28 ideas
| 21752 | Prior to Gödel we thought truth in mathematics consisted in provability [Gödel, by Quine] |
| Full Idea: Gödel's proof wrought an abrupt turn in the philosophy of mathematics. We had supposed that truth, in mathematics, consisted in provability. | |
| From: report of Kurt Gödel (On Formally Undecidable Propositions [1931]) by Willard Quine - Forward to Gödel's Unpublished | |
| A reaction: This explains the crisis in the early 1930s, which Tarski's theory appeared to solve. |
| 17835 | Gödel show that the incompleteness of set theory was a necessity [Gödel, by Hallett,M] |
| Full Idea: Gödel's incompleteness results of 1931 show that all axiom systems precise enough to satisfy Hilbert's conception are necessarily incomplete. | |
| From: report of Kurt Gödel (On Formally Undecidable Propositions [1931]) by Michael Hallett - Introduction to Zermelo's 1930 paper p.1215 | |
| A reaction: [Hallett italicises 'necessarily'] Hilbert axioms have to be recursive - that is, everything in the system must track back to them. |
| 17886 | The limitations of axiomatisation were revealed by the incompleteness theorems [Gödel, by Koellner] |
| Full Idea: The inherent limitations of the axiomatic method were first brought to light by the incompleteness theorems. | |
| From: report of Kurt Gödel (On Formally Undecidable Propositions [1931]) by Peter Koellner - On the Question of Absolute Undecidability 1.1 |
| 10071 | Second Incompleteness: nice theories can't prove their own consistency [Gödel, by Smith,P] |
| Full Idea: Second Incompleteness Theorem: roughly, nice theories that include enough basic arithmetic can't prove their own consistency. | |
| From: report of Kurt Gödel (On Formally Undecidable Propositions [1931]) by Peter Smith - Intro to Gödel's Theorems 1.5 | |
| A reaction: On the face of it, this sounds less surprising than the First Theorem. Philosophers have often noticed that it seems unlikely that you could use reason to prove reason, as when Descartes just relies on 'clear and distinct ideas'. |
| 19123 | If soundness can't be proved internally, 'reflection principles' can be added to assert soundness [Gödel, by Halbach/Leigh] |
| Full Idea: Gödel showed PA cannot be proved consistent from with PA. But 'reflection principles' can be added, which are axioms partially expressing the soundness of PA, by asserting what is provable. A Global Reflection Principle asserts full soundness. | |
| From: report of Kurt Gödel (On Formally Undecidable Propositions [1931]) by Halbach,V/Leigh,G.E. - Axiomatic Theories of Truth (2013 ver) 1.2 | |
| A reaction: The authors point out that this needs a truth predicate within the language, so disquotational truth won't do, and there is a motivation for an axiomatic theory of truth. |
| 10621 | Gödel's First Theorem sabotages logicism, and the Second sabotages Hilbert's Programme [Smith,P on Gödel] |
| Full Idea: Where Gödel's First Theorem sabotages logicist ambitions, the Second Theorem sabotages Hilbert's Programme. | |
| From: comment on Kurt Gödel (On Formally Undecidable Propositions [1931]) by Peter Smith - Intro to Gödel's Theorems 36 | |
| A reaction: Neo-logicism (Crispin Wright etc.) has a strategy for evading the First Theorem. |
| 17888 | The undecidable sentence can be decided at a 'higher' level in the system [Gödel] |
| Full Idea: My undecidable arithmetical sentence ...is not at all absolutely undecidable; rather, one can always pass to 'higher' systems in which the sentence in question is decidable. | |
| From: Kurt Gödel (On Formally Undecidable Propositions [1931]), quoted by Peter Koellner - On the Question of Absolute Undecidability 1.1 | |
| A reaction: [a 1931 MS] He says the reals are 'higher' than the naturals, and the axioms of set theory are higher still. The addition of a truth predicate is part of what makes the sentence become decidable. |
| 10132 | There can be no single consistent theory from which all mathematical truths can be derived [Gödel, by George/Velleman] |
| Full Idea: Gödel's far-reaching work on the nature of logic and formal systems reveals that there can be no single consistent theory from which all mathematical truths can be derived. | |
