hope the universe contains a similar ontogeny: that interstellar space retains a purposeful phylogeny. Should macrocosm be the less than microcosmic being? Myfrage rests, I must confess, on belief without the seeing. IV Meiosis is a pretty way (mutation is another) to alter one man's DNA from that within his brother. Genetic codes are at the root of making men and women, moron, dolt and crazy coot and those with more acumen. Diversity is Nature's aim, selection is her meth—od; if everybody were the same, would not be birth and death odd? Perspectives in Biology and Medicine ¦ Autumn 1977 59 On the Discovery of an Ancient Hominid, or Alas, Poor Yorick! The carbon dating of hisjaw suggests that he is very old, old enough to rouse our awe to think he roamed and ran the veldt so many years ago. Of course, he died quite young, stalked, felled and chewed by hungry carnivores. When Homo sapiens is found, charred by a nuclear bomb and blast, beneath a crumbling rock or mound, his jaw will also show the past and what he was—by a bridge, perhaps an inlay—and a fatal count of fusion-fashioned isotopes; and H. superior may muse, on pondering the fossil's age of 70, what lame excuse he gave himself for the feral rage that made him and his knowledge more deadly than any predator, more vicious than a carnivore. VI Why My Crystal Ball Is Cloudy To what unfathomable future end did stars condense from empty space and simple molecules arise, somehow to make the life which we now grace? It would be simple, if we only knew the purpose of it all, could see ahead, understand why hemoglobins all make different cells just blue or red. 60 Samuel Stearns · Evolution A seer could use a crystal ball, and a famous Greek geometer once said two points would lead unerringly beyond, so past and present point ahead. Perhaps, this way, the Roman augur feigned prophéties from the entrails of a fowl, the oracle from fumes of pot or such, the Chinese ditto, tea-leafed bowl. Alas! the past is full of points—which one? The present is a presbyopic blur: my crystal ball shows many lines, and I, confused, at prophecy demur. VII Selection, Not Election It's not the nice things only: rut, strut, song or color, blush of wattles, or a musky piss against a tree, but fiercer traits which win death battles for a turf, a mate and nature's dot. Aggression is a force of nature, too, like rain, snow, tide, heat and cold. Selection is impartial: to be true it can be neither good nor bad, but bold and pitiless to chose the fittest few. Some philosophers protest, demur: "Man's an angel, not a scowling ape!" They cannot let the truer thought occur: their fondest wish is just a sour grape, and their ideal must to the real defer. Perspectives in Biology and Medicine ¦ Autumn 1977 | 61 Vili On Dropping Out of the University-for Growing Peas in the Back Yard Father Mendel failed at school and turned to raising common peas, some dark, some white, some short, some tall— a hobby for his simple ease. Who really knows why he chose this...