77 ideas
17774 | Definitions make our intuitions mathematically useful [Mayberry] |
17773 | Proof shows that it is true, but also why it must be true [Mayberry] |
9161 | Maybe reasonableness requires circular justifications - that is one coherentist view [Field,H] |
10825 | The notion of truth is to help us make use of the utterances of others [Field,H] |
10820 | In the early 1930s many philosophers thought truth was not scientific [Field,H] |
13499 | Tarski reduced truth to reference or denotation [Field,H, by Hart,WD] |
10818 | Tarski really explained truth in terms of denoting, predicating and satisfied functions [Field,H] |
10817 | Tarski just reduced truth to some other undefined semantic notions [Field,H] |
17795 | Set theory can't be axiomatic, because it is needed to express the very notion of axiomatisation [Mayberry] |
17796 | There is a semi-categorical axiomatisation of set-theory [Mayberry] |
17800 | The misnamed Axiom of Infinity says the natural numbers are finite in size [Mayberry] |
17801 | The set hierarchy doesn't rely on the dubious notion of 'generating' them [Mayberry] |
17803 | Limitation of size is part of the very conception of a set [Mayberry] |
9570 | In Field's Platonist view, set theory is false because it asserts existence for non-existent things [Field,H, by Chihara] |
19369 | Lull's combinatorial art would articulate all the basic concepts, then show how they combine [Lull, by Arthur,R] |
17786 | The mainstream of modern logic sees it as a branch of mathematics [Mayberry] |
17788 | First-order logic only has its main theorems because it is so weak [Mayberry] |
17791 | Only second-order logic can capture mathematical structure up to isomorphism [Mayberry] |
10260 | Logical consequence is defined by the impossibility of P and ¬q [Field,H, by Shapiro] |
17787 | Big logic has one fixed domain, but standard logic has a domain for each interpretation [Mayberry] |
10819 | Tarski gives us the account of truth needed to build a group of true sentences in a model [Field,H] |
10827 | Model theory is unusual in restricting the range of the quantifiers [Field,H] |
17790 | No Löwenheim-Skolem logic can axiomatise real analysis [Mayberry] |
17779 | 'Classificatory' axioms aim at revealing similarity in morphology of structures [Mayberry] |
17778 | Axiomatiation relies on isomorphic structures being essentially the same [Mayberry] |
17780 | 'Eliminatory' axioms get rid of traditional ideal and abstract objects [Mayberry] |
17789 | No logic which can axiomatise arithmetic can be compact or complete [Mayberry] |
9226 | If mathematical theories conflict, it may just be that they have different subject matter [Field,H] |
8958 | In Field's version of science, space-time points replace real numbers [Field,H, by Szabó] |
17784 | Real numbers can be eliminated, by axiom systems for complete ordered fields [Mayberry] |
17782 | Greek quantities were concrete, and ratio and proportion were their science [Mayberry] |
17781 | Real numbers were invented, as objects, to simplify and generalise 'quantity' [Mayberry] |
17799 | Cantor's infinite is an absolute, of all the sets or all the ordinal numbers [Mayberry] |
17797 | Cantor extended the finite (rather than 'taming the infinite') [Mayberry] |
17775 | If proof and definition are central, then mathematics needs and possesses foundations [Mayberry] |
17776 | The ultimate principles and concepts of mathematics are presumed, or grasped directly [Mayberry] |
17777 | Foundations need concepts, definition rules, premises, and proof rules [Mayberry] |
17804 | Axiom theories can't give foundations for mathematics - that's using axioms to explain axioms [Mayberry] |
18221 | 'Metric' axioms uses functions, points and numbers; 'synthetic' axioms give facts about space [Field,H] |
17792 | 1st-order PA is only interesting because of results which use 2nd-order PA [Mayberry] |
17793 | It is only 2nd-order isomorphism which suggested first-order PA completeness [Mayberry] |
17794 | Set theory is not just first-order ZF, because that is inadequate for mathematics [Mayberry] |
17802 | We don't translate mathematics into set theory, because it comes embodied in that way [Mayberry] |
17805 | Set theory is not just another axiomatised part of mathematics [Mayberry] |
8757 | The Indispensability Argument is the only serious ground for the existence of mathematical entities [Field,H] |
18212 | Nominalists try to only refer to physical objects, or language, or mental constructions [Field,H] |
10261 | The application of mathematics only needs its possibility, not its truth [Field,H, by Shapiro] |
18218 | Hilbert explains geometry, by non-numerical facts about space [Field,H] |
9623 | Field needs a semantical notion of second-order consequence, and that needs sets [Brown,JR on Field,H] |
18215 | It seems impossible to explain the idea that the conclusion is contained in the premises [Field,H] |
18216 | Abstractions can form useful counterparts to concrete statements [Field,H] |
18214 | Mathematics is only empirical as regards which theory is useful [Field,H] |
8714 | Fictionalists say 2+2=4 is true in the way that 'Oliver Twist lived in London' is true [Field,H] |
18210 | Why regard standard mathematics as truths, rather than as interesting fictions? [Field,H] |
18211 | You can reduce ontological commitment by expanding the logic [Field,H] |
8959 | Field presumes properties can be eliminated from science [Field,H, by Szabó] |
17785 | Real numbers as abstracted objects are now treated as complete ordered fields [Mayberry] |
18213 | Abstract objects are only applicable to the world if they are impure, and connect to the physical [Field,H] |
9160 | Lots of propositions are default reasonable, but the a priori ones are empirically indefeasible [Field,H] |
9164 | We treat basic rules as if they were indefeasible and a priori, with no interest in counter-evidence [Field,H] |
9165 | Reliability only makes a rule reasonable if we place a value on the truth produced by reliable processes [Field,H] |
9162 | Believing nothing, or only logical truths, is very reliable, but we want a lot more than that [Field,H] |
9166 | People vary in their epistemological standards, and none of them is 'correct' [Field,H] |
9163 | If we only use induction to assess induction, it is empirically indefeasible, and hence a priori [Field,H] |
18222 | Beneath every extrinsic explanation there is an intrinsic explanation [Field,H] |
10826 | 'Valence' and 'gene' had to be reduced to show their compatibility with physicalism [Field,H] |
9917 | 'Abstract' is unclear, but numbers, functions and sets are clearly abstract [Field,H] |
22244 | 'Partial reference' is when the subject thinks two objects are one object [Field,H, by Recanati] |
7615 | Field says reference is a causal physical relation between mental states and objects [Field,H, by Putnam] |
8404 | Explain single events by general rules, or vice versa, or probability explains both, or they are unconnected [Field,H] |
8401 | Physical laws are largely time-symmetric, so they make a poor basis for directional causation [Field,H] |
8400 | Identifying cause and effect is not just conventional; we explain later events by earlier ones [Field,H] |
8402 | The only reason for adding the notion of 'cause' to fundamental physics is directionality [Field,H] |
18223 | In theories of fields, space-time points or regions are causal agents [Field,H] |
18220 | Both philosophy and physics now make substantivalism more attractive [Field,H] |
18219 | Relational space is problematic if you take the idea of a field seriously [Field,H] |
19371 | Nine principles of God: goodness, greatness, eternity, power, wisdom, will, virtue, truth and glory [Lull, by Arthur,R] |