109 ideas
16325 | Analysis rests on natural language, but its ideal is a framework which revises language [Halbach] |
224 | When questions are doubtful we should concentrate not on objects but on ideas of the intellect [Plato] |
232 | Opposites are as unlike as possible [Plato] |
8937 | Plato's 'Parmenides' is the greatest artistic achievement of the ancient dialectic [Hegel on Plato] |
16292 | An explicit definition enables the elimination of what is defined [Halbach] |
16307 | Don't trust analogies; they are no more than a guideline [Halbach] |
16330 | Truth-value 'gluts' allow two truth values together; 'gaps' give a partial conception of truth [Halbach] |
16339 | Truth axioms prove objects exist, so truth doesn't seem to be a logical notion [Halbach] |
16324 | Any definition of truth requires a metalanguage [Halbach] |
16293 | Traditional definitions of truth often make it more obscure, rather than less [Halbach] |
16301 | If people have big doubts about truth, a definition might give it more credibility [Halbach] |
16297 | Semantic theories avoid Tarski's Theorem by sticking to a sublanguage [Halbach] |
16337 | Disquotational truth theories are short of deductive power [Halbach] |
16322 | CT proves PA consistent, which PA can't do on its own, so CT is not conservative over PA [Halbach] |
16294 | Axiomatic truth doesn't presuppose a truth-definition, though it could admit it at a later stage [Halbach] |
16326 | The main semantic theories of truth are Kripke's theory, and revisions semantics [Halbach] |
16311 | To axiomatise Tarski's truth definition, we need a binary predicate for his 'satisfaction' [Halbach] |
16318 | Compositional Truth CT has the truth of a sentence depending of the semantic values of its constituents [Halbach] |
16299 | Gödel numbering means a theory of truth can use Peano Arithmetic as its base theory [Halbach] |
16340 | Truth axioms need a base theory, because that is where truth issues arise [Halbach] |
16305 | We know a complete axiomatisation of truth is not feasible [Halbach] |
16313 | A theory is 'conservative' if it adds no new theorems to its base theory [Halbach, by PG] |
16315 | The Tarski Biconditional theory TB is Peano Arithmetic, plus truth, plus all Tarski bi-conditionals [Halbach] |
16314 | Theories of truth are 'typed' (truth can't apply to sentences containing 'true'), or 'type-free' [Halbach] |
16327 | Friedman-Sheard is type-free Compositional Truth, with two inference rules for truth [Halbach] |
16329 | Kripke-Feferman theory KF axiomatises Kripke fixed-points, with Strong Kleene logic with gluts [Halbach] |
16331 | The KF is much stronger deductively than FS, which relies on classical truth [Halbach] |
16332 | The KF theory is useful, but it is not a theory containing its own truth predicate [Halbach] |
16320 | Some say deflationism is axioms which are conservative over the base theory [Halbach] |
16338 | Deflationism says truth is a disquotation device to express generalisations, adding no new knowledge [Halbach] |
16317 | The main problem for deflationists is they can express generalisations, but not prove them [Halbach] |
16316 | Deflationists say truth is just for expressing infinite conjunctions or generalisations [Halbach] |
16319 | Compositional Truth CT proves generalisations, so is preferred in discussions of deflationism [Halbach] |
16335 | In Strong Kleene logic a disjunction just needs one disjunct to be true [Halbach] |
16334 | In Weak Kleene logic there are 'gaps', neither true nor false if one component lacks a truth value [Halbach] |
16309 | Every attempt at formal rigour uses some set theory [Halbach] |
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] |
16333 | The underestimated costs of giving up classical logic are found in mathematical reasoning [Halbach] |
18168 | 'Propositional functions' are propositions with a variable as subject or predicate [Maddy] |
16310 | A theory is some formulae and all of their consequences [Halbach] |
16342 | You cannot just say all of Peano arithmetic is true, as 'true' isn't part of the system [Halbach] |
16341 | Normally we only endorse a theory if we believe it to be sound [Halbach] |
16344 | Soundness must involve truth; the soundness of PA certainly needs it [Halbach] |
16347 | Many new paradoxes may await us when we study interactions between frameworks [Halbach] |
13986 | Plato found antinomies in ideas, Kant in space and time, and Bradley in relations [Plato, by Ryle] |
14150 | Plato's 'Parmenides' is perhaps the best collection of antinomies ever made [Russell on Plato] |
16336 | The liar paradox applies truth to a negated truth (but the conditional will serve equally) [Halbach] |
18190 | Completed infinities resulted from giving foundations to calculus [Maddy] |
18171 | Cantor and Dedekind brought completed infinities into mathematics [Maddy] |
