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Full Idea
Mathematical proofs are philosophical in method if they do not only demonstrate that a certain mathematical truth holds but if they also disclose why it holds, that is, if they uncover its grounds.
Gist of Idea
Philosophical proofs in mathematics establish truths, and also show their grounds
Source
report of Bernard Bolzano (Theory of Science (Wissenschaftslehre, 4 vols) [1837]) by Correia,F/Schnieder,B - Grounding: an opinionated introduction 2.3
Book Ref
'Metaphysical Grounding', ed/tr. Correia,F/Schnieder,B [CUP 2012], p.6
A Reaction
I aim to defend the role of explanation in mathematics, but this says that this is only if the proofs are 'philosophical', which may be of no interest to mathematicians. Oh well, that's their loss.
| 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] |