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Full Idea
Numbers cannot be mental objects constructed by our own minds: there exists at most a potential infinity of mental constructions, whereas the axioms of mathematics require an actual infinity of numbers.
Gist of Idea
We can only mentally construct potential infinities, but maths needs actual infinities
Source
Keith Hossack (Knowledge and the Philosophy of Number [2020], Intro 2)
Book Ref
Hossack, Keith: 'Knowledge and the Philosophy of Number' [Routledge 2021], p.3
A Reaction
Doubt this, but don't know enough to refute it. Actual infinities were a fairly late addition to maths, I think. I would think treating fictional complete infinities as real would be sufficient for the job. Like journeys which include imagined roads.
Related Idea
Idea 23626 Transfinite ordinals are needed in proof theory, and for recursive functions and computability [Hossack]
| 23621 | Numbers are properties, not sets (because numbers are magnitudes) [Hossack] |
| 23622 | We can only mentally construct potential infinities, but maths needs actual infinities [Hossack] |
| 23623 | Predicativism says only predicated sets exist [Hossack] |
| 23624 | The iterative conception has to appropriate Replacement, to justify the ordinals [Hossack] |
| 23625 | Limitation of Size justifies Replacement, but then has to appropriate Power Set [Hossack] |
| 23626 | Transfinite ordinals are needed in proof theory, and for recursive functions and computability [Hossack] |
| 23627 | 'Before' and 'after' are not two relations, but one relation with two orders [Hossack] |
| 23628 | The connective 'and' can have an order-sensitive meaning, as 'and then' [Hossack] |