232 ideas
| 21916 | Philosophers can't be religious, and don't need to be; philosophy is perilous but free [Schopenhauer] |
| 4171 | Philosophy considers only the universal, in nature as everywhere else [Schopenhauer] |
| 4186 | Everyone is conscious of all philosophical truths, but philosophers bring them to conceptual awareness [Schopenhauer] |
| 12171 | Absurdity is incongruity between correct and false points of view [Schopenhauer] |
| 21366 | Metaphysics must understand the world thoroughly, as a principal source of knowledge [Schopenhauer] |
| 21474 | Metaphysics studies the inexplicable ends of explanation [Schopenhauer] |
| 10237 | Coherence is a primitive, intuitive notion, not reduced to something formal [Shapiro] |
| 21918 | Sufficient Reason can't be proved, because all proof presupposes it [Schopenhauer, by Lewis,PB] |
| 4187 | 'There is nothing without a reason why it should be rather than not be' (a generalisation of 'Why?') [Schopenhauer] |
| 10204 | An 'implicit definition' gives a direct description of the relations of an entity [Shapiro] |
| 13634 | Satisfaction is 'truth in a model', which is a model of 'truth' [Shapiro] |
| 13643 | Aristotelian logic is complete [Shapiro] |
| 10206 | Modal operators are usually treated as quantifiers [Shapiro] |
| 13651 | A set is 'transitive' if contains every member of each of its members [Shapiro] |
| 13647 | Choice is essential for proving downward Löwenheim-Skolem [Shapiro] |
| 10208 | Axiom of Choice: some function has a value for every set in a given set [Shapiro] |
| 10252 | The Axiom of Choice seems to license an infinite amount of choosing [Shapiro] |
| 10301 | The axiom of choice is controversial, but it could be replaced [Shapiro] |
| 13631 | Are sets part of logic, or part of mathematics? [Shapiro] |
| 13654 | It is central to the iterative conception that membership is well-founded, with no infinite descending chains [Shapiro] |
| 13640 | Russell's paradox shows that there are classes which are not iterative sets [Shapiro] |
| 13666 | Iterative sets are not Boolean; the complement of an iterative set is not an iterative sets [Shapiro] |
| 13653 | 'Well-ordering' of a set is an irreflexive, transitive, and binary relation with a least element [Shapiro] |
| 10207 | Anti-realists reject set theory [Shapiro] |
| 13627 | There is no 'correct' logic for natural languages [Shapiro] |
| 13642 | Logic is the ideal for learning new propositions on the basis of others [Shapiro] |
| 13668 | Bernays (1918) formulated and proved the completeness of propositional logic [Shapiro] |
| 13669 | Can one develop set theory first, then derive numbers, or are numbers more basic? [Shapiro] |
| 13667 | Skolem and Gödel championed first-order, and Zermelo, Hilbert, and Bernays championed higher-order [Shapiro] |
| 13662 | First-order logic was an afterthought in the development of modern logic [Shapiro] |
| 13624 | The 'triumph' of first-order logic may be related to logicism and the Hilbert programme, which failed [Shapiro] |
| 13660 | Maybe compactness, semantic effectiveness, and the Löwenheim-Skolem properties are desirable [Shapiro] |
| 13673 | The notion of finitude is actually built into first-order languages [Shapiro] |
| 10588 | First-order logic is Complete, and Compact, with the Löwenheim-Skolem Theorems [Shapiro] |
| 15944 | Second-order logic is better than set theory, since it only adds relations and operations, and nothing else [Shapiro, by Lavine] |
| 13629 | Broad standard semantics, or Henkin semantics with a subclass, or many-sorted first-order semantics? [Shapiro] |
| 13650 | Henkin semantics has separate variables ranging over the relations and over the functions [Shapiro] |
| 10298 | Some say that second-order logic is mathematics, not logic [Shapiro] |
| 13645 | In standard semantics for second-order logic, a single domain fixes the ranges for the variables [Shapiro] |
