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 1973,0807606995]].
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1. Philosophy / D. Nature of Philosophy / 5. Aims of Philosophy / e. Philosophy as reason
17641

Discoveries in mathematics can challenge philosophy, and offer it a new foundation

2. Reason / A. Nature of Reason / 6. Coherence
17638

If one proposition is deduced from another, they are more certain together than alone

2. Reason / B. Laws of Thought / 3. NonContradiction
17632

Noncontradiction was learned from instances, and then found to be indubitable

5. Theory of Logic / K. Features of Logics / 1. Axiomatisation
17630

The sources of a proof are the reasons why we believe its conclusion

17629

Which premises are ultimate varies with context

17640

Finding the axioms may be the only route to some new results

6. Mathematics / B. Foundations for Mathematics / 2. Proof in Mathematics
17627

It seems absurd to prove 2+2=4, where the conclusion is more certain than premises

6. Mathematics / C. Sources of Mathematics / 4. Mathematical Empiricism / a. Mathematical empiricism
17628

Arithmetic was probably inferred from relationships between physical objects

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

13. Knowledge Criteria / B. Internal Justification / 5. Coherentism / a. Coherence as justification
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Believing a whole science is more than believing each of its propositions

14. Science / C. Induction / 2. Aims of Induction
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Induction is inferring premises from consequences

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
