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Single Idea 9610
[filed under theme 6. Mathematics / C. Sources of Mathematics / 4. Mathematical Empiricism / c. Against mathematical empiricism
]
Full Idea
Numbers are not 'abstract' (in the old sense, of universals abstracted from particulars), since each of the integers is a unique individual, a particular, not a universal.
Gist of Idea
Numbers are not abstracted from particulars, because each number is a particular
Source
James Robert Brown (Philosophy of Mathematics [1999], Ch. 2)
Book Ref
Brown,James Robert: 'Philosophy of Mathematics' [Routledge 2002], p.12
A Reaction
An interesting observation which I have not seen directly stated before. Compare Idea 645. I suspect that numbers should be thought of as higher-order abstractions, which don't behave like normal universals (i.e. they're not distributed).
Related Ideas
Idea 645
If two is part of three then numbers aren't Forms, because they would all be intermingled [Aristotle]
Idea 8311
If 2 is a particular, then adding particulars to themselves does nothing, and 2+2=2 [Lowe]
The
33 ideas
from 'Philosophy of Mathematics'
9605
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If a proposition is false, then its negation is true
[Brown,JR]
|
9604
|
Mathematics is the only place where we are sure we are right
[Brown,JR]
|
9606
|
The irrationality of root-2 was achieved by intellect, not experience
[Brown,JR]
|
9612
|
There is an infinity of mathematical objects, so they can't be physical
[Brown,JR]
|
9610
|
Numbers are not abstracted from particulars, because each number is a particular
[Brown,JR]
|
9608
|
There are no constructions for many highly desirable results in mathematics
[Brown,JR]
|
9611
|
'Abstract' nowadays means outside space and time, not concrete, not physical
[Brown,JR]
|
9609
|
The older sense of 'abstract' is where 'redness' or 'group' is abstracted from particulars
[Brown,JR]
|
9613
|
Naïve set theory assumed that there is a set for every condition
[Brown,JR]
|
9615
|
Nowadays conditions are only defined on existing sets
[Brown,JR]
|
9617
|
The 'iterative' view says sets start with the empty set and build up
[Brown,JR]
|
9619
|
David's 'Napoleon' is about something concrete and something abstract
[Brown,JR]
|
9620
|
Empiricists base numbers on objects, Platonists base them on properties
[Brown,JR]
|
9625
|
To see a structure in something, we must already have the idea of the structure
[Brown,JR]
|
9628
|
Sets seem basic to mathematics, but they don't suit structuralism
[Brown,JR]
|
9622
|
'There are two apples' can be expressed logically, with no mention of numbers
[Brown,JR]
|
9621
|
Mathematics represents the world through structurally similar models.
[Brown,JR]
|
9630
|
The most brilliant formalist was Hilbert
[Brown,JR]
|
9638
|
Berry's Paradox finds a contradiction in the naming of huge numbers
[Brown,JR]
|
9634
|
Set theory says that natural numbers are an actual infinity (to accommodate their powerset)
[Brown,JR]
|
9629
|
For nomalists there are no numbers, only numerals
[Brown,JR]
|
9635
|
Given atomism at one end, and a finite universe at the other, there are no physical infinities
[Brown,JR]
|
9640
|
A term can have not only a sense and a reference, but also a 'computational role'
[Brown,JR]
|
9639
|
Does some mathematics depend entirely on notation?
[Brown,JR]
|
9643
|
Set theory may represent all of mathematics, without actually being mathematics
[Brown,JR]
|
9644
|
When graphs are defined set-theoretically, that won't cover unlabelled graphs
[Brown,JR]
|
9641
|
Definitions should be replaceable by primitives, and should not be creative
[Brown,JR]
|
9642
|
A flock of birds is not a set, because a set cannot go anywhere
[Brown,JR]
|
9645
|
Constructivists say p has no value, if the value depends on Goldbach's Conjecture
[Brown,JR]
|
9646
|
There is no limit to how many ways something can be proved in mathematics
[Brown,JR]
|
9647
|
Computers played an essential role in proving the four-colour theorem of maps
[Brown,JR]
|
9649
|
Axioms are either self-evident, or stipulations, or fallible attempts
[Brown,JR]
|
9648
|
π is a 'transcendental' number, because it is not the solution of an equation
[Brown,JR]
|