| 13649 | Completeness, Compactness and Löwenheim-Skolem fail in second-order standard semantics [Shapiro] |
| 10299 | If the aim of logic is to codify inferences, second-order logic is useless [Shapiro] |
| 10300 | Logical consequence can be defined in terms of the logical terminology [Shapiro] |
| 10259 | The two standard explanations of consequence are semantic (in models) and deductive [Shapiro] |
| 13626 | Semantic consequence is ineffective in second-order logic [Shapiro] |
| 13637 | If a logic is incomplete, its semantic consequence relation is not effective [Shapiro] |
| 10257 | Intuitionism only sanctions modus ponens if all three components are proved [Shapiro] |
| 10253 | Either logic determines objects, or objects determine logic, or they are separate [Shapiro] |
| 10251 | The law of excluded middle might be seen as a principle of omniscience [Shapiro] |
| 8729 | Intuitionists deny excluded middle, because it is committed to transcendent truth or objects [Shapiro] |
| 13632 | Finding the logical form of a sentence is difficult, and there are no criteria of correctness [Shapiro] |
| 10212 | Classical connectives differ from their ordinary language counterparts; '∧' is timeless, unlike 'and' [Shapiro] |
| 10209 | A function is just an arbitrary correspondence between collections [Shapiro] |
| 13674 | We might reduce ontology by using truth of sentences and terms, instead of using objects satisfying models [Shapiro] |
| 10290 | Second-order variables also range over properties, sets, relations or functions [Shapiro] |
| 10268 | Maybe plural quantifiers should be understood in terms of classes or sets [Shapiro] |
| 10235 | A sentence is 'satisfiable' if it has a model [Shapiro] |
| 13633 | 'Satisfaction' is a function from models, assignments, and formulas to {true,false} [Shapiro] |
| 10240 | Model theory deals with relations, reference and extensions [Shapiro] |
| 13644 | Semantics for models uses set-theory [Shapiro] |
| 10239 | The central notion of model theory is the relation of 'satisfaction' [Shapiro] |
| 13636 | An axiomatization is 'categorical' if its models are isomorphic, so there is really only one interpretation [Shapiro] |
| 13670 | Categoricity can't be reached in a first-order language [Shapiro] |
| 10214 | Theory ontology is never complete, but is only determined 'up to isomorphism' [Shapiro] |
| 10238 | The set-theoretical hierarchy contains as many isomorphism types as possible [Shapiro] |
| 13648 | The Löwenheim-Skolem theorems show an explosion of infinite models, so 1st-order is useless for infinity [Shapiro] |
| 13675 | Substitutional semantics only has countably many terms, so Upward Löwenheim-Skolem trivially fails [Shapiro] |
| 10292 | Downward Löwenheim-Skolem: if there's an infinite model, there is a countable model [Shapiro] |
| 10234 | Any theory with an infinite model has a model of every infinite cardinality [Shapiro] |
| 10590 | Up Löwenheim-Skolem: if natural numbers satisfy wffs, then an infinite domain satisfies them [Shapiro] |
| 10296 | The Löwenheim-Skolem Theorems fail for second-order languages with standard semantics [Shapiro] |
| 13658 | Downward Löwenheim-Skolem: each satisfiable countable set always has countable models [Shapiro] |
| 13659 | Upward Löwenheim-Skolem: each infinite model has infinite models of all sizes [Shapiro] |
| 10297 | The Löwenheim-Skolem theorem seems to be a defect of first-order logic [Shapiro] |
| 13635 | 'Weakly sound' if every theorem is a logical truth; 'sound' if every deduction is a semantic consequence [Shapiro] |
| 13628 | We can live well without completeness in logic [Shapiro] |
| 13630 | Non-compactness is a strength of second-order logic, enabling characterisation of infinite structures [Shapiro] |
| 13646 | Compactness is derived from soundness and completeness [Shapiro] |
| 13661 | A language is 'semantically effective' if its logical truths are recursively enumerable [Shapiro] |
| 10201 | Virtually all of mathematics can be modeled in set theory [Shapiro] |
