76 ideas
| 8797 | The negation of all my beliefs about my current headache would be fully coherent [Sosa] |
| Full Idea: If I have a headache, I could have a set of beliefs that I do not have a headache, that I am not in pain, that no one is in pain, and so on. The resulting system of beliefs would cohere as fully as does my actual system of beliefs. | |
| From: Ernest Sosa (The Raft and the Pyramid [1980], §9) | |
| A reaction: I think this is a misunderstanding of coherentism. Beliefs are not to be formulated through a process of coherence, but are evaluated that way. A belief that I have headache just arrives; I then see that its denial is incoherent, so I accept it. |
| 10354 | Correspondence could be with other beliefs, rather than external facts [Kusch] |
| Full Idea: The correspondence theory of truth does not commit one to the view the reality is mind-independent. There is no reason why the 'facts' that correspond to true beliefs might not themselves be beliefs or ideas. | |
| From: Martin Kusch (Knowledge by Agreement [2002], Ch.17) | |
| A reaction: This seems important, as it is very easy to assume that espousal of correspondence necessarily goes with realism about the external world. It is surprising to think that a full-blown Idealist might espouse the correspondence theory. |
| 10353 | Tarskians distinguish truth from falsehood by relations between members of sets [Kusch] |
| Full Idea: According to the Tarskians we separate out truths from falsehoods by tracing the relations between members of different sets. | |
| From: Martin Kusch (Knowledge by Agreement [2002], Ch.16) |
| 15901 | Trying to represent curves, we study arbitrary functions, leading to the ordinals, which produces set theory [Cantor, by Lavine] |
| Full Idea: The notion of a function evolved gradually from wanting to see what curves can be represented as trigonometric series. The study of arbitrary functions led Cantor to the ordinal numbers, which led to set theory. | |
| From: report of George Cantor (works [1880]) by Shaughan Lavine - Understanding the Infinite I |
| 18098 | Cantor proved that all sets have more subsets than they have members [Cantor, by Bostock] |
| Full Idea: Cantor's diagonalisation argument generalises to show that any set has more subsets than it has members. | |
| From: report of George Cantor (works [1880]) by David Bostock - Philosophy of Mathematics 4.5 | |
| A reaction: Thus three members will generate seven subsets. This means that 'there is no end to the series of cardinal numbers' (Bostock p.106). |
| 13444 | Cantor's Theorem: for any set x, its power set P(x) has more members than x [Cantor, by Hart,WD] |
| Full Idea: Cantor's Theorem says that for any set x, its power set P(x) has more members than x. | |
| From: report of George Cantor (works [1880]) by William D. Hart - The Evolution of Logic 1 |
| 15505 | If a set is 'a many thought of as one', beginners should protest against singleton sets [Cantor, by Lewis] |
| Full Idea: Cantor taught that a set is 'a many, which can be thought of as one'. ...After a time the unfortunate beginner student is told that some classes - the singletons - have only a single member. Here is a just cause for student protest, if ever there was one. | |
| From: report of George Cantor (works [1880]) by David Lewis - Parts of Classes 2.1 | |
| A reaction: There is a parallel question, almost lost in the mists of time, of whether 'one' is a number. 'Zero' is obviously dubious, but if numbers are for counting, that needs units, so the unit is the precondition of counting, not part of it. |
| 10701 | Cantor showed that supposed contradictions in infinity were just a lack of clarity [Cantor, by Potter] |
| Full Idea: Cantor's theories exhibited the contradictions others had claimed to derive from the supposition of infinite sets as confusions resulting from the failure to mark the necessary distinctions with sufficient clarity. | |
| From: report of George Cantor (works [1880]) by Michael Potter - Set Theory and Its Philosophy Intro 1 |
| 10865 | The continuum is the powerset of the integers, which moves up a level [Cantor, by Clegg] |
| Full Idea: Cantor discovered that the continuum is the powerset of the integers. While adding or multiplying infinities didn't move up a level of complexity, multiplying a number by itself an infinite number of times did. | |
| From: report of George Cantor (works [1880]) by Brian Clegg - Infinity: Quest to Think the Unthinkable Ch.14 |
| 13016 | The Axiom of Union dates from 1899, and seems fairly obvious [Cantor, by Maddy] |
| Full Idea: Cantor first stated the Union Axiom in a letter to Dedekind in 1899. It is nearly too obvious to deserve comment from most commentators. Justifications usually rest on 'limitation of size' or on the 'iterative conception'. | |
| From: report of George Cantor (works [1880]) by Penelope Maddy - Believing the Axioms I §1.3 | |
| A reaction: Surely someone can think of some way to challenge it! An opportunity to become notorious, and get invited to conferences. |
