58 ideas
15945 | Second-order set theory just adds a version of Replacement that quantifies over functions [Lavine] |
Full Idea: Second-order set theory is just like first-order set-theory, except that we use the version of Replacement with a universal second-order quantifier over functions from set to sets. | |
From: Shaughan Lavine (Understanding the Infinite [1994], VII.4) |
15914 | An 'upper bound' is the greatest member of a subset; there may be several of these, so there is a 'least' one [Lavine] |
Full Idea: A member m of M is an 'upper bound' of a subset N of M if m is not less than any member of N. A member m of M is a 'least upper bound' of N if m is an upper bound of N such that if l is any other upper bound of N, then m is less than l. | |
From: Shaughan Lavine (Understanding the Infinite [1994], III.4) | |
A reaction: [if you don't follow that, you'll have to keep rereading it till you do] |
15921 | Collections of things can't be too big, but collections by a rule seem unlimited in size [Lavine] |
Full Idea: Since combinatorial collections are enumerated, some multiplicities may be too large to be gathered into combinatorial collections. But the size of a multiplicity seems quite irrelevant to whether it forms a logical connection. | |
From: Shaughan Lavine (Understanding the Infinite [1994], IV.2) |
15937 | Those who reject infinite collections also want to reject the Axiom of Choice [Lavine] |
Full Idea: Many of those who are skeptical about the existence of infinite combinatorial collections would want to doubt or deny the Axiom of Choice. | |
From: Shaughan Lavine (Understanding the Infinite [1994], VI.2) |
15936 | The Power Set is just the collection of functions from one collection to another [Lavine] |
Full Idea: The Power Set is just he codification of the fact that the collection of functions from a mathematical collection to a mathematical collection is itself a mathematical collection that can serve as a domain of mathematical study. | |
From: Shaughan Lavine (Understanding the Infinite [1994], VI.1) |
15899 | Replacement was immediately accepted, despite having very few implications [Lavine] |
Full Idea: The Axiom of Replacement (of Skolem and Fraenkel) was remarkable for its universal acceptance, though it seemed to have no consequences except for the properties of the higher reaches of the Cantorian infinite. | |
From: Shaughan Lavine (Understanding the Infinite [1994], I) |
15930 | Foundation says descending chains are of finite length, blocking circularity, or ungrounded sets [Lavine] |
Full Idea: The Axiom of Foundation (Zermelo 1930) says 'Every (descending) chain in which each element is a member of the previous one is of finite length'. ..This forbids circles of membership, or ungrounded sets. ..The iterative conception gives this centre stage. | |
From: Shaughan Lavine (Understanding the Infinite [1994], V.4) |
15898 | The controversy was not about the Axiom of Choice, but about functions as arbitrary, or given by rules [Lavine] |
Full Idea: The controversy was not about Choice per se, but about the correct notion of function - between advocates of taking mathematics to be about arbitrary functions and advocates of taking it to be about functions given by rules. | |
From: Shaughan Lavine (Understanding the Infinite [1994], I) |
15920 | Pure collections of things obey Choice, but collections defined by a rule may not [Lavine] |
Full Idea: Combinatorial collections (defined just by the members) obviously obey the Axiom of Choice, while it is at best dubious whether logical connections (defined by a rule) do. | |
From: Shaughan Lavine (Understanding the Infinite [1994], IV.2) |
15919 | The 'logical' notion of class has some kind of definition or rule to characterise the class [Lavine] |
Full Idea: The Peano-Russell notion of class is the 'logical' notion, where each collection is associated with some kind of definition or rule that characterises the members of the collection. | |
From: Shaughan Lavine (Understanding the Infinite [1994], IV.1) |
15900 | The iterative conception of set wasn't suggested until 1947 [Lavine] |
Full Idea: The iterative conception of set was not so much as suggested, let alone advocated by anyone, until 1947. | |
From: Shaughan Lavine (Understanding the Infinite [1994], I) |
15931 | The iterative conception needs the Axiom of Infinity, to show how far we can iterate [Lavine] |
Full Idea: The iterative conception of sets does not tell us how far to iterate, and so we must start with an Axiom of Infinity. It also presupposes the notion of 'transfinite iteration'. | |
From: Shaughan Lavine (Understanding the Infinite [1994], V.5) |
