| 9792 | Mathematicians study quantity and continuity, and remove the perceptible features of things [Aristotle] |
| Full Idea: The mathematician conducts a study into things in abstraction (after the removal of all perceptible features, such as weight and hardness, leaving only quantity and continuity). | |
| From: Aristotle (Metaphysics [c.324 BCE], 1061a26) | |
| A reaction: Frege complained that there is nothing left if you remove the perceptible features, but clearly Aristotle is not an empiricist in this passage, and it is doubtful if even Mill can be totally empirical in his account. We have relations of ideas. |
| 9077 | Mathematicians suppose inseparable aspects to be separable, and study them in isolation [Aristotle] |
| Full Idea: Study things as mathematicians do. Suppose what is not separable to be separable. A man qua man is an indivisible unity, so the arithmetician supposes a man to be an indivisible unity, and investigates the accidental features of man qua indivisible. | |
| From: Aristotle (Metaphysics [c.324 BCE], 1078a19) | |
| A reaction: This is the abstractionist view of mathematics. Qua indivisible, a man will have the same properties as a toothbrush. Aristotle clearly intends the method for scientists as well. It strikes me as common sense, but there is a lot of modern caution. |
| 9788 | Mathematicians study what is conceptually separable, and doesn't lead to error [Aristotle] |
| Full Idea: Mathematicians abstract properties which are conceptually separable from the world of change. It makes no difference if you treat them as separate, in the sense that it does not result in error. | |
| From: Aristotle (Physics [c.337 BCE], 193b33) | |
| A reaction: This strikes me as a crucial point to make against Frege (if Aristotle is right). Frege hates abstractionism precisely because it is psychological, and hence admits subjective error, instead of objective truth. Does 'pure' abstraction avoid error? |
| 9094 | Mathematical objects abstract both from perceived matter, and from particular substance [Aquinas] |
| Full Idea: Objects of mathematics abstract from perceived matter both in particular and in general, though from thought matter (substance as underlying quality) only in particular and not in general. | |
| From: Thomas Aquinas (Summa Theologicae [1265], Ch.5 Q85.1) | |
| A reaction: This appears to be a thoroughly abstractionist view of the way in which humans create mathematics. Aquinas explicitly denies the Platonic view that the numbers already have abstract existence, awaiting our discovery. |
| 10505 | We can just think of an apple's colour, because the apple is not part of the colour's nature [Aquinas] |
| Full Idea: The apple is not part of the nature of the colour, and so nothing prevents one from understanding the colour while understanding nothing of the apple. | |
| From: Thomas Aquinas (Summa Theologicae [1265], Q85 1 Ad 1) | |
| A reaction: This helps to clarify why the procedure of 'ignoring' features is possible. It suggests that some features might be too entangled with the substance (too essential?) to be thus ignored. I can't think of an example, though. Why not?! |
| 10504 | Abstracting either treats something as separate, or thinks of it separately [Aquinas] |
| Full Idea: Abstracting takes place in two ways: by composition and division, understanding something to be not in another or to be separated from it; and by a simple and unconditioned consideration, understanding one thing while not considering the other at all. | |
| From: Thomas Aquinas (Summa Theologicae [1265], Q85 1 Ad 1) | |
| A reaction: The second way is by 'ignoring', which he says cannot contain error. The first seems to be considering some mode of a thing to be actually separate from the thing, which could clearly be erroneous. Ignoring makes to commitment to a unity. |
| 10507 | Numbers and shapes are abstracted by ignoring their sensible qualities [Aquinas] |
| Full Idea: Quantities such as numbers and dimensions, and also shapes (which are the limits of quantities) can be considered without their sensible qualities, which is for them to be abstracted from sensible matter. | |
| From: Thomas Aquinas (Summa Theologicae [1265], Q85 Ad2) | |
| A reaction: His account relies on underlying substance, which is where quantity is to be found (presumably because a substance is the epitome of a unit). |
| 13186 | Universals are just abstractions by concealing some of the circumstances [Leibniz] |
| Full Idea: In forming universals the soul only abstracts certain circumstances by concealing innumerable others. ..A spherical body complete in all respects is nowhere in nature; the soul forms such a notion by concealing aberrations. | |
| From: Gottfried Leibniz (Letters to Burcher De Volder [1706], 1704 or 1705) | |
| A reaction: This is Leibniz's affirmation of traditional 'abstraction by ignoring', which everyone seems to have believed in before Frege, and which I personally think is simply correct, even though it is deeply unfashionable and I keep it to myself. |
