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Single Idea 9802
[filed under theme 6. Mathematics / C. Sources of Mathematics / 4. Mathematical Empiricism / a. Mathematical empiricism
]
Full Idea
The fact asserted in the definition of a number is a physical fact. Each of the numbers two, three, four denotes physical phenomena, and connotes a physical property of those phenomena. Two denotes all pairs of things, and twelve all dozens.
Gist of Idea
Numbers denote physical properties of physical phenomena
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.150
A Reaction
The least plausible part of Mill's thesis. Is the fact that a pair of things is fewer than five things also a property? You see two boots, or you see a pair of boots, depending partly on you. Is pure two a visible property? Courage and an onion?
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]
|
9796
|
Things possess the properties of numbers, as quantity, and as countable parts
[Mill]
|
9794
|
There are no such things as numbers in the abstract
[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]
|
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]
|