Document (#25737)

Author

Heintz, B.

Title

¬Die Innenwelt der Mathematik : Zur Kultur und Praxis einer beweisenden Disziplin

Imprint

Wien : Springer

Year

2000

Pages

318 S

Isbn

3-211-82961-X

Series

Ästhetik und Naturwissenschaften : Bildende Wissenschaften - Zivilisierung der Kulturen

Abstract

Mathematisches Wissen gilt als erfahrungsunabhängig - a priori in Kants M Ausdrucksweise - und damit sicher. Es wird begründet durch Beweise und ist dadurch in aller Regel nicht kontrovers. Das Gebäude der Mathematik wird Stein um Stein kumulativ aufgeführt, unbehelligt von Revolutionen, wie sie andere Wissenschaften heimsuchen. Und ihre Gegenstände existieren außerhalb von Zeit und Raum - soweit die gängige Vorstellung. "Die moderne Mathematik zeichnet sich durch Merkmale aus, die für eine soziologische Analyse tatsächlich kaum Raum mehr lassen", konzediert Bettina Heintz gegen Ende ihres Buches. Dennoch gelingt es ihr, 275 oft hochinteressante Seiten mit einer solchen zu füllen. Auf die Frage "Was ist Mathematik?" gibt Heintz eine informative Übersicht zu den verschiedensten Positionen in der Mathematikphilosophie. Den Anfang machen die altbekannten "ismen": Platonismus, Intuitionismus und Formalismus. Eine markante Schwierigkeit ist das Truth/proof-Problem: Wie lässt es sich erklären, dass Beweisbarkeit und Wahrheit in der Mathematik zusammenfallen? Oder ist das gar nicht so? Im Anschluss an Karl Popper hat Imre Lakatos 1963 in "Beweise und Widerlegungen" die geschichtliche Bedingtheit mathematischen Wissens herausgestellt. Dieses entwickle sich in einem Wechselspiel von Beweisversuchen, Widerlegungen, Präzisierungen und erneuten Beweisversuchen und habe damit die Qualität von Erfahrungswissen: "Ein Beweis ist im Prinzip immer nur wahr auf Zeit." Gegen diesen "Quasi-Empirismus" setzt Saunders MacLane, ein führender Mathematiker unserer Tage, die Behauptung "Mathematics rests on proof and proof is etemal" und gibt damit wohl die Mehrheitsansicht der mathematischen Gemeinschaft wieder. Die Soziologie sieht die Mathematik doch deutlich anders als diese sich selbst. Bettina Heintz hat dazu in Interviews am Bonner Max-Planck-Institut für Mathematik herauszufinden versucht, wie Mathematiker arbeiten und welches Selbstverständnis sie dabei haben. So erfährt man, woran ein Mathematiker noch lange vor dem Beweis zu erkennen glaubt, ob eine Behauptung wahr ist ("Schönheit"), was es mit dem "Aufschreiben" - allgemein mit der symbolischen Dimension von Mathematik - auf sich hat und wie Mathematiker über die Möglichkeiten denken, die der Computer bietet. Interessante Fragen, zu denen bislang wenig Material vorlag

Footnote

Rez. in: Spektrum der Wissenschaft 2000, H.1, S.105-106 (K. Volkert)

Field

Mathematik
Philosophie

RSWK

Mathematik / Wissenschaftssoziologie (21)

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