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6. Mathematics / C. Sources of Mathematics / 4. Mathematical Empiricism / a. Mathematical empiricism

[the view that mathematics is rooted in experience]

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