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Single Idea 9643
[filed under theme 6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / b. Mathematics is not set theory
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Full Idea
Maybe all of mathematics can be represented in set theory, but we should not think that mathematics is set theory. Functions can be represented as order pairs, but perhaps that is not what functions really are.
Gist of Idea
Set theory may represent all of mathematics, without actually being mathematics
Source
James Robert Brown (Philosophy of Mathematics [1999], Ch. 7)
Book Ref
Brown,James Robert: 'Philosophy of Mathematics' [Routledge 2002], p.102
A Reaction
This seems to me to be the correct view of the situation. If 2 is represented as {φ,{φ}}, why is that asymmetrical? The first digit seems to be the senior and original partner, but how could the digits of 2 differ from one another?
The
33 ideas
from James Robert Brown
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9605
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If a proposition is false, then its negation is true
[Brown,JR]
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9604
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Mathematics is the only place where we are sure we are right
[Brown,JR]
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9606
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The irrationality of root-2 was achieved by intellect, not experience
[Brown,JR]
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9612
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There is an infinity of mathematical objects, so they can't be physical
[Brown,JR]
|
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9610
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Numbers are not abstracted from particulars, because each number is a particular
[Brown,JR]
|
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9608
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There are no constructions for many highly desirable results in mathematics
[Brown,JR]
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9611
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'Abstract' nowadays means outside space and time, not concrete, not physical
[Brown,JR]
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9609
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The older sense of 'abstract' is where 'redness' or 'group' is abstracted from particulars
[Brown,JR]
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9613
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Naïve set theory assumed that there is a set for every condition
[Brown,JR]
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9615
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Nowadays conditions are only defined on existing sets
[Brown,JR]
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9617
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The 'iterative' view says sets start with the empty set and build up
[Brown,JR]
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9619
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David's 'Napoleon' is about something concrete and something abstract
[Brown,JR]
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9620
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Empiricists base numbers on objects, Platonists base them on properties
[Brown,JR]
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9625
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To see a structure in something, we must already have the idea of the structure
[Brown,JR]
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9628
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Sets seem basic to mathematics, but they don't suit structuralism
[Brown,JR]
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9622
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'There are two apples' can be expressed logically, with no mention of numbers
[Brown,JR]
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9621
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Mathematics represents the world through structurally similar models.
[Brown,JR]
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9630
|
The most brilliant formalist was Hilbert
[Brown,JR]
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9634
|
Set theory says that natural numbers are an actual infinity (to accommodate their powerset)
[Brown,JR]
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9638
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Berry's Paradox finds a contradiction in the naming of huge numbers
[Brown,JR]
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9629
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For nomalists there are no numbers, only numerals
[Brown,JR]
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9635
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Given atomism at one end, and a finite universe at the other, there are no physical infinities
[Brown,JR]
|
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9640
|
A term can have not only a sense and a reference, but also a 'computational role'
[Brown,JR]
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9639
|
Does some mathematics depend entirely on notation?
[Brown,JR]
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9641
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Definitions should be replaceable by primitives, and should not be creative
[Brown,JR]
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9642
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A flock of birds is not a set, because a set cannot go anywhere
[Brown,JR]
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9643
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Set theory may represent all of mathematics, without actually being mathematics
[Brown,JR]
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9644
|
When graphs are defined set-theoretically, that won't cover unlabelled graphs
[Brown,JR]
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9645
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Constructivists say p has no value, if the value depends on Goldbach's Conjecture
[Brown,JR]
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9646
|
There is no limit to how many ways something can be proved in mathematics
[Brown,JR]
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9647
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Computers played an essential role in proving the four-colour theorem of maps
[Brown,JR]
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9648
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π is a 'transcendental' number, because it is not the solution of an equation
[Brown,JR]
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9649
|
Axioms are either self-evident, or stipulations, or fallible attempts
[Brown,JR]
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