Ideas from 'Regressive Method for Premises in Mathematics' by Bertrand Russell [1907], by Theme Structure
[found in 'Essays in Analysis' by Russell,Bertrand (ed/tr Lackey,Douglas) [George Braziller 1973,0-8076-0699-5]].
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1. Philosophy / D. Nature of Philosophy / 5. Aims of Philosophy / e. Philosophy as reason
17641
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Discoveries in mathematics can challenge philosophy, and offer it a new foundation
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2. Reason / A. Nature of Reason / 6. Coherence
17638
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If one proposition is deduced from another, they are more certain together than alone
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2. Reason / B. Laws of Thought / 3. Non-Contradiction
17632
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Non-contradiction was learned from instances, and then found to be indubitable
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5. Theory of Logic / K. Features of Logics / 1. Axiomatisation
17629
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Which premises are ultimate varies with context
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17630
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The sources of a proof are the reasons why we believe its conclusion
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17640
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Finding the axioms may be the only route to some new results
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6. Mathematics / B. Foundations for Mathematics / 2. Proof in Mathematics
17627
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It seems absurd to prove 2+2=4, where the conclusion is more certain than premises
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6. Mathematics / C. Sources of Mathematics / 4. Mathematical Empiricism / a. Mathematical empiricism
17628
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Arithmetic was probably inferred from relationships between physical objects
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11. Knowledge Aims / B. Certain Knowledge / 3. Fallibilism
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The most obvious beliefs are not infallible, as other obvious beliefs may conflict
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13. Knowledge Criteria / B. Internal Justification / 5. Coherentism / a. Coherence as justification
17639
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Believing a whole science is more than believing each of its propositions
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14. Science / C. Induction / 2. Aims of Induction
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Induction is inferring premises from consequences
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26. Natural Theory / D. Laws of Nature / 1. Laws of Nature
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The law of gravity has many consequences beyond its grounding observations
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