15932 | The iterative conception doesn't unify the axioms, and has had little impact on mathematical proofs [Lavine] |
Full Idea: The iterative conception does not provide a conception that unifies the axioms of set theory, ...and it has had very little impact on what theorems can be proved. | |
From: Shaughan Lavine (Understanding the Infinite [1994], V.5) | |
A reaction: He says he would like to reject the iterative conception, but it may turn out that Foundation enables new proofs in mathematics (though it hasn't so far). |
15933 | Limitation of Size: if it's the same size as a set, it's a set; it uses Replacement [Lavine] |
Full Idea: Limitation of Size has it that if a collection is the same size as a set, then it is a set. The Axiom of Replacement is characteristic of limitation of size. | |
From: Shaughan Lavine (Understanding the Infinite [1994], V.5) |
15913 | A collection is 'well-ordered' if there is a least element, and all of its successors can be identified [Lavine] |
Full Idea: A collection M is 'well-ordered' by a relation < if < linearly orders M with a least element, and every subset of M that has an upper bound not in it has an immediate successor. | |
From: Shaughan Lavine (Understanding the Infinite [1994], III.4) |
6548 | Physicalism requires the naturalisation or rejection of set theory [Lycan] |
Full Idea: Eventually set theory will have to be either naturalised or rejected, if a thoroughgoing physicalism is to be maintained. | |
From: William Lycan (Consciousness [1987], 8.4) | |
A reaction: Personally I regard Platonism as a form of naturalism (though a rather bold and dramatic one). The central issue seems to be the ability of the human main/brain to form 'abstract' notions about the physical world in which it lives. |
15926 | Second-order logic presupposes a set of relations already fixed by the first-order domain [Lavine] |
Full Idea: The distinctive feature of second-order logic is that it presupposes that, given a domain, there is a fact of the matter about what the relations on it are, so that the range of the second-order quantifiers is fixed as soon as the domain is fixed. | |
From: Shaughan Lavine (Understanding the Infinite [1994], V.3) | |
A reaction: This sounds like a rather large assumption, which is open to challenge. I am not sure whether it was the basis of Quine's challenge to second-order logic. He seems to have disliked its vagueness, because it didn't stick with 'objects'. |
15934 | Mathematical proof by contradiction needs the law of excluded middle [Lavine] |
Full Idea: The Law of Excluded Middle is (part of) the foundation of the mathematical practice of employing proofs by contradiction. | |
From: Shaughan Lavine (Understanding the Infinite [1994], VI.1) | |
A reaction: This applies in a lot of logic, as well as in mathematics. Come to think of it, it applies in Sudoku. |
15907 | Mathematics is nowadays (thanks to set theory) regarded as the study of structure, not of quantity [Lavine] |
Full Idea: Mathematics is today thought of as the study of abstract structure, not the study of quantity. That point of view arose directly out of the development of the set-theoretic notion of abstract structure. | |
From: Shaughan Lavine (Understanding the Infinite [1994], III.2) | |
A reaction: It sounds as if Structuralism, which is a controversial view in philosophy, is a fait accompli among mathematicians. |
15942 | Every rational number, unlike every natural number, is divisible by some other number [Lavine] |
Full Idea: One reason to introduce the rational numbers is that it simplifes the theory of division, since every rational number is divisible by every nonzero rational number, while the analogous statement is false for the natural numbers. | |
From: Shaughan Lavine (Understanding the Infinite [1994], VI.3) | |
A reaction: That is, with rations every division operation has an answer. |
15922 | For the real numbers to form a set, we need the Continuum Hypothesis to be true [Lavine] |
Full Idea: The chief importance of the Continuum Hypothesis for Cantor (I believe) was that it would show that the real numbers form a set, and hence that they were encompassed by his theory. | |
From: Shaughan Lavine (Understanding the Infinite [1994], IV.2) |
18250 | Cauchy gave a necessary condition for the convergence of a sequence [Lavine] |
Full Idea: The Cauchy convergence criterion for a sequence: the sequence S0,S1,... has a limit if |S(n+r) - S(n)| is less than any given quantity for every value of r and sufficiently large values of n. He proved this necessary, but not sufficient. | |
From: Shaughan Lavine (Understanding the Infinite [1994], 2.5) |
