95 ideas
| 2797 | As coherence expands its interrelations become steadily tighter, culminating only in necessary truth [Dancy,J] |
| Full Idea: As our system grows in coherence, the interrelations between its parts becomes tighter and tighter;… at the limit contingent truth vanishes, leaving only necessary truth. | |
| From: Jonathan Dancy (Intro to Contemporary Epistemology [1985], 14.7) |
| 8721 | An 'impredicative' definition seems circular, because it uses the term being defined [Friend] |
| Full Idea: An 'impredicative' definition is one that uses the terms being defined in order to give the definition; in some way the definition is then circular. | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], Glossary) | |
| A reaction: There has been a big controversy in the philosophy of mathematics over these. Shapiro gives the definition of 'village idiot' (which probably mentions 'village') as an example. |
| 8680 | Classical definitions attempt to refer, but intuitionist/constructivist definitions actually create objects [Friend] |
| Full Idea: In classical logic definitions are thought of as revealing our attempts to refer to objects, ...but for intuitionist or constructivist logics, if our definitions do not uniquely characterize an object, we are not entitled to discuss the object. | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 2.4) | |
| A reaction: In defining a chess piece we are obviously creating. In defining a 'tree' we are trying to respond to fact, but the borderlines are vague. Philosophical life would be easier if we were allowed a mixture of creation and fact - so let's have that. |
| 3678 | Reductio ad absurdum proves an idea by showing that its denial produces contradiction [Friend] |
| Full Idea: Reductio ad absurdum arguments are ones that start by denying what one wants to prove. We then prove a contradiction from this 'denied' idea and more reasonable ideas in one's theory, showing that we were wrong in denying what we wanted to prove. | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 2.3) | |
| A reaction: This is a mathematical definition, which rests on logical contradiction, but in ordinary life (and philosophy) it would be enough to show that denial led to absurdity, rather than actual contradiction. |
| 8705 | Anti-realists see truth as our servant, and epistemically contrained [Friend] |
| Full Idea: For the anti-realist, truth belongs to us, it is our servant, and as such, it must be 'epistemically constrained'. | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 5.1) | |
| A reaction: Put as clearly as this, it strikes me as being utterly and spectacularly wrong, a complete failure to grasp the elementary meaning of a concept etc. etc. If we aren't the servants of truth then we jolly we ought to be. Truth is above us. |
| 2768 | The correspondence theory also has the problem that two sets of propositions might fit the facts equally well [Dancy,J] |
| Full Idea: The correspondence theory as well as the coherence theory has the problem of more than one set of truths. Why can't two sets of propositions "fit the facts" equally well? | |
| From: Jonathan Dancy (Intro to Contemporary Epistemology [1985], 8.2) |
| 2765 | Rescher says that if coherence requires mutual entailment, this leads to massive logical redundancy [Dancy,J] |
| Full Idea: Rescher complains that if coherence requires mutual entailment, then what is entailed is logically redundant, and the whole system is infected with mutual redundancy. | |
| From: Jonathan Dancy (Intro to Contemporary Epistemology [1985], 8.1) |
| 2769 | If one theory is held to be true, all the other theories appear false, because they can't be added to the true one [Dancy,J] |
| Full Idea: From the point of view of someone with a theory every other theory is false, because it cannot be added to the true theory. | |
| From: Jonathan Dancy (Intro to Contemporary Epistemology [1985], 8.2) |
| 2766 | Even with a tight account of coherence, there is always the possibility of more than one set of coherent propositions [Dancy,J] |
| Full Idea: No matter how tight our account of coherence we have to admit that there may be more than one set of coherent propositions (as Russell pointed out (1907)). | |
| From: Jonathan Dancy (Intro to Contemporary Epistemology [1985], 8.2) |
| 8713 | In classical/realist logic the connectives are defined by truth-tables [Friend] |
| Full Idea: In the classical or realist view of logic the meaning of abstract symbols for logical connectives is given by the truth-tables for the symbol. | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007]) | |
| A reaction: Presumably this is realist because it connects them to 'truth', but only if that involves a fairly 'realist' view of truth. You could, of course, translate 'true' and 'false' in the table to empty (formalist) symbols such a 0 and 1. Logic is electronics. |
| 8708 | Double negation elimination is not valid in intuitionist logic [Friend] |
| Full Idea: In intuitionist logic, if we do not know that we do not know A, it does not follow that we know A, so the inference (and, in general, double negation elimination) is not intuitionistically valid. | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 5.2) | |
| A reaction: That inference had better not be valid in any logic! I am unaware of not knowing the birthday of someone I have never heard of. Propositional attitudes such as 'know' are notoriously difficult to explain in formal logic. |
| 8694 | Free logic was developed for fictional or non-existent objects [Friend] |
| Full Idea: Free logic is especially designed to help regiment our reasoning about fictional objects, or nonexistent objects of some sort. | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 3.7) | |
| A reaction: This makes it sound marginal, but I wonder whether existential commitment shouldn't be eliminated from all logic. Why do fictional objects need a different logic? What logic should we use for Robin Hood, if we aren't sure whether or not he is real? |
| 8665 | A 'proper subset' of A contains only members of A, but not all of them [Friend] |
| Full Idea: A 'subset' of A is a set containing only members of A, and a 'proper subset' is one that does not contain all the members of A. Note that the empty set is a subset of every set, but it is not a member of every set. | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 1.5) | |
| A reaction: Is it the same empty set in each case? 'No pens' is a subset of 'pens', but is it a subset of 'paper'? Idea 8219 should be borne in mind when discussing such things, though I am not saying I agree with it. |
