104 ideas
| 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 |
| 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 |
| 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). |
| 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. |
| 4098 | The theory of descriptions supports internalism, since they are thinkable when the object is non-existent [Crane] |
| Full Idea: The theory of descriptions gives a model of internalist intentionality, in that it describes cases where the thinkability of a belief does not depend on the existence of a specific object. | |
| From: Tim Crane (Elements of Mind [2001], 4.36) | |
| A reaction: So what do externalists say about the theory? Surely a reference to 'water' can't entail the existence of water? |
| 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 Michèle 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. |
| 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 |
| 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 |
| 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 |
| 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. |
| 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. |
| 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. |
| 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 |
| 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. |
| 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? |
| 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 |
| 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'. |
| 4077 | Aesthetic properties of thing supervene on their physical properties [Crane] |
| Full Idea: It is sometimes said that the aesthetic properties of a thing supervene on its physical properties. | |
| From: Tim Crane (Elements of Mind [2001], 2.16) | |
| A reaction: A confusing example, as aesthetic properties only exist if there is an observer. Is 'supervenience' just an empty locution which tries to avoid reduction? |
| 4078 | Constitution (as in a statue constituted by its marble) is supervenience without identity [Crane] |
| Full Idea: A statue is constituted by the marble that makes it up. It is plausible to say that constitution is not the same as identity - since identity is symmetrical and identity is not - but nonetheless constitution is a supervenience relation. | |
| From: Tim Crane (Elements of Mind [2001], 2.16) | |
| A reaction: So what makes it a statue, as opposed to a piece of marble? It may well be an abstraction which only exists relative to observers. |
| 4082 | The distinction between 'resultant' properties (weight) and 'emergent' properties is a bit vague [Crane] |
| Full Idea: The distinction between 'resultant' properties like weight, and 'emergent' properties like colour, seems intuitive enough, but on examination it is very hard to make precise. | |
| From: Tim Crane (Elements of Mind [2001], 2.18) | |
| A reaction: It is no coincidence that the examples are of primary and secondary qualities. If 'the physical entails the mental' then all mental properties are resultant. |
| 4083 | If mental properties are emergent they add a new type of causation, and physics is not complete [Crane] |
| Full Idea: Whatever the causal process is, it remains true that if emergentism is true, the completeness of physics is false; there are some effects which would not have come about if mental things were absent from the world. | |
| From: Tim Crane (Elements of Mind [2001], 2.18) | |
| A reaction: Emergentism looks to me like an incoherent concept, unless it is another word for dualism. |
| 4079 | Properties are causes [Crane] |
| Full Idea: Properties are causes. | |
| From: Tim Crane (Elements of Mind [2001], 2.17) | |
| A reaction: We can't detect properties if they lack causal powers. This may be a deep confusion. Properties are what make causal powers possible, but that isn't what properties are? |
| 4068 | Traditional substance is separate from properties and capable of independent existence [Crane] |
| Full Idea: The traditional concept of substance says substances bear properties which are distinct from them, and substances are capable of independent existence. | |
| From: Tim Crane (Elements of Mind [2001], 2.9) | |
| A reaction: Put like that, it sounds ridiculous as a physical theory. It is hard to dislodge substance, though, from a priori human metaphysics. |
| 4096 | Maybe beliefs don't need to be conscious, if you are not conscious of the beliefs guiding your actions [Crane] |
| Full Idea: The beliefs that are currently guiding your actions do not need to be in your stream of consciousness, which suggests that beliefs do not need to be conscious at all. | |
| From: Tim Crane (Elements of Mind [2001], 4.31) | |
| A reaction: Too bold, I think. Presumably this would eliminate all the other propositional attitudes from consciousness. There would only be qualia left! |
| 4097 | Maybe there are two kinds of belief - 'de re' beliefs and 'de dicto' beliefs [Crane] |
| Full Idea: Some philosophers have claimed that there are two kinds of belief, 'de re' belief and 'de dicto' belief. | |
