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I read somewhere that Euclid's element is a little outdated (sounds normal since it's super old) then i read on wikipedia:

"..Well-known modern axiomatizations of Euclidean geometry are those of Alfred Tarski, George Birkhoff and David Hilbert".. so since there are 3 (and I guess more) "new geometries" I think they are somehow better then classic euclidean geometry (with this I mean the geometry explained by Euclid) so I was wondering what the problem is with the old geometry. Can you guys explain me in easy words please since I don't know more than high school level geometry?

Thank you!

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    $\begingroup$ The History of Science and Math StackExchange might be a better place for this, but ... Euclid's sparse 5-postulate axiomatization leaves holes (such as in the areas of continuity and betweenness) that, eg, Hilbert take 20 statements to fill. Moreover, Euclid gives hopelessly vague definitions of fundamental terms ("a line is breadthless length"(???)) that modern treatments describe in terms of properties ("whatever 'lines' are, there's exactly one of them through two distinct 'points', whatever those are"). $\endgroup$ – Blue Feb 15 at 23:36
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Hilbert and others showed that Euclid's axioms were incomplete and that further axioms were needed to prove the standard results rigorously.

What I don't know is whether or not the increased rigor has allowed any new results to be deduced.

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  • $\begingroup$ "What I don't know is whether or not the increased rigor has allowed any new results to be deduced." I'd argue that it has, or at least that doubt in Euclid's axioms has. Part of the 19th century reluctance to accept non-Euclidean (hyperbolic and elliptic) geometry was that they were rooted in the mindset that Euclidean geometry was the default. While rigour may not necessarily falsify theorems that we have intuitionistically derived from precarious axioms (e.g., Pythagoras' theorem is not per se any better now than it was millennia ago), rigour can show us the validity of alternatives. $\endgroup$ – Jam Feb 16 at 0:14
  • $\begingroup$ Have you studied geometry in the context of automated reasoning, such as Tarskian geometry in that context? Though it might not satisfy the standard for new results, perhaps some of the formulas used to derive the theorems talked about here: arxiv.org/pdf/1606.07095v1.pdf are new results? $\endgroup$ – Doug Spoonwood Feb 16 at 4:11
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    $\begingroup$ The main purpose of rigorous mathematical treatment is not necessarily to prove new results in a classical subject, but to provide the classical theory a solid foundation. Here is one theorem that you, as a computer programmer, might like. It is due to Tarsky: Euclidean geometry is decidable. More precisely, given any statement S in the setting of Euclidean geometry (more precisely, Tarsky's axioms), there is an algorithm (although a very slow one) to determine if S is true or false. Such a theorem would have been impossible without a complete set of axioms. $\endgroup$ – Moishe Kohan Feb 16 at 14:53
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One of the most important aspects of Euclid's elements is that it is an axiomatic logical system, i.e., built by proofs from the ground up. In fact it was the first one. Nothing is true unless assumed or proven via inference (e.g., by constructing and relating quantities). In fact much of Euclid's Elements (c. 3rd century BCE) was nothing new even to the Greeks alone. The Pythagorean theorem (c. 5th century BCE but discovered earlier elsewhere) and Thales' theorem (c. 6th century BCE) predate the Elements by centuries. I'd even argue that a lot of Euclidean geometry can be intuitively seen to be true without rigorous proofs. However, none of these theorems were proven meticulously from a foundation of axioms until Euclid.

So, it is not to say that we are at a total loss if we don't approach logic from the ground up. I'm sure many professional mathematicians are ignorant of minor foundational details that their work is contingent on. But, the boon of axiomatic approaches is that it can show what faulty assumptions we've made (as intuition can be wrong) and whether we're limiting ourselves by overlooking possibilities that could be valid (e.g., alternative non-Euclidean geometries). This is one reason why it is important for the axioms of Euclidean geometry to be well formulated. Hilbert's, Tarski's, etc. formulations of geometry rectify some of the missing pieces that Euclidean geometry was missing; namely the uncertainty of whether the parallel postulate could be proven from other axioms.

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    $\begingroup$ Euclid explicitly assumed the parallel postulate. $\endgroup$ – Eric M. Schmidt Feb 16 at 7:41
  • $\begingroup$ @EricM.Schmidt I've attempted to clarify what I meant. $\endgroup$ – Jam Feb 16 at 14:38

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