#supremum

Ever since I took too many mathematics classes, I started using the concept of "upper bound" literally. It confuses people.

Girlfriend: How long do you think it'll take you to work out?

Me: Oh, I don't know. I'd say less than five hours.

Girlfriend: Five hours?! What are you planning to do there?

Me: I didn't say it would take five hours. I said it would take less than five hours.

Well, it did take me less than five hours. It took me an hour and a half.

If you eat one carrot a day for a entire year, by the end you will have eat over 200 carrots !

— Carrot Facts (@RealCarrotFacts) July 1, 2013

  Same problem with confidence intervals -- I give these very literal answers. My former boss told me a story about a "rationality test" she was given, by a statistician or something. First she was asked to guess some fact that only a 5th grader would know, like how many tons the moon weighs or what's the square mileage of Antarctica. Then the statistician asked her to give 90% confidence bounds. That's, you're ≥90% sure that the value is between these two numbers. Most people fail by saying numbers close to their original guess.

My boss just said, "I'm confident that the number is somewhere between zero and a trillion trillion trillion trillion trillion trillion trillion ... trillion." Well she was correct! And she was one of the only ones.

You can make confidence bounds as wide as you want and be logically correct. People will look at you in bewilderment when you say things like "I'll be gone somewhere between two seconds and seven days," but you will not be a liar.

  Mathematicians are the only people who go around making statements like "I found out that the answer is greater than 6 and less than 3→3→64→2. I'm pretty sure the answer is 13, though."

(3→3→64→2 is bigger than billions of universes.)

It's not even the difference between certainty-of-proof and casual guesstimation. It's the difference between giving an upper bound and giving the least upper bound (supremum).

I can say with complete confidence that I will never earn over 10^18737 pounds in my life, no matter how much hyperinflation or life extending medicines lie in the future. I can say the same about 10^18736 pounds. How low am I willing to go with these statements? Ay, there's the lub.

  No less of an intellect than Paul Graham swaps upper and lower bounds. In describing how he's designing a new programming language (arc) with the goal that useful programs should be as short as possible in it, he writes:

That's part of why I focus on code size. Length is an external constraint. If you start looking at code thinking "what is the lower bound on how long this has to be?" you're one step from discovering the new operator that will make it that short.

I might be the only person who reads this and is confused. When I hear "lower bound" I think "Nothing is lower than the lower bound. It has to be bigger than the lower bound." But then he is talking like putting a lower bound on the code means the code is shorter. Zoinks?

Let us continue our discussion about Riemann Integration!

Now at this point, we have a notion of what our upper and lower sums looks like and what they tell us about a function, but how do we know that these upper and lower sums even tell us anything accurate about our function? Well, we can utilize a very powerful result that comes from a long list of inequalities. Now, you can prove this but for the sake of time, I'm just going to show it. Given a function f and an arbitrary partition, P we have that

Essentially, what this tells us is that if our upper and lower sums approach each other, we can be sure the the Riemann Integral of our function exists. This is a very powerful result in that now we can prove that a function is Riemann Integrable. In other words a function is Riemann Integrable iff:

What a useful result. Now, given any function f, we can show it is Riemann Integrable so long as we can cleverly choose a Partition so that the above inequality is true.