more on this theme
|
more from this text
Single Idea 9647
[filed under theme 6. Mathematics / B. Foundations for Mathematics / 2. Proof in Mathematics
]
Full Idea
The celebrity of the famous proof in 1976 of the four-colour theorem of maps is that a computer played an essential role in the proof.
Clarification
What is the minimum number of colours needed for a good map?
Gist of Idea
Computers played an essential role in proving the four-colour theorem of maps
Source
James Robert Brown (Philosophy of Mathematics [1999], Ch.10)
Book Ref
Brown,James Robert: 'Philosophy of Mathematics' [Routledge 2002], p.154
A Reaction
The problem concerns the reliability of the computers, but then all the people who check a traditional proof might also be unreliable. Quis custodet custodies?
The
33 ideas
from James Robert Brown
|
9605
|
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]
|
|
9610
|
Numbers are not abstracted from particulars, because each number is a particular
[Brown,JR]
|
|
9612
|
There is an infinity of mathematical objects, so they can't be physical
[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]
|
|
9615
|
Nowadays conditions are only defined on existing sets
[Brown,JR]
|
|
9613
|
Naïve set theory assumed that there is a set for every condition
[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]
|
|
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]
|
|
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]
|
|
9634
|
Set theory says that natural numbers are an actual infinity (to accommodate their powerset)
[Brown,JR]
|
|
9638
|
Berry's Paradox finds a contradiction in the naming of huge numbers
[Brown,JR]
|
|
9630
|
The most brilliant formalist was Hilbert
[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]
|
|
9648
|
π is a 'transcendental' number, because it is not the solution of an equation
[Brown,JR]
|
|
9649
|
Axioms are either self-evident, or stipulations, or fallible attempts
[Brown,JR]
|