display all the ideas for this combination of texts
9 ideas
| 9790 | Geometry studies naturally occurring lines, but not as they occur in nature [Aristotle] |
| Full Idea: Geometry studies naturally occurring lines, but not as they occur in nature. | |
| From: Aristotle (Physics [c.337 BCE], 194a09) | |
| A reaction: What a splendid remark. If the only specimen you could find of a very rare animal was maimed, you wouldn't be particularly interested in the nature of its injury, but in the animal. |
| 13861 | Number theory aims at the essence of natural numbers, giving their nature, and the epistemology [Wright,C] |
| Full Idea: In the Fregean view number theory is a science, aimed at those truths furnished by the essential properties of zero and its successors. The two broad question are then the nature of the objects, and the epistemology of those facts. | |
| From: Crispin Wright (Frege's Concept of Numbers as Objects [1983], Intro) | |
| A reaction: [compressed] I pounce on the word 'essence' here (my thing). My first question is about the extent to which the natural numbers all have one generic essence, and the extent to which they are individuals (bless their little cotton socks). |
| 13892 | One could grasp numbers, and name sizes with them, without grasping ordering [Wright,C] |
| Full Idea: Someone could be clear about number identities, and distinguish numbers from other things, without conceiving them as ordered in a progression at all. The point of them would be to make comparisons between sizes of groups. | |
| From: Crispin Wright (Frege's Concept of Numbers as Objects [1983], 3.xv) | |
| A reaction: Hm. Could you grasp size if you couldn't grasp which of two groups was the bigger? What's the point of noting that I have ten pounds and you only have five, if you don't realise that I have more than you? You could have called them Caesar and Brutus. |
| 22962 | Two is the least number, but there is no least magnitude, because it is always divisible [Aristotle] |
| Full Idea: The least number, without qualification, is the two. …but in magnitude there is no least number, for every line always gets divided. | |
| From: Aristotle (Physics [c.337 BCE], 220a27) | |
| A reaction: Showing the geometrical approach of the Greeks to number. Two is the last number because numbers are for counting, and picking out one thing is not counting. |
| 13867 | Instances of a non-sortal concept can only be counted relative to a sortal concept [Wright,C] |
| Full Idea: The invitation to number the instances of some non-sortal concept is intelligible only if it is relativised to a sortal. | |
| From: Crispin Wright (Frege's Concept of Numbers as Objects [1983], 1.i) | |
| A reaction: I take this to be an essentially Fregean idea, as when we count the boots when we have decided whether they fall under the concept 'boot' or the concept 'pair'. I also take this to be the traditional question 'what units are you using'? |
| 18090 | Without infinity time has limits, magnitudes are indivisible, and numbers come to an end [Aristotle] |
| Full Idea: If there is, unqualifiedly, no infinite, it is clear that many impossible things result. For there will be a beginning and an end of time, and magnitudes will not be divisible into magnitudes, and number will not be infinite. | |
| From: Aristotle (Physics [c.337 BCE], 206b09), quoted by David Bostock - Philosophy of Mathematics 1.8 | |
| A reaction: This is a commitment to infinite time, and uncountable real numbers, and infinite ordinals. Dedekind cuts are implied. Nice. |
| 22929 | Aristotle's infinity is a property of the counting process, that it has no natural limit [Aristotle, by Le Poidevin] |
| Full Idea: For Aristotle infinity is not so much a property of some set of objects - the numbers - as of the process of counting, namely of its not having a natural limit. This is 'potential' infinite | |
| From: report of Aristotle (Physics [c.337 BCE]) by Robin Le Poidevin - Travels in Four Dimensions 06 'Illusion' | |
| A reaction: I increasingly favour this view. Mathematicians have foisted fictional objects on us, such as real infinities, limits and zero, because it makes their job easier, but it makes discussion of the natural world very obscure. |
| 22930 | Lengths do not contain infinite parts; parts are created by acts of division [Aristotle, by Le Poidevin] |
| Full Idea: Aristotle says that a length does not already contain, waiting to be discovered, an infinite number of parts; such parts only come into existence once they are defined by an act of division. | |
| From: report of Aristotle (Physics [c.337 BCE]) by Robin Le Poidevin - Travels in Four Dimensions 07 'Two' | |
| A reaction: If that is true of infinite parts then it must also be true of finite parts. So a cake has no parts at all until it is cut. That could play merry hell with discussions of mereology. Wholes are ontologically prior to parts. |
| 18833 | A continuous line cannot be composed of indivisible points [Aristotle] |
| Full Idea: No continuum can be composed of indivisibles: e.g. a line cannot be composed of points, the line being continuous and the points indivisibles. | |
| From: Aristotle (Physics [c.337 BCE], 231a23), quoted by Ian Rumfitt - The Boundary Stones of Thought 7.4 | |
| A reaction: Rumfitt observes that ' the basic problem is to say what the ultimate parts of a continuum are, of they are not points'. Early modern philosophers had lots of proposals. |