9 ideas
| 21554 | Sets always exceed terms, so all the sets must exceed all the sets [Lackey] |
| Full Idea: Cantor proved that the number of sets in a collection of terms is larger than the number of terms. Hence Cantor's Paradox says the number of sets in the collection of all sets must be larger than the number of sets in the collection of all sets. | |
| From: Douglas Lackey (Intros to Russell's 'Essays in Analysis' [1973], p.127) | |
| A reaction: The sets must count as terms in the next iteration, but that is a normal application of the Power Set axiom. |
| 21553 | It seems that the ordinal number of all the ordinals must be bigger than itself [Lackey] |
| Full Idea: The ordinal series is well-ordered and thus has an ordinal number, and a series of ordinals to a given ordinal exceeds that ordinal by 1. So the series of all ordinals has an ordinal number that exceeds its own ordinal number by 1. | |
| From: Douglas Lackey (Intros to Russell's 'Essays in Analysis' [1973], p.127) | |
| A reaction: Formulated by Burali-Forti in 1897. |
| 10245 | One geometry cannot be more true than another [Poincaré] |
| Full Idea: One geometry cannot be more true than another; it can only be more convenient. | |
| From: Henri Poincaré (Science and Method [1908], p.65), quoted by Stewart Shapiro - Philosophy of Mathematics | |
| A reaction: This is the culminating view after new geometries were developed by tinkering with Euclid's parallels postulate. |
| 23681 | The first motion or effect cannot be produced necessarily, so the First Cause must be a free agent [Reid] |
| Full Idea: That the first motion, or the first effect, whatever it be, cannot be produced necessarily, and, consequently, that the First Cause must be a free agent, has been demonstrated clearly and unanswerably. | |
| From: Thomas Reid (Essays on Active Powers 4: Liberty of Agents [1788], 8) | |
| A reaction: He has said that the First Cause can only be conceived by us as an 'agent'. If there is an agential First Cause, then he must be right. It is this need for God to be free which makes scepticism about free will unacceptable to many. |
| 23676 | A willed action needs reasonable understanding of what is to be done [Reid] |
| Full Idea: There can be no will without such a degree of understanding, at least, as gives the conception of that which we will. | |
| From: Thomas Reid (Essays on Active Powers 4: Liberty of Agents [1788], 1) | |
| A reaction: Presumably this 'conception' includes an understanding of the probable consequences, but they are of infinite complexity. I see this as an objection to 'ultimate' free will and responsibility, because there are only ever degrees of understanding. |
| 23680 | We are morally free, because we experience it, we are accountable, and we pursue projects [Reid] |
| Full Idea: I believe in moral liberty first because we have a natural conviction of belief that in many cases we act freely, second because we are accountable, and third because we can prosecute an end by a long series of means adapted. | |
| From: Thomas Reid (Essays on Active Powers 4: Liberty of Agents [1788], 5) | |
| A reaction: This is his final summary of why he believes in free will. Why didn't Plato and Aristotle have this natural belief? He could only believe we are 'accountable' because he believes in free will. Ants and bees pursue lengthy projects. Hm. |
| 23678 | A motive is merely an idea, like advice, and not a force for action [Reid] |
| Full Idea: A motive is equally incapable of action and of passion; because it is not a thing that exists, but a thing that is conceived. …Motives may be compared to advice or exhortation. | |
| From: Thomas Reid (Essays on Active Powers 4: Liberty of Agents [1788], 4) | |
| A reaction: We say people are motivated by greed or anger or love, which seems a bit stronger than mere advice. |
| 23677 | We all know that mere priority or constant conjunction do not have to imply causation [Reid] |
| Full Idea: Every man who understands the language knows that neither priority, nor constant conjunction, nor both taken together, imply efficiency. | |
| From: Thomas Reid (Essays on Active Powers 4: Liberty of Agents [1788], 2) | |
| A reaction: This invites the question of how we do know causal events, if none of our experiences are enough to prove it. Reid says we have an innate knowledge that all events are caused, but that isn't much help. The presence of power? |
| 23679 | The principle of the law of nature is that matter is passive, and is acted upon [Reid] |
| Full Idea: The law of nature respecting matter is grounded upon this principle: That matter is an inert, inactive substance, which does not act, but is acted upon. | |
| From: Thomas Reid (Essays on Active Powers 4: Liberty of Agents [1788], 5) | |
| A reaction: A clear statement (alongside Euler's) of the 18th century view, still with us, but strikes me as entirely wrong. Their view needs the active power of God to drive the laws. Matter has intrinsic primitive powers, and laws describe patterns of behaviour. |