80 ideas
1403 | A rational donkey would starve to death between two totally identical piles of hay [Buridan, by PG] |
15945 | Second-order set theory just adds a version of Replacement that quantifies over functions [Lavine] |
15914 | An 'upper bound' is the greatest member of a subset; there may be several of these, so there is a 'least' one [Lavine] |
15921 | Collections of things can't be too big, but collections by a rule seem unlimited in size [Lavine] |
15937 | Those who reject infinite collections also want to reject the Axiom of Choice [Lavine] |
15936 | The Power Set is just the collection of functions from one collection to another [Lavine] |
15899 | Replacement was immediately accepted, despite having very few implications [Lavine] |
15930 | Foundation says descending chains are of finite length, blocking circularity, or ungrounded sets [Lavine] |
15898 | The controversy was not about the Axiom of Choice, but about functions as arbitrary, or given by rules [Lavine] |
15920 | Pure collections of things obey Choice, but collections defined by a rule may not [Lavine] |
15919 | The 'logical' notion of class has some kind of definition or rule to characterise the class [Lavine] |
15900 | The iterative conception of set wasn't suggested until 1947 [Lavine] |
15931 | The iterative conception needs the Axiom of Infinity, to show how far we can iterate [Lavine] |
15932 | The iterative conception doesn't unify the axioms, and has had little impact on mathematical proofs [Lavine] |
15933 | Limitation of Size: if it's the same size as a set, it's a set; it uses Replacement [Lavine] |
15913 | A collection is 'well-ordered' if there is a least element, and all of its successors can be identified [Lavine] |
15926 | Second-order logic presupposes a set of relations already fixed by the first-order domain [Lavine] |
15934 | Mathematical proof by contradiction needs the law of excluded middle [Lavine] |
15907 | Mathematics is nowadays (thanks to set theory) regarded as the study of structure, not of quantity [Lavine] |
15942 | Every rational number, unlike every natural number, is divisible by some other number [Lavine] |
15922 | For the real numbers to form a set, we need the Continuum Hypothesis to be true [Lavine] |
18250 | Cauchy gave a necessary condition for the convergence of a sequence [Lavine] |
15904 | The two sides of the Cut are, roughly, the bounding commensurable ratios [Lavine] |
15912 | Counting results in well-ordering, and well-ordering makes counting possible [Lavine] |
15947 | The infinite is extrapolation from the experience of indefinitely large size [Lavine] |
15949 | The theory of infinity must rest on our inability to distinguish between very large sizes [Lavine] |
15940 | The intuitionist endorses only the potential infinite [Lavine] |
15909 | 'Aleph-0' is cardinality of the naturals, 'aleph-1' the next cardinal, 'aleph-ω' the ω-th cardinal [Lavine] |
15915 | Ordinals are basic to Cantor's transfinite, to count the sets [Lavine] |
15917 | Paradox: the class of all ordinals is well-ordered, so must have an ordinal as type - giving a bigger ordinal [Lavine] |
15918 | Paradox: there is no largest cardinal, but the class of everything seems to be the largest [Lavine] |
15929 | Set theory will found all of mathematics - except for the notion of proof [Lavine] |
15935 | Modern mathematics works up to isomorphism, and doesn't care what things 'really are' [Lavine] |
15928 | Intuitionism rejects set-theory to found mathematics [Lavine] |
16678 | Without magnitude a thing would retain its parts, but they would have no location [Buridan] |
16793 | A thing is (less properly) the same over time if each part is succeeded by another [Buridan] |
16726 | Why can't we deduce secondary qualities from primary ones, if they cause them? [Buridan] |
16577 | Induction is not demonstration, because not all of the instances can be observed [Buridan] |
16576 | Science is based on induction, for general truths about fire, rhubarb and magnets [Buridan] |
4669 | Persons are conscious, they relate, they think, they feel, and they are self-aware [Glover] |
4656 | A problem arises in any moral system that allows more than one absolute right [Glover] |
4657 | Double Effect: no bad acts with good consequences, but possibly good acts despite bad consequences [Glover] |
4658 | Acts and Omissions: bad consequences are morally better if they result from an omission rather than an act [Glover] |
4659 | It doesn't seem worse to switch off a life-support machine than to forget to switch it on [Glover] |
4660 | Harmful omissions are unavoidable, while most harmful acts can be avoided [Glover] |
4661 | What matters is not intrinsic value of life or rights, but worthwhile and desired life, and avoidance of pain [Glover] |
4648 | 'Death' is best seen as irreversible loss of consciousness, since this is why we care about brain function [Glover] |
3785 | You can't separate acts from the people performing them [Glover] |
3786 | Aggression in defence may be beneficial but morally corrupting [Glover] |
4650 | The quality of a life is not altogether independent of its length [Glover] |
3784 | Duty prohibits some acts, whatever their consequences [Glover] |
3782 | Satisfaction of desires is not at all the same as achieving happiness [Glover, by PG] |
3787 | Rule-utilitarianism is either act-utilitarianism, or not really utilitarian [Glover] |
3783 | How can utilitarianism decide the ideal population size? [Glover] |
4675 | The sanctity of life doctrine implies a serious increase of abnormality among the population [Glover] |
4654 | Autonomy favours present opinions over future ones, and says nothing about the interests of potential people [Glover] |
4655 | If a whole community did not mind death, respect for autonomy suggests that you could kill them all [Glover] |
4680 | Autonomy seems to acquire greater weight when the decision is more important to a person [Glover] |
4670 | Being alive is not intrinsically good, and there is no 'right to life' [Glover] |
4668 | You can't have a right to something you can't desire, so a foetus has no 'right' to life [Glover] |
4649 | If someone's life is 'worth living', that gives one direct reason not to kill him [Glover] |
4651 | Utilitarians object to killing directly (pain, and lost happiness), and to side-effects (loss to others, and precedents) [Glover] |
4671 | What is wrong with killing someone, if another equally worthwhile life is substituted? [Glover] |
4676 | The 'no trade-off' position: killing is only justified if it prevents other deaths [Glover] |
4685 | Societies spend a lot to save known persons, but very little to reduce fatal accidents [Glover] |
4681 | The Nazi mass murders seem to have originated in their euthanasia programme [Glover] |
4682 | Euthanasia is voluntary (patient's wish), or involuntary (ignore wish), or non-voluntary (no wish possible) [Glover] |
4683 | Involuntary euthanasia is wrong because it violates autonomy, and it has appalling side-effects [Glover] |
4684 | Maybe extreme treatment is not saving life, but prolonging the act of dying [Glover] |
4673 | Abortion differs morally from deliberate non-conception only in its side-effects [Glover] |
4652 | If killing is wrong because it destroys future happiness, not conceiving a happy child is also wrong [Glover] |
4665 | Conception isn't the fixed boundary for a person's beginning, because twins are possible within two weeks [Glover] |
4662 | Defenders of abortion focus on early pregnancy, while opponents focus on later stages [Glover] |
4663 | If abortion is wrong, it is because a foetus is a human being or a person (or potentially so) [Glover] |
4664 | If abortion is wrong because of the 'potential' person, that makes contraception wrong too [Glover] |
4666 | If viability is a test or boundary at the beginning of life, it should also be so for frail old people [Glover] |
4672 | Apart from side effects, it seems best to replace an inadequate foetus with one which has a better chance [Glover] |
4674 | It is always right for a qualified person to perform an abortion when requested by the mother [Glover] |
4667 | How would we judge abortion if mothers had transparent wombs? [Glover] |
4679 | One test for a worthwhile life is to assess the amount of life for which you would rather be unconscious [Glover] |