91 ideas
5196 | Philosophy is a department of logic [Ayer] |
5189 | Philosophers should abandon speculation, as philosophy is wholly critical [Ayer] |
7919 | Humeans rejected the a priori synthetic, and so rejected even Kantian metaphysics [Ayer, by Macdonald,C] |
5195 | Critics say analysis can only show the parts, and not their distinctive configuration [Ayer] |
5179 | Philosophy deals with the questions that scientists do not wish to handle [Ayer] |
22353 | One view says objectivity is making a successful claim which captures the facts [Reiss/Sprenger] |
22356 | An absolute scientific picture of reality must not involve sense experience, which is perspectival [Reiss/Sprenger] |
22359 | Topic and application involve values, but can evidence and theory choice avoid them? [Reiss/Sprenger] |
22360 | The Value-Free Ideal in science avoids contextual values, but embraces epistemic values [Reiss/Sprenger] |
22362 | Value-free science needs impartial evaluation, theories asserting facts, and right motivation [Reiss/Sprenger] |
22364 | Thermometers depend on the substance used, and none of them are perfect [Reiss/Sprenger] |
4749 | We cannot analyse the concept of 'truth', because it is simply a mark that a sentence is asserted [Ayer] |
15901 | Trying to represent curves, we study arbitrary functions, leading to the ordinals, which produces set theory [Cantor, by Lavine] |
13444 | Cantor's Theorem: for any set x, its power set P(x) has more members than x [Cantor, by Hart,WD] |
18098 | Cantor proved that all sets have more subsets than they have members [Cantor, by Bostock] |
15505 | If a set is 'a many thought of as one', beginners should protest against singleton sets [Cantor, by Lewis] |
10701 | Cantor showed that supposed contradictions in infinity were just a lack of clarity [Cantor, by Potter] |
10865 | The continuum is the powerset of the integers, which moves up a level [Cantor, by Clegg] |
13016 | The Axiom of Union dates from 1899, and seems fairly obvious [Cantor, by Maddy] |
14199 | Cantor's sets were just collections, but Dedekind's were containers [Cantor, by Oliver/Smiley] |
10082 | There are infinite sets that are not enumerable [Cantor, by Smith,P] |
13483 | Cantor's Paradox: the power set of the universe must be bigger than the universe, yet a subset of it [Cantor, by Hart,WD] |
8710 | The powerset of all the cardinal numbers is required to be greater than itself [Cantor, by Friend] |
15910 | Cantor named the third realm between the finite and the Absolute the 'transfinite' [Cantor, by Lavine] |
15905 | Cantor proved the points on a plane are in one-to-one correspondence to the points on a line [Cantor, by Lavine] |
9983 | Cantor took the ordinal numbers to be primary [Cantor, by Tait] |
17798 | Cantor presented the totality of natural numbers as finite, not infinite [Cantor, by Mayberry] |
9971 | Cantor introduced the distinction between cardinals and ordinals [Cantor, by Tait] |
9892 | Cantor showed that ordinals are more basic than cardinals [Cantor, by Dummett] |
14136 | A cardinal is an abstraction, from the nature of a set's elements, and from their order [Cantor] |
15906 | Cantor tried to prove points on a line matched naturals or reals - but nothing in between [Cantor, by Lavine] |
11015 | Cantor's diagonal argument proved you can't list all decimal numbers between 0 and 1 [Cantor, by Read] |
15903 | A real is associated with an infinite set of infinite Cauchy sequences of rationals [Cantor, by Lavine] |
18251 | Irrational numbers are the limits of Cauchy sequences of rational numbers [Cantor, by Lavine] |
15902 | Irrationals and the Dedekind Cut implied infinite classes, but they seemed to have logical difficulties [Cantor, by Lavine] |
15908 | It was Cantor's diagonal argument which revealed infinities greater than that of the real numbers [Cantor, by Lavine] |
13464 | Cantor proposes that there won't be a potential infinity if there is no actual infinity [Cantor, by Hart,WD] |
10112 | The naturals won't map onto the reals, so there are different sizes of infinity [Cantor, by George/Velleman] |
8733 | The Continuum Hypothesis says there are no sets between the natural numbers and reals [Cantor, by Shapiro] |
17889 | CH: An infinite set of reals corresponds 1-1 either to the naturals or to the reals [Cantor, by Koellner] |
13447 | Cantor: there is no size between naturals and reals, or between a set and its power set [Cantor, by Hart,WD] |
10883 | Cantor's Continuum Hypothesis says there is a gap between the natural and the real numbers [Cantor, by Horsten] |
13528 | Continuum Hypothesis: there are no sets between N and P(N) [Cantor, by Wolf,RS] |
9555 | Continuum Hypothesis: no cardinal greater than aleph-null but less than cardinality of the continuum [Cantor, by Chihara] |
