102 ideas
13466 | We are all post-Kantians, because he set the current agenda for philosophy [Hart,WD] |
6928 | Only that which can be an object of religion is an object of philosophy [Feuerbach] |
13477 | The problems are the monuments of philosophy [Hart,WD] |
6918 | Philosophy should not focus on names, but on the determined nature of things [Feuerbach] |
13515 | To study abstract problems, some knowledge of set theory is essential [Hart,WD] |
6904 | Modern philosophy begins with Descartes' abstraction from sensation and matter [Feuerbach] |
6931 | Empiricism is right about ideas, but forgets man himself as one of our objects [Feuerbach] |
6933 | The laws of reality are also the laws of thought [Feuerbach] |
9641 | Definitions should be replaceable by primitives, and should not be creative [Brown,JR] |
13469 | Tarski showed how we could have a correspondence theory of truth, without using 'facts' [Hart,WD] |
13504 | Truth for sentences is satisfaction of formulae; for sentences, either all sequences satisfy it (true) or none do [Hart,WD] |
13503 | A first-order language has an infinity of T-sentences, which cannot add up to a definition of truth [Hart,WD] |
13500 | Conditional Proof: infer a conditional, if the consequent can be deduced from the antecedent [Hart,WD] |
13502 | ∃y... is read as 'There exists an individual, call it y, such that...', and not 'There exists a y such that...' [Hart,WD] |
13456 | Set theory articulates the concept of order (through relations) [Hart,WD] |
13497 | Nowadays ZFC and NBG are the set theories; types are dead, and NF is only useful for the whole universe [Hart,WD] |
13443 | ∈ relates across layers, while ⊆ relates within layers [Hart,WD] |
13442 | Without the empty set we could not form a∩b without checking that a and b meet [Hart,WD] |
9634 | Set theory says that natural numbers are an actual infinity (to accommodate their powerset) [Brown,JR] |
13493 | In the modern view, foundation is the heart of the way to do set theory [Hart,WD] |
13495 | Foundation Axiom: an nonempty set has a member disjoint from it [Hart,WD] |
13461 | We can choose from finite and evident sets, but not from infinite opaque ones [Hart,WD] |
13462 | With the Axiom of Choice every set can be well-ordered [Hart,WD] |
13516 | If we accept that V=L, it seems to settle all the open questions of set theory [Hart,WD] |
13441 | Naïve set theory has trouble with comprehension, the claim that every predicate has an extension [Hart,WD] |
9613 | Naïve set theory assumed that there is a set for every condition [Brown,JR] |
9615 | Nowadays conditions are only defined on existing sets [Brown,JR] |
13494 | The iterative conception may not be necessary, and may have fixed points or infinitely descending chains [Hart,WD] |
9617 | The 'iterative' view says sets start with the empty set and build up [Brown,JR] |
13457 | A 'partial ordering' is irreflexive and transitive; the sets are ordered, but not the subsets [Hart,WD] |
13458 | A partial ordering becomes 'total' if any two members of its field are comparable [Hart,WD] |
13490 | Von Neumann defines α<β as α∈β [Hart,WD] |
13460 | 'Well-ordering' must have a least member, so it does the natural numbers but not the integers [Hart,WD] |
9642 | A flock of birds is not a set, because a set cannot go anywhere [Brown,JR] |
13481 | Maybe sets should be rethought in terms of the even more basic categories [Hart,WD] |
9605 | If a proposition is false, then its negation is true [Brown,JR] |
13506 | The universal quantifier can't really mean 'all', because there is no universal set [Hart,WD] |
13513 | Models are ways the world might be from a first-order point of view [Hart,WD] |
13505 | Model theory studies how set theory can model sets of sentences [Hart,WD] |
13511 | Model theory is mostly confined to first-order theories [Hart,WD] |
13512 | Modern model theory begins with the proof of Los's Conjecture in 1962 [Hart,WD] |
9649 | Axioms are either self-evident, or stipulations, or fallible attempts [Brown,JR] |
13496 | First-order logic is 'compact': consequences of a set are consequences of a finite subset [Hart,WD] |
13484 | Berry's Paradox: we succeed in referring to a number, with a term which says we can't do that [Hart,WD] |
9638 | Berry's Paradox finds a contradiction in the naming of huge numbers [Brown,JR] |
13482 | The Burali-Forti paradox is a crisis for Cantor's ordinals [Hart,WD] |
13507 | The machinery used to solve the Liar can be rejigged to produce a new Liar [Hart,WD] |
9604 | Mathematics is the only place where we are sure we are right [Brown,JR] |
9622 | 'There are two apples' can be expressed logically, with no mention of numbers [Brown,JR] |
13459 | The less-than relation < well-orders, and partially orders, and totally orders the ordinal numbers [Hart,WD] |
