62 ideas
19250 | Everything interesting should be recorded, with records that can be rearranged [Peirce] |
19228 | Sciences concern existence, but philosophy also concerns potential existence [Peirce] |
19241 | An idea on its own isn't an idea, because they are continuous systems [Peirce] |
19227 | Philosophy is a search for real truth [Peirce] |
19218 | Metaphysics is pointless without exact modern logic [Peirce] |
19229 | Metaphysics is the science of both experience, and its general laws and types [Peirce] |
19219 | Metaphysical reasoning is simple enough, but the concepts are very hard [Peirce] |
19231 | Metaphysics is turning into logic, and logic is becoming mathematics [Peirce] |
19247 | The one unpardonable offence in reasoning is to block the route to further truth [Peirce] |
19246 | 'Holding for true' is either practical commitment, or provisional theory [Peirce] |
21752 | Prior to Gödel we thought truth in mathematics consisted in provability [Gödel, by Quine] |
17835 | Gödel show that the incompleteness of set theory was a necessity [Gödel, by Hallett,M] |
11211 | If a sound conclusion comes from two errors that cancel out, the path of the argument must matter [Rumfitt] |
19237 | Deduction is true when the premises facts necessarily make the conclusion fact true [Peirce] |
19256 | Our research always hopes that reality embodies the logic we are employing [Peirce] |
11212 | The sense of a connective comes from primitively obvious rules of inference [Rumfitt] |
11210 | Standardly 'and' and 'but' are held to have the same sense by having the same truth table [Rumfitt] |
19238 | The logic of relatives relies on objects built of any relations (rather than on classes) [Peirce] |
17886 | The limitations of axiomatisation were revealed by the incompleteness theorems [Gödel, by Koellner] |
10071 | Second Incompleteness: nice theories can't prove their own consistency [Gödel, by Smith,P] |
19123 | If soundness can't be proved internally, 'reflection principles' can be added to assert soundness [Gödel, by Halbach/Leigh] |
10621 | Gödel's First Theorem sabotages logicism, and the Second sabotages Hilbert's Programme [Smith,P on Gödel] |
17888 | The undecidable sentence can be decided at a 'higher' level in the system [Gödel] |
10132 | There can be no single consistent theory from which all mathematical truths can be derived [Gödel, by George/Velleman] |
3198 | Gödel showed that arithmetic is either incomplete or inconsistent [Gödel, by Rey] |
10072 | First Incompleteness: arithmetic must always be incomplete [Gödel, by Smith,P] |
9590 | Arithmetical truth cannot be fully and formally derived from axioms and inference rules [Gödel, by Nagel/Newman] |
11069 | Gödel's Second says that semantic consequence outruns provability [Gödel, by Hanna] |
10118 | First Incompleteness: a decent consistent system is syntactically incomplete [Gödel, by George/Velleman] |
10122 | Second Incompleteness: a decent consistent system can't prove its own consistency [Gödel, by George/Velleman] |
10611 | There is a sentence which a theory can show is true iff it is unprovable [Gödel, by Smith,P] |
10867 | 'This system can't prove this statement' makes it unprovable either way [Gödel, by Clegg] |
19226 | We now know that mathematics only studies hypotheses, not facts [Peirce] |
8747 | Realists are happy with impredicative definitions, which describe entities in terms of other existing entities [Gödel, by Shapiro] |
19240 | Realism is the belief that there is something in the being of things corresponding to our reasoning [Peirce] |
19239 | There may be no reality; it's just our one desperate hope of knowing anything [Peirce] |
19252 | Objective chance is the property of a distribution [Peirce] |
19232 | In ordinary language a conditional statement assumes that the antecedent is true [Peirce] |
19223 | We act on 'full belief' in a crisis, but 'opinion' only operates for trivial actions [Peirce] |
19253 | We talk of 'association by resemblance' but that is wrong: the association constitutes the resemblance [Peirce] |
19224 | Scientists will give up any conclusion, if experience opposes it [Peirce] |
19243 | If each inference slightly reduced our certainty, science would soon be in trouble [Peirce] |
19225 | I classify science by level of abstraction; principles derive from above, and data from below [Peirce] |
19234 | 'Induction' doesn't capture Greek 'epagoge', which is singulars in a mass producing the general [Peirce] |
19235 | How does induction get started? [Peirce] |
19236 | Induction can never prove that laws have no exceptions [Peirce] |
19251 | The worst fallacy in induction is generalising one recondite property from a sample [Peirce] |
19222 | Men often answer inner 'whys' by treating unconscious instincts as if they were reasons [Peirce] |
19220 | We may think animals reason very little, but they hardly ever make mistakes! [Peirce] |
19255 | Generalisation is the great law of mind [Peirce] |
19242 | Generalization is the true end of life [Peirce] |
19249 | 'Know yourself' is not introspection; it is grasping how others see you [Peirce] |
19257 | Whatever is First must be sentient [Peirce] |
3192 | Basic logic can be done by syntax, with no semantics [Gödel, by Rey] |
19248 | Reasoning involves observation, experiment, and habituation [Peirce] |
19221 | Everybody overrates their own reasoning, so it is clearly superficial [Peirce] |
19233 | Indexicals are unusual words, because they stimulate the hearer to look around [Peirce] |
11214 | We learn 'not' along with affirmation, by learning to either affirm or deny a sentence [Rumfitt] |
19230 | People should follow what lies before them, and is within their power [Peirce] |
19245 | We are not inspired by other people's knowledge; a sense of our ignorance motivates study [Peirce] |
19244 | Chemists rely on a single experiment to establish a fact; repetition is pointless [Peirce] |
19254 | Our laws of nature may be the result of evolution [Peirce] |