18174 | Cantor extended ordinals into the transfinite, and they can thus measure infinite cardinalities [Cantor, by Maddy] |
15893 | Cantor's theory concerns collections which can be counted, using the ordinals [Cantor, by Lavine] |
18173 | Cardinality strictly concerns one-one correspondence, to test infinite sameness of size [Cantor, by Maddy] |
10232 | Property extensions outstrip objects, so shortage of objects caused the Caesar problem [Cantor, by Shapiro] |
18176 | Pure mathematics is pure set theory [Cantor] |
8631 | Cantor says that maths originates only by abstraction from objects [Cantor, by Frege] |
5202 | Maths and logic are true universally because they are analytic or tautological [Ayer] |
6523 | Positivists regard ontology as either meaningless or stipulated [Ayer, by Robinson,H] |
5183 | Only tautologies can be certain; other propositions can only be probable [Ayer] |
6525 | Logical positivists could never give the sense-data equivalent of 'there is a table next door' [Robinson,H on Ayer] |
5170 | Material things are constructions from actual and possible occurrences of sense-contents [Ayer] |
5198 | We could verify 'a thing can't be in two places at once' by destroying one of the things [Ierubino on Ayer] |
2619 | Whether geometry can be applied to reality is an empirical question outside of geometry [Ayer] |
5197 | By changing definitions we could make 'a thing can't be in two places at once' a contradiction [Ayer] |
5204 | To say that a proposition is true a priori is to say that it is a tautology [Ayer] |
6524 | Positivists prefer sense-data to objects, because the vocabulary covers both illusions and perceptions [Ayer, by Robinson,H] |
5193 | Causal and representative theories of perception are wrong as they refer to unobservables [Ayer] |
5200 | The main claim of rationalism is that thought is an independent source of knowledge [Ayer] |
4729 | Empiricism lacked a decent account of the a priori, until Ayer said it was entirely analytic [O'Grady on Ayer] |
5180 | All propositions (especially 'metaphysics') must begin with the senses [Ayer] |
5169 | My empiricism logically distinguishes analytic and synthetic propositions, and metaphysical verbiage [Ayer] |
5185 | It is further sense-experience which informs us of the mistakes that arise out of sense-experience [Ayer] |
5199 | Empiricism, it is said, cannot account for our knowledge of necessary truths [Ayer] |
22357 | The 'experimenter's regress' says success needs reliability, which is only tested by success [Reiss/Sprenger] |
5190 | The induction problem is to prove generalisations about the future based on the past [Ayer] |
5191 | We can't use the uniformity of nature to prove induction, as that would be circular [Ayer] |
22365 | The Bayesian approach is explicitly subjective about probabilities [Reiss/Sprenger] |
5177 | Other minds are 'metaphysical' objects, because I can never observe their experiences [Ayer] |
5178 | A conscious object is by definition one that behaves in a certain way, so behaviour proves consciousness [Ayer] |
5172 | If the self is meaningful, it must be constructed from sense-experiences [Ayer] |
5173 | Two experiences belong to one self if their contents belong with one body [Ayer] |
5176 | Empiricists can define personal identity as bodily identity, which consists of sense-contents [Ayer] |
5171 | The supposed 'gulf' between mind and matter is based on the senseless concept of 'substances' [Ayer] |
8715 | Infinities expand the bounds of the conceivable; we explore concepts to explore conceivability [Cantor, by Friend] |
13454 | Cantor says (vaguely) that we abstract numbers from equal sized sets [Hart,WD on Cantor] |
5181 | A sentence is factually significant to someone if they know how to verify its proposition [Ayer] |
5184 | Factual propositions imply (in conjunction with a few other premises) possible experiences [Ayer] |
5186 | Tautologies and empirical hypotheses form the entire class of significant propositions [Ayer] |
5205 | Moral intuition is worthless if there is no criterion to decide between intuitions [Ayer] |
23725 | Ayer defends the emotivist version of expressivism [Ayer, by Smith,M] |
5206 | To say an act is wrong makes no further statement about it, but merely expresses disapproval [Ayer] |
10863 | Cantor proved that three dimensions have the same number of points as one dimension [Cantor, by Clegg] |
13465 | Only God is absolutely infinite [Cantor, by Hart,WD] |
5208 | A person with non-empirical attributes is unintelligible. [Ayer] |
5187 | When we ascribe an attribute to a thing, we covertly assert that it exists [Ayer] |
5207 | If theism is non-sensical, then so is atheism. [Ayer] |
5209 | The 'truths' expressed by theists are not literally significant [Ayer] |