76 ideas
5728 | The concept of truth was originated by the senses [Lucretius] |
15945 | Second-order set theory just adds a version of Replacement that quantifies over functions [Lavine] |
15914 | An 'upper bound' is the greatest member of a subset; there may be several of these, so there is a 'least' one [Lavine] |
15921 | Collections of things can't be too big, but collections by a rule seem unlimited in size [Lavine] |
15937 | Those who reject infinite collections also want to reject the Axiom of Choice [Lavine] |
15936 | The Power Set is just the collection of functions from one collection to another [Lavine] |
15899 | Replacement was immediately accepted, despite having very few implications [Lavine] |
15930 | Foundation says descending chains are of finite length, blocking circularity, or ungrounded sets [Lavine] |
15920 | Pure collections of things obey Choice, but collections defined by a rule may not [Lavine] |
15898 | The controversy was not about the Axiom of Choice, but about functions as arbitrary, or given by rules [Lavine] |
15919 | The 'logical' notion of class has some kind of definition or rule to characterise the class [Lavine] |
15900 | The iterative conception of set wasn't suggested until 1947 [Lavine] |
15931 | The iterative conception needs the Axiom of Infinity, to show how far we can iterate [Lavine] |
15932 | The iterative conception doesn't unify the axioms, and has had little impact on mathematical proofs [Lavine] |
15933 | Limitation of Size: if it's the same size as a set, it's a set; it uses Replacement [Lavine] |
15913 | A collection is 'well-ordered' if there is a least element, and all of its successors can be identified [Lavine] |
15926 | Second-order logic presupposes a set of relations already fixed by the first-order domain [Lavine] |
15934 | Mathematical proof by contradiction needs the law of excluded middle [Lavine] |
15907 | Mathematics is nowadays (thanks to set theory) regarded as the study of structure, not of quantity [Lavine] |
15942 | Every rational number, unlike every natural number, is divisible by some other number [Lavine] |
15922 | For the real numbers to form a set, we need the Continuum Hypothesis to be true [Lavine] |
18250 | Cauchy gave a necessary condition for the convergence of a sequence [Lavine] |
15904 | The two sides of the Cut are, roughly, the bounding commensurable ratios [Lavine] |
15912 | Counting results in well-ordering, and well-ordering makes counting possible [Lavine] |
15949 | The theory of infinity must rest on our inability to distinguish between very large sizes [Lavine] |
15947 | The infinite is extrapolation from the experience of indefinitely large size [Lavine] |
15940 | The intuitionist endorses only the potential infinite [Lavine] |
15909 | 'Aleph-0' is cardinality of the naturals, 'aleph-1' the next cardinal, 'aleph-ω' the ω-th cardinal [Lavine] |
15915 | Ordinals are basic to Cantor's transfinite, to count the sets [Lavine] |
15917 | Paradox: the class of all ordinals is well-ordered, so must have an ordinal as type - giving a bigger ordinal [Lavine] |
15918 | Paradox: there is no largest cardinal, but the class of everything seems to be the largest [Lavine] |
15929 | Set theory will found all of mathematics - except for the notion of proof [Lavine] |
15935 | Modern mathematics works up to isomorphism, and doesn't care what things 'really are' [Lavine] |
15928 | Intuitionism rejects set-theory to found mathematics [Lavine] |
5702 | The senses are much the best way to distinguish true from false [Lucretius] |
5729 | If the senses are deceptive, reason, which rests on them, is even worse [Lucretius] |
5697 | The only possible standard for settling doubts is the foundation of the senses [Lucretius] |
5727 | Most supposed delusions of the senses are really misinterpretations by the mind [Lucretius] |
5714 | Even simple facts are hard to believe at first hearing [Lucretius] |
20653 | Six reduction levels: groups, lives, cells, molecules, atoms, particles [Putnam/Oppenheim, by Watson] |
5718 | The mind is in the middle of the breast, because there we experience fear and joy [Lucretius] |
5717 | The mind is a part of a man, just like a hand or an eye [Lucretius] |
21387 | The separate elements and capacities of a mind cannot be distinguished [Lucretius] |
5709 | The actions of the mind are not determinate and passive, because atoms can swerve [Lucretius] |
5695 | Only bodies can touch one another [Lucretius] |
5711 | The earth is and always has been an insentient being [Lucretius] |
5712 | Particles may have sensation, but eggs turning into chicks suggests otherwise [Lucretius] |
5719 | The mind moves limbs, wakes the body up, changes facial expressions, which involve touch [Lucretius] |
5724 | Lions, foxes and deer have distinct characters because their minds share in their bodies [Lucretius] |
5713 | You needn't be made of laughing particles to laugh, so why not sensation from senseless seeds? [Lucretius] |
6611 | One man's meat is another man's poison [Lucretius] |
5730 | Our bodies weren't created to be used; on the contrary, their creation makes a use possible [Lucretius] |
5726 | The dead are no different from those who were never born [Lucretius] |
5705 | Nature only wants two things: freedom from pain, and pleasure [Lucretius] |
5716 | Nature runs the universe by herself without the aid of gods [Lucretius] |
5704 | There can be no centre in infinity [Lucretius] |
5703 | The universe must be limitless, since there could be nothing outside to limit it [Lucretius] |
5693 | Everything is created and fed by nature from atoms, and they return to atoms in death [Lucretius] |
5701 | If an object is infinitely subdivisible, it will be the same as the whole universe [Lucretius] |
5708 | In downward motion, atoms occasionally swerve slightly for no reason [Lucretius] |
17004 | Nothing can break the binding laws of eternity [Lucretius] |
5696 | If there were no space there could be no movement, or even creation [Lucretius] |
5706 | Atoms move themselves [Lucretius] |
5700 | It is quicker to break things up than to assemble them [Lucretius] |
5698 | We can only sense time by means of movement, or its absence [Lucretius] |
5715 | This earth is very unlikely to be the only one created [Lucretius] |
5694 | Nothing can be created by divine power out of nothing [Lucretius] |
5699 | If matter wasn't everlasting, everything would have disappeared by now [Lucretius] |
5707 | The universe can't have been created by gods, because it is too imperfect [Lucretius] |
5710 | Gods are tranquil and aloof, and have no need of or interest in us [Lucretius] |
5731 | Why does Jupiter never hurl lightning from a blue sky? [Lucretius] |
5722 | For a separated spirit to remain sentient it would need sense organs attached to it [Lucretius] |
5725 | An immortal mind couldn't work harmoniously with a mortal body [Lucretius] |
5720 | Spirit is mortal [Lucretius] |
5721 | The mind is very small smooth particles, which evaporate at death [Lucretius] |
5723 | If spirit is immortal and enters us at birth, why don't we remember a previous existence? [Lucretius] |