| 13641 | Complex numbers can be defined as reals, which are defined as rationals, then integers, then naturals [Shapiro] |
| 8763 | The number 3 is presumably identical as a natural, an integer, a rational, a real, and complex [Shapiro] |
| 13676 | Only higher-order languages can specify that 0,1,2,... are all the natural numbers that there are [Shapiro] |
| 13677 | Natural numbers are the finite ordinals, and integers are equivalence classes of pairs of finite ordinals [Shapiro] |
| 10213 | Real numbers are thought of as either Cauchy sequences or Dedekind cuts [Shapiro] |
| 18243 | Understanding the real-number structure is knowing usage of the axiomatic language of analysis [Shapiro] |
| 18249 | Cauchy gave a formal definition of a converging sequence. [Shapiro] |
| 18245 | Cuts are made by the smallest upper or largest lower number, some of them not rational [Shapiro] |
| 13652 | The 'continuum' is the cardinality of the powerset of a denumerably infinite set [Shapiro] |
| 10236 | There is no grounding for mathematics that is more secure than mathematics [Shapiro] |
| 8764 | Categories are the best foundation for mathematics [Shapiro] |
| 10256 | For intuitionists, proof is inherently informal [Shapiro] |
| 13657 | First-order arithmetic can't even represent basic number theory [Shapiro] |
| 10202 | Natural numbers just need an initial object, successors, and an induction principle [Shapiro] |
| 10294 | Second-order logic has the expressive power for mathematics, but an unworkable model theory [Shapiro] |
| 10205 | Mathematics originally concerned the continuous (geometry) and the discrete (arithmetic) [Shapiro] |
| 8762 | Two definitions of 3 in terms of sets disagree over whether 1 is a member of 3 [Shapiro] |
| 13656 | Some sets of natural numbers are definable in set-theory but not in arithmetic [Shapiro] |
| 10222 | Mathematical foundations may not be sets; categories are a popular rival [Shapiro] |
| 10218 | Baseball positions and chess pieces depend entirely on context [Shapiro] |
| 10224 | The even numbers have the natural-number structure, with 6 playing the role of 3 [Shapiro] |
| 10228 | Could infinite structures be apprehended by pattern recognition? [Shapiro] |
| 10230 | The 4-pattern is the structure common to all collections of four objects [Shapiro] |
| 10249 | The main mathematical structures are algebraic, ordered, and topological [Shapiro] |
| 10273 | Some structures are exemplified by both abstract and concrete [Shapiro] |
| 10276 | Mathematical structures are defined by axioms, or in set theory [Shapiro] |
| 8760 | Numbers do not exist independently; the essence of a number is its relations to other numbers [Shapiro] |
| 8761 | A 'system' is related objects; a 'pattern' or 'structure' abstracts the pure relations from them [Shapiro] |
| 10270 | The main versions of structuralism are all definitionally equivalent [Shapiro] |
| 10221 | Is there is no more to structures than the systems that exemplify them? [Shapiro] |
| 10248 | Number statements are generalizations about number sequences, and are bound variables [Shapiro] |
| 10220 | Because one structure exemplifies several systems, a structure is a one-over-many [Shapiro] |
| 10223 | There is no 'structure of all structures', just as there is no set of all sets [Shapiro] |
| 8703 | Shapiro's structuralism says model theory (comparing structures) is the essence of mathematics [Shapiro, by Friend] |
| 10274 | Does someone using small numbers really need to know the infinite structure of arithmetic? [Shapiro] |
| 10200 | We distinguish realism 'in ontology' (for objects), and 'in truth-value' (for being either true or false) [Shapiro] |
| 10210 | If mathematical objects are accepted, then a number of standard principles will follow [Shapiro] |
| 10215 | Platonists claim we can state the essence of a number without reference to the others [Shapiro] |
| 10233 | Platonism must accept that the Peano Axioms could all be false [Shapiro] |
