128 ideas
8558 | One system has properties, powers, events, similarity and substance [Shoemaker] |
8559 | Analysis aims at internal relationships, not reduction [Shoemaker] |
8594 | People have had good reasons for thinking that the circle has been squared [Shoemaker] |
8721 | An 'impredicative' definition seems circular, because it uses the term being defined [Friend] |
8680 | Classical definitions attempt to refer, but intuitionist/constructivist definitions actually create objects [Friend] |
3678 | Reductio ad absurdum proves an idea by showing that its denial produces contradiction [Friend] |
8705 | Anti-realists see truth as our servant, and epistemically contrained [Friend] |
8713 | In classical/realist logic the connectives are defined by truth-tables [Friend] |
8708 | Double negation elimination is not valid in intuitionist logic [Friend] |
8694 | Free logic was developed for fictional or non-existent objects [Friend] |
8665 | A 'proper subset' of A contains only members of A, but not all of them [Friend] |
8672 | A 'powerset' is all the subsets of a set [Friend] |
8677 | Set theory makes a minimum ontological claim, that the empty set exists [Friend] |
8666 | Infinite sets correspond one-to-one with a subset [Friend] |
8682 | Major set theories differ in their axioms, and also over the additional axioms of choice and infinity [Friend] |
8709 | The law of excluded middle is syntactic; it just says A or not-A, not whether they are true or false [Friend] |
8711 | Intuitionists read the universal quantifier as "we have a procedure for checking every..." [Friend] |
15091 | Restrict 'logical truth' to formal logic, rather than including analytic and metaphysical truths [Shoemaker] |
8675 | Paradoxes can be solved by talking more loosely of 'classes' instead of 'sets' [Friend] |
8674 | The Burali-Forti paradox asks whether the set of all ordinals is itself an ordinal [Friend] |
9935 | Mathematical truth is always compromising between ordinary language and sensible epistemology [Benacerraf] |
13412 | Obtaining numbers by abstraction is impossible - there are too many; only a rule could give them, in order [Benacerraf] |
13413 | We must explain how we know so many numbers, and recognise ones we haven't met before [Benacerraf] |
9912 | There are no such things as numbers [Benacerraf] |
9901 | Numbers can't be sets if there is no agreement on which sets they are [Benacerraf] |
8667 | The 'integers' are the positive and negative natural numbers, plus zero [Friend] |
8668 | The 'rational' numbers are those representable as fractions [Friend] |
8670 | A number is 'irrational' if it cannot be represented as a fraction [Friend] |
13411 | If numbers are basically the cardinals (Frege-Russell view) you could know some numbers in isolation [Benacerraf] |
9151 | Benacerraf says numbers are defined by their natural ordering [Benacerraf, by Fine,K] |
8661 | The natural numbers are primitive, and the ordinals are up one level of abstraction [Friend] |
13891 | To understand finite cardinals, it is necessary and sufficient to understand progressions [Benacerraf, by Wright,C] |
17904 | A set has k members if it one-one corresponds with the numbers less than or equal to k [Benacerraf] |
17906 | To explain numbers you must also explain cardinality, the counting of things [Benacerraf] |
8664 | Cardinal numbers answer 'how many?', with the order being irrelevant [Friend] |
8671 | The 'real' numbers (rationals and irrationals combined) is the Continuum, which has no gaps [Friend] |
9898 | We can count intransitively (reciting numbers) without understanding transitive counting of items [Benacerraf] |
17903 | Someone can recite numbers but not know how to count things; but not vice versa [Benacerraf] |
9897 | The application of a system of numbers is counting and measurement [Benacerraf] |
8663 | Raising omega to successive powers of omega reveal an infinity of infinities [Friend] |
8662 | The first limit ordinal is omega (greater, but without predecessor), and the second is twice-omega [Friend] |
8669 | Between any two rational numbers there is an infinite number of rational numbers [Friend] |
8676 | Is mathematics based on sets, types, categories, models or topology? [Friend] |
9900 | For Zermelo 3 belongs to 17, but for Von Neumann it does not [Benacerraf] |
