Bolzano’s Infinite Quantities

Result code in IS VaVaI
<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F68407700%3A21240%2F18%3A00328600" target="_blank" >RIV/68407700:21240/18:00328600 - isvavai.cz</a>
Result on the web
<a href="https://link.springer.com/article/10.1007/s10699-018-9549-z#citeas" target="_blank" >https://link.springer.com/article/10.1007/s10699-018-9549-z#citeas</a>
DOI - Digital Object Identifier
<a href="http://dx.doi.org/10.1007/s10699-018-9549-z" target="_blank" >10.1007/s10699-018-9549-z</a>

Result language
angličtina
Original language name
Bolzano’s Infinite Quantities
Original language description
In his Foundations of a General Theory of Manifolds, Georg Cantor praised Bernard Bolzano as a clear defender of actual infinity who had the courage to work with infinite numbers. At the same time, he sharply criticized the way Bolzano dealt with them. Cantor’s concept was based on the existence of a one-to-one correspondence, while Bolzano insisted on Euclid’s Axiom of the whole being greater than a part. Cantor’s set theory has eventually prevailed, and became a formal basis of contemporary mathematics, while Bolzano’s approach is generally considered a step in the wrong direction. In the present paper, we demonstrate that a fragment of Bolzano’s theory of infinite quantities retaining the part-whole principle can be extended to a consistent mathematical structure. It can be interpreted in several possible ways. We obtain either a linearly ordered ring of finite and infinitely great quantities or a partially ordered ring containing infinitely small, finite and infinitely great quantities. These structures can be used as a basis of the infinitesimal calculus similarly as in non-standard analysis, whether in its full version employing ultrafilters due to Abraham Robinson or in the recent “cheap version” avoiding ultrafilters due to Terence Tao.
Czech name
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Czech description
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Type
J<sub>imp</sub> - Article in a specialist periodical, which is included in the Web of Science database
CEP classification
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OECD FORD branch
10101 - Pure mathematics

Project
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Continuities
I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Publication year
2018
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

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