15904 | The two sides of the Cut are, roughly, the bounding commensurable ratios [Lavine] |
Full Idea: Roughly speaking, the upper and lower parts of the Dedekind cut correspond to the commensurable ratios greater than and less than a given incommensurable ratio. | |
From: Shaughan Lavine (Understanding the Infinite [1994], II.6) | |
A reaction: Thus there is the problem of whether the contents of the gap are one unique thing, or many. |
15912 | Counting results in well-ordering, and well-ordering makes counting possible [Lavine] |
Full Idea: Counting a set produces a well-ordering of it. Conversely, if one has a well-ordering of a set, one can count it by following the well-ordering. | |
From: Shaughan Lavine (Understanding the Infinite [1994], III.4) | |
A reaction: Cantor didn't mean that you could literally count the set, only in principle. |
15947 | The infinite is extrapolation from the experience of indefinitely large size [Lavine] |
Full Idea: My proposal is that the concept of the infinite began with an extrapolation from the experience of indefinitely large size. | |
From: Shaughan Lavine (Understanding the Infinite [1994], VIII.2) | |
A reaction: I think it might be better to talk of an 'abstraction' than an 'extrapolition', since the latter is just more of the same, which doesn't get you to concept. Lavine spends 100 pages working out his proposal. |
15949 | The theory of infinity must rest on our inability to distinguish between very large sizes [Lavine] |
Full Idea: The indiscernibility of indefinitely large sizes will be a critical part of the theory of indefinitely large sizes. | |
From: Shaughan Lavine (Understanding the Infinite [1994], VIII.2) |
15940 | The intuitionist endorses only the potential infinite [Lavine] |
Full Idea: The intuitionist endorse the actual finite, but only the potential infinite. | |
From: Shaughan Lavine (Understanding the Infinite [1994], VI.2) |
15909 | 'Aleph-0' is cardinality of the naturals, 'aleph-1' the next cardinal, 'aleph-ω' the ω-th cardinal [Lavine] |
Full Idea: The symbol 'aleph-nought' denotes the cardinal number of the set of natural numbers. The symbol 'aleph-one' denotes the next larger cardinal number. 'Aleph-omega' denotes the omega-th cardinal number. | |
From: Shaughan Lavine (Understanding the Infinite [1994], III.3) |
15915 | Ordinals are basic to Cantor's transfinite, to count the sets [Lavine] |
Full Idea: The ordinals are basic because the transfinite sets are those that can be counted, or (equivalently for Cantor), those that can be numbered by an ordinal or are well-ordered. | |
From: Shaughan Lavine (Understanding the Infinite [1994], III.4) | |
A reaction: Lavine observes (p.55) that for Cantor 'countable' meant 'countable by God'! |
15917 | Paradox: the class of all ordinals is well-ordered, so must have an ordinal as type - giving a bigger ordinal [Lavine] |
Full Idea: The paradox of the largest ordinal (the 'Burali-Forti') is that the class of all ordinal numbers is apparently well-ordered, and so it has an ordinal number as order type, which must be the largest ordinal - but all ordinals can be increased by one. | |
From: Shaughan Lavine (Understanding the Infinite [1994], III.5) |
15918 | Paradox: there is no largest cardinal, but the class of everything seems to be the largest [Lavine] |
Full Idea: The paradox of the largest cardinal ('Cantor's Paradox') says the diagonal argument shows there is no largest cardinal, but the class of all individuals (including the classes) must be the largest cardinal number. | |
From: Shaughan Lavine (Understanding the Infinite [1994], III.5) |
15929 | Set theory will found all of mathematics - except for the notion of proof [Lavine] |
Full Idea: Every theorem of mathematics has a counterpart with set theory - ...but that theory cannot serve as a basis for the notion of proof. | |
From: Shaughan Lavine (Understanding the Infinite [1994], V.3) |
15935 | Modern mathematics works up to isomorphism, and doesn't care what things 'really are' [Lavine] |
Full Idea: In modern mathematics virtually all work is only up to isomorphism and no one cares what the numbers or points and lines 'really are'. | |
From: Shaughan Lavine (Understanding the Infinite [1994], VI.1) | |
A reaction: At least that leaves the field open for philosophers, because we do care what things really are. So should everybody else, but there is no persuading some people. |
15928 | Intuitionism rejects set-theory to found mathematics [Lavine] |
Full Idea: Intuitionism in philosophy of mathematics rejects set-theoretic foundations. | |
From: Shaughan Lavine (Understanding the Infinite [1994], V.3 n33) |
