204 ideas
17774 | Definitions make our intuitions mathematically useful [Mayberry] |
17773 | Proof shows that it is true, but also why it must be true [Mayberry] |
17795 | Set theory can't be axiomatic, because it is needed to express the very notion of axiomatisation [Mayberry] |
17796 | There is a semi-categorical axiomatisation of set-theory [Mayberry] |
17800 | The misnamed Axiom of Infinity says the natural numbers are finite in size [Mayberry] |
17801 | The set hierarchy doesn't rely on the dubious notion of 'generating' them [Mayberry] |
17803 | Limitation of size is part of the very conception of a set [Mayberry] |
8625 | What physical facts could underlie 0 or 1, or very large numbers? [Frege on Mill] |
17786 | The mainstream of modern logic sees it as a branch of mathematics [Mayberry] |
17788 | First-order logic only has its main theorems because it is so weak [Mayberry] |
17791 | Only second-order logic can capture mathematical structure up to isomorphism [Mayberry] |
17895 | Combining two distinct assertions does not necessarily lead to a single 'complex proposition' [Mill] |
10427 | All names are names of something, real or imaginary [Mill] |
4944 | Mill says names have denotation but not connotation [Mill, by Kripke] |
7762 | Proper names are just labels for persons or objects, and the meaning is the object [Mill, by Lycan] |
17787 | Big logic has one fixed domain, but standard logic has a domain for each interpretation [Mayberry] |
17790 | No Löwenheim-Skolem logic can axiomatise real analysis [Mayberry] |
17778 | Axiomatiation relies on isomorphic structures being essentially the same [Mayberry] |
17779 | 'Classificatory' axioms aim at revealing similarity in morphology of structures [Mayberry] |
17780 | 'Eliminatory' axioms get rid of traditional ideal and abstract objects [Mayberry] |
17789 | No logic which can axiomatise arithmetic can be compact or complete [Mayberry] |
17784 | Real numbers can be eliminated, by axiom systems for complete ordered fields [Mayberry] |
9801 | Numbers must be assumed to have identical units, as horses are equalised in 'horse-power' [Mill] |
17782 | Greek quantities were concrete, and ratio and proportion were their science [Mayberry] |
17781 | Real numbers were invented, as objects, to simplify and generalise 'quantity' [Mayberry] |
17799 | Cantor's infinite is an absolute, of all the sets or all the ordinal numbers [Mayberry] |
17797 | Cantor extended the finite (rather than 'taming the infinite') [Mayberry] |
17775 | If proof and definition are central, then mathematics needs and possesses foundations [Mayberry] |
17776 | The ultimate principles and concepts of mathematics are presumed, or grasped directly [Mayberry] |
17777 | Foundations need concepts, definition rules, premises, and proof rules [Mayberry] |
17804 | Axiom theories can't give foundations for mathematics - that's using axioms to explain axioms [Mayberry] |
8742 | The only axioms needed are for equality, addition, and successive numbers [Mill, by Shapiro] |
17792 | 1st-order PA is only interesting because of results which use 2nd-order PA [Mayberry] |
17793 | It is only 2nd-order isomorphism which suggested first-order PA completeness [Mayberry] |
9800 | Arithmetic is based on definitions, and Sums of equals are equal, and Differences of equals are equal [Mill] |
17794 | Set theory is not just first-order ZF, because that is inadequate for mathematics [Mayberry] |
17802 | We don't translate mathematics into set theory, because it comes embodied in that way [Mayberry] |
17805 | Set theory is not just another axiomatised part of mathematics [Mayberry] |
5201 | Mill says logic and maths is induction based on a very large number of instances [Mill, by Ayer] |
9360 | If two black and two white objects in practice produced five, what colour is the fifth one? [Lewis,CI on Mill] |
9888 | Mill mistakes particular applications as integral to arithmetic, instead of general patterns [Dummett on Mill] |
9794 | There are no such things as numbers in the abstract [Mill] |
9796 | Things possess the properties of numbers, as quantity, and as countable parts [Mill] |
9795 | Numbers have generalised application to entities (such as bodies or sounds) [Mill] |
9798 | Different parcels made from three pebbles produce different actual sensations [Mill] |
9797 | '2 pebbles and 1 pebble' and '3 pebbles' name the same aggregation, but different facts [Mill] |
9799 | 3=2+1 presupposes collections of objects ('Threes'), which may be divided thus [Mill] |
9802 | Numbers denote physical properties of physical phenomena [Mill] |
9803 | We can't easily distinguish 102 horses from 103, but we could arrange them to make it obvious [Mill] |
9804 | Arithmetical results give a mode of formation of a given number [Mill] |
9805 | 12 is the cube of 1728 means pebbles can be aggregated a certain way [Mill] |
8741 | Numbers must be of something; they don't exist as abstractions [Mill] |
12411 | Mill is too imprecise, and is restricted to simple arithmetic [Kitcher on Mill] |
