Math can be confusing. Everyone knows that. And, actually, that’s what lots of people love about it. Some things in math are more confusing than others. One such thing, in my opinion, is a theorem developed by this kinda creepy-looking guy:

His name is Kurt Gödel, and he’s responsible for a theorem that basically says: You know how you thought we had rules for arithmetic that work, don’t contradict each other, and can answer all kinds of questions with numbers? Well, there are problems with numbers (really strange problems, granted) that our arithmetic cannot answer. And if you try to fix your system so that it can answer those problems, you’ll have issues with other problems. There’s no way to repair your system so that it stays complete and answers all problems.

If this sounds disturbing to you (math doesn’t work?!?!), you’re not alone. Lots of mathematicians were upset by this. They thought, as lots of us do, that math is supposed to be logical. It’s supposed to give us the answers we need. We’re supposed to be able to rely on it. Gödel arrived at this theorem by playing with paradoxes, or statements that self-contradict. (Such as, “Today is opposite day.”) The statement that he came up with really rocked the world of math.

If you’d like to learn more about Gödel and his disturbing theorem, listen to this podcast episode from Radiolab. It talks about Gödel’s life and what his theorem meant for math, with an appearance by everyone’s favorite mathematician, Steve Strogatz!

Gödel’s confusing theorem is only one in a long string of crazy, confusing math paradoxes. Another of my favorites is the Barber Paradox, which mathematician Bertrand Russell came up with. Here it is, in dry-humor video form:

If you like that paradox, you’ll probably also like the Pinocchio Paradox— which was developed by 11-year-old Veronique Eldridge-Smith:

This video comes from the YouTube channel, SpikedMathGames. I suggest you check it out!

Finally, I thought it would be nice to close off this loopy Math Munch post with a loop back to podcasts– and a link to a very large archive of math podcasts called Math Factor. Math Factor is a podcast produced out of the University of Arkansas about all kinds of interesting math. They even have an episode about the topic of this week’s Math Munch! Give it a listen.

Both of these relate really well to what Gödel had discovered by helping explain what had happened when he found that not all equations can give us the answers that we’re looking for in math & that some statements contradict each other. I wonder if its possible to ‘bend the rules’ when trying to figure out paradoxes,like in the barbers case. I wanted to believe that he got his hair shaved elsewhere or that he couldn’t grow a beard, then I remembered that these two would go against the definition of being a barber. In paradoxes, you can’t bend the rules to find an answer and the same goes for math.

This is a very good video on the a paradox. I have somethings new I have learned from you: The paradox asks questions that are confusing to the listener. I had to watch the video more than once and even had someone explain a little part of it to me. One question I have is can you make a paradox out of any ordinary thing? Hope you get back to me.

Pingback: Gödel, Other Crazy Paradoxes, and Math Factor — Math Munch | Mathpresso

Both of these relate really well to what Gödel had discovered by helping explain what had happened when he found that not all equations can give us the answers that we’re looking for in math & that some statements contradict each other. I wonder if its possible to ‘bend the rules’ when trying to figure out paradoxes,like in the barbers case. I wanted to believe that he got his hair shaved elsewhere or that he couldn’t grow a beard, then I remembered that these two would go against the definition of being a barber. In paradoxes, you can’t bend the rules to find an answer and the same goes for math.

I thought that math should be absolute, so can u explain that sometimes it works and sometimes it doesnt?

It’s amazing that definitions can cancel themselves out and make itself invalid.

This is a very good video on the a paradox. I have somethings new I have learned from you: The paradox asks questions that are confusing to the listener. I had to watch the video more than once and even had someone explain a little part of it to me. One question I have is can you make a paradox out of any ordinary thing? Hope you get back to me.