6 ideas
| 18832 | Mathematical statements and entities that result from an infinite process must lack a truth-value [Dummett] |
| Full Idea: On an intuitionistic view, neither the truth-value of a statement nor any other mathematical entity can be given as the final result of an infinite process, since an infinite process is precisely one that does not have a final result. | |
| From: Michael Dummett (Elements of Intuitionism (2nd ed) [2000], p.41), quoted by Ian Rumfitt - The Boundary Stones of Thought 7.3 | |
| A reaction: This is rather a persuasive reason to sympathise with intuitionism. Mathematical tricks about 'limits' have lured us into believing in completed infinities, but actually that idea is incoherent. |
| 15946 | Cantor developed sets from a progression into infinity by addition, multiplication and exponentiation [Cantor, by Lavine] |
| Full Idea: Cantor's development of set theory began with his discovery of the progression 0, 1, ....∞, ∞+1, ∞+2, ..∞x2, ∞x3, ...∞^2, ..∞^3, ...∞^∞, ...∞^∞^∞..... | |
| From: report of George Cantor (Grundlagen (Foundations of Theory of Manifolds) [1883]) by Shaughan Lavine - Understanding the Infinite VIII.2 |
| 15911 | Ordinals are generated by endless succession, followed by a limit ordinal [Cantor, by Lavine] |
| Full Idea: Ordinal numbers are generated by two principles: each ordinal has an immediate successor, and each unending sequence has an ordinal number as its limit (that is, an ordinal that is next after such a sequence). | |
| From: report of George Cantor (Grundlagen (Foundations of Theory of Manifolds) [1883]) by Shaughan Lavine - Understanding the Infinite III.4 |
| 3644 | Two things being joined together doesn't prove they are the same [Descartes] |
| Full Idea: The fact that we often see two things joined together does not license the inference that they are one and the same. | |
| From: René Descartes (Reply to Sixth Objections [1641], 444) | |
| A reaction: Correct. The problem comes when they are never ever apart, and you begin to suspect that they are conjoined in all possible worlds. Why might this be so? It can only be identity or a causal link. |
| 3621 | Only judgement decides which of our senses are reliable [Descartes] |
| Full Idea: Sense alone does not suffice to correct visual error: we also need a degree of reason to tell us that we should believe the judgement based on touch rather than vision. Since we don't have this power in infancy, it must be attributed to the intellect. | |
| From: René Descartes (Reply to Sixth Objections [1641], 439) |
| 3637 | Ideas in God's mind only have value if he makes it so [Descartes] |
| Full Idea: It is impossible to imagine that anything is thought of in the divine intellect as good or true, or worthy of belief or action or omission, prior to the decision of the divine will to make it so. | |
| From: René Descartes (Reply to Sixth Objections [1641], 432) |