| 14199 | Cantor's sets were just collections, but Dedekind's were containers [Cantor, by Oliver/Smiley] |
| Full Idea: Cantor's definition of a set was a collection of its members into a whole, but within a few years Dedekind had the idea of a set as a container, enclosing its members like a sack. | |
| From: report of George Cantor (works [1880]) by Oliver,A/Smiley,T - What are Sets and What are they For? Intro | |
| A reaction: As the article goes on to show, these two view don't seem significantly different until you start to ask about the status of the null set and of singletons. I intuitively vote for Dedekind. Set theory is the study of brackets. |
| 10082 | There are infinite sets that are not enumerable [Cantor, by Smith,P] |
| Full Idea: Cantor's Theorem (1874) says there are infinite sets that are not enumerable. This is proved by his 1891 'diagonal argument'. | |
| From: report of George Cantor (works [1880]) by Peter Smith - Intro to Gödel's Theorems 2.3 | |
| A reaction: [Smith summarises the diagonal argument] |
| 13483 | Cantor's Paradox: the power set of the universe must be bigger than the universe, yet a subset of it [Cantor, by Hart,WD] |
| Full Idea: The problem of Cantor's Paradox is that the power set of the universe has to be both bigger than the universe (by Cantor's theorem) and not bigger (since it is a subset of the universe). | |
| From: report of George Cantor (works [1880]) by William D. Hart - The Evolution of Logic 3 | |
| A reaction: Russell eliminates the 'universe' in his theory of types. I don't see why you can't just say that the members of the set are hypothetical rather than real, and that hypothetically the universe might contain more things than it does. |
| 8710 | The powerset of all the cardinal numbers is required to be greater than itself [Cantor, by Friend] |
| Full Idea: Cantor's Paradox says that the powerset of a set has a cardinal number strictly greater than the original set, but that means that the powerset of the set of all the cardinal numbers is greater than itself. | |
| From: report of George Cantor (works [1880]) by Michčle Friend - Introducing the Philosophy of Mathematics | |
| A reaction: Friend cites this with the Burali-Forti paradox and the Russell paradox as the best examples of the problems of set theory in the early twentieth century. Did this mean that sets misdescribe reality, or that we had constructed them wrongly? |
| 15910 | Cantor named the third realm between the finite and the Absolute the 'transfinite' [Cantor, by Lavine] |
| Full Idea: Cantor believed he had discovered that between the finite and the 'Absolute', which is 'incomprehensible to the human understanding', there is a third category, which he called 'the transfinite'. | |
| From: report of George Cantor (works [1880]) by Shaughan Lavine - Understanding the Infinite III.4 |
| 15905 | Cantor proved the points on a plane are in one-to-one correspondence to the points on a line [Cantor, by Lavine] |
| Full Idea: In 1878 Cantor published the unexpected result that one can put the points on a plane, or indeed any n-dimensional space, into one-to-one correspondence with the points on a line. | |
| From: report of George Cantor (works [1880]) by Shaughan Lavine - Understanding the Infinite III.1 |
| 9983 | Cantor took the ordinal numbers to be primary [Cantor, by Tait] |
| Full Idea: Cantor took the ordinal numbers to be primary: in his generalization of the cardinals and ordinals into the transfinite, it is the ordinals that he calls 'numbers'. | |
| From: report of George Cantor (works [1880]) by William W. Tait - Frege versus Cantor and Dedekind VI | |
| A reaction: [Tait says Dedekind also favours the ordinals] It is unclear how the matter might be settled. Humans cannot give the cardinality of large groups without counting up through the ordinals. A cardinal gets its meaning from its place in the ordinals? |
| 17798 | Cantor presented the totality of natural numbers as finite, not infinite [Cantor, by Mayberry] |
| Full Idea: Cantor taught us to regard the totality of natural numbers, which was formerly thought to be infinite, as really finite after all. | |
| From: report of George Cantor (works [1880]) by John Mayberry - What Required for Foundation for Maths? p.414-2 | |
| A reaction: I presume this is because they are (by definition) countable. |
| 9971 | Cantor introduced the distinction between cardinals and ordinals [Cantor, by Tait] |
| Full Idea: Cantor introduced the distinction between cardinal and ordinal numbers. | |
| From: report of George Cantor (works [1880]) by William W. Tait - Frege versus Cantor and Dedekind Intro | |
| A reaction: This seems remarkably late for what looks like a very significant clarification. The two concepts coincide in finite cases, but come apart in infinite cases (Tait p.58). |
| 9892 | Cantor showed that ordinals are more basic than cardinals [Cantor, by Dummett] |
| Full Idea: Cantor's work revealed that the notion of an ordinal number is more fundamental than that of a cardinal number. | |
| From: report of George Cantor (works [1880]) by Michael Dummett - Frege philosophy of mathematics Ch.23 | |
