95 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 |
| 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 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. |
| 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'. |
| 2392 | Properties supervene if you can't have one without the other [Chalmers] |
| Full Idea: B-properties supervene on A-properties if no two possible situations are identical with respect to their A-properties while differing in their B-properties. | |
| From: David J.Chalmers (The Conscious Mind [1996], 1.2.1) | |
| A reaction: Personally I would have thought that if this condition is achieved, then we could go on to say B-properties supervene on A because A is causing them. We shouldn't be shy about this. Personally I think the Bs are necessary. |
| 2393 | Logical supervenience is when one set of properties must be accompanied by another set [Chalmers] |
| Full Idea: B-properties logically supervene on A-properties if no two logically possible situations are identical with respect to their A-properties but distinct with respect to their B-properties. | |
| From: David J.Chalmers (The Conscious Mind [1996], 1.2.1) | |
| A reaction: This is the gap into which Chalmers wants to slip zombies. He's wrong. He thinks that because he can imagine Bs without As, that this makes their separation logically possible. No doubt he can imagine a bonfire on the moon. |
| 2394 | Natural supervenience is when one set of properties is always accompanied by another set [Chalmers] |
| Full Idea: B-properties supervene naturally on A-properties if any two naturally possible situations with the same A-properties have the same B-properties. | |
| From: David J.Chalmers (The Conscious Mind [1996], 1.2.1) | |
| A reaction: Since it is hard to imagine a healthy working brain failing to produce consciousness, given the current laws of nature, almost everyone (except extreme dualists) must concede that they are naturally supervenient. I wonder why they are. |
| 2398 | Reduction requires logical supervenience [Chalmers] |
| Full Idea: Reductive explanation requires a logical supervenience relation. | |
| From: David J.Chalmers (The Conscious Mind [1996], 1.2.3) | |
| A reaction: Why can't you say that in another world there are zombies, but in this world the mind is explained by its natural supervenience on the brain (given the current natural laws)? Driving on the left in Britain is explained by current laws. |
| 16048 | Physicalism says in any two physically indiscernible worlds the positive facts are the same [Chalmers, by Bennett,K] |
| Full Idea: Chalmers says that physicalism is true in a world w just in case every positive fact that obtains in w also obtains in any world physically indiscernible from w. | |
| From: report of David J.Chalmers (The Conscious Mind [1996], 2.1) by Karen Bennett - Supervenience | |
| A reaction: [Bennett summarises Chalmers' argument on pp.39-40] Chalmers says negative facts depend on the world's limits, which aren't part of the physical facts of the world. |
| 2401 | All facts are either physical, experiential, laws of nature, second-order final facts, or indexical facts about me [Chalmers] |
| Full Idea: Facts about the world are exhausted by physical facts, conscious experiences, laws of nature, a second-order that's-all fact, and perhaps an indexical fact about my location. | |
| From: David J.Chalmers (The Conscious Mind [1996], 1.2.5) | |
| A reaction: A bold claim! I don't think laws of nature are a component of ontology. What would they be made of? Presumably the indexical fact drops out when I do. Personally I (unlike Chalmers) think experience is physical |
| 16424 | Strong metaphysical necessity allows fewer possible worlds than logical necessity [Chalmers] |
| Full Idea: The hypothesized modality of 'strong' metaphysical necessity says there are fewer metaphysically possible worlds than there are logically possible worlds, and the a posteriori necessities can stem from factors independent of the semantics of terms. | |
| From: David J.Chalmers (The Conscious Mind [1996], 2.4.2) | |
| A reaction: Chalmers sets this up in order to reject it. He notes that it involves a big gap between conceivability and possibility. If a world is logically possible but metaphysically impossible, then it is impossible, surely? |
