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Single Idea 10280

[filed under theme 6. Mathematics / C. Sources of Mathematics / 4. Mathematical Empiricism / a. Mathematical empiricism ]

Full Idea

For each stone, there is at least one pattern such that the stone is a position in that pattern. The stone can be treated in terms of places-are-objects, or places-are-offices, to be filled with objects drawn from another ontology.

Clarification

His 'offices' are like offices of government, which can be held by varied persons

Gist of Idea

A stone is a position in some pattern, and can be viewed as an object, or as a location

Source

Stewart Shapiro (Philosophy of Mathematics [1997], 8.4)

Book Ref

Shapiro,Stewart: 'Philosophy of Mathematics:structure and ontology' [OUP 1997], p.259

A Reaction

I believe this is the story J.S. Mill had in mind. His view was that the structures move off into abstraction, but it is only at the empirical and physical level that we can possibly learn the structures.

The 36 ideas with the same theme [the view that mathematics is rooted in experience]:

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]
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]
9795 Numbers have generalised application to entities (such as bodies or sounds) [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]
9802 Numbers denote physical properties of physical phenomena [Mill]
9803 We can't easily distinguish 102 horses from 103, but we could arrange them to make it obvious [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]