31 ideas
| 17610 | The Axiom of Choice paradoxically allows decomposing a sphere into two identical spheres [Maddy] |
| 17620 | Critics of if-thenism say that not all starting points, even consistent ones, are worth studying [Maddy] |
| 17605 | Hilbert's geometry and Dedekind's real numbers were role models for axiomatization [Maddy] |
| 17625 | If two mathematical themes coincide, that suggest a single deep truth [Maddy] |
| 10631 | If 'x is heterological' iff it does not apply to itself, then 'heterological' is heterological if it isn't heterological [Hale/Wright] |
| 17615 | Every infinite set of reals is either countable or of the same size as the full set of reals [Maddy] |
| 10624 | The incompletability of formal arithmetic reveals that logic also cannot be completely characterized [Hale/Wright] |
| 17618 | Set-theory tracks the contours of mathematical depth and fruitfulness [Maddy] |
| 10629 | If structures are relative, this undermines truth-value and objectivity [Hale/Wright] |
| 10628 | The structural view of numbers doesn't fit their usage outside arithmetical contexts [Hale/Wright] |
| 17614 | The connection of arithmetic to perception has been idealised away in modern infinitary mathematics [Maddy] |
| 10622 | The neo-Fregean is more optimistic than Frege about contextual definitions of numbers [Hale/Wright] |
| 8780 | Attributes are functions, not objects; this distinguishes 'square of 2' from 'double of 2' [Geach] |
| 10626 | Objects just are what singular terms refer to [Hale/Wright] |
| 11910 | Being 'the same' is meaningless, unless we specify 'the same X' [Geach] |
| 8775 | A big flea is a small animal, so 'big' and 'small' cannot be acquired by abstraction [Geach] |
| 8776 | We cannot learn relations by abstraction, because their converse must be learned too [Geach] |
| 2567 | You can't define real mental states in terms of behaviour that never happens [Geach] |
| 2568 | Beliefs aren't tied to particular behaviours [Geach] |
| 8781 | The mind does not lift concepts from experience; it creates them, and then applies them [Geach] |
| 8769 | If someone has aphasia but can still play chess, they clearly have concepts [Geach] |
| 8770 | 'Abstractionism' is acquiring a concept by picking out one experience amongst a group [Geach] |
| 10630 | Abstracted objects are not mental creations, but depend on equivalence between given entities [Hale/Wright] |
| 8771 | 'Or' and 'not' are not to be found in the sensible world, or even in the world of inner experience [Geach] |
| 8772 | We can't acquire number-concepts by extracting the number from the things being counted [Geach] |
| 8773 | Abstractionists can't explain counting, because it must precede experience of objects [Geach] |
| 8774 | The numbers don't exist in nature, so they cannot have been abstracted from there into our languages [Geach] |
| 8778 | Blind people can use colour words like 'red' perfectly intelligently [Geach] |
| 8777 | If 'black' and 'cat' can be used in the absence of such objects, how can such usage be abstracted? [Geach] |
| 8779 | We can form two different abstract concepts that apply to a single unified experience [Geach] |
| 10627 | Many conceptual truths ('yellow is extended') are not analytic, as derived from logic and definitions [Hale/Wright] |