86 ideas
| 18137 | Impredicative definitions are wrong, because they change the set that is being defined? [Bostock] |
| 18122 | Classical interdefinitions of logical constants and quantifiers is impossible in intuitionism [Bostock] |
| 18114 | There is no single agreed structure for set theory [Bostock] |
| 18107 | A 'proper class' cannot be a member of anything [Bostock] |
| 18115 | We could add axioms to make sets either as small or as large as possible [Bostock] |
| 18139 | The Axiom of Choice relies on reference to sets that we are unable to describe [Bostock] |
| 18105 | Replacement enforces a 'limitation of size' test for the existence of sets [Bostock] |
| 18108 | First-order logic is not decidable: there is no test of whether any formula is valid [Bostock] |
| 18109 | The completeness of first-order logic implies its compactness [Bostock] |
| 18123 | Substitutional quantification is just standard if all objects in the domain have a name [Bostock] |
| 18120 | The Deduction Theorem is what licenses a system of natural deduction [Bostock] |
| 18125 | Berry's Paradox considers the meaning of 'The least number not named by this name' [Bostock] |
| 18100 | ω + 1 is a new ordinal, but its cardinality is unchanged [Bostock] |
| 18101 | Each addition changes the ordinality but not the cardinality, prior to aleph-1 [Bostock] |
| 18102 | A cardinal is the earliest ordinal that has that number of predecessors [Bostock] |
| 18106 | Aleph-1 is the first ordinal that exceeds aleph-0 [Bostock] |
| 18095 | Instead of by cuts or series convergence, real numbers could be defined by axioms [Bostock] |
| 18099 | The number of reals is the number of subsets of the natural numbers [Bostock] |
| 18093 | For Eudoxus cuts in rationals are unique, but not every cut makes a real number [Bostock] |
| 18110 | Infinitesimals are not actually contradictory, because they can be non-standard real numbers [Bostock] |
| 18156 | Modern axioms of geometry do not need the real numbers [Bostock] |
| 18097 | The Peano Axioms describe a unique structure [Bostock] |
| 18148 | Hume's Principle is a definition with existential claims, and won't explain numbers [Bostock] |
| 18149 | There are many criteria for the identity of numbers [Bostock] |
| 18145 | Many things will satisfy Hume's Principle, so there are many interpretations of it [Bostock] |
| 18143 | Frege makes numbers sets to solve the Caesar problem, but maybe Caesar is a set! [Bostock] |
| 18116 | Numbers can't be positions, if nothing decides what position a given number has [Bostock] |
| 18117 | Structuralism falsely assumes relations to other numbers are numbers' only properties [Bostock] |
| 18157 | Nominalism as based on application of numbers is no good, because there are too many applications [Bostock] |
| 18141 | Nominalism about mathematics is either reductionist, or fictionalist [Bostock] |
| 18150 | Actual measurement could never require the precision of the real numbers [Bostock] |
| 18158 | Ordinals are mainly used adjectively, as in 'the first', 'the second'... [Bostock] |
| 18127 | Simple type theory has 'levels', but ramified type theory has 'orders' [Bostock] |
| 18147 | Neo-logicists meet the Caesar problem by saying Hume's Principle is unique to number [Bostock] |
| 18144 | Neo-logicists agree that HP introduces number, but also claim that it suffices for the job [Bostock] |
| 18111 | Treating numbers as objects doesn't seem like logic, since arithmetic fixes their totality [Bostock] |
| 18146 | If Hume's Principle is the whole story, that implies structuralism [Bostock] |
| 18129 | Many crucial logicist definitions are in fact impredicative [Bostock] |
| 18159 | Higher cardinalities in sets are just fairy stories [Bostock] |
| 18155 | A fairy tale may give predictions, but only a true theory can give explanations [Bostock] |
| 18138 | Conceptualism fails to grasp mathematical properties, infinity, and objective truth values [Bostock] |
| 18140 | The best version of conceptualism is predicativism [Bostock] |
| 18131 | If abstracta only exist if they are expressible, there can only be denumerably many of them [Bostock] |
| 18132 | The predicativity restriction makes a difference with the real numbers [Bostock] |