5656 | Empirical theories of arithmetic ignore zero, limit our maths, and need probability to get started [Frege on Mill] |
9624 | Numbers are a very general property of objects [Mill, by Brown,JR] |
22355 | In the realist view, the real external world explains how it (and perceptions of it) are possible [Williams,B] |
17785 | Real numbers as abstracted objects are now treated as complete ordered fields [Mayberry] |
9806 | Whatever is made up of parts is made up of parts of those parts [Mill] |
11156 | The essence is that without which a thing can neither be, nor be conceived to be [Mill] |
12190 | Necessity is what will be, despite any alternative suppositions whatever [Mill] |
22623 | Necessity can only mean what must be, without conditions of any kind [Mill] |
23283 | Necessity implies possibility, but in experience it matters which comes first [Williams,B] |
3583 | External objects are permanent possibilities of sensation [Mill] |
16859 | Most perception is one-tenth observation and nine-tenths inference [Mill] |
9082 | Clear concepts result from good observation, extensive experience, and accurate memory [Mill] |
4243 | Our ability to react to an alien culture shows that ethical thought extends beyond cultural boundaries [Williams,B] |
4244 | It is very confused to deduce a nonrelativist morality of universal toleration from relativism [Williams,B] |
16860 | Inductive generalisation is more reliable than one of its instances; they can't all be wrong [Mill] |
16843 | Mill's methods (Difference,Agreement,Residues,Concomitance,Hypothesis) don't nail induction [Mill, by Lipton] |
16845 | The whole theory of induction rests on causes [Mill] |
17086 | Surprisingly, empiricists before Mill ignore explanation, which seems to transcend experience [Mill, by Ruben] |
17091 | Explanation is fitting of facts into ever more general patterns of regularity [Mill, by Ruben] |
16805 | Causal inference is by spotting either Agreements or Differences [Mill, by Lipton] |
16835 | The Methods of Difference and of Agreement are forms of inference to the best explanation [Mill, by Lipton] |
3537 | I judge others' feeling by analogy with my body and behaviour [Mill] |
9079 | We can focus our minds on what is common to a whole class, neglecting other aspects [Mill] |
9081 | We don't recognise comparisons by something in our minds; the concepts result from the comparisons [Mill] |
3238 | 'Dead person' isn't a contradiction, so 'person' is somewhat vague [Williams,B] |
3239 | You can only really love a person as a token, not as a type [Williams,B] |
7946 | The memory criterion has a problem when one thing branches into two things [Williams,B, by Macdonald,C] |
2176 | There is only a problem of free will if you think the notion of 'voluntary' can be metaphysically deepened [Williams,B] |
2181 | It is an absurd Kantian idea that at the limit rationality and freedom coincide [Williams,B] |
24008 | Reference to a person's emotions is often essential to understanding their actions [Williams,B] |
24009 | Moral education must involve learning about various types of feeling towards things [Williams,B] |
9078 | The study of the nature of Abstract Ideas does not belong to logic, but to a different science [Mill] |
9080 | General conceptions are a necessary preliminary to Induction [Mill] |
3772 | The will, in the beginning, is entirely produced by desire [Mill] |
4317 | We judge weakness of will by an assessment after the event is concluded [Williams,B, by Cottingham] |
9284 | Reasons are 'internal' if they give a person a motive to act, but 'external' otherwise [Williams,B] |
2174 | Responsibility involves cause, intention, state of mind, and response after the event [Williams,B] |
22455 | Many ethical theories neglect the power of regretting the ought not acted upon [Williams,B] |
4114 | Philosophers try to produce ethical theories because they falsely assume that ethics can be simple [Williams,B] |
22453 | Moral conflicts have a different feeling and structure from belief conflicts [Williams,B, by Foot] |
22454 | We tolerate inconsistency in ethics but not in other beliefs (which reflect an independent order) [Williams,B, by Foot] |
22450 | If moral systems can't judge other moral systems, then moral relativism is true [Williams,B, by Foot] |
7222 | It is a crime for someone with a violent disposition to get drunk [Mill] |
2178 | In bad actions, guilt points towards victims, and shame to the agent [Williams,B] |
20168 | Blame usually has no effect if the recipient thinks it unjustified [Williams,B] |
20167 | Blame partly rests on the fiction that blamed agents always know their obligations [Williams,B] |
3769 | With early training, any absurdity or evil may be given the power of conscience [Mill] |
4128 | Intuitionism has been demolished by critics, and no longer looks interesting [Williams,B] |
4366 | We can't accept Aristotle's naturalism about persons, because it is normative and unscientific [Williams,B, by Hursthouse] |
4132 | The category of person is a weak basis for ethics, because it is not fixed but comes in degrees [Williams,B] |