| 16425 | Metaphysical necessity is a bizarre, brute and inexplicable constraint on possibilities [Chalmers] |
| Full Idea: Strong metaphysical necessities will put constraints on the space of possible worlds that are brute and inexplicable. That's fine for our world, but bizarre for possible worlds. The realm of the possible has no room for such arbitrary constraint. | |
| From: David J.Chalmers (The Conscious Mind [1996], 2.4.2) | |
| A reaction: He would say this, given that he wants zombies to be possible, just because he thinks he can conceive of them. Presumably he thinks a raging bonfire with no flames is also possible. His objection here is weak. |
| 16426 | How can we know the metaphysical impossibilities; the a posteriori only concerns this world [Chalmers] |
| Full Idea: If some worlds are metaphysically impossible, it seems that we could never know it. By assumption the information is not available a priori, and a posteriori information only tells us about our world. | |
| From: David J.Chalmers (The Conscious Mind [1996], 2.4.2) | |
| A reaction: You need essentialism to reply to this. If you discover the essence of something, you can predict its possibilities. You discover the natures of the powers and dispositions of actuality. |
| 13956 | Kripke is often taken to be challenging a priori insights into necessity [Chalmers] |
| Full Idea: At various points in this book, I use a priori methods to gain insight into necessity; this is the sort of thing that Kripke's account is often taken to challenge. | |
| From: David J.Chalmers (The Conscious Mind [1996], 1.2.4) | |
| A reaction: Chalmers uses his 2-D approach to split off an a priori part from Kripke's a posterior part of our insight into necessity. |
| 13963 | Maybe logical possibility does imply conceivability - by an ideal mind [Chalmers] |
| Full Idea: If we understand conceivability as conceivability-in-principle (by a superbeing?) then it is plausible that logical possibility of a world implies conceivability of the world, so logical possibility of a statement implies its conceivability. | |
| From: David J.Chalmers (The Conscious Mind [1996], 1.2.4) | |
| A reaction: I see nothing incoherent in the possibility that there might be aspects of existence which are utterly inconceivable to any conscious mind. Infinity might be a start, if an 'infinite' mind were impossible. |
| 2407 | One can wrongly imagine two things being non-identical even though they are the same (morning/evening star) [Chalmers] |
| Full Idea: Just because one can imagine that A and B are not identical, it does not follow that A and B are not identical (think of the morning star and the evening star). | |
| From: David J.Chalmers (The Conscious Mind [1996], 2.4.1) |
| 2390 | We attribute beliefs to people in order to explain their behaviour [Chalmers] |
| Full Idea: Belief is something of an explanatory construct: we attribute beliefs to others largely in order to explain their behaviour. | |
| From: David J.Chalmers (The Conscious Mind [1996], 1.1.3) |
| 2397 | 'Perception' means either an action or a mental state [Chalmers] |
| Full Idea: 'Perception' can be used to refer either to the act of perceiving, or the internal state that arises as a result. | |
| From: David J.Chalmers (The Conscious Mind [1996], 1.2.2) |
| 2422 | The structure of the retina has already simplified the colour information which hits it [Chalmers] |
| Full Idea: In vision three varieties of cones abstract out information according to the amount of light present in various overlapping wavelength ranges. Immediately, many distinctions present in the original light wave are lost. | |
| From: David J.Chalmers (The Conscious Mind [1996], 3.8.3) |
| 2396 | Reductive explanation is not the be-all and the end-all of explanation [Chalmers] |
| Full Idea: Reductive explanation is not the be-all and the end-all of explanation. | |
| From: David J.Chalmers (The Conscious Mind [1996], 1.2.2) |
| 2426 | Why are minds homogeneous and brains fine-grained? [Chalmers] |
| Full Idea: The 'grain problem' for materialism was raised by Sellars: how could an experience be identical with a vast collection of physiological events, given the homogeneity of the former, and the fine-grainedness of the latter? | |
| From: David J.Chalmers (The Conscious Mind [1996], 3.8.5) | |