| 9189 | Dedekind said numbers were abstracted from systems of objects, leaving only their position [Dedekind, by Dummett] |
| Full Idea: By applying the operation of abstraction to a system of objects isomorphic to the natural numbers, Dedekind believed that we obtained the abstract system of natural numbers, each member having only properties consequent upon its position. | |
| From: report of Richard Dedekind (Nature and Meaning of Numbers [1888]) by Michael Dummett - The Philosophy of Mathematics | |
| A reaction: Dummett is scornful of the abstractionism. He cites Benacerraf as a modern non-abstractionist follower of Dedekind's view. There seems to be a suspicion of circularity in it. How many objects will you abstract from to get seven? |
| 9827 | We derive the natural numbers, by neglecting everything of a system except distinctness and order [Dedekind] |
| Full Idea: If in an infinite system, set in order, we neglect the special character of the elements, simply retaining their distinguishability and their order-relations to one another, then the elements are the natural numbers, created by the human mind. | |
| From: Richard Dedekind (Nature and Meaning of Numbers [1888], VI.73) | |
| A reaction: [compressed] This is the classic abstractionist view of the origin of number, but with the added feature that the order is first imposed, so that ordinals remain after the abstraction. This, of course, sounds a bit circular, as well as subjective. |
| 9890 | The modern account of real numbers detaches a ratio from its geometrical origins [Frege] |
| Full Idea: From geometry we retain the interpretation of a real number as a ratio of quantities or measurement-number; but in more recent times we detach it from geometrical quantities, and from all particular types of quantity. | |
| From: Gottlob Frege (Grundgesetze der Arithmetik 2 (Basic Laws) [1903], §159), quoted by Michael Dummett - Frege philosophy of mathematics | |
| A reaction: Dummett glosses the 'recent' version as by Cantor and Dedekind in 1872. This use of 'detach' seems to me startlingly like the sort of psychological abstractionism which Frege was so desperate to avoid. |
| 10803 | Frege himself abstracts away from tone and color [Yablo on Frege] |
| Full Idea: Frege himself abstracts away from tone and color. | |
| From: comment on Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884]) by Stephen Yablo - Carving Content at the Joints §3 | |
| A reaction: Gotcha! I have been searching for instances where Frege perpetrates psychological abstraction right in the heart of his theory. No one can avoid it, if they are in the business of trying to formulate new concepts. Reference ignores sense, and vice versa. |
| 9988 | If we abstract 'from' two cats, the units are not black or white, or cats [Tait on Frege] |
| Full Idea: When from a set of two cats, one black and one white, we 'abstract' the number two as a set of pure units, the units are not black and white, respectively, and they are not cats. | |
| From: comment on Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884], §34) by William W. Tait - Frege versus Cantor and Dedekind XI | |
| A reaction: Well said. Frege is contemptuous of this approach, as if we were incapable of thinking of a black cat as anything other than as black or cat, when we can catch cats as 'food', or 'objects', or just plain 'countables'. |
| 9579 | Disregarding properties of two cats still leaves different objects, but what is now the difference? [Frege] |
| Full Idea: If from a black cat and a white cat we disregard colour, then posture, then location, ..we finally derive something which is completely without restrictions on content; but what is derived from the objects does differ, although it is not easy to say how. | |
| From: Gottlob Frege (Review of Husserl's 'Phil of Arithmetic' [1894], p.324) | |
| A reaction: This is a key objection to abstractionism for Frege - we are counting two cats, not two substrata of essential catness, or whatever. But what makes a cat countable? (Key question!) It isn't its colour, or posture or location. |
| 9587 | How do you find the right level of inattention; you eliminate too many or too few characteristics [Frege] |
| Full Idea: Inattention is a very strong lye which must not be too concentrated, or it dissolves everything (such as the connection between the objects), but must not be too weak, to produce sufficient change. Personally I cannot find the proper dilution. | |
| From: Gottlob Frege (Review of Husserl's 'Phil of Arithmetic' [1894], p.330) | |
| A reaction: We may sympathise with the lack of precision here (frustrating for a logician), but it is not difficult to say of a baseball defence 'just concentrate on the relations, and ignore the individuals who implement it'. You retain basic baseball skills. |
| 8770 | 'Abstractionism' is acquiring a concept by picking out one experience amongst a group [Geach] |
| Full Idea: I call 'abstractionism' the doctrine that a concept is acquired by a process of singling out in attention some one feature given in direct experience - abstracting it - and ignoring the other features simultaneously given - abstracting from them. | |
