Euclid is widely touted as the "inventor" of the axiomatic method, as his Elements are build around it. But there are earlier reported Elements (seemingly completely lost), and parts of the Elements are attributed to earlier mathematicians. So the question is if he was the inventor of this approach, or just the great systematizer/organizer of earlier work, which his Elements so thoroughly incorporated that it all but faded from view?
$\begingroup$
$\endgroup$
2
-
1$\begingroup$ Even aside from predecessors Euclid is not quite the inventor of the axiomatic method, and Elements are not axiomatic in the modern sense of Hilbert and Pasch. Euclid does not treat his axioms as implicit definitions of formal terms that can refer to anything, nor are his axioms sufficient to derive his theorems on such a reading. Euclid's method is rather synthetic, in the old sense of synthetic geometry, with diagrams playing an essential role in demostrations. $\endgroup$– ConifoldNov 3, 2015 at 23:19
-
1$\begingroup$ @ Conifold: What is axiomatic is not to be defined by Hilbert and Pasch and their disciples! And there is no virtue in the possibility of applying axioms of geometry to Bierseideln and other rubbish. Further diagrams are a way of communication which is at least as valuable as written formal language. $\endgroup$– Franz KurzJun 11, 2017 at 15:17
Add a comment
|
1 Answer
$\begingroup$
$\endgroup$
See :
- Thomas Heath (editor), The Thirteen Books of Euclid's Elements (Dover reprint, Volume 1) : Introduction, Ch.IX, §2 : ELEMENTS ANTERIOR TO EUCLID'S, page 116 :
for references to earlier Elements now lost :
Hippocrates of Chios, Leon, Eudoxus of Cnidus, Theudius of Magnesia.
See :
- Thomas Heath, A History of Greek Mathematics. Volume I (1921), page 319, lists :
[the] names mentioned in the summary of Proclus [in his influential commentary on the first book of Euclid's Elements, §66] :
"Younger than Leodamas were Neoclides and his pupil Leon, who added many things to what was known before their time, so that Leon was actually able to make a collection of the elements more carefully designed in respect both of the number of proposition proved and of their utility, besides which he invented diorismi (the object of which is to determine) when the problem under investigation is possible of solution and when impossible."
[...] Eudoxus of Cnidos [...]was the first to increase the number of the so-called general theorems [...]. Amyclas [more correctly Amyntas] of Heraclea [...] Menaechmus, a pupil of Eudoxus [...] and Dinostratus, his brother, made the whole of geometry still more perfect. Theudius of Magnesia [...] put together the elements admirably and made many partials theorems more general.
See in the complete translation of Proclus, Prologue, §65-on, Proclus' historical account, from Eudemus of Rhodes' history of Greek mathematics, concluding with :
[...] Not long after these men came Euclid, who brought together the Elements, systematizing many of the theorems of Eudoxus, perfecting many of those of Theaetetus, and putting in irrefutably demonstrable form propositions that had been rather loosely established by his predecessors.
In conclusion, you are right :
he was not the inventor of this approach, but just the great systematizer/ organizer of earlier work.
answered Nov 2, 2015 at 16:58