36 ideas
| 1642 | We must fight fiercely for knowledge, understanding and intelligence [Plato] |
| Full Idea: We need to use every argument we can to fight against anyone who does away with knowledge, understanding, and intelligence, but at the same time asserts anything at all about anything. | |
| From: Plato (The Sophist [c.359 BCE], 249c) | |
| A reaction: Thus showing that reason is only central if you want to put a high value on it? |
| 1645 | The desire to split everything into its parts is unpleasant and unphilosophical [Plato] |
| Full Idea: To try to set apart everything from everything is not only especially jangling, but it is the mark of someone altogether unmusical and unphilosophic. | |
| From: Plato (The Sophist [c.359 BCE], 259e) |
| 287 | Good analysis involves dividing things into appropriate forms without confusion [Plato] |
| Full Idea: It takes expertise in dialectic to divide things by kinds and not to think that the same form is a different one or that a different form is the same. | |
| From: Plato (The Sophist [c.359 BCE], 253d) |
| 1644 | Dialectic should only be taught to those who already philosophise well [Plato] |
| Full Idea: The dialectical capacity - you won't give it to anyone else, I suspect, except to whoever philosophises purely and justly. | |
| From: Plato (The Sophist [c.359 BCE], 253e) |
| 20478 | In discussion a person's opinions are shown to be in conflict, leading to calm self-criticism [Plato] |
| Full Idea: They collect someone's opinions together during the discussion, put them side by side, and show that they conflict with each other at the same time on the same subjects.... The person sees this, gets angry at themselves, and calmer towards others. | |
| From: Plato (The Sophist [c.359 BCE], 230b) | |
| A reaction: He goes on to say that the process is like a doctor purging a patient of internal harms. If anyone talks for long enough (even a good philosopher), their opinions will probably be seen to be in conflict. But which opinions do you abandon? |
| 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| From: report of Kurt Gödel (On Formally Undecidable Propositions [1931]) by Willard Quine - Forward to Gödel's Unpublished | |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. |
| 11278 | What does 'that which is not' refer to? [Plato] |
| Full Idea: What should the name 'that which is not' be applied to? | |
| From: Plato (The Sophist [c.359 BCE], 237c) | |
| A reaction: This leads into a discussion of the problem, in The Sophist. It became a large issue when modern logic was being developed by Frege and Russell. |
| 1643 | If statements about non-existence are logically puzzling, so are statements about existence [Plato] |
| Full Idea: When the question was put to us as to the name of 'that which is not', to whatever one must apply it, we got stuck in every kind of perplexity. Are we now in any less perplexity about 'that which is'? | |
| From: Plato (The Sophist [c.359 BCE], 250d) | |
| A reaction: Nice. This precapitulates the whole story of modern philosophy of language. What started as a nagging doubt about reference to non-existents ends as bewilderment about everything we say. |
| 7022 | To be is to have a capacity, to act on other things, or to receive actions [Plato] |
| Full Idea: A thing really is if it has any capacity, either by nature to do something to something else or to have even the smallest thing done to it by the most trivial thing, even if it only happens once. I'll define those which are as nothing other than capacity. | |
| From: Plato (The Sophist [c.359 BCE], 247e) | |
| A reaction: If philosophy is footnotes to Plato, this should be the foundational remark in all discussions of existence (though Parmenides might claim priority). It seems to say 'to be is to have a causal role (active or passive)'. It also seems essentialist. |
| 1641 | Some alarming thinkers think that only things which you can touch exist [Plato] |
| Full Idea: One group drags everything down to earth, insisting that only what offers tangible contact is, since they define being as the same as body, despising anyone who says that something without a body is. These are frightening men. | |
| From: Plato (The Sophist [c.359 BCE], 246b) |
| 10784 | Whenever there's speech it has to be about something [Plato] |
| Full Idea: Whenever there's speech it has to be about something. It's impossible for it not to be about something. | |
| From: Plato (The Sophist [c.359 BCE], 262e) | |
| A reaction: [Quoted by Marcus about ontological commitment] The interesting test case would be speech about the existence of circular squares. |
| 16122 | Good thinkers spot forms spread through things, or included within some larger form [Plato] |
| Full Idea: It takes dialectic to divide things by kinds...such a person can discriminate a single form spread through a lot of separate things…and forms included in a single outside form…or a form connected as a unit through many wholes. | |
| From: Plato (The Sophist [c.359 BCE], 253d) | |
| A reaction: [compressed] This is very helpful in indicating the complex structure of the Forms that Plato envisages. If you talk of the meanings of words (other than names), though, it comes to the same thing. Wise people fully understand their language. |
| 10422 | The not-beautiful is part of the beautiful, though opposed to it, and is just as real [Plato] |
| Full Idea: So 'the not beautiful' turns out to be ..both marked off within one kind of those that are, and also set over against one of those that are, ..and the beautiful is no more a being than the not beautiful. | |
| From: Plato (The Sophist [c.359 BCE], 257d) | |
| A reaction: [dialogue eliminated] This is a highly significant passage, for two reasons. It suggests that the Form of the beautiful can have parts, and also that the negations of Forms are Forms themselves (both of which come as a surprise). |
| 15855 | If we see everything as separate, we can then give no account of it [Plato] |
| Full Idea: To dissociate each thing from everything else is to destroy totally everything there is to say. The weaving together of forms is what makes speech [logos] possible for us. | |
| From: Plato (The Sophist [c.359 BCE], 259e) | |
| A reaction: This I take to be the lynchpin of metaphysics. We are forced to see the world in a way which enables us to give some sort of account of it. Our metaphysics is 'inference to the best logos'. |
| 1637 | A soul without understanding is ugly [Plato] |
| Full Idea: The soul that lacks understanding must be set down as ugly. | |
| From: Plato (The Sophist [c.359 BCE], 228d) | |
| A reaction: The teleological view of things understands their nature in things of their perfection. and the essence of beauty is perfection. It is the mind's nature to know. Failing to know is as ugly as allowing your crops to die. |
| 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. |
| 1636 | Wickedness is an illness of the soul [Plato] |
| Full Idea: Wickedness is a sedition and illness of the soul. | |
| From: Plato (The Sophist [c.359 BCE], 228b) |
| 541 | Virtue comes more from habit than character [Critias] |
| Full Idea: More men are good through habit than through character. | |
| From: Critias (fragments/reports [c.440 BCE], B09), quoted by John Stobaeus - Anthology 3.29.41 |
| 1638 | Didactic education is hard work and achieves little [Plato] |
| Full Idea: With a lot of effort the admonitory species of education accomplishes little. | |
| From: Plato (The Sophist [c.359 BCE], 230a) |
| 542 | Fear of the gods was invented to discourage secret sin [Critias] |
| Full Idea: When the laws forbade men to commit open crimes of violence, and they began to do them in secret, a wise and clever man invented fear of the gods for mortals, to frighten the wicked, even if they sin in secret. | |
| From: Critias (fragments/reports [c.440 BCE], B25), quoted by Sextus Empiricus - Against the Professors (six books) 9.54 |