display all the ideas for this combination of texts
8 ideas
| 8764 | Categories are the best foundation for mathematics [Shapiro] |
| Full Idea: There is a dedicated contingent who hold that the category of 'categories' is the proper foundation for mathematics. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 10.3 n7) | |
| A reaction: He cites Lawvere (1966) and McLarty (1993), the latter presenting the view as a form of structuralism. I would say that the concept of a category will need further explication, and probably reduce to either sets or relations or properties. |
| 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) |
| 8762 | Two definitions of 3 in terms of sets disagree over whether 1 is a member of 3 [Shapiro] |
| Full Idea: Zermelo said that for each number n, its successor is the singleton of n, so 3 is {{{null}}}, and 1 is not a member of 3. Von Neumann said each number n is the set of numbers less than n, so 3 is {null,{null},{null,{null}}}, and 1 is a member of 3. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 10.2) | |
| A reaction: See Idea 645 - Zermelo could save Plato from the criticisms of Aristotle! These two accounts are cited by opponents of the set-theoretical account of numbers, because it seems impossible to arbitrate between them. |
| 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'). |
| 8760 | Numbers do not exist independently; the essence of a number is its relations to other numbers [Shapiro] |
| Full Idea: The structuralist vigorously rejects any sort of ontological independence among the natural numbers; the essence of a natural number is its relations to other natural numbers. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 10.1) | |
| A reaction: This seems to place the emphasis on ordinals (what order?) rather than on cardinality (how many?). I am strongly inclined to think that this is the correct view, though you can't really have relations if there is nothing to relate. |
| 8761 | A 'system' is related objects; a 'pattern' or 'structure' abstracts the pure relations from them [Shapiro] |
| Full Idea: A 'system' is a collection of objects with certain relations among them; a 'pattern' or 'structure' is the abstract form of a system, highlighting the interrelationships and ignoring any features they do not affect how they relate to other objects. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 10.1) | |
| A reaction: Note that 'ignoring' features is a psychological account of abstraction, which (thanks to Frege and Geach) is supposed to be taboo - but which I suspect is actually indispensable in any proper account of thought and concepts. |