David

Consider the following two statements:

1. If the second statement is true, then all numbers are even.

2. This statement is true.

why does this not prove all numbers are even?

Theorem:

Everyone has the same name.

Proof (by induction on the number of people in a group):

for n=1, a group of only one person, they obviously have the same name as everyone else in the group.

now suppose that the theorem is true for all groups of people of size k < n.

suppose we take any one person out of the group. then, by our induction hypothesis, everyone left has the same name (since we now have n-1 < n people). now add the person we took out, and take out another person. since we still have a group of n-1 < n people, they all have the same name, which shows that the person we excluded originally has the same name as all the others. thus all n people have the same name, from which we conclude that any group of of any number of people must all have the same name. therefore, everyone has the same name.

see if you can figure out what is wrong with this proof.

Time to step up and say what you believe. Is mathematics just a game we play we symbols, or does it describe something that actually exists, independent of what we humans choose to believe?

Elaborations of Platonists and fictionalists welcomed. Will give full marks for best answer.