| From: report of Kurt Gödel (On Formally Undecidable Propositions [1931]) by A.George / D.J.Velleman - Philosophies of Mathematics Ch.8 |
| 3198 | Gödel showed that arithmetic is either incomplete or inconsistent [Gödel, by Rey] |
| Full Idea: Gödel's theorem states that either arithmetic is incomplete, or it is inconsistent. | |
| From: report of Kurt Gödel (On Formally Undecidable Propositions [1931]) by Georges Rey - Contemporary Philosophy of Mind 8.7 |
| 10072 | First Incompleteness: arithmetic must always be incomplete [Gödel, by Smith,P] |
| Full Idea: First Incompleteness Theorem: any properly axiomatised and consistent theory of basic arithmetic must remain incomplete, whatever our efforts to complete it by throwing further axioms into the mix. | |
| From: report of Kurt Gödel (On Formally Undecidable Propositions [1931]) by Peter Smith - Intro to Gödel's Theorems 1.2 | |
| A reaction: This is because it is always possible to formulate a well-formed sentence which is not provable within the theory. |
| 9590 | Arithmetical truth cannot be fully and formally derived from axioms and inference rules [Gödel, by Nagel/Newman] |
| Full Idea: The vast continent of arithmetical truth cannot be brought into systematic order by laying down a fixed set of axioms and rules of inference from which every true mathematical statement can be formally derived. For some this was a shocking revelation. | |
| From: report of Kurt Gödel (On Formally Undecidable Propositions [1931]) by E Nagel / JR Newman - Gödel's Proof VII.C | |
| A reaction: Good news for philosophy, I'd say. The truth cannot be worked out by mechanical procedures, so it needs the subtle and intuitive intelligence of your proper philosopher (Parmenides is the role model) to actually understand reality. |
| 11069 | Gödel's Second says that semantic consequence outruns provability [Gödel, by Hanna] |
| Full Idea: Gödel's Second Incompleteness Theorem says that true unprovable sentences are clearly semantic consequences of the axioms in the sense that they are necessarily true if the axioms are true. So semantic consequence outruns provability. | |
| From: report of Kurt Gödel (On Formally Undecidable Propositions [1931]) by Robert Hanna - Rationality and Logic 5.3 |
| 10118 | First Incompleteness: a decent consistent system is syntactically incomplete [Gödel, by George/Velleman] |
| Full Idea: First Incompleteness Theorem: If S is a sufficiently powerful formal system, then if S is consistent then S is syntactically incomplete. | |
| From: report of Kurt Gödel (On Formally Undecidable Propositions [1931]) by A.George / D.J.Velleman - Philosophies of Mathematics Ch.6 | |
| A reaction: Gödel found a single sentence, effectively saying 'I am unprovable in S', which is neither provable nor refutable in S. |
| 10122 | Second Incompleteness: a decent consistent system can't prove its own consistency [Gödel, by George/Velleman] |
| Full Idea: Second Incompleteness Theorem: If S is a sufficiently powerful formal system, then if S is consistent then S cannot prove its own consistency | |
| From: report of Kurt Gödel (On Formally Undecidable Propositions [1931]) by A.George / D.J.Velleman - Philosophies of Mathematics Ch.6 | |
| A reaction: This seems much less surprising than the First Theorem (though it derives from it). It was always kind of obvious that you couldn't use reason to prove that reason works (see, for example, the Cartesian Circle). |
| 10611 | There is a sentence which a theory can show is true iff it is unprovable [Gödel, by Smith,P] |
| Full Idea: The original Gödel construction gives us a sentence that a theory shows is true if and only if it satisfies the condition of being unprovable-in-that-theory. | |
| From: report of Kurt Gödel (On Formally Undecidable Propositions [1931]) by Peter Smith - Intro to Gödel's Theorems 20.5 |
| 10867 | 'This system can't prove this statement' makes it unprovable either way [Gödel, by Clegg] |
| Full Idea: An approximation of Gödel's Theorem imagines a statement 'This system of mathematics can't prove this statement true'. If the system proves the statement, then it can't prove it. If the statement can't prove the statement, clearly it still can't prove it. | |
