49 ideas
| 19648 | Since Kant we think we can only access 'correlations' between thinking and being [Meillassoux] |
| Full Idea: The central notion of philosophy since Kant is 'correlation' - that we only ever have access to the correlation between thinking and being, and never to either term considered apart from the other. | |
| From: Quentin Meillassoux (After Finitude; the necessity of contingency [2006], 1) | |
| A reaction: Meillassoux's charge is that philosophy has thereby completely failed to grasp the scientific revolution, which has used mathematics to make objectivity possible. Quine and Putnam would be good examples of what he has in mind. |
| 19674 | The Copernican Revolution decentres the Earth, but also decentres thinking from reality [Meillassoux] |
| Full Idea: The Copernican Revolution is not so much the decentring of observers in the solar system, but (by the mathematizing of nature) the decentring of thought relative to the world within the process of knowledge. | |
| From: Quentin Meillassoux (After Finitude; the necessity of contingency [2006], 5) | |
| A reaction: In other words, I take it, the Copernican Revolution was the discovery of objectivity. That is a very nice addition to my History of Ideas collection. |
| 19657 | In Kant the thing-in-itself is unknowable, but for us it has become unthinkable [Meillassoux] |
| Full Idea: The major shift that has occurred in the conception of thought from Kant's time to ours is from the unknowability of the thing-in-itself to its unthinkability. | |
| From: Quentin Meillassoux (After Finitude; the necessity of contingency [2006], 2) | |
| A reaction: Meillassoux is making the case that philosophy is alienating us more and more from the triumphant realism of the scientific revolution. He says thinking has split from being. He's right. Modern American pragmatists are the worst (not Peirce!). |
| 19675 | Since Kant, philosophers have claimed to understand science better than scientists do [Meillassoux] |
| Full Idea: Ever since Kant, to think science as a philosopher has been to claim that science harbours a meaning other than the one delivered by science itself. | |
| From: Quentin Meillassoux (After Finitude; the necessity of contingency [2006], 5) | |
| A reaction: The point is that science discovered objectivity (via the mathematising of nature), and Kant utterly rejected objectivity, by enmeshing the human mind in every possible scientific claim. This makes Meillassoux and I very cross. |
| 19649 | Since Kant, objectivity is defined not by the object, but by the statement's potential universality [Meillassoux] |
| Full Idea: Since Kant, objectivity is no longer defined with reference to the object in itself, but rather with reference to the possible universality of an objective statement. | |
| From: Quentin Meillassoux (After Finitude; the necessity of contingency [2006], 1) | |
| A reaction: Meillassoux disapproves of this, as a betrayal by philosophers of the scientific revolution, which gave us true objectivity (e.g. about how the world was before humanity). |
| 19666 | If we insist on Sufficient Reason the world will always be a mystery to us [Meillassoux] |
| Full Idea: So long as we continue to believe that there is a reason why things are the way they are rather than some other way, we will construe this world is a mystery, since no such reason will every be vouchsafed to us. | |
| From: Quentin Meillassoux (After Finitude; the necessity of contingency [2006], 4) | |
| A reaction: Giving up sufficient reason sounds like a rather drastic response to this. Put it like this: Will we ever be able to explain absolutely everything? No. So will the world always be a little mysterious to us? Yes, obviously. Is that a problem? No! |
| 19656 | Non-contradiction is unjustified, so it only reveals a fact about thinking, not about reality? [Meillassoux] |
| Full Idea: The principle of non-contradiction itself is without reason, and consequently it can only be the norm for what is thinkable by us, rather than for what is possible in the absolute sense. | |
| From: Quentin Meillassoux (After Finitude; the necessity of contingency [2006], 2) | |
| A reaction: This is not Meillassoux's view, but describes the modern heresy of 'correlationism', which ties all assessments of how reality is to our capacity to think about it. Personally I take logical non-contradiction to derive from non-contradiction in nature. |
| 13520 | A 'tautology' must include connectives [Wolf,RS] |
| Full Idea: 'For every number x, x = x' is not a tautology, because it includes no connectives. | |
