Continuity

Jerzy Mioduszewski

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ISBN/ISSN:

978-83-8012-950-4

DOI:

Wydawnictwo:

Uniwersytet Śląski

Rok wydania:

2016

Liczba stron:

202

XML:ISBN/ISSN:

ONIX MARC21

Bibliografia:

Mioduszewski, Jerzy. Continuity. Red. . Katowice: Uniwersytet Śląski, 2016, 202 s. ISBN 978-83-8012-950-4

Bibliografia refWorks:

RT Book, Whole

SR Electronic(1)

A1 Mioduszewski, J.

T1 Continuity

PB Uniwersytet Śląski

PP Katowice

YR 2016

SN 978-83-8012-950-4

Bibliografia BibTex:

@Book{ 165201, author = "Mioduszewski, Jerzy", editor = "", title = "Continuity", publisher = "Uniwersytet Śląski", year = "2016", address = "Katowice", isbn = "978-83-8012-950-4" }

Bibliografia endNote:

TY - BOOK

AU - Mioduszewski, Jerzy

ED -

TI - Continuity

PB - Uniwersytet Śląski

CY - Katowice

PY - 2016

SN - 978-83-8012-950-4

ER -

Netografia (standard APA):

Mioduszewski, Jerzy. Continuity [baza danych online] Katowice: Uniwersytet Śląski, 2016 [dostęp: 07 07 2024]. Dostęp w Ibuk Libra: https://libra.ibuk.pl/reader/continuity-jerzy-mioduszewski-165201.

matematyka, przeszłość, szkice

The book was written in the eighties of the last century. Being encouraged by the editorial board of monthly Delta in the person of Professor Marek Kordos, the author’s first aim was a collection of essays about Peano maps, lakes of Wada, an...

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ISBN/ISSN:

978-83-8012-950-4

DOI:

Wydawnictwo:

Uniwersytet Śląski

Rok wydania:

2016

Liczba stron:

202

XML:ISBN/ISSN:

ONIX MARC21

Table of contentsFrom the author /7
Introduction /9
Chapter IThe flying arrow • Aristotle’s view of the aporia of Zeno • Its influence on the evolution of geometry and on the science of motion • Democritus’s version of this aporia • On mathematical atomism /15
Chapter IIAporia of the wanderer • The Archimedean postulate • The Eudoxian exhaustion lemma • Non-Archimedean continua • Another Zeno’s difficulty: Stadium /27
Chapter IIINumber • On idealism in mathematics • Its two varieties: Pythagoreism and Platonism • Discovery of incommensurable segments • The Euclidean algorithm • On some possible meanings of the proportion of segments /35
Chapter IVOn geometric magnitudes • Comparison of polygons from the point of view of area • Comparison through complementation • Comparison through finite decomposition • The role of Archimedean postulate • On quadratures /51
Chapter VThe Eudoxian theory of proportions • The role in it of the Archimedean postulate • The theorem on interchanging terms in a proportion • On Tales’s theorem • Comparison with Dedekind theory • Inequality of proportions • On the area of a circle • On Greek geometric algebra • The Elements as an attempt to geometrize arithmetic /63
Chapter VIThe Arab Middle Ages • Euclid’s parallel postulate: mistakes and progress • Singular features of the philosophy of nature of the Arab East /81
Chapter VIIThe European Middle Ages • Disputes about the structure of the continuum • Oresme and the Calculators on the intensity of changes • The 1 : 3 : 5 : 7 : … sequence • The theory of impetus • Quies media • The ballistics of Albert of Saxony and of Tartaglia • Galileo • Supplements /95
Chapter VIIIThe method of indivisibles • Three ways of computing the area of a circle • Kepler: the principle of fields and a barrel • The Cavalieri’s principle • The Roberval cycloid • Need we explain it by undivisibles? • Towards magic thinking • Descartes /123
Chapter IXCalculus • Derivative of xn • Barrow’s observation as motivation for this computation • The role of the Calculators • The impetum theory as motivation • Philosophiae Naturalis Principia Mathematica and the theory of fluxions • Leibniz • Voltaire on Newton • Hypotheses non fingo? /141
Chapter XEuler • A century of computations and undisturbed progress • Discussion on the term of arbitrary function • The number e • Infinitesimals and infinities • Lagrange’s objection • Barriers of growth /167
Chapter XIA debt redeemed • The beginning of the new analysis: Cauchy and Bolzano • Weierstrass • The arithmetization of analysis • Cantor • The role of Dedekind • Set theory did not come into being accidentally • Nothing is ever completely settled /177
Epilogue /189
Author’s reminiscences /195

