Why proof is important! (1 Viewer)

[seanieg89]

A lot of friends have asked me why the standard of rigour in mathematical proof is so high.

Why it is necessary to fiddle with things like epsilons and deltas?

Why do people waste their time looking for proofs of theorems that we have verified up until massive numbers using computers? Eg. Goldbach's conjecture is the statement that every even number greater than 2 can be written as the sum of two primes. This has been verified by computers for numbers up to 4×10^18 !

Here is a neat collection of trippy examples in mathematics of apparent patterns that break down when you get to large enough numbers.

http://math.stackexchange.com/questions/111440/examples-of-apparent-patterns-that-eventually-fail

http://www.math.sjsu.edu/~hsu/courses/126/Law-of-Small-Numbers.pdf

Feel free to post your own examples.



As a taste, one of my favourite examples is the following:

If you mark n points on a circle, and connect each pair of them by a line segment, what is the largest number of regions p(n) you can partition the circle into?

p(1)=1
p(2)=2
p(3)=4
p(4)=8
p(5)=16
p(6)=31!!!

 

[Carrotsticks]

Great post! Something I have also been asked often by fellow staff and students. I use the partition example you gave as well.

Other examples I usually give are polynomials such as . Or in general which will yield P(k)=k for the first n integers only.

P(1) = 1

P(2) = 2

P(3) = 3

P(4) = 4

P(5) = ???