21 ideas
| 10482 | The logic of ZF is classical first-order predicate logic with identity [Boolos] |
| Full Idea: The logic of ZF Set Theory is classical first-order predicate logic with identity. | |
| From: George Boolos (Must We Believe in Set Theory? [1997], p.121) | |
| A reaction: This logic seems to be unable to deal with very large cardinals, precisely those that are implied by set theory, so there is some sort of major problem hovering here. Boolos is fairly neutral. |
| 10492 | A few axioms of set theory 'force themselves on us', but most of them don't [Boolos] |
| Full Idea: Maybe the axioms of extensionality and the pair set axiom 'force themselves on us' (Gödel's phrase), but I am not convinced about the axioms of infinity, union, power or replacement. | |
| From: George Boolos (Must We Believe in Set Theory? [1997], p.130) | |
| A reaction: Boolos is perfectly happy with basic set theory, but rather dubious when very large cardinals come into the picture. |
| 10485 | Naïve sets are inconsistent: there is no set for things that do not belong to themselves [Boolos] |
| Full Idea: The naïve view of set theory (that any zero or more things form a set) is natural, but inconsistent: the things that do not belong to themselves are some things that do not form a set. | |
| From: George Boolos (Must We Believe in Set Theory? [1997], p.127) | |
| A reaction: As clear a summary of Russell's Paradox as you could ever hope for. |
| 10484 | The iterative conception says sets are formed at stages; some are 'earlier', and must be formed first [Boolos] |
| Full Idea: According to the iterative conception, every set is formed at some stage. There is a relation among stages, 'earlier than', which is transitive. A set is formed at a stage if and only if its members are all formed before that stage. | |
| From: George Boolos (Must We Believe in Set Theory? [1997], p.126) | |
| A reaction: He gives examples of the early stages, and says the conception is supposed to 'justify' Zermelo set theory. It is also supposed to make the axioms 'natural', rather than just being selected for convenience. And it is consistent. |
| 10491 | Infinite natural numbers is as obvious as infinite sentences in English [Boolos] |
| Full Idea: The existence of infinitely many natural numbers seems to me no more troubling than that of infinitely many computer programs or sentences of English. There is, for example, no longest sentence, since any number of 'very's can be inserted. | |
| From: George Boolos (Must We Believe in Set Theory? [1997], p.129) | |
| A reaction: If you really resisted an infinity of natural numbers, presumably you would also resist an actual infinity of 'very's. The fact that it is unclear what could ever stop a process doesn't guarantee that the process is actually endless. |
| 10483 | Mathematics and science do not require very high orders of infinity [Boolos] |
| Full Idea: To the best of my knowledge nothing in mathematics or science requires the existence of very high orders of infinity. | |
| From: George Boolos (Must We Believe in Set Theory? [1997], p.122) | |
| A reaction: He is referring to particular high orders of infinity implied by set theory. Personally I want to wield Ockham's Razor. Is being implied by set theory a sufficient reason to accept such outrageous entities into our ontology? |
| 10490 | Mathematics isn't surprising, given that we experience many objects as abstract [Boolos] |
| Full Idea: It is no surprise that we should be able to reason mathematically about many of the things we experience, for they are already 'abstract'. | |
| From: George Boolos (Must We Believe in Set Theory? [1997], p.129) | |
| A reaction: He has just given a list of exemplary abstract objects (Idea 10489), but I think there is a more interesting idea here - that our experience of actual physical objects is to some extent abstract, as soon as it is conceptualised. |
| 10488 | It is lunacy to think we only see ink-marks, and not word-types [Boolos] |
| Full Idea: It's a kind of lunacy to think that sound scientific philosophy demands that we think that we see ink-tracks but not words, i.e. word-types. | |
| From: George Boolos (Must We Believe in Set Theory? [1997], p.128) | |
| A reaction: This seems to link him with Armstrong's mockery of 'ostrich nominalism'. There seems to be some ambiguity with the word 'see' in this disagreement. When we look at very ancient scratches on stones, why don't we always 'see' if it is words? |
| 10487 | I am a fan of abstract objects, and confident of their existence [Boolos] |
| Full Idea: I am rather a fan of abstract objects, and confident of their existence. Smaller numbers, sets and functions don't offend my sense of reality. | |
| From: George Boolos (Must We Believe in Set Theory? [1997], p.128) | |
| A reaction: The great Boolos is rather hard to disagree with, but I disagree. Logicians love abstract objects, indeed they would almost be out of a job without them. It seems to me they smuggle them into our ontology by redefining either 'object' or 'exists'. |
| 10489 | We deal with abstract objects all the time: software, poems, mistakes, triangles.. [Boolos] |
| Full Idea: We twentieth century city dwellers deal with abstract objects all the time, such as bank balances, radio programs, software, newspaper articles, poems, mistakes, triangles. | |
| From: George Boolos (Must We Believe in Set Theory? [1997], p.129) | |
| A reaction: I find this claim to be totally question-begging, and typical of a logician. The word 'object' gets horribly stretched in these discussions. We can create concepts which have all the logical properties of objects. Maybe they just 'subsist'? |
