5 ideas
8923 | Numbers are identified by their main properties and relations, involving the successor function [MacBride] |
Full Idea: The mathematically significant properties and relations of natural numbers arise from the successor function that orders them; the natural numbers are identified simply as the objects that answer to this basic function. | |
From: Fraser MacBride (Structuralism Reconsidered [2007], §1) | |
A reaction: So Julius Caesar would be a number if he was the successor of Pompey the Great? I would have thought that counting should be mentioned - cardinality as well as ordinality. Presumably Peano's Axioms are being referred to. |
8926 | For mathematical objects to be positions, positions themselves must exist first [MacBride] |
Full Idea: The identification of mathematical objects with positions in structures rests upon the prior credibility of the thesis that positions are objects in their own right. | |
From: Fraser MacBride (Structuralism Reconsidered [2007], §3) | |
A reaction: Sounds devastating, but something has to get the whole thing off the ground. This is why Resnik's word 'patterns' is so appealing. Patterns stare you in the face, and they don't change if all the objects making it up are replaced by others. |
19413 | If we know what is good or rational, our knowledge is extended, and our free will restricted [Leibniz] |
Full Idea: The more perfect one is, the more one is determined to the good, and so is more free at the same time. ...Our power and knowledge are more extended, and our will much the more limited within the bounds of perfect reason. | |
From: Gottfried Leibniz (Letters to Pierre Bayle [1702], 1702) | |
A reaction: I like this idea, which seems to me to derive from Aquinas. When I choose to eat and drink each day, or agree that 7+5 is 12, I don't complain about my lack of freedom in the choices. Goodness and reason are constraints I welcome. |
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. |