24007 | Emotivism saw morality as expressing emotions, and influencing others' emotions [Williams,B] |
4134 | The weakness of prescriptivism is shown by "I simply don't like staying at good hotels" [Williams,B] |
4135 | Some ethical ideas, such as 'treachery' and 'promise', seem to express a union of facts and values [Williams,B] |
22410 | Maybe the unthinkable is a moral category, and considering some options is dishonourable or absurd [Williams,B] |
3767 | Motive shows the worth of the agent, but not of the action [Mill] |
22408 | Consequentialism assumes that situations can be compared [Williams,B] |
22411 | For a consequentialist massacring 7 million must be better than massacring 7 million and one [Williams,B] |
4120 | It is an error of consequentialism to think we just aim at certain states of affairs; we also want to act [Williams,B] |
23282 | If all that matters in morality is motive and intention, that makes moral luck irrelevant [Williams,B] |
7076 | Mill wondered if he would be happy if all his aims were realised, and answered no [Mill, by Critchley] |
4252 | Promise keeping increases reliability, by making deliberation focus on something which would be overlooked [Williams,B] |
4116 | A weakness of contractual theories is the position of a person of superior ability and power [Williams,B] |
2169 | Greek moral progress came when 'virtue' was freed from social status [Williams,B] |
3771 | Virtues only have value because they achieve some further end [Mill] |
4112 | A crucial feature of moral thought is second-order desire - the desire to have certain desires [Williams,B] |
24010 | An admirable human being should have certain kinds of emotional responses [Williams,B] |
23279 | It is important that a person can change their character, and not just be successive 'selves' [Williams,B] |
23280 | Kantians have an poor account of individuals, and insist on impartiality, because they ignore character [Williams,B] |
3236 | Equality of opportunity without equality of respect would create a very inhuman society [Williams,B] |
4113 | 'Deon' in Greek means what one must do; there was no word meaning 'duty' [Williams,B] |
4110 | Obligation and duty look backwards (because of a promise or job), although the acts are in the future [Williams,B] |
4249 | "Ought implies can" is a famous formula in connection with moral obligation [Williams,B] |
4248 | Not all moral deliberations lead to obligations; some merely reveal what 'may' be done [Williams,B] |
2172 | The modern idea of duty is unknown in archaic Greece [Williams,B] |
3768 | Orthodox morality is the only one which feels obligatory [Mill] |
4250 | The concept of a 'duty to myself' is fraudulent [Williams,B] |
4121 | Why should I think of myself as both the legislator and the citizen who follows the laws? [Williams,B] |
22409 | We don't have a duty to ensure that others do their duty [Williams,B] |
4122 | If the self becomes completely impartial, it no longer has enough identity to worry about its interests [Williams,B] |
2179 | If the moral self is seen as characterless, then other people have a very limited role in our moral lives [Williams,B] |
2180 | If reason cannot lead people to good, we must hope they have an internal voice [Williams,B] |
24012 | Kant's love of consistency is too rigid, and it even overrides normal fairness [Williams,B] |
7214 | Ethics rests on utility, which is the permanent progressive interests of people [Mill] |
7202 | The English believe in the task of annihilating evil for the victory of good [Nietzsche on Mill] |
5935 | Mill's qualities of pleasure is an admission that there are other good states of mind than pleasure [Ross on Mill] |
3764 | Actions are right if they promote pleasure, wrong if they promote pain [Mill] |
23278 | For utilitarians states of affairs are what have value, not matter who produced them [Williams,B] |
22407 | Utilitarianism cannot make any serious sense of integrity [Williams,B] |
3776 | Utilitarianism only works if everybody has a totally equal right to happiness [Mill] |
3766 | Better to be Socrates dissatisfied than a fool satisfied [Mill] |
3765 | Only pleasure and freedom from pain are desirable as ends [Mill] |
3763 | Ultimate goods such as pleasure can never be proved to be good [Mill] |
3770 | General happiness is only desirable because individuals desire their own happiness [Mill] |
4124 | Utilitarian benevolence involves no particular attachments, and is immune to the inverse square law [Williams,B] |
6697 | Moral rules protecting human welfare are more vital than local maxims [Mill] |
4245 | Ethical conviction must be to some extent passive, and can't just depend on the will and decisions [Williams,B] |
4246 | Taking responsibility won't cure ethical uncertainty by; we are uncertain what to decide [Williams,B] |
7212 | Individuals have sovereignty over their own bodies and minds [Mill] |
3774 | Rights are a matter of justice, not of benevolence [Mill] |
3773 | No individual has the right to receive our benevolence [Mill] |
