44 ideas
| 162 | Can we understand an individual soul without knowing the soul in general? [Plato] |
| Full Idea: Do you think it possible to form an adequate conception of the nature of an individual soul without considering the nature of soul in general? | |
| From: Plato (Phaedrus [c.366 BCE], 270c) | |
| A reaction: Do animals understand anything (as opposed to simply being aware of things)? |
| 160 | The highest ability in man is the ability to discuss unity and plurality in the nature of things [Plato] |
| Full Idea: When I believe that I have found in anyone the ability to discuss unity and plurality as they exist in the nature of things, I follow his footsteps as if he was a god. | |
| From: Plato (Phaedrus [c.366 BCE], 266b) | |
| A reaction: This sounds like the problem of identity, which is at the heart of modern metaphysics. |
| 166 | A speaker should be able to divide a subject, right down to the limits of divisibility [Plato] |
| Full Idea: A speaker must be able to define a subject generically, and then to divide it into its various specific kinds until he reaches the limits of divisibility. | |
| From: Plato (Phaedrus [c.366 BCE], 277b) |
| 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| From: report of Kurt Gödel (On Formally Undecidable Propositions [1931]) by Willard Quine - Forward to Gödel's Unpublished | |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 |
| 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. |
| 7953 | Reasoning needs to cut nature accurately at the joints [Plato] |
| Full Idea: In our reasoning we need a clear view of the ability to divide a genus into species, observing the natural joints, not mangling any of the parts, like an unskilful butcher. | |
| From: Plato (Phaedrus [c.366 BCE], 265d) | |
| A reaction: In modern times this Platonic idea has become the standard metaphor for realism. I endorse it. I think nature has joints, and we should hunt for them. There are natural sets. The joints may exist in abstract concepts, as well as in objects. |
| 16121 | I revere anyone who can discern a single thing that encompasses many things [Plato] |
| Full Idea: If I believe that someone is capable of discerning a single thing that is also by nature capable of encompassing many, I follow 'straight behind, in his footsteps, as if he were a god'. | |
| From: Plato (Phaedrus [c.366 BCE], 266b) | |
| A reaction: [Plato quote Odyssey 2.406] This is the sort of simple but profound general observation which only the early philosophers bothered to make, and no one comments on now. Encompassing many under one is the very essence of thinking. |
| 153 | It takes a person to understand, by using universals, and by using reason to create a unity out of sense-impressions [Plato] |
| Full Idea: It takes a man to understand by the use of universals, and to collect out of the multiplicity of sense-impressions a unity arrived at by a process of reason. | |
| From: Plato (Phaedrus [c.366 BCE], 249b) |
| 154 | We would have an overpowering love of knowledge if we had a pure idea of it - as with the other Forms [Plato] |
| Full Idea: What overpowering love knowledge would inspire if it could bring a clear image of itself before our sight, and the same may be said of the other forms. | |
| From: Plato (Phaedrus [c.366 BCE], 250d) | |
| A reaction: the motivation in Plato's theory |
| 151 | True knowledge is of the reality behind sense experience [Plato] |
| Full Idea: True knowledge is concerned with the abode of true reality, without colour or shape, intangible but utterly real, apprehensible only to the intellect. | |
| From: Plato (Phaedrus [c.366 BCE], 247c) |
| 165 | If the apparent facts strongly conflict with probability, it is in everyone's interests to suppress the facts [Plato] |
| Full Idea: There are some occasions when both prosecution and defence should positively suppress the facts in favour of probability, if the facts are improbable. | |
| From: Plato (Phaedrus [c.366 BCE], 272e) |
| 9296 | The soul is self-motion [Plato] |
| Full Idea: Self-motion is of the very nature of the soul. | |
| From: Plato (Phaedrus [c.366 BCE], 245e) | |
| A reaction: This culminates a length discussion of the soul. He gives an implausible argument that the soul is immortal, because it could never cease its self-motion. Why are we so unimpressed by motion, when the Greeks were amazed by it? |
| 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. |
| 23997 | Plato saw emotions and appetites as wild horses, in need of taming [Plato, by Goldie] |
| Full Idea: Plato had a conception of the emotions and our bodily appetites as being like wild horses, to be harnassed and controlled by reason. | |
| From: report of Plato (Phaedrus [c.366 BCE]) by Peter Goldie - The Emotions 4 'Education' | |
| A reaction: This seems to make Plato the patriarch of puritanism. See Symposium, as well as Phaedrus. But bringing up children can often seem like taming wild beasts. |