| From: Robert S. Wolf (A Tour through Mathematical Logic [2005], 1.2) |
| 13524 | Deduction Theorem: T∪{P}|-Q, then T|-(P→Q), which justifies Conditional Proof [Wolf,RS] |
| Full Idea: Deduction Theorem: If T ∪ {P} |- Q, then T |- (P → Q). This is the formal justification of the method of conditional proof (CPP). Its converse holds, and is essentially modus ponens. | |
| From: Robert S. Wolf (A Tour through Mathematical Logic [2005], 1.3) |
| 13522 | Universal Generalization: If we prove P(x) with no special assumptions, we can conclude ∀xP(x) [Wolf,RS] |
| Full Idea: Universal Generalization: If we can prove P(x), only assuming what sort of object x is, we may conclude ∀xP(x) for the same x. | |
| From: Robert S. Wolf (A Tour through Mathematical Logic [2005], 1.3) | |
| A reaction: This principle needs watching closely. If you pick one person in London, with no presuppositions, and it happens to be a woman, can you conclude that all the people in London are women? Fine in logic and mathematics, suspect in life. |
| 13521 | Universal Specification: ∀xP(x) implies P(t). True for all? Then true for an instance [Wolf,RS] |
| Full Idea: Universal Specification: from ∀xP(x) we may conclude P(t), where t is an appropriate term. If something is true for all members of a domain, then it is true for some particular one that we specify. | |
| From: Robert S. Wolf (A Tour through Mathematical Logic [2005], 1.3) |
| 13523 | Existential Generalization (or 'proof by example'): if we can say P(t), then we can say something is P [Wolf,RS] |
| Full Idea: Existential Generalization (or 'proof by example'): From P(t), where t is an appropriate term, we may conclude ∃xP(x). | |
| From: Robert S. Wolf (A Tour through Mathematical Logic [2005], 1.3) | |
| A reaction: It is amazing how often this vacuous-sounding principles finds itself being employed in discussions of ontology, but I don't quite understand why. |
| 19663 | We can allow contradictions in thought, but not inconsistency [Meillassoux] |
| Full Idea: For contemporary logicians, it is not non-contradiction that provides the criterion for what is thinkable, but rather inconsistency. | |
| From: Quentin Meillassoux (After Finitude; the necessity of contingency [2006], 3) | |
| A reaction: The point is that para-consistent logic might permit isolated contradictions (as true) within a system, but it is only contradiction across the system (inconsistencies) which make the system untenable. |
| 19664 | Paraconsistent logics are to prevent computers crashing when data conflicts [Meillassoux] |
| Full Idea: Paraconsistent logics were only developed in order to prevent computers, such as expert medical systems, from deducing anything whatsoever from contradictory data, because of the principle of 'ex falso quodlibet'. | |
| From: Quentin Meillassoux (After Finitude; the necessity of contingency [2006], 3) |
| 19665 | Paraconsistent logic is about statements, not about contradictions in reality [Meillassoux] |
| Full Idea: Paraconsistent logics are only ever dealing with contradictions inherent in statements about the world, never with the real contradictions in the world. | |
| From: Quentin Meillassoux (After Finitude; the necessity of contingency [2006], 3) | |
| A reaction: Thank goodness for that! I can accept that someone in a doorway is both in the room and not in the room, but not that they are existing in a real state of contradiction. I fear that a few daft people embrace the logic as confirming contradictory reality. |
| 13529 | Empty Set: ∃x∀y ¬(y∈x). The unique empty set exists [Wolf,RS] |
| Full Idea: Empty Set Axiom: ∃x ∀y ¬ (y ∈ x). There is a set x which has no members (no y's). The empty set exists. There is a set with no members, and by extensionality this set is unique. | |
| From: Robert S. Wolf (A Tour through Mathematical Logic [2005], 2.3) | |
| A reaction: A bit bewildering for novices. It says there is a box with nothing in it, or a pair of curly brackets with nothing between them. It seems to be the key idea in set theory, because it asserts the idea of a set over and above any possible members. |
| 13526 | Comprehension Axiom: if a collection is clearly specified, it is a set [Wolf,RS] |
| Full Idea: The comprehension axiom says that any collection of objects that can be clearly specified can be considered to be a set. | |
| From: Robert S. Wolf (A Tour through Mathematical Logic [2005], 2.2) | |
| A reaction: This is virtually tautological, since I presume that 'clearly specified' means pinning down exact which items are the members, which is what a set is (by extensionality). The naïve version is, of course, not so hot. |
