55 ideas
| 2352 | The job of the philosopher is to distinguish facts about the world from conventions [Putnam] |
| 17774 | Definitions make our intuitions mathematically useful [Mayberry] |
| 17773 | Proof shows that it is true, but also why it must be true [Mayberry] |
| 2345 | Semantic notions do not occur in Tarski's definitions, but assessing their correctness involves translation [Putnam] |
| 2347 | Asserting the truth of an indexical statement is not the same as uttering the statement [Putnam] |
| 17796 | There is a semi-categorical axiomatisation of set-theory [Mayberry] |
| 17795 | Set theory can't be axiomatic, because it is needed to express the very notion of axiomatisation [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] |
| 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] |
| 17787 | Big logic has one fixed domain, but standard logic has a domain for each interpretation [Mayberry] |
| 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] |
| 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] |
| 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] |
| 2349 | Realists believe truth is correspondence, independent of humans, is bivalent, and is unique [Putnam] |
| 2351 | Aristotle says an object (e.g. a lamp) has identity if its parts stay together when it is moved [Putnam] |
| 17785 | Real numbers as abstracted objects are now treated as complete ordered fields [Mayberry] |
| 2331 | Functionalism says robots and people are the same at one level of abstraction [Putnam] |
| 2348 | Is there just one computational state for each specific belief? [Putnam] |
| 2332 | Functionalism can't explain reference and truth, which are needed for logic [Putnam] |
| 2071 | If concepts have external meaning, computational states won't explain psychology [Putnam] |
| 2344 | If we are going to eliminate folk psychology, we must also eliminate folk logic [Putnam] |
| 2074 | Can we give a scientific, computational account of folk psychology? [Putnam] |
| 2343 | Reference may be different while mental representation is the same [Putnam] |
| 2346 | Meaning and translation (which are needed to define truth) both presuppose the notion of reference [Putnam] |
| 2354 | "Meaning is use" is not a definition of meaning [Putnam] |
| 2336 | Holism seems to make fixed definition more or less impossible [Putnam] |
| 2334 | Meaning holism tried to show that you can't get fixed meanings built out of observation terms [Putnam] |
| 2335 | Understanding a sentence involves background knowledge and can't be done in isolation [Putnam] |
| 2340 | We should separate how the reference of 'gold' is fixed from its conceptual content [Putnam] |
| 2341 | Like names, natural kind terms have their meaning fixed by extension and reference [Putnam] |
| 2339 | Aristotle implies that we have the complete concepts of a language in our heads, but we don't [Putnam] |
| 2338 | Reference (say to 'elms') is a social phenomenon which we can leave to experts [Putnam] |
| 13304 | Learned men gain more in one day than others do in a lifetime [Posidonius] |
| 2342 | "Water" is a natural kind term, but "H2O" is a description [Putnam] |
| 20820 | Time is an interval of motion, or the measure of speed [Posidonius, by Stobaeus] |