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6. Mathematics / A. Nature of Mathematics / 4. The Infinite / d. Actual infinite

[treating an infinite collection as a complete thing]

11 ideas
If there were real infinities, you could add two together, which is ridiculous [Locke]
I strongly believe in the actual infinite, which indicates the perfections of its author [Leibniz]
I don't admit infinite numbers, and consider infinitesimals to be useful fictions [Leibniz]
Actual infinities are not allowed in mathematics - only limits which may increase without bound [Gauss]
Cantor proposes that there won't be a potential infinity if there is no actual infinity [Hart,WD on Cantor]
Poincaré rejected the actual infinite, claiming definitions gave apparent infinity to finite objects [Lavine on Poincaré]
The idea of an infinite totality is an illusion [Hilbert]
Cantor and Dedekind brought completed infinities into mathematics [Maddy]
Completed infinities resulted from giving foundations to calculus [Maddy]
Infinite cuts and successors seems to suggest an actual infinity there waiting for us [Read]
The classical mathematician believes the real numbers form an actual set [George/Velleman]