131 ideas
| 2196 | The observation of human blindness and weakness is the result of all philosophy [Hume] |
| 2187 | If we suspect that a philosophical term is meaningless, we should ask what impression it derives from [Hume] |
| 2200 | All experimental conclusions assume that the future will be like the past [Hume] |
| 9955 | Contextual definitions replace a complete sentence containing the expression [George/Velleman] |
| 10031 | Impredicative definitions quantify over the thing being defined [George/Velleman] |
| 4636 | All reasoning concerning matters of fact is based on analogy (with similar results of similar causes) [Hume] |
| 10098 | The 'power set' of A is all the subsets of A [George/Velleman] |
| 10101 |
Cartesian Product A x B: the set of all ordered pairs |
| 10099 | The 'ordered pair' <a, b>, for two sets a and b, is the set {{a, b},{a}} [George/Velleman] |
| 10103 | Grouping by property is common in mathematics, usually using equivalence [George/Velleman] |
| 10104 | 'Equivalence' is a reflexive, symmetric and transitive relation; 'same first letter' partitions English words [George/Velleman] |
| 10096 | Even the elements of sets in ZFC are sets, resting on the pure empty set [George/Velleman] |
| 10097 | Axiom of Extensionality: for all sets x and y, if x and y have the same elements then x = y [George/Velleman] |
| 10100 | Axiom of Pairing: for all sets x and y, there is a set z containing just x and y [George/Velleman] |
| 17900 | The Axiom of Reducibility made impredicative definitions possible [George/Velleman] |
| 10109 | ZFC can prove that there is no set corresponding to the concept 'set' [George/Velleman] |
| 10108 | As a reduction of arithmetic, set theory is not fully general, and so not logical [George/Velleman] |
| 12394 | If the result is bad, we change the rule; if we like the rule, we reject the result [Goodman] |
| 10111 | Asserting Excluded Middle is a hallmark of realism about the natural world [George/Velleman] |
| 10129 | A 'model' is a meaning-assignment which makes all the axioms true [George/Velleman] |
| 10105 | Differences between isomorphic structures seem unimportant [George/Velleman] |
| 10119 | Consistency is a purely syntactic property, unlike the semantic property of soundness [George/Velleman] |
| 10126 | A 'consistent' theory cannot contain both a sentence and its negation [George/Velleman] |
| 10120 | Soundness is a semantic property, unlike the purely syntactic property of consistency [George/Velleman] |
| 10127 | A 'complete' theory contains either any sentence or its negation [George/Velleman] |
| 10106 | Rational numbers give answers to division problems with integers [George/Velleman] |
| 10102 | The integers are answers to subtraction problems involving natural numbers [George/Velleman] |
| 10107 | Real numbers provide answers to square root problems [George/Velleman] |
| 9946 | Logicists say mathematics is applicable because it is totally general [George/Velleman] |
| 10125 | The classical mathematician believes the real numbers form an actual set [George/Velleman] |
| 17899 | Second-order induction is stronger as it covers all concepts, not just first-order definable ones [George/Velleman] |
| 10128 | The Incompleteness proofs use arithmetic to talk about formal arithmetic [George/Velleman] |
| 17902 | A successor is the union of a set with its singleton [George/Velleman] |
| 10133 | Frege's Theorem shows the Peano Postulates can be derived from Hume's Principle [George/Velleman] |
| 10130 | Set theory can prove the Peano Postulates [George/Velleman] |
| 10089 | Talk of 'abstract entities' is more a label for the problem than a solution to it [George/Velleman] |
| 2197 | Reason assists experience in discovering laws, and in measuring their application [Hume] |
| 10131 | If mathematics is not about particulars, observing particulars must be irrelevant [George/Velleman] |
| 10092 | In the unramified theory of types, the types are objects, then sets of objects, sets of sets etc. [George/Velleman] |
| 10094 | The theory of types seems to rule out harmless sets as well as paradoxical ones. [George/Velleman] |
| 10095 | Type theory has only finitely many items at each level, which is a problem for mathematics [George/Velleman] |
| 17901 | Type theory prohibits (oddly) a set containing an individual and a set of individuals [George/Velleman] |
