Communicating the Continuum Hypothesis

I have a friend who is shall we say, pessimistic, about science communication. He thinks it’s too much risk for too little gain, too many misunderstandings while the most important stuff is so abstract the public will never understand it anyway. When I asked him for an example, he started telling me about a professor who works on the continuum hypothesis.

The continuum hypothesis is about different types of infinity. You might have thought there was only one type of infinity, but in the nineteenth century the mathematician Georg Cantor showed there were more, the most familiar of which are countable and uncountable. If you have a countably infinite number of things, then you can “count” them, “one, two, three…”, assigning a number to each one (even if, since they’re still infinite, you never actually finish). To imagine something uncountably infinite, think of a continuum, like distance on a meter stick, where you can always look at smaller and smaller distances. Cantor proved, using various ingenious arguments, that these two types of infinity are different: the continuum is “bigger” than a mere countable infinity.

Cantor wondered if there could be something in between, a type of infinity bigger than countable and smaller than uncountable. His hypothesis (now called the continuum hypothesis) was that there wasn’t: he thought there was no type of infinite between countable and uncountable.

(If you think you have an easy counterexample, you’re wrong. In particular, fractions are countable.)

Kurt Gödel didn’t prove the continuum hypothesis, but in 1940 he showed that at least it couldn’t be disproved, which you’d think would be good enough. In 1964, though, another mathematician named Paul Cohen showed that the continuum hypothesis also can’t be proved, at least with mathematicians’ usual axioms.

In science, if something can’t be proved or disproved, then we shrug our shoulders and say we don’t know. Math is different. In math, we choose the axioms. All we have to do is make sure they’re consistent.

What Cohen and Gödel really showed is that mathematics is consistent either way: if the continuum hypothesis is true or false, the rest of mathematics still works just as well. You can add it as an extra axiom, and add-on that gives you different types of infinity but doesn’t change everyday arithmetic.

You might think that this, finally, would be the end of the story. Instead, it was the beginning of a lively debate that continues to this day. It’s a debate that touches on what mathematics is for, whether infinity is merely a concept or something out there in the world, whether some axioms are right or wrong and what happens when you change them. It involves attempts to codify intuition, arguments about which rules “make sense” that blur the boundary between philosophy and mathematics. It also involves the professor my friend mentioned, W. H. Woodin.

Now, can I explain Woodin’s research to you?

No. I don’t understand it myself, it’s far more abstract and weird than any mathematics I’ve ever touched.

Despite that, I can tell you something about it. I can tell you about the quest he’s on, its history and its relevance, what is and is not at stake. I can get you excited, for the same reasons that I’m excited, I can show you it’s important for the same reasons I think it’s important. I can give you the “flavor” of the topic, and broaden your view of the world you live in, one containing a hundred-year conversation about the nature of infinity.

My friend is right that the public will never understand everything. I’ll never understand everything either. But what we can do, what I strive to do, is to appreciate this wide weird world in all its glory. That, more than anything, is why I communicate science.

5 thoughts on “Communicating the Continuum Hypothesis

  1. Ray Elberson

    Great response. Couldn’t agree more. There’s a lot of what you say about amplitudes that I can’t understand. But at least I know it’s there. I know it has a function. I know a lot of people work on it. I know its evolving. I may not understand it, but I can still appreciate it.

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  2. itaibn

    “[I]t’s far more abstract and weird than any mathematics I’ve ever touched.”

    I think that’s not quite right, or at least that you’re not justified in saying this. After all, quantum field theory and amplitudes research is also very weird and abstract. Rather, what’s going on is that Woodin is studying stuff that is so utterly different from what you study that most of the effort you spent understanding and getting used to the stuff you study won’t help you make sense of the stuff Woodin is studying.

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    1. 4gravitons Post author

      I agree that “weird” is subjective in that way, that it’s really just a measure of familiarity.

      I do think Woodin’s work is meaningfully more “abstract” than what I do, though. I compute functions and numbers. These are abstractions, but I think it’s reasonable to call them “less abstract” than universes of axioms and large cardinals. They’re closer to everyday experience, even if admittedly they’re not that close.

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  3. Oss Ickle

    Ray Elberson speaks for me, word for word. I just shares this post on FB, commenting:

    “I don’t often link to this physics blog, though the writer is very good-natured and clearly well-intentioned. He seems like a really good guy. However, his posts are often too technical for me to get anything out of them or, for that reason, sharing them.

    This short post is an exception, and I encourage you to read it. He’s posting about something he doesn’t understand, a very very very airy area of mathematics research. The post is about why he thinks its worth him at least telling us about the research. I like the post on its own merits (i.e., if I’d never read anything by him before), but I also really like that it’s such an expression of his personality. Like I said, he seems like a really good guy.”

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