Proof by Contradiction

I’ve written elsewhere about why higher mathematics is essential for maintaining a free society. I’m going to extend that idea by talking about a method of proof used by many mathematicians:  proof by contradiction.

Here’s an analogy that I give to my students:  Suppose you know that the bus you take to get home does not cross a river.  One day, you get on a bus to go home, and after awhile, you look out the window and see that you’re crossing the river.  You can immediately conclude you got on the wrong bus.

In proof by contradiction, we make an assumption:  we get on a bus.  We then follow the consequences of that assumption, wherever they take us.  If we ever obtain a result that contradicts something we know to be true, then we know that our original assumption was wrong.

For example, here’s a proof by contradiction:  If we multiply two numbers and get an even number, then at least one of the numbers had to be even.

The art of proof involves picking the right assumption to start with; in general, students will spend years learning how to navigate bus schedules.  I’ll shortcut that by sending you to the bus “Two numbers are odd.”  In this case, the destination of the bus is “The product of the two numbers is even.”

Let’s ride this bus for awhile.  If both numbers are odd, then the product of the two numbers is odd.  But the bus is supposed to end up at the destination “The product of the two numbers is even.”  It is impossible to get there on this bus!

So…we shouldn’t have boarded the bus “Two numbers are odd.”  Instead, we had to take any of the other buses:  in this case, “At least one of the numbers is even.”

How does this apply to maintaining a free society?  A key skill in building a proof by contradiction is entertaining a hypothetical situation, then considering all possible consequences of this situation.  If you don’t like the consequences, you’ll do everything you can to avoid getting on the bus.

So consider any political issue you want:  gun control, LGBTQ rights, reproductive rights, global warming, etc.  Consider any possible solution you want.  Then follow the consequences, not just to the point that you’re satisfied with its results, but as far as you possibly can.  If your solution leads to undesirable consequences, you might reconsider boarding the bus.  Conversely, if no consequence of your solution is objectionable, then it may very well be the right solution.



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