Aristotleisawarethathisargumentsandconclusionsmayprovokethe

chargeofputtingrestrictionsonmathematics.Heanticipatesthechargeand

writesthat“ourreasoningwhichrejectstherealizationofinfinitydoesnot

deprivemathematiciansoftheirtheories;afterall,theyhavenoneedofsuch

aninfinityandtheydonotuseit;allmathematiciansneedisthatthesegment

theyconsider,whilebounded,beaslargeasnecessary,”7and,ofcourse,as

smallasnecessary,whichheaddsinmanyotherplaces.

Theancientsdistinguishedtwokindsofinfinity,potentialandactual.

Apotentialinfinityisthepossibilityofconsideringarbitrarilylargefinitesets.

Anactualinfinityisgreaterthananyfinite.

We,today,canunderstandthisdifference,butweforgetaboutitorignore

itmostofthetime.Euclidsaythat“foranygivennumberofprimenumbers

thereisalargerone.”Wesaythatthesetofprimesisinfinitewithoutpaying

muchattentiontothefactthatinthiswayweintroducenewentity,theactu-

allyinfinitesetofprimes.Ifitweren3tforthefactthatwedonotfearinfinity

becauseofourlackofsensivity,alackacquiredovercenturies,wemightenjoy

thislackoffearoftheinfinite.Inadditiontohavinggottenridofthefearofthe

infinitewehavealsogottenridofthefearoftreatingspaceasasetofpoints

andtimeasasetofmoments.Hencetheaporiaoftheflyingarrowdoesnot

reachourimaginationswiththeacutenesswishwhichitreachedtheimagina-

tionsoftheancients.Thislackofsensitivityisoneoftheclearlyperceivable

characteristicsofthemathematicsofourtime.

Ithinkthat1likeZenobeforehim1Aristotlewantedtoprojectthelogi-

calcontradictionwhichwouldpresumablyappearifweadmittedinourreason-

ingtheactualdivisionofthecontinuumintopoints.Wenowknowthat,from

alogicalpointofview,thisargumentdidnotsuffice;weknowthisbecause,

followingDedekind,wecanbuildcontinuamadeupofpointsandcanevengive

themthestatusofnumbers.ButwethinkthatforAristotleandhiscontempo-

rariesthedifficultiespresentedintheirreasoningswereenoughofadeterrent

tomakethemabandondoubtfulnotionsandtoadoptotherconventions.

Aconsequenceofthisstepwastheexclusionofthescienceofmotion

fromrigorousmathematicalarguments.Greekmathematicswasbydefinition

static.AristotlewroteinMetaphysics:“Itisasciencedealingwithmotionless

existences.”8EuclidleftussuchamathematicsinhisElements.

Euclidgeometrydealswithstraightlinesandcircles.Butapointdoesnot

moveonacircle.Nordoesitmoveonastraightline.True,thereissuperposition

ofsegmentsbutthisisaone-stepactivityratherthanapointbypointactivity.

Asimilarremarkappliestoarotationthroughadefiniteangle.

7Arystoteles,Fizyka,p.91.TranslatedbyAbeShenitzer.

8Arystoteles,Metafizyka[Metaphysics].Warszawa1985,p.284.TranslatedbyAbeShen-

itzer.

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