| From: Tim Crane (Elements of Mind [2001], 4.35) | |
| A reaction: Interesting, though it may only distinguish two objects of belief, not two types. Internalist and externalist views are implied. |
| 4093 | Many cases of knowing how can be expressed in propositional terms (like how to get somewhere) [Crane] |
| Full Idea: There are plenty of cases of knowing how to do something, where that knowledge can also be expressed - without remainder, as it were - in propositional terms (such as knowing how to get to the Albert Hall). | |
| From: Tim Crane (Elements of Mind [2001], 3.28) | |
| A reaction: Presumably all knowing how could be expressed propositionally by God. |
| 4108 | Phenol-thio-urea tastes bitter to three-quarters of people, but to the rest it is tasteless, so which is it? [Crane] |
| Full Idea: Phenol-thio-urea tastes bitter to three-quarters of people, but to the rest it is tasteless. Is it really bitter, or really tasteless? | |
| From: Tim Crane (Elements of Mind [2001], 5.44) | |
| A reaction: A nice reinforcement of a classic Greek question. Good support for the primary/secondary distinction. Common sense, really. |
| 4105 | The traditional supports for the sense datum theory were seeing double and specks before one's eyes [Crane] |
| Full Idea: The traditional examples used to support the sense datum theory were seeing double and specks before one's eyes. | |
| From: Tim Crane (Elements of Mind [2001], 5.43) | |
| A reaction: Presumably, though, direct realists can move one eye, or having something wrong with a retina. |
| 4104 | One can taste that the wine is sour, and one can also taste the sourness of the wine [Crane] |
| Full Idea: One can taste that the wine is sour, and one can also taste the sourness of the wine. | |
| From: Tim Crane (Elements of Mind [2001], 5.42) | |
| A reaction: …so sense data are optional? We create sense data by objectifying them, but animals just taste the wine, and are direct realists. Tasting the sourness seems to be a case of abstraction. |
| 4101 | If we smell something we are aware of the smell separately, but we don't perceive a 'look' when we see [Crane] |
| Full Idea: Visual perception seems to differ from some of the other senses; when we become aware of burning toast, we become aware of the smell, ...but we don't see a garden by seeing a 'look' of the garden. | |
| From: Tim Crane (Elements of Mind [2001], 5.40) | |
| A reaction: Interesting. Do blind people transfer this more direct perception to a different sense (e.g. the one they rely on most)? |
| 4102 | The problems of perception disappear if it is a relation to an intentional state, not to an object or sense datum [Crane] |
| Full Idea: The solution to the problem of perception is to deny that it is related to real objects (things or sense-data); rather, perception is an intentional state (with a subject, mode and content), a relation to the intentional content. | |
| From: Tim Crane (Elements of Mind [2001], 5.42) | |
| A reaction: Not clear. This definition makes it sound like a propositional attitude. |
| 4109 | If perception is much richer than our powers of description, this suggests that it is non-conceptual [Crane] |
| Full Idea: The richness in information of perceptual experience outruns our modes of description of it, which has led some philosophers to claim that the content of perceptual experience is non-conceptual. | |
| From: Tim Crane (Elements of Mind [2001], 5.45) | |
| A reaction: It certainly implies that it can't be entirely conceptual, but it still may be that in humans concepts are always involved. Not when I'm waking up in the morning, though. |
| 4103 | The adverbial theory of perceptions says it is the experiences which have properties, not the objects [Crane] |
| Full Idea: The Adverbial Theory of perception holds that the predicates which other theories take as picking out the properties of objects are really adverbs of the perceptual verb; ..instead of strange objects, we just have properties of experiences. | |
| From: Tim Crane (Elements of Mind [2001], 5.42) | |
| A reaction: Promising. It fits secondary qualities all right, but what about primary? I 'see bluely', but can I 'see squarely'? |
| 4065 | Is knowledge just a state of mind, or does it also involve the existence of external things? [Crane] |
| Full Idea: It is controversial whether knowledge is a state of mind, or a composite state involving a thought about something, plus its existence. | |
| From: Tim Crane (Elements of Mind [2001], 1.5) | |
| A reaction: Pinpoints the internalism/externalism problem. Knowledge is a special type of belief (but maybe belief with external links!). Tricky. I vote for internalism. |
| 4092 | The core of the consciousness problem is the case of Mary, zombies, and the Hard Question [Crane] |
| Full Idea: The three arguments that have been used to articulate the problem of consciousness are the knowledge argument ('Mary'), the possibility of 'zombies' (creatures like us but lacking phenomenal consciousness), and the explanatory gap (the Hard Question). | |
