116 ideas
| 6779 | Instrumentalists say distinctions between observation and theory vanish with ostensive definition [Bird] |
| 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] |
| 17453 | The meaning of a number isn't just the numerals leading up to it [Heck] |
| 17457 | A basic grasp of cardinal numbers needs an understanding of equinumerosity [Heck] |
| 10610 | The reals contain the naturals, but the theory of reals doesn't contain the theory of naturals [Smith,P] |
| 17448 | In counting, numerals are used, not mentioned (as objects that have to correlated) [Heck] |
| 17456 | Counting is the assignment of successively larger cardinal numbers to collections [Heck] |
| 17455 | Is counting basically mindless, and independent of the cardinality involved? [Heck] |
| 17450 | Understanding 'just as many' needn't involve grasping one-one correspondence [Heck] |
| 17451 | We can know 'just as many' without the concepts of equinumerosity or numbers [Heck] |
| 10619 | The truths of arithmetic are just true equations and their universally quantified versions [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] |
| 10618 | All numbers are related to zero by the ancestral of the successor relation [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] |
| 10852 | Robinson Arithmetic (Q) is not negation complete [Smith,P] |
| 10851 | Robinson Arithmetic 'Q' has basic axioms, quantifiers and first-order logic [Smith,P] |
| 10068 | Natural numbers have zero, unique successors, unending, no circling back, and no strays [Smith,P] |
| 17459 | Frege's Theorem explains why the numbers satisfy the Peano axioms [Heck] |
| 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] |
| 17454 | Children can use numbers, without a concept of them as countable objects [Heck] |
| 17458 | Equinumerosity is not the same concept as one-one correspondence [Heck] |
| 17449 | We can understand cardinality without the idea of one-one correspondence [Heck] |
| 6780 | Anti-realism is more plausible about laws than about entities and theories [Bird] |
| 10617 | The 'ancestral' of a relation is a new relation which creates a long chain of the original relation [Smith,P] |
| 6796 | Subjective probability measures personal beliefs; objective probability measures the chance of an event happening [Bird] |
| 6797 | Objective probability of tails measures the bias of the coin, not our beliefs about it [Bird] |
| 6800 | Many philosophers rate justification as a more important concept than knowledge [Bird] |
| 6786 | As science investigates more phenomena, the theories it needs decreases [Bird] |
| 6792 | If theories need observation, and observations need theories, how do we start? [Bird] |
| 6757 | Explanation predicts after the event; prediction explains before the event [Bird] |
| 6805 | Relativity ousted Newtonian mechanics despite a loss of simplicity [Bird] |
| 6777 | Realists say their theories involve truth and the existence of their phenomena [Bird] |
| 6804 | There is no agreement on scientific method - because there is no such thing [Bird] |
| 6778 | Instrumentalists regard theories as tools for prediction, with truth being irrelevant [Bird] |
| 6775 | Induction is inference to the best explanation, where the explanation is a law [Bird] |
| 6791 | If Hume is right about induction, there is no scientific knowledge [Bird] |
| 6790 | Anything justifying inferences from observed to unobserved must itself do that [Bird] |
| 6738 | Any conclusion can be drawn from an induction, if we use grue-like predicates [Bird] |
| 6739 | Several months of observing beech trees supports the deciduous and evergreen hypotheses [Bird] |
| 6799 | We normally learn natural kinds from laws, but Goodman shows laws require prior natural kinds [Bird] |
| 6798 | Bayesianism claims to find rationality and truth in induction, and show how science works [Bird] |
| 6752 | The objective component of explanations is the things that must exist for the explanation [Bird] |
| 6754 | We talk both of 'people' explaining things, and of 'facts' explaining things [Bird] |
| 6750 | Explanations are causal, nomic, psychological, psychoanalytic, Darwinian or functional [Bird] |
| 6761 | Contrastive explanations say why one thing happened but not another [Bird] |
| 6758 | 'Covering law' explanations only work if no other explanations are to be found [Bird] |
| 6759 | Livers always accompany hearts, but they don't explain hearts [Bird] |
| 6756 | Probabilistic-statistical explanations don't entail the explanandum, but makes it more likely [Bird] |
| 6760 | An operation might reduce the probability of death, yet explain a death [Bird] |
| 6785 | Inference to the Best Explanation is done with facts, so it has to be realist [Bird] |
| 6788 | Maybe bad explanations are the true ones, in this messy world [Bird] |
| 6787 | Which explanation is 'best' is bound to be subjective, and no guide to truth [Bird] |
| 6751 | Maybe explanation is so subjective that it cannot be a part of science [Bird] |
| 6776 | Natural kinds are those that we use in induction [Bird] |
| 6767 | Rubies and sapphires are both corundum, with traces of metals varying their colours [Bird] |
| 6768 | Tin is not one natural kind, but appears to be 21, depending on isotope [Bird] |
| 6771 | Natural kinds may overlap, or be sub-kinds of one another [Bird] |
| 6770 | Membership of a purely random collection cannot be used as an explanation [Bird] |
| 6773 | If F is a universal appearing in a natural law, then Fs form a natural kind [Bird] |
| 6769 | In the Kripke-Putnam view only nuclear physicists can know natural kinds [Bird] |
| 6774 | Darwinism suggests that we should have a native ability to detect natural kinds [Bird] |
| 6764 | Nominal essence of a natural kind is the features that make it fit its name [Bird] |
| 6766 | Jadeite and nephrite are superficially identical, but have different composition [Bird] |
| 6808 | Reference to scientific terms is by explanatory role, not by descriptions [Bird] |
| 6753 | Laws are more fundamental in science than causes, and laws will explain causes [Bird] |
| 6762 | Newton's laws cannot be confirmed individually, but only in combinations [Bird] |
| 6763 | Parapsychology is mere speculation, because it offers no mechanisms for its working [Bird] |
| 6772 | Existence requires laws, as inertia or gravity are needed for mass or matter [Bird] |
| 6746 | There may be many laws, each with only a few instances [Bird] |
| 6740 | 'All uranium lumps are small' is a law, but 'all gold lumps are small' is not [Bird] |
| 6741 | There can be remarkable uniformities in nature that are purely coincidental [Bird] |
| 6742 | A law might have no instances, if it was about things that only exist momentarily [Bird] |
| 6743 | If laws are just instances, the law should either have gaps, or join the instances arbitrarily [Bird] |
| 6744 | Where is the regularity in a law predicting nuclear decay? [Bird] |
| 6747 | Laws cannot explain instances if they are regularities, as something can't explain itself [Bird] |
| 6748 | Similar appearance of siblings is a regularity, but shared parents is what links them [Bird] |
| 6749 | We can only infer a true regularity if something binds the instances together [Bird] |
| 6803 | If we only infer laws from regularities among observations, we can't infer unobservable entities. [Bird] |
| 6801 | Accidental regularities are not laws, and an apparent regularity may not be actual [Bird] |
| 6745 | A regularity is only a law if it is part of a complete system which is simple and strong [Bird] |
| 6802 | With strange enough predicates, anything could be made out to be a regularity [Bird] |
| 6789 | If flame colour is characteristic of a metal, that is an empirical claim needing justification [Bird] |
| 6807 | In Newton mass is conserved, but in Einstein it can convert into energy [Bird] |