93 ideas
21267 | Supposing many principles is superfluous if a few will do it [Aquinas] |
17774 | Definitions make our intuitions mathematically useful [Mayberry] |
17773 | Proof shows that it is true, but also why it must be true [Mayberry] |
23176 | Truth is universal, but knowledge of it is not [Aquinas] |
20621 | Types of lying: Speak lies, intend lies, intend deception, aim at deceptive goal? [Aquinas, by Tuckness/Wolf] |
21248 | If the existence of truth is denied, the 'Truth does not exist' must be true! [Aquinas] |
17795 | Set theory can't be axiomatic, because it is needed to express the very notion of axiomatisation [Mayberry] |
17796 | There is a semi-categorical axiomatisation of set-theory [Mayberry] |
17800 | The misnamed Axiom of Infinity says the natural numbers are finite in size [Mayberry] |
17801 | The set hierarchy doesn't rely on the dubious notion of 'generating' them [Mayberry] |
17803 | Limitation of size is part of the very conception of a set [Mayberry] |
17786 | The mainstream of modern logic sees it as a branch of mathematics [Mayberry] |
17788 | First-order logic only has its main theorems because it is so weak [Mayberry] |
17791 | Only second-order logic can capture mathematical structure up to isomorphism [Mayberry] |
23173 | If a syllogism admits one absurdity, others must follow [Aquinas] |
17787 | Big logic has one fixed domain, but standard logic has a domain for each interpretation [Mayberry] |
17790 | No Löwenheim-Skolem logic can axiomatise real analysis [Mayberry] |
17779 | 'Classificatory' axioms aim at revealing similarity in morphology of structures [Mayberry] |
17778 | Axiomatiation relies on isomorphic structures being essentially the same [Mayberry] |
17780 | 'Eliminatory' axioms get rid of traditional ideal and abstract objects [Mayberry] |
17789 | No logic which can axiomatise arithmetic can be compact or complete [Mayberry] |
17784 | Real numbers can be eliminated, by axiom systems for complete ordered fields [Mayberry] |
17782 | Greek quantities were concrete, and ratio and proportion were their science [Mayberry] |
17781 | Real numbers were invented, as objects, to simplify and generalise 'quantity' [Mayberry] |
17447 | Parsons says counting is tagging as first, second, third..., and converting the last to a cardinal [Parsons,C, by Heck] |
17799 | Cantor's infinite is an absolute, of all the sets or all the ordinal numbers [Mayberry] |
17797 | Cantor extended the finite (rather than 'taming the infinite') [Mayberry] |
17775 | If proof and definition are central, then mathematics needs and possesses foundations [Mayberry] |
17776 | The ultimate principles and concepts of mathematics are presumed, or grasped directly [Mayberry] |
17777 | Foundations need concepts, definition rules, premises, and proof rules [Mayberry] |
17804 | Axiom theories can't give foundations for mathematics - that's using axioms to explain axioms [Mayberry] |
17792 | 1st-order PA is only interesting because of results which use 2nd-order PA [Mayberry] |
17793 | It is only 2nd-order isomorphism which suggested first-order PA completeness [Mayberry] |
17794 | Set theory is not just first-order ZF, because that is inadequate for mathematics [Mayberry] |
17802 | We don't translate mathematics into set theory, because it comes embodied in that way [Mayberry] |
17805 | Set theory is not just another axiomatised part of mathematics [Mayberry] |
15812 | Being implies distinctness, which implies division, unity, and multitude [Aquinas] |
21268 | Non-human things are explicable naturally, and voluntary things by the will, so God is not needed [Aquinas] |
17785 | Real numbers as abstracted objects are now treated as complete ordered fields [Mayberry] |
16765 | Humans only have a single substantial form, which contains the others and acts for them [Aquinas] |
23175 | The conclusions of speculative reason about necessities are certain [Aquinas] |
21337 | A knowing being possesses a further reality, the 'presence' of the thing known [Aquinas] |
21249 | Some things are self-evident to us; others are only self-evident in themselves [Aquinas] |
21250 | A proposition is self-evident if the predicate is included in the essence of the subject [Aquinas] |
20224 | Sensation prepares the way for intellectual knowledge, which needs the virtues of reason [Aquinas] |
22107 | Sensations are transmitted to 'internal senses' in the brain, chiefly to 'phantasia' and 'imagination' [Aquinas, by Kretzmann/Stump] |
9098 | Mental activity combines what we sense with imagination of what is not present [Aquinas] |
