call for papers by Robert Hahn
Dialogues d'histoire ancienne, 2022
In order to take a fresh look at Thales geometrical insights, I will place him in the broader, te... more In order to take a fresh look at Thales geometrical insights, I will place him in the broader, technological context of 6th century BCE Ionia. I start with Aristotle’s report about Thales and the early philosophers who he claims were – to use my terms – source and substance monists. I argue that this view is testimony of their modular thinking. I defend the reliability of Aristotle’s report by a new set of arguments pointing out that examples of modular thinking were all around Thales, Anaximander, Anaximenes, Heraclitus, Pythagoras/Pythagoreans and interpenetrated their thoughts -- in monumental stone temple architecture, in the production of coins in the monetization of their society, in the use of the gnomon, and even connected with the production of industrial textiles. Could it be that Thales inferred an underlying module of all things – he called it ‘water’ [ὕδωρ] -- by analogy with these other examples surrounding him? Everything that appears, according to Aristotle, are only alterations or modifications of this underlying nature or unity, because this unity always persists – this is the principle of modular thinking.
Now, had Thales held such a view, it seems difficult to avoid the question “How does this happen – how does ‘water’ flow shapelessly in a cup at one moment, then invisible as air, as fire at the stove, and yet sometimes hard as marble?” And what I propose to explore is the possibility that Thales’ forays in geometry sought to identify the underlying structure of ‘water’ out of which all other appearances were built, re-packaged and re-combined, and that Thales’ plausibly reached the conclusion it was the right triangle. Accordingly, I will invite us to imagine the diagrams corresponding to the reports that Thales in Egypt measured the height of a pyramid, measured the distance of a ship at sea, and place them alongside the diagrams that display the geometrical propositions with which Thales is also credited. Having done this, the reader can see that all, or mostly all, deal not only with right triangles but with similar right-angled triangles. And since one line of proof, preserved by Euclid [VI.31], demonstrates the so-called Pythagorean theorem by similar right triangles, I argue it is plausible that Thales visualized, in a less sophisticated form, this line of reasoning for the famous theorem because it is a consequence of similar right triangles. And the ‘Pythagorean theorem’ by this line of reasoning shows that the right triangle is the fundamental geometrical figure. So, the case I explore is what I have been calling the “Lost Narrative,” the one that connects the reports about Thales’ geometrical insights with his speculations about the underlying unity of nature.
This essay appears in the forthcoming volume:
Heraklit im Kontext
[Heraclitus in Context]
Ed. by... more This essay appears in the forthcoming volume:
Heraklit im Kontext
[Heraclitus in Context]
Ed. by Fantino, Enrica / Muss, Ulrike / Schubert, Charlotte / Sier, Kurt
Series:Studia Praesocratica 8, Walter DeGruyter
This essay will appear in the 2nd edition of LOGOS AND MUTHOS, W. Wians editor
This is a call for papers for an international conference devoted to early Greek philosophy and t... more This is a call for papers for an international conference devoted to early Greek philosophy and thought, taking place at the National and Kapodistrian University of Athens/Greece, in December 2016.
Feel free to apply when you have original things to say!
Talks/Conférences et cours by Robert Hahn
Please visit our new website with a preliminary program as well as abstracts:
http://enlightened... more Please visit our new website with a preliminary program as well as abstracts:
http://enlightenedionia.siu.edu/
Drafts by Robert Hahn
WHY THALES KNEW THE PYTHAGOREAN THEOREM
Once we accept that Thales introduced geometry into Gree... more WHY THALES KNEW THE PYTHAGOREAN THEOREM
Once we accept that Thales introduced geometry into Greece, having traveled to Egypt, as Proclus reports on the authority of Eudemus, who also credits Thales with a number of theorems, we understand that Thales was making geometrical diagrams. From where did he see such diagrams? Egypt is one place, having measured the height of a pyramid there. Diagrams that reflect measurements when the shadow was equal to its height, and un-equal but proportional – the doxography credits him with both techniques -- suggest that Thales understood similar triangles – ratios, proportions, and similarity.
We begin with the diagrams associated with the theorems, and place them next to the ones that reflect the measurements of the pyramid and distance of a ship at sea, Then, we introduce, on the authority of Aristotle, that Thales posited an archê, a principle, from which all things come, and back into which all things return upon dissolution – there is no change, only alteration – a big picture begins to form. Suppose, then, Thales investigated geometry, whether or not they started as practical exercises, as a way to solve the metaphysical problem of explaining HOW this one underlying unity could appear so divergently, modified but not changing? Geometry offered a way to find the basic figure into which all other figures resolve, that re-packed and re-combined, was the building block of all other appearances. We might see a lost narrative of the relation between philosophy and geometry. That narrative is preserved later by Plato at Timaeus 53C and following: the construction of the cosmos out of right triangles.
