display all the ideas for this combination of philosophers
18 ideas
| 2521 | 'Real' maths objects have no causal role, no determinate reference, and no abstract/concrete distinction [Katz] |
| Full Idea: Three objections to realism in philosophy of mathematics: mathematical objects have no space/time location, and so no causal role; that such objects are determinate, but reference to numbers aren't; and that there is no abstract/concrete distinction. | |
| From: Jerrold J. Katz (Realistic Rationalism [2000], Int.xxix) |
| 9798 | Different parcels made from three pebbles produce different actual sensations [Mill] |
| Full Idea: Three pebbles make different sense impressions in one parcel or in two. That the same pebbles by an alteration of place and arrangement may be made to produce either sensation is not the identical proposition. | |
| From: John Stuart Mill (System of Logic [1843], 2.6.2) | |
| A reaction: [compressed] Not quite clear, but Mill seems to be adamant that we really must experience the separation, and not just think what 'may' happen, so Frege is right that Mill is lucky that everything is not 'nailed down'. |
| 9797 | '2 pebbles and 1 pebble' and '3 pebbles' name the same aggregation, but different facts [Mill] |
| Full Idea: The expressions '2 pebbles and 1 pebble' and '3 pebbles' stand for the same aggregation of objects, but do not stand for the same physical fact. They name the same objects in different states, 'denoting' the same things, with different 'connotations'. | |
| From: John Stuart Mill (System of Logic [1843], 2.6.2) | |
| A reaction: Nothing in this would convert me from the analytic view to the empirical view of simple arithmetic, if I were that way inclined. Personally I think of three pebbles as 4 minus 1, because I am haunted by the thought of a missing stone. |
| 9799 | 3=2+1 presupposes collections of objects ('Threes'), which may be divided thus [Mill] |
| Full Idea: 'Three is two and one' presupposes that collections of objects exist, which while they impress the senses thus, ¶¶¶, may be separated into two parts, thus, ¶¶ ¶. This being granted, we term all such parcels Threes. | |
| From: John Stuart Mill (System of Logic [1843], 2.6.2) | |
| A reaction: Mill is clearly in trouble here because he sticks to simple arithmetic. He must deal with parcels too big for humans to count, and parcels so big that they could not naturally exist, and that is before you even reach infinite parcels. |
| 9802 | Numbers denote physical properties of physical phenomena [Mill] |
| Full Idea: The fact asserted in the definition of a number is a physical fact. Each of the numbers two, three, four denotes physical phenomena, and connotes a physical property of those phenomena. Two denotes all pairs of things, and twelve all dozens. | |
| From: John Stuart Mill (System of Logic [1843], 3.24.5) | |
| A reaction: The least plausible part of Mill's thesis. Is the fact that a pair of things is fewer than five things also a property? You see two boots, or you see a pair of boots, depending partly on you. Is pure two a visible property? Courage and an onion? |
| 9803 | We can't easily distinguish 102 horses from 103, but we could arrange them to make it obvious [Mill] |
| Full Idea: 102 horses are not as easily distinguished from 103 as two are from three, yet the horses may be so placed that a difference will be perceptible. | |
| From: John Stuart Mill (System of Logic [1843], 3.24.5) | |
| A reaction: More trouble for Mill. We are now moving from the claim that we actually perceive numbers to the claim that we could if we arranged things right. But we would still only see which group of horses was bigger by one, not how many horses there were. |
| 9804 | Arithmetical results give a mode of formation of a given number [Mill] |
| Full Idea: Every statement of the result of an arithmetical operation is a statement of one of the modes of formation of a given number. | |
| From: John Stuart Mill (System of Logic [1843], 3.24.5) | |
| A reaction: Although Mill sticks cautiously to very simple arithmetic, inviting empirical accounts of much higher mathematics, I think the phrase 'modes of formation' of numbers is very helpful. It could take us either into structuralism, or into constructivism. |
| 9805 | 12 is the cube of 1728 means pebbles can be aggregated a certain way [Mill] |
| Full Idea: When we say 12 is the cube of 1728, we affirm that if we had sufficient pebbles, we put them into parcels or aggregates called twelves, and put those twelves into similar collections, and make twelve of these largests parcels, we have the aggregate 1728. | |
| From: John Stuart Mill (System of Logic [1843], 3.24.5) | |
| A reaction: There is always hidden modal thinking in Mill's proposals, despite his longing to stick to actual experience. Imagination actually plays a much bigger role in his theory than sense experience does. |
| 8741 | Numbers must be of something; they don't exist as abstractions [Mill] |
| Full Idea: All numbers must be numbers of something: there are no such things as numbers in the abstract. | |
| From: John Stuart Mill (System of Logic [1843], p.245?), quoted by Stewart Shapiro - Thinking About Mathematics 4.3 | |
| A reaction: This shows why the concept of 'abstraction' is such a deep problem. Numbers can't be properties of objects, because two boots can become one boot without changing the surviving boot. But why should abstractions have to 'exist'? |
| 5201 | Mill says logic and maths is induction based on a very large number of instances [Mill, by Ayer] |