| A reaction: Dummett makes it sound like a proof, which I find hard to believe. Is the notion that I have 'more' sheep than you logically prior to how many sheep we have? If I have one more, that implies the next number, whatever that number may be. Hm. |
| 14136 | A cardinal is an abstraction, from the nature of a set's elements, and from their order [Cantor] |
| Full Idea: The cardinal number of M is the general idea which, by means of our active faculty of thought, is deduced from the collection M, by abstracting from the nature of its diverse elements and from the order in which they are given. | |
| From: George Cantor (works [1880]), quoted by Bertrand Russell - The Principles of Mathematics §284 | |
| A reaction: [Russell cites 'Math. Annalen, XLVI, §1'] See Fine 1998 on this. |
| 11015 | Cantor's diagonal argument proved you can't list all decimal numbers between 0 and 1 [Cantor, by Read] |
| Full Idea: Cantor's diagonal argument showed that all the infinite decimals between 0 and 1 cannot be written down even in a single never-ending list. | |
| From: report of George Cantor (works [1880]) by Stephen Read - Thinking About Logic Ch.6 |
| 15906 | Cantor tried to prove points on a line matched naturals or reals - but nothing in between [Cantor, by Lavine] |
| Full Idea: Cantor said he could show that every infinite set of points on the line could be placed into one-to-one correspondence with either the natural numbers or the real numbers - with no intermediate possibilies (the Continuum hypothesis). His proof failed. | |
| From: report of George Cantor (works [1880]) by Shaughan Lavine - Understanding the Infinite III.1 |
| 15903 | A real is associated with an infinite set of infinite Cauchy sequences of rationals [Cantor, by Lavine] |
| Full Idea: Cantor's theory of Cauchy sequences defines a real number to be associated with an infinite set of infinite sequences of rational numbers. | |
| From: report of George Cantor (works [1880]) by Shaughan Lavine - Understanding the Infinite II.6 | |
| A reaction: This sounds remarkably like the endless decimals we use when we try to write down an actual real number. |
| 18251 | Irrational numbers are the limits of Cauchy sequences of rational numbers [Cantor, by Lavine] |
| Full Idea: Cantor introduced irrationals to play the role of limits of Cauchy sequences of rational numbers. | |
| From: report of George Cantor (works [1880]) by Shaughan Lavine - Understanding the Infinite 4.2 |
| 15902 | Irrationals and the Dedekind Cut implied infinite classes, but they seemed to have logical difficulties [Cantor, by Lavine] |
| Full Idea: From the very nature of an irrational number, it seems necessary to understand the mathematical infinite thoroughly before an adequate theory of irrationals is possible. Infinite classes are obvious in the Dedekind Cut, but have logical difficulties | |
| From: report of George Cantor (works [1880]) by Shaughan Lavine - Understanding the Infinite II Intro | |
| A reaction: Almost the whole theory of analysis (calculus) rested on the irrationals, so a theory of the infinite was suddenly (in the 1870s) vital for mathematics. Cantor wasn't just being eccentric or mystical. |
| 15908 | It was Cantor's diagonal argument which revealed infinities greater than that of the real numbers [Cantor, by Lavine] |
| Full Idea: Cantor's 1891 diagonal argument revealed there are infinitely many infinite powers. Indeed, it showed more: it shows that given any set there is another of greater power. Hence there is an infinite power strictly greater than that of the set of the reals. | |
| From: report of George Cantor (works [1880]) by Shaughan Lavine - Understanding the Infinite III.2 |
| 13464 | Cantor proposes that there won't be a potential infinity if there is no actual infinity [Cantor, by Hart,WD] |
| Full Idea: What we might call 'Cantor's Thesis' is that there won't be a potential infinity of any sort unless there is an actual infinity of some sort. | |
| From: report of George Cantor (works [1880]) by William D. Hart - The Evolution of Logic 1 | |
| A reaction: This idea is nicely calculated to stop Aristotle in his tracks. |
| 10112 | The naturals won't map onto the reals, so there are different sizes of infinity [Cantor, by George/Velleman] |
| Full Idea: Cantor showed that the complete totality of natural numbers cannot be mapped 1-1 onto the complete totality of the real numbers - so there are different sizes of infinity. | |
| From: report of George Cantor (works [1880]) by A.George / D.J.Velleman - Philosophies of Mathematics Ch.4 |
| 17889 | CH: An infinite set of reals corresponds 1-1 either to the naturals or to the reals [Cantor, by Koellner] |
| Full Idea: Cantor's Continuum Hypothesis (CH) says that for every infinite set X of reals there is either a one-to-one correspondence between X and the natural numbers, or between X and the real numbers. | |
| From: report of George Cantor (works [1880]) by Peter Koellner - On the Question of Absolute Undecidability 1.2 | |
