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 / 4. 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
17629

Which premises are ultimate varies with context

17630

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

17640

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

6. Mathematics / A. Nature of Mathematics / 6. Proof in Mathematics
17627

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

6. Mathematics / B. Foundations for Mathematics / 3. Axioms for Number / d. Peano arithmetic
17634

Peano axioms not only support arithmetic, but are also fairly obvious

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

Arithmetic was probably inferred from relationships between physical objects

6. Mathematics / C. Sources of Mathematics / 6. Logicism / a. Early logicism
17635

Arithmetic can have even simpler logical premises than the Peano Axioms

11. Knowledge Aims / B. Certain Knowledge / 4. Fallibilism
17637

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
17639

Believing a whole science is more than believing each of its propositions

14. Science / C. Induction / 2. Aims of Induction
17631

Induction is inferring premises from consequences

26. Natural Theory / D. Laws of Nature / 1. Laws of Nature
17633

The law of gravity has many consequences beyond its grounding observations