| 8672 | A 'powerset' is all the subsets of a set [Friend] |
| Full Idea: The 'powerset' of a set is a set made up of all the subsets of a set. For example, the powerset of {3,7,9} is {null, {3}, {7}, {9}, {3,7}, {3,9}, {7,9}, {3,7,9}}. Taking the powerset of an infinite set gets us from one infinite cardinality to the next. | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 1.5) | |
| A reaction: Note that the null (empty) set occurs once, but not in the combinations. I begin to have queasy sympathies with the constructivist view of mathematics at this point, since no one has the time, space or energy to 'take' an infinite powerset. |
| 8677 | Set theory makes a minimum ontological claim, that the empty set exists [Friend] |
| Full Idea: As a realist choice of what is basic in mathematics, set theory is rather clever, because it only makes a very simple ontological claim: that, independent of us, there exists the empty set. The whole hierarchy of finite and infinite sets then follows. | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 2.3) | |
| A reaction: Even so, for non-logicians the existence of the empty set is rather counterintuitive. "There was nobody on the road, so I overtook him". See Ideas 7035 and 8322. You might work back to the empty set, but how do you start from it? |
| 8666 | Infinite sets correspond one-to-one with a subset [Friend] |
| Full Idea: Two sets are the same size if they can be placed in one-to-one correspondence. But even numbers have one-to-one correspondence with the natural numbers. So a set is infinite if it has one-one correspondence with a proper subset. | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 1.5) | |
| A reaction: Dedekind's definition. We can match 1 with 2, 2 with 4, 3 with 6, 4 with 8, etc. Logicians seem happy to give as a definition anything which fixes the target uniquely, even if it doesn't give the essence. See Frege on 0 and 1, Ideas 8653/4. |
| 8682 | Major set theories differ in their axioms, and also over the additional axioms of choice and infinity [Friend] |
| Full Idea: Zermelo-Fraenkel and Gödel-Bernays set theory differ over the notions of ordinal construction and over the notion of class, among other things. Then there are optional axioms which can be attached, such as the axiom of choice and the axiom of infinity. | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 2.6) | |
| A reaction: This summarises the reasons why we cannot just talk about 'set theory' as if it was a single concept. The philosophical interest I would take to be found in disentangling the ontological commitments of each version. |
| 8709 | The law of excluded middle is syntactic; it just says A or not-A, not whether they are true or false [Friend] |
| Full Idea: The law of excluded middle is purely syntactic: it says for any well-formed formula A, either A or not-A. It is not a semantic law; it does not say that either A is true or A is false. The semantic version (true or false) is the law of bivalence. | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 5.2) | |
| A reaction: No wonder these two are confusing, sufficiently so for a lot of professional philosophers to blur the distinction. Presumably the 'or' is exclusive. So A-and-not-A is a contradiction; but how do you explain a contradiction without mentioning truth? |
| 8711 | Intuitionists read the universal quantifier as "we have a procedure for checking every..." [Friend] |
| Full Idea: In the intuitionist version of quantification, the universal quantifier (normally read as "all") is understood as "we have a procedure for checking every" or "we have checked every". | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 5.5) | |
| A reaction: It seems better to describe this as 'verificationist' (or, as Dummett prefers, 'justificationist'). Intuition suggests an ability to 'see' beyond the evidence. It strikes me as bizarre to say that you can't discuss things you can't check. |
| 8675 | Paradoxes can be solved by talking more loosely of 'classes' instead of 'sets' [Friend] |
| Full Idea: The realist meets the Burali-Forti paradox by saying that all the ordinals are a 'class', not a set. A proper class is what we discuss when we say "all" the so-and-sos when they cannot be reached by normal set-construction. Grammar is their only limit. | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 2.3) | |
| A reaction: This strategy would be useful for Class Nominalism, which tries to define properties in terms of classes, but gets tangled in paradoxes. But why bother with strict sets if easy-going classes will do just as well? Descartes's Dream: everything is rational. |
| 8674 | The Burali-Forti paradox asks whether the set of all ordinals is itself an ordinal [Friend] |
| Full Idea: The Burali-Forti paradox says that if ordinals are defined by 'gathering' all their predecessors with the empty set, then is the set of all ordinals an ordinal? It is created the same way, so it should be a further member of this 'complete' set! | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 2.3) | |
| A reaction: This is an example (along with Russell's more famous paradox) of the problems that began to appear in set theory in the early twentieth century. See Idea 8675 for a modern solution. |
| 8667 | The 'integers' are the positive and negative natural numbers, plus zero [Friend] |
| Full Idea: The set of 'integers' is all of the negative natural numbers, and zero, together with the positive natural numbers. | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 1.5) | |
| A reaction: Zero always looks like a misfit at this party. Credit and debit explain positive and negative nicely, but what is the difference between having no money, and money being irrelevant? I can be 'broke', but can the North Pole be broke? |
| 8668 | The 'rational' numbers are those representable as fractions [Friend] |
| Full Idea: The 'rational' numbers are all those that can be represented in the form m/n (i.e. as fractions), where m and n are natural numbers different from zero. | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 1.5) | |
| A reaction: Pythagoreans needed numbers to stop there, in order to represent the whole of reality numerically. See irrational numbers for the ensuing disaster. How can a universe with a finite number of particles contain numbers that are not 'rational'? |
| 8670 | A number is 'irrational' if it cannot be represented as a fraction [Friend] |
| Full Idea: A number is 'irrational' just in case it cannot be represented as a fraction. An irrational number has an infinite non-repeating decimal expansion. Famous examples are pi and e. | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 1.5) | |