| 18133 | The usual definitions of identity and of natural numbers are impredicative [Bostock] |
| 18134 | Predicativism makes theories of huge cardinals impossible [Bostock] |
| 18135 | If mathematics rests on science, predicativism may be the best approach [Bostock] |
| 18136 | If we can only think of what we can describe, predicativism may be implied [Bostock] |
| 3942 | I do not believe in the existence of anything, if I see no reason to believe it [Berkeley] |
| 3952 | I know that nothing inconsistent can exist [Berkeley] |
| 3959 | There is no other substance, in a strict sense, than spirit [Berkeley] |
| 3946 | A thing is shown to be impossible if a contradiction is demonstrated within its definition [Berkeley] |
| 3958 | Since our ideas vary when the real things are said to be unchanged, they cannot be true copies [Berkeley] |
| 3943 | If existence is perceived directly, by which sense; if indirectly, how is it inferred from direct perception? [Berkeley] |
| 3931 | Sensible objects are just sets of sensible qualities [Berkeley] |
| 5192 | Berkeley did not deny material things; he merely said they must be defined through sensations [Berkeley, by Ayer] |
| 5174 | Berkeley needed a phenomenalist account of the self, as well as of material things [Ayer on Berkeley] |
| 1103 | 'To be is to be perceived' is a simple confusion of experience with its objects [Russell on Berkeley] |
| 6403 | For Berkelely, reality is ideas and a community of minds, including God's [Berkeley, by Grayling] |
| 3945 | There is nothing in nature which needs the concept of matter to explain it [Berkeley] |
| 3947 | Perceptions are ideas, and ideas exist in the mind, so objects only exist in the mind [Berkeley] |
| 3936 | Time is measured by the succession of ideas in our minds [Berkeley] |
| 3930 | There is no such thing as 'material substance' [Berkeley] |
| 3939 | I conceive a tree in my mind, but I cannot prove that its existence can be conceived outside a mind [Berkeley] |
| 3933 | Primary qualities (such as shape, solidity, mass) are held to really exist, unlike secondary qualities [Berkeley] |
| 3934 | A mite would see its own foot as large, though we would see it as tiny [Berkeley] |
| 3935 | The apparent size of an object varies with its distance away, so that can't be a property of the object [Berkeley] |
| 3937 | 'Solidity' is either not a sensible quality at all, or it is clearly relative to our senses [Berkeley] |
| 3940 | Distance is not directly perceived by sight [Berkeley] |
| 3957 | Immediate objects of perception, which some treat as appearances, I treat as the real things themselves [Berkeley] |
| 3953 | Real things and imaginary or dreamed things differ because the latter are much fainter [Berkeley] |
| 3938 | Geometry is originally perceived by senses, and so is not purely intellectual [Berkeley] |
| 3944 | It is possible that we could perceive everything as we do now, but nothing actually existed. [Berkeley] |
| 3932 | A hot hand and a cold hand will have different experiences in the same tepid water [Berkeley] |
| 3948 | Experience tells me that other minds exist independently from my own [Berkeley] |
| 3941 | How can that which is unthinking be a cause of thought? [Berkeley] |
| 5374 | Berkeley probably used 'idea' to mean both the act of apprehension and the thing apprehended [Russell on Berkeley] |
| 18121 | In logic a proposition means the same when it is and when it is not asserted [Bostock] |
| 3954 | Immorality is not in the action, but in the deviation of the will from moral law [Berkeley] |
| 541 | Virtue comes more from habit than character [Critias] |
| 3950 | There must be a God, because all sensible things must be perceived by him [Berkeley] |
| 3951 | There must be a God, because I and my ideas are not independent [Berkeley] |
| 3949 | It has been proved that creation is the workmanship of God, from its beauty and usefulness [Berkeley] |
| 542 | Fear of the gods was invented to discourage secret sin [Critias] |
| 3956 | People are responsible because they have limited power, though this ultimately derives from God [Berkeley] |
| 3955 | If sin is not just physical, we don't consider God the origin of sin because he causes physical events [Berkeley] |