| A reaction: Every single writer I read defines this differently, which drives me crazy, but is also helpfully illuminating. There is a moral there somewhere. |
| 10883 | Cantor's Continuum Hypothesis says there is a gap between the natural and the real numbers [Cantor, by Horsten] |
| Full Idea: Cantor's Continuum Hypothesis states that there are no sets which are too large for there to be a one-to-one correspondence between the set and the natural numbers, but too small for there to exist a one-to-one correspondence with the real numbers. | |
| From: report of George Cantor (works [1880]) by Leon Horsten - Philosophy of Mathematics §5.1 |
| 8733 | The Continuum Hypothesis says there are no sets between the natural numbers and reals [Cantor, by Shapiro] |
| Full Idea: Cantor's 'continuum hypothesis' is the assertion that there are no infinite cardinalities strictly between the size of the natural numbers and the size of the real numbers. | |
| From: report of George Cantor (works [1880]) by Stewart Shapiro - Thinking About Mathematics 2.4 | |
| A reaction: The tricky question is whether this hypothesis can be proved. |
| 13528 | Continuum Hypothesis: there are no sets between N and P(N) [Cantor, by Wolf,RS] |
| Full Idea: Cantor's conjecture (the Continuum Hypothesis) is that there are no sets between N and P(N). The 'generalized' version replaces N with an arbitrary infinite set. | |
| From: report of George Cantor (works [1880]) by Robert S. Wolf - A Tour through Mathematical Logic 2.2 | |
| A reaction: The initial impression is that there is a single gap in the numbers, like a hole in ozone layer, but the generalised version implies an infinity of gaps. How can there be gaps in the numbers? Weird. |
| 9555 | Continuum Hypothesis: no cardinal greater than aleph-null but less than cardinality of the continuum [Cantor, by Chihara] |
| Full Idea: Cantor's Continuum Hypothesis was that there is no cardinal number greater than aleph-null but less than the cardinality of the continuum. | |
| From: report of George Cantor (works [1880]) by Charles Chihara - A Structural Account of Mathematics 05.1 | |
| A reaction: I have no view on this (have you?), but the proposal that there are gaps in the number sequences has to excite all philosophers. |
| 13447 | Cantor: there is no size between naturals and reals, or between a set and its power set [Cantor, by Hart,WD] |
| Full Idea: Cantor conjectured that there is no size between those of the naturals and the reals - called the 'continuum hypothesis'. The generalized version says that for no infinite set A is there a set larger than A but smaller than P(A). | |
| From: report of George Cantor (works [1880]) by William D. Hart - The Evolution of Logic 1 | |
| A reaction: Thus there are gaps between infinite numbers, and the power set is the next size up from any infinity. Much discussion as ensued about whether these two can be proved. |
| 15893 | Cantor's theory concerns collections which can be counted, using the ordinals [Cantor, by Lavine] |
| Full Idea: Cantor's set theory was not of collections in some familiar sense, but of collections that can be counted using the indexes - the finite and transfinite ordinal numbers. ..He treated infinite collections as if they were finite. | |
| From: report of George Cantor (works [1880]) by Shaughan Lavine - Understanding the Infinite I |
| 18174 | Cantor extended ordinals into the transfinite, and they can thus measure infinite cardinalities [Cantor, by Maddy] |
| Full Idea: Cantor's second innovation was to extend the sequence of ordinal numbers into the transfinite, forming a handy scale for measuring infinite cardinalities. | |
| From: report of George Cantor (works [1880]) by Penelope Maddy - Naturalism in Mathematics I.1 | |
| A reaction: Struggling with this. The ordinals seem to locate the cardinals, but in what sense do they 'measure' them? |
| 18173 | Cardinality strictly concerns one-one correspondence, to test infinite sameness of size [Cantor, by Maddy] |
| Full Idea: Cantor's first innovation was to treat cardinality as strictly a matter of one-to-one correspondence, so that the question of whether two infinite sets are or aren't of the same size suddenly makes sense. | |
| From: report of George Cantor (works [1880]) by Penelope Maddy - Naturalism in Mathematics I.1 | |
| A reaction: It makes sense, except that all sets which are infinite but countable can be put into one-to-one correspondence with one another. What's that all about, then? |
| 10232 | Property extensions outstrip objects, so shortage of objects caused the Caesar problem [Cantor, by Shapiro] |
| Full Idea: Cantor's theorem entails that there are more property extensions than objects. So there are not enough objects in any domain to serve as extensions for that domain. So Frege's view that numbers are objects led to the Caesar problem. | |
| From: report of George Cantor (works [1880]) by Stewart Shapiro - Philosophy of Mathematics 4.6 | |
| A reaction: So the possibility that Caesar might have to be a number arises because otherwise we are threatening to run out of numbers? Is that really the problem? |