| A reaction: There must be an infinite number of irrational numbers. You could, for example, take the expansion of pi, and change just one digit to produce a new irrational number, and pi has an infinity of digits to tinker with. |
| 8661 | The natural numbers are primitive, and the ordinals are up one level of abstraction [Friend] |
| Full Idea: The natural numbers are quite primitive, and are what we first learn about. The order of objects (the 'ordinals') is one level of abstraction up from the natural numbers: we impose an order on objects. | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 1.4) | |
| A reaction: Note the talk of 'levels of abstraction'. So is there a first level of abstraction? Dedekind disagrees with Friend (Idea 7524). I would say that natural numbers are abstracted from something, but I'm not sure what. See Structuralism in maths. |
| 8664 | Cardinal numbers answer 'how many?', with the order being irrelevant [Friend] |
| Full Idea: The 'cardinal' numbers answer the question 'How many?'; the order of presentation of the objects being counted as immaterial. Def: the cardinality of a set is the number of members of the set. | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 1.5) | |
| A reaction: If one asks whether cardinals or ordinals are logically prior (see Ideas 7524 and 8661), I am inclined to answer 'neither'. Presenting them as answers to the questions 'how many?' and 'which comes first?' is illuminating. |
| 8671 | The 'real' numbers (rationals and irrationals combined) is the Continuum, which has no gaps [Friend] |
| Full Idea: The set of 'real' numbers, which consists of the rational numbers and the irrational numbers together, represents "the continuum", since it is like a smooth line which has no gaps (unlike the rational numbers, which have the irrationals missing). | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 1.5) | |
| A reaction: The Continuum is the perfect abstract object, because a series of abstractions has arrived at a vast limit in its nature. It still has dizzying infinities contained within it, and at either end of the line. It makes you feel humble. |
| 8663 | Raising omega to successive powers of omega reveal an infinity of infinities [Friend] |
| Full Idea: After the multiples of omega, we can successively raise omega to powers of omega, and after that is done an infinite number of times we arrive at a new limit ordinal, which is called 'epsilon'. We have an infinite number of infinite ordinals. | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 1.4) | |
| A reaction: When most people are dumbstruck by the idea of a single infinity, Cantor unleashes an infinity of infinities, which must be the highest into the stratosphere of abstract thought that any human being has ever gone. |
| 8662 | The first limit ordinal is omega (greater, but without predecessor), and the second is twice-omega [Friend] |
| Full Idea: The first 'limit ordinal' is called 'omega', which is ordinal because it is greater than other numbers, but it has no immediate predecessor. But it has successors, and after all of those we come to twice-omega, which is the next limit ordinal. | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 1.4) | |
| A reaction: This is the gateway to Cantor's paradise of infinities, which Hilbert loved and defended. Who could resist the pleasure of being totally boggled (like Aristotle) by a concept such as infinity, only to have someone draw a map of it? See 8663 for sequel. |
| 8669 | Between any two rational numbers there is an infinite number of rational numbers [Friend] |
| Full Idea: Since between any two rational numbers there is an infinite number of rational numbers, we could consider that we have infinity in three dimensions: positive numbers, negative numbers, and the 'depth' of infinite numbers between any rational numbers. | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 1.5) | |
| A reaction: This is before we even reach Cantor's staggering infinities (Ideas 8662 and 8663), which presumably reside at the outer reaches of all three of these dimensions of infinity. The 'deep' infinities come from fractions with huge denominators. |
| 8676 | Is mathematics based on sets, types, categories, models or topology? [Friend] |
| Full Idea: Successful competing founding disciplines in mathematics include: the various set theories, type theory, category theory, model theory and topology. | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 2.3) | |
| A reaction: Or none of the above? Set theories are very popular. Type theory is, apparently, discredited. Shapiro has a version of structuralism based on model theory (which sound promising). Topology is the one that intrigues me... |
| 8678 | Most mathematical theories can be translated into the language of set theory [Friend] |
| Full Idea: Most of mathematics can be faithfully redescribed by classical (realist) set theory. More precisely, we can translate other mathematical theories - such as group theory, analysis, calculus, arithmetic, geometry and so on - into the language of set theory. | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 2.3) | |
| A reaction: This is why most mathematicians seem to regard set theory as foundational. We could also translate football matches into the language of atomic physics. |
| 8701 | The number 8 in isolation from the other numbers is of no interest [Friend] |
| Full Idea: There is no interest for the mathematician in studying the number 8 in isolation from the other numbers. | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 4.4) | |
| A reaction: This is a crucial and simple point (arising during a discussion of Shapiro's structuralism). Most things are interesting in themselves, as well as for their relationships, but mathematical 'objects' just are relationships. |
| 8702 | In structuralism the number 8 is not quite the same in different structures, only equivalent [Friend] |
| Full Idea: Structuralists give a historical account of why the 'same' number occupies different structures. Numbers are equivalent rather than identical. 8 is the immediate predecessor of 9 in the whole numbers, but in the rationals 9 has no predecessor. | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 4.4) | |
| A reaction: I don't become a different person if I move from a detached house to a terraced house. This suggests that 8 can't be entirely defined by its relations, and yet it is hard to see what its intrinsic nature could be, apart from the units which compose it. |