| From: report of Kurt Gödel (On Formally Undecidable Propositions [1931]) by Brian Clegg - Infinity: Quest to Think the Unthinkable Ch.15 | |
| A reaction: Gödel's contribution to this simple idea seems to be a demonstration that formal arithmetic is capable of expressing such a statement. |
| 8747 | Realists are happy with impredicative definitions, which describe entities in terms of other existing entities [Gödel, by Shapiro] |
| Full Idea: Gödel defended impredicative definitions on grounds of ontological realism. From that perspective, an impredicative definition is a description of an existing entity with reference to other existing entities. | |
| From: report of Kurt Gödel (On Formally Undecidable Propositions [1931]) by Stewart Shapiro - Thinking About Mathematics 5.3 | |
| A reaction: This is why constructivists must be absolutely precise about definition, where realists only have to do their best. Compare building a car with painting a landscape. |
| 23029 | Knowledge is secured by the relations between its parts, through differences and identities [Green,TH, by Muirhead] |
| Full Idea: What gives reality and stability to our knowledge is the reality and stability of the relations established between its parts..…by the differences and identities with other things which have similarly achieved comparative fixity and substantiality. | |
| From: report of T.H. Green (Lectures on the Principles of Political Obligation [1882]) by John H. Muirhead - The Service of the State I | |
| A reaction: Although I don't sympathise with Green's idealist metaphysics, and nevertheless think that this internalist account of knowledge is correct. |
| 3192 | Basic logic can be done by syntax, with no semantics [Gödel, by Rey] |
| Full Idea: Gödel in his completeness theorem for first-order logic showed that a certain set of syntactically specifiable rules was adequate to capture all first-order valid arguments. No semantics (e.g. reference, truth, validity) was necessary. | |
| From: report of Kurt Gödel (On Formally Undecidable Propositions [1931]) by Georges Rey - Contemporary Philosophy of Mind 8.2 | |
| A reaction: This implies that a logic machine is possible, but we shouldn't raise our hopes for proper rationality. Validity can be shown for purely algebraic arguments, but rationality requires truth as well as validity, and that needs propositions and semantics. |
| 7903 | The six perfections are giving, morality, patience, vigour, meditation, and wisdom [Nagarjuna] |
| Full Idea: The six perfections are of giving, morality, patience, vigour, meditation, and wisdom. | |
| From: Nagarjuna (Mahaprajnaparamitashastra [c.120], 88) | |
| A reaction: What is 'morality', if giving is not part of it? I like patience and vigour being two of the virtues, which immediately implies an Aristotelian mean (which is always what is 'appropriate'). |
| 23046 | States only have full authority if they heed the claims of human fellowship [Green,TH] |
| Full Idea: The claim of the state is only absolutely paramount on the supposition that in its commands and prohibitions it takes account of all the claims that arise out of human fellowship. | |
| From: T.H. Green (Lectures on the Principles of Political Obligation [1882], §146), quoted by John H. Muirhead - The Service of the State III | |
| A reaction: He rejects the idea of the general will in ordinary political activity, so it is not clear how this condition could ever be met in practice. Hideous governments just pay lip service to 'human fellowship'. How could you tell whether they believe it? |
| 23051 | Equality also implies liberty, because equality must be of opportunity as well as possessions [Green,TH] |
| Full Idea: Liberty was essential, not only as a means to equality, but as part of it. …because the opportunity which was to be equalised was not merely to have and to be happy, but to do and to realise. It was 'the right of man to make the best of himself'. | |
| From: T.H. Green (Lectures on the Principles of Political Obligation [1882]), quoted by John H. Muirhead - The Service of the State IV | |
