278 ideas
20771 | Six parts: dialectic, rhetoric, ethics, politics, physics, theology [Cleanthes, by Diog. Laertius] |
10237 | Coherence is a primitive, intuitive notion, not reduced to something formal [Shapiro] |
19767 | Reason leads to prudent selfishness, which overrules natural compassion [Rousseau] |
19807 | Both nature and reason require that everything has a cause [Rousseau] |
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] |
13644 | Semantics for models uses set-theory [Shapiro] |
10240 | Model theory deals with relations, reference and extensions [Shapiro] |
10239 | The central notion of model theory is the relation of 'satisfaction' [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] |
13636 | An axiomatization is 'categorical' if its models are isomorphic, so there is really only one interpretation [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] |
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] |
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] |
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] |
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] |
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] |
10258 | Logical modalities may be acceptable, because they are reducible to satisfaction in models [Shapiro] |
10266 | Why does the 'myth' of possible worlds produce correct modal logic? [Shapiro] |
19757 | No one would bother to reason, and try to know things, without a desire for enjoyment [Rousseau] |
8725 | Rationalism tries to apply mathematical methodology to all of knowledge [Shapiro] |
10203 | We apprehend small, finite mathematical structures by abstraction from patterns [Shapiro] |
19760 | General ideas are purely intellectual; imagining them is immediately particular [Rousseau] |
19759 | Only words can introduce general ideas into the mind [Rousseau] |
6028 | Bodies interact with other bodies, and cuts cause pain, and shame causes blushing, so the soul is a body [Cleanthes, by Nemesius] |
20831 | The soul suffers when the body hurts, creates redness from shame, and pallor from fear [Cleanthes] |
19758 | Language may aid thinking, but powerful thought was needed to produce language [Rousseau] |
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] |
19773 | Without love, what use is beauty? [Rousseau] |
7235 | Without freedom of will actions lack moral significance [Rousseau] |
19769 | Rational morality is OK for brainy people, but ordinary life can't rely on that [Rousseau] |
19752 | If we should not mistreat humans, it is mainly because of sentience, not rationality [Rousseau] |
19768 | The better Golden Rule is 'do good for yourself without harming others' [Rousseau] |
19766 | The fact that we weep (e.g. in theatres) shows that we are naturally compassionate [Rousseau] |
20759 | Feelings are prior to intelligence; we should be content to live with our simplest feelings [Rousseau] |
19756 | Humans are less distinguished from other animals by understanding, than by being free agents [Rousseau] |
19755 | Most human ills are self-inflicted; the simple, solitary, regular natural life is good [Rousseau] |
19762 | Is language a pre-requisite for society, or might it emerge afterwards? [Rousseau] |
19763 | I doubt whether a savage person ever complains of life, or considers suicide [Rousseau] |
19778 | Leisure led to envy, inequality, vice and revenge, which we now see in savages [Rousseau] |
19779 | Primitive man was very gentle [Rousseau] |
19751 | Our two starting principles are concern for self-interest, and compassion for others [Rousseau] |
19791 | Natural mankind is too fragmented for states of peace, or of war and enmity [Rousseau] |
19765 | Savages avoid evil because they are calm, and never think of it (not because they know goodness) [Rousseau] |
19771 | Savage men quietly pursue desires, without the havoc of modern frenzied imagination [Rousseau] |
20501 | Rousseau assumes that laws need a people united by custom and tradition [Rousseau, by Wolff,J] |
7237 | The act of becoming 'a people' is the real foundation of society [Rousseau] |
19792 | To overcome obstacles, people must unite their forces into a single unified power [Rousseau] |
19812 | Human nature changes among a people, into a moral and partial existence [Rousseau] |
19814 | A state must be big enough to preserve itself, but small enough to be governable [Rousseau] |
19815 | Too much land is a struggle, producing defensive war; too little makes dependence, and offensive war [Rousseau] |
19822 | If the state enlarges, the creators of the general will become less individually powerful [Rousseau] |
19823 | If the population is larger, the government needs to be more powerful [Rousseau] |
19774 | A savage can steal fruit or a home, but there is no means of achieving obedience [Rousseau] |
7232 | Man is born free, and he is everywhere in chains [Rousseau] |
7234 | No man has any natural authority over his fellows [Rousseau] |