6532 | Types cannot be reduced, but levels of reduction are varied groupings of the same tokens [Lycan] |
Full Idea: If types cannot be reduced to more physical levels, this is not an embarrassment, as long as our institutional categories, our physiological categories, and our physical categories are just alternative groupings of the same tokens. | |
From: William Lycan (Consciousness [1987], 4.3) | |
A reaction: This is a self-evident truth about a car engine, so I don't see why it wouldn't apply equally to a brain. Lycan's identification of the type as the thing which cannot be reduced seems a promising explanation of much confusion among philosophers. |
6531 | Institutions are not reducible as types, but they are as tokens [Lycan] |
Full Idea: Institutional types are irreducible, though I assume that institutional tokens are reducible in the sense of strict identity, all the way down to the subatomic level. | |
From: William Lycan (Consciousness [1987], 4.3) | |
A reaction: This seems a promising distinction, as the boundaries of 'institutions' disappear when you begin to reduce them to lower levels (cf. Idea 4601), and yet plenty of institutions are self-evidently no more than physics. Plants are invisible as physics. |
6534 | One location may contain molecules, a metal strip, a key, an opener of doors, and a human tragedy [Lycan] |
Full Idea: One space-time slice may be occupied by a collection of molecules, a metal strip, a key, an allower of entry to hotel rooms, a facilitator of adultery, and a destroyer souls. | |
From: William Lycan (Consciousness [1987], 4.3) | |
A reaction: Desdemona's handkerchief is a nice example. This sort of remark seems to be felt by some philosophers to be heartless wickedness, and yet it so screamingly self-evident that it is impossible to deny. |
6529 | I see the 'role'/'occupant' distinction as fundamental to metaphysics [Lycan] |
Full Idea: I see the 'role'/'occupant' distinction as fundamental to metaphysics. | |
From: William Lycan (Consciousness [1987], 4.0) | |
A reaction: A passing remark in a discussion of functionalism about the mind, but I find it appealing. Causation is basic to materialistic metaphysics, and it creates networks of regular causes. It leaves open the essentialist question of WHY it has that role. |
6549 | I think greenness is a complex microphysical property of green objects [Lycan] |
Full Idea: Personally I favour direct realism regarding secondary qualities, and identify greenness with some complex microphysical property exemplified by green physical objects. | |
From: William Lycan (Consciousness [1987], 8.4) | |
A reaction: He cites D.M.Armstrong (1981) as his source. Personally I find this a bewildering proposal. Does he think there is greenness in grass AS WELL AS the emission of that wavelength of electro-magnetic radiation? Is greenness zooming through the air? |
6543 | Intentionality comes in degrees [Lycan] |
Full Idea: Intentionality comes in degrees. | |
From: William Lycan (Consciousness [1987], 5.4) | |
A reaction: I agree. A footprint is 'about' a foot, in the sense of containing concentrated information about it. Can we, though, envisage a higher degree than human thought? Is there a maximum degree? Everything is 'about' everything, in some respect. |
6537 | Teleological views allow for false intentional content, unlike causal and nomological theories [Lycan] |
Full Idea: The teleological view begins to explain intentionality, and in particular allows brain states and events to have false intentional content; causal and nomological theories of intentionality tend to falter on this last task. | |
From: William Lycan (Consciousness [1987], 4.4) | |
A reaction: Certainly if you say thought is 'caused' by the world, false thought become puzzling. I'm not sure I understand the rest of this, but it is an intriguing remark about a significant issue… |
6546 | Pain is composed of urges, desires, impulses etc, at different levels of abstraction [Lycan] |
Full Idea: Our phenomenal experience of pain has components - it is a complex, consisting (perhaps) of urges, desires, impulses, and beliefs, probably occurring at quite different levels of institutional abstraction. | |
From: William Lycan (Consciousness [1987], 5.5) | |
A reaction: This seems to be true, and offers the reductionist a strategy for making inroads into the supposed irreducable and fundamental nature of qualia. What's it like to be a complex hierarchically structured multi-functional organism? |
6547 | The right 'level' for qualia is uncertain, though top (behaviourism) and bottom (particles) are false [Lycan] |