| 10244 | Intuition is an outright hindrance to five-dimensional geometry [Shapiro] |
| 10280 | A stone is a position in some pattern, and can be viewed as an object, or as a location [Shapiro] |
| 13664 | Logicism is distinctive in seeking a universal language, and denying that logic is a series of abstractions [Shapiro] |
| 13625 | Mathematics and logic have no border, and logic must involve mathematics and its ontology [Shapiro] |
| 8744 | Logicism seems to be a non-starter if (as is widely held) logic has no ontology of its own [Shapiro] |
| 8749 | Term Formalism says mathematics is just about symbols - but real numbers have no names [Shapiro] |
| 8750 | Game Formalism is just a matter of rules, like chess - but then why is it useful in science? [Shapiro] |
| 8752 | Deductivism says mathematics is logical consequences of uninterpreted axioms [Shapiro] |
| 10254 | Can the ideal constructor also destroy objects? [Shapiro] |
| 10255 | Presumably nothing can block a possible dynamic operation? [Shapiro] |
| 8753 | Critics resent the way intuitionism cripples mathematics, but it allows new important distinctions [Shapiro] |
| 8731 | Conceptualist are just realists or idealist or nominalists, depending on their view of concepts [Shapiro] |
| 13663 | Some reject formal properties if they are not defined, or defined impredicatively [Shapiro] |
| 8730 | 'Impredicative' definitions refer to the thing being described [Shapiro] |
| 10279 | Can we discover whether a deck is fifty-two cards, or a person is time-slices or molecules? [Shapiro] |
| 4168 | Matter and intellect are inseparable correlatives which only exist relatively, and for each other [Schopenhauer] |
| 10227 | The abstract/concrete boundary now seems blurred, and would need a defence [Shapiro] |
| 10226 | Mathematicians regard arithmetic as concrete, and group theory as abstract [Shapiro] |
| 21470 | For me the objective thing-in-itself is the will [Schopenhauer] |
| 21926 | Schopenhauer, unlike other idealists, says reality is irrational [Schopenhauer, by Lewis,PB] |
| 4167 | The knowing subject and the crude matter of the world are both in themselves unknowable [Schopenhauer] |
| 10262 | Fictionalism eschews the abstract, but it still needs the possible (without model theory) [Shapiro] |
| 10277 | Structuralism blurs the distinction between mathematical and ordinary objects [Shapiro] |
| 21920 | No need for a priori categories, since sufficient reason shows the interrelations [Schopenhauer, by Lewis,PB] |
| 13638 | Properties are often seen as intensional; equiangular and equilateral are different, despite identity of objects [Shapiro] |
| 10591 | Logicians use 'property' and 'set' interchangeably, with little hanging on it [Shapiro] |
| 10272 | The notion of 'object' is at least partially structural and mathematical [Shapiro] |
| 10275 | A blurry border is still a border [Shapiro] |
| 21362 | Necessity is physical, logical, mathematical or moral [Schopenhauer, by Janaway] |
| 10258 | Logical modalities may be acceptable, because they are reducible to satisfaction in models [Shapiro] |
| 4192 | All necessity arises from causation, which is conditioned; there is no absolute or unconditioned necessity [Schopenhauer] |
| 10266 | Why does the 'myth' of possible worlds produce correct modal logic? [Shapiro] |
| 4190 | All understanding is an immediate apprehension of the causal relation [Schopenhauer] |
| 21479 | Knowledge is not power! Ignorant people possess supreme authority [Schopenhauer] |
| 4165 | Descartes found the true beginning of philosophy with the Cogito, in the consciousness of the individual [Schopenhauer] |
| 21361 | For Schopenhauer, material things would not exist without the mind [Schopenhauer, by Janaway] |
| 21923 | Schopenhauer can't use force/energy instead of 'will', because he is not a materialist [Lewis,PB on Schopenhauer] |