| 13534 | In first-order logic syntactic and semantic consequence (|- and |=) nicely coincide [Wolf,RS] |
| Full Idea: One of the most appealing features of first-order logic is that the two 'turnstiles' (the syntactic single |-, and the semantic double |=), which are the two reasonable notions of logical consequence, actually coincide. | |
| From: Robert S. Wolf (A Tour through Mathematical Logic [2005], 5.3) | |
| A reaction: In the excitement about the possibility of second-order logic, plural quantification etc., it seems easy to forget the virtues of the basic system that is the target of the rebellion. The issue is how much can be 'expressed' in first-order logic. |
| 13535 | First-order logic is weakly complete (valid sentences are provable); we can't prove every sentence or its negation [Wolf,RS] |
| Full Idea: The 'completeness' of first order-logic does not mean that every sentence or its negation is provable in first-order logic. We have instead the weaker result that every valid sentence is provable. | |
| From: Robert S. Wolf (A Tour through Mathematical Logic [2005], 5.3) | |
| A reaction: Peter Smith calls the stronger version 'negation completeness'. |
| 13531 | Model theory reveals the structures of mathematics [Wolf,RS] |
| Full Idea: Model theory helps one to understand what it takes to specify a mathematical structure uniquely. | |
| From: Robert S. Wolf (A Tour through Mathematical Logic [2005], 5.1) | |
| A reaction: Thus it is the development of model theory which has led to the 'structuralist' view of mathematics. |
| 13532 | Model theory 'structures' have a 'universe', some 'relations', some 'functions', and some 'constants' [Wolf,RS] |
| Full Idea: A 'structure' in model theory has a non-empty set, the 'universe', as domain of variables, a subset for each 'relation', some 'functions', and 'constants'. | |
| From: Robert S. Wolf (A Tour through Mathematical Logic [2005], 5.2) |
| 13519 | Model theory uses sets to show that mathematical deduction fits mathematical truth [Wolf,RS] |
| Full Idea: Model theory uses set theory to show that the theorem-proving power of the usual methods of deduction in mathematics corresponds perfectly to what must be true in actual mathematical structures. | |
| From: Robert S. Wolf (A Tour through Mathematical Logic [2005], Pref) | |
| A reaction: That more or less says that model theory demonstrates the 'soundness' of mathematics (though normal arithmetic is famously not 'complete'). Of course, he says they 'correspond' to the truths, rather than entailing them. |
| 13533 | First-order model theory rests on completeness, compactness, and the Löwenheim-Skolem-Tarski theorem [Wolf,RS] |
| Full Idea: The three foundations of first-order model theory are the Completeness theorem, the Compactness theorem, and the Löwenheim-Skolem-Tarski theorem. | |
| From: Robert S. Wolf (A Tour through Mathematical Logic [2005], 5.3) | |
| A reaction: On p.180 he notes that Compactness and LST make no mention of |- and are purely semantic, where Completeness shows the equivalence of |- and |=. All three fail for second-order logic (p.223). |
| 13537 | An 'isomorphism' is a bijection that preserves all structural components [Wolf,RS] |
| Full Idea: An 'isomorphism' is a bijection between two sets that preserves all structural components. The interpretations of each constant symbol are mapped across, and functions map the relation and function symbols. | |
| From: Robert S. Wolf (A Tour through Mathematical Logic [2005], 5.4) |
| 13539 | The LST Theorem is a serious limitation of first-order logic [Wolf,RS] |
| Full Idea: The Löwenheim-Skolem-Tarski theorem demonstrates a serious limitation of first-order logic, and is one of primary reasons for considering stronger logics. | |
| From: Robert S. Wolf (A Tour through Mathematical Logic [2005], 5.7) |
| 13538 | If a theory is complete, only a more powerful language can strengthen it [Wolf,RS] |
| Full Idea: It is valuable to know that a theory is complete, because then we know it cannot be strengthened without passing to a more powerful language. | |
| From: Robert S. Wolf (A Tour through Mathematical Logic [2005], 5.5) |
| 13525 | Most deductive logic (unlike ordinary reasoning) is 'monotonic' - we don't retract after new givens [Wolf,RS] |
| Full Idea: Deductive logic, including first-order logic and other types of logic used in mathematics, is 'monotonic'. This means that we never retract a theorem on the basis of new givens. If T|-φ and T⊆SW, then S|-φ. Ordinary reasoning is nonmonotonic. | |