| 10114 | Bounded quantification is originally finitary, as conjunctions and disjunctions [George/Velleman] |
| 10134 | Much infinite mathematics can still be justified finitely [George/Velleman] |
| 10123 | The intuitionists are the idealists of mathematics [George/Velleman] |
| 10124 | Gödel's First Theorem suggests there are truths which are independent of proof [George/Velleman] |
| 7700 | We can't think about the abstract idea of triangles, but only of particular triangles [Hume] |
| 14292 | Dispositions seem more ethereal than behaviour; a non-occult account of them would be nice [Goodman] |
| 13602 | We cannot form an idea of a 'power', and the word is without meaning [Hume] |
| 2216 | We transfer the frequency of past observations to our future predictions [Hume] |
| 2215 | There is no such thing as chance [Hume] |
| 2209 | Belief is stronger, clearer and steadier than imagination [Hume] |
| 2208 | Belief is just a particular feeling attached to ideas of objects [Hume] |
| 2207 | Belief can't be a concept plus an idea, or we could add the idea to fictions [Hume] |
| 3661 | 'Natural beliefs' are unavoidable, whatever our judgements [Hume, by Strawson,G] |
| 2213 | Beliefs are built up by resemblance, contiguity and causation [Hume] |
| 2191 | Relations of ideas are known by thought, independently from the world [Hume] |
| 2239 | If secondary qualities (e.g. hardness) are in the mind, so are primary qualities like extension [Hume] |
| 2237 | It never occurs to people that they only experience representations, not the real objects [Hume] |
| 2184 | All ideas are copies of impressions [Hume] |
| 2190 | All objects of enquiry are Relations of Ideas, or Matters of Fact [Hume] |
| 2182 | Impressions are our livelier perceptions, Ideas the less lively ones [Hume] |
| 2192 | All reasoning about facts is causal; nothing else goes beyond memory and senses [Hume] |
| 2246 | If books don't relate ideas or explain facts, commit them to the flames [Hume] |
| 23631 | Hume is loose when he says perceptions of different strength are different species [Reid on Hume] |
| 2189 | All ideas are connected by Resemblance, Contiguity in time or place, and Cause and Effect [Hume] |
| 2702 | Only madmen dispute the authority of experience [Hume] |
| 2183 | We can only invent a golden mountain by combining experiences [Hume] |
| 2194 | How could Adam predict he would drown in water or burn in fire? [Hume] |
| 2205 | You couldn't reason at all if you lacked experience [Hume] |
| 2186 | We cannot form the idea of something we haven't experienced [Hume] |
| 2217 | When definitions are pushed to the limit, only experience can make them precise [Hume] |
| 3902 | Hume mistakenly lumps sensations and perceptions together as 'impressions' [Scruton on Hume] |
| 23421 | If a person had a gap in their experience of blue shades, they could imaginatively fill it in [Hume] |
| 2206 | Reasons for belief must eventually terminate in experience, or they are without foundation [Hume] |
| 2235 | There is no certain supreme principle, or infallible rule of inference [Hume] |
| 10328 | We think testimony matches reality because of experience, not some a priori connection [Hume] |
| 2230 | Good testimony needs education, integrity, motive and agreement [Hume, by PG] |
| 2238 | Reason can never show that experiences are connected to external objects [Hume] |
| 2242 | Mitigated scepticism draws attention to the limitations of human reason, and encourages modesty [Hume] |
| 2243 | Mitigated scepticism sensibly confines our enquiries to the narrow capacity of human understanding [Hume] |
| 2236 | Examples of illusion only show that sense experience needs correction by reason [Hume] |
| 2240 | It is a very extravagant aim of the sceptics to destroy reason and argument by means of reason and argument [Hume] |
| 2241 | The main objection to scepticism is that no good can come of it [Hume] |
| 2198 | We assume similar secret powers behind similar experiences, such as the nourishment of bread [Hume] |
| 2202 | Fools, children and animals all learn from experience [Hume] |
| 2204 | All inferences from experience are effects of custom, not reasoning [Hume] |
| 2199 | Reason cannot show why reliable past experience should extend to future times and remote places [Hume] |
| 2201 | Induction can't prove that the future will be like the past, since induction assumes this [Hume] |