| 4162 | The world only exists in relation to something else, as an idea of the one who conceives it [Schopenhauer] |
| 21922 | We know reality because we know our own bodies and actions [Schopenhauer] |
| 21913 | Kant rightly separates appearance and thing-in-itself [Schopenhauer] |
| 21919 | Object for a subject and representation are the same thing [Schopenhauer] |
| 21476 | A priori propositions are those we could never be seriously motivated to challenge [Schopenhauer] |
| 4164 | Direct feeling of the senses are merely data; perception of the world comes with understanding causes [Schopenhauer] |
| 4163 | All perception is intellectual [Schopenhauer] |
| 8725 | Rationalism tries to apply mathematical methodology to all of knowledge [Shapiro] |
| 19508 | Contextualism needs a semantics for knowledge sentences that are partly indexical [Schiffer,S] |
| 19509 | The indexical aspect of contextual knowledge might be hidden, or it might be in what 'know' means [Schiffer,S] |
| 21473 | All knowledge and explanation rests on the inexplicable [Schopenhauer] |
| 21917 | The four explanations: objects by causes, concepts by ground, maths by spacetime, ethics by motive [Schopenhauer, by Lewis,PB] |
| 4166 | A consciousness without an object is no consciousness [Schopenhauer] |
| 21369 | We have hidden and unadmitted desires and fears, suppressed because of vanity [Schopenhauer] |
| 21478 | Half our thinking is unconscious, and we reach conclusions while unaware of premises [Schopenhauer] |
| 10203 | We apprehend small, finite mathematical structures by abstraction from patterns [Shapiro] |
| 4191 | What we know in ourselves is not a knower but a will [Schopenhauer] |
| 21367 | I know both aspects of my body, as representation, and as will [Schopenhauer] |
| 21368 | The knot of the world is the use of 'I' to refer to both willing and knowing [Schopenhauer] |
| 4175 | It is as perverse to resent our individuality being replaced by others, as to resent the body renewing itself [Schopenhauer] |
| 4176 | We all regard ourselves a priori as free, but see from experience that character and motive compel us [Schopenhauer] |
| 21477 | We don't control our own thinking [Schopenhauer] |
| 4170 | Man's actions are not free, because they follow strictly from impact of motive on character [Schopenhauer] |
| 21921 | Concepts are abstracted from perceptions [Schopenhauer, by Lewis,PB] |
| 21475 | All of our concepts are borrowed from perceptual knowledge [Schopenhauer] |
| 10229 | Simple types can be apprehended through their tokens, via abstraction [Shapiro] |
| 9626 | A structure is an abstraction, focussing on relationships, and ignoring other features [Shapiro] |
| 10217 | We can apprehend structures by focusing on or ignoring features of patterns [Shapiro] |
| 9554 | We can focus on relations between objects (like baseballers), ignoring their other features [Shapiro] |
| 10231 | Abstract objects might come by abstraction over an equivalence class of base entities [Shapiro] |
| 4169 | Every true act of will is also at once and without exception a movement of the body [Schopenhauer] |
| 7187 | Schopenhauer was caught in Christian ideals, because he didn't deify his 'will' [Nietzsche on Schopenhauer] |
| 21365 | Only the will is thing-in-itself, seen both in blind nature and in human action [Schopenhauer] |
| 21924 | As the subject of willing I am wretched, but absorption in knowledge is bliss [Schopenhauer] |
| 4173 | If we were essentially intellect rather than will, our moral worth would depend on imagined motives [Schopenhauer] |
| 21363 | Motivation is causality seen from within [Schopenhauer] |
| 21372 | Aesthetics concerns how we can take pleasure in an object, with no reference to the will [Schopenhauer] |
| 21370 | Schopenhauer is a chief proponent of aesthetic experience as 'disinterested' [Schopenhauer, by Janaway] |
| 4182 | A principal pleasure of the beautiful is that it momentarily silences the will [Schopenhauer] |
