46 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| From: report of Kurt Gödel (On Formally Undecidable Propositions [1931]) by Willard Quine - Forward to Gödel's Unpublished | |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. |
| 3005 | 'Jocasta' needs to be distinguished from 'Oedipus's mother' because they are connected by different properties [Fodor] |
| Full Idea: If the concept 'Jocasta' needs to be distinguished from the concept 'Oedipus's mother', that's all right because the two concepts are connected with different properties. | |
| From: Jerry A. Fodor (Psychosemantics [1987], p. 84) |
| 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. |
| 7014 | A particle and a coin heads-or-tails pick out to perfectly well-defined predicates and properties [Fodor] |
| Full Idea: 'Is a particle and my coin is heads' and 'is a particle and my coin is tails' are perfectly well defined predicates and they pick out perfectly well defined (relational) properties of physical particles. | |
| From: Jerry A. Fodor (Psychosemantics [1987], Ch.2) | |
| A reaction: (Somewhat paraphrased). This is a very nice offering for the case that all predicates are properties, and hence that 'properties' is an entirely conventional category. It strikes me as self-evident that Fodor is not picking out 'natural' properties. |
| 3008 | Evolution suggests that innate knowledge of human psychology would be beneficial [Fodor] |
| Full Idea: If I had to design homo sapiens, I would have made commonsense knowledge of homo sapiens psychology innate; that way nobody would have to spend time learning it. | |
| From: Jerry A. Fodor (Psychosemantics [1987], p.132) |
| 2990 | Contrary to commonsense, most of what is in the mind seems to be unlearned [Fodor] |
| Full Idea: Contrary to commonsense, it looks as though much of what is in the mind is unlearned. | |
| From: Jerry A. Fodor (Psychosemantics [1987], p. 15) |
| 3009 | Sticklebacks have an innate idea that red things are rivals [Fodor] |
| Full Idea: God gave the male stickleback the idea that whatever is red is a rival. | |
| From: Jerry A. Fodor (Psychosemantics [1987], p.133) |
| 2994 | In CRTT thought may be represented, content must be [Fodor] |
| Full Idea: In the Representation Theory of Mind, programs (the 'laws of thought') may be explicitly represented, but data structures (the 'contents of thought') have to be. | |
| From: Jerry A. Fodor (Psychosemantics [1987], p. 25) | |
| A reaction: Presumably this is because content is where mental events actually meet up with the reality being considered. It may be an abstract procedure, but if it doesn't plug into reality then it isn't thought, but merely activity, like that of the liver. |
| 15494 | We can't use propositions to explain intentional attitudes, because they would need explaining [Fodor] |
| Full Idea: It's not clear what the point would be of an explanation of the intentionality of attitudes which presupposes objects that are intentional intrinsically. Why not just say that the attitudes are? | |
| From: Jerry A. Fodor (Psychosemantics [1987], Ch.3) |
| 7326 | Intentionality doesn't go deep enough to appear on the physicists' ultimate list of things [Fodor] |
| Full Idea: Sooner or later the physicists will complete the catalogue of ultimate and irreducible things, with the likes of spin, charm and charge. But aboutness won't be on the list; intentionality simply doesn't go that deep. | |
| From: Jerry A. Fodor (Psychosemantics [1987], 4 Intro) | |
| A reaction: I totally agree with this, which I take to be a warning to John Searle against including something called 'intrinsic intentionality' into his ontology. Intentionality 'emerges' out of certain complex brain activity. |
| 3001 | Behaviourism has no theory of mental causation [Fodor] |
| Full Idea: Behaviourists had trouble providing a robust construal of mental causation (and hence had no logical space for a psychology of mental processes). | |
| From: Jerry A. Fodor (Psychosemantics [1987], p. 67) | |
| A reaction: If they could reduce all mental events to stimulus-response, that seems to fall within the normal procedures of physical causation. There is no problem of mental causation if your ontology is entirely physical. |
| 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. |
| 2993 | Any piece of software can always be hard-wired [Fodor] |
| Full Idea: For any machine that computes a function by executing an explicit algorithm, there exists a hard-wired machine that computes the same function by not executing an explicit algorithm. | |