| 18176 | Pure mathematics is pure set theory [Cantor] |
| Full Idea: Pure mathematics ...according to my conception is nothing other than pure set theory. | |
| From: George Cantor (works [1880], I.1), quoted by Penelope Maddy - Naturalism in Mathematics I.1 | |
| A reaction: [an unpublished paper of 1884] So right at the beginning of set theory this claim was being made, before it was axiomatised, and so on. Zermelo endorsed the view, and it flourished unchallenged until Benacerraf (1965). |
| 8631 | Cantor says that maths originates only by abstraction from objects [Cantor, by Frege] |
| Full Idea: Cantor calls mathematics an empirical science in so far as it begins with consideration of things in the external world; on his view, number originates only by abstraction from objects. | |
| From: report of George Cantor (works [1880]) by Gottlob Frege - Grundlagen der Arithmetik (Foundations) §21 | |
| A reaction: Frege utterly opposed this view, and he seems to have won the day, but I am rather thrilled to find the great Cantor endorsing my own intuitions on the subject. The difficulty is to explain 'abstraction'. |
| 10337 | We can have knowledge without belief, if others credit us with knowledge [Kusch] |
| Full Idea: We can have knowledge that p without believing that p. It is enough that others credit us with the knowledge. | |
| From: Martin Kusch (Knowledge by Agreement [2002], Ch. 5) | |
| A reaction: [He is discussing Welbourne 1993] This is an extreme of the communitarian view. |
| 10357 | Methodological Solipsism assumes all ideas could be derived from one mind [Kusch] |
| Full Idea: 'Methodological solipsism' says merely that everyone can conceive of themselves as the only subject. Everyone can construct all referents of their thought and talk out of complexes of their very own experience. | |
| From: Martin Kusch (Knowledge by Agreement [2002], Ch.19) | |
| A reaction: The possibility of this can be denied (e.g. by Putnam 1983, dating back to Wittgenstein). I too would doubt it, though finding a good argument seems a forlorn hope. |
| 8794 | There are very few really obvious truths, and not much can be proved from them [Sosa] |
| Full Idea: Radical foundationalism suffers from two weaknesses: there are not so many perfectly obvious truths as Descartes thought; and if we restrict ourselves to what it truly obvious, very little supposed common sense knowledge can be proved. | |
| From: Ernest Sosa (The Raft and the Pyramid [1980], §3) | |
| A reaction: It is striking how few examples can ever be found of self-evident a priori truths. However, if there are self-evident truths about direct experience (pace Descartes), that would give us more than enough. |
| 8796 | A single belief can trail two regresses, one terminating and one not [Sosa] |
| Full Idea: A single belief can trail at once regresses of both sorts: one terminating and one not. | |
| From: Ernest Sosa (The Raft and the Pyramid [1980], §6) | |
| A reaction: This makes foundationalism possible, while admitting the existence of regresses. It is a good point, and triumphalist anti-foundationalists can't just point out a regress and then smugly troop off to the pub. |
| 8799 | If mental states are not propositional, they are logically dumb, and cannot be foundations [Sosa] |
| Full Idea: If a mental state is not propositional, then how can it possibly serve as a foundation for belief? How can one infer or justify anything on the basis of a state that, having no propositional content, must be logically dumb? | |
| From: Ernest Sosa (The Raft and the Pyramid [1980], §11) | |
| A reaction: This may be the best objection to foundationalism. McDowell tries to argue that conceptual content is inherent in perception, thus giving the beginnings of inbuilt propositional content. But an organism awash with bare experiences knows nothing. |
| 8795 | Mental states cannot be foundational if they are not immune to error [Sosa] |
| Full Idea: If a mental state provides no guarantee against error, then it cannot serve as a foundation for knowledge. | |
| From: Ernest Sosa (The Raft and the Pyramid [1980], §4) | |
| A reaction: That assumes that knowledge entails certainty, which I am sure it should not. On a fallibilist account, a foundation could be incredibly secure, despite a barely imaginable scenario in which it turned out to be false. |
| 10339 | Foundations seem utterly private, even from oneself at a later time [Kusch] |
| Full Idea: Foundationalists place the foundations of knowledge at a point where they are in principle accessible only to the individual knower. They cannot be 'shared' with another person, or with oneself at a later time. | |
| From: Martin Kusch (Knowledge by Agreement [2002], Ch. 8) | |
| A reaction: Kusch is defending an extremely social view of knowledge. Being private to an individual may just he an unfortunate epistemological fact. Being unavailable even to one's later self seems a real problem for foundational certainty. |
| 10331 | Testimony is reliable if it coheres with evidence for a belief, and with other beliefs [Kusch] |