| A reaction: An interesting question, but it doesn't sound like a huge problem, given the number of connections in the brain. If the brain were expanded (as Leibniz suggested), the 'grains' might start to appear. We can't propose a 'deceived homunculus' to solve it. |
| 2391 | Can we be aware but not conscious? [Chalmers] |
| Full Idea: Consciousness is always accompanied by awareness, but awareness as I have described it need not be accompanied by consciousness. | |
| From: David J.Chalmers (The Conscious Mind [1996], 1.1.5) | |
| A reaction: One should consult Chalmers, but he is stretching the English word 'awareness' rather far. This road leads to saying that thermostats are 'aware', and information is aware of its content, which is probably very wrong indeed. Compare Idea 2415. |
| 2412 | Can we explain behaviour without consciousness? [Chalmers] |
| Full Idea: However the metaphysics of causation turns out, it seems relatively straightforward that a physical explanation of behaviour can be given that neither appeals to nor implies the existence of consciousness. | |
| From: David J.Chalmers (The Conscious Mind [1996], 2.5.2) | |
| A reaction: Chalmers needs this to support his idea that zombies are possible, but it strikes me as implausible. I find it inconceivable that our behaviour would be unchanged if we retained 'awareness' but lost consciousness. Try visiting an art gallery. |
| 2386 | Hard Problem: why brains experience things [Chalmers] |
| Full Idea: The Hard Problem is: why is all this brain processing accompanied by an experienced inner life? | |
| From: David J.Chalmers (The Conscious Mind [1996], Intro) | |
| A reaction: The word 'accompanied' is interesting. A very epiphenomenal word! The answer to this neo-dualist question may be: if you do enough complex representational brain processing at high speed, it adds up to some which we call 'experience'. |
| 2416 | What turns awareness into consciousness? [Chalmers] |
| Full Idea: Given the necessity of awareness, any candidate for an underlying law will have the form "Awareness plus something gives rise to consciousness" (…but simplicity suggests leaving out the 'something'). | |
| From: David J.Chalmers (The Conscious Mind [1996], 3.6.5) | |
| A reaction: You can't leave out the 'something' if you think awareness without consciousness is possible. The phenomenon of blindsight suggests that a whole extra brain area must come into play to produce the consciousness. It may not have a distinct ontology. |
| 2423 | Going down the scale, where would consciousness vanish? [Chalmers] |
| Full Idea: Moving down the scale from lizards to slugs, there doesn't seem much reason to suppose that phenomenology should wink out while a reasonably complex perceptual psychology persists….and if you move on down to thermostats, where would it wink out? | |
| From: David J.Chalmers (The Conscious Mind [1996], 3.8.4) | |
| A reaction: This doesn't seem much of an argument, particularly if its conclusion is that there is phenomenology in thermostats. When day changes into night, where does it 'wink out'? Are we to conclude that night doesn't exist, or that day doesn't exist? |
| 2403 | Nothing in physics even suggests consciousness [Chalmers] |
| Full Idea: Even if we knew every last detail about the physics of the universe, that information would not lead us to postulate the existence of conscious experience. | |
| From: David J.Chalmers (The Conscious Mind [1996], 2.3.1.3) | |
| A reaction: I find this a very strange claim. Given that the biggest gap in our physical knowledge is that concerning the brain and consciousness, Chalmer is no position to say this. Why shouldn't a physical revelation suddenly make consciousness inevitable? |
| 2400 | Is intentionality just causal connections? [Chalmers] |
| Full Idea: Intentional properties should be analyzable in terms of causal connections to behaviour and the environment….so there is no separate ontological problem of intentionality. | |
| From: David J.Chalmers (The Conscious Mind [1996], 1.2.5) | |
| A reaction: There could only be no ontological problem if intentional states were purely physical. Everything is made of something (I presume). |
| 2419 | Why should qualia fade during silicon replacement? [Chalmers] |
