56 ideas
| 3600 | Slow and accurate thought makes the greatest progress [Descartes] |
| 3601 | Most things in human life seem vain and useless [Descartes] |
| 3602 | Almost every daft idea has been expressed by some philosopher [Descartes] |
| 3603 | Methodical thinking is cautious, analytical, systematic, and panoramic [Descartes, by PG] |
| 3612 | Clear and distinct conceptions are true because a perfect God exists [Descartes] |
| 3610 | Truth is clear and distinct conception - of which it is hard to be sure [Descartes] |
| 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] |
| 3605 | We can believe a thing without knowing we believe it [Descartes] |
| 1583 | In morals Descartes accepts the conventional, but rejects it in epistemology [Roochnik on Descartes] |
| 3607 | In thinking everything else false, my own existence remains totally certain [Descartes] |
| 3617 | I aim to find the principles and causes of everything, using the seeds within my mind [Descartes] |
| 3611 | Understanding, rather than imagination or senses, gives knowledge [Descartes] |
| 3606 | I was searching for reliable rock under the shifting sand [Descartes] |
| 3604 | When rebuilding a house, one needs alternative lodgings [Descartes] |
| 3618 | Only experiments can settle disagreements between rival explanations [Descartes] |
| 3615 | Little reason is needed to speak, so animals have no reason at all [Descartes] |
| 3609 | I am a thinking substance, which doesn't need a place or material support [Descartes] |
| 3608 | I can deny my body and the world, but not my own existence [Descartes] |
| 3613 | Reason is universal in its responses, but a physical machine is constrained by its organs [Descartes] |
| 3616 | The soul must unite with the body to have appetites and sensations [Descartes] |
| 3614 | A machine could speak in response to physical stimulus, but not hold a conversation [Descartes] |
| 1581 | Greeks elevate virtues enormously, but never explain them [Descartes] |
| 7811 | Sophoclean heroes die terrible deaths when they oppose the new Athenian values [Sophocles, by Grayling] |
| 16686 | God has established laws throughout nature, and implanted ideas of them within us [Descartes] |