56 ideas
| 7113 | Phenomenology assumes that all consciousness is of something [Sartre] |
| Full Idea: The essential principle of phenomenology is that 'all consciousness is consciousness of something'. | |
| From: Jean-Paul Sartre (Transcendence of the Ego [1937], I (B)) | |
| A reaction: This idea is found well before Husserl, in Schopenhauer (Idea 4166). It seems to contradict a thought such as Locke's (Idea 1202), that self-awareness is a separate and distinct criterion for personal identity. Sartre gives a nice account. |
| 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) |
| 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) |
| 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) |
| 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) |
| 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. |
| 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) |
| 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. |
| 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) |
| 7112 | The Cogito depends on a second-order experience, of being conscious of consciousness [Sartre] |
| Full Idea: We must remember that all authors who have described the Cogito have presented it as a reflective operation, i.e. as second-order. This Cogito is performed by a consciousness directed towards consciousness, which takes consciousness as its object. | |
| From: Jean-Paul Sartre (Transcendence of the Ego [1937], I (B)) | |
| A reaction: Sartre is raising the nice question of whether the Cogito still works for first-order consciousness, which attends totally to external objects. He claims that it doesn't. Contrast Russell, who says (Idea 5380) that it only works when it is first-order! |
| 7114 | The consciousness that says 'I think' is not the consciousness that thinks [Sartre] |
| Full Idea: The consciousness that says 'I think' is precisely not the consciousness that thinks. | |
| From: Jean-Paul Sartre (Transcendence of the Ego [1937], I (B)) | |
| A reaction: All parties seem to be agreed that if we are going to introspect in search of our own ego, we must distinguish between the mental act of instrospection and the mental act of applying the mind to the world. Each gives a different result. |
| 7119 | Is the Cogito reporting an immediate experience of doubting, or the whole enterprise of doubting? [Sartre] |
| Full Idea: When Descartes says 'I doubt therefore I am', is he talking about the spontaneous doubt that reflective consciousness grasps in its instantaneous character, or is he talking of the enterprise of doubting? This ambiguity can lead to serious errors. | |
| From: Jean-Paul Sartre (Transcendence of the Ego [1937], II (B)) | |
| A reaction: Interesting. The obvious response is that it is about the immediate experience, but that leads to the problem of an instantaneous ego, which can't be justified over time. The 'enterprise' gives an enduring ego, but it is a more intellectual concept. |
| 7122 | We can never, even in principle, grasp other minds, because the Ego is self-conceiving [Sartre] |
| Full Idea: The Ego can be conceived only through itself and this is why we cannot grasp the consciousness of another (for this reason alone, and not because bodies separate us). | |
| From: Jean-Paul Sartre (Transcendence of the Ego [1937], II (D)) | |
| A reaction: Interesting. This makes telepathy a logical impossibility, and the body the only possible route for the communication between two minds. But, is Sartre is right, how do bodily events penetrate the inturned world of the Ego? |
| 7125 | A consciousness can conceive of no other consciousness than itself [Sartre] |
| Full Idea: A consciousness can conceive of no other consciousness than itself. | |
| From: Jean-Paul Sartre (Transcendence of the Ego [1937], Conc (1)) | |
| A reaction: This is why we don't know what it is like to be a bat. This seems right, though it looks like a contingent truth, and yet Sartre seems to offer it as a necessary truth. Can God conceive of my consciousness? |
| 7108 | The eternal truth of 2+2=4 is what gives unity to the mind which regularly thinks it [Sartre] |
| Full Idea: The unity of the thousand active consciousnesses through which I have added two and two to make four, is the transcendent object '2+2=4'. Without the permanence of this eternal truth, it would be impossible to conceive of a real unity of mind. | |
| From: Jean-Paul Sartre (Transcendence of the Ego [1937], I (A)) | |
| A reaction: This is the germ of externalism, here presented as a Platonic attitude to arithmetic, rather than being about water or gold. He claims that internalist attitudes to unity are fictions. I am inclined to think he is wrong, and that unity is biological. |
