display all the ideas for this combination of philosophers
25 ideas
| 560 | Mathematical precision is only possible in immaterial things [Aristotle] |
| Full Idea: We should not see mathematical exactitude in all things, but only for things that do not have matter. | |
| From: Aristotle (Metaphysics [c.324 BCE], 0995a14) |
| 12377 | Mathematics is concerned with forms, not with superficial properties [Aristotle] |
| Full Idea: Mathematics is concerned with forms [eide]: its objects are not said of any underlying subject - for even if geometrical objects are said of some underlying subject, still it is not as being said of an underlying subject that they are studied. | |
| From: Aristotle (Posterior Analytics [c.327 BCE], 79a08) | |
| A reaction: Since forms turn out to be essences, in 'Metaphysics', this indicates an essentialist view of mathematics. |
| 9076 | Mathematics studies the domain of perceptible entities, but its subject-matter is not perceptible [Aristotle] |
| Full Idea: Mathematics does not take perceptible entities as its domain just because its subject-matter is accidentally perceptible; but neither does it take as its domain some other entities separable from the perceptible ones. | |
| From: Aristotle (Metaphysics [c.324 BCE], 1078a03) | |
| A reaction: This implies a very naturalistic view of mathematics, with his very empiricist account of abstraction deriving the mathematical concepts within the process of perceiving the physical world. And quite right too. |
| 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. |
| 12372 | The essence of a triangle comes from the line, mentioned in any account of triangles [Aristotle] |
| Full Idea: Something holds of an item in itself if it holds of it in what it is - e.g., line of triangles and point of lines (their essence comes from these items, which inhere in the account which says what they are). | |
| From: Aristotle (Posterior Analytics [c.327 BCE], 73a35) | |
| A reaction: A helpful illustration of how a definition gives us the essence of something. You could not define triangles without mentioning straight lines. The lines are necessary features, but they are essential for any explanation, and for proper understanding. |
| 1729 | We perceive number by the denial of continuity [Aristotle] |
| Full Idea: Number we perceive by the denial of continuity. | |
| From: Aristotle (De Anima [c.329 BCE], 425a19) | |
| A reaction: This is a key thought. A being (call it 'Parmenides') which sees all Being as One would make no distinctions of identity, and so could not count anything. Why would they want numbers? |
| 10958 | Perhaps numbers are substances? [Aristotle] |
| Full Idea: We should consider whether there is some other sort of substance, such as, perhaps, numbers. | |
| From: Aristotle (Metaphysics [c.324 BCE], 1037a11) | |
| A reaction: I don't think Aristotle considers numbers to be substances, but Pythagoreans seem to think that way, if they think the world is literally made of numbers. |
| 13273 | Pluralities divide into discontinous countables; magnitudes divide into continuous things [Aristotle] |
| Full Idea: A plurality is a denumerable quantity, and a magnitude is a measurable quantity. A plurality is what is potentially divisible into things that are not continuous, whereas what is said to be a magnitude is divisible into continuous things. | |
| From: Aristotle (Metaphysics [c.324 BCE], 1020a09) | |
| A reaction: This illuminating distinction is basic to the Greek attitude to number, and echoes the distinction between natural and real numbers. |
| 11044 | One is prior to two, because its existence is implied by two [Aristotle] |
| Full Idea: One is prior to two because if there are two it follows at once that there is one, whereas if there is one there is not necessarily two. | |
| From: Aristotle (Categories [c.331 BCE], 14a29) | |
| A reaction: The axiomatic introduction of a 'successor' to a number does not seem to introduce this notion of priority, based on inclusiveness. Introducing order by '>' also does not seem to indicate any logical priority. |
| 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. |
| 11042 | Parts of a line join at a point, so it is continuous [Aristotle] |
| Full Idea: A line is a continuous quantity. For it is possible to find a common boundary at which its parts join together, a point. | |
| From: Aristotle (Categories [c.331 BCE], 04b33) | |
| A reaction: This appears to be the essential concept of a Dedekind cut. It seems to be an open question whether a cut defines a unique number, but a boundary seems to be intrinsically unique. Aristotle wins again. |
