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Single Idea 9344

[filed under theme 12. Knowledge Sources / A. A Priori Knowledge / 4. A Priori as Necessities ]

Full Idea

At the end of the analytical method in mathematics there are simple ideas of which no definition can be given. Moreover there are axioms and postulates, in short, primitive principles, which cannot be demonstrated and do not need demonstration.

Gist of Idea

Mathematical analysis ends in primitive principles, which cannot be and need not be demonstrated

Source

Gottfried Leibniz (Monadology [1716], §35)

Book Ref

Marcus Aurelius: 'The Meditations', ed/tr. Grube,G.M.A. [Hackett 1983], p.153

A Reaction

My view is that we do not know such principles when we apprehend them in isolation. I would call them 'intuitions'. They only ascend to the status of knowledge when the mathematics is extended and derived from them, and found to work.

The 19 ideas with the same theme [a priori knowledge is an insight into necessary truths]:

2279 A triangle has a separate non-invented nature, shown by my ability to prove facts about it [Descartes]
2602 What experience could prove 'If a=c and b=c then a=b'? [Descartes]
5012 'Nothing comes from nothing' is an eternal truth found within the mind [Descartes]
9344 Mathematical analysis ends in primitive principles, which cannot be and need not be demonstrated [Leibniz]
9155 An a priori proof is independent of experience [Leibniz]
5404 Two plus two objects make four objects even if experience is impossible, so Kant is wrong [Russell on Kant]
9345 Propositions involving necessity are a priori, and pure a priori if they only derive from other necessities [Kant]
16893 The apriori is independent of its sources, and marked by necessity and generality [Kant, by Burge]
9347 A priori knowledge is indispensable for the possibility and certainty of experience [Kant]
9352 An a priori truth is one derived from general laws which do not require proof [Frege]
16889 A truth is a priori if it can be proved entirely from general unproven laws [Frege]
16894 An apriori truth is grounded in generality, which is universal quantification [Frege, by Burge]
5397 The rationalists were right, because we know logical principles without experience [Russell]
5198 We could verify 'a thing can't be in two places at once' by destroying one of the things [Ierubino on Ayer]
9354 Why should necessities only be knowable a priori? That Hesperus is Phosporus is known empirically [Devitt]
19565 How could the mind have a link to the necessary character of reality? [Devitt]
20472 Analysis of the a priori by necessity or analyticity addresses the proposition, not the justification [Casullo]
14712 A sentence is a priori if no possible way the world might actually be could make it false [Chalmers]
9369 'Snow is white or it isn't' is just true, not made true by stipulation [Boghossian]