28 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'? |
| 4871 | A thing is free if it acts only by the necessity of its own nature [Spinoza] |
| Full Idea: I say that a thing is free, which exists and acts solely by the necessity of its own nature. | |
| From: Baruch de Spinoza (Letter to G.H. Schaller [1674], 1674.10) | |
| A reaction: Of course, this isn't 'freedom' at all, but it seems to exactly right as an account of so-called freedom. In the case of a human being the 'necessity of our own nature' is character, and virtue and vice are the expressions of the necessities of character. |
| 3772 | The will, in the beginning, is entirely produced by desire [Mill] |
| Full Idea: The will, in the beginning, is entirely produced by desire. | |
| From: John Stuart Mill (Utilitarianism [1861], Ch.4) | |
| A reaction: This is the sort of simplistic psychology that modern philosophers tend to avoid. Personally I am more Kantian. I will and desire that the answer to 3+2=? is 5, simply because it is true. Mill must realise we can will ourselves to desire something. |
| 3769 | With early training, any absurdity or evil may be given the power of conscience [Mill] |
| Full Idea: There is hardly anything so absurd or so mischievous that it may not, by means of early sanctions and influence, be made to act on the human mind with all the influence of conscience. | |
| From: John Stuart Mill (Utilitarianism [1861], Ch.3) | |
| A reaction: Like this! Think of all the people who have had weird upbringings, and end up feeling guilty about absurd things. Conscience just summarise upbringing and social conventions. |
| 3767 | Motive shows the worth of the agent, but not of the action [Mill] |
| Full Idea: The motive has nothing to do with the morality of the action, though much with the worth of the agent. | |
| From: John Stuart Mill (Utilitarianism [1861], Ch.2) | |
| A reaction: I think it is an error to try to separate these too sharply. Morality can't be purely consequential, because it would make earthquakes immoral. Actions indicate the worth of agents. |
| 3771 | Virtues only have value because they achieve some further end [Mill] |
| Full Idea: Utilitarians believe that actions and dispositions are only virtuous because they promote another end than virtue. | |
| From: John Stuart Mill (Utilitarianism [1861], Ch.4) |
| 3768 | Orthodox morality is the only one which feels obligatory [Mill] |
| Full Idea: The customary morality, that which education and opinion have consecrated, is the only one which presents itself to the mind with the feeling of being in itself obligatory. | |
| From: John Stuart Mill (Utilitarianism [1861], Ch.3) |
| 3776 | Utilitarianism only works if everybody has a totally equal right to happiness [Mill] |
| Full Idea: The Greatest Happiness Principle is a mere form of empty words unless one person's happiness, supposed equal in degree, is counted for exactly as much as another's (Bentham's "everybody to count for one, nobody for more than one"). | |
| From: John Stuart Mill (Utilitarianism [1861], Ch.5) |
| 7202 | The English believe in the task of annihilating evil for the victory of good [Nietzsche on Mill] |
| Full Idea: One continues to believe in good and evil: in such a way that one feels the victory of good and the annihilation of evil to be a task (- this is English; a typical case is that shallow-headed John Stuart Mill). | |
| From: comment on John Stuart Mill (Utilitarianism [1861]) by Friedrich Nietzsche - Writings from Late Notebooks 11[148]e | |
| A reaction: The poor old English try very hard to be clear, sensible, practical and realistic, and get branded as 'shallow' for their pains. Nietzsche was a deeper thinker than Mill, but I would prefer Mill to Heidegger any day. |
| 5935 | Mill's qualities of pleasure is an admission that there are other good states of mind than pleasure [Ross on Mill] |
| Full Idea: Mill's introduction of quality of pleasures into the hedonistic calculus is an unconscious departure from hedonism and a half-hearted admission that there are other qualities than pleasantness in virtue of which states of mind are good. | |
| From: comment on John Stuart Mill (Utilitarianism [1861], Ch.2) by W. David Ross - The Right and the Good §VI | |
| A reaction: Mill argues that experienced people prefer some pleasures to others, but ducks the question of why they might prefer them. It can only be because they have some further desirable quality on top of the equal amount of pleasure. |
| 3764 | Actions are right if they promote pleasure, wrong if they promote pain [Mill] |
| Full Idea: The Greatest Happiness Principle holds that actions are right in proportion as they tend to promote happiness, wrong as they tend to produce the reverse of happiness. By happiness is intended pleasure, and the absence of pain. | |
| From: John Stuart Mill (Utilitarianism [1861], Ch.2) |
| 3763 | Ultimate goods such as pleasure can never be proved to be good [Mill] |
| Full Idea: What can be proved good must be so by being shown to be a means to something admitted to be good without proof. Music is good because it produces pleasure, but what proof is it possible to give that pleasure is good? | |
| From: John Stuart Mill (Utilitarianism [1861], Ch.1) |
| 3765 | Only pleasure and freedom from pain are desirable as ends [Mill] |
| Full Idea: Pleasure and freedom from pain are the only things desirable as ends. | |
| From: John Stuart Mill (Utilitarianism [1861], Ch.2) |
| 3766 | Better to be Socrates dissatisfied than a fool satisfied [Mill] |
| Full Idea: Better to be Socrates dissatisfied than a fool satisfied. | |
| From: John Stuart Mill (Utilitarianism [1861], Ch.2) |
| 3770 | General happiness is only desirable because individuals desire their own happiness [Mill] |
| Full Idea: No reason can be given why the general happiness is desirable, except that each person, so far as he believes it to be attainable, desires his own happiness. | |
| From: John Stuart Mill (Utilitarianism [1861], Ch.4) |
| 6697 | Moral rules protecting human welfare are more vital than local maxims [Mill] |
| Full Idea: Moral rules which forbid mankind to hurt one another are more vital to human well-being than any maxims about some department of human affairs; ..though in particular cases a social duty is so important, as to overrule any general maxim of justice. | |
| From: John Stuart Mill (Utilitarianism [1861]), quoted by Gordon Graham - Eight Theories of Ethics Ch.7 | |
| A reaction: The qualification is realistic, but raises the question of whether an 'act' calculation will always overrule any 'rule'. Maybe rule utilitirianism is just act utilitarianism, but ensuring that the calculations are long-term and extensive. (1871 edn) |
| 3774 | Rights are a matter of justice, not of benevolence [Mill] |
| Full Idea: Wherever there is a right, the case is one of justice, and not of the virtue of benevolence. | |
| From: John Stuart Mill (Utilitarianism [1861], Ch.5) |
| 3773 | No individual has the right to receive our benevolence [Mill] |
| Full Idea: No one has a moral right to our generosity or beneficence, because we are not morally bound to practise those virtues towards any given individual. | |
| From: John Stuart Mill (Utilitarianism [1861], Ch.5) |
| 3775 | A right is a valid claim to society's protection [Mill] |
| Full Idea: When we call anything a person's right, we mean that he has a valid claim on society to protect him in the possession of it. | |
| From: John Stuart Mill (Utilitarianism [1861], Ch.5) |