Linguistic formulae as cognitive tools

1999 ◽  
Vol 7 (1) ◽  
pp. 147-176 ◽  
Author(s):  
Reviel Netz
Keyword(s):  
Cognitive Tools ◽  
Cognitive Level ◽  
Ancient Greek ◽  
Partial Explanation ◽  
Logical Tool ◽  
Greek Mathematics ◽  
Fixed System ◽  
Mathematical Texts ◽  
Original Feature

Ancient Greek mathematics developed the original feature of being deductive mathematics. This article attempts to give a (partial) explanation f or this achievement. The focus is on the use of a fixed system of linguistic formulae (expressions used repetitively) in Greek mathematical texts. It is shown that (a) the structure of this system was especially adapted for the easy computation of operations of substitution on such formulae, that is, of replacing one element in a fixed formula by another, and it is further argued that (b) such operations of substitution were the main logical tool required by Greek mathematical deduction. The conclusion explains why, assuming the validity of the description above, this historical level (as against the universal cognitive level) is the best explanatory level for the phenomenon of Greek mathematical deduction.

Ancient Greek Mathematics

2015 ◽  
Author(s):  
José Ferreirós
Keyword(s):  
Elementary Mathematics ◽  
Euclidean Geometry ◽  
Advanced Mathematics ◽  
Ancient Greek ◽  
Greek Tradition ◽  
The Common ◽  
Greek Mathematics ◽  
Joint Consideration ◽  
Practical Geometry ◽  
Further Development

This chapter focuses on the ancient Greek tradition of geometrical proof in light of recent studies by Kenneth Manders and others. It advances the view that the borderline of elementary mathematics is strictly linked with the adoption of hypotheses. To this end, the chapter considers Euclidean geometry, which elaborates on both the problems and the proof methods based on diagrams. It argues that Euclidean geometry can be understood as a theoretical, idealized analysis (and further development) of practical geometry; that by way of the idealizations introduced, Euclid's Elements builds on hypotheses that turn them into advanced mathematics; and that the axioms or “postulates” of Book I of the Elements mainly regiment diagrammatic constructions, while the “common notions” are general principles of a theory of quantities. The chapter concludes by discussing how the proposed approach, based on joint consideration of agents and frameworks, can be applied to the case of Greek geometry.

Historically Speaking—: The Bernoulli Family

1966 ◽  
Vol 59 (3) ◽  
pp. 276-278
Author(s):  
Howard Eves
Keyword(s):  
United States ◽  
Eighteenth Century ◽  
General Rule ◽  
The United States ◽  
Great Mathematician ◽  
Ancient Greek ◽  
Isaac Newton ◽  
Greek Mathematics ◽  
Successive Generations ◽  
Father And Son

There is a general rule to the effect that any given family possesses at most one outstanding mathematician and that, in fact, most families possess none. Thus a search through the ancestors, descendents, and relatives of Isaac Newton fails to turn up any other great mathematician. There are exceptions to this general rule. For example we have, here in the United States, the two Lehmers (father and son) and the two Birkhoffs (father and son). One also recalls the two Cassinis (father and son) of the late seventeenth and early eighteenth centuries, and perhaps one can build a case for the two Clairaut children of the eighteenth century. And of course there were Theon and Hypatia (father and daughter), who lived during the closing years of ancient Greek mathematics. But such cases are relatively rare. All the more striking, then, is the Bernoulli family of Switzerland, which in three successive generations produced no less than eight noted mathematicians.

scholarly journals On a Collection of Geometrical Riddles and their Role in the Shaping of Four to Six “Algebras”

2001 ◽  
Vol 14 (1-2) ◽  
pp. 85-131 ◽  
Author(s):  
Jens Høyrup
Keyword(s):  
Theoretical Investigation ◽  
Ancient Greek ◽  
New Approach ◽  
Islamic World ◽  
Near Eastern ◽  
Old Babylonian ◽  
Greek Mathematics ◽  
Practical Geometry

For more than a century, there has been some discussion about whether medieval Arabic al-jabr (and hence also later European algebra) has its roots in Indian or Greek mathematics. Since the 1930s, the possibility of Babylonian ultimate roots has entered the debate. This article presents a new approach to the problem, pointing to a set of quasi-algebraic riddles that appear to have circulated among Near Eastern practical geometers since c. 2000 BCE, and which inspired first the so-called “algebra” of the Old Babylonian scribal school and later the geometry of Elements II (where the techniques are submitted to theoretical investigation). The riddles also turn up in ancient Greek practical geometry and Jaina mathematics. Eventually they reached European (Latin and abbaco) mathematics via the Islamic world. However, no evidence supports a derivation of medieval Indian algebra or the original core of al-jabr from the riddles.

