115 ideas
| 3722 | Metaphysics goes beyond the empirical, so doesn't need examples [Kant] |
| 3738 | The hallmark of rationality is setting itself an end [Kant] |
| 8721 | An 'impredicative' definition seems circular, because it uses the term being defined [Friend] |
| 8680 | Classical definitions attempt to refer, but intuitionist/constructivist definitions actually create objects [Friend] |
| 3678 | Reductio ad absurdum proves an idea by showing that its denial produces contradiction [Friend] |
| 8705 | Anti-realists see truth as our servant, and epistemically contrained [Friend] |
| 8713 | In classical/realist logic the connectives are defined by truth-tables [Friend] |
| 8708 | Double negation elimination is not valid in intuitionist logic [Friend] |
| 8694 | Free logic was developed for fictional or non-existent objects [Friend] |
| 8665 | A 'proper subset' of A contains only members of A, but not all of them [Friend] |
| 8672 | A 'powerset' is all the subsets of a set [Friend] |
| 8677 | Set theory makes a minimum ontological claim, that the empty set exists [Friend] |
| 8666 | Infinite sets correspond one-to-one with a subset [Friend] |
| 8682 | Major set theories differ in their axioms, and also over the additional axioms of choice and infinity [Friend] |
| 8709 | The law of excluded middle is syntactic; it just says A or not-A, not whether they are true or false [Friend] |
| 8711 | Intuitionists read the universal quantifier as "we have a procedure for checking every..." [Friend] |
| 8675 | Paradoxes can be solved by talking more loosely of 'classes' instead of 'sets' [Friend] |
| 8674 | The Burali-Forti paradox asks whether the set of all ordinals is itself an ordinal [Friend] |
| 8667 | The 'integers' are the positive and negative natural numbers, plus zero [Friend] |
| 8668 | The 'rational' numbers are those representable as fractions [Friend] |
| 8670 | A number is 'irrational' if it cannot be represented as a fraction [Friend] |
| 8661 | The natural numbers are primitive, and the ordinals are up one level of abstraction [Friend] |
| 8664 | Cardinal numbers answer 'how many?', with the order being irrelevant [Friend] |
| 8671 | The 'real' numbers (rationals and irrationals combined) is the Continuum, which has no gaps [Friend] |
| 8663 | Raising omega to successive powers of omega reveal an infinity of infinities [Friend] |
| 8662 | The first limit ordinal is omega (greater, but without predecessor), and the second is twice-omega [Friend] |
| 8669 | Between any two rational numbers there is an infinite number of rational numbers [Friend] |
| 8676 | Is mathematics based on sets, types, categories, models or topology? [Friend] |
| 8678 | Most mathematical theories can be translated into the language of set theory [Friend] |
| 8701 | The number 8 in isolation from the other numbers is of no interest [Friend] |
| 8702 | In structuralism the number 8 is not quite the same in different structures, only equivalent [Friend] |
| 8699 | Are structures 'ante rem' (before reality), or are they 'in re' (grounded in physics)? [Friend] |
| 8696 | Structuralist says maths concerns concepts about base objects, not base objects themselves [Friend] |
| 8695 | Structuralism focuses on relations, predicates and functions, with objects being inessential [Friend] |
| 8700 | 'In re' structuralism says that the process of abstraction is pattern-spotting [Friend] |
| 8681 | The big problem for platonists is epistemic: how do we perceive, intuit, know or detect mathematical facts? [Friend] |
| 8712 | Mathematics should be treated as true whenever it is indispensable to our best physical theory [Friend] |
| 8716 | Formalism is unconstrained, so cannot indicate importance, or directions for research [Friend] |
| 8706 | Constructivism rejects too much mathematics [Friend] |
| 8707 | Intuitionists typically retain bivalence but reject the law of excluded middle [Friend] |
| 8704 | Structuralists call a mathematical 'object' simply a 'place in a structure' [Friend] |
| 13768 | Validity can preserve certainty in mathematics, but conditionals about contingents are another matter [Edgington] |
| 13770 | There are many different conditional mental states, and different conditional speech acts [Edgington] |
| 13764 | Are conditionals truth-functional - do the truth values of A and B determine the truth value of 'If A, B'? [Edgington] |
| 13765 | 'If A,B' must entail ¬(A & ¬B); otherwise we could have A true, B false, and If A,B true, invalidating modus ponens [Edgington] |
| 3726 | The categorical imperative is a practical synthetic a priori proposition [Kant] |
| 3739 | Free will is a kind of causality which works independently of other causes [Kant] |
| 3741 | We shall never be able to comprehend how freedom is possible [Kant] |
| 3740 | We cannot conceive of reason as being externally controlled [Kant] |
| 5296 | Kant made the political will into a pure self-determined "free" will [Kant, by Marx/Engels] |
| 8685 | Studying biology presumes the laws of chemistry, and it could never contradict them [Friend] |
| 24011 | Kant thought emotions are too random and passive to be part of morality [Kant, by Williams,B] |
| 8688 | Concepts can be presented extensionally (as objects) or intensionally (as a characterization) [Friend] |
| 5074 | Kant united religion and philosophy, by basing obedience to law on reason instead of faith [Taylor,R on Kant] |
| 8024 | The categorical imperative says nothing about what our activities and ends should be [MacIntyre on Kant] |
| 22390 | Kant thought human nature was pure hedonism, so virtue is only possible via the categorical imperative [Foot on Kant] |
| 9750 | We must only value what others find acceptable [Kant, by Korsgaard] |
