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 / 4. Aims of Philosophy / e. Philosophy as reason
Discoveries in mathematics can challenge philosophy, and offer it a new foundation
2. Reason / A. Nature of Reason / 6. Coherence
If one proposition is deduced from another, they are more certain together than alone
2. Reason / B. Laws of Thought / 3. Non-Contradiction
Non-contradiction was learned from instances, and then found to be indubitable
5. Theory of Logic / K. Features of Logics / 1. Axiomatisation
Which premises are ultimate varies with context
The sources of a proof are the reasons why we believe its conclusion
Finding the axioms may be the only route to some new results
6. Mathematics / A. Nature of Mathematics / 6. Proof in Mathematics
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
Peano axioms not only support arithmetic, but are also fairly obvious
6. Mathematics / C. Sources of Mathematics / 4. Mathematical Empiricism / a. Mathematical empiricism
Arithmetic was probably inferred from relationships between physical objects
6. Mathematics / C. Sources of Mathematics / 6. Logicism / a. Early logicism
Arithmetic can have even simpler logical premises than the Peano Axioms
11. Knowledge Aims / B. Certain Knowledge / 4. Fallibilism
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
Believing a whole science is more than believing each of its propositions
14. Science / C. Induction / 2. Aims of Induction
Induction is inferring premises from consequences
26. Natural Theory / D. Laws of Nature / 1. Laws of Nature
The law of gravity has many consequences beyond its grounding observations