| From: Jerry A. Fodor (Psychosemantics [1987], p. 23) | |
| A reaction: It is certainly vital for functionalists to understand that software can be hardwired. Presumably we should understand a hardwired alogirthm as 'implicit'? |
| 3011 | Causal powers must be a crucial feature of mental states [Fodor] |
| Full Idea: Everybody is a functionalist, in that we all hold that mental states are individuated, at least in part, by reference to their causal powers. | |
| From: Jerry A. Fodor (Psychosemantics [1987], p.138) | |
| A reaction: I might individuate the Prime Minister by the carnation in his buttonhole. However, even a dualist must concede that we individuate mental faculties by their role within the mind. |
| 5498 | Mind is a set of hierarchical 'homunculi', which are made up in turn from subcomponents [Fodor, by Lycan] |
| Full Idea: Fodor sees behaviour as manifestations of psychological capacities, which result from the subject being a set of interconnected 'homunculi', which in turn have subcomponents, all of it arranged in a hierarchy. | |
| From: report of Jerry A. Fodor (Psychosemantics [1987]) by William Lycan - Introduction - Ontology p.9 | |
| A reaction: This may well miss out the most interesting parts of a mind (such as awareness, and personal identity), but it sounds basically right, especially when an evolutionary history is added to the system. Parts of my mind intrude into my trains of thought. |
| 2995 | Supervenience gives good support for mental causation [Fodor] |
| Full Idea: Mind/brain supervenience is the best idea anyone has had so far about how mental causation is possible. | |
| From: Jerry A. Fodor (Psychosemantics [1987], p. 30) | |
| A reaction: I would have thought that mind brain identity was a much better idea (see Idea 3440). Supervenience seems to prove that 'mental causation' occurs, but doesn't explain it. |
| 2991 | Hume's associationism offers no explanation at all of rational thought [Fodor] |
| Full Idea: With Associationism there proved to be no way to get a rational mental life to emerge from the sorts of causal relations among thoughts that the 'laws of association' recognised. | |
| From: Jerry A. Fodor (Psychosemantics [1987], p. 18) | |
| A reaction: This might not be true if you add the concept of evolution, which has refined the associations to generate truth (which is vital for survival). |
| 3002 | If mind is just physical, how can it follow the rules required for intelligent thought? [Fodor] |
| Full Idea: Central state identity theorists had trouble providing for the nomological possibility of rational machines (and hence no space for a non-biological, e.g. computational, theory of intelligence). | |
| From: Jerry A. Fodor (Psychosemantics [1987], p. 67) | |
| A reaction: I surmise that a more externalist account of the physical mind might do the trick, by explaining intelligence in terms of an evolved relationship between brain and environment. |
| 2992 | We may be able to explain rationality mechanically [Fodor] |
| Full Idea: We are on the verge of solving a great mystery about the mind: how is rationality mechanically possible? | |
| From: Jerry A. Fodor (Psychosemantics [1987], p. 20) | |
| A reaction: Optimistic, given that AI has struggled to implement natural languages, mainly because common sense knowledge seems too complex to encode. Can a machine determine logical forms of sentences? |
| 2988 | Folk psychology is the only explanation of behaviour we have [Fodor] |
| Full Idea: Commonsense belief/desire psychology explains vastly more of the facts about behaviour than any of the alternative theories available. It could hardly fail to; there are no alternative theories available. | |
| From: Jerry A. Fodor (Psychosemantics [1987], p.x) | |
| A reaction: The alternative view wouldn't expect a clear-cut theory, because it deals with the endless complexity of brain events. The charge is that Fodor and co oversimplify their account, in their desperation for a 'theory'. |
| 3010 | Belief and desire are structured states, which need mentalese [Fodor] |
| Full Idea: A defence of the language of thought has to be an argument that believing and desiring are typically structured states. | |
| From: Jerry A. Fodor (Psychosemantics [1987], p.136) | |
| A reaction: A structure is one thing, and a language is another. Both believings and desirings can be extremely vague, to the point where the owner is unsure what is believed or desired. They can, of course, be extremely precise. |