scholarly journals Plato’s notion of hypothesis in dialogues Meno, Phaedo and The Republic

2017 ◽  
Vol 60 (2) ◽  
pp. 120-144
Author(s):  
Visnja Knezevic
Keyword(s):  
Critical Thinking ◽  
Mathematical Methods ◽  
Ancient Greek ◽  
Metaphysical Theory ◽  
Methods Of Analysis ◽  
Greek Mathematics ◽  
The Republic ◽  
Deductive Science ◽  
Theory Of Forms

The author analyses Plato?s use of the hypothesis notion in connection with his hypotheses method, as it was articulated in Meno and Phaedo, and later criticized in The Republic. It is shown that, at first, Plato?s use of this notion was identical to its use in ancient Greek mathematics, and that the same stands in regards with his method of inquiry - this, too, was at first modeled after ancient Greek mathematical methods of analysis and diorismos. Later, as he developed the metaphysical theory of forms, Plato distanced himself from ideal of building philosophy on the model of ancient Greek deductive science and established it as auto reflexive, critical thinking instead, with dialectics as method in its own right.

scholarly journals The problem of incommensurability and the crisis of foundations of the Ancient Greek mathematics

2021 ◽  
pp. 54-65
Author(s):  
Denis Aleksandrovich Kiryanov
Keyword(s):  
Numerical Characteristic ◽  
Ancient Greek ◽  
Special Contribution ◽  
Line Segments ◽  
Science Culture ◽  
History Of ◽  
The Subject ◽  
Greek Mathematics ◽  
Small Parts

The subject of this research is the problem of incommensurability and the crisis of foundations of the Ancient Greek mathematics. The article describes that the crisis of foundations was caused by the discovery of irrationality by the Pythagorean Hippasus of Metapontum, which resulted in the theoretical instability of mathematics of the Pythagoreans, who believed that everything could be expressed through numbers. The discovery of incommensurable line segments demonstrated that the relations between rational numbers cannot express any variable, for example the diagonal of a square with one side equal to one. Analysis is conducted on the achievements of the Pythagorean School in the field of mathematics. Special attention is given to the role of a number in the philosophy of this school. The article explores the main ways for overcoming this crisis, philosophical explanation of the unfolded situation, based on which the Pythagoreans formulate the methodological ways out of the discovered problem of incommensurability. It is noted that the Pythagoreans were actively elaborating on their philosophy and mathematical apparatus intending to find the answer to the discovery of incommensurability. The author’s special contribution lies in the statement that the discovery of irrationality was not critical for the Pythagoreans: they continued working towards the answer to the problem of incommensurability, as well as refined the mathematical theory of proportions, reconsidered the representation of infiniteness as a certain numerical characteristic of the things and processes. This article is first to advance a hypothesis on the possibility of dividing the object into an infinitely large number of infinitely small parts, which is now understood as the limit of function, which contributes to the development and application of dialectics. The problem of incommensurability led to the creation of new, complex theories in the history of science, culture, architecture, and art.

scholarly journals Leopardi and the ancient Greek mathematics

2002 ◽  
Vol 01 (02) ◽  
pp. A01
Author(s):  
Annalisa Reggi
Keyword(s):  
Bertrand Russell ◽  
Close Connection ◽  
Scientific Culture ◽  
Ancient Greek ◽  
The World ◽  
Greek Mathematics

"I consider Leopardi's poetry and pessimism to be the best expression of what a scientist's credo should be". This quotation is from Bertrand Russell, no less. With these very emblematic words, the greatest man of letters, the supreme icon of the Italian Parnasse, the author of such collections of poems as Canti (Poems) and Operette Morali (The Moral Essays) and philosophical thoughts as Zibaldone (Miscellany) has been associated to the world of science. This relationship, very intense and to a certain extent new, was greatly emphasised on the occasion of the poet's birth bicentenary. During the celebration in 1996, an exhibition with the name of Giacomo and Science was organized in his birthplace to underline the close connection between the poet and the scientific culture of his epoch. This point has also been stressed recently.

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