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Full Idea
A truly infinite quantity (for example, the length of a straight line, unbounded in either direction) does not by any means need to be a variable.
Gist of Idea
A truly infinite quantity does not need to be a variable
Source
Bernard Bolzano (Paradoxes of the Infinite [1846]), quoted by Brian Clegg - Infinity: Quest to Think the Unthinkable §10
Book Ref
Clegg,Brian: 'Infinity' [Robinson 2003], p.131
A Reaction
This is an important idea, followed up by Cantor, which relegated to the sidelines the view of infinity as simply something that could increase without limit. Personally I like the old view, but there is something mathematically stable about infinity.
| 10856 | A truly infinite quantity does not need to be a variable [Bolzano] |
| 9987 | An aggregate in which order does not matter I call a 'set' [Bolzano] |
| 9618 | Bolzano wanted to reduce all of geometry to arithmetic [Bolzano, by Brown,JR] |
| 9830 | Bolzano began the elimination of intuition, by proving something which seemed obvious [Bolzano, by Dummett] |
| 17265 | Philosophical proofs in mathematics establish truths, and also show their grounds [Bolzano, by Correia/Schnieder] |
| 9185 | Bolzano wanted to avoid Kantian intuitions, and prove everything that could be proved [Bolzano, by Dummett] |
| 22276 | Bolzano saw propositions as objective entities, existing independently of us [Bolzano, by Potter] |
| 17264 | Propositions are abstract structures of concepts, ready for judgement or assertion [Bolzano, by Correia/Schnieder] |
| 12233 | The ground of a pure conceptual truth is only in other conceptual truths [Bolzano] |
| 7807 | The laws of thought are true, but they are not the axioms of logic [Bolzano, by George/Van Evra] |
| 12232 | A 'proposition' is the sense of a linguistic expression, and can be true or false [Bolzano] |