| 16146 | Two can't be a self-contained unit, because it would need to be one to do that [Democritus, by Aristotle] |
| Full Idea: Democritus claimed that one substance could not be composed from two nor two from one. …The same will clearly go for number, on the popular assumption that number is a combination of units. Unless two is one, it cannot contain a unit in actuality. | |
| From: report of Democritus (fragments/reports [c.431 BCE]) by Aristotle - Metaphysics 1039a15 | |
| A reaction: Chrysippus followed this up the first part with the memorable example of Dion and Theon. The problem with the second part is that 2, 3 and 4 are three numbers, so they can count as meta-units. |
| 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. |
| 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? |
| 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 |
| 24035 | Unity is something shared by many things, so in that respect they are equals [Descartes] |
| Full Idea: Unity is that common nature in which all things that are compared with each other must participate equally. | |
| From: René Descartes (Rules for the Direction of the Mind [1628], 14) | |
| A reaction: A lovely explanation of the concept of 'units' for counting. Fregeans hate units, but we Grecian thinkers love them. |
| 24036 | I can only see the proportion of two to three if there is a common measure - their unity [Descartes] |
| Full Idea: I do not recognise what the proportion of magnitude is between two and three, unless I consider a third term, namely unity, which is the common measure of the one and the other. | |
| From: René Descartes (Rules for the Direction of the Mind [1628], 14) | |
| A reaction: A striking defence of the concept of the need for the unit in arithmetic. To say 'three is half as big again', you must be discussing the same size of 'half' in each instance. |
| 12956 | Only whole numbers are multitudes of units [Leibniz] |
| Full Idea: The definition of 'number' as a multitude of units is appropriate only for whole numbers. | |
| From: Gottfried Leibniz (New Essays on Human Understanding [1704], 2.15) | |
| A reaction: One can also define rational numbers by making use of units, but the strategy breaks down with irrational numbers like root-2 and pi. I still say the concept of a unit is the basis of numbers. Without whole numbers, we wouldn't call the real 'numbers'. |
| 12920 | There is no multiplicity without true units [Leibniz] |
| Full Idea: There is no multiplicity without true units. | |
| From: Gottfried Leibniz (Letters to Antoine Arnauld [1686], 1687.04.30) | |
| A reaction: Hence real numbers do not embody 'multiplicity'. So either they don't 'embody' anything, or they embody 'magnitudes'. Does this give two entirely different notions, of measure of multiplicity and measures of magnitude? |
| 9147 | Number cannot be defined as addition of ones, since that needs the number; it is a single act of abstraction [Fine,K on Leibniz] |
| Full Idea: Leibniz's talk of the addition of ones cannot define number, since it cannot be specified how often they are added without using the number itself. Number must be an organic unity of ones, achieved by a single act of abstraction. | |
| From: comment on Gottfried Leibniz (works [1690]) by Kit Fine - Cantorian Abstraction: Recon. and Defence §1 | |
| A reaction: I doubt whether 'abstraction' is the right word for this part of the process. It seems more like a 'gestalt'. The first point is clearly right, that it is the wrong way round if you try to define number by means of addition. |
| 9801 | Numbers must be assumed to have identical units, as horses are equalised in 'horse-power' [Mill] |
| Full Idea: There is one hypothetical element in the basis of arithmetic, without which none of it would be true: all the numbers are numbers of the same or of equal units. When we talk of forty horse-power, we assume all horses are of equal strength. | |
| From: John Stuart Mill (System of Logic [1843], 2.6.3) | |
| A reaction: Of course, horses are not all of equal strength, so there is a problem here for your hard-line empiricist. Mill needs processes of idealisation and abstraction before his empirical arithmetic can get off the ground. |
| 8641 | You can abstract concepts from the moon, but the number one is not among them [Frege] |
| Full Idea: What are we supposed to abstract from to get from the moon to the number 1? We do get certain concepts, such as satellite, but 1 is not to be met with. In the case of 0 we have no objects at all. ..The essence of number must work for 0 and 1. | |
| From: Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884], §44) | |