7210 | The will of the people is that of the largest or most active part of the people [Mill] |
7227 | It is evil to give a government any more power than is necessary [Mill] |
7228 | Individuals often do things better than governments [Mill] |
7230 | Aim for the maximum dissemination of power consistent with efficiency [Mill] |
20515 | Maximise happiness by an area of strict privacy, and an area of utilitarian interventions [Mill, by Wolff,J] |
7229 | People who transact their own business will also have the initiative to control their government [Mill] |
20508 | How people vote should be on public record, so they can be held accountable [Mill, by Wolff,J] |
20507 | Voting is a strict duty, like jury service, and must only be aimed at the public good [Mill] |
20505 | Direct democracy is inexperience judging experience, and ignorance judging knowledge [Mill] |
20504 | People can only participate in decisions in small communities, so representatives are needed [Mill] |
7211 | Prevention of harm to others is the only justification for exercising power over people [Mill] |
7231 | The worth of a State, in the long run, is the worth of the individuals composing it [Mill] |
7217 | The main argument for freedom is that interference with it is usually misguided [Mill] |
7213 | Liberty arises at the point where people can freely and equally discuss things [Mill] |
20516 | Mill defends freedom as increasing happiness, but maybe it is an intrinsic good [Wolff,J on Mill] |
20517 | Utilitarianism values liberty, but guides us on which ones we should have or not have [Mill, by Wolff,J] |
7215 | True freedom is pursuing our own good, while not impeding others [Mill] |
7220 | Restraint for its own sake is an evil [Mill] |
7218 | Individuals are not accountable for actions which only concern themselves [Mill] |
7221 | Blocking entry to an unsafe bridge does not infringe liberty, since no one wants unsafe bridges [Mill] |
7223 | Pimping and running a gambling-house are on the border between toleration and restraint [Mill] |
3233 | Equality implies that people are alike in potential as well as in needs [Williams,B] |
3234 | Equality seems to require that each person be acknowledged as having a significant point of view [Williams,B] |
3235 | It is a mark of extreme exploitation that the sufferers do not realise their plight [Williams,B] |
3775 | A right is a valid claim to society's protection [Mill] |
7219 | Society can punish actions which it believes to be prejudicial to others [Mill] |
7226 | Benefits performed by individuals, not by government, help also to educate them [Mill] |
7224 | We need individual opinions and conduct, and State education is a means to prevent that [Mill] |
4247 | It is a mark of our having ethical values that we aim to reproduce them in our children [Williams,B] |
4131 | Most women see an early miscarriage and a late stillbirth as being very different in character [Williams,B] |
7225 | It is a crime to create a being who lacks the ordinary chances of a desirable existence [Mill] |
4133 | Speciesism isn't like racism, because the former implies a viewpoint which belongs to no one [Williams,B] |
8345 | A cause is the total of all the conditions which inevitably produce the result [Mill] |
10391 | Causes and conditions are not distinct, because we select capriciously from among them [Mill] |
14547 | The strict cause is the total positive and negative conditions which ensure the consequent [Mill] |
8377 | Causation is just invariability of succession between every natural fact and a preceding fact [Mill] |
14545 | A cause is an antecedent which invariably and unconditionally leads to a phenomenon [Mill] |
4773 | Mill's regularity theory of causation is based on an effect preceded by a conjunction of causes [Mill, by Psillos] |
4775 | In Mill's 'Method of Agreement' cause is the common factor in a range of different cases [Mill, by Psillos] |
4776 | In Mill's 'Method of Difference' the cause is what stops the effect when it is removed [Mill, by Psillos] |
9417 | What are the fewest propositions from which all natural uniformities could be inferred? [Mill] |
21332 | We don't get a love of 'order' from nature - which is thoroughly chaotic [Mill] |
7216 | The ethics of the Gospel has been supplemented by barbarous Old Testament values [Mill] |
21333 | Evil comes from good just as often as good comes from evil [Mill] |
21335 | Belief that an afterlife is required for justice is an admission that this life is very unjust [Mill] |
21334 | No necessity ties an omnipotent Creator, so he evidently wills human misery [Mill] |
2175 | There is a problem of evil only if you expect the world to be good [Williams,B] |
21329 | Nature dispenses cruelty with no concern for either mercy or justice [Mill] |
21328 | Killing is a human crime, but nature kills everyone, and often with great tortures [Mill] |
21330 | Nature makes childbirth a miserable experience, often leading to the death of the mother [Mill] |
21331 | Hurricanes, locusts, floods and blight can starve a million people to death [Mill] |