19772 | In a state of nature people are much more equal; it is society which increases inequalities [Rousseau] |
19789 | It is against nature for children to rule old men, fools to rule the wise, and the rich to hog resources [Rousseau] |
19816 | A state's purpose is liberty and equality - liberty for strength, and equality for liberty [Rousseau] |
7247 | The greatest social good comes down to freedom and equality [Rousseau] |
19838 | The measure of a successful state is increase in its population [Rousseau] |
19848 | The sovereignty does not appoint the leaders [Rousseau] |
19787 | People accept the right to be commanded, because they themselves wish to command [Rousseau] |
20567 | Rousseau insists that popular sovereignty needs a means of expressing consent [Rousseau, by Oksala] |
19801 | Sovereignty is the exercise of the general will, which can never be delegated [Rousseau] |
19805 | Just as people control their limbs, the general-will state has total control of its members [Rousseau] |
19818 | Political laws are fundamental, as they firmly organise the state - but they could still be changed [Rousseau] |
19790 | Force can only dominate if it is seen as a right, and obedience as a duty [Rousseau] |
7233 | The social order is a sacred right, but based on covenants, not nature [Rousseau] |
19842 | The government is instituted by a law, not by a contract [Rousseau] |
7239 | The social pact is the total subjection of individuals to the general will [Rousseau] |
19793 | We need a protective association which unites forces, but retains individual freedom [Rousseau] |
7240 | To foreign powers a state is seen as a simple individual [Rousseau] |
19795 | The act of association commits citizens to the state, and the state to its citizens [Rousseau] |
19797 | Citizens must ultimately for forced to accept the general will (so freedom is compulsory!) [Rousseau] |
19796 | Individual citizens still retain a private will, which may be contrary to the general will [Rousseau] |
7244 | The general will is common interest; the will of all is the sum of individual desires [Rousseau] |
19802 | The general will is always right, but the will of all can err, because it includes private interests [Rousseau] |
19803 | If the state contains associations there are fewer opinions, undermining the general will [Rousseau] |
19804 | If a large knowledgeable population votes in isolation, their many choices will have good results [Rousseau] |
19808 | The general will changes its nature when it focuses on particulars [Rousseau] |
7246 | The general will is always good, but sometimes misunderstood [Rousseau] |
7250 | Laws are authentic acts of the general will [Rousseau] |
19844 | Assemblies must always confirm the form of government, and the current administration [Rousseau] |
19846 | The more unanimous the assembly, the stronger the general will becomes [Rousseau] |
19854 | We all owe labour in return for our keep, and every idle citizen is a thief [Rousseau] |
19817 | Citizens should be independent of each other, and very dependent on the state [Rousseau] |
19840 | A citizen is a subject who is also sovereign [Rousseau] |
19780 | We seem to have made individual progress since savagery, but actually the species has decayed [Rousseau] |
19839 | The flourishing of arts and letters is too much admired [Rousseau] |
19798 | Ancient monarchs were kings of peoples; modern monarchs more cleverly rule a land [Rousseau] |
19831 | The highest officers under a monarchy are normally useless; the public could choose much better [Rousseau] |
19833 | Hereditary monarchy is easier, but can lead to dreadful monarchs [Rousseau] |
19834 | Attempts to train future kings don't usually work, and the best have been unprepared [Rousseau] |
19829 | Natural aristocracy is primitive, and hereditary is dreadful, but elective aristocracy is best [Rousseau] |
7249 | Natural aristocracy is primitive, hereditary is bad, and elective aristocracy is the best [Rousseau] |
19830 | Large states need a nobility to fill the gap between a single prince and the people [Rousseau] |
19827 | Law makers and law implementers should be separate [Rousseau] |
19820 | The state has a legislature and an executive, just like the will and physical power in a person [Rousseau] |
19821 | I call the executive power the 'government', which is the 'prince' - a single person, or a group [Rousseau] |
19824 | Large populations needs stronger control, which means power should be concentrated [Rousseau] |
19826 | Democracy for small states, aristocracy for intermediate, monarchy for large [Rousseau] |
19747 | Revolutionaries usually confuse liberty with total freedom, and end up with heavier chains [Rousseau] |
19837 | If inhabitants are widely dispersed, organising a revolt is much more difficult [Rousseau] |
19843 | The state is not bound to leave civil authority to its leaders [Rousseau] |