| Full Idea: Testimony must be reliable since its deliveries cohere both with input from other information routes in the formation of single beliefs, and with other types of beliefs in the formation of systems of belief. | |
| From: Martin Kusch (Knowledge by Agreement [2002], Ch. 4) | |
| A reaction: Kusch criticises this view (credited to C.A.J. Coady 1992) as too individualistic , but it sounds to me dead right. I take a major appeal of the coherence account of justification to be its capacity to extend seamlessly out into external testimony. |
| 10338 | The coherentist restricts the space of reasons to the realm of beliefs [Kusch] |
| Full Idea: The coherentist restricts the space of reasons to the realm of beliefs. | |
| From: Martin Kusch (Knowledge by Agreement [2002], Ch. 8) | |
| A reaction: I endorse this idea, which endorses Davidson's slogan on the subject. The key thought is that a 'pure' sensation is uninterpreted, and so cannot justify anything. It is only once it generates a proposition that it can justify. But McDowell 1994. |
| 10340 | Individualistic coherentism lacks access to all of my beliefs, or critical judgement of my assessment [Kusch] |
| Full Idea: Individualistic versions of coherentism assume that a belief is justified if it fits with all, or most, of my contemporaneous beliefs. But who has access to that totality? Who can judge my assessment? From what position could it be judged? | |
| From: Martin Kusch (Knowledge by Agreement [2002], Ch. 8) | |
| A reaction: [compressed] Though I agree with Kusch on the social aspect of coherence, I don't think these are major criticisms. Who can access, or critically evaluate a society's body of supposedly coherent beliefs? We just do our best. |
| 10345 | Individual coherentism cannot generate the necessary normativity [Kusch] |
| Full Idea: Standard forms of coherentism are unable to account for normativity, because of their common individualism. Normativity cannot be generated within the isolated individual, or in the causal interaction between world and individual mind. | |
| From: Martin Kusch (Knowledge by Agreement [2002], Ch.10) | |
| A reaction: This thought leads to belief in rationalism and the a priori, not (as Kusch hopes) to the social dimension. How can social normativity get off the ground if there is none of it to be found in individuals? The criteria of coherence seem to be given. |
| 8798 | Vision causes and justifies beliefs; but to some extent the cause is the justification [Sosa] |
| Full Idea: Visual experience is recognized as both the cause and the justification of our visual beliefs. But these are not wholly independent. Presumably the justification that something is red derives partly from the fact that it originates in visual experience. | |
| From: Ernest Sosa (The Raft and the Pyramid [1980], §10) | |
| A reaction: Yes, but the fact that certain visual experiences originate in dreams is taken as grounds for denying their truth, not affirming it. So why do we distinguish them? I am thinking that only in the 'space of reasons' can a cause become a justification. |
| 10350 | Cultures decide causal routes, and they can be critically assessed [Kusch] |
| Full Idea: Assessments of causal routes are specific to cultures, and thus not beyond dialectical justification. | |
| From: Martin Kusch (Knowledge by Agreement [2002], Ch.11) | |
| A reaction: This is a good defence of the social and communitarian view against those who are trying to be thoroughly naturalistic and physicalist by relying entirely on causal processes for all explanation, even though I sympathise with such naturalism. |
| 10343 | Process reliabilism has been called 'virtue epistemology', resting on perception, memory, reason [Kusch] |
| Full Idea: Process reliabilism is sometimes subsumed under the label 'virtue epistemology', so that processes are 'epistemically virtuous' if they lead mostly to true beliefs. The 'intellectual virtues' here are perception, memory or reasoning. | |
| From: Martin Kusch (Knowledge by Agreement [2002], Ch. 9) | |
| A reaction: I am shocked that 'intellectual virtue' should be hijacked by reliabilists, suggesting that it even applies to a good clock. I like the Aristotelian idea that sound knowledge rests on qualities of character in the knower - including social qualities. |
| 10341 | Justification depends on the audience and one's social role [Kusch] |
| Full Idea: How a claim (about an X-ray) needs to be justified depends on whether one is confronted by a group of laypersons, or of experts, and is prescribed by one's social role. | |
| From: Martin Kusch (Knowledge by Agreement [2002], Ch. 8) | |
| A reaction: I think this is exactly right. I cannot think of any absolute criterion for justification which doesn't play straight into the hands of sceptics. Final and certain justification is an incoherent notion. But I am a little more individualistic than Kusch. |
| 10324 | Testimony does not just transmit knowledge between individuals - it actually generates knowledge [Kusch] |