| A reaction: This nicely identifies the core idea of civilised liberalism (as opposed to the crazy self-seeking kind). I think 'give people the right to make the best of themselves' makes a good slogan, because it implies ensuring that they have the means. |
| 23028 | The highest political efforts express our deeper social spirit [Green,TH, by Muirhead] |
| Full Idea: Political effort in all its highest forms is the expression of a belief in the reality of the social spirit as the deeper element in the individual. | |
| From: report of T.H. Green (Lectures on the Principles of Political Obligation [1882]) by John H. Muirhead - The Service of the State I | |
| A reaction: Although Green is rather literally spiritual, if we express it as a central aspect of human nature, this idea strikes me as correct. Writing in 2021, I am totally bewildered by the entire absence of any 'higher' forms of political expression. |
| 23054 | Communism is wrong because it restricts the freedom of individuals to contribute to the community [Green,TH, by Muirhead] |
| Full Idea: Green condemned pure communism, not in the name of any abstract rights of the individual, but of the right of the community itself to the best that individuals can contribute through the free and spontaneous exercise of their powers of self-expression. | |
| From: report of T.H. Green (Lectures on the Principles of Political Obligation [1882]) by John H. Muirhead - The Service of the State IV | |
| A reaction: Interesting. In a very authoritarian communist state it does seem that citizens are less able to contribute to the general good. But extreme liberty seems also to undermine the general good. Hm. |
| 23047 | Original common ownership is securing private property, not denying it [Green,TH, by Muirhead] |
| Full Idea: Common ownership in early societies is not the denial of a man's private property in the products of his own labour, but the only way under the circumstances of securing it. | |
| From: report of T.H. Green (Lectures on the Principles of Political Obligation [1882], §218) by John H. Muirhead - The Service of the State III | |
| A reaction: This is announced with some confidence, but it is very speculative. I think there is some truth in Locke's thought that putting work into a creation creates natural ownership. But who owns the raw materials? Why is work valued highly? |
| 23042 | National spirit only exists in the individuals who embody it [Green,TH, by Muirhead] |
| Full Idea: A national spirit cannot exist apart from the individuals who embody it. | |
| From: report of T.H. Green (Lectures on the Principles of Political Obligation [1882]) by John H. Muirhead - The Service of the State II | |
| A reaction: We see this in football supporters. They are thrilled by the glory of a great victory, but the reality is just the thrill of the players, and the exuberance in each supporter's mind. There is no further entity called the 'glory'. Green was a liberal. |
| 23048 | The ground of property ownership is not force but the power to use it for social ends [Green,TH, by Muirhead] |
| Full Idea: It is not the power of forcible tenure but the power of utilisation for social ends that is the ground of the permanent recognition that constitutes a right to property. | |
| From: report of T.H. Green (Lectures on the Principles of Political Obligation [1882]) by John H. Muirhead - The Service of the State III | |
| A reaction: Tell that to the aristocratic owners of British grouse moors! This just seems to be wishful thinking. Does that mean that I have no right to property if my ends are not 'social'? |
| 23049 | Property is needed by all citizens, to empower them to achieve social goods [Green,TH] |
| Full Idea: The rationale of property is that every one should be secured by society in the power of getting and keeping the means of realising a will which in possibility is a will directed to social good. | |
| From: T.H. Green (Lectures on the Principles of Political Obligation [1882], §220), quoted by John H. Muirhead - The Service of the State III | |
| A reaction: An interesting argument. If you want free citizens in a liberal society to be capable of achieving social good, you must allow them the right to acquire the means of doing so. |