| From: Tim Crane (Elements of Mind [2001], 3.26) | |
| A reaction: All of these push towards the implausible claim that there could never be a physical explanation of why we experience things. Zombies are impossible, in my opinion. |
| 4087 | Intentionalism does not require that all mental states be propositional attitudes [Crane] |
| Full Idea: Intentionalism (the doctrine that all mental states are intentional) need not be the thesis that all mental states are propositional attitudes. | |
| From: Tim Crane (Elements of Mind [2001], 3.22) | |
| A reaction: This points to the requirement for an intentionalist to prove that so-called 'qualia' states are essentially intentional, which is not implausible. |
| 4095 | Object-directed attitudes like love are just as significant as propositional attitudes [Crane] |
| Full Idea: Love, hate, and the other object-directed attitudes have as much of a role in explaining behaviour as the propositional attitudes. | |
| From: Tim Crane (Elements of Mind [2001], 4.34) | |
| A reaction: A good clarification of the range of intentional states. Objects seem to be external, where propositions are clearly internal. |
| 4106 | If someone removes their glasses the content of experience remains, but the quality changes [Crane] |
| Full Idea: There is a phenomenal difference between a short-sighted person wearing glasses and not; they do not judge that the world is different, but the properties of the experience (the qualia) have changed. | |
| From: Tim Crane (Elements of Mind [2001], 5.43) | |
| A reaction: Could be challenged. If a notice becomes unreadable, that is more than the qualia changing. |
| 4089 | Pains have a region of the body as their intentional content, not some pain object [Crane] |
| Full Idea: The intentional object of a pain-state is a part or region of the body, not a pain-object. | |
| From: Tim Crane (Elements of Mind [2001], 3.24) | |
| A reaction: Plausible. Has anyone ever suffered from pain without some sense of what part of the body is actually in pain? |
| 4090 | Weak intentionalism says qualia are extra properties; strong intentionalism says they are intentional [Crane] |
| Full Idea: Weak intentionalism says all mental states are intentional, but qualia are higher-order properties of these states. ..Strong intentionalists say the phenomenal character of a sensation consists purely in that state's intentionality. | |
| From: Tim Crane (Elements of Mind [2001], 3.25) | |
| A reaction: The weak version sounds better. Asking 'how could a thought have a quality of experience just by being about something?' is a restatement of the traditional problem, which won't go away. The Hard Question. |
| 4107 | With inverted qualia a person's experiences would change, but their beliefs remain the same [Crane] |
| Full Idea: The right thing to say about inverted qualia is that the person's experiences are different from other people's, but their beliefs are the same. | |
| From: Tim Crane (Elements of Mind [2001], 5.44) | |
| A reaction: Right - which reinforces the idea that all beliefs are the result of judgement, and none come directly from perception. |
| 4069 | Descartes did not think of minds as made of a substance, because they are not divisible [Crane] |
| Full Idea: It would be wrong to represent Descartes' view as the idea that bodies are made of one kind of stuff and minds of another; he did not think minds are made of stuff at all, because then they would be divisible. | |
| From: Tim Crane (Elements of Mind [2001], 2.10) | |
| A reaction: I'm not convinced. It could be an indivisible substance. Without a mental substance, Descartes may have to say the mind is an abstraction, perhaps a pattern of Platonic forms. |
| 4074 | Functionalism defines mental states by their causal properties, which rules out epiphenomenalism [Crane] |
| Full Idea: Functionalism holds that it is in the nature of certain mental states to have certain effects; therefore there can be no mental epiphenomena. | |
| From: Tim Crane (Elements of Mind [2001], 2.14) | |
| A reaction: I strongly resist the idea that a thing's identity is its function. Functionalism may not say that. Mind is an abstraction referring to a causal nexus of unknowable components. |
| 4091 | The problems of misrepresentation and error have dogged physicalist reductions of intentionality [Crane] |
| Full Idea: The fundamental problems of misrepresentation and error have dogged physicalist reductions of intentionality. | |
| From: Tim Crane (Elements of Mind [2001], 3.26) | |
| A reaction: If footprints or tree-rings are the model for reductions of intentionality, there doesn't seem much scope in them for giving false information, except by some freak event. |
| 4070 | Properties dualism says mental properties are distinct from physical, despite a single underlying substance [Crane] |
| Full Idea: According to property dualism, mental properties are distinct from physical properties, even though they are properties of one substance. | |
