Thursday, June 23, 2022

Certainty and uncertainty

Michael - Thursday

When I was at high school and later at university, I was attracted to science and mathematics because of the beauty of the theories, their elegant structures, and their certainty. I was very naïve.

As I learned more, I began to understand that science is about theories that have to be tested, and possibly rejected, on the basis of experimental evidence. Crudely put, that’s the scientific method. A theory is as good as the evidence that doesn’t contradict it, and preferably supports it. Newton tested his theories of mechanics and gravity against astronomical observations. They held up. Tick. Move on. Until Einstein comes along with a better theory. It doesn’t contradict Newton’s theory, it just says that it is a really good approximation under the sort of conditions we deal with every day – when things are beetling along well below the speed of light. So far Einstein’s theories don’t contradict any evidence – in fact experiments and observations unavailable in his time support them – black holes, for example. Are his theories Truth? Almost certainly not. It’s the current theory.

To make it more confusing, Heisenberg came up with his Uncertainty Principle which implies that position and velocity can't both be measured exactly at the same time. Hmm.

It seems that in the political sphere these days, it’s not necessary to test theories against observed evidence. If one does so, many just fall apart at once. I won’t even try to list them. Unfortunately that doesn’t seem to shake people’s belief in them at all.

I was in search of certainty. Religion offers that, but requires belief. I didn’t have that, so I turned to mathematics. Here at last, I felt, was an endeavour that had beautiful constructions, deep theories and implications, and certainty. Each step could be proved to follow the previous one. Nothing was guessed. Nothing had to be tested against evidence. The evidence had to fit or the evidence was wrong. I was very naïve.

The foundations of mathematics lie with logic and set theory. At the end of the nineteenth century, Georg Cantor – a very famous mathematician considered the father of modern set theory – published a book explaining the area and building the theory from obvious axioms. Before the book was published, he sent a copy to British philosopher and mathematician Bertrand Russell. Russell was a brilliant man and thought deeply about many things, so he thought about Cantor’s constructions.  Sets are nothing more than collections of things or numbers or anything you like. The set of all fruits. The set of all even numbers. Even the set of all sets. So that means a set can contain itself, since the set of all sets must contain itself. But most sets do not contain themselves. The set of fruits or the set of all even numbers, for example. Okay, thought Russell, what about the set of all sets that do not contain themselves? Think about that set. Does it contain itself? Well, if it does, then it doesn’t meet the definition, so it can not contain itself. That’s false then. So then it does not contain itself. But in that case, it is in itself by the definition. So neither of the two possible options can be true.

BR had a few logic issues also...

This is a bit like submitting your new murder mystery and your editor pointing out that early in the book the murderer was on stage with a group of people at the time of the murder, and worse that that scene is essential to the rest of the novel.

Known as Russell’s paradox, it collapsed the set theory of the day. Now there had to be things called classes and various work-arounds. And were those “obvious axioms” so obvious after all? Was it possible that they also led to a contradiction like Russell’s paradox? No one has ever been able to answer that question.

But worse was to come. To the original “obvious” axioms of set theory, a new one needed to be added called the Axiom of Choice. (If you are interested in what it is, and probably you are not, you can find out at It also seems obvious, but it isn’t. In fact it’s been proved that set theory works (or indeed doesn’t work) whether or not you include it. But almost all of mathematics turns out to require it. All those beautiful proofs. All those important implications. All that certainty.

Okay. I admit this one is for maths geeks.
There's a hint in the labels at the end...

I still love mathematics, and I now understand that it is a science like the others. The evidence supports all its implications. Nothing has ever implied that anything about it is wrong. Maybe it's based on faith, but it remains an incredibly powerful tool, and an elegant one.

So forget the certainty. We don’t find it in life and we don’t find it in science and we don’t find it in mathematics. Uncertainty is the nature of things. Live with it.



  1. A Squared + B Squared=C how Scared I am of today's math.

  2. I find those kinds of contradictions in math to point out that they're probably related to infinity, where EVERYTHING breaks down. Divide anything by smaller and smaller numbers, and the closer you get to zero, the result APPROACHES infinity. But you never really reach infinity, because when you divide by 0, the result is 'undefined'. One could think of it as infinite, or one could think of it as having an infinite number of results. Or one could think of it as proving the truth of the multi-worlds hypotheses. Or one could be talking out of their zero region...

    1. Absolutely spot on. Any finite system is well-behaved and can be easily managed. (It may take a while. There are a lot of atoms in the universe.) On the other hand, anything infinite needs axiomatic foundations. And there's the rub...

  3. Certainties....death, tax, dog hair on good trousers.