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Single Idea 9804
[filed under theme 6. Mathematics / C. Sources of Mathematics / 4. Mathematical Empiricism / a. Mathematical empiricism
]
Full Idea
Every statement of the result of an arithmetical operation is a statement of one of the modes of formation of a given number.
Gist of Idea
Arithmetical results give a mode of formation of a given number
Source
John Stuart Mill (System of Logic [1843], 3.24.5)
Book Ref
Mill,John Stuart: 'System of Logic (9th ed, 2 vols)' [Longmans, Green etc 1875], p.151
A Reaction
Although Mill sticks cautiously to very simple arithmetic, inviting empirical accounts of much higher mathematics, I think the phrase 'modes of formation' of numbers is very helpful. It could take us either into structuralism, or into constructivism.
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]
|