9974 | Ten sheep and ten dogs are the same numerically, but it is not the same ten [Aristotle] |
7782 | Every simple idea we ever have brings the idea of unity along with it [Locke] |
2197 | Reason assists experience in discovering laws, and in measuring their application [Hume] |
17617 | Maths is a priori, but without its relation to empirical objects it is meaningless [Kant] |
5201 | Mill says logic and maths is induction based on a very large number of instances [Mill, by Ayer] |
9360 | If two black and two white objects in practice produced five, what colour is the fifth one? [Lewis,CI on Mill] |
9888 | Mill mistakes particular applications as integral to arithmetic, instead of general patterns [Dummett on Mill] |
9795 | Numbers have generalised application to entities (such as bodies or sounds) [Mill] |
9794 | There are no such things as numbers in the abstract [Mill] |
9796 | Things possess the properties of numbers, as quantity, and as countable parts [Mill] |
9798 | Different parcels made from three pebbles produce different actual sensations [Mill] |
9797 | '2 pebbles and 1 pebble' and '3 pebbles' name the same aggregation, but different facts [Mill] |
9799 | 3=2+1 presupposes collections of objects ('Threes'), which may be divided thus [Mill] |
9803 | We can't easily distinguish 102 horses from 103, but we could arrange them to make it obvious [Mill] |
9802 | Numbers denote physical properties of physical phenomena [Mill] |
9804 | Arithmetical results give a mode of formation of a given number [Mill] |
9805 | 12 is the cube of 1728 means pebbles can be aggregated a certain way [Mill] |
8741 | Numbers must be of something; they don't exist as abstractions [Mill] |
8631 | Cantor says that maths originates only by abstraction from objects [Cantor, by Frege] |
17628 | Arithmetic was probably inferred from relationships between physical objects [Russell] |
10271 | Basic mathematics is related to abstract elements of our empirical ideas [Gödel] |
17738 | Quine blurs the difference between knowledge of arithmetic and of physics [Jenkins on Quine] |
9940 | Maybe mathematics is empirical in that we could try to change it [Putnam] |
9914 | It is unfashionable, but most mathematical intuitions come from nature [Putnam] |
4044 | Rat behaviour reveals a considerable ability to count [Goldman] |
12387 | Mathematical knowledge arises from basic perception [Kitcher] |
12412 | My constructivism is mathematics as an idealization of collecting and ordering objects [Kitcher] |
18065 | We derive limited mathematics from ordinary things, and erect powerful theories on their basis [Kitcher] |
18077 | The defenders of complex numbers had to show that they could be expressed in physical terms [Kitcher] |
13870 | We can't use empiricism to dismiss numbers, if numbers are our main evidence against empiricism [Wright,C] |
12209 | The indispensability argument shows that nature is non-numerical, not the denial of numbers [Fine,K] |
10280 | A stone is a position in some pattern, and can be viewed as an object, or as a location [Shapiro] |
17733 | We know mind-independent mathematical truths through sets, which rest on experience [Maddy, by Jenkins] |
17716 | Mathematics is relations between properties we abstract from experience [Mares] |
17719 | Arithmetic concepts are indispensable because they accurately map the world [Jenkins] |
17717 | Senses produce concepts that map the world, and arithmetic is known through these concepts [Jenkins] |