| From: Tim Crane (Elements of Mind [2001], 2.10) | |
| A reaction: Two properties may be phenomenologically different (transparent and magnetic), but that doesn't put them in different ontological categories. |
| 4084 | Non-reductive physicalism seeks an explanation of supervenience, but emergentists accept it as basic [Crane] |
| Full Idea: While the non-reductive physicalist believes that mental/physical supervenience must be explained, the emergentist is willing to accept it as a fact of nature. | |
| From: Tim Crane (Elements of Mind [2001], 2.18) | |
| A reaction: A good reason not to be an emergentist. No philosopher should abandon the principle of sufficient reason. |
| 4080 | If mental supervenes on the physical, then every physical cause will be accompanied by a mental one [Crane] |
| Full Idea: If the mental supervenes on the physical, then whenever a physical cause brings about some effect, a mental cause comes along for the ride. | |
| From: Tim Crane (Elements of Mind [2001], 2.17) | |
| A reaction: This is why supervenience seems to imply epiphenomenalism. The very concept of supervenience is dubious. |
| 4075 | Identity theory is either of particular events, or of properties, depending on your theory of causation [Crane] |
| Full Idea: If causation concerns events, then we have an identity theory of mental and physical events (particulars) [Davidson]. If causation is by properties, then it is mental and physical properties which are identical [Lewis and Armstrong]. | |
| From: Tim Crane (Elements of Mind [2001], 2.14) | |
| A reaction: Events are tokens, and properties are types. Tricky. Events are dynamic, but properties can be static. |
| 4085 | Physicalism may be the source of the mind-body problem, rather than its solution [Crane] |
| Full Idea: Physicalism may be the source of the mind-body problem, rather than its solution. | |
| From: Tim Crane (Elements of Mind [2001], 2.19) | |
| A reaction: Certainly if the physical is seen as just a pile of atoms, it is hard to see how they could ever think (see idea 1909). |
| 4073 | Overdetermination occurs if two events cause an effect, when each would have caused it alone [Crane] |
| Full Idea: Causal overdetermination is when an effect has more than one cause, and each event would have caused the effect if the other one had not done so. | |
| From: Tim Crane (Elements of Mind [2001], 2.13) | |
| A reaction: Overdetermination is a symptom that an explanation is questionable, but it can occur. Two strong people can join to push over a light hatstand. |
| 4072 | The completeness of physics must be an essential component of any physicalist view of mind [Crane] |
| Full Idea: I claim that the completeness of physics must be an essential component of any physicalist view of mind. | |
| From: Tim Crane (Elements of Mind [2001], 2.12) | |
| A reaction: He does not convince me of this. The mind may be within physics, but why should we say a priori that no exceptions to physical law will ever be discovered. Crane is setting up straw men. |
| 4094 | Experience teaches us propositions, because we can reason about our phenomenal experience [Crane] |
| Full Idea: In experience we learn propositions, since someone can reason using the sentence 'Red looks like this' (e.g. 'If red looks like this, then either it looks like this to dogs or it doesn't'). | |
| From: Tim Crane (Elements of Mind [2001], 3.28) | |
| A reaction: The fact that we can create propositions about experiences doesn't prove that experience is inherently propositional. |
| 4100 | The Twin Earth argument depends on reference being determined by content, which may be false. [Crane] |
| Full Idea: The Twin Earth argument does not refute internalism, since it depends on the 'Content-Determines-Reference' principle, which internalists can reject. | |
| From: Tim Crane (Elements of Mind [2001], 4.37) | |
| A reaction: The idea is that content should be understood in a context (e.g. on a particular planet). Indexicals count against a totally narrow view of content (Twins thinking 'I am here'). |
| 4067 | Broad content entails the existence of the object of the thought [Crane] |
| Full Idea: If a mental state is broad, then the existence of the mental state entails the existence of its object. | |
| From: Tim Crane (Elements of Mind [2001], 1.7) | |
| A reaction: Hence thinking of non-existent things like unicorns is problematic for externalists. However, externalists can think about numbers or Platonic ideals. |
| 4063 | In intensional contexts, truth depends on how extensions are conceived. [Crane] |
| Full Idea: Intensional contexts are those where truth or falsehood depends on the way the extensions are conceived. | |
| From: Tim Crane (Elements of Mind [2001], 1.4) | |
| A reaction: An important distinction for anyone defending an internalist view of concepts or of knowledge |
| 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 Michèle 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? |
| 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. |