| 8699 | Are structures 'ante rem' (before reality), or are they 'in re' (grounded in physics)? [Friend] |
| Full Idea: Structuralists disagree over whether objects in structures are 'ante rem' (before reality, existing independently of whether the objects exist) or 'in re' (in reality, grounded in the real world, usually in our theories of physics). | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 4.4) | |
| A reaction: Shapiro holds the first view, Hellman and Resnik the second. The first view sounds too platonist and ontologically extravagant; the second sounds too contingent and limited. The correct account is somewhere in abstractions from the real. |
| 8696 | Structuralist says maths concerns concepts about base objects, not base objects themselves [Friend] |
| Full Idea: According to the structuralist, mathematicians study the concepts (objects of study) such as variable, greater, real, add, similar, infinite set, which are one level of abstraction up from prima facie base objects such as numbers, shapes and lines. | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 4.1) | |
| A reaction: This still seems to imply an ontology in which numbers, shapes and lines exist. I would have thought you could eliminate the 'base objects', and just say that the concepts are one level of abstraction up from the physical world. |
| 8695 | Structuralism focuses on relations, predicates and functions, with objects being inessential [Friend] |
| Full Idea: Structuralism says we study whole structures: objects together with their predicates, relations that bear between them, and functions that take us from one domain of objects to a range of other objects. The objects can even be eliminated. | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 4.1) | |
| A reaction: The unity of object and predicate is a Quinean idea. The idea that objects are inessential is the dramatic move. To me the proposal has very strong intuitive appeal. 'Eight' is meaningless out of context. Ordinality precedes cardinality? Ideas 7524/8661. |
| 8700 | 'In re' structuralism says that the process of abstraction is pattern-spotting [Friend] |
| Full Idea: In the 'in re' version of mathematical structuralism, pattern-spotting is the process of abstraction. | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 4.4) | |
| A reaction: This might work for non-mathematical abstraction as well, if we are allowed to spot patterns within sensual experience, and patterns within abstractions. Properties are causal patterns in the world? No - properties cause patterns. |
| 8681 | The big problem for platonists is epistemic: how do we perceive, intuit, know or detect mathematical facts? [Friend] |
| Full Idea: The main philosophical problem with the position of platonism or realism is the epistemic problem: of explaining what perception or intuition consists in; how it is possible that we should accurately detect whatever it is we are realists about. | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 2.5) | |
| A reaction: The best bet, I suppose, is that the mind directly perceives concepts just as eyes perceive the physical (see Idea 8679), but it strikes me as implausible. If we have to come up with a special mental faculty for an area of knowledge, we are in trouble. |
| 8712 | Mathematics should be treated as true whenever it is indispensable to our best physical theory [Friend] |
| Full Idea: Central to naturalism about mathematics are 'indispensability arguments', to the effect that some part of mathematics is indispensable to our best physical theory, and therefore we ought to take that part of mathematics to be true. | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 6.1) | |
| A reaction: Quine and Putnam hold this view; Field challenges it. It has the odd consequence that the dispensable parts (if they can be identified!) do not need to be treated as true (even though they might follow logically from the dispensable parts!). Wrong! |
| 8716 | Formalism is unconstrained, so cannot indicate importance, or directions for research [Friend] |
| Full Idea: There are not enough constraints in the Formalist view of mathematics, so there is no way to select a direction for trying to develop mathematics. There is no part of mathematics that is more important than another. | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 6.6) | |
| A reaction: One might reply that an area of maths could be 'important' if lots of other areas depended on it, and big developments would ripple big changes through the interior of the subject. Formalism does, though, seem to reduce maths to a game. |
| 8706 | Constructivism rejects too much mathematics [Friend] |
| Full Idea: Too much of mathematics is rejected by the constructivist. | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 5.1) | |
| A reaction: This was Hilbert's view. This seems to be generally true of verificationism. My favourite example is that legitimate speculations can be labelled as meaningless. |
| 8707 | Intuitionists typically retain bivalence but reject the law of excluded middle [Friend] |
| Full Idea: An intuitionist typically retains bivalence, but rejects the law of excluded middle. | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 5.2) | |
| A reaction: The idea would be to say that only T and F are available as truth-values, but failing to be T does not ensure being F, but merely not-T. 'Unproven' is not-T, but may not be F. |
| 8978 | Events are made of other things, and are not fundamental to ontology [Bennett] |
| Full Idea: Events are not basic items in the universe; they should not be included in any fundamental ontology...all the truths about them are entailed by and explained and made true by truths that do not involve the event concept. | |
| From: Jonathan Bennett (Events and Their Names [1988], p.12), quoted by Peter Simons - Events 3.1 | |
| A reaction: Given the variable time spans of events, their ability to coincide, their ability to contain no motion, their blatantly conventional component, and their recalcitrance to individuation, I say Bennett is right. |
| 2781 | Realism says that most perceived objects exist, and have some of their perceived properties [Dancy,J] |
| Full Idea: Realism in the theory of perception is that objects we perceive usually do exist, and retain some at least of the properties we perceive them as having. | |
| From: Jonathan Dancy (Intro to Contemporary Epistemology [1985], 10.2) |
| 8704 | Structuralists call a mathematical 'object' simply a 'place in a structure' [Friend] |
| Full Idea: What the mathematician labels an 'object' in her discipline, is called 'a place in a structure' by the structuralist. | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 4.5) | |