| 7111 | Consciousness exists as consciousness of itself [Sartre] |
| Full Idea: The existence of consciousness is an absolute, because consciousness is consciousness of itself; the type of existence that consciousness has is that it is consciousness of itself. | |
| From: Jean-Paul Sartre (Transcendence of the Ego [1937], I (A)) | |
| A reaction: I find this unconvincing. Anyone analysis the nature of the mind should think as much about animal minds as human minds. It seems obvious to me that there is likely to be an animal consciousness which is entirely of environment and its body. |
| 22226 | Since we are a consciousness, Sartre entirely rejected the unconscious mind [Sartre, by Daigle] |
| Full Idea: Sartre refused, denied and fought against the unconscious. Since we are consciousness, there cannot be such a thing as unconsciousness. | |
| From: report of Jean-Paul Sartre (Transcendence of the Ego [1937]) by Christine Daigle - Jean-Paul Sartre 2.1 | |
| A reaction: The modern view is increasingly opposed to this, as neuroscience and psychology uncover hidden motives etc. Sartre's view is still legitimate, though. An unconscious motive is not more my motive than a law of the land is part of me? |
| 7107 | Intentionality defines, transcends and unites consciousness [Sartre] |
| Full Idea: Consciousness is defined by intentionality. Through intentionality it transcends itself, it unifies itself by going outside itself. | |
| From: Jean-Paul Sartre (Transcendence of the Ego [1937], I (A)) | |
| A reaction: The standard view for a hundred years was Brentano's idea that intentionality defines the mind. Qualia are the modern rival. If I had to choose I think I would go for intentionality, but they may be naturally and metaphysically inseparable. |
| 7109 | If you think of '2+2=4' as the content of thought, the self must be united transcendentally [Sartre] |
| Full Idea: It is possible that those who think that '2 and 2 make 4' is the content of my representations may be forced to resort to a transcendental and subjective principle of unification - in other words, the I. | |
| From: Jean-Paul Sartre (Transcendence of the Ego [1937], I (A)) | |
| A reaction: He suggests that thoughts themselves unite the mind, externally. If you think of thoughts as internal, you must resort to a transcendental fiction to unify the mind. Personally I think the mind is inherently unified by brain structures. |
| 7106 | The Ego is not formally or materially part of consciousness, but is outside in the world [Sartre] |
| Full Idea: I should like to show here that the Ego is neither formally nor materially in consciousness; it is outside, in the world; it is a being in the world, like the Ego of another. | |
| From: Jean-Paul Sartre (Transcendence of the Ego [1937], Intro) | |
| A reaction: This idea is the germ of what has got modern externalists about the mind (see quotations from Mark Rowlands) interested in Sartre. Personally I think he is wrong, and the Ego is a part of consciousness. It doesn't, though, have sharp boundaries. |
| 7117 | How could two I's, the reflective and the reflected, communicate with each other? [Sartre] |
| Full Idea: If the 'I' is part of consciousness, there will be two I's: the reflective and the reflected. ...but it is unacceptable for any communication to be established between the reflective I and the reflected I, if they are real elements of consciousness. | |
| From: Jean-Paul Sartre (Transcendence of the Ego [1937], I (B)) | |
| A reaction: If we accept that there are two orders of consciousness (reflective, about itself, and reflected, about the world) it seems reasonable to say that there cannot be an 'I' in both of them. A nice, and intriguing, argument. |
| 7123 | Knowing yourself requires an exterior viewpoint, which is necessarily false [Sartre] |
| Full Idea: 'To know oneself well' is inevitably to look at oneself from the point of view of someone else, in other words from a point of view that is necessarily false. | |
| From: Jean-Paul Sartre (Transcendence of the Ego [1937], II (D)) | |
| A reaction: (This is because the Ego cannot be known from the outside). I agree with Russell that the self is most evident when we are engaged with the world, which implies that you can only acquire self-knowledge by studying those engagements. |
| 22225 | My ego is more intimate to me, but not more certain than other egos [Sartre] |
| Full Idea: My I, in efffect, is no more certain for consciousness than the I of other men. It is only more intimate. | |