| 4071 | Causation can be seen in counterfactual terms, or as increased probability, or as energy flow [Crane] |
| Full Idea: A theory of causation might say 'If A had not existed, B would not have existed' (counterfactual theory), or 'B is more likely if A occurs' (probabilistic), or 'energy flows from A to B'. | |
| From: Tim Crane (Elements of Mind [2001], 2.11) | |
| A reaction: As always, it is vital to separate epistemology from ontology. Energy won't cover agents. Whisper "Fire!" in a theatre. |
| 4076 | Causes are properties, not events, because properties are what make a difference in a situation [Crane] |
| Full Idea: My view is that causes are properties (not events); when we look for causes, we look for the aspect of a situation which made a difference, and aspects are properties or qualities. | |
| From: Tim Crane (Elements of Mind [2001], 2.14) | |
| A reaction: He is talking about explanations, which may not be causes, or at least they have a different emphasis. Don't events 'make a difference'? Events are ontologically weird |
| 21181 | Relativity and Quantum theory give very different accounts of forces [Hesketh] |
| Full Idea: General Relativity and quantum mechanics are the two great theories in physics today but they give two very different ideas for how forces work. | |
| From: Gavin Hesketh (The Particle Zoo [2016], 01) | |
| A reaction: Relativity says it is space curvature, and quantum theory says it is particle exchange? But is there a Relativity account of the strong nuclear force? |
| 21183 | Thermodynamics introduced work and entropy, to understand steam engine efficiency [Hesketh] |
| Full Idea: The Laws of Thermodynamics introduced the concepts of entropy and work; put simply, how much useful energy you can really get out of a steam engine. | |
| From: Gavin Hesketh (The Particle Zoo [2016], 03) | |
| A reaction: The point of science by this stage was to introduce measurable and quantifiable concepts |
| 21191 | Photons are B and W° bosons, linked by the Higgs mechanism [Hesketh] |
| Full Idea: The photon is actually a mix of two deeper things, the B and the W°, tied together by the Higgs mechanism. | |
| From: Gavin Hesketh (The Particle Zoo [2016], 06) | |
| A reaction: The B (for 'Boson') transmits a force associated with the 'winding symmetry'. (I record this without properly understanding it.) |
| 21199 | Spinning electric charge produces magnetism, so all fermions are magnets [Hesketh] |
| Full Idea: The muon, like all fermions, spins - and because a spinning electric charge generates a magnetic field all fermions act like tiny bar magnets. | |
| From: Gavin Hesketh (The Particle Zoo [2016], 11) |
| 21189 | Electrons may have smaller components, bound by a new force [Hesketh] |
| Full Idea: Quarks, leptons or bosons may actually be made up of something even smaller, bound together by a conjectural new force. | |
| From: Gavin Hesketh (The Particle Zoo [2016], 05) | |
| A reaction: Electrons are a type of lepton. Compare Idea 21180, from the same book. If electrons are not fundamental, what matters is not some 'stuff' they are made of, but a different force that would bind the ingredients. |
| 21180 | Electrons are fundamental and are not made of anything; they are properties without size [Hesketh] |
| Full Idea: As far as we can tell, electrons (and quarks) are fundamental. They are not small lumps of material, because we could always ask what the material is. The electron just ...is. They are collections of properties, with no apparent size. | |
| From: Gavin Hesketh (The Particle Zoo [2016], 01) | |
| A reaction: This idea from physics HAS to be of interest to philosophers! The bundle theory is discredited for normal objects and for minds, and so is the substrate idea for supporting properties. But rigorous physics accepts a bundle theory. |
| 21182 | Quantum mechanics is our only theory, and is very precise, and repeatedly confirmed [Hesketh] |
| Full Idea: Quantum mechanics is the only working description of the universe that we have. It is amazingly precise, and so far every experimental test has verified its predictions. | |
| From: Gavin Hesketh (The Particle Zoo [2016], 02) | |
| A reaction: I take it from this that quantum mechanics is simply TRUE. Get over it! It will never turn out to be wrong, but may be subsumed within some more fine-grained or extensive theory. |
| 21184 | Physics was rewritten to explain stable electron orbits [Hesketh] |
| Full Idea: Explaining the stable electron orbits would require a complete rewriting of the physics of subatomic particles. | |
| From: Gavin Hesketh (The Particle Zoo [2016], 03) | |
| A reaction: This really looks like a simple and major landmark moment. You can ignore a single anomaly, but not a central feature of your entire theory. |
| 21187 | Virtual particles can't be measured, and can ignore the laws of physics [Hesketh] |
| Full Idea: We can never measure these virtual (transitory) particles directly, and it turns out that they don't even have to obey the laws of physics. | |
| From: Gavin Hesketh (The Particle Zoo [2016], 05) | |
