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
Discoveries in mathematics can challenge philosophy, and offer it a new foundation
                        Full Idea: Any new discovery as to mathematical method and principles is likely to upset a great deal of otherwise plausible philosophising, as well as to suggest a new philosophy which will be solid in proportion as its foundations in mathematics are securely laid.
                        From: Bertrand Russell (Regressive Method for Premises in Mathematics [1907], p.283)
                        A reaction: This is a manifesto for modern analytic philosophy. I'm not convinced, especially if a fictionalist view of maths is plausible. What Russell wants is rigour, but there are other ways of getting that. Currently I favour artificial intelligence.
2. Reason / A. Nature of Reason / 6. Coherence
If one proposition is deduced from another, they are more certain together than alone
                        Full Idea: Two obvious propositions of which one can be deduced from the other both become more certain than either in isolation; thus in a complicated deductive system, many parts of which are obvious, the total probability may become all but absolute certainty.
                        From: Bertrand Russell (Regressive Method for Premises in Mathematics [1907], p.279)
                        A reaction: Thagard picked this remark out, in support of his work on coherence.
2. Reason / B. Laws of Thought / 3. Non-Contradiction
Non-contradiction was learned from instances, and then found to be indubitable
                        Full Idea: The law of contradiction must have been originally discovered by generalising from instances, though, once discovered, it was found to be quite as indubitable as the instances.
                        From: Bertrand Russell (Regressive Method for Premises in Mathematics [1907], p.274)
5. Theory of Logic / K. Features of Logics / 1. Axiomatisation
Which premises are ultimate varies with context
                        Full Idea: Premises which are ultimate in one investigation may cease to be so in another.
                        From: Bertrand Russell (Regressive Method for Premises in Mathematics [1907], p.273)
The sources of a proof are the reasons why we believe its conclusion
                        Full Idea: In mathematics, except in the earliest parts, the propositions from which a given proposition is deduced generally give the reason why we believe the given proposition.
                        From: Bertrand Russell (Regressive Method for Premises in Mathematics [1907], p.273)
Finding the axioms may be the only route to some new results
                        Full Idea: The premises [of a science] ...are pretty certain to lead to a number of new results which could not otherwise have been known.
                        From: Bertrand Russell (Regressive Method for Premises in Mathematics [1907], p.282)
                        A reaction: I identify this as the 'fruitfulness' that results when the essence of something is discovered.
6. Mathematics / B. Foundations for Mathematics / 2. Proof in Mathematics
It seems absurd to prove 2+2=4, where the conclusion is more certain than premises
                        Full Idea: It is an apparent absurdity in proceeding ...through many rather recondite propositions of symbolic logic, to the 'proof' of such truisms as 2+2=4: for it is plain that the conclusion is more certain than the premises, and the supposed proof seems futile.
                        From: Bertrand Russell (Regressive Method for Premises in Mathematics [1907], p.272)
                        A reaction: Famously, 'Principia Mathematica' proved this fact at enormous length. I wonder if this thought led Moore to his common sense view of his own hand - the conclusion being better than the sceptical arguments?
6. Mathematics / C. Sources of Mathematics / 4. Mathematical Empiricism / a. Mathematical empiricism
Arithmetic was probably inferred from relationships between physical objects
                        Full Idea: When 2 + 2 =4 was first discovered, it was probably inferred from the case of sheep and other concrete cases.
                        From: Bertrand Russell (Regressive Method for Premises in Mathematics [1907], p.272)
11. Knowledge Aims / B. Certain Knowledge / 3. Fallibilism
The most obvious beliefs are not infallible, as other obvious beliefs may conflict
                        Full Idea: Even where there is the highest degree of obviousness, we cannot assume that we are infallible - a sufficient conflict with other obvious propositions may lead us to abandon our belief, as in the case of a hallucination afterwards recognised as such.
                        From: Bertrand Russell (Regressive Method for Premises in Mathematics [1907], p.279)
                        A reaction: This approach to fallibilism seems to arise from the paradox that undermined Frege's rather obvious looking axioms. After Peirce and Russell, fallibilism has become a secure norm of modern thought.
13. Knowledge Criteria / B. Internal Justification / 5. Coherentism / a. Coherence as justification
Believing a whole science is more than believing each of its propositions
                        Full Idea: Although intrinsic obviousness is the basis of every science, it is never, in a fairly advanced science, the whole of our reason for believing any one proposition of the science.
                        From: Bertrand Russell (Regressive Method for Premises in Mathematics [1907], p.279)
14. Science / C. Induction / 2. Aims of Induction
Induction is inferring premises from consequences
                        Full Idea: The inferring of premises from consequences is the essence of induction.
                        From: Bertrand Russell (Regressive Method for Premises in Mathematics [1907], p.274)
                        A reaction: So induction is just deduction in reverse? Induction is transcendental deduction? Do I deduce the premises from observing a lot of white swans? Hm.
26. Natural Theory / D. Laws of Nature / 1. Laws of Nature
The law of gravity has many consequences beyond its grounding observations
                        Full Idea: The law of gravitation leads to many consequences which could not be discovered merely from the apparent motions of the heavenly bodies.
                        From: Bertrand Russell (Regressive Method for Premises in Mathematics [1907], p.275)