110 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] |
15945 | Second-order set theory just adds a version of Replacement that quantifies over functions [Lavine] |
15914 | An 'upper bound' is the greatest member of a subset; there may be several of these, so there is a 'least' one [Lavine] |
15921 | Collections of things can't be too big, but collections by a rule seem unlimited in size [Lavine] |
15937 | Those who reject infinite collections also want to reject the Axiom of Choice [Lavine] |
15936 | The Power Set is just the collection of functions from one collection to another [Lavine] |
15899 | Replacement was immediately accepted, despite having very few implications [Lavine] |
15930 | Foundation says descending chains are of finite length, blocking circularity, or ungrounded sets [Lavine] |
15920 | Pure collections of things obey Choice, but collections defined by a rule may not [Lavine] |
15898 | The controversy was not about the Axiom of Choice, but about functions as arbitrary, or given by rules [Lavine] |
15919 | The 'logical' notion of class has some kind of definition or rule to characterise the class [Lavine] |
15900 | The iterative conception of set wasn't suggested until 1947 [Lavine] |
15931 | The iterative conception needs the Axiom of Infinity, to show how far we can iterate [Lavine] |
15932 | The iterative conception doesn't unify the axioms, and has had little impact on mathematical proofs [Lavine] |
15933 | Limitation of Size: if it's the same size as a set, it's a set; it uses Replacement [Lavine] |
15913 | A collection is 'well-ordered' if there is a least element, and all of its successors can be identified [Lavine] |
15926 | Second-order logic presupposes a set of relations already fixed by the first-order domain [Lavine] |
15934 | Mathematical proof by contradiction needs the law of excluded middle [Lavine] |
15091 | Restrict 'logical truth' to formal logic, rather than including analytic and metaphysical truths [Shoemaker] |
15907 | Mathematics is nowadays (thanks to set theory) regarded as the study of structure, not of quantity [Lavine] |
15942 | Every rational number, unlike every natural number, is divisible by some other number [Lavine] |
15922 | For the real numbers to form a set, we need the Continuum Hypothesis to be true [Lavine] |
18250 | Cauchy gave a necessary condition for the convergence of a sequence [Lavine] |
15904 | The two sides of the Cut are, roughly, the bounding commensurable ratios [Lavine] |
15912 | Counting results in well-ordering, and well-ordering makes counting possible [Lavine] |
15949 | The theory of infinity must rest on our inability to distinguish between very large sizes [Lavine] |
15947 | The infinite is extrapolation from the experience of indefinitely large size [Lavine] |
15940 | The intuitionist endorses only the potential infinite [Lavine] |
15909 | 'Aleph-0' is cardinality of the naturals, 'aleph-1' the next cardinal, 'aleph-ω' the ω-th cardinal [Lavine] |
15915 | Ordinals are basic to Cantor's transfinite, to count the sets [Lavine] |
15917 | Paradox: the class of all ordinals is well-ordered, so must have an ordinal as type - giving a bigger ordinal [Lavine] |
15918 | Paradox: there is no largest cardinal, but the class of everything seems to be the largest [Lavine] |
15929 | Set theory will found all of mathematics - except for the notion of proof [Lavine] |
15935 | Modern mathematics works up to isomorphism, and doesn't care what things 'really are' [Lavine] |
15928 | Intuitionism rejects set-theory to found mathematics [Lavine] |
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] |
4938 | Prior to language, concepts are universals created by self-mapping of brain activity [Edelman/Tononi] |
8550 | Triangular and trilateral are coextensive, but different concepts; but powers and properties are the same [Shoemaker] |
8555 | There is no subset of properties which guarantee a thing's identity [Shoemaker] |
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] |
4934 | Cultures have a common core of colour naming, based on three axes of colour pairs [Edelman/Tononi] |
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] |
4924 | A conscious human being rapidly reunifies its mind after any damage to the brain [Edelman/Tononi] |
4932 | A conscious state endures for about 100 milliseconds, known as the 'specious present' [Edelman/Tononi] |
4931 | Consciousness is a process (of neural interactions), not a location, thing, property, connectivity, or activity [Edelman/Tononi] |
4923 | The three essentials of conscious experience are privateness, unity and informativeness [Edelman/Tononi] |
4941 | Consciousness can create new axioms, but computers can't do that [Edelman/Tononi] |
4930 | Consciousness arises from high speed interactions between clusters of neurons [Edelman/Tononi] |
4929 | Dreams and imagery show the brain can generate awareness and meaning without input [Edelman/Tononi] |
4940 | Physicists see information as a measure of order, but for biologists it is symbolic exchange between animals [Edelman/Tononi] |
4935 | The sensation of red is a point in neural space created by dimensions of neuronal activity [Edelman/Tononi] |
4936 | The self is founded on bodily awareness centred in the brain stem [Edelman/Tononi] |
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] |
4939 | A sense of self begins either internally, or externally through language and society [Edelman/Tononi] |
4925 | Brains can initiate free actions before the person is aware of their own decision [Edelman/Tononi] |
4933 | Consciousness is a process, not a thing, as it maintains unity as its composition changes [Edelman/Tononi] |
4928 | Brain complexity balances segregation and integration, like a good team of specialists [Edelman/Tononi] |
4927 | Information-processing views of the brain assume the existence of 'information', and dubious brain codes [Edelman/Tononi] |
4922 | Consciousness involves interaction with persons and the world, as well as brain functions [Edelman/Tononi] |
5793 | Concepts and generalisations result from brain 'global mapping' by 'reentry' [Edelman/Tononi, by Searle] |
4926 | Concepts arise when the brain maps its own activities [Edelman/Tononi] |
4937 | Systems that generate a sense of value are basic to the primitive brain [Edelman/Tononi] |
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] |