display all the ideas for this combination of texts
27 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. |
| 8958 | In Field's version of science, space-time points replace real numbers [Field,H, by Szabó] |
| Full Idea: Field's nominalist version of science develops a version of Newtonian gravitational theory, where no quantifiers range over mathematical entities, and space-time points and regions play the role of surrogates for real numbers. | |
| From: report of Hartry Field (Science without Numbers [1980]) by Zoltán Gendler Szabó - Nominalism 5.1 | |
| A reaction: This seems to be a very artificial contrivance, but Field has launched a programme for rewriting science so that numbers can be omitted. All of this is Field's rebellion against the Indispensability Argument for mathematics. I sympathise. |
| 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. |
| 18221 | 'Metric' axioms uses functions, points and numbers; 'synthetic' axioms give facts about space [Field,H] |
| Full Idea: There are two approaches to axiomatising geometry. The 'metric' approach uses a function which maps a pair of points into the real numbers. The 'synthetic' approach is that of Euclid and Hilbert, which does without real numbers and functions. | |
| From: Hartry Field (Science without Numbers [1980], 5) |
| 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'). |
| 8757 | The Indispensability Argument is the only serious ground for the existence of mathematical entities [Field,H] |
| Full Idea: There is one and only one serious argument for the existence of mathematical entities, and that is the Indispensability Argument of Putnam and Quine. | |
| From: Hartry Field (Science without Numbers [1980], p.5), quoted by Stewart Shapiro - Thinking About Mathematics 9.1 | |
| A reaction: Personally I don't believe (and nor does Field) that this gives a good enough reason to believe in such things. Quine (who likes 'desert landscapes' in ontology) ends up believing that sets are real because of his argument. Not for me. |
| 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 |
| 18212 | Nominalists try to only refer to physical objects, or language, or mental constructions [Field,H] |
| Full Idea: The most popular approach of nominalistically inclined philosophers is to try to reinterpret mathematics, so that its terms and quantifiers only make reference to, say, physical objects, or linguistic expressions, or mental constructions. | |
| From: Hartry Field (Science without Numbers [1980], Prelim) | |
| A reaction: I am keen on naturalism and empiricism, but only referring to physical objects is a non-starter. I think I favour constructions, derived from the experience of patterns, and abstracted, idealised and generalised. Field says application is the problem. |
| 10261 | The application of mathematics only needs its possibility, not its truth [Field,H, by Shapiro] |
| Full Idea: Field argues that to account for the applicability of mathematics, we need to assume little more than the possibility of the mathematics, not its truth. | |
| From: report of Hartry Field (Science without Numbers [1980]) by Stewart Shapiro - Philosophy of Mathematics 7.2 | |
| A reaction: Very persuasive. We can apply chess to real military situations, provided that chess isn't self-contradictory (or even naturally impossible?). |
| 18218 | Hilbert explains geometry, by non-numerical facts about space [Field,H] |
| Full Idea: Facts about geometric laws receive satisfying explanations, by the intrinsic facts about physical space, i.e. those laid down without reference to numbers in Hilbert's axioms. | |
| From: Hartry Field (Science without Numbers [1980], 3) | |
| A reaction: Hilbert's axioms mention points, betweenness, segment-congruence and angle-congruence (Field 25-26). Field cites arithmetic and geometry (as well as Newtonian mechanics) as not being dependent on number. |
| 9623 | Field needs a semantical notion of second-order consequence, and that needs sets [Brown,JR on Field,H] |
| Full Idea: Field needs the notion of logical consequence in second-order logic, but (since this is not recursively axiomatizable) this is a semantical notion, which involves the idea of 'true in all models', a set-theoretic idea if there ever was one. | |
| From: comment on Hartry Field (Science without Numbers [1980], Ch.4) by James Robert Brown - Philosophy of Mathematics | |
| A reaction: Brown here summarises a group of critics. Field was arguing for modern nominalism, that actual numbers could (in principle) be written out of the story, as useful fictions. Popper's attempt to dump induction seemed to need induction. |
| 18215 | It seems impossible to explain the idea that the conclusion is contained in the premises [Field,H] |
| Full Idea: No clear explanation of the idea that the conclusion was 'implicitly contained in' the premises was ever given, and I do not believe that any clear explanation is possible. | |
| From: Hartry Field (Science without Numbers [1980], 1) |
| 18216 | Abstractions can form useful counterparts to concrete statements [Field,H] |
| Full Idea: Abstract entities are useful because we can use them to formulate abstract counterparts of concrete statements. | |
| From: Hartry Field (Science without Numbers [1980], 3) | |
| A reaction: He defends the abstract statements as short cuts. If the concrete statements were 'true', then it seems likely that the abstract counterparts will also be true, which is not what fictionalism claims. |
| 18214 | Mathematics is only empirical as regards which theory is useful [Field,H] |
| Full Idea: Mathematics is in a sense empirical, but only in the rather Pickwickian sense that is an empirical question as to which mathematical theory is useful. | |
| From: Hartry Field (Science without Numbers [1980], 1) | |
| A reaction: Field wants mathematics to be fictions, and not to be truths. But can he give an account of 'useful' that does not imply truth? Only in a rather dubiously pragmatist way. A novel is not useful. |
| 18210 | Why regard standard mathematics as truths, rather than as interesting fictions? [Field,H] |
| Full Idea: Why regard the axioms of standard mathematics as truths, rather than as fictions that for a variety of reasons mathematicians have become interested in? | |
| From: Hartry Field (Science without Numbers [1980], p.viii) |