| A reaction: Note that Frege seems to be conceding psychological abstraction for most other concepts. But why can't you abstract from your abstractions, to reach high-level abstractions? And why should numbers not emerge at those higher levels? |
| 9989 | Units can be equal without being identical [Tait on Frege] |
| Full Idea: The fact that units are equal does not mean that they are identical. The units can be equal just in the sense that once can be substituted for any other without altering the name assigned, i.e. the number. | |
| From: comment on Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884], §54) by William W. Tait - Frege versus Cantor and Dedekind XI | |
| A reaction: [this is in reference to Thomae 1880] Presumably this might mean that units have type-identity, rather than token-dentity. 'This' unit might be a token, but 'a' unit would be a type. I am extremely reluctant to ditch the old concept of a unit. |
| 17429 | Frege says only concepts which isolate and avoid arbitrary division can give units [Frege, by Koslicki] |
| Full Idea: It is Frege's view that only concepts which satisfy isolation and non-arbitrary division can play the role of dividing up what falls under them into countable units. | |
| From: report of Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884], §54) by Kathrin Koslicki - Isolation and Non-arbitrary Division 2.1 | |
| A reaction: Compare Idea 17429. If I count out a 'team of players', I need this unit concept to get what a 'player' is, but then need the 'team' concept to do the counting. Number doesn't attach to the unit concept. |
| 20361 | We need 'unities' for reckoning, but that does not mean they exist [Nietzsche] |
| Full Idea: We need 'unities' in order to be able to reckon: that does not mean we must suppose that such unities exist. | |
| From: Friedrich Nietzsche (The Will to Power (notebooks) [1888], §635) | |
| A reaction: True. I takes this thought to be important in the Psychology of Metaphysics (an unfashionable branch). |
| 9576 | Multiplicity in general is just one and one and one, etc. [Husserl] |
| Full Idea: Multiplicity in general is no more than something and something and something, etc.; ..or more briefly, one and one and one, etc. | |
| From: Edmund Husserl (Philosophy of Arithmetic [1894], p.85), quoted by Gottlob Frege - Review of Husserl's 'Phil of Arithmetic' | |
| A reaction: Frege goes on to attack this idea fairly convincingly. It seems obvious that it is hard to say that you have seventeen items, if the only numberical concept in your possession is 'one'. How would you distinguish 17 from 16? What makes the ones 'multiple'? |
| 18392 | Classes have cardinalities, so their members must all be treated as units [Armstrong] |
| Full Idea: Classes, because they have a particular cardinality, are essentially a certain number of ones, things that, within the particular class, are each taken as a unit. | |
| From: David M. Armstrong (Truth and Truthmakers [2004], 09.1) | |
| A reaction: [Singletons are exceptions] So units are basic to set theory, which is the foundations of technical analytic philosophy (as well as, for many, of mathematics). If you can't treat something as a unit, it won't go into set theory. Vagueness... |
| 9895 | A number is a multitude composed of units [Dummett] |
| Full Idea: A number is a multitude composed of units. | |
| From: Michael Dummett (Frege philosophy of mathematics [1991], 7 Def 2) | |
| A reaction: This is outdated by the assumption that 0 and 1 are also numbers, but if we say one is really just the 'unit' which is preliminary to numbers, and 0 is as bogus a number as i is, we might stick with the original Greek distinction. |
| 18071 | A one-operation is the segregation of a single object [Kitcher] |
| Full Idea: We perform a one-operation when we perform a segregative operation in which a single object is segregated. | |
| From: Philip Kitcher (The Nature of Mathematical Knowledge [1984], 06.3) | |
| A reaction: This is part of Kitcher's empirical but constructive account of arithmetic, which I find very congenial. He avoids the word 'unit', and goes straight to the concept of 'one' (which he treats as more primitive than zero). |
| 17435 | Objects do not naturally form countable units [Koslicki] |
| Full Idea: Objects do not by themselves naturally fall into countable units. | |
| From: Kathrin Koslicki (Isolation and Non-arbitrary Division [1997], 2.2) | |
| A reaction: Hm. This seems to be modern Fregean orthodoxy. Why did the institution of counting ever get started if the things in the world didn't demand counting? Even birds are aware of the number of eggs in their nest (because they miss a stolen one). |