| A reaction: These seems to be the real significance of the Uncertainty Principle. Such particles 'borrow' huge amounts of energy for very short times. |
| 21185 | Colour charge is positive or negative, and also has red, green or blue direction [Hesketh] |
| Full Idea: Colour charge is 'three-dimensional'. As well as the charge having a positive or negative sign, it can also have a direction, and for convenience these three different directions (pointing like a weather vane) are labelled 'red', 'green' and 'blue'. | |
| From: Gavin Hesketh (The Particle Zoo [2016], 04) |
| 21194 | The Standard Model omits gravity, because there are no particles involved [Hesketh] |
| Full Idea: Gravity is not included in the Standard Model because we simply cannot study it using particles. | |
| From: Gavin Hesketh (The Particle Zoo [2016], 09) | |
| A reaction: I'm guessing that Einstein describes how gravity behaves, but not what it is. |
| 21195 | In Supersymmetry the Standard Model simplifies at high energies [Hesketh] |
| Full Idea: Supersymmetry suggest that the Standard Model becomes much simpler at high energies. | |
| From: Gavin Hesketh (The Particle Zoo [2016], 10) |
| 21197 | Standard Model forces are one- two- and three-dimensional [Hesketh] |
| Full Idea: The forces in the Standard Model are built on gauge symmetries, with a one-dimensional charge (like electromagnetism), a two-dimensional charge (the weak force), and a three dimensional charge (the strong force). | |
| From: Gavin Hesketh (The Particle Zoo [2016], 10) | |
| A reaction: See also Idea 21185. |
| 21188 | Quarks and leptons have a weak charge, for the weak force [Hesketh] |
| Full Idea: For the weak force there must be a corresponding 'weak charge', and all the fermions, all the quarks and leptons carry it. | |
| From: Gavin Hesketh (The Particle Zoo [2016], 05) | |
| A reaction: So electrons carry a weak charge, as well as an electromagnetic charge. Like owning several passports. |
| 21186 | Quarks rush wildly around in protons, restrained by the gluons [Hesketh] |
| Full Idea: Inside a proton the quarks are rushing around like caged animals, free to move until they push against the bars to try to escape, when the gluons pull them back in. | |
| From: Gavin Hesketh (The Particle Zoo [2016], 04) |
| 21192 | Neutrinos only interact with the weak force, but decays produce them in huge numbers [Hesketh] |
| Full Idea: Neutrinos only interact with the weak force, which means they barely interact at all, but because the weak force is crucial in the decays of so many other particles, neutrinos are still produced in huge numbers. | |
| From: Gavin Hesketh (The Particle Zoo [2016], 08) | |
| A reaction: They only interact with the W and Z bosons. |
| 21196 | To combine the forces, they must all be the same strength at some point [Hesketh] |
| Full Idea: If all the forces are to combine, at some point they must all be the same strength, and Supersymmetry (SuSy) makes this happen. | |
| From: Gavin Hesketh (The Particle Zoo [2016], 10) | |
| A reaction: This sounds like an impressive reason for favouring supersymmetry - as long as you have an a priori preference for everything combining. |
| 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 |
| 21190 | 'Space' in physics just means location [Hesketh] |
| Full Idea: 'Space' in physics really just means location. | |
| From: Gavin Hesketh (The Particle Zoo [2016], 06) | |
| A reaction: Location can, of course, only be specified relative to something else. Space is really an abstraction, but at least it means there is some sort of background to locate all the fundamental fields. |
| 21193 | The universe is 68% dark energy, 27% dark matter, 5% regular matter [Hesketh] |
| Full Idea: The most precise surveys of the stars and galaxies tell us that the universe is made up of 68% dark energy, 27% dark matter, and just 5% regular matter (the stuff of the Standard Model of particle physics). | |
| From: Gavin Hesketh (The Particle Zoo [2016], 09) | |
| A reaction: Regular matter - that's me, that is. |
| 21198 | If a cosmic theory relies a great deal on fine-tuning basic values, it is probably wrong [Hesketh] |
| Full Idea: If a theory has to rely on excessive 'fine-tuning', a series of extremely unlikely events in order to produce the universe we see around us, then it is extremely unlikely that this theory is correct. | |
| From: Gavin Hesketh (The Particle Zoo [2016], 10) | |
| A reaction: He says the Standard Model has 26 parameters which are only known by experiment, rather than by theory. So instead of saying '...so there is a God', we should say '...so our theory isn't very good'. |
| 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'. |
| 4066 | It seems that 'exists' could sometimes be a predicate [Crane] |
| Full Idea: The view that 'exists' is never a predicate is not plausible. | |
| From: Tim Crane (Elements of Mind [2001], 1.7) | |
| A reaction: He doesn't enlarge. Russell says 'exists' is a quantifier. 'Your very existence offends me - I hope it is confiscated'. |