George Pólya was a Martian.

… At least in the sense that he was a mathematician from Hungary. This may make some more sense in the context of this quote:

In an answer to the question of why there is no evidence of intelligent life beyond Earth despite the high probability of it existing, Szilárd responded:

They are already here among us – they just call themselves Hungarians.

In the first half of the 20th century there was a raft of brilliant people who emigrated from Hungary to the United States, including Bela Lugosi, John von Neumann, Halmos, Erdos, and Polanyi to name a few. And of course, George Pólya.

Pólya probably wasn’t the first to notice “The Inventor’s Paradox” but he was the first to put his finger on it, and give it a name. This is the phenomenon that, sometimes in mathematics, it is actually easier to prove a stronger result.

Let’s see an example. Suppose that you want to sum the numbers from 1 to 100. That is to say, for some reason (and in fact there are good reasons, but we won’t name them right now) you want to compute the very long sum 1+2+3+…+100. There is an obvious solution. It is: get to work. Ok, 1+2 is 3, then 3+3 is 6, then …

Well, that’s enough of that. Mathematicians are famously lazy, in the sense that we do not want to spend too long doing uninteresting things. And boy oh boy, that’s this. Let’s find another way.

Let’s reset our focus not on finding the sum 1+2+…+100, but instead let’s try to find the sum 1+2+…+n for any natural number n. This is a harder problem! It’s a stronger result! It seems like, whatever logic we might apply to 1+2+…+n, we ought instead to just directly apply it to 1+2+…+100, and perhaps the more specific problem which ends at 100 could be a simpler logic since we might imagine that there’s a way to exploit the knowledge of where the sum ends.

But in fact, there are many good ways to prove that 1+2+…+n has a certain formula, and in some sense it is actually easier to see it with a general n. One way to get started is just to write out some of the results and see if you can detect a pattern. For instance

1+2 = 3

1+2+3 = 6

1+2+3+4 = 10

1+2+3+4+5 = 15

See the pattern? Well, you couldn’t be blamed if you don’t—it’s not easy to see. One thing that is noticeable though, is that whatever the pattern is, it’s not linear. That is to say, you can’t just add a fixed number over and over to get the sequence 3, 6, 10, 15. I mean, well, duh. If the sequence is built by adding a growing number, then obviously you couldn’t also get it by adding a constant number. But it’s at least worth logging in our brains that the solution cannot look like an+b for any constant values a and c.

But let’s not entirely give up on just noticing a pattern. Maybe a good picture of the pattern could help. What about this?

The shape might be evocative! It looks like …

… yeah a pixelated triangle, but that’s not super helpful because we don’t really know useful things about pixelated triangles …

It’s “like” a square. What might we do with that observation? Maybe close off the negative space and regard the quantity we want as a sub-region of a 3x3 square? Well, if we did then we couldn’t say the yellow sub-region is half of the square. It looks close, but it’s not quite.

… Ok, what if we didn’t make it a sub-region of a square? What if we do something close, like make a copy of the yellow region, and make the copy “interlock” with the original?

Now by construction the region we’re interested in is half the total region, making a much easier analysis. What is the area of the total region? A rectangle is so easy to find its area. Width times height.

The width is obviously 3, and the height is pretty clearly 4. Or more generally the width will be n if we sum the first n numbers. The height will more generally be n+1. The total area is n(n+1). Therefore the answer to our question is, finally, \(\frac{n(n+1)}2\).

We can then use this general formula to find the sum 1+2+…+100. It is, according to the formula above, \(\frac{100(100+1)}{2} = 5050\).

Returning to the main point, it was in fact easier to find this general formula than it was to go directly for the specific problem that we started with. But why?

It is a paradox, and the point of a paradox is that it is counter-intuitive, but presumably has some kind of resolution. My suggestion is that the abstract helps you to focus on precisely the features of the problem that are actually important. When summing 1 through 100, all the details of all the various terms shouldn’t be important. Rather what should be important is the way in which the next term adds onto the previous sum, in a sort of homogeneous way. The fact of this homogeneity is what allowed us to know that the yellow region of the rectangle filled precisely half the rectangle. If the quantity fluctuated in some more complicated way, we might not have been able to use this solution method.

And I think more generally, this is why it is sometimes easier to prove a stronger result. By removing some details from a problem, you seem like you are losing tools to solve the problem. But if you remove the right details then you are still able to solve the problem, and as an added benefit you can use this result to solve other problems which do not share those details.

In some sense that may be the most important task in all of mathematics. For any given problem, it is a challenge to know which details are the right details to remove.

Polya's Inventor's Paradox.