| Full Idea: Testimony is not just a means of transmission of complete items of knowledge from and to an individual. Testimony is almost always generative of knowledge. | |
| From: Martin Kusch (Knowledge by Agreement [2002], Intro) | |
| A reaction: I'm not clear how my testimony could fail to be knowledge for me, but become knowledge just because I pass it to you. I might understand what I say better than you did. When fools pool their testimony, presumably not much knowledge results. |
| 10327 | Some want to reduce testimony to foundations of perceptions, memories and inferences [Kusch] |
| Full Idea: Reductionalists about testimony are foundationalists by temperament. ...Their project amounts to justifying our testimonial beliefs in terms of perceptions, memories and inferences. | |
| From: Martin Kusch (Knowledge by Agreement [2002], Ch. 4) | |
| A reaction: Kusch wants to claim that the sharing of testimony is the means by which knowledge is created. My line is something like knowledge being founded on a social coherence, which is an extension of internal individual coherence. |
| 10329 | Testimony won't reduce to perception, if perception depends on social concepts and categories [Kusch] |
| Full Idea: How can we hope to reduce testimony to perception if the way we perceive the world is to a considerable extent shaped by concepts and categories that we have learned from others? | |
| From: Martin Kusch (Knowledge by Agreement [2002], Ch. 4) | |
| A reaction: To me this sounds like good support for coherentism, the benign circle between my reason, my experience, and the testimony and reason of others. Asking how the circle could get started shows ignorance of biology. |
| 10330 | A foundation is what is intelligible, hence from a rational source, and tending towards truth [Kusch] |
| Full Idea: It can be argued that testimony is non-reductive because it relies on the fact that whatever is intelligible is likely to come from a rational source, and that rational sources, by their very nature, tend towards the truth. | |
| From: Martin Kusch (Knowledge by Agreement [2002], Ch. 4 n7) | |
| A reaction: [He cites Tyler Burge 1993, 1997] If this makes testimony non-reductive, how would one assess whether the testimony is 'intelligible'? |
| 10334 | Testimony is an area in which epistemology meets ethics [Kusch] |
| Full Idea: Testimony is an area in which epistemology meets ethics. | |
| From: Martin Kusch (Knowledge by Agreement [2002], Ch. 5) | |
| A reaction: This is very thought-provoking. A key concept linking the two would be 'respect'. Consider also 'experts'. |
| 10336 | Powerless people are assumed to be unreliable, even about their own lives [Kusch] |
| Full Idea: The powerless in society are not usually taken to be trustworthy witnesses even when it comes to providing information about their own lives. | |
| From: Martin Kusch (Knowledge by Agreement [2002], Ch. 5) | |
| A reaction: This is where epistemology shades off into politics and the writings of Foucault. |
| 10325 | Vindicating testimony is an expression of individualism [Kusch] |
| Full Idea: To believe that testimony needs a general vindication is itself an expression of individualism. | |
| From: Martin Kusch (Knowledge by Agreement [2002], Intro) | |
| A reaction: Kusch is a spokesman for Communitarian Epistemology. Surely we are allowed to identify the criteria for what makes a good witness? Ask a policeman. |
| 10335 | Myths about lonely genius are based on epistemological individualism [Kusch] |
| Full Idea: Many myths about the lonely scientific genius underwrite epistemological individualism. | |
| From: Martin Kusch (Knowledge by Agreement [2002], Ch. 5) | |
| A reaction: They all actually say that they 'stood on the shoulders of giants', and they are invariably immersed in the contemporary researches of teams of like-minded people. How surprised were the really expert contemporaries by Newton, Einstein, Gödel? |
| 10323 | Communitarian Epistemology says 'knowledge' is a social status granted to groups of people [Kusch] |
| Full Idea: I propose 'communitarian epistemology' - claiming first that the term 'knowledge' marks a social status, and is dependent on the existence of communities, and second that this social status is typically granted to groups of people. | |
| From: Martin Kusch (Knowledge by Agreement [2002], Intro) | |
| A reaction: I find this very congenial, though Kusch goes a little far when he claims that knowledge is largely created by social groups. He allows that Robinson Crusoe might have knowledge of his island, but can't give a decent account of it. |
| 10348 | Private justification is justification to imagined other people [Kusch] |
| Full Idea: Coming to convince myself is actually to form a pretend communal belief with pretend others, ..which is clearly parasitic on the case where the others are real. | |
| From: Martin Kusch (Knowledge by Agreement [2002], Ch.11) | |
| A reaction: This slightly desperate move is a way for 'communitarian' epistemologists to deal with Robinson Crusoe cases. I think Kusch is right, but it is a bit hard to prove that this is what is 'actually' going on. |