13463 | There are at least as many infinite cardinals as transfinite ordinals (because they will map) [Hart,WD] |
13491 | The axiom of infinity with separation gives a least limit ordinal ω [Hart,WD] |
13492 | Von Neumann's ordinals generalise into the transfinite better, because Zermelo's ω is a singleton [Hart,WD] |
13446 | 19th century arithmetization of analysis isolated the real numbers from geometry [Hart,WD] |
9648 | π is a 'transcendental' number, because it is not the solution of an equation [Brown,JR] |
9621 | Mathematics represents the world through structurally similar models. [Brown,JR] |
13509 | We can establish truths about infinite numbers by means of induction [Hart,WD] |
9646 | There is no limit to how many ways something can be proved in mathematics [Brown,JR] |
9647 | Computers played an essential role in proving the four-colour theorem of maps [Brown,JR] |
13474 | Euclid has a unique parallel, spherical geometry has none, and saddle geometry has several [Hart,WD] |
9643 | Set theory may represent all of mathematics, without actually being mathematics [Brown,JR] |
9644 | When graphs are defined set-theoretically, that won't cover unlabelled graphs [Brown,JR] |
9625 | To see a structure in something, we must already have the idea of the structure [Brown,JR] |
9628 | Sets seem basic to mathematics, but they don't suit structuralism [Brown,JR] |
9606 | The irrationality of root-2 was achieved by intellect, not experience [Brown,JR] |
9612 | There is an infinity of mathematical objects, so they can't be physical [Brown,JR] |
9610 | Numbers are not abstracted from particulars, because each number is a particular [Brown,JR] |
9620 | Empiricists base numbers on objects, Platonists base them on properties [Brown,JR] |
13471 | Mathematics makes existence claims, but philosophers usually say those are never analytic [Hart,WD] |
9630 | The most brilliant formalist was Hilbert [Brown,JR] |
9629 | For nomalists there are no numbers, only numerals [Brown,JR] |
9639 | Does some mathematics depend entirely on notation? [Brown,JR] |
9608 | There are no constructions for many highly desirable results in mathematics [Brown,JR] |
9645 | Constructivists say p has no value, if the value depends on Goldbach's Conjecture [Brown,JR] |
6919 | Absolute thought remains in another world from being [Feuerbach] |
19457 | Being is what is undetermined, and hence indistinguishable [Feuerbach] |
6920 | Being posits essence, and my essence is my being [Feuerbach] |
6921 | Particularity belongs to being, whereas generality belongs to thought [Feuerbach] |
6926 | The only true being is of the senses, perception, feeling and love [Feuerbach] |
9619 | David's 'Napoleon' is about something concrete and something abstract [Brown,JR] |
13488 | Mass words do not have plurals, or numerical adjectives, or use 'fewer' [Hart,WD] |
6908 | Consciousness is absolute reality, and everything exists through consciousness [Feuerbach] |
6932 | Ideas arise through communication, and reason is reached through community [Feuerbach] |
13480 | Fregean self-evidence is an intrinsic property of basic truths, rules and definitions [Hart,WD] |
13476 | The failure of key assumptions in geometry, mereology and set theory throw doubt on the a priori [Hart,WD] |
6935 | In man the lowest senses of smell and taste elevate themselves to intellectual acts [Feuerbach] |
13475 | The Fregean concept of GREEN is a function assigning true to green things, and false to the rest [Hart,WD] |
9611 | 'Abstract' nowadays means outside space and time, not concrete, not physical [Brown,JR] |
6925 | The new philosophy thinks of the concrete in a concrete (not a abstract) manner [Feuerbach] |
9609 | The older sense of 'abstract' is where 'redness' or 'group' is abstracted from particulars [Brown,JR] |
9640 | A term can have not only a sense and a reference, but also a 'computational role' [Brown,JR] |
6924 | Plotinus was ashamed to have a body [Feuerbach] |
6927 | If you love nothing, it doesn't matter whether something exists or not [Feuerbach] |
6934 | Man is not a particular being, like animals, but a universal being [Feuerbach] |
6936 | The essence of man is in community, but with distinct individuals [Feuerbach] |
9635 | Given atomism at one end, and a finite universe at the other, there are no physical infinities [Brown,JR] |
6913 | God's existence cannot be separated from essence and concept, which can only be thought as existing [Feuerbach] |
6903 | If God is only an object for man, then only the essence of man is revealed in God [Feuerbach] |
6923 | God is what man would like to be [Feuerbach] |
6911 | God is for us a mere empty idea, which we fill with our own ego and essence [Feuerbach] |
6902 | Catholicism concerns God in himself, Protestantism what God is for man [Feuerbach] |
6905 | Absolute idealism is the realized divine mind of Leibnizian theism [Feuerbach] |