Bolzano, continuum, and the Part-Whole Principle

Kateřina Trlifajová (Czech Technical University, Prague)

Abstract 

Normal 0 21 false false false NL-BE X-NONE X-NONE MicrosoftInternetExplorer4 /* Style Definitions */ table.MsoNormalTable {mso-style-name:"Table Normal"; mso-tstyle-rowband-size:0; mso-tstyle-colband-size:0; mso-style-noshow:yes; mso-style-priority:99; mso-style-parent:""; mso-padding-alt:0cm 5.4pt 0cm 5.4pt; mso-para-margin:0cm; mso-pagination:widow-orphan; font-size:10.0pt; font-family:"Times New Roman",serif;}

Bolzano's concept of the continuum, proposed in his last work Paradoxes of the Infinite, has long been considered erroneous or even inconsistent. In this talk, I will argue that Bolzano’s framework provides a plausible theory with a clear analogy in contemporary mathematics. He defines two quantities associated with continuous extension: magnitude and multitude. Magnitude corresponds to the Lebesgue measure. As for multitudes, Bolzano insisted on the Part-Whole Principle, in contrast to Cantor's notion of cardinality. Nevertheless, multitudes can also be consistently interpreted within the framework of Numerosity theory. The relationship between Bolzano's magnitude and multitude corresponds to the relationship between the Lebesgue measure and numerosity.