120 ideas
6779 | Instrumentalists say distinctions between observation and theory vanish with ostensive definition [Bird] |
3646 | There is always a reason why things are thus rather than otherwise [Leibniz] |
2104 | No reason could limit the quantity of matter, so there is no limit [Leibniz] |
2098 | The principle of sufficient reason is needed if we are to proceed from maths to physics [Leibniz] |
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] |
10075 | A 'partial function' maps only some elements to another set [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] |
10076 | The 'range' of a function is the set of elements in the output set created by the function [Smith,P] |
10605 | Two functions are the same if they have the same extension [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] |
10078 | An 'injective' ('one-to-one') function creates a distinct output element from each original [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] |
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] |
10080 | 'Effective' means simple, unintuitive, independent, controlled, dumb, and terminating [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] |
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] |
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] |
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] |
10848 | Multiplication only generates incompleteness if combined with addition and successor [Smith,P] |
10604 | Incompleteness results in arithmetic from combining addition and successor with multiplication [Smith,P] |
19385 | All simply substances are in harmony, because they all represent the one universe [Leibniz] |
6780 | Anti-realism is more plausible about laws than about entities and theories [Bird] |
21346 | The ratio between two lines can't be a feature of one, and cannot be in both [Leibniz] |
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] |
2105 | Things are infinitely subdivisible and contain new worlds, which atoms would make impossible [Leibniz] |
2106 | The only simple things are monads, with no parts or extension [Leibniz] |
2102 | Atomism is irrational because it suggests that two atoms can be indistinguishable [Leibniz] |
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] |
6770 | Membership of a purely random collection cannot be used as an explanation [Bird] |
6771 | Natural kinds may overlap, or be sub-kinds of one another [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] |
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] |
6746 | There may be many laws, each with only a few instances [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] |
20965 | Leibniz upheld conservations of momentum and energy [Leibniz, by Papineau] |
6807 | In Newton mass is conserved, but in Einstein it can convert into energy [Bird] |
2103 | The idea that the universe could be moved forward with no other change is just a fantasy [Leibniz] |
2100 | Space and time are purely relative [Leibniz] |
2107 | No time exists except instants, and instants are not even a part of time, so time does not exist [Leibniz] |
2101 | If everything in the universe happened a year earlier, there would be no discernible difference [Leibniz] |
22894 | If time were absolute that would make God's existence dependent on it [Leibniz, by Bardon] |
2099 | The existence of God, and all metaphysics, follows from the Principle of Sufficient Reason [Leibniz] |