display all the ideas for this combination of texts
26 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. |
| 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. |
| 13459 | The less-than relation < well-orders, and partially orders, and totally orders the ordinal numbers [Hart,WD] |
| Full Idea: We can show (using the axiom of choice) that the less-than relation, <, well-orders the ordinals, ...and that it partially orders the ordinals, ...and that it totally orders the ordinals. | |
| From: William D. Hart (The Evolution of Logic [2010], 1) |
| 13491 | The axiom of infinity with separation gives a least limit ordinal ω [Hart,WD] |
| Full Idea: The axiom of infinity with separation yields a least limit ordinal, which is called ω. | |
| From: William D. Hart (The Evolution of Logic [2010], 3) |
| 13463 | There are at least as many infinite cardinals as transfinite ordinals (because they will map) [Hart,WD] |
| Full Idea: Since we can map the transfinite ordinals one-one into the infinite cardinals, there are at least as many infinite cardinals as transfinite ordinals. | |
| From: William D. Hart (The Evolution of Logic [2010], 1) |
| 13492 | Von Neumann's ordinals generalise into the transfinite better, because Zermelo's ω is a singleton [Hart,WD] |
| Full Idea: It is easier to generalize von Neumann's finite ordinals into the transfinite. All Zermelo's nonzero finite ordinals are singletons, but if ω were a singleton it is hard to see how if could fail to be the successor of its member and so not a limit. | |
| From: William D. Hart (The Evolution of Logic [2010], 3) |
| 18253 | I wish to go straight from cardinals to reals (as ratios), leaving out the rationals [Frege] |
| Full Idea: You need a double transition, from cardinal numbes (Anzahlen) to the rational numbers, and from the latter to the real numbers generally. I wish to go straight from the cardinal numbers to the real numbers as ratios of quantities. | |
| From: Gottlob Frege (Letters to Russell [1902], 1903.05.21), quoted by Michael Dummett - Frege philosophy of mathematics 21 'Frege's' | |
| A reaction: Note that Frege's real numbers are not quantities, but ratios of quantities. In this way the same real number can refer to lengths, masses, intensities etc. |
| 13446 | 19th century arithmetization of analysis isolated the real numbers from geometry [Hart,WD] |
| Full Idea: The real numbers were not isolated from geometry until the arithmetization of analysis during the nineteenth century. | |
| From: William D. Hart (The Evolution of Logic [2010], 1) |
| 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. |
| 13509 | We can establish truths about infinite numbers by means of induction [Hart,WD] |
| Full Idea: Mathematical induction is a way to establish truths about the infinity of natural numbers by a finite proof. | |
| From: William D. Hart (The Evolution of Logic [2010], 5) | |
| A reaction: If there are truths about infinities, it is very tempting to infer that the infinities must therefore 'exist'. A nice, and large, question in philosophy is whether there can be truths without corresponding implications of existence. |
| 13474 | Euclid has a unique parallel, spherical geometry has none, and saddle geometry has several [Hart,WD] |
| Full Idea: There is a familiar comparison between Euclid (unique parallel) and 'spherical' geometry (no parallel) and 'saddle' geometry (several parallels). | |
| From: William D. Hart (The Evolution of Logic [2010], 2) |
| 17843 | The idea of 'one' is the foundation of number [Aristotle] |
| Full Idea: One is the principle of number qua number. | |
| From: Aristotle (Metaphysics [c.324 BCE], 1052b21) |
| 17850 | Each many is just ones, and is measured by the one [Aristotle] |
| Full Idea: The reason for saying of each number that it is many is just that it is ones and that each number is measured by the one. | |
| From: Aristotle (Metaphysics [c.324 BCE], 1056b16) |
| 17851 | Number is plurality measured by unity [Aristotle] |
| Full Idea: Number is plurality as measured by unity. | |
| From: Aristotle (Metaphysics [c.324 BCE], 1057a04) |
| 9793 | Mathematics studies abstracted relations, commensurability and proportion [Aristotle] |
| Full Idea: Mathematicians abstract perceptible features to study quantity and continuity ...and examine the mutual relations of some and the features of those relations, and commensurabilities of others, and of yet others the proportions. | |
| From: Aristotle (Metaphysics [c.324 BCE], 1061a32) | |
| A reaction: This sounds very much like the intuition of structuralism to me - that the subject is entirely about relations between things, with very little interest in the things themselves. See Aristotle on abstraction (under 'Thought'). |
| 13738 | It is a simple truth that the objects of mathematics have being, of some sort [Aristotle] |
| Full Idea: Since there are not only separable things but also inseparable things (such as, for instance, things which are moving), it is also true to say simpliciter that the objects of mathematic have being and that they are of such a sort as is claimed. | |
| From: Aristotle (Metaphysics [c.324 BCE], 1077b31) | |
| A reaction: This is almost Aristotle's only discussion of whether mathematical entities exist. They seem to have an 'inseparable' existence (the way properties do), but he evidently regards a denial of their existence (Field-style) as daft. |
| 12339 | Aristotle removes ontology from mathematics, and replaces the true with the beautiful [Aristotle, by Badiou] |
| Full Idea: For Aristotle, the de-ontologization of mathematics draws the beautiful into the place of the true. | |
| From: report of Aristotle (Metaphysics [c.324 BCE]) by Alain Badiou - Briefings on Existence 14 |
| 18166 | The loss of my Rule V seems to make foundations for arithmetic impossible [Frege] |
| Full Idea: With the loss of my Rule V, not only the foundations of arithmetic, but also the sole possible foundations of arithmetic, seem to vanish. | |
| From: Gottlob Frege (Letters to Russell [1902], 1902.06.22) | |
| A reaction: Obviously he was stressed, but did he really mean that there could be no foundation for arithmetic, suggesting that the subject might vanish into thin air? |
| 13471 | Mathematics makes existence claims, but philosophers usually say those are never analytic [Hart,WD] |
| Full Idea: The thesis that no existence proposition is analytic is one of the few constants in philosophical consciences, but there are many existence claims in mathematics, such as the infinity of primes, five regular solids, and certain undecidable propositions. | |
| From: William D. Hart (The Evolution of Logic [2010], 2) |