13416
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Mathematics must be based on axioms, which are true because they are axioms, not vice versa [Tait, by Parsons,C]
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Full Idea:
The axiomatic conception of mathematics is the only viable one. ...But they are true because they are axioms, in contrast to the view advanced by Frege (to Hilbert) that to be a candidate for axiomhood a statement must be true.
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From:
report of William W. Tait (Intro to 'Provenance of Pure Reason' [2005], p.4) by Charles Parsons - Review of Tait 'Provenance of Pure Reason' §2
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A reaction:
This looks like the classic twentieth century shift in the attitude to axioms. The Greek idea is that they must be self-evident truths, but the Tait-style view is that they are just the first steps in establishing a logical structure. I prefer the Greeks.
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21756
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All revolutions result from spirit changing its categories, to achieve a deeper understanding [Hegel]
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Full Idea:
All revolutions ...originate solely from the fact that spirit, in order to understand and comprehend itself with a view to possessing itself, has changed its categories, comprehending itself more truly, more deeply, more intimately in unity with itself.
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From:
Georg W.F.Hegel (Philosophy of Nature (Encylopedia II) [1817], §246), quoted by Stephen Houlgate - An Introduction to Hegel 01
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A reaction:
Some Hegelian waffle here, but it focuses on what seems important, which is how societal thinking has shifted, so that what was previously tolerated now triggers a revolution.
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