# Was Euclid's method of proof axiomatic?

Euclid's method of proof has often been described in textbooks as axiomatic, but was it really so? And if not, how else can Euclid's method be characterized?

• @virmaior, I see your point. I mean - by philosophers in light of the history of philosophy. Am versed with Euclid's period but not with prior era and wondered of whether one could possibly provide me here with example of non-axiomatic proof prior to the systematization of geometry by Euclid. I am not familiar with history of the philosophy of proof but it is why I ask what I ask.
– user18096
Jun 28, 2018 at 0:06
• I don't mean this is a "your question isn't on topic here" because I do think it's interesting and probably appropriate for this site, but I think it might be even more explicitly an appropriate question to have on the History of Science and Mathematics.SE instead. Jun 28, 2018 at 1:08
• The "standard" of proof has evolved during time (and will change in the future) but the idea of demonstrative poof (an argument that "necessarily concludes" by way of necessity-preserving steps) as we can find into Euclid's Elements evolved in Ancient Greece from the inetrplay of philosophy (the Sophists, Socrates, Plato, Aristotle) and mathematics. Ancient "proofs" by way of pebbles and diagrams (see Netz) were transformed into full arguments. Jun 28, 2018 at 8:34
• @DavidThornley No, he did not, and no, they are not. The logical reasoning in the accompanying text is rudimentary and trivial, it does not establish his propositions without inferences from the diagram. This is acknowledged even by those, like Leibniz, who thought it should be. Ironically, what he selected to prove was not based on intuition either, it was handed down by the tradition. He systematized, yes, but logical form, no. Generally, formal inference is rarely used in mathematics. I am not sure what your source is (older geometry textbooks?), but it is not very good. Dec 1, 2018 at 5:01

Reading page 27 of Science Without Numbers by Hartry Field, it says Hilbert did an axiomatization of Euclidean geometry in 1905, leading one to believe Euclid's theories were not originally axiomatic. So no, they were not axiomatic.

• Euclid made some theories from his perception of physical space(also on pg 27) Mar 6, 2019 at 21:34