| From: Robert S. Wolf (A Tour through Mathematical Logic [2005], 1.7) | |
| A reaction: The classic example of nonmonotonic reasoning is the induction that 'all birds can fly', which is retracted when the bird turns out to be a penguin. He says nonmonotonic logic is a rich field in computer science. |
| 13530 | An ordinal is an equivalence class of well-orderings, or a transitive set whose members are transitive [Wolf,RS] |
| Full Idea: Less theoretically, an ordinal is an equivalence class of well-orderings. Formally, we say a set is 'transitive' if every member of it is a subset of it, and an ordinal is a transitive set, all of whose members are transitive. | |
| From: Robert S. Wolf (A Tour through Mathematical Logic [2005], 2.4) | |
| A reaction: He glosses 'transitive' as 'every member of a member of it is a member of it'. So it's membership all the way down. This is the von Neumann rather than the Zermelo approach (which is based on singletons). |
| 19677 | What is mathematically conceivable is absolutely possible [Meillassoux] |
| Full Idea: We must establish the thesis that what is mathematically conceivable is absolutely possible. | |
| From: Quentin Meillassoux (After Finitude; the necessity of contingency [2006], 5) | |
| A reaction: The truth of this thesis would permanently establish mathematics as the only possible language of science. Personally I have no idea how you could prove or assess such a thesis. It is a lovely speculation, though. 'The structure of the possible' (p,127) |
| 13518 | Modern mathematics has unified all of its objects within set theory [Wolf,RS] |
| Full Idea: One of the great achievements of modern mathematics has been the unification of its many types of objects. It began with showing geometric objects numerically or algebraically, and culminated with set theory representing all the normal objects. | |
| From: Robert S. Wolf (A Tour through Mathematical Logic [2005], Pref) | |
| A reaction: His use of the word 'object' begs all sorts of questions, if you are arriving from the street, where an object is something which can cause a bruise - but get used to it, because the word 'object' has been borrowed for new uses. |
| 19659 | The absolute is the impossibility of there being a necessary existent [Meillassoux] |
| Full Idea: We maintain that it is absolutely necessary that every entity might not exist. ...The absolute is the absolute impossibility of a necessary being. | |
| From: Quentin Meillassoux (After Finitude; the necessity of contingency [2006], 3) | |
| A reaction: This is the main thesis of his book. The usual candidates for necessary existence are God, and mathematical objects. I am inclined to agree with Meillassoux. |
| 19662 | It is necessarily contingent that there is one thing rather than another - so something must exist [Meillassoux] |
| Full Idea: It is necessary that there be something rather than nothing because it is necessarily contingent that there is something rather than something else. | |
| From: Quentin Meillassoux (After Finitude; the necessity of contingency [2006], 3) | |
| A reaction: The great charm of metaphysics is the array of serious answers to the question of why there is something rather than nothing. You'll need to read Meillassoux's book to understand this one. |
| 19654 | We must give up the modern criterion of existence, which is a correlation between thought and being [Meillassoux] |
| Full Idea: It is incumbent upon us to break with the ontological requisite of the moderns, according to which 'to be is to be a correlate'. | |
| From: Quentin Meillassoux (After Finitude; the necessity of contingency [2006], 2) | |
| A reaction: He blames Kant for this pernicious idea, which has driven philosophy away from realist science, when it should be supporting and joining it. As a realist I agree, and find Meillassoux very illuminating on the subject. |
| 5500 | Biologists see many organic levels, 'abstract' if seen from below, 'structural' if seen from above [Lycan] |
| Full Idea: Biologists don't split living things into a 'structural' level and an 'abstract' level; ..rather, they are organised at many levels, each level 'abstract' with respect to those beneath it, but 'structural' as it realises those levels above it. | |
| From: William Lycan (Introduction - Ontology [1999], p.9) | |
| A reaction: This is a very helpful distinction. Compare Idea 4601. It seems to fit well with the 'homuncular' picture of a hierarchical mind, and explains why there are so many levels of description available for mental life. |