Full Idea: It is just arbitrary to choose a level of nature a priori as the locus of qualia, even though we can agree that high levels (such as behaviourism) and low-levels (such as the subatomic) can be ruled out as totally improbable. | |
From: William Lycan (Consciousness [1987], 5.6) | |
A reaction: Very good. People scream 'qualia!' whenever the behaviour level or the atomic level are proposed as the locations of the mind, but the suggestion that they are complex, and are spread across many functional levels in the middle sounds good. |
6527 | If energy in the brain disappears into thin air, this breaches physical conservation laws [Lycan] |
Full Idea: By interacting causally, Cartesian dualism seems to violate the conservation laws of physics (concerning matter and energy). This seems testable, and afferent and efferent pathways disappearing into thin air would suggest energy is not conserved. | |
From: William Lycan (Consciousness [1987], 1.1) | |
A reaction: It would seem to be no problem as long as outputs were identical in energy to inputs. If the experiment could actually be done, the result might astonish us. |
6528 | In lower animals, psychology is continuous with chemistry, and humans are continuous with animals [Lycan] |
Full Idea: Evolution has proceeded in all other known species by increasingly complex configurations of molecules and organs, which support primitive psychologies; our human psychologies are more advanced, but undeniably continuous with lower animals. | |
From: William Lycan (Consciousness [1987], 1.1) | |
A reaction: Personally I find the evolution objection to dualism highly persuasive. I don't see how anyone can take evolution seriously and be a dualist. If there is a dramatic ontological break at some point, a plausible reason would be needed for that. |
6554 | Two behaviourists meet. The first says,"You're fine; how am I?" [Lycan] |
Full Idea: Old joke: two Behaviourists meet in the street, and the first says,"You're fine; how am I?" | |
From: William Lycan (Consciousness [1987], n1.6) | |
A reaction: This invites the response that introspection is uniquely authoritative about 'how we are', but this has been challenged quite a lot recently, which pushes us to consider whether these stupid behaviourists might actually have a good point. |
6545 | If functionalism focuses on folk psychology, it ignores lower levels of function [Lycan] |
Full Idea: 'Analytical functionalists', who hold that meanings of mental terms are determined by the causal roles associated with them by 'folk psychology', deny themselves appeals to lower levels of functional organisation. | |
From: William Lycan (Consciousness [1987], 5.4) | |
A reaction: Presumably folk psychology can fit into the kind of empirical methodology favoured by behaviourists, whereas 'lower levels' are going to become rather speculative and unscientific. |
6541 | Functionalism must not be too abstract to allow inverted spectrum, or so structural that it becomes chauvinistic [Lycan] |
Full Idea: The functionalist must find a level of characterisation of mental states that is not so abstract or behaviouristic as to rule out the possibility of inverted spectrum etc., nor so specific and structural as to fall into chauvinism. | |
From: William Lycan (Consciousness [1987], 5.4) | |
A reaction: If too specific then animals and aliens won't be able to implement the necessary functions; if the theory becomes very behaviouristic, then it loses interest in the possibility of an inverted spectrum. He is certainly right to hunt for a middle ground. |
6539 | The distinction between software and hardware is not clear in computing [Lycan] |
Full Idea: Even the software/hardware distinction as it is literally applied within computer science is philosophically unclear. | |
From: William Lycan (Consciousness [1987], 4.4) | |
A reaction: This is true, and very important for functionalist theories of the mind. Even very volatile software is realised in 'hard' physics, and rewritable discs etc blur the distinction between 'programmable' and 'hardwired'. |
6535 | Teleological characterisations shade off smoothly into brutely physical ones [Lycan] |
Full Idea: Highly teleological characterisations, unlike naïve and explicated mental characterisations, have the virtue of shading off fairly smoothly into (more) brutely physical ones. | |
From: William Lycan (Consciousness [1987], 4.3) | |
A reaction: Thus the purpose of a car engine, and a spark plug, and the spark, and the temperature, and the vibration of molecules show a fading away of the overt purpose, disappearing into the pointless activity of electrons and quantum levels. |