9899 | The successor of x is either x and all its members, or just the unit set of x [Benacerraf] |
8678 | Most mathematical theories can be translated into the language of set theory [Friend] |
8697 | Disputes about mathematical objects seem irrelevant, and mathematicians cannot resolve them [Benacerraf, by Friend] |
8304 | No particular pair of sets can tell us what 'two' is, just by one-to-one correlation [Benacerraf, by Lowe] |
9906 | If ordinal numbers are 'reducible to' some set-theory, then which is which? [Benacerraf] |
13415 | An adequate account of a number must relate it to its series [Benacerraf] |
9907 | If any recursive sequence will explain ordinals, then it seems to be the structure which matters [Benacerraf] |
9908 | The job is done by the whole system of numbers, so numbers are not objects [Benacerraf] |
9909 | The number 3 defines the role of being third in a progression [Benacerraf] |
9911 | Number words no more have referents than do the parts of a ruler [Benacerraf] |
8925 | Mathematical objects only have properties relating them to other 'elements' of the same structure [Benacerraf] |
9938 | How can numbers be objects if order is their only property? [Benacerraf, by Putnam] |
8701 | The number 8 in isolation from the other numbers is of no interest [Friend] |
8702 | In structuralism the number 8 is not quite the same in different structures, only equivalent [Friend] |
8699 | Are structures 'ante rem' (before reality), or are they 'in re' (grounded in physics)? [Friend] |
8696 | Structuralist says maths concerns concepts about base objects, not base objects themselves [Friend] |
8695 | Structuralism focuses on relations, predicates and functions, with objects being inessential [Friend] |
8700 | 'In re' structuralism says that the process of abstraction is pattern-spotting [Friend] |
9910 | Number-as-objects works wholesale, but fails utterly object by object [Benacerraf] |
17927 | Realists have semantics without epistemology, anti-realists epistemology but bad semantics [Benacerraf, by Colyvan] |
9936 | The platonist view of mathematics doesn't fit our epistemology very well [Benacerraf] |
8681 | The big problem for platonists is epistemic: how do we perceive, intuit, know or detect mathematical facts? [Friend] |
8712 | Mathematics should be treated as true whenever it is indispensable to our best physical theory [Friend] |
9903 | Number words are not predicates, as they function very differently from adjectives [Benacerraf] |
9904 | The set-theory paradoxes mean that 17 can't be the class of all classes with 17 members [Benacerraf] |
8716 | Formalism is unconstrained, so cannot indicate importance, or directions for research [Friend] |
8706 | Constructivism rejects too much mathematics [Friend] |
8707 | Intuitionists typically retain bivalence but reject the law of excluded middle [Friend] |
8596 | Inability to measure equality doesn't make all lengths unequal [Shoemaker] |
8597 | We couldn't verify the earth's rotation if everyone simultaneously fell asleep [Shoemaker] |
15092 | Formerly I said properties are individuated by essential causal powers and causing instantiation [Shoemaker, by Shoemaker] |
15095 | A property's causal features are essential, and only they fix its identity [Shoemaker] |
15097 | I claim that a property has its causal features in all possible worlds [Shoemaker] |
8543 | Genuine properties are closely related to genuine changes [Shoemaker] |
8551 | Properties must be essentially causal if we can know and speak about them [Shoemaker] |
8557 | To ascertain genuine properties, examine the object directly [Shoemaker] |
15761 | We should abandon the idea that properties are the meanings of predicate expressions [Shoemaker] |
15756 | Some truths are not because of a thing's properties, but because of the properties of related things [Shoemaker] |
15758 | Things have powers in virtue of (which are entailed by) their properties [Shoemaker] |
8547 | One power can come from different properties; a thing's powers come from its properties [Shoemaker] |
8549 | Properties are functions producing powers, and powers are functions producing effects [Shoemaker] |
15094 | I now deny that properties are cluster of powers, and take causal properties as basic [Shoemaker] |
