display all the ideas for this combination of texts
14 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) |
| 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. |
| 6297 | Euclid's geometry is synthetic, but Descartes produced an analytic version of it [Euclid, by Resnik] |
| Full Idea: Euclid's geometry is a synthetic geometry; Descartes supplied an analytic version of Euclid's geometry, and we now have analytic versions of the early non-Euclidean geometries. | |
| From: report of Euclid (Elements of Geometry [c.290 BCE]) by Michael D. Resnik - Maths as a Science of Patterns One.4 | |
| A reaction: I take it that the original Euclidean axioms were observations about the nature of space, but Descartes turned them into a set of pure interlocking definitions which could still function if space ceased to exist. |
| 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. |
| 9603 | An assumption that there is a largest prime leads to a contradiction [Euclid, by Brown,JR] |
| Full Idea: Assume a largest prime, then multiply the primes together and add one. The new number isn't prime, because we assumed a largest prime; but it can't be divided by a prime, because the remainder is one. So only a larger prime could divide it. Contradiction. | |
| From: report of Euclid (Elements of Geometry [c.290 BCE]) by James Robert Brown - Philosophy of Mathematics Ch.1 | |
| A reaction: Not only a very elegant mathematical argument, but a model for how much modern logic proceeds, by assuming that the proposition is false, and then deducing a contradiction from it. |
| 9894 | A unit is that according to which each existing thing is said to be one [Euclid] |
| Full Idea: A unit is that according to which each existing thing is said to be one. | |
| From: Euclid (Elements of Geometry [c.290 BCE], 7 Def 1) | |
| A reaction: See Frege's 'Grundlagen' §29-44 for a sustained critique of this. Frege is good, but there must be something right about the Euclid idea. If I count stone, paper and scissors as three, each must first qualify to be counted as one. Psychology creeps in. |
| 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. |
| 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. |
| 8738 | Postulate 2 says a line can be extended continuously [Euclid, by Shapiro] |
| Full Idea: Euclid's Postulate 2 says the geometer can 'produce a finite straight line continuously in a straight line'. | |
| From: report of Euclid (Elements of Geometry [c.290 BCE]) by Stewart Shapiro - Thinking About Mathematics 4.2 | |
| A reaction: The point being that this takes infinity for granted, especially if you start counting how many points there are on the line. The Einstein idea that it might eventually come round and hit you on the back of the head would have charmed Euclid. |