19825 | If the sovereign entrusts government to at least half the citizens, that is 'democracy' [Rousseau] |
19832 | Democratic elections are dangerous intervals in government [Rousseau] |
19748 | Plebiscites are bad, because they exclude the leaders from crucial decisions [Rousseau] |
7243 | Silence of the people implies their consent [Rousseau] |
19749 | In a direct democracy, only the leaders should be able to propose new laws [Rousseau] |
7251 | The English are actually slaves in between elections [Rousseau] |
7238 | Minorities only accept majority-voting because of a prior unanimous agreement [Rousseau] |
19828 | Democracy leads to internal strife, as people struggle to maintain or change ways of ruling [Rousseau] |
19835 | When ministers change the state changes, because they always reverse policies [Rousseau] |
19745 | The nature of people is decided by the government and politics of their society [Rousseau] |
19849 | In early theocracies the god was the king, and there were as many gods as nations [Rousseau] |
19784 | Enslaved peoples often boast of their condition, calling it a state of 'peace' [Rousseau] |
19785 | If the child of a slave woman is born a slave, then a man is not born a man [Rousseau] |
19841 | Sometimes full liberty is only possible at the expense of some complete enslavement [Rousseau] |
19847 | We can never assume that the son of a slave is a slave [Rousseau] |
19775 | People must be made dependent before they can be enslaved [Rousseau] |
7242 | Appetite alone is slavery, and self-prescribed laws are freedom [Rousseau] |
19746 | Like rich food, liberty can ruin people who are too weak to cope with it [Rousseau] |
19786 | Three stages of the state produce inequalities of wealth, power, and enslavement [Rousseau] |
19800 | The social compact imposes conventional equality of rights on people who may start unequally [Rousseau] |
19788 | The pleasure of wealth and power is largely seeing others deprived of them [Rousseau] |
7248 | No citizen should be rich enough to buy another, and none so poor as forced to sell himself [Rousseau] |
19794 | If we all give up all of our rights together to the community, we will always support one another [Rousseau] |
7241 | In society man loses natural liberty, but gains a right to civil liberty and property [Rousseau] |
19806 | We alienate to society only what society needs - but society judges that, not us [Rousseau] |
19777 | Persuading other people that some land was 'owned' was the beginning of society [Rousseau] |
19782 | What else could property arise from, but the labour people add to it? [Rousseau] |
19781 | Land cultivation led to a general right of ownership, administered justly [Rousseau] |
19754 | If we have a natural right to property, what exactly does 'belonging to' mean? [Rousseau] |
19799 | Private property must always be subordinate to ownership by the whole community [Rousseau] |
19819 | The state ensures liberty, so civil law separates citizens, and binds them to the state [Rousseau] |
19750 | Writers just propose natural law as the likely useful agreements among people [Rousseau] |
7245 | Natural justice, without sanctions, benefits the wicked, who exploit it [Rousseau] |
19809 | We accept the death penalty to prevent assassinations, so we must submit to it if necessary [Rousseau] |
19810 | A trial proves that a criminal has broken the social treaty, and is no longer a member of the state [Rousseau] |
19770 | Primitive people simply redressed the evil caused by violence, without thought of punishing [Rousseau] |
19811 | Only people who are actually dangerous should be executed, even as an example [Rousseau] |
7236 | War gives no right to inflict more destruction than is necessary for victory [Rousseau] |
23607 | Wars are between States, not people, and the individuals are enemies by accident [Rousseau] |
19783 | A state of war remains after a conquest, if the losers don't accept the winners [Rousseau] |
19850 | By separating theological and political systems, Jesus caused divisions in the state [Rousseau] |
19852 | Civil religion needs one supreme god, an afterlife, justice, and the sanctity of the social contract [Rousseau] |
19853 | All religions should be tolerated, if they tolerate each other, and support citizenship [Rousseau] |
19851 | Every society has a religion as its base [Rousseau] |
19836 | The amount of taxation doesn't matter, if it quickly circulates back to the citizens [Rousseau] |
19753 | Both men and animals are sentient, which should give the latter the right not to be mistreated [Rousseau] |
19761 | Men started with too few particular names, but later had too few natural kind names [Rousseau] |
19776 | Small uninterrupted causes can have big effects [Rousseau] |
5993 | The ascending scale of living creatures requires a perfect being [Cleanthes, by Tieleman] |
7252 | A tyrant exploits Christians because they don't value this life, and are made to be slaves [Rousseau] |