| 158 | An excellent speech seems to imply a knowledge of the truth in the mind of the speaker [Plato] |
| Full Idea: If a speech is to be classed as excellent, does that not presuppose knowledge of the truth about the subject of the speech in the mind of the speaker. | |
| From: Plato (Phaedrus [c.366 BCE], 259e) | |
| A reaction: I like the thought that Plato's main interest was rhetoric, but with the view that the only good rhetoric is truth-speaking. It would be hard to admire a speech if you disagreed with it. |
| 5946 | 'Phaedrus' pioneers the notion of philosophical rhetoric [Lawson-Tancred on Plato] |
| Full Idea: The purpose of the 'Phaedrus' is to pioneer the notion of philosophical rhetoric. | |
| From: comment on Plato (Phaedrus [c.366 BCE], Ch.10) by Hugh Lawson-Tancred - Plato's Republic and Greek Enlightenment | |
| A reaction: This is a wonderfully challenging view of what Plato was up to. One might connect it with Rorty's claim that philosophy should move away from epistemology and analysis, towards hermeneutics, which sounds to me like rhetoric. 'Phaedrus' is beautiful. |
| 159 | Only a good philosopher can be a good speaker [Plato] |
| Full Idea: Unless a man becomes an adequate philosopher he will never be an adequate speaker on any subject. | |
| From: Plato (Phaedrus [c.366 BCE], 261a) | |
| A reaction: Depends. Hitler showed little sign of clear philosophical thinking, but the addition of lights and uniforms seemed to sweep reasonably intelligent people along with him. |
| 155 | Beauty is the clearest and most lovely of the Forms [Plato] |
| Full Idea: Only beauty has the privilege of being the most clearly discerned and the most lovely of the forms. | |
| From: Plato (Phaedrus [c.366 BCE], 250e) | |
| A reaction: the motivation in Plato's theory |
| 143 | The two ruling human principles are the natural desire for pleasure, and an acquired love of virtue [Plato] |
| Full Idea: In each one of us there are two ruling and impelling principles: a desire for pleasure, which is innate, and an acquired conviction which causes us to aim at excellence. | |
| From: Plato (Phaedrus [c.366 BCE], 237d) | |
| A reaction: This division is too neat and simple. An obsession with pleasure I would take to be acquired. If you set out to do something, I think there is an innate desire to do it well. |
| 157 | Most pleasure is release from pain, and is therefore not worthwhile [Plato] |
| Full Idea: Life is not worth living for pleasures whose enjoyment entirely depends on previous sensation of pain, like almost all physical pleasures. | |
| From: Plato (Phaedrus [c.366 BCE], 258e) | |
| A reaction: Eating exotic food which is hard to obtain? (Pay someone to obtain it). Rock climbing. Training for sport. |
| 144 | Reason impels us towards excellence, which teaches us self-control [Plato] |
| Full Idea: The conviction which impels us towards excellence is rational, and the power by which it masters us we call self-control. | |
| From: Plato (Phaedrus [c.366 BCE], 237e) |
| 156 | Bad people are never really friends with one another [Plato] |
| Full Idea: It is not ordained that bad men should be friends with one another. | |
| From: Plato (Phaedrus [c.366 BCE], 255b) |
| 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 |
| 148 | If the prime origin is destroyed, it will not come into being again out of anything [Plato] |
| Full Idea: If the prime origin is destroyed, it will not come into being again out of anything. | |
| From: Plato (Phaedrus [c.366 BCE], 245d) | |
| A reaction: This is the essence of Aquinas's Third Way of proving God's existence. |
| 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. |
| 152 | The mind of God is fully satisfied and happy with a vision of reality and truth [Plato] |
| Full Idea: The mind of a god, sustained by pure intelligence and knowledge, is satisfied with the vision of reality, and nourished and made happy by the vision of truth. | |
| From: Plato (Phaedrus [c.366 BCE], 247d) |
| 150 | We cannot conceive of God, so we have to think of Him as an immortal version of ourselves [Plato] |
| Full Idea: Because we have never seen or formed an adequate idea of a god, we picture him to ourselves as a being of the same kind as ourselves but immortal. | |
| From: Plato (Phaedrus [c.366 BCE], 246d) |
| 149 | There isn't a single reason for positing the existence of immortal beings [Plato] |
| Full Idea: There is not a single sound reason for positing the existence of such a being who is immortal | |
| From: Plato (Phaedrus [c.366 BCE], 246d) |
| 146 | Soul is always in motion, so it must be self-moving and immortal [Plato] |
| Full Idea: All soul is immortal, for what is always in motion is immortal. Only that which moves itself never ceases to be in motion. | |
| From: Plato (Phaedrus [c.366 BCE], 245c) |