152 ideas
10237 | Coherence is a primitive, intuitive notion, not reduced to something formal [Shapiro] |
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] |
6710 | You can only define a statement that something is 'true' by referring to its functional possibilities [James] |
22305 | If the hypothesis of God is widely successful, it is true [James] |
10206 | Modal operators are usually treated as quantifiers [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] |
10207 | Anti-realists reject set theory [Shapiro] |
10259 | The two standard explanations of consequence are semantic (in models) and deductive [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] |
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] |
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] |
10240 | Model theory deals with relations, reference and extensions [Shapiro] |
10239 | The central notion of model theory is the relation of 'satisfaction' [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] |
10234 | Any theory with an infinite model has a model of every infinite cardinality [Shapiro] |
10201 | Virtually all of mathematics can be modeled in set theory [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] |
18245 | Cuts are made by the smallest upper or largest lower number, some of them not rational [Shapiro] |
10236 | There is no grounding for mathematics that is more secure than mathematics [Shapiro] |
10256 | For intuitionists, proof is inherently informal [Shapiro] |
10202 | Natural numbers just need an initial object, successors, and an induction principle [Shapiro] |
10205 | Mathematics originally concerned the continuous (geometry) and the discrete (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] |
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] |
10254 | Can the ideal constructor also destroy objects? [Shapiro] |
10255 | Presumably nothing can block a possible dynamic operation? [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] |
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] |
10203 | We apprehend small, finite mathematical structures by abstraction from patterns [Shapiro] |
10229 | Simple types can be apprehended through their tokens, via abstraction [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] |
7235 | Without freedom of will actions lack moral significance [Rousseau] |
19791 | Natural mankind is too fragmented for states of peace, or of war and enmity [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] |
7232 | Man is born free, and he is everywhere in chains [Rousseau] |
7234 | No man has any natural authority over his fellows [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] |
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] |
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] |
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] |
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] |
7243 | Silence of the people implies their consent [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] |
19849 | In early theocracies the god was the king, and there were as many gods as nations [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] |
7242 | Appetite alone is slavery, and self-prescribed laws are freedom [Rousseau] |
19800 | The social compact imposes conventional equality of rights on people who may start unequally [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] |
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] |
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] |
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] |
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] |
7252 | A tyrant exploits Christians because they don't value this life, and are made to be slaves [Rousseau] |