| 2203 | If we infer causes from repetition, this explains why we infer from a thousand objects what we couldn't infer from one [Hume] |
| 6350 | Premises can support an argument without entailing it [Pollock/Cruz on Hume] |
| 3598 | Hume just shows induction isn't deduction [Williams,M on Hume] |
| 18749 | Goodman argued that the confirmation relation can never be formalised [Goodman, by Horsten/Pettigrew] |
| 17646 | Goodman showed that every sound inductive argument has an unsound one of the same form [Goodman, by Putnam] |
| 2210 | A picture of a friend strengthens our idea of him, by resemblance [Hume] |
| 17712 | General ideas are the connection by resemblance to some particular [Hume] |
| 8544 | Hume does not distinguish real resemblances among degrees of resemblance [Shoemaker on Hume] |
| 2211 | When I am close to (contiguous with) home, I feel its presence more nearly [Hume] |
| 2214 | Our awareness of patterns of causation is too important to be left to slow and uncertain reasoning [Hume] |
| 2212 | An object made by a saint is the best way to produce thoughts of him [Hume] |
| 2222 | The doctrine of free will arises from a false sensation we have of freedom in many actions [Hume] |
| 2223 | Liberty is merely acting according to the will, which anyone can do if they are not in chains [Hume] |
| 3655 | Hume makes determinism less rigid by removing the necessity from causation [Trusted on Hume] |
| 10110 | Corresponding to every concept there is a class (some of them sets) [George/Velleman] |
| 2220 | Only experience teaches us about our wills [Hume] |
| 2224 | Praise and blame can only be given if an action proceeds from a person's character and disposition [Hume] |
| 2225 | If you deny all necessity and causation, then our character is not responsible for our crime [Hume] |
| 2226 | Repentance gets rid of guilt, which shows that responsibility arose from the criminal principles in the mind [Hume] |
| 2233 | No government has ever suffered by being too tolerant of philosophy [Hume] |
| 2195 | We can discover some laws of nature, but never its ultimate principles and causes [Hume] |
| 2245 | A priori it looks as if a cause could have absolutely any effect [Hume] |
| 4772 | If a singular effect is studied, its cause can only be inferred from the types of events involved [Hume] |
| 8341 | Hume never even suggests that there is no such thing as causation [Hume, by Strawson,G] |
| 8344 | At first Hume said qualities are the causal entities, but later he said events [Hume, by Davidson] |
| 2234 | It is only when two species of thing are constantly conjoined that we can infer one from the other [Hume] |
| 3662 | Hume says we can only know constant conjunctions, not that that's what causation IS [Hume, by Strawson,G] |
| 4771 | In both of Hume's definitions, causation is extrinsic to the sequence of events [Psillos on Hume] |
| 5194 | Hume's definition of cause as constantly joined thoughts can't cover undiscovered laws [Ayer on Hume] |
| 2221 | A cause is either similar events following one another, or an experience always suggesting a second experience [Hume] |
| 2193 | No causes can be known a priori, but only from experience of constant conjunctions [Hume] |
| 8422 | Cause is where if the first object had not been, the second had not existed [Hume] |
| 2218 | In observing causes we can never observe any necessary connections or binding qualities [Hume] |
| 15249 | Hume never shows how a strong habit could generate the concept of necessity [Harré/Madden on Hume] |
| 8339 | Hume's regularity theory of causation is epistemological; he believed in some sort of natural necessity [Hume, by Strawson,G] |
| 4794 | We don't use laws to make predictions, we call things laws if we make predictions with them [Goodman] |
| 2244 | It can never be a logical contradiction to assert the non-existence of something thought to exist [Hume] |
| 2232 | You can't infer the cause to be any greater than its effect [Hume] |
| 2229 | To establish a miracle the falseness of the evidence must be a greater miracle than the claimed miraculous event [Hume] |
| 2227 | A miracle violates laws which have been established by continuous unchanging experience, so should be ignored [Hume] |
| 2228 | All experience must be against a supposed miracle, or it wouldn't be called 'a miracle' [Hume] |
| 2185 | The idea of an infinite, intelligent, wise and good God arises from augmenting the best qualities of our own minds [Hume] |