30 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| From: report of Kurt Gödel (On Formally Undecidable Propositions [1931]) by Willard Quine - Forward to Gödel's Unpublished | |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. |
| 14329 | Some dispositional properties (such as mental ones) may have no categorical base [Price,HH] |
| Full Idea: There is no a priori necessity for supposing that all disposition properties must have a 'categorical base'. In particular, there may be some mental dispositions which are ultimate. | |
| From: H.H. Price (Thinking and Experience [1953], Ch.XI) | |
| A reaction: I take the notion that mental dispositions could be ultimate as rather old-fashioned, but I agree with the notion that dispositions might be more fundamental that categorical (actual) properties. Personally I like 'powers'. |
| 9032 | Before we can abstract from an instance of violet, we must first recognise it [Price,HH] |
| Full Idea: Abstraction is preceded by an earlier stage, in which we learn to recognize instances; before I can conceive of the colour violet in abstracto, I must learn to recognize instances of this colour when I see them. | |
| From: H.H. Price (Thinking and Experience [1953], Ch.II) | |
| A reaction: The problem here might be one of circularity. If you are actually going to identify something as violet, you seem to need the abstract concept of 'violet' in advance. See Idea 9034 for Price's attempt to deal with the problem. |
| 9035 | If judgement of a characteristic is possible, that part of abstraction must be complete [Price,HH] |
| Full Idea: If we are to 'judge' - rightly or not - that this object has a specific characteristic, it would seem that so far as the characteristic is concerned the process of abstraction must already be completed. | |
| From: H.H. Price (Thinking and Experience [1953], Ch.III) | |
| A reaction: Personally I think Price is right, despite the vicious attack from Geach that looms. We all know the experiences of familiarity, recognition, and identification that go on when see a person or picture. 'What animal is that, in the distance?' |
| 9034 | There may be degrees of abstraction which allow recognition by signs, without full concepts [Price,HH] |
| Full Idea: If abstraction is a matter of degree, and the first faint beginnings of it are already present as soon as anything has begun to feel familiar to us, then recognition by means of signs can occur long before the process of abstraction has been completed. | |
| From: H.H. Price (Thinking and Experience [1953], Ch.III) | |
| A reaction: I like this, even though it is unscientific introspective psychology, for which no proper evidence can be adduced - because it is right. Neuroscience confirms that hardly any mental life has an all-or-nothing form. |
| 9036 | There is pre-verbal sign-based abstraction, as when ice actually looks cold [Price,HH] |
| Full Idea: We must still insist that some degree of abstraction, and even a very considerable degree of it, is present in sign-cognition, pre-verbal as it is. ...To us, who are familiar with northern winters, the ice actually looks cold. | |
| From: H.H. Price (Thinking and Experience [1953], Ch.IV) | |
| A reaction: Price may be in the weak position of doing armchair psychology, but something like his proposal strikes me as correct. I'm much happier with accounts of thought that talk of 'degrees' of an activity, than with all-or-nothing cut-and-dried pictures. |
| 9037 | Intelligent behaviour, even in animals, has something abstract about it [Price,HH] |
| Full Idea: Though it may sound odd to say so, intelligent behaviour has something abstract about it no less than intelligent cognition; and indeed at the animal level it is unrealistic to separate the two. | |
| From: H.H. Price (Thinking and Experience [1953], Ch.IV) | |
| A reaction: This elusive thought strikes me as being a key one for understanding human existence. To think is to abstract. Brains are abstraction machines. Resemblance and recognition require abstaction. |
| 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. |
| 9033 | Recognition must precede the acquisition of basic concepts, so it is the fundamental intellectual process [Price,HH] |
| Full Idea: Recognition is the first stage towards the acquisition of a primary or basic concept. It is, therefore, the most fundamental of all intellectual processes. | |
| From: H.H. Price (Thinking and Experience [1953], Ch.II) | |
| A reaction: An interesting question is whether it is an 'intellectual' process. Animals evidently recognise things, though it is a moot point whether slugs 'recognise' tasty leaves. |
| 9030 | Abstractions can be interpreted dispositionally, as the ability to recognise or imagine an item [Price,HH] |
| Full Idea: An abstract idea may have a dispositional as well as an occurrent interpretation. ..A man who possesses the concept Dog, when he is actually perceiving a dog can recognize that it is one, and can think about dogs when he is not perceiving any dog. | |
| From: H.H. Price (Thinking and Experience [1953], Ch.IX) | |
| A reaction: Ryle had just popularised the 'dispositional' account of mental events. Price is obviously right. The man may also be able to use the word 'dog' in sentences, but presumably dogs recognise dogs, and probably dream about dogs too. |
| 9029 | If ideas have to be images, then abstract ideas become a paradoxical problem [Price,HH] |
| Full Idea: There used to be a 'problem of Abstract Ideas' because it was assumed that an idea ought, somehow, to be a mental image; if some of our ideas appeared not to be images, this was a paradox and some solution must be found. | |
| From: H.H. Price (Thinking and Experience [1953], Ch.VIII) | |
| A reaction: Berkeley in particular seems to be struck by the fact that we are incapable of thinking of a general triangle, simply because there is no image related to it. Most conversations go too fast for images to form even of very visual things. |
| 9031 | The basic concepts of conceptual cognition are acquired by direct abstraction from instances [Price,HH] |
| Full Idea: Basic concepts are acquired by direct abstraction from instances; unless there were some concepts acquired in this way by direct abstraction, there would be no conceptual cognition at all. | |
| From: H.H. Price (Thinking and Experience [1953], Ch.II) | |
| A reaction: This seems to me to be correct. A key point is that not only will I acquire the concept of 'dog' in this direct way, from instances, but also the concept of 'my dog Spot' - that is I can acquire the abstract concept of an instance from an instance. |
| 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. |