62 ideas
18194 | 'Forcing' can produce new models of ZFC from old models [Maddy] |
18195 | A Large Cardinal Axiom would assert ever-increasing stages in the hierarchy [Maddy] |
18191 | Axiom of Infinity: completed infinite collections can be treated mathematically [Maddy] |
18193 | The Axiom of Foundation says every set exists at a level in the set hierarchy [Maddy] |
18169 | Axiom of Reducibility: propositional functions are extensionally predicative [Maddy] |
18168 | 'Propositional functions' are propositions with a variable as subject or predicate [Maddy] |
18190 | Completed infinities resulted from giving foundations to calculus [Maddy] |
18171 | Cantor and Dedekind brought completed infinities into mathematics [Maddy] |
18172 | Infinity has degrees, and large cardinals are the heart of set theory [Maddy] |
18175 | For any cardinal there is always a larger one (so there is no set of all sets) [Maddy] |
18196 | An 'inaccessible' cardinal cannot be reached by union sets or power sets [Maddy] |
18187 | Theorems about limits could only be proved once the real numbers were understood [Maddy] |
13007 | Archimedes defined a straight line as the shortest distance between two points [Archimedes, by Leibniz] |
18182 | The extension of concepts is not important to me [Maddy] |
18177 | In the ZFC hierarchy it is impossible to form Frege's set of all three-element sets [Maddy] |
18164 | Frege solves the Caesar problem by explicitly defining each number [Maddy] |
18184 | Making set theory foundational to mathematics leads to very fruitful axioms [Maddy] |
18185 | Unified set theory gives a final court of appeal for mathematics [Maddy] |
18183 | Set theory brings mathematics into one arena, where interrelations become clearer [Maddy] |
18163 | Mathematics rests on the logic of proofs, and on the set theoretic axioms [Maddy] |
18186 | Identifying geometric points with real numbers revealed the power of set theory [Maddy] |
18188 | The line of rationals has gaps, but set theory provided an ordered continuum [Maddy] |
18207 | Maybe applications of continuum mathematics are all idealisations [Maddy] |
18204 | Scientists posit as few entities as possible, but set theorist posit as many as possible [Maddy] |
18167 | We can get arithmetic directly from HP; Law V was used to get HP from the definition of number [Maddy] |
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] |
18205 | The theoretical indispensability of atoms did not at first convince scientists that they were real [Maddy] |
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] |
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] |
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] |
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] |
18206 | Science idealises the earth's surface, the oceans, continuities, and liquids [Maddy] |
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] |
3954 | Immorality is not in the action, but in the deviation of the will from moral law [Berkeley] |
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] |
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] |