| 20160 | Kant focuses exclusively on human values, and neglects cultural and personal values [Kekes on Kant] |
| 9749 | Our rational choices confer value, arising from the sense that we ourselves are important [Kant, by Korsgaard] |
| 7671 | Values are created by human choices, and are not some intrinsic quality, out there [Kant, by Berlin] |
| 3720 | We may claim noble motives, but we cannot penetrate our secret impulses [Kant] |
| 3717 | Reverence is awareness of a value which demolishes my self-love [Kant] |
| 3712 | A good will is not good because of what it achieves [Kant] |
| 3725 | The good of an action is in the mind of the doer, not the consequences [Kant] |
| 3733 | The 'golden rule' cannot be a universal law as it implies no duties [Kant] |
| 3736 | Virtue lets a rational being make universal law, and share in the kingdom of ends [Kant] |
| 3544 | Kant thinks virtue becomes passive, and hence morally unaccountable [Kant, by Annas] |
| 7674 | Generosity and pity are vices, because they falsely imply one person's superiority to another [Kant, by Berlin] |
| 21029 | Kantian respect is for humanity and reason (not from love or sympathy or solidarity) [Kant, by Sandel] |
| 7105 | If 'maxims' are deeper underlying intentions, Kant can be read as a virtue theorist [Kant, by Statman] |
| 7625 | We can ask how rational goodness is, but also why is rationality good [Putnam on Kant] |
| 3710 | The only purely good thing is a good will [Kant] |
| 3737 | The will is good if its universalised maxim is never in conflict with itself [Kant] |
| 3715 | Other causes can produce nice results, so morality must consist in the law, found only in rational beings [Kant] |
| 20715 | It is basic that moral actions must be done from duty [Kant] |
| 4024 | Kant follows Rousseau in defining freedom and morality in terms of each other [Taylor,C on Kant] |
| 3735 | Men are subject to laws which are both self-made and universal [Kant] |
| 3718 | Telling the truth from duty is quite different from doing so to avoid inconvenience [Kant] |
| 3723 | There are no imperatives for a holy will, as the will is in harmony with moral law [Kant] |
| 3714 | Dutiful actions are judged not by purpose, but by the maxim followed [Kant] |
| 5295 | Kant was happy with 'good will', even if it had no result [Kant, by Marx/Engels] |
| 6695 | Kant has to attribute high moral worth to some deeply unattractive human lives [Kant, by Graham] |
| 8028 | Kantian duty seems to imply conformism with authority [MacIntyre on Kant] |
| 3724 | A categorical imperative sees an action as necessary purely for its own sake [Kant] |
| 8026 | Almost any precept can be consistently universalized [MacIntyre on Kant] |
| 15673 | The intuition behind the categorical imperative is that one ought not to make an exception of oneself [Kant, by Finlayson] |
| 8068 | Universalising a maxim needs to first stipulate the right description for the action [Anscombe on Kant] |
| 8025 | The categorical imperative will not suggest maxims suitable for testing [MacIntyre on Kant] |
| 8027 | I can universalize a selfish maxim, if it is expressed in a way that only applies to me [MacIntyre on Kant] |
| 3728 | Suicide, false promises, neglected talent, and lack of charity all involve contradictions of principle [Kant, by PG] |
| 22008 | Always treat yourself and others as an end, and never simply as a means [Kant] |
| 3719 | If lying were the universal law it would make promises impossible [Kant] |
| 3762 | Why couldn't all rational beings accept outrageously immoral rules of conduct? [Mill on Kant] |
| 4413 | The categorical imperative smells of cruelty [Nietzsche on Kant] |
| 3716 | Act according to a maxim you can will as a universal law [Kant] |
| 3727 | Act as if your maxim were to become a universal law of nature [Kant] |
| 22009 | Morality is the creation of the laws that enable a Kingdom of Ends [Kant] |
| 22050 | The maxim of an action is chosen, and not externally imposed [Kant, by Bowie] |
| 6694 | Always treat humanity as an end and never as a means only [Kant] |
| 3731 | Rational beings necessarily conceive their own existence as an end in itself [Kant] |
| 4345 | For Kant, even a person who lacks all sympathy for others still has a motive for benevolence [Kant, by Hursthouse] |
| 4251 | If we are required to give moral thought the highest priority, this gives morality no content [Williams,B on Kant] |
| 16004 | If Kant lives by self-administered laws, this is as feeble as self-administered punishments [Kierkegaard on Kant] |
| 3711 | Only a good will makes us worthy of happiness [Kant] |
| 3713 | The function of reason is to produce a good will [Kant] |
| 3729 | Our inclinations are not innately desirable; in fact most rational beings would like to be rid of them [Kant] |
| 4344 | Actions where people spread happiness because they enjoy it have no genuine moral worth [Kant] |
| 3732 | Rational beings have a right to share in the end of an action, not just be part of the means [Kant] |
| 7670 | Kant is the father of the notion of exploitation as an evil [Kant, by Berlin] |
| 7591 | Kant completed Grotius's project of a non-religious basis for natural law [Scruton on Kant] |
| 7673 | Retributive punishment is better than being sent to hospital for your crimes [Kant, by Berlin] |
| 3730 | Non-rational beings only have a relative value, as means rather than as ends [Kant] |
| 3721 | We judge God to be good by a priori standards of moral perfection [Kant] |
| 8046 | We can only know we should obey God if we already have moral standards for judging God [Kant, by MacIntyre] |
| 20714 | God is not proved by reason, but is a postulate of moral thinking [Kant, by Davies,B] |