| 21488 | The beautiful is a perception of Plato's Forms, which eliminates the will [Schopenhauer] |
| 21928 | The Sublime fights for will-less knowing, when faced with a beautiful threat to humanity [Schopenhauer, by Lewis,PB] |
| 21927 | Schopenhauer emphasises Ideas in art, unlike most romantics [Schopenhauer, by Lewis,PB] |
| 8116 | The will-less contemplation of art brings a liberation from selfhood [Schopenhauer, by Gardner] |
| 4174 | Man is more beautiful than anything else, and the loftiest purpose of art is to reveal his nature [Schopenhauer] |
| 21380 | The only aim of our existence is to grasp that non-existence would be better [Schopenhauer] |
| 21374 | We should no more expect ethical theory to produce good people than aesthetics to produce artists [Schopenhauer] |
| 21378 | We clearly feel responsible for our deeds, because we are quite certain that we did them [Schopenhauer] |
| 21375 | Reason can be vicious, and great crimes have to be rational [Schopenhauer] |
| 21915 | To deduce morality from reason is blasphemy, because it is holy, and far above reason [Schopenhauer] |
| 21483 | Man is essentially a dreadful wild animal [Schopenhauer] |
| 21379 | Man's three basic ethical incentives are egoism, malice and compassion [Schopenhauer] |
| 4181 | Every good is essentially relative, for it has its essential nature only in its relation to a desiring will [Schopenhauer] |
| 5649 | Will casts aside each of its temporary fulfilments, so human life has no ultimate aim [Schopenhauer, by Scruton] |
| 4177 | Most people would probably choose non-existence at the end of their life, rather than relive the whole thing [Schopenhauer] |
| 4185 | Altruistic people make less distinction than usual between themselves and others [Schopenhauer] |
| 4183 | Only self-love can motivate morality, but that also makes it worthless [Schopenhauer] |
| 4172 | Happiness is the swift movement from desire to satisfaction, and then again on to desire [Schopenhauer] |
| 21371 | We can never attain happiness while our will is pursuing desires [Schopenhauer] |
| 21466 | Pleasure is weaker, and pain stronger, than we expect [Schopenhauer] |
| 4184 | Virtue must spring from an intuitive recognition that other people are essentially like us [Schopenhauer] |
| 21484 | A man's character can be learned from a single characteristic action [Schopenhauer] |
| 21482 | The five Chinese virtues: pity, justice, politeness, wisdom, honesty [Schopenhauer] |
| 21481 | Buddhists wisely start with the cardinal vices [Schopenhauer] |
| 21480 | Boredom is only felt by those clever enough to need activity [Schopenhauer] |
| 21469 | Human life is a mistake, shown by boredom, which is direct awareness of the fact [Schopenhauer] |
| 21485 | The state only exists to defend citizens, from exterior threats, and from one another [Schopenhauer] |
| 21486 | Poverty and slavery are virtually two words for the same thing [Schopenhauer] |
| 21487 | The freedom of the press to sell poison outweighs its usefulness [Schopenhauer] |
| 21471 | If suicide was quick and easy, most people would have done it by now [Schopenhauer] |
| 21467 | Would humanity still exist if sex wasn't both desired and pleasurable? [Schopenhauer] |
| 21376 | Philosophy treats animals as exploitable things, ignoring the significance of their lives [Schopenhauer] |
| 4179 | The essence of nature is the will to life itself [Schopenhauer] |
| 4189 | Time may be defined as the possibility of mutually exclusive conditions of the same thing [Schopenhauer] |
| 4178 | Christianity is a pessimistic religion, in which the world is equated with evil [Schopenhauer] |
| 4180 | Religion is the mythical clothing of the truth which is inaccessible to the crude human intellect [Schopenhauer] |
| 21472 | Only religion introduces serious issues to uneducated people [Schopenhauer] |
| 21468 | The Creator created the possibilities for worlds, so should have made a better one than this possible [Schopenhauer] |