| 2999 | Obsession with narrow content leads to various sorts of hopeless anti-realism [Fodor] |
| Full Idea: People who ask what the narrow content of the thought that water is wet is (for example) get what they deserve: phenomenalism, verificationism, 'procedural' semantics, or scepticism, according to temperament. | |
| From: Jerry A. Fodor (Psychosemantics [1987], p. 51) | |
| A reaction: The question is whether content IS narrow. We could opt for broad content because then we wouldn't have to worry about scepticism, but I doubt whether we would then sleep well at night. |
| 3012 | Do identical thoughts have identical causal roles? [Fodor] |
| Full Idea: If thoughts have their causal roles in virtue of their contents, then two thoughts with identical contents ought to be identical in their causal roles. | |
| From: Jerry A. Fodor (Psychosemantics [1987], p.140) | |
| A reaction: A pencil would presumably have the same causal role if it wrote a love poem or hate mail. But a pencil is also good for scratching your back. 'Causal role' can be a rather vacuous idea. |
| 2998 | Grice thinks meaning is inherited from the propositional attitudes which sentences express [Fodor] |
| Full Idea: According to Gricean theories of meaning, the meaning of a sentence is inherited from the propositional attitudes that the sentence is conventionally used to express. | |
| From: Jerry A. Fodor (Psychosemantics [1987], p. 50) | |
| A reaction: Since the propositional attitudes contain propositions, this seems like a very plausible idea. If an indexical like 'I' is involved, the meaning of the sentence is not the same as its 'conventional' use. |
| 3006 | Whatever in the mind delivers falsehood is parasitic on what delivers truth [Fodor] |
| Full Idea: The mechanisms that deliver falsehoods are somehow parasitic on the ones that deliver truths. | |
| From: Jerry A. Fodor (Psychosemantics [1987], p.107) | |
| A reaction: In the case of a sentence and its negation it is not clear which one is 'parasitic', because that can usually be reversed by paraphrasing. Historically, I very much hope that truth-speaking came first. |
| 3007 | Many different verification procedures can reach 'star', but it only has one semantic value [Fodor] |
| Full Idea: Verification procedures connect terms with their denotations in too many ways. Different routes to 'star' do not determine different semantic values for 'star'. | |
| From: Jerry A. Fodor (Psychosemantics [1987], p.125) | |
| A reaction: This fairly conclusively shows that meaning is not 'the method of verification' - but that wasn't a difficult target to hit. |
| 3004 | The meaning of a sentence derives from its use in expressing an attitude [Fodor] |
| Full Idea: The meaning of a sentence derives from its use in expressing an attitude. | |
| From: Jerry A. Fodor (Psychosemantics [1987], p. 79) | |
| A reaction: Among other things. It can also arrive from a desire to remember something. A sentence can also acquire meaning compositionally (by assembling) with no use or aim. |
| 3000 | Meaning holism is a crazy doctrine [Fodor] |
| Full Idea: Meaning holism really is a crazy doctrine. | |
| From: Jerry A. Fodor (Psychosemantics [1987], p. 60) | |
| A reaction: Yes. What is not crazy is a contextualist account of utterances, and a recognition of the contextual and relational ingredient in the meanings of most of our sentences. |
| 3003 | Very different mental states can share their contents, so content doesn't seem to be constructed from functional role [Fodor] |
| Full Idea: It's an embarrassment for attempts to construct content from functional role that quite different sorts of mental states can nevertheless share their contents. | |
| From: Jerry A. Fodor (Psychosemantics [1987], p. 70) | |
| A reaction: That is, presumably, one content having two different roles. Two contents with the same role is 'multiple realisability'. Pain can tell me I'm damaged, or reveal that my damaged nerves are healing. Problem? |
| 2996 | Mental states may have the same content but different extensions [Fodor] |
| Full Idea: The identity of the content of mental states does not ensure the identity of their extensions. | |
| From: Jerry A. Fodor (Psychosemantics [1987], p. 45) | |
| A reaction: Obviously if I am thinking each day about 'my sheep', that won't change if I am unaware that one of them died this morning. …Because I didn’t have the precise number of sheep in mind. |
| 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? |