In the comments section of my last post, Jack in Danville writes:
I may have misunderstood [an offhand comment about the "irrelevance" of the Continuum Hypothesis] … Intuitively I’ve thought the Continuum Hypothesis describes an aspect of the real world.
I know we’ve touched on similar topics before, but something tells me many of you are hungerin’ for a metamathematical foodfight, and Jack’s perplexity seemed as good a pretext as any for starting a new thread.
So, Jack: this is a Deep Question, but let me try to summarize my view in a few paragraphs.
It’s easy to imagine a “physical process” whose outcome could depend on whether Goldbach’s Conjecture is true or false. (For example, a computer program that tests even numbers successively and halts if it finds one that’s not a sum of two primes.) Likewise for P versus NP, the Riemann Hypothesis, and even considerably more abstract questions.
But can you imagine a “physical process” whose outcome could depend on whether there’s a set larger than the set of integers but smaller than the set of real numbers? If so, what would it look like?
I submit that the key distinction is between
- questions that are ultimately about Turing machines and finite sets of integers (even if they’re not phrased that way), and
- questions that aren’t.
We need to assume that we have a “direct intuition” about integers and finite processes, which precedes formal reasoning — since without such an intuition, we couldn’t even do formal reasoning in the first place. By contrast, for me the great lesson of Gödel and Cohen’s independence results is that we don’t have a similar intuition about transfinite sets, even if we sometimes fool ourselves into thinking we do. Sure, we might say we’re talking about arbitrary subsets of real numbers, but on closer inspection, it turns out we’re just talking about consequences of the ZFC axioms, and those axioms will happily admit models with intermediate cardinalities and other models without them, the same way the axioms of group theory admit both abelian and non-abelian groups. (Incidentally, Gödel’s models of ZFC+CH and Cohen’s models of ZFC+not(CH) both involve only countably many elements, which makes the notion that they’re telling us about some external reality even harder to understand.)
Of course, everything I’ve said is consistent with the possibility that there’s a “truth” about CH floating in Platonic heaven, or even that a plausible axiom system other than ZFC could prove or disprove CH (which was Gödel’s hope). But the “truth” of CH is not going to have consequences for human beings or the physical universe independent of its provability, in the same way that the truth of P=NP could conceivably have consequences for us even if we weren’t able to prove or disprove it.
For mathematicians, this distinction between “CH-like questions” and “Goldbach/Riemann/Pvs.NP-like questions” is a cringingly obvious one, probably even too obvious to point out. But I’ve seen so many people argue about Platonism versus formalism as if this distinction didn’t exist — as if one can’t be a Platonist about integers but a formalist about transfinite sets — that I think it’s worth hammering home.
To summarize, Kronecker had it backwards. Man and Woman deal with the integers; all else is the province of God.