117 ideas
| 21405 | Cicero sees wisdom in terms of knowledge, but earlier Stoics saw it as moral [Cicero, by Long] |
| 20871 | Unfortunately we choose a way of life before we are old enough to think clearly [Cicero] |
| 5893 | A wise man has integrity, firmness of will, nobility, consistency, sobriety, patience [Cicero] |
| 5891 | Philosophy is the collection of rational arguments [Cicero] |
| 2661 | Dialectic is speech cast in the form of logical argument [Cicero] |
| 2653 | If the parts of the universe are subject to the law of nature, the whole universe must also be subject to it [Cicero] |
| 2673 | There cannot be more than one truth [Cicero] |
| 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] |
| 13011 | New axioms are being sought, to determine the size of the continuum [Maddy] |
| 13014 | Extensional sets are clearer, simpler, unique and expressive [Maddy] |
| 13013 | The Axiom of Extensionality seems to be analytic [Maddy] |
| 13021 | The Axiom of Infinity states Cantor's breakthrough that launched modern mathematics [Maddy] |
| 13022 | Infinite sets are essential for giving an account of the real numbers [Maddy] |
| 18191 | Axiom of Infinity: completed infinite collections can be treated mathematically [Maddy] |
| 13023 | The Power Set Axiom is needed for, and supported by, accounts of the continuum [Maddy] |
| 18193 | The Axiom of Foundation says every set exists at a level in the set hierarchy [Maddy] |
| 13024 | Efforts to prove the Axiom of Choice have failed [Maddy] |
| 13025 | Modern views say the Choice set exists, even if it can't be constructed [Maddy] |
| 13026 | A large array of theorems depend on the Axiom of Choice [Maddy] |
| 17610 | The Axiom of Choice paradoxically allows decomposing a sphere into two identical spheres [Maddy] |
| 18169 | Axiom of Reducibility: propositional functions are extensionally predicative [Maddy] |
| 13019 | The Iterative Conception says everything appears at a stage, derived from the preceding appearances [Maddy] |
| 13018 | Limitation of Size is a vague intuition that over-large sets may generate paradoxes [Maddy] |
| 17824 | The master science is physical objects divided into sets [Maddy] |
| 8755 | Maddy replaces pure sets with just objects and perceived sets of objects [Maddy, by Shapiro] |
| 10594 | Henkin semantics is more plausible for plural logic than for second-order logic [Maddy] |
| 17620 | Critics of if-thenism say that not all starting points, even consistent ones, are worth studying [Maddy] |
| 21677 | How can the not-true fail to be false, or the not-false fail to be true? [Cicero] |
| 2669 | Dialectic assumes that all statements are either true or false, but self-referential paradoxes are a big problem [Cicero] |
| 18168 | 'Propositional functions' are propositions with a variable as subject or predicate [Maddy] |
| 17605 | Hilbert's geometry and Dedekind's real numbers were role models for axiomatization [Maddy] |
| 17625 | If two mathematical themes coincide, that suggest a single deep truth [Maddy] |
| 18171 | Cantor and Dedekind brought completed infinities into mathematics [Maddy] |
| 18190 | Completed infinities resulted from giving foundations to calculus [Maddy] |
| 17615 | Every infinite set of reals is either countable or of the same size as the full set of reals [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] |
| 17825 | Set theory (unlike the Peano postulates) can explain why multiplication is commutative [Maddy] |
| 17826 | Standardly, numbers are said to be sets, which is neat ontology and epistemology [Maddy] |
| 17828 | Numbers are properties of sets, just as lengths are properties of physical objects [Maddy] |
| 10718 | A natural number is a property of sets [Maddy, by Oliver] |
| 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] |
| 17618 | Set-theory tracks the contours of mathematical depth and fruitfulness [Maddy] |
| 17830 | Number theory doesn't 'reduce' to set theory, because sets have number properties [Maddy] |
| 17827 | Sets exist where their elements are, but numbers are more like universals [Maddy] |
| 17823 | If mathematical objects exist, how can we know them, and which objects are they? [Maddy] |
| 8756 | Intuition doesn't support much mathematics, and we should question its reliability [Maddy, by Shapiro] |
| 17733 | We know mind-independent mathematical truths through sets, which rest on experience [Maddy, by Jenkins] |
| 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] |
| 17614 | The connection of arithmetic to perception has been idealised away in modern infinitary mathematics [Maddy] |
| 17829 | Number words are unusual as adjectives; we don't say 'is five', and numbers always come first [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] |
| 1212 | Replacing timbers on Theseus' ship was the classic illustration of the problem of growth and change [Plutarch] |