| Full Idea: If parts of the brain are gradually replaced, perhaps by silicon chips, ...the most reasonable hypothesis is that qualia do not fade at all. | |
| From: David J.Chalmers (The Conscious Mind [1996], 3.7.3) | |
| A reaction: As it stands this could either assert dualism or functionalism. Personally I think the most reasonable hypothesis is that qualia would fade. Chalmers needs more imagination (or less?). What is it like to experience Alzheimer's Disease? |
| 2389 | Sometimes we don't notice our pains [Chalmers] |
| Full Idea: What of the fact that we speak of pains that last for a day, even though there are times that they are not conscious? | |
| From: David J.Chalmers (The Conscious Mind [1996], 1.1.3) | |
| A reaction: This is hardly proof that there are non-conscious pains. Otherwise we might say we have a pain even after it has left us for good (because it might return), which seems daft. Not a crucial issue. The word 'pain' has two uses… |
| 2402 | It seems possible to invert qualia [Chalmers] |
| Full Idea: It seems entirely coherent that experiences could be inverted while physical structure is duplicated exactly. | |
| From: David J.Chalmers (The Conscious Mind [1996], 2.3.1.2) | |
| A reaction: Strange how what seems 'entirely coherent' to a leading philosopher strikes me as totally incoherent. I would have thought it was only coherent to a dualist. I don't believe God makes the physics on Thursday, and adds experiences on Friday. |
| 2415 | In blindsight both qualia and intentionality are missing [Chalmers] |
| Full Idea: In blindsight, the information does not qualify as directly available for global control, and subjects are not truly aware of the information. The lack of experience corresponds directly to a lack of awareness. | |
| From: David J.Chalmers (The Conscious Mind [1996], 3.6.3) | |
| A reaction: Blindsight patients give correct answers about objects in their visual field, and you need 'global control' to speak the truth, even if you lack confidence in what you are saying. Philosophers should not be frightened of blindsight. Cf Idea 2391. |
| 2414 | When distracted we can totally misjudge our own experiences [Chalmers] |
| Full Idea: If one is distracted one may make judgements about one's experiences that are quite false. | |
| From: David J.Chalmers (The Conscious Mind [1996], 2.5.5) | |
| A reaction: Of course, when one is distracted one can make mistakes about anything. This does imply that if there is indeed infallible knowledge to be had from introspection, it will at least require full concentration to achieve it. Cf Idea 8883. |
| 2409 | Maybe dualist interaction is possible at the quantum level? [Chalmers] |
| Full Idea: The only form of interactionist dualism that has seemed even remotely tenable in the contemporary picture is one that exploits certain properties of quantum mechanics. | |
| From: David J.Chalmers (The Conscious Mind [1996], 2.4.4) | |
| A reaction: I think he is bluffing. No doubt quantum mechanics offers many intriguing possibilities, such as the interaction of many worlds within the mind, but I am not aware that anything non-physical is ever postulated. Physicists don't deal in the non-physical. |
| 2411 | Supervenience makes interaction laws possible [Chalmers] |
| Full Idea: There is an objection to dualism that it cannot explain how the physical and the nonphysical interact, but the answer is simple on a natural supervenience framework - they interact by virtue of psychophysical laws (…which are as eternal as physics). | |
| From: David J.Chalmers (The Conscious Mind [1996], 2.4.6) | |
| A reaction: There are different sorts of laws. What Chalmers is hoping for would be a mere regularity, like the connection of cancer to smoking, but the objection is that the discovery of causal mechanisms, to give truly explanatory laws, is simply impossible. |
| 2424 | It is odd if experience is a very recent development [Chalmers] |
| Full Idea: It would be odd for a fundamental property like experience to be instantiated for the first time only relatively late in the history of the universe, and even then only in occasional complex systems. | |
| From: David J.Chalmers (The Conscious Mind [1996], 3.8.4) | |
