17265
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Philosophical proofs in mathematics establish truths, and also show their grounds [Bolzano, by Correia/Schnieder]
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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.
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From:
report of Bernard Bolzano (Theory of Science (Wissenschaftslehre, 4 vols) [1837]) by Correia,F/Schnieder,B - Grounding: an opinionated introduction 2.3
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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.
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21673
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There are simple and complex facts; the latter depend on further facts [Chrysippus, by Cicero]
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Full Idea:
Chrysippus says there are two classes of facts, simple and complex. An instance of a simple fact is 'Socrates will die at a given date', ...but 'Milo will wrestle at Olympia' is a complex statement, because there can be no wrestling without an opponent.
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From:
report of Chrysippus (fragments/reports [c.240 BCE]) by M. Tullius Cicero - On Fate ('De fato') 13.30
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A reaction:
We might say that there are atomic and complex facts, but our atomic facts tend to be much simpler, usually just saying some object has some property.
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