| 468 | Musical performance can reveal a range of virtues [Damon of Ath.] |
| Full Idea: In singing and playing the lyre, a boy will be likely to reveal not only courage and moderation, but also justice. | |
| From: Damon (fragments/reports [c.460 BCE], B4), quoted by (who?) - where? |
| 21075 | The state of nature always involves the threat of war [Kant] |
| Full Idea: The state of nature is a state of war. For even if it does not involve active hostilities, it involves a constant threat of their breaking out. | |
| From: Immanuel Kant (Perpetual Peace [1795], 2) | |
| A reaction: Kant is siding with Hobbes against Rousseau, despite Rousseau's claim that Hobbes's pessimism concerns a more advanced situation that the true (and peaceful) state of nature. |
| 20569 | Kant made the social contract international and cosmopolitan [Kant, by Oksala] |
| Full Idea: Kant developed the social contract theory into an international and cosmopolitan idea. | |
| From: report of Immanuel Kant (Perpetual Peace [1795]) by Johanna Oksala - Political Philosophy: all that matters Ch.6 | |
| A reaction: That is, the contract both operates between states, and rises above them. I found this idea rather thrilling when I first met it (listening to Onora O'Neill). But then I remain a child of the Englightenment. |
| 21079 | The a priori general will of a people shows what is right [Kant] |
| Full Idea: It is precisely the general will as it is given a priori, within a single people or in the mutual relationships of various peoples, which alone determines what is right among men. | |
| From: Immanuel Kant (Perpetual Peace [1795], App 1) | |
| A reaction: The clearest quotation for showing Kant's debt to Rousseau. Why should Rousseau bother to have a real assembly of the people, if the General Will can be worked out a priori? Indeed, the a priori version must be deemed superior to any meeting. |
| 21077 | Each nation should, from self-interest, join an international security constitution [Kant] |
| Full Idea: Each nation, for the sake of its own security, can and ought to demand of the others that they should enter along with it into a constitution, similar to the civil one, within which the rights of each could be secured. | |
| From: Immanuel Kant (Perpetual Peace [1795], 2.2nd) | |
| A reaction: Not sure how close the United Nations takes us to this. You have to admire Kant for this one. |
| 21078 | A constitution must always be improved when necessary [Kant] |
| Full Idea: Changes for the better are necessary, in order that the constitution may constantly approach the optimum end prescribed by laws of right. | |
| From: Immanuel Kant (Perpetual Peace [1795], App 1) | |
| A reaction: This should be a clause in every constitution. It is crazy to feel trapped by a misjudgement or outdated view of your ancestors. |
| 21076 | Equality is where you cannot impose a legal obligation you yourself wouldn't endure [Kant] |
| Full Idea: Rightful equality within a state is a relationship among citizens where no-one can put anyone else under a legal obligation without submitting simultaneously to a law which requires that he can be put under the same kind of obligation by the other person. | |
| From: Immanuel Kant (Perpetual Peace [1795], 2.1st n) | |
| A reaction: This appears only to be legal equality, rather than political or economic or social equality. |
| 20570 | There is now a growing universal community, and violations of rights are felt everywhere [Kant] |
| Full Idea: The peoples of the earth have entered in varying degrees into a universal community, and it has developed to the point where a violation of rights in one part of the world is felt everywhere. | |
| From: Immanuel Kant (Perpetual Peace [1795], 'Third') | |
| A reaction: I hope slavery was at the forefront of his mind when he wrote that. It is only in very recent times (since about 1960?) that major violations of rights are felt to matter to the whole human race. A long way to go, though. |
| 20571 | There are political and inter-national rights, but also universal cosmopolitan rights [Kant] |
| Full Idea: The idea of a cosmopolitan right is not fantastic and overstrained; it is a necessary complement to the unwritten code of political and international right, transforming it into a universal right of humanity. | |
| From: Immanuel Kant (Perpetual Peace [1795], 'Third') | |
| A reaction: The interesting thought is that there are no 'natural rights', but there can be universal rights insofar as there exists a universal community. See the UN Declaration of Human Rights c.1948. |
| 21073 | Hiring soldiers is to use them as instruments, ignoring their personal rights [Kant] |
| Full Idea: The hiring of men to kill or be killed seems to mean using them as mere machines and insturments in the hands of someone else (the state), which cannot easily be reconciled with the rights of man in one's own person. | |
| From: Immanuel Kant (Perpetual Peace [1795], 1.3) | |
| A reaction: Kant was not a pacificist, though this makes him sound like one. Some men go off to war with enthusiasm, and then regret it. Exploitation of rational beings may be the worst sin in Kant's Enlightenment world. |
| 21074 | Some trust in the enemy is needed during wartime, or peace would be impossible [Kant] |
| Full Idea: It must remain possible, even in wartime, to have some sort of trust in the attitude of the enemy, otherwise peace could not be concluded and the hostilities would turn into a war of extermination. | |
| From: Immanuel Kant (Perpetual Peace [1795], 1.6) | |
| A reaction: Consider the 'unconditional surrender' approach to the Nazis in 1944, and the peace of May 1945, made with very different Germans. How do you make peace with an enemy you cannot trust? |