9 ideas
7557 | To solve Zeno's paradox, reject the axiom that the whole has more terms than the parts [Russell] |
Full Idea: Presumably Zeno appealed to the axiom that the whole has more terms than the parts; so if Achilles were to overtake the tortoise, he would have been in more places than the tortoise, which he can't be; but the conclusion is absurd, so reject the axiom. | |
From: Bertrand Russell (Mathematics and the Metaphysicians [1901], p.89) | |
A reaction: The point is that the axiom is normally acceptable (a statue contains more particles than the arm of the statue), but it breaks down when discussing infinity (Idea 7556). Modern theories of infinity are needed to solve Zeno's Paradoxes. |
10059 | In mathematic we are ignorant of both subject-matter and truth [Russell] |
Full Idea: Mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true. | |
From: Bertrand Russell (Mathematics and the Metaphysicians [1901], p.76) | |
A reaction: A famous remark, though Musgrave is rather disparaging about Russell's underlying reasoning here. |
7556 | A collection is infinite if you can remove some terms without diminishing its number [Russell] |
Full Idea: A collection of terms is infinite if it contains as parts other collections which have as many terms as it has; that is, you can take away some terms of the collection without diminishing its number; there are as many even numbers as numbers all together. | |
From: Bertrand Russell (Mathematics and the Metaphysicians [1901], p.86) | |
A reaction: He cites Dedekind and Cantor as source for these ideas. If it won't obey the rule that subtraction makes it smaller, then it clearly isn't a number, and really it should be banned from all mathematics. |
9089 | Knowledge is a quality existing subjectively in the soul [William of Ockham] |
Full Idea: Knowledge is a certain quality which exists in the soul as its subject ('existens subiective in anima'). | |
From: William of Ockham (Expositio super viii libros [1340], Prologue) | |
A reaction: One might say here that knowledge is a property, and so it might not be susceptible to further analysis. It invites the question of how you could know by introspection that you have got it, which would be an extreme internalist view. |
9091 | Sometimes 'knowledge' just concerns the conclusion, sometimes the whole demonstration [William of Ockham] |
Full Idea: Sometimes 'knowledge' means evident cognition of the conclusion alone, sometimes of the demonstration as a whole. | |
From: William of Ockham (Expositio super viii libros [1340], Prologue) | |
A reaction: 'Demonstration' will be something like Greek 'logos' - full understanding, ability to explain and give reasons. William is certainly right about normal usage. I know the answer in a quiz, without any requirement for justifications. |
9090 | Knowledge is certain cognition of something that is true [William of Ockham] |
Full Idea: Knowledge is certain cognition of something that is true. | |
From: William of Ockham (Expositio super viii libros [1340], Prologue) | |
A reaction: This view has problems. William is not facing up to the sceptical questions which can shake any degree of certainty, and also that someone who lacked self-confidence might know many things while always feeling uncertain about them. 'Cognition' must go! |
7554 | Self-evidence is often a mere will-o'-the-wisp [Russell] |
Full Idea: Self-evidence is often a mere will-o'-the-wisp, which is sure to lead us astray if we take it as our guide. | |
From: Bertrand Russell (Mathematics and the Metaphysicians [1901], p.78) | |
A reaction: The sort of nice crisp remark you would expect from a good empiricist philosopher. Compare Idea 4948. However Russell qualifies it with the word 'often', and all philosophers eventually realise that you have to start somewhere. |
13304 | Learned men gain more in one day than others do in a lifetime [Posidonius] |
Full Idea: In a single day there lies open to men of learning more than there ever does to the unenlightened in the longest of lifetimes. | |
From: Posidonius (fragments/reports [c.95 BCE]), quoted by Seneca the Younger - Letters from a Stoic 078 | |
A reaction: These remarks endorsing the infinite superiority of the educated to the uneducated seem to have been popular in late antiquity. It tends to be the religions which discourage great learning, especially in their emphasis on a single book. |
20820 | Time is an interval of motion, or the measure of speed [Posidonius, by Stobaeus] |
Full Idea: Posidonius defined time thus: it is an interval of motion, or the measure of speed and slowness. | |
From: report of Posidonius (fragments/reports [c.95 BCE]) by John Stobaeus - Anthology 1.08.42 | |
A reaction: Hm. Can we define motion or speed without alluding to time? Looks like we have to define them as a conjoined pair, which means we cannot fully understand either of them. |