38 ideas
16943 | Philosophy is continuous with science, and has no external vantage point [Quine] |
Full Idea: I see philosophy not as an a priori propaedeutic or groundwork for science, but as continuous with science. I see philosophy and science as in the same boat. …There is no external vantage point, no first philosophy. | |
From: Willard Quine (Natural Kinds [1969], p.126) | |
A reaction: Philosophy is generalisation. Science holds the upper hand, because it settles the subject-matter to be generalised. |
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From: report of Kurt Gödel (On Formally Undecidable Propositions [1931]) by Willard Quine - Forward to Gödel's Unpublished | |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. |
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. |
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. |
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 |
16949 | Klein summarised geometry as grouped together by transformations [Quine] |
Full Idea: Felix Klein's so-called 'Erlangerprogramm' in geometry involved characterizing the various branches of geometry by what transformations were irrelevant to each. | |
From: Willard Quine (Natural Kinds [1969], p.137) |
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. |
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 |
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. |
16939 | Mass terms just concern spread, but other terms involve both spread and individuation [Quine] |
Full Idea: 'Yellow' and 'water' are mass terms, concerned only with spread; 'apple' and 'square' are terms of divided reference, concerned with both spread and individuation. | |
From: Willard Quine (Natural Kinds [1969], p.124) | |
A reaction: Would you like some apple? Pass me that water. It is helpful to see that it is a requirement of 'individuation' that is missing from terms for stuff. |
16948 | Once we know the mechanism of a disposition, we can eliminate 'similarity' [Quine] |
Full Idea: Once we can legitimize a disposition term by defining the relevant similarity standard, we are apt to know the mechanism of the disposition, and so by-pass the similarity. | |
From: Willard Quine (Natural Kinds [1969], p.135) | |
A reaction: I love mechanisms, but can we characterise mechanisms without mentioning powers and dispositions? Quine's dream is to eliminate 'similarity'. |
16945 | We judge things to be soluble if they are the same kind as, or similar to, things that do dissolve [Quine] |
Full Idea: Intuitively, what qualifies a thing as soluble though it never gets into water is that it is of the same kind as the things that actually did or will dissolve; it is similar to them. | |
From: Willard Quine (Natural Kinds [1969], p.130) | |
A reaction: If you can judge that the similar things 'will' dissolve, you can cut to the chase and judge that this thing will dissolve. |
16944 | Science is common sense, with a sophisticated method [Quine] |
Full Idea: Sciences differ from common sense only in the degree of methodological sophistication. | |
From: Willard Quine (Natural Kinds [1969], p.129) | |
A reaction: Science is normal thinking about the world, but it is teamwork, with the bar set very high. |
16941 | Induction relies on similar effects following from each cause [Quine] |
Full Idea: Induction expresses our hopes that similar causes will have similar effects. | |
From: Willard Quine (Natural Kinds [1969], p.125) | |
A reaction: Some top philosophers are also top teachers, and Quine was one of them, in his writings. He boils it down for the layman. Once again, he is pointing to the fundamental role of the similarity relation. |
16940 | Induction is just more of the same: animal expectations [Quine] |
Full Idea: Induction is essentially only more of the same: animal expectation or habit formation. | |
From: Willard Quine (Natural Kinds [1969], p.125) | |
A reaction: My working definition of induction is 'learning from experience', but that doesn't disagree with Quine. Lipton has a richer account of different types of induction. Quine's point is that it rests on resemblance. |
16933 | Grue is a puzzle because the notions of similarity and kind are dubious in science [Quine] |
Full Idea: What makes Goodman's example a puzzle is the dubious scientific standing of a general notion of similarity, or of kind. | |
From: Willard Quine (Natural Kinds [1969], p.116) | |
A reaction: Illuminating. It might be best expressed as revealing a problem with sortal terms, as employed by Geach, or by Wiggins. Grue is a bit silly, but sortals are subject to convention and culture. 'Natural' properties seem needed. |
16934 | General terms depend on similarities among things [Quine] |
Full Idea: The usual general term, whether a common noun or a verb or an adjective, owes its generality to some resemblance among the things referred to. | |
From: Willard Quine (Natural Kinds [1969], p.116) | |
A reaction: Quine has a nice analysis of the basic role of similarity in a huge amount of supposedly strict scientific thought. |
16938 | To learn yellow by observation, must we be told to look at the colour? [Quine] |
Full Idea: According to the 'respects' view, our learning of yellow by ostension would have depended on our first having been told or somehow apprised that it was going to be a question of color. | |
From: Willard Quine (Natural Kinds [1969], p.122) | |
