In the late 1800s George Cantor stunned the world by proving the existence of the unknown: he showed that there are uncountable numbers and even sets of numbers.
His work laid the foundation for a wide range of paradoxes that lie at the edge of human knowledge, like Godels incompleteness theorems, Russel’s paradox, and the halting problem. For mathematicians, it was the start of over a hundred years of work that continues to this day. For you and me, it serves as a reminder that there is more to the world than we take for granted.
At the heart of Cantor’s work is the “diagonal proof”, which is simple, elegant and surprisingly easy to understand. I’ll just brush up the Wikipedia article a bit and let it do the explaining:
The proof starts with a family of sets, for example these sequences of binary numbers:
Next, a new sequence s is constructed by choosing the 1st digit as complementary to the 1st digit of s1 (swapping 0s for 1s and vice versa), the 2nd digit as complementary to the 2nd digit of s2, and so on. For every n, the nth digit is made complementary to the nth digit of the set before it. This yields:
By construction, the new set s differs from every other sn, since their nth digits differ (highlighted above on the diagonal). Hence, set s cannot occur in the enumeration.
So by moving diagonally through the sets, we have generated a new set that isn’t in the original. And now when we add it to the original, we will usher into existence another new constructive set that did not exist! This will go on and on forever. In other words, the set sn is uncountable.
The diagonal proof is a simple way of revealing the existence of something underneath the obvious. It is a reminder for us that the answers we are looking for are not always forthcoming, and that we may be asking the wrong questions.
As we go about our daily lives building software products and delivering value to customers, we need to think diagonally. A famous quote by Steve Jobs illuminates this sentiment well with an apocryphal quote from Henry Ford:
“Some people say, “Give the customers what they want.” But that’s not my approach. Our job is to figure out what they’re going to want before they do. I think Henry Ford once said, “If I’d asked customers what they wanted, they would have told me, ‘A faster horse!'” People don’t know what they want until you show it to them.”Steve Jobs
This captures that idea of “something out there” that hasn’t been found yet, and won’t be found by merely counting what already exists. And this awakening to the unknown is a key part of diagonal thinking.
But there is another part that is captured well by a quote from Henry Ford, this time a real one:
“If there is any one secret of success, it lies in the ability to get the other person’s point of view and see things from that person’s angle as well as your own.”Henry Ford
This captures the idea that it is worth taking the time to look at those unrevealed sets. By moving diagonally through perspectives, we can generate new ones that don’t exist yet. Steve Jobs may disagree, but there is value in doing this for many different reasons, e.g. to develop empathy and to create diversity.
In both cases, diagonal thinking means we do not merely accept our own point of view. We use our creative energy to build a vision and look beyond: to find the questions no one knew to ask, to uncover a new answers previously unseen, to go after the unanswerable.
“Trying to find the solution, we may repeatedly change our point of view, our way of looking at the problem. We have to shift our position again and again.”How To Solve It, Polya
Whether you are trying to figure out what customers want, or trying to figure out innovative ways to build it, it helps to remember that there is always another perspective on your problem waiting to be uncovered, and possibilities that remain undiscovered.
Are you counting what’s already there? or are you looking for the uncountable?
A visual representation of algebraic numbers on the complex plane coloured by polynomial degree. Points become smaller as the integer polynomial coefficients become larger. (source)