4

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?

  • 1
    To be regarded by *whom* as a mathematical proof? By philosophers? By contemporary mathematicians? – virmaior Jun 27 '18 at 23:37
  • 2
    @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. –  Jun 28 '18 at 00:06
  • 3
    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](https://hsm.stackexchange.com/) instead. – Not_Here Jun 28 '18 at 01:08
  • 1
    I don't know if this will answer your question because I have not read it all, but it looks interesting. PDF Greek Mathematics before Euclid, Univ. Cal. Riverside: http://math.ucr.edu/~res/math153/history02.pdf – Gordon Jun 28 '18 at 01:18
  • 3
    Several books : A.Szabò, [The Beginnings of Greek Mathematics](https://books.google.it/books?id=VCj2CAAAQBAJ&printsec=frontcover), W.R.Knorr, [The Evolution of the Euclidean Elements](https://books.google.it/books?id=_1H6BwAAQBAJ&pg=PP4), R.Netz, [The Shaping of Deduction in Greek Mathematics](https://books.google.it/books?id=VwggGX0ORLkC&printsec=frontcover), K.Chemla, [The History of Mathematical Proof in Ancient Traditions](https://books.google.it/books?id=1ML3CwAAQBAJ&printsec=frontcover). – Mauro ALLEGRANZA Jun 28 '18 at 06:01
  • 2
    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](https://en.wikipedia.org/wiki/Greek_mathematics#Origins_of_Greek_mathematics) from the inetrplay of philosophy (the Sophists, Socrates, Plato, Aristotle) and mathematics. Ancient "proofs" by way of [pebbles](http://www.dm.uniba.it/~psiche/bas2/node3.html) and diagrams (see Netz) were transformed into full arguments. – Mauro ALLEGRANZA Jun 28 '18 at 08:34
  • 2
    That Euclid's method is axiomatic is a modern misconception. Recent in-depth research on his diagrammatic approach, which became the standard in antiquity, is referenced under [What caused or contributed to Euclid's Elements and Synthetic Geometry falling into disfavor?](https://hsm.stackexchange.com/a/7104/55) on hsm SE. Euclid's contribution was to systematize and codify arguments that predated him, unfortunately we have almost no independent sources for them. – Conifold Jun 28 '18 at 18:45
  • 1
    @Conifold Euclid attempted to prove everything from his axioms, postulates, and definitions. The diagrams used are backed by logical reasoning from those. A proposition is not considered true because the drawing looks good, but because it follows from the premises through logical deduction. Euclid certainly selected what to prove based on geometrical intuition, but that's true of math through the present day. It's very reasonable to intuitively come up with something and attempt to prove it. Euclid's work to systematize and codify arguments put them in logical form. – David Thornley Dec 01 '18 at 00:07
  • 3
    @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. – Conifold Dec 01 '18 at 05:01

1 Answers1

1

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.

Math Bob
  • 364
  • 2
  • 13