| From: Jean-Paul Sartre (Transcendence of the Ego [1937], p.104), quoted by Christine Daigle - Jean-Paul Sartre 2.1 | |
| A reaction: Not sure how to assess this. Other people seem just as real as I do, when I encounter them, as friend or as foe. And in dealing with them we act as if dealing with their Self (rather than their legs, say). So this idea seems a good one. |
| 7124 | The Ego never appears except when we are not looking for it [Sartre] |
| Full Idea: The Ego never appears except when we are not looking for it. | |
| From: Jean-Paul Sartre (Transcendence of the Ego [1937], II (D)) | |
| A reaction: He denies that we know the Ego when engaged with the world, and agrees with Hume that the ego can't be directly known. All that is left is this, which seems to be introspection 'out of the corner of your eye'. Not persuasive. |
| 7116 | When we are unreflective (as when chasing a tram) there is no 'I' [Sartre] |
| Full Idea: There is no 'I' on the unreflected level. When I run after a tram, ...there is no I. There is a consciousness of the tram-needing-to-be-caught, and a non-positional consciousness of consciousness. | |
| From: Jean-Paul Sartre (Transcendence of the Ego [1937], I (B)) | |
| A reaction: Russell (Idea 5380) says exactly the opposite. My sympathies are more with Russell. I don't just focus on the tram, I focus on the relation between myself and the tram, and that includes my need to catch it, as well as my body. |
| 7120 | It is theoretically possible that the Ego consists entirely of false memories [Sartre] |
| Full Idea: One cannot rule out the metaphysical hypothesis that my Ego is not composed of elements that have existed in reality (ten years or one second ago), but is merely constituted by false memories. | |
| From: Jean-Paul Sartre (Transcendence of the Ego [1937], II (D)) | |
| A reaction: (He mentions the evil demon as a source). The problem that false memories (such as George IV 'remembering' he was at Waterloo, when he wasn't) is well known. But this raises the possibility of all memories being false, yet constituting the person. |
| 7110 | If the 'I' is transcendental, it unnecessarily splits consciousness in two [Sartre] |
| Full Idea: The superfluous transcendental 'I' is actually a hindrance. If it existed, it would violently separate consciousness from itself, it would divide it, slicing through consciousness like an opaque blade. | |
| From: Jean-Paul Sartre (Transcendence of the Ego [1937], I (A)) | |
| A reaction: I see no a priori reason why consciousness should not be split in two, if that's how it is. Personally I am happy with a fairly traditional Cartesian view, that the self is the will and understanding, and the rest of consciousness is its working material. |
| 7115 | Maybe it is the act of reflection that brings 'me' into existence [Sartre] |
| Full Idea: Might it not be precisely the reflective act that brings the me into being in reflected consciousness? | |
| From: Jean-Paul Sartre (Transcendence of the Ego [1937], I (B)) | |
| A reaction: He admits some sort of self a second-order entity, but this is 'transcendental', and essentially an illusion. This elimination of the first-order self clears the way for the existential view, that we can create whatever self we want. I disagree. |
| 7121 | The Ego only appears to reflection, so it is cut off from the World [Sartre] |
| Full Idea: The Ego is an object that appears only to reflection, and is thereby radically cut off from the World. | |
| From: Jean-Paul Sartre (Transcendence of the Ego [1937], II (D)) | |
| A reaction: This is the culmination of Sartre's attack (in 1937) on the Ego, paving the way for the freedom of existentialism. Personally I don't accept this picture of the Ego as a second-order fiction. My Ego is part of my relationship with the World. |
| 16713 | Philosophers are the forefathers of heretics [Tertullian] |
| Full Idea: Philosophers are the forefathers of heretics. | |
| From: Tertullian (works [c.200]), quoted by Robert Pasnau - Metaphysical Themes 1274-1671 20.2 |
| 6610 | I believe because it is absurd [Tertullian] |
| Full Idea: I believe because it is absurd ('Credo quia absurdum est'). | |
| From: Tertullian (works [c.200]), quoted by Robert Fogelin - Walking the Tightrope of Reason n4.2 | |
| A reaction: This seems to be a rather desperate remark, in response to what must have been rather good hostile arguments. No one would abandon the support of reason if it was easy to acquire. You can't deny its engaging romantic defiance, though. |