| A reaction: This is a strategy for dispersing the idea of an object in the world of thought, parallel to attempts to eliminate them from physical ontology (e.g. Idea 614). |
| 2745 | A pupil who lacks confidence may clearly know something but not be certain of it [Dancy,J] |
| Full Idea: Why isn't certainty required for knowledge? Because we are often prepared to allow that someone does in fact have knowledge when the person is so uncertain they would not claim knowledge for themselves (the 'diffident schoolboy'). | |
| From: Jonathan Dancy (Intro to Contemporary Epistemology [1985], 2.1) |
| 2755 | If senses are fallible, then being open to correction is an epistemological virtue [Dancy,J] |
| Full Idea: In my view, once we admit that our beliefs about our sensory states are not infallible, incorrigibility would be a vice rather than a virtue. | |
| From: Jonathan Dancy (Intro to Contemporary Epistemology [1985], 4.3) | |
| A reaction: This seems to be axiomatic among modern philosophers, and I certainly agree with it. |
| 5677 | Naïve direct realists hold that objects retain all of their properties when unperceived [Dancy,J] |
| Full Idea: The naïve direct realist holds that unperceived objects are able to retain properties of all the types we perceive them as having, which includes not only a shape and a size, but also a colour, a taste and a smell. | |
| From: Jonathan Dancy (Intro to Contemporary Epistemology [1985], 10.3) | |
| A reaction: This I take to be a completely untenable view, if we are including the qualia of red, sweet or pungent among the properties. It seems uncontroversial that objects retain the capacity to cause redness etc. when they are unperceived. |
| 5678 | Scientific direct realism says we know some properties of objects directly [Dancy,J] |
| Full Idea: The scientific direct realist accepts the directness of our perception of the world, but restricts his realism to a special group of properties, ..not including those which are dependent for their existence upon the existence of a perceiver. | |
| From: Jonathan Dancy (Intro to Contemporary Epistemology [1985], 10.3) | |
| A reaction: Dancy goes on to say that this distinction is a 'close relative' of Locke's primary/secondary distinction. Am I a direct realist or a representative realist about primary properties? Maybe the distinction dissolves as we unravel the true process. |
| 5681 | Maybe we are forced from direct into indirect realism by the need to explain perceptual error [Dancy,J] |
| Full Idea: Direct realism is unlikely to be able to provide an explanation of perceptual error without collapsing into indirect realism. | |
| From: Jonathan Dancy (Intro to Contemporary Epistemology [1985], 10.3) | |
| A reaction: If there is an error, there must be two things which don't match: the perception, and the reality. This seems to me a powerful reason for preferring indirect or representative realism. I like the idea that we make mental 'models' (rather than inferences). |
| 5682 | Internal realism holds that we perceive physical objects via mental objects [Dancy,J] |
| Full Idea: Indirect realism holds that in perception we are indirectly aware of the physical objects around us in virtue of a direct awareness of internal, non-physical objects. | |
| From: Jonathan Dancy (Intro to Contemporary Epistemology [1985], 10.4) | |
| A reaction: This may be a slightly prejudicial definition which invites insoluble questions about the ontological status of the internal 'objects'. It seems to me obvious that we create some sort of inner 'models' or constructions in the process of perception. |
| 5683 | Indirect realism depends on introspection, the time-lag, illusions, and neuroscience [Dancy,J, by PG] |
| Full Idea: The four standard reasons for preferring indirect to direct realism are introspection of our mental processes, the time-lag argument, the argument from illusion, and the findings of neuroscience. | |
| From: report of Jonathan Dancy (Intro to Contemporary Epistemology [1985], 10.4) by PG - Db (ideas) | |
| A reaction: Ultimately one's views about realism depend on one's views of the mind/brain, and it is the last of the four reasons that sways me. We know enough about the complexity of the brain to accept that it represents reality, with no additional ontology. |
| 2778 | Phenomenalism includes possible experiences, but idealism only refers to actual experiences [Dancy,J] |
| Full Idea: Phenomenalism talks about actual and possible experiences, whereas idealism confines itself to the actual experiences. | |
| From: Jonathan Dancy (Intro to Contemporary Epistemology [1985], 9.5) |
| 5684 | Eliminative idealists say there are no objects; reductive idealists say objects exist as complex experiences [Dancy,J] |
| Full Idea: The eliminativist idealist holds that there is no such thing as a material object; there is nothing but experience (idea, sensation). The reductive idealist holds that there are material objects, but they are nothing other than complexes of experience. | |
| From: Jonathan Dancy (Intro to Contemporary Epistemology [1985], 10.6) | |
| A reaction: Dancy says Berkeley was of the latter type. The distinction doesn't strike me as entirely clear. I can't make much sense of the words 'are' or 'exist' in the second theory. To say it is only experiences translates (to me) as 'doesn't exist'. |
| 2777 | Extreme solipsism only concerns current experience, but it might include past and future [Dancy,J] |
| Full Idea: Extreme solipsism only considers present experiences, but more relaxed solipsism may include past and possible future experiences. | |
| From: Jonathan Dancy (Intro to Contemporary Epistemology [1985], 9.5) |
| 2794 | Knowing that a cow is not a horse seems to be a synthetic a priori truth [Dancy,J] |
| Full Idea: The fact that a cow is not a horse is a candidate for a priori synthetic truth. It doesn't seem to be analytic, because you can know what a cow is without knowing what a horse is. | |
| From: Jonathan Dancy (Intro to Contemporary Epistemology [1985], 14.3) |
| 2780 | Perception is either direct realism, indirect realism, or phenomenalism [Dancy,J] |
| Full Idea: There are three main families of theories of perception: direct realism, indirect realism, and phenomenalism. | |
| From: Jonathan Dancy (Intro to Contemporary Epistemology [1985], 10.2) |
| 5679 | We can't grasp the separation of quality types, or what a primary-quality world would be like [Dancy,J] |