A reaction: Quine suggests there is just one notion of similarity, and respects can be 'abstracted' afterwards. Even the ontologically ruthless Quine admits psychological abstraction! |
8486 | Standards of similarity are innate, and the spacing of qualities such as colours can be mapped [Quine] |
Full Idea: A standard of similarity is in some sense innate. The spacing of qualities (such as red, pink and blue) can be explored and mapped in the laboratory by experiments. They are needed for all learning. | |
From: Willard Quine (Natural Kinds [1969], p.123) | |
A reaction: This reasserts Hume's original point in more scientific terms. It is one of the undeniable facts about our perceptions of qualities and properties, no matter how platonist your view of universals may be. |
16947 | Similarity is just interchangeability in the cosmic machine [Quine] |
Full Idea: Things are similar to the extent that they are interchangeable parts of the cosmic machine. | |
From: Willard Quine (Natural Kinds [1969], p.134) | |
A reaction: This is a major idea for Quine, because it is a means to gradually eliminate the fuzzy ideas of 'resemblance' or 'similarity' or 'natural kind' from science. I love it! Two tigers are same insofar as they are substitutable. |
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. |
16932 | Projectible predicates can be universalised about the kind to which they refer [Quine] |
Full Idea: 'Projectible' predicates are predicates F and G whose shared instances all do count, for whatever reason, towards confirmation of 'All F are G'. ….A projectible predicate is one that is true of all and only the things of a kind. | |
From: Willard Quine (Natural Kinds [1969], p.115-6) | |
A reaction: Both Quine and Goodman are infuriatingly brief about the introduction of this concept. 'Red' is true of all ripe tomatoes, but not 'only' of them. Hardly any predicates are true only of one kind. Is that a scholastic 'proprium'? |
7375 | Quine probably regrets natural kinds now being treated as essences [Quine, by Dennett] |
Full Idea: The concept of natural kinds was reintroduced by Quine, who may now regret the way it has become a stand-in for the dubious but covertly popular concept of essences. | |
From: report of Willard Quine (Natural Kinds [1969]) by Daniel C. Dennett - Consciousness Explained 12.2 n2 | |
A reaction: He is right that Quine would regret it, and he is right that we can't assume that there are necessary essences just because there seem to be stable natural kinds, but personally I am an essentialist, so I'm not that bothered. |
16935 | If similarity has no degrees, kinds cannot be contained within one another [Quine] |
Full Idea: If similarity has no degrees there is no containing of kinds within broader kinds. If colored things are a kind, they are similar, but red things are too narrow for a kind. If red things are a kind, colored things are not similar, and it's too broad. | |
From: Willard Quine (Natural Kinds [1969], p.118) | |
A reaction: [compressed] I'm on Quine's side with this. We glibly talk of 'kinds', but the criteria for sorting things into kinds seems to be a mess. Quine goes on to offer a better account than the (diadic, yes-no) one rejected here. |
16936 | Comparative similarity allows the kind 'colored' to contain the kind 'red' [Quine] |
Full Idea: With the triadic relation of comparative similarity, kinds can contain one another, as well as overlapping. Red and colored things can both count as kinds. Colored things all resemble one another, even though less than red things do. | |
From: Willard Quine (Natural Kinds [1969], p.119) | |
A reaction: [compressed] Quine claims that comparative similarity is necessary for kinds - that there be some 'foil' in a similarity - that A is more like C than B is. |
16937 | You can't base kinds just on resemblance, because chains of resemblance are a muddle [Quine] |
Full Idea: If kinds are based on similarity, this has the Imperfect Community problem. Red round, red wooden and round wooden things all resemble one another somehow. There may be nothing outside the set resembling them, so it meets the definition of kind. | |
From: Willard Quine (Natural Kinds [1969], p.120) | |
A reaction: [ref. to Goodman 'Structure' 2nd 163- , which attacks Carnap on this] This suggests an invocation of Wittgenstein's family resemblance, which won't be much help for natural kinds. |
16942 | It is hard to see how regularities could be explained [Quine] |
Full Idea: Why there have been regularities is an obscure question, for it is hard to see what would count as an answer. | |
From: Willard Quine (Natural Kinds [1969], p.126) | |
A reaction: This is the standard pessimism of the 20th century Humeans, but it strikes me as comparable to the pessimism about science found in Locke and Hume. Regularities are explained all the time by scientists, though the lowest level may be hopeless. |
7417 | God can't have silly perfections, but how do we decide which ones are 'silly'? [Joslin] |
Full Idea: It is clear that God cannot have all conceivable perfections, because otherwise he would have absurd perfections (like being the perfect prawn sandwich), so a line must be drawn, and how are we to decide which perfections are appropriate and essential? | |
From: Jack Joslin (talk [2006]), quoted by PG - Db (ideas) | |
A reaction: This is an excellent question for curbing the absurdities of those who want to load God with every good thing that can possibly be conceived. Is the God who is also a perfect prawn sandwich more perfect than the one who isn't? |