| A reaction: The assumption of this remark is that experience is 'fundamental', which seems to claim that it is a separate ontological category. Maybe, but experience doesn't seem to be a thing. 'Process' seems a better term, and that is not a novelty in the universe. |
| 2413 | If I can have a zombie twin, my own behaviour doesn't need consciousness [Chalmers] |
| Full Idea: The explanation of my zombie twin's claims does not depend on consciousness, as there is none in his world. It follows that the explanation of my claims is also independent of the existence of consciousness. | |
| From: David J.Chalmers (The Conscious Mind [1996], 2.5.2) | |
| A reaction: Epiphenomenalism says my accounts of my consciousness are NOT because of my consciousness (which seems daft). Chalmers here gives a very good reason why we should not be a friend of philosophical zombies. |
| 2417 | Does consciousness arise from fine-grained non-reductive functional organisation? [Chalmers] |
| Full Idea: I claim that conscious experience arises from fine-grained functional organisation….. we might call it 'non-reductive functionalism'. | |
| From: David J.Chalmers (The Conscious Mind [1996], 3.7.1) | |
| A reaction: This is Chalmers' final position. If consciousness is 'emergent' and cannot be reduced, what has fine-grained got to do with it? I take 'fine-grained' to be a hint at why the brain becomes conscious. Fine-grained functions cause something. |
| 2428 | Maybe the whole Chinese Room understands Chinese, though the person doesn't [Chalmers] |
| Full Idea: Opponents typically reply to Searle's argument by conceding that the person in the room does not understand Chinese, and arguing that the understanding should instead be attributed to the system consisting of the person and the pieces of paper. | |
| From: David J.Chalmers (The Conscious Mind [1996], 4.9.4) | |
| A reaction: Searle himself spotted this reply. It seems plausible to say that a book contains 'understanding', so the translation dictionary may have it. A good Room would cope with surprise questions. |
| 2418 | The Chinese Mind doesn't seem conscious, but then nor do brains from outside [Chalmers] |
| Full Idea: While it may be intuitively implausible that Block's 'mind' made of the population of China would give rise to conscious experience, it is equally intuitively implausible that a brain should give rise to experience. | |
| From: David J.Chalmers (The Conscious Mind [1996], 3.7.2) | |
| A reaction: This sounds like good support for functionalism, but I am more inclined to see it as a critique of 'intuition' as a route to truth where minds are concerned. Intuition isn't designed for that sort of work. |
| 2406 | H2O causes liquidity, but no one is a dualist about that [Chalmers] |
| Full Idea: Searle argues that H2O causes liquidity, but no one is a dualist about liquidity. | |
| From: David J.Chalmers (The Conscious Mind [1996], 2.4.1) | |
| A reaction: Good! |
| 2405 | Perhaps consciousness is physically based, but not logically required by that base [Chalmers] |
| Full Idea: It remains plausible that consciousness arises from a physical basis, even though it is not entailed by that basis. | |
| From: David J.Chalmers (The Conscious Mind [1996], 2.4.1) | |
| A reaction: Personally I find this totally implausible. Since every other property or process in the known universe seems to be entailed by its physical basis, I don't expect the mind to be an exception. |
| 2395 | Zombies imply natural but not logical supervenience [Chalmers] |
| Full Idea: It seems logically possible that a creature physically identical to a conscious creature might have no conscious experiences (a zombie)…so conscious experience supervenes naturally but not logically on the physical. | |
| From: David J.Chalmers (The Conscious Mind [1996], 1.2.1) | |
| A reaction: "It seems possible" isn't much of an argument. This claim by Chalmers has been a great incentive to reassess what is or isn't possible. Can a brain lack consciousness? Can a tree fall over silently? Can cyanide stop poisoning us? |
| 9318 | Phenomenal consciousness is fundamental, with no possible nonphenomenal explanation [Chalmers, by Kriegel/Williford] |
| Full Idea: In Chalmers's non-reductive theory, phenomenal consciousness is treated as a fundamental feature of the world, that cannot be explained in nonphenomenal terms. Theory is still possible, in the regularities of interaction. | |
