92 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] |
| 10073 | There cannot be a set theory which is complete [Smith,P] |
| 10616 | Second-order arithmetic can prove new sentences of first-order [Smith,P] |
| 10074 | A 'total function' maps every element to one element in another set [Smith,P] |
| 10612 | An argument is a 'fixed point' for a function if it is mapped back to itself [Smith,P] |
| 10605 | Two functions are the same if they have the same extension [Smith,P] |
| 10075 | A 'partial function' maps only some elements to another set [Smith,P] |
| 10076 | The 'range' of a function is the set of elements in the output set created by the function [Smith,P] |
| 10615 | The Comprehension Schema says there is a property only had by things satisfying a condition [Smith,P] |
| 10595 | A 'theorem' of a theory is a sentence derived from the axioms using the proof system [Smith,P] |
| 10602 | A 'natural deduction system' has no axioms but many rules [Smith,P] |
| 10613 | No nice theory can define truth for its own language [Smith,P] |
| 10079 | A 'bijective' function has one-to-one correspondence in both directions [Smith,P] |
| 10077 | A 'surjective' ('onto') function creates every element of the output set [Smith,P] |
| 10078 | An 'injective' ('one-to-one') function creates a distinct output element from each original [Smith,P] |
| 10070 | If everything that a theory proves is true, then it is 'sound' [Smith,P] |
| 10086 | Soundness is true axioms and a truth-preserving proof system [Smith,P] |
| 10596 | A theory is 'sound' iff every theorem is true (usually from true axioms and truth-preservation) [Smith,P] |
| 10598 | A theory is 'negation complete' if it proves all sentences or their negation [Smith,P] |
| 10597 | 'Complete' applies both to whole logics, and to theories within them [Smith,P] |
| 10069 | A theory is 'negation complete' if one of its sentences or its negation can always be proved [Smith,P] |
| 10609 | Two routes to Incompleteness: semantics of sound/expressible, or syntax of consistency/proof [Smith,P] |
| 10087 | A theory is 'decidable' if all of its sentences could be mechanically proved [Smith,P] |
| 10088 | Any consistent, axiomatized, negation-complete formal theory is decidable [Smith,P] |
| 10080 | 'Effective' means simple, unintuitive, independent, controlled, dumb, and terminating [Smith,P] |
| 10081 | A set is 'enumerable' is all of its elements can result from a natural number function [Smith,P] |
| 10083 | A set is 'effectively enumerable' if a computer could eventually list every member [Smith,P] |
| 10084 | A finite set of finitely specifiable objects is always effectively enumerable (e.g. primes) [Smith,P] |
| 10085 | The set of ordered pairs of natural numbers <i,j> is effectively enumerable [Smith,P] |
| 10601 | The thorems of a nice arithmetic can be enumerated, but not the truths (so they're diffferent) [Smith,P] |
| 10600 | Being 'expressible' depends on language; being 'capture/represented' depends on axioms and proof system [Smith,P] |
| 10599 | For primes we write (x not= 1 ∧ ∀u∀v(u x v = x → (u = 1 ∨ v = 1))) [Smith,P] |
| 10610 | The reals contain the naturals, but the theory of reals doesn't contain the theory of naturals [Smith,P] |
| 10619 | The truths of arithmetic are just true equations and their universally quantified versions [Smith,P] |
| 10618 | All numbers are related to zero by the ancestral of the successor relation [Smith,P] |
| 10608 | The number of Fs is the 'successor' of the Gs if there is a single F that isn't G [Smith,P] |
| 10849 | Baby arithmetic covers addition and multiplication, but no general facts about numbers [Smith,P] |
| 10850 | Baby Arithmetic is complete, but not very expressive [Smith,P] |
| 10851 | Robinson Arithmetic 'Q' has basic axioms, quantifiers and first-order logic [Smith,P] |
| 10852 | Robinson Arithmetic (Q) is not negation complete [Smith,P] |
| 10068 | Natural numbers have zero, unique successors, unending, no circling back, and no strays [Smith,P] |
| 10603 | The logic of arithmetic must quantify over properties of numbers to handle induction [Smith,P] |
| 10604 | Incompleteness results in arithmetic from combining addition and successor with multiplication [Smith,P] |
| 10848 | Multiplication only generates incompleteness if combined with addition and successor [Smith,P] |
| 10617 | The 'ancestral' of a relation is a new relation which creates a long chain of the original relation [Smith,P] |
| 2730 | Because 'gold is malleable' is necessary does not mean that it is analytic [Audi,R] |
| 3605 | We can believe a thing without knowing we believe it [Descartes] |
| 2715 | Beliefs are based on perception, memory, introspection or reason [Audi,R] |
| 2735 | Could you have a single belief on its own? [Audi,R] |
| 1583 | In morals Descartes accepts the conventional, but rejects it in epistemology [Roochnik on Descartes] |
| 2736 | We can make certain of what we know, so knowing does not entail certainty [Audi,R] |
| 3607 | In thinking everything else false, my own existence remains totally certain [Descartes] |
| 2722 | Sense-data theory is indirect realism, but phenomenalism is direct irrealism [Audi,R] |
| 2721 | If you gradually remove a book's sensory properties, what is left at the end? [Audi,R] |
| 3617 | I aim to find the principles and causes of everything, using the seeds within my mind [Descartes] |
| 2727 | Red and green being exclusive colours seems to be rationally graspable but not analytic [Audi,R] |
| 2728 | The concepts needed for a priori thought may come from experience [Audi,R] |
| 2716 | To see something as a field, I obviously need the concept of a field [Audi,R] |
| 2717 | How could I see a field and believe nothing regarding it? [Audi,R] |
| 2720 | Sense-data (and the rival 'adverbial' theory) are to explain illusions and hallucinations [Audi,R] |
| 2719 | Sense data imply representative realism, possibly only representing primary qualities [Audi,R] |
| 2718 | Perception is first simple, then objectual (with concepts) and then propositional [Audi,R] |
| 2741 | The principles of justification have to be a priori [Audi,R] |
| 2729 | Virtually all rationalists assert that we can have knowledge of synthetic a priori truths [Audi,R] |
| 3611 | Understanding, rather than imagination or senses, gives knowledge [Descartes] |
| 2725 | To remember something is to know it [Audi,R] |
| 2724 | I might remember someone I can't recall or image, by recognising them on meeting [Audi,R] |
| 2731 | Justification is either unanchored (infinite or circular), or anchored (in knowledge or non-knowledge) [Audi,R] |
| 2739 | Internalism about justification implies that there is a right to believe something [Audi,R] |
| 3606 | I was searching for reliable rock under the shifting sand [Descartes] |
| 2732 | Maths may be consistent with observations, but not coherent [Audi,R] |
| 2733 | It is very hard to show how much coherence is needed for justification [Audi,R] |
| 2734 | A consistent madman could have a very coherent belief system [Audi,R] |
| 2738 | Consistent accurate prediction looks like knowledge without justified belief [Audi,R] |
| 2740 | A reliability theory of knowledge seems to involve truth as correspondence [Audi,R] |
| 2737 | 'Reliable' is a very imprecise term, and may even mean 'justified' [Audi,R] |
| 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] |
| 2726 | We can be ignorant about ourselves, for example, our desires and motives [Audi,R] |
| 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] |
| 16686 | God has established laws throughout nature, and implanted ideas of them within us [Descartes] |