| 5494 | 'Lightning is electric discharge' and 'Phosphorus is Venus' are synthetic a posteriori identities [Lycan] |
| Full Idea: There is such a thing as synthetic and a posteriori identity that is nonetheless genuine identity, as in lightning being electrical discharge, and the Morning Star being Venus. | |
| From: William Lycan (Introduction - Ontology [1999], p.5) | |
| A reaction: It is important to note that although these identities are synthetic a posteriori, that doesn't make them contingent. The early identity theorists like Smart seemed to think that it did. Kripke must be right that they are necessary identities. |
| 19660 | Possible non-being which must be realised is 'precariousness'; absolute contingency might never not-be [Meillassoux] |
| Full Idea: My term 'precariousness' designates a possibility of not-being which must eventually be realised. By contrast, absolute contingency designates a pure possibility; one which may never be realised. | |
| From: Quentin Meillassoux (After Finitude; the necessity of contingency [2006], 3) | |
| A reaction: I thoroughly approve of this distinction, because I have often enountered the assumption that all contingency is precariousness, and I have never seen why that should be so. In Aquinas's Third Way, for example. The 6 on a die may never come up. |
| 19671 | The idea of chance relies on unalterable physical laws [Meillassoux] |
| Full Idea: The very notion of chance is only conceivable on condition that there are unalterable physical laws. | |
| From: Quentin Meillassoux (After Finitude; the necessity of contingency [2006], 4) | |
| A reaction: Laws might be contingent, even though they never alter. Chance in horse racing relies on the stability of whole institution of horse racing. |
| 19651 | Unlike speculative idealism, transcendental idealism assumes the mind is embodied [Meillassoux] |
| Full Idea: What distinguishes transcendental idealism from speculative idealism is the fact that the former does not posit the existence of the transcendental subject apart from its bodily individuation. | |
| From: Quentin Meillassoux (After Finitude; the necessity of contingency [2006], 1) | |
| A reaction: These modern French philosophers explain things so much more clearly than the English! The 'speculative' version is seen in Berkeley. On p.17 he says transcendental idealism is 'civilised', and speculative idealism is 'uncouth'. |
| 19647 | The aspects of objects that can be mathematical allow it to have objective properties [Meillassoux] |
| Full Idea: All aspects of the object that can give rise to a mathematical thought rather than to a perception or a sensation can be meaningfully turned into the properties of the thing not only as it is with me, but also as it is without me. | |
| From: Quentin Meillassoux (After Finitude; the necessity of contingency [2006], 1) | |
| A reaction: This is Meillassoux's spin on the primary/secondary distinction, which he places at the heart of the scientific revolution. Cartesian dualism offers a separate space for the secondary qualities. He is appalled when philosophers reject the distinction. |
| 19652 | How can we mathematically describe a world that lacks humans? [Meillassoux] |
| Full Idea: How is mathematical discourse able to describe a reality where humanity is absent? | |
| From: Quentin Meillassoux (After Finitude; the necessity of contingency [2006], 1) | |
| A reaction: He is referring to the prehistoric world. He takes this to be a key question about the laws of nature. We extrapolate mathematically from the experienced world, relying on the stability of the laws. Must they be necessary to be stable? No, it seems. |
| 19668 | Hume's question is whether experimental science will still be valid tomorrow [Meillassoux] |
| Full Idea: Hume's question can be formulated as follows: can we demonstrate that the experimental science which is possible today will still be possible tomorrow? | |
| From: Quentin Meillassoux (After Finitude; the necessity of contingency [2006], 4) | |
| A reaction: Could there be deep universal changes going on in nature which science could never, even in principle, detect? |
| 19650 | The transcendental subject is not an entity, but a set of conditions making science possible [Meillassoux] |
| Full Idea: The transcendental subject simply cannot be said to exist; which is to say that the subject is not an entity, but rather a set of conditions rendering objective scientific knowledge of entities possible. | |
