Editing Euclidean geometry (section)

==Logical basis==
{{Expand section|date=June 2010}}
{{See also|Hilbert's axioms|Axiomatic system|Real closed field}}

===Classical logic===
Euclid frequently used the method of [[proof by contradiction]], and therefore the traditional presentation of Euclidean geometry assumes [[classical logic]], in which every proposition is either true or false, i.e., for any proposition P, the proposition "P or not P" is automatically true.<ref>{{Cite book |last=Matthews |first=Bennie |title=Statics and Analytical Geometry |publisher=Edtech |year=2019 |isbn=9781839473333 |edition=1st |publication-date=June 21, 2019 |page=27 |language=EN}}</ref> The proof by contradiction (or [[reductio ad absurdum]] method) rests on two cardinal principles of classical logic: the [[law of contradiction]] and the [[law of the excluded middle]]. In simple terms, the law of contradiction says ''that if S is any statement, then S and a contradiction'' (that is, the denial) ''of S cannot both hold''. And the law of the excluded middle states, that ''either S or the denial of S must hold'' (that is, there is no third, or middle, possibility). This method therefore consists of assuming (by way of hypothesis) that a proposition that is to be established is false; if an absurdity follows, one concludes that the hypothesis is untenable and that the original proposition must then be true.<ref>{{Cite book |last=Eves |first=Howard Whitley |title=Foundations and fundamental concepts of mathematics |date=1997 |publisher=Dover Publications |isbn=978-0-486-69609-6 |edition=3rd |series=Dover books on mathematics |location=Mineola, NY |page=55}}</ref>

===Modern standards of rigor===
Placing Euclidean geometry on a solid axiomatic basis was a preoccupation of mathematicians for centuries.<ref name=Smith>A detailed discussion can be found in {{cite book |title=Methods of geometry |author= James T. Smith |chapter-url=https://books.google.com/books?id=mWpWplOVQ6MC&pg=RA1-PA19 |chapter=Chapter 2: Foundations |pages=19 ''ff'' |isbn=0-471-25183-6 |publisher=Wiley |year=2000}}</ref> The role of [[primitive notion]]s, or undefined concepts, was clearly put forward by [[Alessandro Padoa]] of the [[Giuseppe Peano|Peano]] delegation at the 1900 Paris conference:<ref name = Smith/><ref name="revue">{{cite book |author= |author-link=Société française de philosophie |url=https://books.google.com/books?id=4aoLAAAAIAAJ&pg=PA592 |title=Revue de métaphysique et de morale, Volume 8 |publisher=Hachette |year=1900 |page=592}}</ref> {{blockquote|text=...when we begin to formulate the theory, we can imagine that the undefined symbols are ''completely devoid of meaning'' and that the unproved propositions are simply ''conditions'' imposed upon the undefined symbols.

Then, the ''system of ideas'' that we have initially chosen is simply ''one interpretation'' of the undefined symbols; but..this interpretation can be ignored by the reader, who is free to replace it in his mind by ''another interpretation''.. that satisfies the conditions...

''Logical'' questions thus become completely independent of ''empirical'' or ''psychological'' questions...

The system of undefined symbols can then be regarded as the ''abstraction'' obtained from the ''specialized theories'' that result when...the system of undefined symbols is successively replaced by each of the interpretations...|source=''Essai d'une théorie algébrique des nombre entiers, avec une Introduction logique à une théorie déductive quelconque'' |sign=Padoa}}

That is, mathematics is context-independent knowledge within a hierarchical framework. As said by [[Bertrand Russell]]:<ref name=Newman>{{cite book |title= The world of mathematics |volume=3 |editor=James Roy Newman |author=Bertrand Russell |chapter=Mathematics and the metaphysicians |isbn=0-486-41151-6 |year=2000|chapter-url=https://books.google.com/books?id=_b2ShqRj8YMC&pg=PA1577 |page=1577 |edition=Reprint of Simon and Schuster 1956 |publisher=Courier Dover Publications }}</ref>
{{blockquote|text=If our hypothesis is about ''anything'', and not about some one or more particular things, then our deductions constitute mathematics. Thus, mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true. |source =''Mathematics and the metaphysicians'' |sign=Bertrand Russell}}