There are two proofs of the Pythagorean theorem, not one, preserved by Euclid. The one we learned in school, if we learned it at all, was I.47 that Proclus reports was Euclid’s own invention. But, the other one, in book VI.31, by similar figures, by ratios and proportions, plausibly points back to Thales himself, perhaps taken up and perfected in proof by Pythagoras and the Pythagoreans. That proof shows that the right triangle is the fundamental geometrical figure, that expands or contracts in a pattern that came to be called continuous proportions. The argument that Thales knew the hypotenuse theorem is that, surprisingly, this was what he was looking for to explain HOW a single unity could appear so divergently, altering without changing. The plausibility rests on following the diagrams as evidence.
Papers by Robert Hahn
Philosophy East and West, 2006
Page 1. REMEMBERING LEWIS E. HAHN The following testimonials were offered on the occasion of a me... more Page 1. REMEMBERING LEWIS E. HAHN The following testimonials were offered on the occasion of a memorial gathering for Dr. Lewis E. Hahn held on February 19, 2005, and were compiled for presentation here by Sharon (Hahn) Crowell. ...
Revista Archai, 2020
In this essay on ancient architectural technologies, I propose to challenge the largely conventio... more In this essay on ancient architectural technologies, I propose to challenge the largely conventional idea of the transcendent origins of philosophy, that philosophy dawned only when the mind turned inside, away from the world grasped by the body and senses. By focusing on one premier episode in the history of western thinking – the emergence of Greek philosophical thought in the cosmic architecture of Anaximander of Miletus – I am arguing that the abstract, speculative, rationalising thinking characteristic of philosophy, is indeed rooted in practical activities, and emerges by means of them rather than in repudiation of them. The spirit of rational inquiry emerged from several factors but the contributing role of monumental architecture and building technologies has been vastly under-appreciated. In the process of figuring out how to build on an enormous scale that the eastern Greeks had never before tried, the architects discovered and revealed nature’s order in their thaumata, th...
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call for papers by Robert Hahn
Now, had Thales held such a view, it seems difficult to avoid the question “How does this happen – how does ‘water’ flow shapelessly in a cup at one moment, then invisible as air, as fire at the stove, and yet sometimes hard as marble?” And what I propose to explore is the possibility that Thales’ forays in geometry sought to identify the underlying structure of ‘water’ out of which all other appearances were built, re-packaged and re-combined, and that Thales’ plausibly reached the conclusion it was the right triangle. Accordingly, I will invite us to imagine the diagrams corresponding to the reports that Thales in Egypt measured the height of a pyramid, measured the distance of a ship at sea, and place them alongside the diagrams that display the geometrical propositions with which Thales is also credited. Having done this, the reader can see that all, or mostly all, deal not only with right triangles but with similar right-angled triangles. And since one line of proof, preserved by Euclid [VI.31], demonstrates the so-called Pythagorean theorem by similar right triangles, I argue it is plausible that Thales visualized, in a less sophisticated form, this line of reasoning for the famous theorem because it is a consequence of similar right triangles. And the ‘Pythagorean theorem’ by this line of reasoning shows that the right triangle is the fundamental geometrical figure. So, the case I explore is what I have been calling the “Lost Narrative,” the one that connects the reports about Thales’ geometrical insights with his speculations about the underlying unity of nature.
Heraklit im Kontext
[Heraclitus in Context]
Ed. by Fantino, Enrica / Muss, Ulrike / Schubert, Charlotte / Sier, Kurt
Series:Studia Praesocratica 8, Walter DeGruyter
Feel free to apply when you have original things to say!
Talks/Conférences et cours by Robert Hahn
http://enlightenedionia.siu.edu/
Drafts by Robert Hahn
Once we accept that Thales introduced geometry into Greece, having traveled to Egypt, as Proclus reports on the authority of Eudemus, who also credits Thales with a number of theorems, we understand that Thales was making geometrical diagrams. From where did he see such diagrams? Egypt is one place, having measured the height of a pyramid there. Diagrams that reflect measurements when the shadow was equal to its height, and un-equal but proportional – the doxography credits him with both techniques -- suggest that Thales understood similar triangles – ratios, proportions, and similarity.