| From: Quentin Meillassoux (After Finitude; the necessity of contingency [2006], 1) | |
| A reaction: Meillassoux treats this as part of the Kantian Disaster, which made an accurate account of the scientific revolution impossible for philosophers. Kant's ego seems to have primarily an epistemological role. |
| 5496 | Functionalism has three linked levels: physical, functional, and mental [Lycan] |
| Full Idea: Functionalism has three distinct levels of description: a neurophysiological description, a functional description (relative to a program which the brain is realising), and it may have a further mental description. | |
| From: William Lycan (Introduction - Ontology [1999], p.6) | |
| A reaction: I have always thought that the 'levels of description' idea was very helpful in describing the mind/brain. I feel certain that we are dealing with a single thing, so this is the only way we can account for the diverse ways in which we discuss it. |
| 5499 | A mental state is a functional realisation of a brain state when it serves the purpose of the organism [Lycan] |
| Full Idea: Some theorists have said that the one-to-one correspondence between the organism and parts of its 'program' is too liberal, and suggest that the state and its functional role are seen teleologically, as functioning 'for' the organism. | |
| From: William Lycan (Introduction - Ontology [1999], p.9) | |
| A reaction: This seems an inevitable development, once the notion of a 'function' is considered. It has to be fitted into some sort of Aristotelian teleological picture, even if the functions are seen subjectively (by what?). Purpose is usually seen as evolutionary. |
| 5501 | People are trying to explain biological teleology in naturalistic causal terms [Lycan] |
| Full Idea: There is now a small but vigorous industry whose purpose is to explicate biological teleology in naturalistic terms, typically in terms of causes. | |
| From: William Lycan (Introduction - Ontology [1999], p.10) | |
| A reaction: This looks like a good strategy. In some sense, it seems clear that the moon has no purpose, but an eyeball has one. Via evolution, one would expect to reduce this to causation. Purposes are real (not subjective), but they are reducible. |
| 19667 | If the laws of nature are contingent, shouldn't we already have noticed it? [Meillassoux] |
| Full Idea: The standard objection is that if the laws of nature were actually contingent, we would already have noticed it. | |
| From: Quentin Meillassoux (After Finitude; the necessity of contingency [2006], 4) | |
| A reaction: Meillassoux offers a sustained argument that the laws of nature are necessarily contingent. In Idea 19660 he distinguishes contingencies that must change from those that merely could change. |
| 19670 | Why are contingent laws of nature stable? [Meillassoux] |
| Full Idea: We must ask how we are to explain the manifest stability of physical laws, given that we take these to be contingent? | |
| From: Quentin Meillassoux (After Finitude; the necessity of contingency [2006], 4) | |
| A reaction: Meissalloux offers a very deep and subtle answer to this question... It is based on the possibilities of chaos being an uncountable infinity... It is a very nice question, which physicists might be able to answer, without help from philosophy. |
| 19653 | The ontological proof of a necessary God ensures a reality external to the mind [Meillassoux] |
| Full Idea: Since Descartes conceives of God as existing necessarily, whether I exist to think of him or not, Descartes assures me of a possible access to an absolute reality - a Great Outdoors that is not a correlate of my thought. | |
| From: Quentin Meillassoux (After Finitude; the necessity of contingency [2006], 2) | |
| A reaction: His point is that the ontological argument should be seen as part of the scientific revolution, and not an anomaly within it. Interesting. |
| 19658 | Now that the absolute is unthinkable, even atheism is just another religious belief (though nihilist) [Meillassoux] |
| Full Idea: Once the absolute has become unthinkable, even atheism, which also targets God's inexistence in the manner of an absolute, is reduced to a mere belief, and hence to a religion, albeit of the nihilist kind. | |
| From: Quentin Meillassoux (After Finitude; the necessity of contingency [2006], 2) | |
| A reaction: An interesting claim. Rather hard to agree or disagree, though the idea that atheism must qualify as a religion seems odd. If it is unqualified it does have the grand quality of a religion, but if it is fallibilist it just seems like an attitude. |