| From: report of David J.Chalmers (The Conscious Mind [1996]) by U Kriegel / K Williford - Intro to 'Self-Representational Consciousness' n2 | |
| A reaction: I can't make much sense of this view without a backing of panpsychism. How could a 'fundamental' feature of reality only begin to appear when life evolves on one particular planet? But 'panpsychism' is a warning of big misunderstandings. See Idea 2424. |
| 2404 | Nothing external shows whether a mouse is conscious [Chalmers] |
| Full Idea: It is consistent with the physical facts about a mouse that it has conscious experiences, and it is consistent with the physical facts that it does not. | |
| From: David J.Chalmers (The Conscious Mind [1996], 2.3.1.4) | |
| A reaction: No. It is consistent with our KNOWLEDGE of a mouse that it may or may not be conscious. I take this to be the key error of Chalmers, which led him to the mistaken idea that zombies are possible. The usual confusion of ontology and epistemology…. |
| 2429 | Temperature (etc.) is agreed to be reducible, but it is multiply realisable [Chalmers] |
| Full Idea: Many physical phenomena that are often taken to be paradigms of reducibility (e.g. temperature) are in fact multiply realizable. | |
| From: David J.Chalmers (The Conscious Mind [1996], n 2.20) | |
| A reaction: So multiple realisability isn't such a big problem for physicalism. I take it, though, that all hot things have some physical type of event in common (a level of molecular energy). Finding the level of commonality is the challenge. |
| 18403 | Indexicals may not be objective, but they are a fact about the world as I see it [Chalmers] |
| Full Idea: Even if the indexical is not an objective fact about the world, it is a fact about the world as I find it, and it is the world as I find it that needs explanation. | |
| From: David J.Chalmers (The Conscious Mind [1996], 1.2.5) | |
| A reaction: Chalmers treats them as important, whereas the way he expresses it could make them eliminable, if the world seen by him is eliminable. |
| 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. |
| 14708 | Rationalist 2D semantics posits necessary relations between meaning, apriority, and possibility [Chalmers, by Schroeter] |
| Full Idea: Chalmers seeks a rationalist interpretation of the 2D framework, situated in the tradition which posits a golden triangle of necessary constitutive relations between meaning, apriority, and possibility. | |
| From: report of David J.Chalmers (The Conscious Mind [1996]) by Laura Schroeter - Two-Dimensional Semantics 2.3.1 | |
| A reaction: The first prize of the project is to get some sort of apriori knowledge about these crucial relations. I suppose the superduper prize is to get apriori knowledge of the possibilities of the world, but I wouldn't hold your breath waiting for that. |
| 13958 | The 'primary intension' is non-empirical, and fixes extensions based on the actual-world reference [Chalmers] |
| Full Idea: The 'primary intension' of a concept is a function from worlds to extensions reflecting the way the actual-world reference is fixed, ...which is independent of empirical factors. | |
| From: David J.Chalmers (The Conscious Mind [1996], 1.2.4) | |
| A reaction: This bit is a priori because the concept picks out something, no matter what its essence turns out to be. I take it to be a priori because it is stipulative. |
| 2399 | Meaning has split into primary ("watery stuff"), and secondary counterfactual meaning ("H2O") [Chalmers] |
| Full Idea: The single Fregean intension has fragmented into two: a primary intension ("watery stuff") that fixes reference in the actual world, and a secondary intension ("H2O") that picks out reference in counterfactual possible worlds. | |
| From: David J.Chalmers (The Conscious Mind [1996], 1.2.4) | |
| A reaction: No one actually performs this schizoid double operation, so this is theory disconnected from life. What is the role of 'H2O' in the actual world, and 'watery stuff' in the others? |
| 13959 | The 'secondary intension' is determined by rigidifying (as H2O) the 'water' picked out in the actual world [Chalmers] |
| Full Idea: The 'secondary intension' of 'water' picks out the water (H2O) in all worlds. ..It is determined by first evaluating the primary intension at the actual world, and then rigidifying it so that the same sort of thing is picked out in all possible worlds. | |