We begin with the diagrams associated with the theorems, and place them next to the ones that reflect the measurements of the pyramid and distance of a ship at sea, Then, we introduce, on the authority of Aristotle, that Thales posited an archê, a principle, from which all things come, and back into which all things return upon dissolution – there is no change, only alteration – a big picture begins to form. Suppose, then, Thales investigated geometry, whether or not they started as practical exercises, as a way to solve the metaphysical problem of explaining HOW this one underlying unity could appear so divergently, modified but not changing? Geometry offered a way to find the basic figure into which all other figures resolve, that re-packed and re-combined, was the building block of all other appearances. We might see a lost narrative of the relation between philosophy and geometry. That narrative is preserved later by Plato at Timaeus 53C and following: the construction of the cosmos out of right triangles.
There are two proofs of the Pythagorean theorem, not one, preserved by Euclid. The one we learned in school, if we learned it at all, was I.47 that Proclus reports was Euclid’s own invention. But, the other one, in book VI.31, by similar figures, by ratios and proportions, plausibly points back to Thales himself, perhaps taken up and perfected in proof by Pythagoras and the Pythagoreans. That proof shows that the right triangle is the fundamental geometrical figure, that expands or contracts in a pattern that came to be called continuous proportions. The argument that Thales knew the hypotenuse theorem is that, surprisingly, this was what he was looking for to explain HOW a single unity could appear so divergently, altering without changing. The plausibility rests on following the diagrams as evidence.
Papers by Robert Hahn
Now, had Thales held such a view, it seems difficult to avoid the question “How does this happen – how does ‘water’ flow shapelessly in a cup at one moment, then invisible as air, as fire at the stove, and yet sometimes hard as marble?” And what I propose to explore is the possibility that Thales’ forays in geometry sought to identify the underlying structure of ‘water’ out of which all other appearances were built, re-packaged and re-combined, and that Thales’ plausibly reached the conclusion it was the right triangle. Accordingly, I will invite us to imagine the diagrams corresponding to the reports that Thales in Egypt measured the height of a pyramid, measured the distance of a ship at sea, and place them alongside the diagrams that display the geometrical propositions with which Thales is also credited. Having done this, the reader can see that all, or mostly all, deal not only with right triangles but with similar right-angled triangles. And since one line of proof, preserved by Euclid [VI.31], demonstrates the so-called Pythagorean theorem by similar right triangles, I argue it is plausible that Thales visualized, in a less sophisticated form, this line of reasoning for the famous theorem because it is a consequence of similar right triangles. And the ‘Pythagorean theorem’ by this line of reasoning shows that the right triangle is the fundamental geometrical figure. So, the case I explore is what I have been calling the “Lost Narrative,” the one that connects the reports about Thales’ geometrical insights with his speculations about the underlying unity of nature.
Heraklit im Kontext
[Heraclitus in Context]
Ed. by Fantino, Enrica / Muss, Ulrike / Schubert, Charlotte / Sier, Kurt
Series:Studia Praesocratica 8, Walter DeGruyter
Feel free to apply when you have original things to say!
http://enlightenedionia.siu.edu/
Once we accept that Thales introduced geometry into Greece, having traveled to Egypt, as Proclus reports on the authority of Eudemus, who also credits Thales with a number of theorems, we understand that Thales was making geometrical diagrams. From where did he see such diagrams? Egypt is one place, having measured the height of a pyramid there. Diagrams that reflect measurements when the shadow was equal to its height, and un-equal but proportional – the doxography credits him with both techniques -- suggest that Thales understood similar triangles – ratios, proportions, and similarity.
We begin with the diagrams associated with the theorems, and place them next to the ones that reflect the measurements of the pyramid and distance of a ship at sea, Then, we introduce, on the authority of Aristotle, that Thales posited an archê, a principle, from which all things come, and back into which all things return upon dissolution – there is no change, only alteration – a big picture begins to form. Suppose, then, Thales investigated geometry, whether or not they started as practical exercises, as a way to solve the metaphysical problem of explaining HOW this one underlying unity could appear so divergently, modified but not changing? Geometry offered a way to find the basic figure into which all other figures resolve, that re-packed and re-combined, was the building block of all other appearances. We might see a lost narrative of the relation between philosophy and geometry. That narrative is preserved later by Plato at Timaeus 53C and following: the construction of the cosmos out of right triangles.
There are two proofs of the Pythagorean theorem, not one, preserved by Euclid. The one we learned in school, if we learned it at all, was I.47 that Proclus reports was Euclid’s own invention. But, the other one, in book VI.31, by similar figures, by ratios and proportions, plausibly points back to Thales himself, perhaps taken up and perfected in proof by Pythagoras and the Pythagoreans. That proof shows that the right triangle is the fundamental geometrical figure, that expands or contracts in a pattern that came to be called continuous proportions. The argument that Thales knew the hypotenuse theorem is that, surprisingly, this was what he was looking for to explain HOW a single unity could appear so divergently, altering without changing. The plausibility rests on following the diagrams as evidence.