51 ideas
| 17641 | Discoveries in mathematics can challenge philosophy, and offer it a new foundation [Russell] |
| 17638 | If one proposition is deduced from another, they are more certain together than alone [Russell] |
| 4642 | No fact can be real and no proposition true unless there is a Sufficient Reason (even if we can't know it) [Leibniz] |
| 17632 | Non-contradiction was learned from instances, and then found to be indubitable [Russell] |
| 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] |
| 2115 | Everything in the universe is interconnected, so potentially a mind could know everything [Leibniz] |
| 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] |
| 2111 | Falsehood involves a contradiction, and truth is contradictory of falsehood [Leibniz] |
| 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] |
| 16871 | A truth can be an axiom in one system and not in another [Frege] |
| 16870 | Axioms are truths which cannot be doubted, and for which no proof is needed [Frege] |
| 17629 | Which premises are ultimate varies with context [Russell] |
| 17630 | The sources of a proof are the reasons why we believe its conclusion [Russell] |
| 17640 | Finding the axioms may be the only route to some new results [Russell] |
| 16869 | To create order in mathematics we need a full system, guided by patterns of inference [Frege] |
| 17627 | It seems absurd to prove 2+2=4, where the conclusion is more certain than premises [Russell] |
| 16864 | If principles are provable, they are theorems; if not, they are axioms [Frege] |
| 17628 | Arithmetic was probably inferred from relationships between physical objects [Russell] |
| 7644 | The monad idea incomprehensibly spiritualises matter, instead of materialising soul [La Mettrie on Leibniz] |
| 11857 | He replaced Aristotelian continuants with monads [Leibniz, by Wiggins] |
| 7843 | Is a drop of urine really an infinity of thinking monads? [Voltaire on Leibniz] |
| 12751 | It is unclear in 'Monadology' how extended bodies relate to mind-like monads. [Garber on Leibniz] |
| 19363 | Changes in a monad come from an internal principle, and the diversity within its substance [Leibniz] |
| 19352 | A 'monad' has basic perception and appetite; a 'soul' has distinct perception and memory [Leibniz] |
| 7931 | If a substance is just a thing that has properties, it seems to be a characterless non-entity [Leibniz, by Macdonald,C] |
| 9388 | Every concept must have a sharp boundary; we cannot allow an indeterminate third case [Frege] |
| 17554 | There must be some internal difference between any two beings in nature [Leibniz] |
| 2112 | Truths of reason are known by analysis, and are necessary; facts are contingent, and their opposites possible [Leibniz] |
| 17637 | The most obvious beliefs are not infallible, as other obvious beliefs may conflict [Russell] |
| 9344 | Mathematical analysis ends in primitive principles, which cannot be and need not be demonstrated [Leibniz] |
| 2110 | We all expect the sun to rise tomorrow by experience, but astronomers expect it by reason [Leibniz] |
| 17639 | Believing a whole science is more than believing each of its propositions [Russell] |
| 17631 | Induction is inferring premises from consequences [Russell] |
| 2109 | Increase a conscious machine to the size of a mill - you still won't see perceptions in it [Leibniz] |
| 19362 | We know the 'I' and its contents by abstraction from awareness of necessary truths [Leibniz] |
| 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] |
| 12707 | The true elements are atomic monads [Leibniz] |
| 17633 | The law of gravity has many consequences beyond its grounding observations [Russell] |
| 2114 | This is the most perfect possible universe, in its combination of variety with order [Leibniz] |
| 2113 | God alone (the Necessary Being) has the privilege that He must exist if He is possible [Leibniz] |