| Full Idea: There is doubt about whether our experience of the world is such that we can conceive of the sort of separation of primary and secondary qualities which the scientific view calls for, and can understand what the world is like with no secondary qualities. | |
| From: Jonathan Dancy (Intro to Contemporary Epistemology [1985], 10.3) | |
| A reaction: Dancy attributes these doubts to Berkeley (e.g. Idea 3837). I think what is claimed here is false. Obviously we spend our whole lives immersed in secondary qualities, but separating the different aspects is precisely what scientists (and philosophers) do. |
| 5680 | For direct realists the secondary and primary qualities seem equally direct [Dancy,J] |
| Full Idea: For a direct realist our awareness of colour and heat can hardly be of a different order from our awareness of shape and size. Both sorts of properties are presented with equal directness. | |
| From: Jonathan Dancy (Intro to Contemporary Epistemology [1985], 10.3) | |
| A reaction: This is a good objection to 'direct scientific realism', which claims direct apprehension of primary qualities alongside a totally relative view of secondary qualities. The best response seems to be to move to a representative view of primary properties. |
| 2782 | We can be looking at distant stars which no longer actually exist [Dancy,J] |
| Full Idea: An object such as a distant star can have ceased to exist by the moment at which we are directly aware of it. | |
| From: Jonathan Dancy (Intro to Contemporary Epistemology [1985], 10.2) |
| 2775 | It is not clear from the nature of sense data whether we should accept them as facts [Dancy,J] |
| Full Idea: The question whether something which appears as datum should remain as accepted fact is one which is not even partially determined by its origin as datum. | |
| From: Jonathan Dancy (Intro to Contemporary Epistemology [1985], 8.5) |
| 2784 | Appearances don't guarantee reality, unless the appearance is actually caused by the reality [Dancy,J] |
| Full Idea: If I stare at a white wall with my brain wired to a virtual reality computer, and it generates a white wall, we wouldn't say I am seeing reality. It seems that the wall itself must in some way cause my perception of it. | |
| From: Jonathan Dancy (Intro to Contemporary Epistemology [1985], 11.4) | |
| A reaction: But suppose the computer generated in my mind an image of the wall which was actually in front of me? And suppose the computer got its image from the identical wall next door, not from mine? And it was only judged identical because the architect said so |
| 2785 | Perceptual beliefs may be directly caused, but generalisations can't be [Dancy,J] |
| Full Idea: A perceptual belief that p can have as its main cause the fact that p. More general facts (all men are mortal; e=mc2) cannot be the main cause of my belief, even if they do function causally in some way. | |
| From: Jonathan Dancy (Intro to Contemporary Epistemology [1985], 11.5) | |
| A reaction: Note that the perceptual belief can be the "main" cause; it seems to me that most beliefs are caused by judgements, though I may normally accept beliefs which are directly caused by perception, if I have no reason to challenge them. |
| 2788 | If perception and memory are indirect, then two things stand between mind and reality [Dancy,J] |
| Full Idea: If perception is indirect as well as memory, this means there are two direct objects of awareness between the remembering mind and the original object. | |
| From: Jonathan Dancy (Intro to Contemporary Epistemology [1985], 12.2) |
| 2787 | Memories aren't directly about the past, because time-lags and illusions suggest representation [Dancy,J] |
| Full Idea: Direct realism about memory believes the memory is the past. But the time-lag argument and various illusions are powerful here, suggesting indirect realism involving a memory image. | |
| From: Jonathan Dancy (Intro to Contemporary Epistemology [1985], 12.2) |
| 2791 | Phenomenalism about memory denies the past, or reduces it to present experience [Dancy,J] |
| Full Idea: Eliminative phenomenalism about memory holds that there is no such thing as the past, just certain present experiences; reductive phenomenalism holds that there is a past, but it is no more than a complex of those present experiences. | |
| From: Jonathan Dancy (Intro to Contemporary Epistemology [1985], 12.4) |
| 2790 | I can remember plans about the future, and images aren't essential (2+3=5) [Dancy,J] |
| Full Idea: Memory is not solely concerned with the past, let alone one's own past (I remember that I must be in London next week), and need not involve images (2+2=4). | |
| From: Jonathan Dancy (Intro to Contemporary Epistemology [1985], 12.3) | |
| A reaction: I can hardly remember the future, so I presume I am remembering my past commitment to go to London, even if I visualise the future with me in London. The non-necessity of images seems right. I can remember the Mona Lisa without a precise image. |
| 2754 | Foundations are justified by non-beliefs, or circularly, or they need no justification [Dancy,J] |
| Full Idea: Foundationalism can get rid of the regress argument with one of three types of belief: those justified by something other than beliefs, those which justify themselves, or those which need no justification. | |
| From: Jonathan Dancy (Intro to Contemporary Epistemology [1985], 4.3) | |
| A reaction: A nice clear trilemma, and none of them will do, which is why foundationalism is false. I vote for Davidson's view, that only a belief can justify another belief. |
| 2749 | For internalists we must actually know that the fact caused the belief [Dancy,J] |
| Full Idea: The internalist would claim that even if the belief is caused by the true fact to which it refers, it is also necessary that the believer believes that this is how their belief arose, and not some other way. | |
| From: Jonathan Dancy (Intro to Contemporary Epistemology [1985], 3.5) | |
| A reaction: I'm converted to internalism. If the belief is externally supported in the right way, then it may well be a true belief, but knowledge needs critical faculties, and justifications which can be articulated. |
| 2770 | Internalists tend to favour coherent justification, but not the coherence theory of truth [Dancy,J] |
| Full Idea: Internalists such as Keith Lehrer tend to suggest that we adopt a coherence theory of justification but reject the coherence theory of truth. | |
| From: Jonathan Dancy (Intro to Contemporary Epistemology [1985], 8.3) | |
| A reaction: I agree with Lehrer. Truth just isn't coherence, for all sorts of well known reasons (found in this database!). High coherence can be totally false. For justification, though, it is the best we have. |