6533 | Mental types are a subclass of teleological types at a high level of functional abstraction [Lycan] |
Full Idea: I am taking mental types to form a small subclass of teleological types occurring for the most part at a high level of functional abstraction. | |
From: William Lycan (Consciousness [1987], 4.3) | |
A reaction: He goes on to say that he understand teleology in evolutionary terms. There is always a gap between how you characterise or individuate something, and what it actually is. To say spanners are 'a small subclass of tools' is not enough. |
6544 | Identity theory is functionalism, but located at the lowest level of abstraction [Lycan] |
Full Idea: 'Neuron' may be understood as a physiological term or a functional term, so even the Identity Theorist is a Functionalist - one who locates mental entities at a very low level of abstraction. | |
From: William Lycan (Consciousness [1987], 5.4) | |
A reaction: This is a striking observation, and somewhat inclines me to switch from identity theory to functionalism. If you ask what is the correct level of abstraction, Lycan's teleological-homuncular version refers you to all the levels. |
6536 | Teleological functionalism helps us to understand psycho-biological laws [Lycan] |
Full Idea: Teleological functionalism helps us to understand the nature of biological and psychological laws, particularly in the face of Davidsonian scepticism about the latter. | |
From: William Lycan (Consciousness [1987], 4.4) | |
A reaction: Personally I doubt the existence of psycho-physical laws, but only because of the vast complexity. They would be like the laws of weather. 'Psycho-physical' laws seem to presuppose some sort of dualism. |
6530 | We reduce the mind through homuncular groups, described abstractly by purpose [Lycan] |
Full Idea: I am explicating the mental in a reductive way, by reducing mental characterizations to homuncular institutional ones, which are teleological characterizations at various levels of functional abstraction. | |
From: William Lycan (Consciousness [1987], 4.3) | |
A reaction: I think this is the germ of a very good physicalist account of the mind. More is needed than a mere assertion about what the mind reduces to at the very lowest level; this offers a decent account of the descending stages of reduction. |
6542 | A Martian may exhibit human-like behaviour while having very different sensations [Lycan] |
Full Idea: Quite possibly a Martian's humanoid behaviour is prompted by his having sensations somewhat unlike ours, despite his superficial behavioural similarities to us. | |
From: William Lycan (Consciousness [1987], 5.4) | |
A reaction: I think this firmly refutes the multiple realisability objection to type-type physicalism. Mental events are individuated by their phenomenal features (known only to the user), and by their causal role (publicly available). These are separate. |
6538 | We need a notion of teleology that comes in degrees [Lycan] |
Full Idea: We need a notion of teleology that comes in degrees. | |
From: William Lycan (Consciousness [1987], 4.4) | |
A reaction: Anyone who says that key concepts, such as those concerning the mind, should come 'in degrees' wins my instant support. A whole car engine requires a very teleological explanation, the spark in the sparkplug far less so. |
6551 | 'Physical' means either figuring in physics descriptions, or just located in space-time [Lycan] |
Full Idea: An object is specifically physical if it figures in explanations and descriptions of features of ordinary non-living matter, as in current physics; it is more generally physical if it is simply located in space-time. | |
From: William Lycan (Consciousness [1987], 8.5) | |
A reaction: This gives a useful distinction when trying to formulate a 'physicalist' account of the mind, where type-type physicalism says only the 'postulates of physics' can be used, whereas 'naturalism' about the mind uses the more general concept. |
7417 | God can't have silly perfections, but how do we decide which ones are 'silly'? [Joslin] |
Full Idea: It is clear that God cannot have all conceivable perfections, because otherwise he would have absurd perfections (like being the perfect prawn sandwich), so a line must be drawn, and how are we to decide which perfections are appropriate and essential? | |
From: Jack Joslin (talk [2006]), quoted by PG - Db (ideas) | |
A reaction: This is an excellent question for curbing the absurdities of those who want to load God with every good thing that can possibly be conceived. Is the God who is also a perfect prawn sandwich more perfect than the one who isn't? |