Infinity Warning: You are not logged in. Your IP address will be publicly visible if you make any edits. If you log in or create an account, your edits will be attributed to your username, along with other benefits.Anti-spam check. Do not fill this in! ==History== {{Further|Infinity (philosophy)}} Ancient cultures had various ideas about the nature of infinity. The [[Vedic period|ancient Indians]] and the [[ancient Greece|Greeks]] did not define infinity in precise formalism as does modern mathematics, and instead approached infinity as a philosophical concept. ===Early Greek=== The earliest recorded idea of infinity in Greece may be that of [[Anaximander]] (c. 610 – c. 546 BC) a [[Pre-Socratic philosophy|pre-Socratic]] Greek philosopher. He used the word ''[[apeiron]]'', which means "unbounded", "indefinite", and perhaps can be translated as "infinite".<ref name=":1" /><ref>{{harvnb|Wallace|2004|p=44}}</ref> [[Aristotle]] (350 BC) distinguished ''potential infinity'' from ''[[actual infinity]]'', which he regarded as impossible due to the various paradoxes it seemed to produce.<ref>{{cite book |author=Aristotle |url=http://classics.mit.edu/Aristotle/physics.3.iii.html |translator-last1=Hardie|translator-first1=R. P. |translator-last2=Gaye|translator-first2=R. K. |at=Book 3, Chapters 5–8|title=Physics|publisher=The Internet Classics Archive}}</ref> It has been argued that, in line with this view, the [[Hellenistic]] Greeks had a "horror of the infinite"<ref>{{cite book |author=Goodman |first=Nicolas D. |title=Constructive Mathematics |chapter=Reflections on Bishop's philosophy of mathematics |year=1981 |editor1-last=Richman |editor1-first=F. |series=Lecture Notes in Mathematics |publisher=Springer |volume=873|pages=135–145 |doi=10.1007/BFb0090732 |isbn=978-3-540-10850-4 }}</ref><ref>Maor, p. 3</ref> which would, for example, explain why [[Euclid]] (c. 300 BC) did not say that there are an infinity of primes but rather "Prime numbers are more than any assigned multitude of prime numbers."<ref>{{Cite journal |last=Sarton |first=George |date=March 1928 |title=''The Thirteen Books of Euclid's Elements''. Thomas L. Heath, Heiberg |url=https://www.journals.uchicago.edu/doi/10.1086/346308 |journal=Isis |volume=10 |issue=1 |pages=60–62 |doi=10.1086/346308 |issn=0021-1753 |via=The University of Chicago Press Journals}}</ref> It has also been maintained, that, in proving the [[infinitude of the prime numbers]], Euclid "was the first to overcome the horror of the infinite".<ref>{{Cite book |last=Hutten |first=Ernest Hirschlaff |url=https://archive.org/details/originsofscience0000hutt_n9u7 |title=The origins of science; an inquiry into the foundations of Western thought |date=1962 |publisher=London, Allen and Unwin |others=Internet Archive |isbn=978-0-04-946007-2 |pages=1–241 |language=en |access-date=2020-01-09}}</ref> There is a similar controversy concerning Euclid's [[parallel postulate]], sometimes translated: {{blockquote|If a straight line falling across two [other] straight lines makes internal angles on the same side [of itself whose sum is] less than two right angles, then the two [other] straight lines, being produced to infinity, meet on that side [of the original straight line] that the [sum of the internal angles] is less than two right angles.<ref>{{cite book|author=Euclid |orig-year=c. 300 BC|translator-last1=Fitzpatrick |translator-first1=Richard |title=Euclid's Elements of Geometry |url=http://farside.ph.utexas.edu/Books/Euclid/Elements.pdf|year=2008 |isbn=978-0-6151-7984-1 |page=6 (Book I, Postulate 5)|publisher=Lulu.com }}</ref>}} Other translators, however, prefer the translation "the two straight lines, if produced indefinitely ...",<ref>{{cite book|last1=Heath|first1=Sir Thomas Little|last2=Heiberg|first2=Johan Ludvig|author-link1=Thomas Heath (classicist)|title=The Thirteen Books of Euclid's Elements|volume=v. 1|publisher=The University Press|year=1908|url=https://books.google.com/books?id=dkk6AQAAMAAJ&q=right+angles+infinite&pg=PR8|page=212}}</ref> thus avoiding the implication that Euclid was comfortable with the notion of infinity. Finally, it has been maintained that a reflection on infinity, far from eliciting a "horror of the infinite", underlay all of early Greek philosophy and that Aristotle's "potential infinity" is an aberration from the general trend of this period.<ref>{{cite book|last=Drozdek|first=Adam|title=''In the Beginning Was the'' Apeiron'': Infinity in Greek Philosophy''|year=2008|isbn=978-3-515-09258-6|publisher=Franz Steiner Verlag|location=Stuttgart, Germany}} </ref> ===Zeno: Achilles and the tortoise=== {{Main|Zeno's paradoxes#Achilles and the tortoise}} [[Zeno of Elea]] ({{c.}} 495 – {{c.