81 ideas
16000 | Fixed ideas should be tackled aggressively [Kierkegaard] |
7578 | I conceived it my task to create difficulties everywhere [Kierkegaard] |
22087 | Philosophy fails to articulate the continual becoming of existence [Kierkegaard, by Carlisle] |
22047 | Wherever there is painless contradiction there is also comedy [Kierkegaard] |
16012 | Philosophy can't be unbiased if it ignores language, as that is no more independent than individuals are [Kierkegaard] |
22092 | Kierkegaard's truth draws on authenticity, fidelity and honesty [Kierkegaard, by Carlisle] |
15999 | Pure truth is for infinite beings only; I prefer endless striving for truth [Kierkegaard] |
22094 | Subjective truth can only be sustained by repetition [Kierkegaard, by Carlisle] |
16005 | I recognise knowledge, but it is the truth by which I can live and die that really matters [Kierkegaard] |
5651 | Traditional views of truth are tautologies, and truth is empty without a subject [Kierkegaard, by Scruton] |
20313 | The highest truth we can get is uncertainty held fast by an inward passion [Kierkegaard] |
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] |
15920 | Pure collections of things obey Choice, but collections defined by a rule may not [Lavine] |
15898 | The controversy was not about the Axiom of Choice, but about functions as arbitrary, or given by rules [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] |
15949 | The theory of infinity must rest on our inability to distinguish between very large sizes [Lavine] |
15947 | The infinite is extrapolation from the experience of indefinitely large size [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] |
16007 | I assume existence, rather than reasoning towards it [Kierkegaard] |
16013 | Nothing necessary can come into existence, since it already 'is' [Kierkegaard] |
3061 | Anaxarchus said that he was not even sure that he knew nothing [Anaxarchus, by Diog. Laertius] |
20742 | The real subject is ethical, not cognitive [Kierkegaard] |
16002 | The self is a combination of pairs of attributes: freedom/necessity, infinite/finite, temporal/eternal [Kierkegaard] |
22098 | Socrates neglects the gap between knowing what is good and doing good [Kierkegaard, by Carlisle] |
22086 | The most important aspect of a human being is not reason, but passion [Kierkegaard, by Carlisle] |
15998 | Perfect love is not in spite of imperfections; the imperfections must be loved as well [Kierkegaard] |
16003 | If people marry just because they are lonely, that is self-love, not love [Kierkegaard] |
7579 | While big metaphysics is complete without ethics, personal philosophy emphasises ethics [Kierkegaard] |
7581 | Speculative philosophy loses the individual in a vast vision of humanity [Kierkegaard] |
22090 | For me time stands still, and I with it [Kierkegaard, by Carlisle] |
22096 | Anxiety is not a passing mood, but a response to human freedom [Kierkegaard, by Carlisle] |
22097 | The ultimate in life is learning to be anxious in the right way [Kierkegaard] |
21909 | Ultimate knowledge is being anxious in the right way [Kierkegaard] |
20758 | Anxiety is staring into the yawning abyss of freedom [Kierkegaard] |
21910 | Our destiny is the highest pitch of world-weariness [Kierkegaard] |
9305 | The plebeians bore others; only the nobility bore themselves [Kierkegaard] |
5650 | Reason is just abstractions, so our essence needs a subjective 'leap of faith' [Kierkegaard, by Scruton] |
22095 | There are aesthetic, ethical and religious subjectivity [Kierkegaard, by Carlisle] |
20314 | People want to lose themselves in movements and history, instead of being individuals [Kierkegaard] |
7582 | Becoming what one is is a huge difficulty, because we strongly aspire to be something else [Kierkegaard] |
20747 | What matters is not right choice, but energy, earnestness and pathos in the choosing [Kierkegaard] |
16001 | Life may be understood backwards, but it has to be lived forwards [Kierkegaard] |
22093 | Life is a repetition when what has been now becomes [Kierkegaard] |
16009 | When we seek our own 'freedom' we are just trying to avoid responsibility [Kierkegaard] |
22091 | Kierkegaard prioritises the inward individual, rather than community [Kierkegaard, by Carlisle] |
7586 | God does not think or exist; God creates, and is eternal [Kierkegaard] |
16006 | Either Abraham rises higher than universal ethics, or he is a mere murderer [Kierkegaard] |
7577 | Abraham was willing to suspend ethics, for a higher idea [Kierkegaard] |
20312 | God cannot be demonstrated objectively, because God is a subject, only existing inwardly [Kierkegaard] |
7580 | Pantheism destroys the distinction between good and evil [Kierkegaard] |
16008 | The best way to be a Christian is without 'Christianity' [Kierkegaard] |
20735 | We need to see that Christianity cannot be understood [Kierkegaard] |
22088 | Faith is like a dancer's leap, going up to God, but also back to earth [Kierkegaard, by Carlisle] |
7583 | Faith is the highest passion in the sphere of human subjectivity [Kierkegaard] |
7584 | Without risk there is no faith [Kierkegaard] |