18172 | Infinity has degrees, and large cardinals are the heart of set theory [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] |
18187 | Theorems about limits could only be proved once the real numbers were understood [Maddy] |
16321 | The compactness theorem can prove nonstandard models of PA [Halbach] |
16343 | The global reflection principle seems to express the soundness of Peano Arithmetic [Halbach] |
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] |
18184 | Making set theory foundational to mathematics leads to very fruitful 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] |
16312 | To reduce PA to ZF, we represent the non-negative integers with von Neumann ordinals [Halbach] |
18186 | Identifying geometric points with real numbers revealed the power of set theory [Maddy] |
18188 | The line of rationals has gaps, but set theory provided an ordered continuum [Maddy] |
18163 | Mathematics rests on the logic of proofs, and on the set theoretic axioms [Maddy] |
16150 | One is, so numbers exist, so endless numbers exist, and each one must partake of being [Plato] |
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] |
16308 | Set theory was liberated early from types, and recent truth-theories are exploring type-free [Halbach] |
18167 | We can get arithmetic directly from HP; Law V was used to get HP from the definition of number [Maddy] |
229 | The one was and is and will be and was becoming and is becoming and will become [Plato] |
21821 | Plato's Parmenides has a three-part theory, of Primal One, a One-Many, and a One-and-Many [Plato, by Plotinus] |
16345 | That Peano arithmetic is interpretable in ZF set theory is taken by philosophers as a reduction [Halbach] |
221 | Absolute ideas, such as the Good and the Beautiful, cannot be known by us [Plato] |
18205 | The theoretical indispensability of atoms did not at first convince scientists that they were real [Maddy] |
223 | If you deny that each thing always stays the same, you destroy the possibility of discussion [Plato] |
227 | You must always mean the same thing when you utter the same name [Plato] |
210 | It would be absurd to think there were abstract Forms for vile things like hair, mud and dirt [Plato] |
220 | The concept of a master includes the concept of a slave [Plato] |
211 | If admirable things have Forms, maybe everything else does as well [Plato] |
219 | If absolute ideas existed in us, they would cease to be absolute [Plato] |
228 | Greatness and smallness must exist, to be opposed to one another, and come into being in things [Plato] |
16151 | Plato moves from Forms to a theory of genera and principles in his later work [Plato, by Frede,M] |
218 | Participation is not by means of similarity, so we are looking for some other method of participation [Plato] |
215 | If things partake of ideas, this implies either that everything thinks, or that everything actually is thought [Plato] |
212 | The whole idea of each Form must be found in each thing which participates in it [Plato] |
213 | Each idea is in all its participants at once, just as daytime is a unity but in many separate places at once [Plato] |
216 | If things are made alike by participating in something, that thing will be the absolute idea [Plato] |
217 | Nothing can be like an absolute idea, because a third idea intervenes to make them alike (leading to a regress) [Plato] |
214 | If absolute greatness and great things are seen as the same, another thing appears which makes them seem great [Plato] |
15851 | Parts must belong to a created thing with a distinct form [Plato] |
15846 | In Parmenides, if composition is identity, a whole is nothing more than its parts [Plato, by Harte,V] |
15849 | Plato says only a one has parts, and a many does not [Plato, by Harte,V] |
15850 | Anything which has parts must be one thing, and parts are of a one, not of a many [Plato] |
13259 | It seems that the One must be composed of parts, which contradicts its being one [Plato] |
15847 | Two things relate either as same or different, or part of a whole, or the whole of the part [Plato] |
16346 | Maybe necessity is a predicate, not the usual operator, to make it more like truth [Halbach] |
18206 | Science idealises the earth's surface, the oceans, continuities, and liquids [Maddy] |
16298 | We need propositions to ascribe the same beliefs to people with different languages [Halbach] |
222 | Only a great person can understand the essence of things, and an even greater person can teach it [Plato] |
225 | The unlimited has no shape and is endless [Plato] |
233 | Some things do not partake of the One [Plato] |
2062 | The only movement possible for the One is in space or in alteration [Plato] |
231 | Everything partakes of the One in some way [Plato] |
234 | We couldn't discuss the non-existence of the One without knowledge of it [Plato] |