77 ideas
4767 | Traditionally, rational beliefs are those which are justified by reasons [Psillos] |
15924 | Predicative definitions are acceptable in mathematics if they distinguish objects, rather than creating them? [Zermelo, by Lavine] |
17608 | We take set theory as given, and retain everything valuable, while avoiding contradictions [Zermelo] |
17607 | Set theory investigates number, order and function, showing logical foundations for mathematics [Zermelo] |
10870 | ZFC: Existence, Extension, Specification, Pairing, Unions, Powers, Infinity, Choice [Zermelo, by Clegg] |
13012 | Zermelo published his axioms in 1908, to secure a controversial proof [Zermelo, by Maddy] |
17609 | Set theory can be reduced to a few definitions and seven independent axioms [Zermelo] |
9565 | Zermelo made 'set' and 'member' undefined axioms [Zermelo, by Chihara] |
3339 | For Zermelo's set theory the empty set is zero and the successor of each number is its unit set [Zermelo, by Blackburn] |
17832 | Zermelo showed that the ZF axioms in 1930 were non-categorical [Zermelo, by Hallett,M] |
13017 | Zermelo introduced Pairing in 1930, and it seems fairly obvious [Zermelo, by Maddy] |
13028 | Replacement was added when some advanced theorems seemed to need it [Zermelo, by Maddy] |
13015 | Zermelo used Foundation to block paradox, but then decided that only Separation was needed [Zermelo, by Maddy] |
13020 | The Axiom of Separation requires set generation up to one step back from contradiction [Zermelo, by Maddy] |
13486 | Not every predicate has an extension, but Separation picks the members that satisfy a predicate [Zermelo, by Hart,WD] |
4810 | Valid deduction is monotonic - that is, it remains valid if further premises are added [Psillos] |
17626 | The antinomy of endless advance and of completion is resolved in well-ordered transfinite numbers [Zermelo] |
13487 | In ZF, the Burali-Forti Paradox proves that there is no set of all ordinals [Zermelo, by Hart,WD] |
15897 | Zermelo realised that Choice would facilitate the sort of 'counting' Cantor needed [Zermelo, by Lavine] |
18178 | For Zermelo the successor of n is {n} (rather than n U {n}) [Zermelo, by Maddy] |
13027 | Zermelo believed, and Von Neumann seemed to confirm, that numbers are sets [Zermelo, by Maddy] |
9627 | Different versions of set theory result in different underlying structures for numbers [Zermelo, by Brown,JR] |
458 | Nothing could come out of nothing, and existence could never completely cease [Empedocles] |
4768 | The 'epistemic fallacy' is inferring what does exist from what can be known to exist [Psillos] |
5112 | Empedocles says things are at rest, unless love unites them, or hatred splits them [Empedocles, by Aristotle] |
14933 | Scientific properties are defined by the laws that embody them [Psillos, by Ladyman/Ross] |
17996 | Powers are claimed to be basic because fundamental particles lack internal structure [Psillos] |
13209 | There is no coming-to-be of anything, but only mixing and separating [Empedocles, by Aristotle] |
457 | Substance is not created or destroyed in mortals, but there is only mixing and exchange [Empedocles] |
462 | One vision is produced by both eyes [Empedocles] |
4807 | A good barometer will predict a storm, but not explain it [Psillos] |
4808 | If we say where Mars was two months ago, we offer an explanation without a prediction [Psillos] |
4811 | Induction (unlike deduction) is non-monotonic - it can be invalidated by new premises [Psillos] |
4812 | Explanation is either showing predictability, or showing necessity, or showing causal relations [Psillos] |
4802 | Just citing a cause does not enable us to understand an event; we also need a relevant law [Psillos] |
4804 | The 'covering law model' says only laws can explain the occurrence of single events [Psillos] |
4805 | If laws explain the length of a flagpole's shadow, then the shadow also explains the length of the pole [Psillos] |
4395 | There are non-causal explanations, most typically mathematical explanations [Psillos] |
