87 ideas
11051 | Frege's logical approach dominates the analytical tradition [Hanna] |
3870 | The real problem of science is how to choose between possible explanations [Newton-Smith] |
3859 | We do not wish merely to predict, we also want to explain [Newton-Smith] |
3853 | For science to be rational, we must explain scientific change rationally [Newton-Smith] |
3854 | Positivists hold that theoretical terms change, but observation terms don't [Newton-Smith] |
3855 | Critics attack positivist division between theory and observation [Newton-Smith] |
11054 | Scientism says most knowledge comes from the exact sciences [Hanna] |
11070 | 'Denying the antecedent' fallacy: φ→ψ, ¬φ, so ¬ψ [Hanna] |
11071 | 'Affirming the consequent' fallacy: φ→ψ, ψ, so φ [Hanna] |
11088 | We can list at least fourteen informal fallacies [Hanna] |
11059 | Circular arguments are formally valid, though informally inadmissible [Hanna] |
11089 | Formally, composition and division fallacies occur in mereology [Hanna] |
3869 | More truthful theories have greater predictive power [Newton-Smith] |
3861 | Theories generate infinite truths and falsehoods, so they cannot be used to assess probability [Newton-Smith] |
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] |
11058 | Logic is explanatorily and ontologically dependent on rational animals [Hanna] |
11072 | Logic is personal and variable, but it has a universal core [Hanna] |
11061 | Intensional consequence is based on the content of the concepts [Hanna] |
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] |
11063 | Logicism struggles because there is no decent theory of analyticity [Hanna] |
11055 | Supervenience can add covariation, upward dependence, and nomological connection [Hanna] |
11083 | A sentence is necessary if it is true in a set of worlds, and nonfalse in the other worlds [Hanna] |
11086 | Metaphysical necessity can be 'weak' (same as logical) and 'strong' (based on essences) [Hanna] |
11084 | Logical necessity is truth in all logically possible worlds, because of laws and concepts [Hanna] |
11085 | Nomological necessity is truth in all logically possible worlds with our laws [Hanna] |
3867 | De re necessity arises from the way the world is [Newton-Smith] |
3872 | We must assess the truth of beliefs in identifying them [Newton-Smith] |
11077 | Intuition includes apriority, clarity, modality, authority, fallibility and no inferences [Hanna] |
11080 | Intuition is more like memory, imagination or understanding, than like perception [Hanna] |
11078 | Intuition is only outside the 'space of reasons' if all reasons are inferential [Hanna] |
3857 | Defeat relativism by emphasising truth and reference, not meaning [Newton-Smith] |
3858 | A full understanding of 'yellow' involves some theory [Newton-Smith] |
3862 | All theories contain anomalies, and so are falsified! [Newton-Smith] |
3863 | The anomaly of Uranus didn't destroy Newton's mechanics - it led to Neptune's discovery [Newton-Smith] |
3864 | Anomalies are judged against rival theories, and support for the current theory [Newton-Smith] |
3865 | Why should it matter whether or not a theory is scientific? [Newton-Smith] |
3866 | If theories are really incommensurable, we could believe them all [Newton-Smith] |
11053 | Explanatory reduction is stronger than ontological reduction [Hanna] |
11081 | Imagination grasps abstracta, generates images, and has its own correctness conditions [Hanna] |
11082 | Should we take the 'depictivist' or the 'descriptivist/propositionalist' view of mental imagery? [Hanna] |
11067 | Rational animals have a normative concept of necessity [Hanna] |
11068 | One tradition says talking is the essence of rationality; the other says the essence is logic [Hanna] |
11047 | Hegelian holistic rationality is the capacity to seek coherence [Hanna] |
11048 | Humean Instrumental rationality is the capacity to seek contingent truths [Hanna] |
11046 | Kantian principled rationality is recognition of a priori universal truths [Hanna] |
11045 | Most psychologists are now cognitivists [Hanna] |
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] |
3871 | Explaining an action is showing that it is rational [Newton-Smith] |
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] |