Bolzanos Lehre von den me�baren Zahlen 1830?1989

Spalt, Detlef D.

doi:10.1007/bf00384332

Cited by 7 publications

(3 citation statements)

References 11 publications

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“…We cannot give here any comprehensive study of the debates but should refer at least, further to those already mentioned, the detailed studies in (Spalt 1991), (Sebestik 1992) and (Rusnock 2000). The last-cited work sums up the difficulties well:…”

Section: Are Measurable Numbers Really Real?mentioning

confidence: 91%

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Bolzano’s measurable numbers: are they real?

Russ

Trlifajová

2016

Proceedings of the Canadian Society for History and Philosophy of Mathematics/La Société Canadienne D’Histoire Et De Philosophi

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During the early 1830's Bernard Bolzano, working in Prague, wrote a manuscript giving a foundational account of numbers and their properties. In the final section of his work he described what he called 'infinite number expressions' and 'measurable numbers'. This work was evidently an attempt to provide an improved proof of the sufficiency of the criterion usually known as the 'Cauchy criterion' for the convergence of an infinite sequence. Bolzano had in fact published this criterion four years earlier than Cauchy who, in his work of 1821, made no attempt at a proof. Any such proof required the construction or definition of real numbers and this, in essence, was what Bolzano achieved in his work on measurable numbers. It therefore pre-dates the well-known constructions of Dedekind, Cantor and many others by several decades. Bolzano's manuscript was partially published in 1962 and more fully published in 1976. We give an account of measurable numbers, the properties Bolzano proved about them, and the controversial reception they have prompted since their publication.

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Section: Are Measurable Numbers Really Real?mentioning

confidence: 91%

“…This is an unpublished dissertation which we have not seen but rely on the report of it in(Spalt 1991). …”

mentioning

confidence: 99%

Bolzano’s measurable numbers: are they real?

Russ

Trlifajová

2016

Proceedings of the Canadian Society for History and Philosophy of Mathematics/La Société Canadienne D’Histoire Et De Philosophi

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“…In particular, he developed a theory of measurable numbers (Bolzano 1976, Part VII) which is often seen as an attempt to define the real numbers (see e.g. Rootselaar 1964;Spalt 1991;Russ and Trlifajová 2016).…”

Section: Comparisons With Related Workmentioning

confidence: 99%

Bolzano’s Mathematical Infinite

Bellomo¹,

Massas²

2021

The Review of Symbolic Logic

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Bernard Bolzano (1781-1848) is commonly thought to have attempted to develop a theory of size for infinite collections that follows the so-called part-whole principle, according to which the whole is always greater than any of its proper parts. In this paper, we develop a novel interpretation of Bolzano's mature theory of the infinite and show that, contrary to mainstream interpretations, it is best understood as a theory of infinite sums. Our formal results show that Bolzano's infinite sums can be equipped with the rich and original structure of a non-commutative ordered ring, and that Bolzano's views on the mathematical infinite are, after all, consistent.

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Bolzano’s Theory of meßbare Zahlen: Insights and Uncertainties Regarding the Number Continuum

Guillén¹

2021

Handbook of the History and Philosophy of Mathematical Practice

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Bolzanos Lehre von den me�baren Zahlen 1830?1989

Cited by 7 publications

References 11 publications

Bolzano’s measurable numbers: are they real?

Bolzano’s measurable numbers: are they real?

Bolzano’s Mathematical Infinite

Bolzano’s Theory of meßbare Zahlen: Insights and Uncertainties Regarding the Number Continuum

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