4806 | An explanation can just be a 'causal story', without laws, as when I knock over some ink [Psillos] |
4404 | Maybe explanation is entirely relative to the interests and presuppositions of the questioner [Psillos] |
4803 | An explanation is the removal of the surprise caused by the event [Psillos] |
4769 | It is hard to analyse causation, if it is presupposed in our theory of the functioning of the mind [Psillos] |
22765 | Wisdom and thought are shared by all things [Empedocles] |
1524 | For Empedocles thinking is almost identical to perception [Empedocles, by Theophrastus] |
17613 | We should judge principles by the science, not science by some fixed principles [Zermelo] |
4770 | Nothing is more usual than to apply to external bodies every internal sensation which they occasion [Psillos] |
552 | Empedocles said good and evil were the basic principles [Empedocles, by Aristotle] |
589 | 'Nature' is just a word invented by people [Empedocles] |
21823 | The principle of 'Friendship' in Empedocles is the One, and is bodiless [Empedocles, by Plotinus] |
2680 | Empedocles said that there are four material elements, and two further creative elements [Empedocles, by Aristotle] |
6002 | Empedocles says bone is water, fire and earth in ratio 2:4:2 [Empedocles, by Inwood] |
13207 | Fire, Water, Air and Earth are elements, being simple as well as homoeomerous [Empedocles, by Aristotle] |
459 | All change is unity through love or division through hate [Empedocles] |
13218 | The elements combine in coming-to-be, but how do the elements themselves come-to-be? [Aristotle on Empedocles] |
13225 | Love and Strife only explain movement if their effects are distinctive [Aristotle on Empedocles] |
460 | If the one Being ever diminishes it would no longer exist, and what could ever increase it? [Empedocles] |
4399 | Causes clearly make a difference, are recipes for events, explain effects, and are evidence [Psillos] |
4400 | Theories of causation are based either on regularity, or on intrinsic relations of properties [Psillos] |
4403 | We can't base our account of causation on explanation, because it is the wrong way round [Psillos] |
4789 | Three divisions of causal theories: generalist/singularist, intrinsic/extrinsic, reductive/non-reductive [Psillos] |
4790 | If causation is 'intrinsic' it depends entirely on the properties and relations of the cause and effect [Psillos] |
4402 | Empiricists tried to reduce causation to explanation, which they reduced to logic-plus-a-law [Psillos] |
4774 | Counterfactual claims about causation imply that it is more than just regular succession [Psillos] |
4793 | "All gold cubes are smaller than one cubic mile" is a true universal generalisation, but not a law [Psillos] |
4397 | Regularity doesn't seem sufficient for causation [Psillos] |
4792 | A Humean view of causation says it is regularities, and causal facts supervene on non-causal facts [Psillos] |
4801 | The regularity of a cock's crow is used to predict dawn, even though it doesn't cause it [Psillos] |
4401 | It is not a law of nature that all the coins in my pocket are euros, though it is a regularity [Psillos] |
4796 | Laws are sets of regularities within a simple and strong coherent system of wider regularities [Psillos] |
4799 | Dispositional essentialism can't explain its key distinction between essential and non-essential properties [Psillos] |
4780 | In some counterfactuals, the counterfactual event happens later than its consequent [Psillos] |
4791 | Counterfactual theories say causes make a difference - if c hadn't occurred, then e wouldn't occur [Psillos] |
5090 | Maybe bodies are designed by accident, and the creatures that don't work are destroyed [Empedocles, by Aristotle] |
466 | God is pure mind permeating the universe [Empedocles] |
461 | God is a pure, solitary, and eternal sphere [Empedocles] |
1719 | In Empedocles' theory God is ignorant because, unlike humans, he doesn't know one of the elements (strife) [Aristotle on Empedocles] |
1522 | It is wretched not to want to think clearly about the gods [Empedocles] |