| From: David J.Chalmers (The Conscious Mind [1996], 1.2.4) | |
| A reaction: No wonder Soames calls 2-D semantics 'Byzantine'. If we don't actually do this psychologically, what exactly is Chalmers describing? Is this revisionary semantics - i.e. how we ought to do it if we want to talk about the world properly? |
| 13957 | Primary and secondary intensions are the a priori (actual) and a posteriori (counterfactual) aspects of meaning [Chalmers] |
| Full Idea: Primary intension picks out a referent in a world considered as actual; secondary considers it as counterfactual. ...(62) We can think of the primary and secondary intensions as the a priori and a posteriori aspects of meaning, respectively. | |
| From: David J.Chalmers (The Conscious Mind [1996], 1.2.4) | |
| A reaction: Primary intension is a priori because, it seems, it is stipulative ('water' means 'the watery stuff'), whereas the secondary intension (in counterfactual worlds) is empirical ('water' is used to refer to H2O/XYZ). We get internalism and externalism. |
| 13961 | We have 'primary' truth-conditions for the actual world, and derived 'secondary' ones for counterfactual worlds [Chalmers] |
| Full Idea: 'Primary' truth-conditions tell us how the actual world has to be for an utterance of the statement to be true in that world; ....'secondary' truth-conditions give the truth-value in counterfactual worlds, given that the actual world turned out some way. | |
| From: David J.Chalmers (The Conscious Mind [1996], 1.2.4) | |
| A reaction: This is the reinterpretation of the truth-conditions account in terms of two-dimensional semantics. My first reaction is not very positive. Why can't we fix our references in counterfactual worlds, and then apply them to the actual (like inventions)? |
| 13962 | Two-dimensional semantics gives a 'primary' and 'secondary' proposition for each statement [Chalmers] |
| Full Idea: If we see a proposition as a function from possible worlds to truth-values, then the two sets of truth-conditions yield two propositions associated with any statement. A 'primary' for those which express a truth, and 'secondary' for counterfactual truth. | |
| From: David J.Chalmers (The Conscious Mind [1996], 1.2.4) | |
| A reaction: This is where 2-D semantics becomes increasingly 'Byzantine'. Intuition and introspection don't seem to offer me two different propositions for every sentence I utter. I can't see this theory catching on, even if it is technically beautiful. |
| 13960 | In two-dimensional semantics we have two aspects to truth in virtue of meaning [Chalmers] |
| Full Idea: Both the 'primary' and 'secondary' intension qualify as truths in virtue of meaning; they are simply true in virtue of different aspects of meaning. | |
| From: David J.Chalmers (The Conscious Mind [1996], 1.2.4) | |
| A reaction: This is the view of two-dimensional semantics, which has split Fregean sense into an a priori and an a posterior part. Chalmers is trying to hang onto the idea that we might see necessity as largely analytic. |
| 22001 | The real will of the cooperative will replace the 'will of the people' [Marx] |
| Full Idea: Under collective property, the so called will of the people disappears in order to make way for the real will of the cooperative. | |
| From: Karl Marx (Grundrisse [1876], p.563), quoted by Peter Singer - Marx 10 | |
| A reaction: [from an 1874 note on Bakunin's 'Statism and Anarchy'] So how do you settle on the 'real' will of a cooperative? The travesty is when a ruling elite decide that, without consultation. An institution is needed. This is still a social contract. |
| 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'. |
| 16427 | Presumably God can do anything which is logically possible [Chalmers] |
| Full Idea: Presumably it is in God's powers, when creating the world, to do anything that is logically possible. | |
| From: David J.Chalmers (The Conscious Mind [1996], 2.4.2) | |
| A reaction: I don't really understand why anyone would say that the only constraint on God is logic. Presumably no logic is breached if God places in object simultaneously in two spacetime locations, but it would be an impressive achievement. |