| 5958 | The sun is always bright; it doesn't become bright when it emerges [Plutarch] |
| 2664 | If we have complete healthy senses, what more could the gods give us? [Cicero] |
| 2665 | How can there be a memory of what is false? [Cicero] |
| 20800 | Every true presentation can have a false one of the same quality [Cicero] |
| 5879 | The soul is the heart, or blood in the heart, or part of the brain, of something living in heart or brain, or breath [Cicero] |
| 5960 | When the soul is intelligent and harmonious, it is part of god and derives from god [Plutarch] |
| 5959 | Some philosophers say the soul is light [Plutarch] |
| 5884 | How can one mind perceive so many dissimilar sensations? [Cicero] |
| 5887 | The soul has a single nature, so it cannot be divided, and hence it cannot perish [Cicero] |
| 18206 | Science idealises the earth's surface, the oceans, continuities, and liquids [Maddy] |
| 5952 | Rather than being the whole soul, maybe I am its chief part? [Plutarch] |
| 5886 | Like the eye, the soul has no power to see itself, but sees other things [Cicero] |
| 6029 | Whoever knows future causes knows everything that will be [Cicero] |
| 2628 | Why would mind mix with matter if it didn't need it? [Cicero] |
| 5885 | Souls contain no properties of elements, and elements contain no properties of souls [Cicero] |
| 5951 | If atoms have no qualities, they cannot possibly produce a mind [Plutarch] |
| 5963 | Some say emotion is a sort of reason, and others say virtue concerns emotion [Plutarch] |
| 21667 | Oratory and philosophy are closely allied; orators borrow from philosophy, and ornament it [Cicero] |
| 20814 | Eloquence educates, exhorts, comforts, distracts and unites us, and raises us from savagery [Cicero] |
| 20796 | Action needs an affinity for a presentation, and an impulse toward the affinity [Plutarch] |
| 21678 | If desire is not in our power then neither are choices, so we should not be praised or punished [Cicero] |
| 1477 | Being manly and brave is the result of convention, not of human nature [Plutarch] |
| 1478 | Animals don't value pleasure, as they cease sexual intercourse after impregnation [Plutarch] |
| 2672 | Virtues must be very detached, to avoid being motivated by pleasure [Cicero] |
| 5948 | The good life involves social participation, loyalty, temperance and honesty [Plutarch] |
| 5890 | We should not share the distress of others, but simply try to relieve it [Cicero] |
| 5894 | All men except philosophers fear poverty [Cicero] |
| 6031 | The essence of propriety is consistency [Cicero] |
| 5895 | If one despises illiterate mechanics individually, they are not worth more collectively [Cicero] |
| 2640 | We have the death penalty, but still have thousands of robbers [Cicero] |
| 1479 | Animals have not been led into homosexuality, because they value pleasure very little [Plutarch] |
| 2652 | Some regard nature simply as an irrational force that imparts movement [Cicero] |
| 5950 | If only atoms exist, how do qualities arise when the atoms come together? [Plutarch] |
| 2645 | Why shouldn't the gods fear their own destruction? [Cicero] |
| 2627 | I wonder whether loss of reverence for the gods would mean the end of all virtue [Cicero] |
| 2651 | God doesn't obey the laws of nature; they are subject to the law of God [Cicero] |
| 2634 | It seems clear to me that we have an innate idea of the divine [Cicero] |
| 2636 | Many primitive people know nothing of the gods [Cicero] |
| 2647 | It is obvious from order that someone is in charge, as when we visit a gymnasium [Cicero] |
| 2650 | If a person cannot feel the power of God when looking at the stars, they are probably incapable of feeling [Cicero] |
| 2655 | If the barbarians of Britain saw a complex machine, they would be baffled, but would know it was designed [Cicero] |
| 2656 | Chance is no more likely to create the world than spilling lots of letters is likely to create a famous poem [Cicero] |
| 2657 | If everything with regular movement and order is divine, then recurrent illnesses must be divine [Cicero] |
| 5974 | People report seeing through rocks, or over the horizon, or impossibly small works [Plutarch] |
| 2638 | Either the gods are identical, or one is more beautiful than another [Cicero] |
| 2635 | The gods are happy, so virtuous, so rational, so must have human shape [Cicero] |
| 2641 | Why believe in gods if you have never seen them? [Cicero] |
| 5957 | Absurd superstitions make people atheist, not disharmony in nature [Plutarch] |
| 5955 | No one will ever find a city that lacks religious practices [Plutarch] |
| 2659 | The lists of good men who have suffered and bad men who have prospered are endless [Cicero] |
| 2658 | The gods blame men for having vices, but they could have given us enough reason to avoid them [Cicero] |