The History of Mathematical Proof in Ancient Traditions

Th e pluralism of Greek ‘mathematics’ 303 Aristotle was, of course, the fi rst to propose an explicit defi nition of rigor- ous ...

304 g e o f f r e y L l o y d Analytics. But mathematics suff ers from a diff erent shortcoming, in his view, which relates to t ...

Th e pluralism of Greek ‘mathematics’ 305 from premisses that are themselves indemonstrable – to avoid the twin fl aws of circul ...

306 g e o f f r e y L l o y d although attempts to provide a proof all turned out to be circular. Yet the controversial characte ...

Th e pluralism of Greek ‘mathematics’ 307 aesthetics that began to be cultivated in the Hellenistic period. 19 I n t h e c a s e ...

308 g e o f f r e y L l o y d that he promoted in part to create a gap between demonstrative reasoning and the merely plausible ...

Th e pluralism of Greek ‘mathematics’ 309 Burkert , W. ( 1972 ) Lore and Science in Ancient Pythagoreanism. Cambridge, MA. Burny ...

310 g e o f f r e y L l o y d Szabó , Á. ( 1978 ) Th e Beginnings of Greek Mathematics , trans. A. M. Ungar. Budapest. (Original ...

311 9 Generalizing about polygonal numbers in ancient Greek mathematics Ian Mueller Introduction Th e main source for our inform ...

312 ian mueller unit α, t w o αα, t h r e e ααα, four αααα, fi v e ααααα, and so on. Nicomachus’ “natural” representation of num ...

Polygonal numbers in ancient Greek mathematics 313 Nicomachus also mentions hexagonal, heptagonal, and octagonal numbers, and th ...

314 ian mueller If we ignore the distinction between a unit and a number, 5 we may express Nicomachus’ claim here as: Th e side ...

Polygonal numbers in ancient Greek mathematics 315 He proceeds to show in the same way that: Th e n th square number is the sum ...

316 ian mueller Nicomachus proceeds through the octagonal numbers without fi gures, making clear that: [Nic*]. Th e sum of the f ...

Polygonal numbers in ancient Greek mathematics 317 from them. And again if any triangle is joined to any square fi gure 6 it pro ...

318 ian mueller Introduction to be, not a scientifi c treatise, but a popular treatment of the subject calculated to awaken in a ...

Polygonal numbers in ancient Greek mathematics 319 of their content; in footnotes I give simple algebraic proofs of the results ...

320 ian mueller multiplied by a certain number which, when a unit is added to it, is double of the multitude of all the numbers ...

Polygonal numbers in ancient Greek mathematics 321 As I have said, the material described thus far in this section is purely ari ...

322 ian mueller What I will try to explain is the specifi c reason why Diophantus’ attempt to do so fails, emphasizing that, alt ...