28 ideas
16877 | A 'constructive' (as opposed to 'analytic') definition creates a new sign [Frege] |
11219 | Frege suggested that mathematics should only accept stipulative definitions [Frege, by Gupta] |
16878 | We must be clear about every premise and every law used in a proof [Frege] |
17610 | The Axiom of Choice paradoxically allows decomposing a sphere into two identical spheres [Maddy] |
16867 | Logic not only proves things, but also reveals logical relations between them [Frege] |
16863 | Does some mathematical reasoning (such as mathematical induction) not belong to logic? [Frege] |
16862 | The closest subject to logic is mathematics, which does little apart from drawing inferences [Frege] |
17620 | Critics of if-thenism say that not all starting points, even consistent ones, are worth studying [Maddy] |
16865 | 'Theorems' are both proved, and used in proofs [Frege] |
16866 | Tracing inference backwards closes in on a small set of axioms and postulates [Frege] |
16868 | The essence of mathematics is the kernel of primitive truths on which it rests [Frege] |
16870 | Axioms are truths which cannot be doubted, and for which no proof is needed [Frege] |
16871 | A truth can be an axiom in one system and not in another [Frege] |
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] |
16869 | To create order in mathematics we need a full system, guided by patterns of inference [Frege] |
17615 | Every infinite set of reals is either countable or of the same size as the full set of reals [Maddy] |
16864 | If principles are provable, they are theorems; if not, they are axioms [Frege] |
17618 | Set-theory tracks the contours of mathematical depth and fruitfulness [Maddy] |
17614 | The connection of arithmetic to perception has been idealised away in modern infinitary mathematics [Maddy] |
9388 | Every concept must have a sharp boundary; we cannot allow an indeterminate third case [Frege] |
16876 | We need definitions to cram retrievable sense into a signed receptacle [Frege] |
16875 | We use signs to mark receptacles for complex senses [Frege] |
16879 | A sign won't gain sense just from being used in sentences with familiar components [Frege] |
16873 | Thoughts are not subjective or psychological, because some thoughts are the same for us all [Frege] |
16872 | A thought is the sense expressed by a sentence, and is what we prove [Frege] |
16874 | The parts of a thought map onto the parts of a sentence [Frege] |
12709 | Motion is not absolute, but consists in relation [Leibniz] |