| Full Idea: Mill maintained that the truths of logic and mathematics are not necessary or certain, by saying these propositions are inductive generalisations based on an extremely large number of instances. | |
| From: report of John Stuart Mill (System of Logic [1843]) by A.J. Ayer - Language,Truth and Logic Ch.4 | |
| A reaction: Ayer asserts that they are necessary (but only because they are tautological). I like the idea that maths is the 'science of patterns', but that might lead from an empirical start to a rationalist belief in a priori synthetic truths. |
| 9360 | If two black and two white objects in practice produced five, what colour is the fifth one? [Lewis,CI on Mill] |
| Full Idea: If Mill has a demon who, every time two things are brought together with two other things, always introduces a fifth, then if two black marbles and two white ones are put in an urn, the demon could choose his color, but there would be more of one colour. | |
| From: comment on John Stuart Mill (System of Logic [1843]) by C.I. Lewis - A Pragmatic Conception of the A Priori p.367 | |
| A reaction: Nice to see philosophers fighting back against demons. This is a lovely argument against the absurdity of thinking that experience could ever controvert a priori knowledge (though Lewis is no great fan of the latter). |
| 9888 | Mill mistakes particular applications as integral to arithmetic, instead of general patterns [Dummett on Mill] |
| Full Idea: Mill's mistake is taking particular applications as integral to the sense of arithmetical propositions. But what is integral to arithmetic is the general principle that explains its applicability, and determines the pattern of particular applications. | |
| From: comment on John Stuart Mill (System of Logic [1843], 2.6) by Michael Dummett - Frege philosophy of mathematics Ch.20 | |
| A reaction: [Dummett is summarising Frege's view] Sounds like a tidy objection, but you still have to connect the general principles and patterns to the physical world. 'Structure' could be the magic word to achieve this. |
| 9794 | There are no such things as numbers in the abstract [Mill] |
| Full Idea: There are no such things as numbers in the abstract. | |
| From: John Stuart Mill (System of Logic [1843], 2.6.2) | |
| A reaction: Depends. Would we want to say that 'horses don't exist' (although each individual horse does exist)? It sounds odd to say of an idea that it doesn't exist, when you are currently thinking about it. I am, however, sympathetic to Mill. |
| 9796 | Things possess the properties of numbers, as quantity, and as countable parts [Mill] |
| Full Idea: All things possess quantity; consist of parts which can be numbered; and in that character possess all the properties which are called properties of numbers. | |
| From: John Stuart Mill (System of Logic [1843], 2.6.2) | |
| A reaction: Here Mill is skating on the very thinnest of ice, and I find myself reluctantly siding with Frege. It is a very optimistic empiricist who hopes to find the numbers actually occurring as properties of experienced objects. A pack of cards, for example. |
| 9795 | Numbers have generalised application to entities (such as bodies or sounds) [Mill] |
| Full Idea: 'Ten' must mean ten bodies, or ten sounds, or ten beatings of the pulse. But though numbers must be numbers of something, they may be numbers of anything. | |
| From: John Stuart Mill (System of Logic [1843], 2.6.2) | |
| A reaction: Mill always prefers things in close proximity, in space or time. 'I've had ten headaches in the last year'. 'There are ten reasons for doubting p'. His second point puts him very close to Aristotle in his view. |
| 12411 | Mill is too imprecise, and is restricted to simple arithmetic [Kitcher on Mill] |
| Full Idea: The problem with Mill is that many of his formulations are imprecise, and he only considers the most rudimentary parts of arithmetic. | |
| From: comment on John Stuart Mill (System of Logic [1843]) by Philip Kitcher - The Nature of Mathematical Knowledge Intro | |
| A reaction: This is from a fan of Mill, trying to restore his approach in the face of the authoritative and crushing criticisms offered by Frege. I too am a fan of Mill's approach. Patterns can be discerned in arrangements of pebbles. Infinities are a problem. |
| 5656 | Empirical theories of arithmetic ignore zero, limit our maths, and need probability to get started [Frege on Mill] |
| Full Idea: Mill does not give us a clue as to how to understand the number zero, he limits our mathematical knowledge to the limits of our experience, ..and induction can only give you probability, but that presupposes arithmetical laws. | |
| From: comment on John Stuart Mill (System of Logic [1843]) by Gottlob Frege - Grundlagen der Arithmetik (Foundations) | |
| A reaction: This summarises Frege's criticisms of Mill's empirical account of maths. I like 'maths is the science of patterns', in which case zero is just a late-introduced trick (it is hardly a Platonic Form!), and induction is the wrong account to give. |
| 9624 | Numbers are a very general property of objects [Mill, by Brown,JR] |
| Full Idea: Mill held that numbers are a kind of very general property that objects possess. | |
| From: report of John Stuart Mill (System of Logic [1843], Ch.4) by James Robert Brown - Philosophy of Mathematics | |
| A reaction: Intuitively this sounds hopeless, because if you place one apple next to another you introduce 'two', but which apple has changed its property? Both? It seems to be a Cambridge change. It isn't a change that would bother the apples. Kitcher pursues this. |