Editing Foundations of mathematics (section)

== Ancient Greece ==
{{Further|Greek mathematics|Greek philosophy}}

Most civilisations developed some mathematics, mainly for practical purposes, such as counting (merchants), [[surveying]] (delimitation of fields), [[Sanskrit prosody|prosody]], [[astronomy]], and [[astrology]]. It seems that [[ancient Greek philosophers]] were the first to study the nature of mathematics and its relation with the real world.

[[Zeno of Elea]] ({{circa|490|430&nbsp;BC}}) produced several paradoxes he used to support his thesis that movement does not exist. These paradoxes involve [[mathematical infinity]], a concept that was outside the mathematical foundations of that time and was not well understood before the end of the 19th century.

The [[Pythagoreanism|Pythagorean school of mathematics]] originally insisted that the only numbers are natural numbers and ratios of natural numbers. The discovery ({{circa|5th century&nbsp;BC|lk=no}}) that the ratio of the diagonal of a square to its side is not the ratio of two natural numbers was a shock to them which they only reluctantly accepted. A testimony of this is the modern terminology of [[irrational number]] for referring to a number that is not the quotient of two integers, since "irrational" means originally "not reasonable" or "not accessible with reason".{{efn|See {{slink|Rational number#Etymology}} for this unusual terminology: "ratio" is derived from "rational", which is itself derived from "irrational".}}

The fact that length ratios are not represented by rational numbers was resolved by [[Eudoxus of Cnidus]] (408–355&nbsp;BC), a student of [[Plato]], who reduced the comparison of two irrational ratios to comparisons of integer multiples of the magnitudes involved. His method anticipated that of [[Dedekind cut]]s in the modern definition of real numbers by [[Richard Dedekind]] (1831&ndash;1916);<ref>{{Cite book|url=https://archive.org/details/thirteenbookseu00heibgoog/page/n136/mode/2up|title=The thirteen books of Euclid's Elements, edited by Sir Thomas Heath|publisher=[[Dover Publications]]|year=1956|isbn=0-486-60089-0|volume=2 (Book V)|location=New York|pages=124–126|translator-last=Heiberg}}</ref> see {{slink|Eudoxus of Cnidus#Eudoxus' proportions}}.

In the ''[[Posterior Analytics]]'', [[Aristotle]] (384–322&nbsp;BC) laid down the [[logic]] for organizing a field of knowledge by means of primitive concepts, axioms, postulates, definitions, and theorems. Aristotle took a majority of his examples for this from arithmetic and from geometry, and his logic served as the foundation of mathematics for centuries. This method resembles the modern [[axiomatic method]] but with a big philosophical difference: axioms and postulates were supposed to be true, being either self-evident or resulting from [[experiment]]s, while no other truth than the correctness of the proof is involved in the axiomatic method. So, for Aristotle, a proved theorem is true, while in the axiomatic methods, the proof says only that the axioms imply the statement of the theorem.
 
Aristotle's logic reached its high point with [[Euclid]]'s [[Euclid's Elements|''Elements'']] (300&nbsp;BC), a treatise on mathematics structured with very high standards of rigor: Euclid justifies each proposition by a demonstration in the form of chains of [[syllogisms]] (though they do not always conform strictly to Aristotelian templates).
Aristotle's [[syllogistic logic]], together with its exemplification by Euclid's ''Elements'', are recognized as scientific achievements of ancient Greece, and remained as the foundations of mathematics for centuries.