12678 | Shoemaker says all genuine properties are dispositional [Shoemaker, by Ellis] |
8545 | A causal theory of properties focuses on change, not (say) on abstract properties of numbers [Shoemaker] |
15757 | 'Square', 'round' and 'made of copper' show that not all properties are dispositional [Shoemaker] |
15759 | The identity of a property concerns its causal powers [Shoemaker] |
15760 | Properties are clusters of conditional powers [Shoemaker] |
15762 | Could properties change without the powers changing, or powers change without the properties changing? [Shoemaker] |
8552 | If properties are separated from causal powers, this invites total elimination [Shoemaker] |
4040 | The notions of property and of causal power are parts of a single system of related concepts [Shoemaker] |
15765 | Actually, properties are individuated by causes as well as effects [Shoemaker] |
14534 | Shoemaker moved from properties as powers to properties bestowing powers [Shoemaker, by Mumford/Anjum] |
8548 | Dispositional predicates ascribe powers, and the rest ascribe properties [Shoemaker] |
9485 | Universals concern how things are, and how they could be [Shoemaker, by Bird] |
8550 | Triangular and trilateral are coextensive, but different concepts; but powers and properties are the same [Shoemaker] |
8704 | Structuralists call a mathematical 'object' simply a 'place in a structure' [Friend] |
8555 | There is no subset of properties which guarantee a thing's identity [Shoemaker] |
9905 | Identity statements make sense only if there are possible individuating conditions [Benacerraf] |
15099 | If something is possible, but not nomologically possible, we need metaphysical possibility [Shoemaker] |
8554 | Possible difference across worlds depends on difference across time in the actual world [Shoemaker] |
15101 | Once you give up necessity as a priori, causal necessity becomes the main type of necessity [Shoemaker] |
15764 | 'Conceivable' is either not-provably-false, or compatible with what we know? [Shoemaker] |
15098 | Empirical evidence shows that imagining a phenomenon can show it is possible [Shoemaker] |
15100 | Imagination reveals conceptual possibility, where descriptions avoid contradiction or incoherence [Shoemaker] |
8562 | It is possible to conceive what is not possible [Shoemaker] |
5691 | The adverbial account of sensation says not 'see a red image' but be 'appeared to redly' [Shoemaker] |
8593 | Maybe billions of changeless years have elapsed since my last meal [Shoemaker] |
15096 | 'Grue' only has causal features because of its relation to green [Shoemaker] |
8556 | Grueness is not, unlike green and blue, associated with causal potential [Shoemaker] |
5687 | For true introspection, must we be aware that we are aware of our mental events? [Shoemaker] |
5688 | Empirical foundationalism says basic knowledge is self-intimating, and incorrigible or infallible [Shoemaker] |
1389 | If memory is the sole criterion of identity, we ought to use it for other people too [Shoemaker] |
1390 | Bodily identity is one criterion and memory another, for personal identity [Shoemaker, by PG] |
8685 | Studying biology presumes the laws of chemistry, and it could never contradict them [Friend] |
8688 | Concepts can be presented extensionally (as objects) or intensionally (as a characterization) [Friend] |
8542 | If causality is between events, there must be reference to the properties involved [Shoemaker] |
8598 | If things turn red for an hour and then explode, we wouldn't say the redness was the cause [Shoemaker] |
15093 | We might say laws are necessary by combining causal properties with Armstrong-Dretske-Tooley laws [Shoemaker] |
8560 | If causal laws describe causal potentialities, the same laws govern properties in all possible worlds [Shoemaker] |
15763 | If properties are causal, then causal necessity is a species of logical necessity [Shoemaker] |
8561 | If a world has different causal laws, it must have different properties [Shoemaker] |
8553 | It looks as if the immutability of the powers of a property imply essentiality [Shoemaker] |
4226 | If three regions 'freeze' every three, four and five years, after sixty years everything stops for a year [Shoemaker, by Lowe] |
8595 | If three regions freeze every 3rd, 4th and 5th year, they all freeze together every 60 years [Shoemaker] |