9092 | Abstracting A from B generates truth, as long as the connection is not denied [Aquinas] |
9093 | We understand the general nature of things by ignoring individual peculiarities [Aquinas] |
9097 | The mind abstracts generalities from images, but also uses images for understanding [Aquinas] |
9095 | Very general ideas (being, oneness, potentiality) can be abstracted from thought matter in general [Aquinas] |
9099 | Particular instances come first, and (pace Plato) generalisations are abstracted from them [Aquinas] |
10508 | Species are abstracted from appearances by ignoring individual conditions [Aquinas] |
22111 | Aquinas attributes freedom to decisions and judgements, and not to the will alone [Aquinas, by Kretzmann/Stump] |
22105 | The human intellectual soul is an incorporeal, subsistent principle [Aquinas] |
22108 | First grasp what it is, then its essential features; judgement is their compounding and division [Aquinas] |
10503 | We abstract forms from appearances, and acquire knowledge of immaterial things [Aquinas] |
10509 | Understanding consists entirely of grasping abstracted species [Aquinas] |
10506 | Mathematics can be abstracted from sensible matter, and from individual intelligible matter [Aquinas] |
9094 | Mathematical objects abstract both from perceived matter, and from particular substance [Aquinas] |
10505 | We can just think of an apple's colour, because the apple is not part of the colour's nature [Aquinas] |
10504 | Abstracting either treats something as separate, or thinks of it separately [Aquinas] |
10507 | Numbers and shapes are abstracted by ignoring their sensible qualities [Aquinas] |
9096 | The mind must produce by its own power an image of the individual species [Aquinas] |
23180 | The will is the rational appetite [Aquinas] |
22112 | For humans good is accordance with reason, and bad is contrary to reason [Aquinas] |
22494 | We must know the end, know that it is the end, and know how to attain it [Aquinas] |
23181 | All acts of virtue relate to justice, which is directed towards the common good [Aquinas] |
8009 | Aquinas wanted, not to escape desire, but to transform it for moral ends [Aquinas, by MacIntyre] |
23182 | Legal justice is supreme, because it directs the other virtues to the common good [Aquinas] |
22399 | Temperance prevents our passions from acting against reason [Aquinas] |
23177 | Justice directs our relations with others, because it denotes a kind of equality [Aquinas] |
23179 | People differ in their social degrees, and a particular type of right applies to each [Aquinas] |
23174 | Natural law is a rational creature's participation in eternal law [Aquinas] |
22113 | Right and wrong actions pertain to natural law, as perceived by practical reason [Aquinas] |
22114 | Tyrannical laws are irrational, and so not really laws [Aquinas] |
7291 | For Aquinas a war must be in a just cause, have proper authority, and aim at good [Aquinas, by Grayling] |
5508 | Aquinas says a fertilized egg is not human, and has no immortal soul [Aquinas, by Martin/Barresi] |
16687 | Bodies are three-dimensional substances [Aquinas] |
23178 | Divine law commands some things because they are good, while others are good because commanded [Aquinas] |
21251 | We can't know God's essence, so his existence can't be self-evident for us [Aquinas] |
5614 | If you assume that there must be a necessary being, you can't say which being has this quality [Kant on Aquinas] |
21269 | Way 1: the infinite chain of potential-to-actual movement has to have a first mover [Aquinas] |
21270 | Way 2: no effect without a cause, and this cannot go back to infinity, so there is First Cause [Aquinas] |
21271 | Way 3: contingent beings eventually vanish, so continuity needs a necessary being [Aquinas] |
21272 | Way 4: the source of all qualities is their maximum, so something (God) causes all perfections [Aquinas] |
21273 | Way 5: mindless things act towards an obvious end, so there is an intelligent director [Aquinas] |
20211 | Life aims at the Beatific Vision - of perfect happiness, and revealed truth [Aquinas, by Zagzebski] |
22106 | Aquinas saw angels as separated forms, rather than as made of 'spiritual matter' [Aquinas, by Kretzmann/Stump] |
23306 | Humans have a non-physical faculty of reason, so they can be immortal [Aquinas, by Sorabji] |
4412 | Those in bliss have their happiness increased by seeing the damned punished [Aquinas] |
21266 | God does not exist, because He is infinite and good, and so no evil should be discoverable [Aquinas] |
21274 | It is part of God's supreme goodness that He brings good even out of evil [Aquinas] |