===Axiomatic formulations===
{{blockquote|text=Geometry is the science of correct reasoning on incorrect figures.|source=''How to Solve It'', p.&nbsp;208 |sign=[[George Pólya]]}}
* Euclid's axioms: In his dissertation to Trinity College, Cambridge, Bertrand Russell summarized the changing role of Euclid's geometry in the minds of philosophers up to that time.<ref name= Russell>{{cite book |title=An essay on the foundations of geometry |author=Bertrand Russell |publisher=Cambridge University Press |year=1897 |url=https://books.google.com/books?id=NecGAAAAYAAJ&pg=PA1 |chapter=Introduction}}</ref> It was a conflict between certain knowledge, independent of experiment, and empiricism, requiring experimental input. This issue became clear as it was discovered that the [[parallel postulate]] was not necessarily valid and its applicability was an empirical matter, deciding whether the applicable geometry was Euclidean or [[Non-Euclidean geometry|non-Euclidean]].
* [[Hilbert's axioms]]: Hilbert's axioms had the goal of identifying a ''simple'' and ''complete'' set of ''independent'' axioms from which the most important geometric theorems could be deduced. The outstanding objectives were to make Euclidean geometry rigorous (avoiding hidden assumptions) and to make clear the ramifications of the parallel postulate.
* [[Birkhoff's axioms]]: Birkhoff proposed four postulates for Euclidean geometry that can be confirmed experimentally with scale and protractor. This system relies heavily on the properties of the [[real numbers]].<ref name=Brikhoff>{{cite book |title=Basic Geometry |author1=George David Birkhoff |author2=Ralph Beatley |chapter-url=https://books.google.com/books?id=TB6xYdomdjQC&pg=PA38 |chapter=Chapter 2: The five fundamental principles |isbn=0-8218-2101-6 |publisher=AMS Bookstore |year=1999 |pages=38 ''ff'' |edition=3rd}}</ref><ref name=Smith2>{{cite book |title=Cited work |author=James T. Smith |pages=84 ''ff'' |chapter-url=https://books.google.com/books?id=mWpWplOVQ6MC&pg=RA1-PA84 |chapter=Chapter 3: Elementary Euclidean Geometry |date=10 January 2000 |publisher=John Wiley & Sons |isbn=9780471251835 }}</ref><ref name=Moise>{{cite book |title=Elementary geometry from an advanced standpoint |author=Edwin E. Moise |url=https://books.google.com/books?id=3UjvAAAAMAAJ&q=Birkhoff
|isbn=0-201-50867-2 |year=1990 |publisher=Addison–Wesley |edition=3rd}}</ref> The notions of ''angle'' and ''distance'' become primitive concepts.<ref name=Silvester>{{cite book |title=Geometry: ancient and modern |author=John R. Silvester |chapter-url=https://books.google.com/books?id=VtH_QG6scSUC&pg=PA5 |chapter=§1.4 Hilbert and Birkhoff |isbn=0-19-850825-5 |publisher=Oxford University Press |year=2001}}</ref>
* [[Tarski's axioms]]: [[Alfred Tarski]] (1902–1983) and his students defined ''elementary'' Euclidean geometry as the geometry that can be expressed in [[first-order logic]] and does not depend on [[set theory]] for its logical basis,<ref name=Tarski0>{{cite book |chapter=What is elementary geometry |author=Alfred Tarski |quote=We regard as elementary that part of Euclidean geometry which can be formulated and established without the help of any set-theoretical devices|chapter-url=https://books.google.com/books?id=eVVKtnKzfnUC&pg=PA16 |page=16 |isbn=978-1-4067-5355-4 |editor1=Leon Henkin |editor2=Patrick Suppes |editor3=Alfred Tarski |publisher=Brouwer Press |year=2007 |title=Studies in Logic and the Foundations of Mathematics – The Axiomatic Method with Special Reference to Geometry and Physics |edition=Proceedings of International Symposium at Berkeley 1957–8; Reprint}}</ref> in contrast to Hilbert's axioms, which involve point sets.<ref name=Simmons>{{cite book |title=Logic from Russell to Church |editor=Dov M. Gabbay |editor2=John Woods|chapter=Tarski's logic |author=Keith Simmons |page=574 |chapter-url=https://books.google.com/books?id=K5dU9bEKencC&pg=PA574 |isbn=978-0-444-51620-6 |year=2009 |publisher=Elsevier}}</ref> Tarski proved that his axiomatic formulation of elementary Euclidean geometry is consistent and complete in a certain [[Decidability (logic)|sense]]: there is an algorithm that, for every proposition, can be shown either true or false.<ref name="Tarski 1951"/> (This does not violate [[Gödel's incompleteness theorems|Gödel's theorem]], because Euclidean geometry cannot describe a sufficient amount of [[Peano arithmetic|arithmetic]] for the theorem to apply.<ref>Franzén, Torkel (2005). Gödel's Theorem: An Incomplete Guide to its Use and Abuse. AK Peters. {{ISBN|1-56881-238-8}}. Pp. 25–26.</ref>) This is equivalent to the decidability of [[real closed fields]], of which elementary Euclidean geometry is a model.