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Full Idea
Both Cantor's real number (Cauchy sequences of rationals) and Dedekind's cuts involved regarding infinite items (sequences or sets) as completed and subject to further manipulation, bringing the completed infinite into mathematics unambiguously.
Gist of Idea
Cantor and Dedekind brought completed infinities into mathematics
Source
Penelope Maddy (Naturalism in Mathematics [1997], I.1 n39)
Book Ref
Maddy,Penelope: 'Naturalism in Mathematics' [OUP 2000], p.15
A Reaction
So it is the arrival of the real numbers which is the culprit for lumbering us with weird completed infinites, which can then be the subject of addition, multiplication and exponentiation. Maybe this was a silly mistake?
Related Idea
Idea 18150 Actual measurement could never require the precision of the real numbers [Bostock]
| 12489 | If there were real infinities, you could add two together, which is ridiculous [Locke] |
| 19406 | I strongly believe in the actual infinite, which indicates the perfections of its author [Leibniz] |
| 13190 | I don't admit infinite numbers, and consider infinitesimals to be useful fictions [Leibniz] |
| 10855 | Actual infinities are not allowed in mathematics - only limits which may increase without bound [Gauss] |
| 13464 | Cantor proposes that there won't be a potential infinity if there is no actual infinity [Cantor, by Hart,WD] |
| 15923 | Poincaré rejected the actual infinite, claiming definitions gave apparent infinity to finite objects [Poincaré, by Lavine] |
| 12455 | The idea of an infinite totality is an illusion [Hilbert] |
| 18171 | Cantor and Dedekind brought completed infinities into mathematics [Maddy] |
| 18190 | Completed infinities resulted from giving foundations to calculus [Maddy] |
| 11025 | Infinite cuts and successors seems to suggest an actual infinity there waiting for us [Read] |
| 10125 | The classical mathematician believes the real numbers form an actual set [George/Velleman] |