31 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| From: report of Kurt Gödel (On Formally Undecidable Propositions [1931]) by Willard Quine - Forward to Gödel's Unpublished | |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. |
| 16435 | Plantinga proposes necessary existent essences as surrogates for the nonexistent things [Plantinga, by Stalnaker] |
| Full Idea: Plantinga proposes surrogates for nonexistent things - individual essences that are themselves necessary existents and that correspond one-to-one with all the 'things' that might exist. | |
| From: report of Alvin Plantinga (World and Essence [1970]) by Robert C. Stalnaker - Mere Possibilities 1 | |
| A reaction: There are an awful lot of competing concepts of essence flying around these days. This one seems to require some abstract 'third realm' (or worse) in which these essences can exist, awaiting the arrival of thinkers. Not for me. |
| 14655 | The 'identity criteria' of a name are a group of essential and established facts [Plantinga] |
| Full Idea: What we might call 'identity criteria' associated with a name such as 'Aristotle' are what the users of the name regard as essential and established facts about him. | |
| From: Alvin Plantinga (World and Essence [1970], I) | |
| A reaction: The problem here is that identifying something is superficial, whereas essences run deep. Plantinga is, in fact, talking about Lockean 'nominal essence' (and seems unaware of the fact, and never mentions the Lockean real/nominal distinction). |
| 14658 | 'Being Socrates' and 'being identical with Socrates' characterise Socrates, so they are among his properties [Plantinga] |
| Full Idea: Surely it is true of Socrates that he is Socrates and he is identical with Socrates. If these are true of him, then 'being Socrates' and 'being identical with Socrates' characterize him; they are among his properties or attributes. | |
| From: Alvin Plantinga (World and Essence [1970], II) | |
| A reaction: As far as I can see (if you insist on accepting self-identity as meaningful) the most you get here is that these are predicates that can attach to Socrates. If you identify predicates with properties you are in deep metaphysical trouble. |
| 14656 | Does Socrates have essential properties, plus a unique essence (or 'haecceity') which entails them? [Plantinga] |
| Full Idea: Does Socrates have, in addition to his essential properties, an 'essence' or 'haecceity' - a property essential to him that entails each of his essential properties and that nothing distinct from him has in the world? | |
| From: Alvin Plantinga (World and Essence [1970], II) | |
| A reaction: Plantinga says yes, and offers 'Socrateity' (borrowed from Boethius) as his candidate. This is a very odd use of the word 'essence'. I take an essence to be a complex set of fundamental properties. I am also puzzled by his use of the word 'entails'. |
| 14654 | Properties are 'trivially essential' if they are instantiated by every object in every possible world [Plantinga] |
| Full Idea: Let us call properties that enjoy the distinction of being instantiated by every object in every possible world 'trivially essential properties'. | |
| From: Alvin Plantinga (World and Essence [1970], I) | |
| A reaction: These would appear to be trivially 'necessary' rather than 'essential'. This continual need for the qualifier 'trivial' shows that they are not talking about proper essences. |
| 14653 | X is essentially P if it is P in every world, or in every X-world, or in the actual world (and not ¬P elsewhere) [Plantinga] |
| Full Idea: Socrates has P essentially if he has P in every world, or has it in every world in which he exists, or - most plausible of all - has P in the actual world and has its complement [non-P] in no world. | |
| From: Alvin Plantinga (World and Essence [1970], Intro) | |
| A reaction: These strike me as mere necessary properties, which are not the same thing at all. Essences give rise to the other properties, but Plantinga offers nothing to do the job (and especially not 'Socrateity'!). Essences must explain, say I! |
| 14660 | If a property is ever essential, can it only ever be an essential property? [Plantinga] |
| Full Idea: Is it the case that any property had essentially by anything is had essentially by everything that has it? | |
| From: Alvin Plantinga (World and Essence [1970], III) | |
| A reaction: Plantinga says it is not true, but the only example he can give is Socrates having the property of 'being Socrates or Greek'. I take it to be universally false. There are not two types of property here. Properties sometimes play an essential role. |
| 14661 | Essences are instantiated, and are what entails a thing's properties and lack of properties [Plantinga] |
| Full Idea: E is an essence if and only if (a) 'has E essentially' is instantiated in some world or other, and (b) for any world W and property P, E entails 'has P in W' or 'does not have P in W'. | |
| From: Alvin Plantinga (World and Essence [1970], IV) | |
| A reaction: 'Entail' strikes me as a very odd word when you are talking about the structure of the physical world (or are we??). Why would a unique self-identity (his candidate for essence) do the necessary entailing? |
| 14657 | Does 'being identical with Socrates' name a property? I can think of no objections to it [Plantinga] |
| Full Idea: Is there any reason to suppose that 'being identical with Socrates' names a property? Well, is there any reason to suppose that it does not? I cannot think of any, nor have I heard any that are at all impressive. | |
| From: Alvin Plantinga (World and Essence [1970], II) | |
| A reaction: Is there any reason to think that a planet somewhere is entirely under the control of white mice? Extraordinary. No wonder Plantinga believes in God and the Ontological Argument, as well as the existence of 'Socrateity' etc. |
| 14652 | 'De re' modality is as clear as 'de dicto' modality, because they are logically equivalent [Plantinga] |
| Full Idea: The idea of modality 'de re' is no more (although no less) obscure that the idea of modality 'de dicto'; for I think we can see that any statement of the former type is logically equivalent to some statement of the latter. | |
| From: Alvin Plantinga (World and Essence [1970], Intro) | |
| A reaction: If two things are logically equivalent, that doesn't ensure that they are equally clear! Personally I am on the side of de re modality. |
| 14659 | We can imagine being beetles or alligators, so it is possible we might have such bodies [Plantinga] |
| Full Idea: We easily understand Kafka's story about the man who wakes up to discover that he now has the body of a beetle; and in fact the state of affairs depicted is entirely possible. I can imagine being an alligator, so Socrates could have had an alligator body. | |
| From: Alvin Plantinga (World and Essence [1970], III) | |
| A reaction: This really is going the whole hog with accepting whatever is conceivable as being possible. I take this to be shocking nonsense, and it greatly reduces Plantinga in my esteem, despite his displays of intelligence and erudition. |
| 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. |
| 1748 | Archelaus was the first person to say that the universe is boundless [Archelaus, by Diog. Laertius] |
| Full Idea: Archelaus was the first person to say that the universe is boundless. | |
| From: report of Archelaus (fragments/reports [c.450 BCE]) by Diogenes Laertius - Lives of Eminent Philosophers 02.Ar.3 |
| 5989 | Archelaus said life began in a primeval slime [Archelaus, by Schofield] |
| Full Idea: Archelaus wrote that life on Earth began in a primeval slime. | |
| From: report of Archelaus (fragments/reports [c.450 BCE]) by Malcolm Schofield - Archelaus | |
| A reaction: This sounds like a fairly clearcut assertion of the production of life by evolution. Darwin's contribution was to propose the mechanism for achieving it. We should honour the name of Archelaus for this idea. |