| 12074 | The one in number just is the particular [Aristotle] |
| Full Idea: It makes no difference whether we speak of the particular or the one in number. For by the one in number we mean the particular. | |
| From: Aristotle (Metaphysics [c.324 BCE], 0999b33) | |
| A reaction: This is the Greek view of 'one', quite different from the Frege or Dedekind view. I prefer the Greek view, because 'one' is the place where numbers plug into the world, and the one indispensable feature of numbers is that they can count particulars. |
| 12273 | Unit is the starting point of number [Aristotle] |
| Full Idea: They say that the unit [monada] is the starting point of number (and the point the starting-point of a line). | |
| From: Aristotle (Topics [c.331 BCE], 108b30) | |
| A reaction: Yes, despite Frege's objections in the early part of the 'Grundlagen' (1884). I take arithmetic to be rooted in counting, despite all abstract definitions of number by Frege and Dedekind. Identity gives the unit, which is countable. See also Topics 141b9 |
| 12369 | A unit is what is quantitatively indivisible [Aristotle] |
| Full Idea: Arithmeticians posit that a unit is what is quantitatively indivisible. | |
| From: Aristotle (Posterior Analytics [c.327 BCE], 72a22) | |
| A reaction: Presumably indeterminate stuff like water is non-quantitatively divisible (e.g. Moses divides the Red Sea), as are general abstracta (curved shapes from rectilinear ones). Does 'quantitative' presupposes units, making the idea circular? |
| 17844 | The unit is stipulated to be indivisible [Aristotle] |
| Full Idea: The unit is stipulated to be indivisible in every respect. | |
| From: Aristotle (Metaphysics [c.324 BCE], 1052b35) |
| 17845 | If only rectilinear figures existed, then unity would be the triangle [Aristotle] |
| Full Idea: Suppose that all things that are ...were rectilinear figures - they would be a number of figures, and unity the triangle. | |
| From: Aristotle (Metaphysics [c.324 BCE], 1054a03) | |
| A reaction: This is how they program graphics for computer games, with profusions of triangles. The thought that geometry might be treated numerically is an obvious glimpse of Descartes' co-ordinate geometry. |
| 17859 | Units came about when the unequals were equalised [Aristotle] |
| Full Idea: The original holder of the theory claimed ...that units came about when the unequals were equalised. | |
| From: Aristotle (Metaphysics [c.324 BCE], 1081a24) | |
| A reaction: Presumably you could count the things that were already equal. You can count days and count raindrops. The genius is to see that you can add the days to the raindrops, by treating them as equal, in respect of number. |
| 17861 | Two men do not make one thing, as well as themselves [Aristotle] |
| Full Idea: A pair of men do not make some one thing in addition to themselves. | |
| From: Aristotle (Metaphysics [c.324 BCE], 1082a18) | |
| A reaction: This seems to contrast nicely with Frege's claim about whether two boots are two things or one pair. |
| 646 | When we count, are we adding, or naming numbers? [Aristotle] |
| Full Idea: It is a vexed question whether, when we count and say 'one, two, three…', we are doing so by addition or by separate modules. We are, of course, doing both. | |
| From: Aristotle (Metaphysics [c.324 BCE], 1082b32) | |
| A reaction: Note that this is almost Benacerraf's famous problem about whether or not 3 is a member of 4. |
| 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. |
| 17809 | Gödel showed that the syntactic approach to the infinite is of limited value [Kreisel] |
| Full Idea: Usually Gödel's incompleteness theorems are taken as showing a limitation on the syntactic approach to an understanding of the concept of infinity. | |
| From: Georg Kreisel (Hilbert's Programme [1958], 05) |
| 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. |
| 13212 | Infinity is only potential, never actual [Aristotle] |
| Full Idea: Nothing is actually infinite. A thing is infinite only potentially. | |
| From: Aristotle (Coming-to-be and Passing-away (Gen/Corr) [c.335 BCE], 318a21) | |
| A reaction: Aristotle is the famous spokesman for this view, though it reappeared somewhat in early twentieth century discussions (e.g. Hilbert). I sympathise with this unfashionable view. Multiple infinites are good fun, but no one knows what they really are. |
| 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. |