135 ideas
3722 | Metaphysics goes beyond the empirical, so doesn't need examples [Kant] |
19335 | Reasonings have a natural ordering in God's understanding, but only a temporal order in ours [Leibniz] |
3738 | The hallmark of rationality is setting itself an end [Kant] |
18137 | Impredicative definitions are wrong, because they change the set that is being defined? [Bostock] |
18122 | Classical interdefinitions of logical constants and quantifiers is impossible in intuitionism [Bostock] |
18114 | There is no single agreed structure for set theory [Bostock] |
18107 | A 'proper class' cannot be a member of anything [Bostock] |
18115 | We could add axioms to make sets either as small or as large as possible [Bostock] |
18139 | The Axiom of Choice relies on reference to sets that we are unable to describe [Bostock] |
18105 | Replacement enforces a 'limitation of size' test for the existence of sets [Bostock] |
18108 | First-order logic is not decidable: there is no test of whether any formula is valid [Bostock] |
18109 | The completeness of first-order logic implies its compactness [Bostock] |
18123 | Substitutional quantification is just standard if all objects in the domain have a name [Bostock] |
18120 | The Deduction Theorem is what licenses a system of natural deduction [Bostock] |
18125 | Berry's Paradox considers the meaning of 'The least number not named by this name' [Bostock] |
18101 | Each addition changes the ordinality but not the cardinality, prior to aleph-1 [Bostock] |
18100 | ω + 1 is a new ordinal, but its cardinality is unchanged [Bostock] |
18102 | A cardinal is the earliest ordinal that has that number of predecessors [Bostock] |
18106 | Aleph-1 is the first ordinal that exceeds aleph-0 [Bostock] |
18095 | Instead of by cuts or series convergence, real numbers could be defined by axioms [Bostock] |
18099 | The number of reals is the number of subsets of the natural numbers [Bostock] |
18093 | For Eudoxus cuts in rationals are unique, but not every cut makes a real number [Bostock] |
18110 | Infinitesimals are not actually contradictory, because they can be non-standard real numbers [Bostock] |
18156 | Modern axioms of geometry do not need the real numbers [Bostock] |
18097 | The Peano Axioms describe a unique structure [Bostock] |
18148 | Hume's Principle is a definition with existential claims, and won't explain numbers [Bostock] |
18145 | Many things will satisfy Hume's Principle, so there are many interpretations of it [Bostock] |
18149 | There are many criteria for the identity of numbers [Bostock] |
18143 | Frege makes numbers sets to solve the Caesar problem, but maybe Caesar is a set! [Bostock] |
18116 | Numbers can't be positions, if nothing decides what position a given number has [Bostock] |
18117 | Structuralism falsely assumes relations to other numbers are numbers' only properties [Bostock] |
18141 | Nominalism about mathematics is either reductionist, or fictionalist [Bostock] |
18157 | Nominalism as based on application of numbers is no good, because there are too many applications [Bostock] |
18150 | Actual measurement could never require the precision of the real numbers [Bostock] |
18158 | Ordinals are mainly used adjectively, as in 'the first', 'the second'... [Bostock] |
18127 | Simple type theory has 'levels', but ramified type theory has 'orders' [Bostock] |
18144 | Neo-logicists agree that HP introduces number, but also claim that it suffices for the job [Bostock] |
18147 | Neo-logicists meet the Caesar problem by saying Hume's Principle is unique to number [Bostock] |
18146 | If Hume's Principle is the whole story, that implies structuralism [Bostock] |
18129 | Many crucial logicist definitions are in fact impredicative [Bostock] |
18111 | Treating numbers as objects doesn't seem like logic, since arithmetic fixes their totality [Bostock] |
18159 | Higher cardinalities in sets are just fairy stories [Bostock] |
18155 | A fairy tale may give predictions, but only a true theory can give explanations [Bostock] |
18140 | The best version of conceptualism is predicativism [Bostock] |
18138 | Conceptualism fails to grasp mathematical properties, infinity, and objective truth values [Bostock] |
18131 | If abstracta only exist if they are expressible, there can only be denumerably many of them [Bostock] |
18134 | Predicativism makes theories of huge cardinals impossible [Bostock] |
18135 | If mathematics rests on science, predicativism may be the best approach [Bostock] |
18136 | If we can only think of what we can describe, predicativism may be implied [Bostock] |
18132 | The predicativity restriction makes a difference with the real numbers [Bostock] |
18133 | The usual definitions of identity and of natural numbers are impredicative [Bostock] |
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] |
19367 | Saying we must will whatever we decide to will leads to an infinite regress [Leibniz] |
5296 | Kant made the political will into a pure self-determined "free" will [Kant, by Marx/Engels] |
19351 | Perfections of soul subordinate the body, but imperfections of soul submit to the body [Leibniz] |
24011 | Kant thought emotions are too random and passive to be part of morality [Kant, by Williams,B] |
18121 | In logic a proposition means the same when it is and when it is not asserted [Bostock] |
19331 | Will is an inclination to pursue something good [Leibniz] |
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] |
19346 | Most people facing death would happily re-live a similar life, with just a bit of variety [Leibniz] |
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] |
19340 | Metaphysical evil is imperfection; physical evil is suffering; moral evil is sin [Leibniz] |
3712 | A good will is not good because of what it achieves [Kant] |
19366 | You can't assess moral actions without referring to the qualities of character that produce them [Leibniz] |
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] |
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] |
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] |
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] |
3724 | A categorical imperative sees an action as necessary purely for its own sake [Kant] |
3735 | Men are subject to laws which are both self-made and universal [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] |
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] |
22009 | Morality is the creation of the laws that enable a Kingdom of Ends [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] |
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] |
19326 | God must be intelligible, to select the actual world from the possibilities [Leibniz] |
19327 | The intelligent cause must be unique and all-perfect, to handle all the interconnected possibilities [Leibniz] |
19344 | God prefers men to lions, but might not exterminate lions to save one man [Leibniz] |
19330 | If justice is arbitrary, or fixed but not observed, or not human justice, this undermines God [Leibniz] |
8046 | We can only know we should obey God if we already have moral standards for judging God [Kant, by MacIntyre] |
3721 | We judge God to be good by a priori standards of moral perfection [Kant] |
19325 | God is the first reason of things; our experiences are contingent, and contain no necessity [Leibniz] |
20714 | God is not proved by reason, but is a postulate of moral thinking [Kant, by Davies,B] |
19329 | The laws of physics are wonderful evidence of an intelligent and free being [Leibniz] |
19437 | Prayers are useful, because God foresaw them in his great plan [Leibniz] |
19337 | How can an all-good, wise and powerful being allow evil, sin and apparent injustice? [Leibniz] |
19345 | Being confident of God's goodness, we disregard the apparent local evils in the visible world [Leibniz] |