}} 430 BC) did not advance any views concerning the infinite. Nevertheless, his paradoxes,<ref name="Zeno's paradoxes">{{cite web|url=https://plato.stanford.edu/entries/paradox-zeno/ |title=Zeno's Paradoxes |date=October 15, 2010 |website=Stanford University |access-date=April 3, 2017}}</ref> especially "Achilles and the Tortoise", were important contributions in that they made clear the inadequacy of popular conceptions. The paradoxes were described by [[Bertrand Russell]] as "immeasurably subtle and profound".<ref>{{harvnb|Russell|1996|p=347}}</ref> [[Achilles]] races a tortoise, giving the latter a head start. *Step #1: Achilles runs to the tortoise's starting point while the tortoise walks forward. *Step #2: Achilles advances to where the tortoise was at the end of Step #1 while the tortoise goes yet further. *Step #3: Achilles advances to where the tortoise was at the end of Step #2 while the tortoise goes yet further. *Step #4: Achilles advances to where the tortoise was at the end of Step #3 while the tortoise goes yet further. Etc. Apparently, Achilles never overtakes the tortoise, since however many steps he completes, the tortoise remains ahead of him. Zeno was not attempting to make a point about infinity. As a member of the [[Eleatic]]s school which regarded motion as an illusion, he saw it as a mistake to suppose that Achilles could run at all. Subsequent thinkers, finding this solution unacceptable, struggled for over two millennia to find other weaknesses in the argument. Finally, in 1821, [[Augustin-Louis Cauchy]] provided both a satisfactory definition of a limit and a proof that, for {{math|0 < ''x'' < 1}},<ref>{{cite book|last=Cauchy|first=Augustin-Louis|author-link=Augustin-Louis Cauchy|access-date=October 12, 2019|title=Cours d'Analyse de l'École Royale Polytechnique|year=1821|publisher=Libraires du Roi & de la Bibliothèque du Roi|url=https://books.google.com/books?id=UrT0KsbDmDwC&pg=PA1|page=124}}</ref> <math display="block">a+ax+ax^2+ax^3+ax^4+ax^5+\cdots=\frac{a}{1-x}.</math> Suppose that Achilles is running at 10 meters per second, the tortoise is walking at 0.1 meters per second, and the latter has a 100-meter head start. The duration of the chase fits Cauchy's pattern with {{math|1=''a'' = 10 seconds}} and {{math|1=''x'' = 0.01}}. Achilles does overtake the tortoise; it takes him <math display="block">10+0.1+0.001+0.00001+\cdots=\frac {10}{1-0.01}= \frac {10}{0.99}=10.10101\ldots\text{ seconds}.</math> ===Early Indian=== The [[Indian mathematics|Jain mathematical]] text Surya Prajnapti (c. 4th–3rd century BCE) classifies all numbers into three sets: [[enumerable]], innumerable, and infinite. Each of these was further subdivided into three orders:<ref>{{cite book|author=Ian Stewart|title=Infinity: a Very Short Introduction|url=https://books.google.com/books?id=iewwDgAAQBAJ&pg=PA117|year=2017|publisher=Oxford University Press|isbn=978-0-19-875523-4|page=117|url-status=live|archive-url=https://web.archive.org/web/20170403200429/https://books.google.com/books?id=iewwDgAAQBAJ&pg=PA117|archive-date=April 3, 2017}}</ref> * Enumerable: lowest, intermediate, and highest * Innumerable: nearly innumerable, truly innumerable, and innumerably innumerable * Infinite: nearly infinite, truly infinite, infinitely infinite ===17th century=== In the 17th century, European mathematicians started using infinite numbers and infinite expressions in a systematic fashion. In 1655, [[John Wallis]] first used the notation <math>\infty</math> for such a number in his ''De sectionibus conicis'',<ref>{{Cite book|url=https://books.google.com/books?id=OQZxHpG2y3UC&q=infinity|title=A History of Mathematical Notations|last=Cajori|first=Florian|publisher=Cosimo, Inc.|year=2007|isbn=9781602066854|volume=1|pages=214|language=en}}</ref> and exploited it in area calculations by dividing the region into [[infinitesimal]] strips of width on the order of <math>\tfrac{1}{\infty}.</math><ref>{{harvnb|Cajori|1993|loc=Sec. 421, Vol. II, p. 44}}</ref> But in ''Arithmetica infinitorum'' (1656),<ref>{{Cite web |title=Arithmetica Infinitorum |url=https://archive.org/details/ArithmeticaInfinitorum/page/n5/mode/2up}}</ref> he indicates infinite series, infinite products and infinite continued fractions by writing down a few terms or factors and then appending "&c.", as in "1, 6, 12, 18, 24, &c."<ref>{{harvnb|Cajori|1993|loc=Sec. 435, Vol. II, p. 58}}</ref> In 1699, [[Isaac Newton]] wrote about equations with an infinite number of terms in his work ''[[De analysi per aequationes numero terminorum infinitas]]''.<ref>{{cite book |title=Landmark Writings in Western Mathematics 1640-1940 |first1=Ivor |last1=Grattan-Guinness |publisher=Elsevier |year=2005 |isbn=978-0-08-045744-4 |page=62 |url=https://books.google.com/books?id=UdGBy8iLpocC |url-status=live |archive-url=https://web.archive.org/web/20160603085825/https://books.google.com/books?id=UdGBy8iLpocC |archive-date=2016-06-03 }} [https://books.google.com/books?id=UdGBy8iLpocC&pg=PA62 Extract of p. 62]</ref> Summary: Please note that all contributions to Christianpedia may be edited, altered, or removed by other contributors. 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