| 2752 | Foundationalism requires inferential and non-inferential justification [Dancy,J] |
| Full Idea: The core of any form of foundationalism is the view that there are two forms of justification - inferential and non-inferential - and that non-inferential justification must be possible to avoid a sceptical regress. | |
| From: Jonathan Dancy (Intro to Contemporary Epistemology [1985], 4.1) | |
| A reaction: The foundation may be non-inferential, but is it also non-evidential, or devoid of any support at all, apart from its own eloquent self? I can't buy that, I'm afraid. |
| 2771 | Foundationalists must accept not only the basic beliefs, but also rules of inference for further progress [Dancy,J] |
| Full Idea: Foundationalists suppose we need not only basic beliefs, but also principles of inference to move to the more sophisticated superstructure. We may understand what justifies the basic beliefs, but what about the inference principles? | |
| From: Jonathan Dancy (Intro to Contemporary Epistemology [1985], 8.3) | |
| A reaction: Very nice question. Of course, you can't justify everything, but each part of a system can be scrutinised in turn by the other parts (with scrutinising principles tested pragmatically). |
| 2756 | If basic beliefs can be false, falsehood in non-basic beliefs might by a symptom [Dancy,J] |
| Full Idea: Falsehood in a non-basic belief would be a reason to doubt the basic beliefs which support it, once we have admitted that basic beliefs can be false. | |
| From: Jonathan Dancy (Intro to Contemporary Epistemology [1985], 4.3) | |
| A reaction: The yearning for foundations arises from the yearning for certainty. If one embraces the fallibilist view of knowledge, as I do, then there is little motivation for foundationalism. |
| 2753 | Beliefs can only be infallible by having almost no content [Dancy,J] |
| Full Idea: Infallible beliefs must have vanishingly small content. No belief with enough content to support the superstructure in which we are really interested is going to be infallible. | |
| From: Jonathan Dancy (Intro to Contemporary Epistemology [1985], 4.2) | |
| A reaction: I see no reason why a foundationalist should not be a fallibilist, rather than insisting on the infallibility of their basic beliefs. I don't, though, see how basic beliefs can count as knowledge. |
| 2773 | Coherentism gives a possible justification of induction, and opposes scepticism [Dancy,J] |
| Full Idea: Coherentists feel that their approach provides a possible justification for induction, and offers a general stance from which the sceptic can be defused, if not rebutted. | |
| From: Jonathan Dancy (Intro to Contemporary Epistemology [1985], 8.3) | |
| A reaction: These are two good reasons why I vote for coherentism (about justification, NOT about truth). Coherence is the main tool for leading us to the best explanation. |
| 2779 | Idealists must be coherentists, but coherentists needn't be idealists [Dancy,J] |
| Full Idea: An idealist should perhaps be a coherentist, but there seems to be no reason why the coherentist should be an idealist; the link between the two is all one-way. | |
| From: Jonathan Dancy (Intro to Contemporary Epistemology [1985], 9.5) | |
| A reaction: I don't see why an idealist shouldn't be a rationalist foundationalist, with a private reality full of certainties founded on simple a priori truths. Personally I'm an empiricist coherentist, this week. |
| 2786 | For coherentists justification and truth are not radically different things [Dancy,J] |
| Full Idea: The coherentist idea is that justification and truth are not properties of radically different types. | |
| From: Jonathan Dancy (Intro to Contemporary Epistemology [1985], 11.6) | |
| A reaction: Oh. And I thought I was a coherentist. It take it that keeping coherence for foundations separate from coherence as truth is absolutely basic. The latter is nonsense. |
| 2767 | If it is empirical propositions which have to be coherent, this eliminates coherent fiction [Dancy,J] |
| Full Idea: If coherence is grounded in, and is trying to make sense of, a set of empirical propositions, this will eliminate some of the more fanciful sets of coherent propositions, such as the complete Sherlock Holmes stories. | |
| From: Jonathan Dancy (Intro to Contemporary Epistemology [1985], 8.2) | |
| A reaction: Interestingly, I suspect that embracing the coherence view of justification drives one back to empiricisim (pace Bonjour), because that is the most authoritative part of the pattern of beliefs. |
| 2776 | Externalism could even make belief unnecessary (e.g. in animals) [Dancy,J] |
| Full Idea: One reading of the externalist approach may lead to a rejection of the belief condition for knowledge (in animals, perhaps). | |
| From: Jonathan Dancy (Intro to Contemporary Epistemology [1985], 9.3) | |
| A reaction: At this point the concept of 'knowledge' seems to disperse into the mist. This pushes me to a 'setting the bar high' view of knowledge. Otherwise plants will have it, and we don't want that. |
| 2746 | How can a causal theory of justification show that all men die? [Dancy,J] |
| Full Idea: How can a causal analysis of justification show that I know that all men die? | |
| From: Jonathan Dancy (Intro to Contemporary Epistemology [1985], 2.3) | |
| A reaction: I presume he means that inductive generalisations can't be purely causal. The claim that men are immortal is absurd because it is 'unconnected' to what actually happens. |
| 2747 | Causal theories don't allow for errors in justification [Dancy,J] |
| Full Idea: Causal accounts of justification do not allow for the possibility that a false belief may still be justified. | |
| From: Jonathan Dancy (Intro to Contemporary Epistemology [1985], 2.4) | |
| A reaction: Good. If you switch to what you only think is the cause of your belief, you have gone internalist and ruined the party. You can't deny that a falsehood can be justified, or justification is vacuous. |
| 2772 | Coherentism moves us towards a more social, shared view of knowledge [Dancy,J] |
| Full Idea: An advantage of coherentism is that it directs attention away from the individual's struggle to achieve knowledge (the classical conception), and points to knowledge as a social phenomenon, shared, and increased by means of sharing. | |
| From: Jonathan Dancy (Intro to Contemporary Epistemology [1985], 8.3) | |
| A reaction: This is exactly the view which I now embrace. Internal coherence is the basis, but that spills out into the community, and into books, and into the relativity of social acceptance. |