| From: Peter Geach (Mental Acts: their content and their objects [1957], §6) | |
| A reaction: Locke seems to be the best known ancestor of this view, and Geach launches a vigorous attack against it. However, contemporary philosophers still refer to the process, and I think Geach should be crushed and this theory revived. |
| 8904 | The Way of Abstraction says an incomplete description of a concrete entity is the complete abstraction [Lewis] |
| Full Idea: The Way of Abstraction says abstract entities are abstractions from concrete entities; they result from somehow subtracting specificity, so that an incomplete description of the original concrete entity is a complete description of the abstraction. | |
| From: David Lewis (On the Plurality of Worlds [1986], 1.7) | |
| A reaction: Defined like this, it rather looks as if abstractions would be entirely verbal - which may well be the correct situation, except that higher animals seem capable of minimal levels of abstraction. This Way is denigrated by Frege and Geach. |
| 9567 | Maths deals with quantities of physical significance, ignoring irrelevant features [Geroch] |
| Full Idea: Mathematics can serve to provide a framework within which one deals only with quantities of physical significance, ignoring other, irrelevant things. | |
| From: Robert Geroch (Mathematical Physics [1985], p.1), quoted by Charles Chihara - A Structural Account of Mathematics 9.8 | |
| A reaction: This is a modern physicist espousing abstractionism, as derided and dismissed by Frege and Geach. It's common sense, really. |
| 8512 | Abstractions come before the mind by concentrating on a part of what is presented [Campbell,K] |
| Full Idea: An item is abstract if it is got before the mind by an act of abstraction, that is, by concentrating attention on some, but not all, of what is presented. | |
| From: Keith Campbell (The Metaphysic of Abstract Particulars [1981], §1) | |
| A reaction: I think this point is incredibly important. Pure Fregean semantics tries to leave out the psychological component, and yet all the problems in semantics concern various sorts of abstraction. Imagination is the focus of the whole operation. |
| 10217 | We can apprehend structures by focusing on or ignoring features of patterns [Shapiro] |
| Full Idea: One way to apprehend a particular structure is through a process of pattern recognition, or abstraction. One observes systems in a structure, and focuses attention on the relations among the objects - ignoring features irrelevant to their relations. | |
| From: Stewart Shapiro (Philosophy of Mathematics [1997], 3.1) | |
| A reaction: A lovely statement of the classic Aristotelian abstractionist approach of focusing-and-ignoring. But this is made in 1997, long after Frege and Geach ridiculed it. It just won't go away - not if you want a full and unified account of what is going on. |
| 9554 | We can focus on relations between objects (like baseballers), ignoring their other features [Shapiro] |
| Full Idea: One can observe a system and focus attention on the relations among the objects - ignoring those features of the objects not relevant to the system. For example, we can understand a baseball defense system by going to several games. | |
| From: Stewart Shapiro (Philosophy of Mathematics [1997], p.74), quoted by Charles Chihara - A Structural Account of Mathematics | |
| A reaction: This is Shapiro perpetrating precisely the wicked abstractionism which Frege and Geach claim is ridiculous. Frege objects that abstract concepts then become private, but baseball defences are discussed in national newspapers. |
| 9626 | A structure is an abstraction, focussing on relationships, and ignoring other features [Shapiro] |
| Full Idea: A structure is the abstract form of a system, focussing on the interrelationships among the objects, and ignoring any features of them that do not affect how they relate to other objects in the system. | |
| From: Stewart Shapiro (Structure and Ontology [1989], 146), quoted by James Robert Brown - Philosophy of Mathematics Ch.4 | |
| A reaction: I find this account very attractive, even though it appeals to supposedly outmoded psychological abstractionism. It seems pretty close to Aristotle's view of things. Shapiro's account must face up to Frege's worries about these matters. |
| 17316 | We can abstract to a dependent entity by blocking out features of its bearer [Koslicki] |
| Full Idea: In 'feature dependence', the ontologically dependent entity may be thought of as the result of a process of abstraction which takes the 'bearer' as its starting point and arrives at the abstracted entity by blocking out all the irrelevant features. | |
| From: Kathrin Koslicki (Varieties of Ontological Dependence [2012], 7.6) | |
| A reaction: She seems unaware that this is traditional abstraction, found in Aristotle, and a commonplace of thought until Frege got his evil hands on abstraction and stole it for other purposes. I'm a fan. |