24 ideas
| 17641 | Discoveries in mathematics can challenge philosophy, and offer it a new foundation [Russell] |
| 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. |
| 13939 | No possible evidence could decide the reality of numbers, so it is a pseudo-question [Carnap] |
| Full Idea: I cannot think of any possible evidence that would be regarded as relevant by both nominalists and realists about numbers, and would decide the controversy, or make one side more probable. Hence I regard the external questions as pseudo-questions. | |
| From: Rudolph Carnap (Empiricism, Semantics and Ontology [1950], 4) |
| 17638 | If one proposition is deduced from another, they are more certain together than alone [Russell] |
| 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. |
| 17632 | Non-contradiction was learned from instances, and then found to be indubitable [Russell] |
| 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) |
| 17629 | Which premises are ultimate varies with context [Russell] |
| 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) |
| 17630 | The sources of a proof are the reasons why we believe its conclusion [Russell] |
| 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) |
| 17640 | Finding the axioms may be the only route to some new results [Russell] |
| 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. |
| 17627 | It seems absurd to prove 2+2=4, where the conclusion is more certain than premises [Russell] |
| 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? |
| 17628 | Arithmetic was probably inferred from relationships between physical objects [Russell] |
| 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) |
| 13936 | Questions about numbers are answered by analysis, and are analytic, and hence logically true [Carnap] |
| Full Idea: For the internal question like 'is there a prime number greater than a hundred?' the answers are found by logical analysis based on the rules for the new expressions. The answers here are analytic, i.e., logically true. | |
| From: Rudolph Carnap (Empiricism, Semantics and Ontology [1950], 2) |
| 8748 | Logical positivists incorporated geometry into logicism, saying axioms are just definitions [Carnap, by Shapiro] |
| Full Idea: The logical positivists brought geometry into the fold of logicism. The axioms of, say, Euclidean geometry are simply definitions of primitive terms like 'point' and 'line'. | |
| From: report of Rudolph Carnap (Empiricism, Semantics and Ontology [1950]) by Stewart Shapiro - Thinking About Mathematics 5.3 | |
| A reaction: If the concept of 'line' is actually created by its definition, then we need to know exactly what (say) 'shortest' means. If we are merely describing a line, then our definition can be 'impredicative', using other accepted concepts. |
| 8960 | Internal questions about abstractions are trivial, and external ones deeply problematic [Carnap, by Szabó] |
| Full Idea: Carnap's verdict is that questions regarding the existence of abstracta tend to be trivial when taken as internal and deeply problematic when taken as external. | |
| From: report of Rudolph Carnap (Empiricism, Semantics and Ontology [1950]) by Zoltán Gendler Szabó - Nominalism 6 | |
| A reaction: If the internal aspect of the problem is 'trivial', this would put Carnap in league with fictionalists, who are only committed to entities while playing the current game. What is the status of the theory? Carnap wanted flowers to bloom. |
| 13933 | Existence questions are 'internal' (within a framework) or 'external' (concerning the whole framework) [Carnap] |
| Full Idea: We distinguish two kinds of existence questions: first, entities of a new kind within the framework; we call them 'internal questions'. Second, 'external questions', concerning the existence or reality of the system of entities as a whole. | |
| From: Rudolph Carnap (Empiricism, Semantics and Ontology [1950], 2) | |
| A reaction: This nicely disposes of many ontological difficulties, but at the price of labelling most external questions as meaningless, so that the internal answers have very little commitment, and the external (big) questions are now banned. Not for me. |
| 13934 | To be 'real' is to be an element of a system, so we cannot ask reality questions about the system itself [Carnap] |
| Full Idea: To be real in the scientific sense means to be an element of the system; hence this concept cannot be meaningfully applied to the system itself. | |
| From: Rudolph Carnap (Empiricism, Semantics and Ontology [1950], 2) |
| 13938 | A linguistic framework involves commitment to entities, so only commitment to the framework is in question [Carnap] |
| Full Idea: If someone accepts a framework for a kind of entities, then he must admit the entities as possible designata. Thus the question of the admissibility of entities is reduced to the question of the acceptability of the linguistic framework for the entities. | |
| From: Rudolph Carnap (Empiricism, Semantics and Ontology [1950], 4) | |
| A reaction: Despite the many differences of opinion between Quine and Carnap, this appears to be a straight endorsement by Carnap of the Quinean conception of ontological commitment. |
| 13935 | We only accept 'things' within a language with formation, testing and acceptance rules [Carnap] |
| Full Idea: To accept the thing world means nothing more than to accept a certain form of language, in other words, to accept rules for forming statements and for testing, accepting, or rejecting them. | |
| From: Rudolph Carnap (Empiricism, Semantics and Ontology [1950], 2) | |
| A reaction: If you derive your metaphysics from your language, then objects are linguistic conventions. But why do we accept conventions about objects? |
| 17637 | The most obvious beliefs are not infallible, as other obvious beliefs may conflict [Russell] |
| 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. |
| 13932 | Empiricists tend to reject abstract entities, and to feel sympathy with nominalism [Carnap] |
| Full Idea: Empiricists are in general rather suspicious with respect to any kind of abstract entities like properties, classes, relations, numbers, propositions etc. They usually feel more sympathy with nominalists than with realists (in the medieval sense). | |
| From: Rudolph Carnap (Empiricism, Semantics and Ontology [1950], 1) | |
| A reaction: The obvious reason is that you can't have sense experiences of abstract entities. I like the question 'what are they made of?' rather than the question 'how can I experience them?'. |
| 13937 | New linguistic claims about entities are not true or false, but just expedient, fruitful or successful [Carnap] |
| Full Idea: The acceptance of new linguistic forms about entities cannot be judged as being either true or false because it is not an assertion. It can only be judged as being more or less expedient, fruitful, conducive to the aim for which the language is intended. | |
| From: Rudolph Carnap (Empiricism, Semantics and Ontology [1950], 3) | |
| A reaction: The obvious problem seems to be that a complete pack of lies might be successful for a very long time, if it plugged a critical hole in a major theory. Is success judged financially? How do we judge success without mentioning truth? |
| 17639 | Believing a whole science is more than believing each of its propositions [Russell] |
| 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) |
| 13940 | All linguistic forms in science are merely judged by their efficiency as instruments [Carnap] |
| Full Idea: The acceptance or rejection of abstract (or any other) linguistic forms in any branch of science will finally be decided by their efficiency as instruments. | |
| From: Rudolph Carnap (Empiricism, Semantics and Ontology [1950], 5) |
| 17631 | Induction is inferring premises from consequences [Russell] |
| 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. |
| 468 | Musical performance can reveal a range of virtues [Damon of Ath.] |
| Full Idea: In singing and playing the lyre, a boy will be likely to reveal not only courage and moderation, but also justice. | |
| From: Damon (fragments/reports [c.460 BCE], B4), quoted by (who?) - where? |
| 17633 | The law of gravity has many consequences beyond its grounding observations [Russell] |
| 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) |