| 10349 | To be considered 'an individual' is performed by a society [Kusch] |
| Full Idea: One cannot even have the social status of 'being an individual' unless it has been conferred on one by a communal performative belief. | |
| From: Martin Kusch (Knowledge by Agreement [2002], Ch.11) | |
| A reaction: This sounds crazy until you think of the mentality of a tenth generation slave in a fully slave-owning society. |
| 8715 | Infinities expand the bounds of the conceivable; we explore concepts to explore conceivability [Cantor, by Friend] |
| Full Idea: Cantor (in his exploration of infinities) pushed the bounds of conceivability further than anyone before him. To discover what is conceivable, we have to enquire into the concept. | |
| From: report of George Cantor (works [1880]) by Michčle Friend - Introducing the Philosophy of Mathematics 6.5 | |
| A reaction: This remark comes during a discussion of Husserl's phenomenology. Intuitionists challenge Cantor's claim, and restrict what is conceivable to what is provable. Does possibility depend on conceivability? |
| 10344 | Our experience may be conceptual, but surely not the world itself? [Kusch] |
| Full Idea: I am unconvinced by McDowell's arguments in favour of treating the world as itself conceptual. Granted that our experience is conceptual in quality; it still does not follow that the world itself is conceptual. | |
| From: Martin Kusch (Knowledge by Agreement [2002], Ch. 9) | |
| A reaction: I would take Kusch's point to be a given in any discussion of concepts, and McDowell as a non-starter on this one. I am inclined to believe that we do have non-conceptual experiences, but I take them to be epistemologically useless. |
| 13454 | Cantor says (vaguely) that we abstract numbers from equal sized sets [Hart,WD on Cantor] |
| Full Idea: Cantor thought that we abstract a number as something common to all and only those sets any one of which has as many members as any other. ...However one wants to see the logic of the inference. The irony is that set theory lays out this logic. | |
| From: comment on George Cantor (works [1880]) by William D. Hart - The Evolution of Logic 1 | |
| A reaction: The logic Hart has in mind is the notion of an equivalence relation between sets. This idea sums up the older and more modern concepts of abstraction, the first as psychological, the second as logical (or trying very hard to be!). Cf Idea 9145. |
| 10358 | Often socialising people is the only way to persuade them [Kusch] |
| Full Idea: Often we can convince members of other cultures only by socializing them into our culture. | |
| From: Martin Kusch (Knowledge by Agreement [2002], Ch.19) | |
| A reaction: This looks both true and interesting, and is good support for Kusch's communitarian epistemology. What actually persuades certainly doesn't have to be reasons, and may be almost entirely social. |
| 10333 | Communitarianism in epistemology sees the community as the primary knower [Kusch] |
| Full Idea: Communitarianism in epistemology sees the community as the primary knower. | |
| From: Martin Kusch (Knowledge by Agreement [2002], Ch. 5) | |
| A reaction: This thought offers an account of epistemology which could fit in with communitarian political views. See the ideas of Martin Kusch in this database. |
| 10351 | Natural kinds are social institutions [Kusch] |
| Full Idea: Natural kinds are social institutions. | |
| From: Martin Kusch (Knowledge by Agreement [2002], Ch.11) | |
| A reaction: I can see what he means, but I take this to be deeply wrong. A clarification of what exactly is meant by a 'natural kind' is needed before we can make any progress with this one. Is a village a natural kind? Or a poodle? Or a shoal? |
| 10863 | Cantor proved that three dimensions have the same number of points as one dimension [Cantor, by Clegg] |
| Full Idea: Cantor proved that one-dimensional space has exactly the same number of points as does two dimensions, or our familiar three-dimensional space. | |
| From: report of George Cantor (works [1880]) by Brian Clegg - Infinity: Quest to Think the Unthinkable Ch.14 |
| 13465 | Only God is absolutely infinite [Cantor, by Hart,WD] |
| Full Idea: Cantor said that only God is absolutely infinite. | |
| From: report of George Cantor (works [1880]) by William D. Hart - The Evolution of Logic 1 | |
| A reaction: We are used to the austere 'God of the philosophers', but this gives us an even more austere 'God of the mathematicians'. |
| 10332 | Omniscience is incoherent, since knowledge is a social concept [Kusch] |
| Full Idea: The very idea of omniscience is dubious, at least for the communitarian epistemologist, since knowing is a social state, and knowledge is a social status, needing a position in a social network. | |
| From: Martin Kusch (Knowledge by Agreement [2002], Ch. 4) | |
| A reaction: A nice test case. Would an omniscient mind have evidence for its beliefs? Would it continually check for coherence? Is it open to criticism? Does it even entertain the possibility of error? Could another 'omniscient' mind challenge it? |