| 2743 | What is the point of arguing against knowledge, if being right undermines your own argument? [Dancy,J] |
| Full Idea: What is the point of arguing that justified belief is impossible, for if you were right there could be no reasons for your conclusion? | |
| From: Jonathan Dancy (Intro to Contemporary Epistemology [1985], 1.3) |
| 2751 | Probabilities can only be assessed relative to some evidence [Dancy,J] |
| Full Idea: In Probability Calculus probability is only assessed relative to some evidence. | |
| From: Jonathan Dancy (Intro to Contemporary Epistemology [1985], 4.1) |
| 2757 | The argument from analogy rests on one instance alone [Dancy,J] |
| Full Idea: As an inductive argument Mill's argument from analogy (other people have inputs and outputs like mine, so the intermediate explanation must be the same) is weak because it is based on a single instance. | |
| From: Jonathan Dancy (Intro to Contemporary Epistemology [1985], 5.3) | |
| A reaction: The argument may be 'weak' as a piece of pure logic, but when faced with a strange situation, one's own case seems like crucial evidence, like a single eye-witness to a crime. |
| 2758 | You can't separate mind and behaviour, as the analogy argument attempts [Dancy,J] |
| Full Idea: The analogy argument makes the error (as Wittgenstein showed) of assuming that mind is quite separate from behaviour, and yet I can understand what it is for others to have mental states, which is contradictory. | |
| From: Jonathan Dancy (Intro to Contemporary Epistemology [1985], 5.3) | |
| A reaction: It has always seemed to me that Wittgenstein is excessively behaviourist, and he always seems to be flirting with eliminative views of mind, so he was never bothered about other minds. Minds aren't separate from behaviour, but they are distinct. |
| 8685 | Studying biology presumes the laws of chemistry, and it could never contradict them [Friend] |
| Full Idea: In the hierarchy of reduction, when we investigate questions in biology, we have to assume the laws of chemistry but not of economics. We could never find a law of biology that contradicted something in physics or in chemistry. | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 3.1) | |
| A reaction: This spells out the idea that there is a direction of dependence between aspects of the world, though we should be cautious of talking about 'levels' (see Idea 7003). We cannot choose the direction in which reduction must go. |
| 8688 | Concepts can be presented extensionally (as objects) or intensionally (as a characterization) [Friend] |
| Full Idea: The extensional presentation of a concept is just a list of the objects falling under the concept. In contrast, an intensional presentation of a concept gives a characterization of the concept, which allows us to pick out which objects fall under it. | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 3.4) | |
| A reaction: Logicians seem to favour the extensional view, because (in the standard view) sets are defined simply by their members, so concepts can be explained using sets. I take this to be a mistake. The intensional view seems obviously prior. |
| 2744 | Verificationism (the 'verification principle') is an earlier form of anti-realism [Dancy,J] |
| Full Idea: Verificationism (the 'verification principle') is an earlier form of anti-realism. | |
| From: Jonathan Dancy (Intro to Contemporary Epistemology [1985], 1.note) | |
| A reaction: If the one true God announced that there is a real world out there, I might take that as a verification of the fact. |
| 2760 | Logical positivism implies foundationalism, by dividing weak from strong verifications [Dancy,J] |
| Full Idea: The foundationalist claim that there are inferential and non-inferential justifications is mirrored by the claim of logical empiricism (the verification principle) that all significant statements are either strongly or weakly verifiable. | |
| From: Jonathan Dancy (Intro to Contemporary Epistemology [1985], 6.2) | |
| A reaction: I take it to be characteristic of both to divide the support for something into two types, one of which is basic, and the other built up on the basics. The first step is to decide what is basic. |
| 2761 | If the meanings of sentences depend on other sentences, how did we learn language? [Dancy,J] |
| Full Idea: It is clearly possible to learn a language from scratch, because we have all done it, but if holism is true and the meaning of each sentence depends on the meanings of others, how did we do it? | |
| From: Jonathan Dancy (Intro to Contemporary Epistemology [1985], 7.2) | |
| A reaction: The question of 'how did it ever get started?' actually seems to block almost every explanation of everything that ever happens. How do I begin to move my hand? |
| 2763 | There is an indeterminacy in juggling apparent meanings against probable beliefs [Dancy,J] |
| Full Idea: Indeterminacy stems from an interplay between belief and meaning, as with a man who tells you he keeps two rhinoceri in the fridge and squeezes the juice of one for a drink each morning. | |
| From: Jonathan Dancy (Intro to Contemporary Epistemology [1985], 7.4) | |
| A reaction: I don't understand why an 'interplay' is called an 'indeterminacy'. Typical philosophers. Close examination will usually show whether the change is just in belief, or just in meaning, or in both. |
| 2762 | Charity makes native beliefs largely true, and Humanity makes them similar to ours [Dancy,J] |
| Full Idea: One criterion for successful translation is that it show native beliefs to be largely true (Principle of Charity), and another is that it imputes to natives beliefs we can make sense of them having (Principle of Humanity). | |
| From: Jonathan Dancy (Intro to Contemporary Epistemology [1985], 7.4) | |
| A reaction: The trouble with such guidelines is that they always have to be 'all things being equal'. Sometimes the natives are really idiotic, and sometimes their attitudes seem quite inhuman. |
| 10364 | Facts are about the world, not in it, so they can't cause anything [Bennett] |
| Full Idea: Facts are not the sort of item that can cause anything. A fact is a true proposition (they say); it is not something in the world but is rather something about the world. | |
| From: Jonathan Bennett (Events and Their Names [1988], p.22), quoted by Jonathan Schaffer - The Metaphysics of Causation 1.1 | |
| A reaction: Compare 10361. Good argument, but maybe 'fact' is ambiguous. See Idea 10365. Events are said to be more concrete, and so can do the job, but their individuation also seems to depend on a description (as Davidson has pointed out). |