68 ideas
| 1597 | Thales was the first western thinker to believe the arché was intelligible [Roochnik on Thales] |
| 1887 | You cannot divide anything into many parts, because after the first division you are no longer dividing the original [Sext.Empiricus] |
| 1885 | Proof moves from agreed premises to a non-evident inference [Sext.Empiricus] |
| 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] |
| 12196 | A valid hypothetical syllogism is 'that which does not begin with a truth and end with a falsehood' [Sext.Empiricus] |
| 18168 | 'Propositional functions' are propositions with a variable as subject or predicate [Maddy] |
| 1902 | Since Socrates either died when he was alive (a contradiction) or died when he was dead (meaningless), he didn't die [Sext.Empiricus] |
| 18171 | Cantor and Dedekind brought completed infinities into mathematics [Maddy] |
| 18190 | Completed infinities resulted from giving foundations to calculus [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] |
| 18172 | Infinity has degrees, and large cardinals are the heart of set theory [Maddy] |
| 18187 | Theorems about limits could only be proved once the real numbers were understood [Maddy] |
| 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] |
| 18163 | Mathematics rests on the logic of proofs, and on the set theoretic 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] |
| 18186 | Identifying geometric points with real numbers revealed the power of set theory [Maddy] |
| 18184 | Making set theory foundational to mathematics leads to very fruitful axioms [Maddy] |
| 18188 | The line of rationals has gaps, but set theory provided an ordered continuum [Maddy] |
| 18204 | Scientists posit as few entities as possible, but set theorist posit as many as possible [Maddy] |
| 18207 | Maybe applications of continuum mathematics are all idealisations [Maddy] |
| 18167 | We can get arithmetic directly from HP; Law V was used to get HP from the definition of number [Maddy] |
| 18205 | The theoretical indispensability of atoms did not at first convince scientists that they were real [Maddy] |
| 3013 | Nothing is stronger than necessity, which rules everything [Thales, by Diog. Laertius] |
| 1889 | If an argument has an absurd conclusion, we should not assent to the absurdity, but avoid the absurd argument [Sext.Empiricus] |
| 1871 | Whether honey is essentially sweet may be doubted, as it is a matter of judgement rather than appearance [Sext.Empiricus] |
| 1883 | How can the intellect know if sensation is reliable if it doesn't directly see external objects? [Sext.Empiricus] |
| 1890 | We distinguish ambiguities by seeing what is useful [Sext.Empiricus] |
| 1870 | The basis of scepticism is the claim that every proposition has an equal opposing proposition [Sext.Empiricus] |
| 1882 | The necks of doves appear different in colour depending on the angle of viewing [Sext.Empiricus] |
| 1881 | The same oar seems bent in water and straight when out of it [Sext.Empiricus] |
| 1872 | The same tower appears round from a distance, but square close at hand [Sext.Empiricus] |
| 1873 | If we press the side of an eyeball, objects appear a different shape [Sext.Empiricus] |
| 1874 | How can we judge between our impressions and those of other animals, when we ourselves are involved? [Sext.Empiricus] |
| 1876 | If we enjoy different things, presumably we receive different impressions [Sext.Empiricus] |
| 1878 | Water that seems lukewarm can seem very hot on inflamed skin [Sext.Empiricus] |
| 1880 | Some actions seem shameful when sober but not when drunk [Sext.Empiricus] |
| 1879 | Sickness is perfectly natural to the sick, so their natural perceptions should carry some weight [Sext.Empiricus] |
| 1877 | If we had no hearing or sight, we would assume no sound or sight exists, so there may be unsensed qualities [Sext.Empiricus] |
| 1910 | With us it is shameful for men to wear earrings, but among Syrians it is considered noble [Sext.Empiricus] |
| 1911 | Even if all known nations agree on a practice, there may be unknown nations which disagree [Sext.Empiricus] |
| 1886 | If you don't view every particular, you may miss the one which disproves your universal induction [Sext.Empiricus] |
| 18206 | Science idealises the earth's surface, the oceans, continuities, and liquids [Maddy] |
| 1884 | If we utter three steps of a logical argument, they never exist together [Sext.Empiricus] |
| 1494 | Thales said water is the first principle, perhaps from observing that food is moist [Thales, by Aristotle] |
| 1894 | Some say that causes are physical, some say not [Sext.Empiricus] |
| 1896 | If there were no causes then everything would have been randomly produced by everything [Sext.Empiricus] |
| 1897 | Knowing an effect results from a cause means knowing that the cause belongs with the effect, which is circular [Sext.Empiricus] |
| 1898 | Cause can't exist before effect, or exist at the same time, so it doesn't exist [Sext.Empiricus] |
| 1895 | Causes are either equal to the effect, or they link equally with other causes, or they contribute slightly [Sext.Empiricus] |
| 1713 | Thales must have thought soul causes movement, since he thought magnets have soul [Thales, by Aristotle] |
| 1899 | Does the original self-mover push itself from behind, or pull itself from in front? [Sext.Empiricus] |
| 1900 | If time and place are infinitely divided, it becomes impossible for movement ever to begin [Sext.Empiricus] |
| 1901 | If all atoms, times and places are the same, everything should move with equal velocity [Sext.Empiricus] |
| 1903 | If motion and rest are abolished, so is time [Sext.Empiricus] |
| 1904 | Time must be unlimited, but past and present can't be non-existent, and can't be now, so time does not exist [Sext.Empiricus] |
| 1905 | How can time be divisible if we can't compare one length of time with another? [Sext.Empiricus] |
| 1891 | How can we agree on the concept of God, unless we agree on his substance or form or place? [Sext.Empiricus] |
| 1892 | The existence of God can't be self-evident or everyone would have agreed on it, so it needs demonstration [Sext.Empiricus] |
| 1742 | Thales said the gods know our wrong thoughts as well as our evil actions [Thales, by Diog. Laertius] |
| 1893 | If God foresaw evil he would presumably prevent it, and if he only foresees some things, why those things? [Sext.Empiricus] |