54 ideas
4642 | No fact can be real and no proposition true unless there is a Sufficient Reason (even if we can't know it) [Leibniz] |
2115 | Everything in the universe is interconnected, so potentially a mind could know everything [Leibniz] |
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] |
2111 | Falsehood involves a contradiction, and truth is contradictory of falsehood [Leibniz] |
18168 | 'Propositional functions' are propositions with a variable as subject or predicate [Maddy] |
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] |
7644 | The monad idea incomprehensibly spiritualises matter, instead of materialising soul [La Mettrie on Leibniz] |
11857 | He replaced Aristotelian continuants with monads [Leibniz, by Wiggins] |
7843 | Is a drop of urine really an infinity of thinking monads? [Voltaire on Leibniz] |
12751 | It is unclear in 'Monadology' how extended bodies relate to mind-like monads. [Garber on Leibniz] |
19363 | Changes in a monad come from an internal principle, and the diversity within its substance [Leibniz] |
19352 | A 'monad' has basic perception and appetite; a 'soul' has distinct perception and memory [Leibniz] |
18205 | The theoretical indispensability of atoms did not at first convince scientists that they were real [Maddy] |
14193 | 'Substance theorists' take modal properties as primitive, without structure, just falling under a sortal [Paul,LA] |
14195 | If an object's sort determines its properties, we need to ask what determines its sort [Paul,LA] |
14196 | Substance essentialism says an object is multiple, as falling under various different sortals [Paul,LA] |
7931 | If a substance is just a thing that has properties, it seems to be a characterless non-entity [Leibniz, by Macdonald,C] |
14198 | Absolutely unrestricted qualitative composition would allow things with incompatible properties [Paul,LA] |
14190 | Deep essentialist objects have intrinsic properties that fix their nature; the shallow version makes it contextual [Paul,LA] |
14191 | Deep essentialists say essences constrain how things could change; modal profiles fix natures [Paul,LA] |
14192 | Essentialism must deal with charges of arbitrariness, and failure to reduce de re modality [Paul,LA] |
14197 | An object's modal properties don't determine its possibilities [Paul,LA] |
17554 | There must be some internal difference between any two beings in nature [Leibniz] |
2112 | Truths of reason are known by analysis, and are necessary; facts are contingent, and their opposites possible [Leibniz] |
14189 | 'Modal realists' believe in many concrete worlds, 'actualists' in just this world, 'ersatzists' in abstract other worlds [Paul,LA] |
9344 | Mathematical analysis ends in primitive principles, which cannot be and need not be demonstrated [Leibniz] |
2110 | We all expect the sun to rise tomorrow by experience, but astronomers expect it by reason [Leibniz] |
2109 | Increase a conscious machine to the size of a mill - you still won't see perceptions in it [Leibniz] |
18206 | Science idealises the earth's surface, the oceans, continuities, and liquids [Maddy] |
19362 | We know the 'I' and its contents by abstraction from awareness of necessary truths [Leibniz] |
12707 | The true elements are atomic monads [Leibniz] |
2114 | This is the most perfect possible universe, in its combination of variety with order [Leibniz] |
2113 | God alone (the Necessary Being) has the privilege that He must exist if He is possible [Leibniz] |