Christmas week seems like a good time to share this video of my talk on “Truth, Provability and the Fabric of the Universe” delivered in March, 2018 in Madison, Wisconsin. The venue was the Free Thought Festival sponsored by a student group that goes by the umbrella title of “Atheists, Humanists and Agnostics” (AHA for short).
The video is below (or will be soon if you’re patient; it might take a minute or two to load). (Edited to add: I believe I’ve fixed things so it loads quickly now; please let me know if there are any problems.)
Or you can either:
- click here for a larger display of the same video
- click here for a (far) higher quality video that might (or might not) take a bit longer to load.
Or:
A big hat tip to Lisa Talpey for cleaning up the video and making it possible to see both my face and the slides at the same time.
Wow! This really stretched my brain early on a Thursday morning.
Two things:
1. I think that guy at about the 15 minute point was wrong. 2 IS the sum of two prime numbers. Those prime numbers are 1 and 1.
2. I didn’t get that a stupid Hercules could defeat the hydra. Every time, he cuts a head, more exist. Isn’t that true ad infinitum?
Re previous comment: 1 is not a prime number. By definition, primes must be greater than 1.
David Henderson:
1) The usual convention in mathematics is that 1 does not count as a prime number. One reason for this is that we want to be able to say that every number is uniquely a product of primes, so that once we know 7429 = 17 x 19 x 23, we can be sure it’s not also equal to some other product of primes, like 13 x 571. If we counted 1 as a prime, we’d have to say that 7429 = 17 x 19 x 23 AND ALSO 7429 = 1 x 17 x 19 x 23, which would be annoying.
2) It does look at first like Hercules playing randomly can never win, and it’s very surprising that he can. But the key is that if you cut off a head just one level above the root, no new heads sprout (because that head has no “grandfather”, and you’re required to go back to the grandfather to see what new heads are going to grow). It’s not obvious that you can cut back your hydra to one with all non-grandfathered heads (and in fact it generally takes a kazillion steps to do this), but once you’ve managed it, you can proceed to cut off the remaining heads one by one.
Thanks, Steve, on #2.
On #1, now I’m really confused. The question you had on the screen was: “Is every even number the sum of two primes?” Your answer was “Dunno.” Someone in the audience said 2 is not. You agreed. And you and Martin above explained that 1 is not a prime.
So in your talk, you revised the answer to “We don’t know except for the number 2, which is not.”
But what about 4? It’s the sum of 1 and 3. 3 is a prime. 1 is not. So 4 is not the sum of two primes. So your answer to the question should be “We do know. The answer is no.”
David Henderson : 4 is equal to 2 plus 2. Notoriously so.
I’m surprised how easy it was to get around the incompleteness theorem. Is that generally the case, or specific to the Hydra problem?
Henri Hein: You can always “get around” the incompleteness theorem by adopting one extra axiom — namely, take as an axiom the theorem you’re trying to prove.
In the case of the hydra, one adopts a different axiom (consistency of the Peano axioms) which at first blush appears to have nothing to do with the hydra, so there’s some substantial mathematics involved in getting from the axiom to the desired theorem, and this is particularly interesting because a great many people (including me) believe that the new axiom is self-evidently true.
But the cheap way to prove the hydra theorem is to adopt the theorem itself as an axiom. That’s always available, though as Bertrand Russell would have said, it has all the advantages of theft over honest labor.
Henri Hein: To follow up my earlier reply:
Of course, in the most important sense, we have NOT gotten around the incompleteness theorem. We have shown that, by adopting the right axiom, we can prove ONE true theorem that we couldn’t prove before. But this stragtegem will never allow us to prove ALL true theorems, which is what Godel precludes and which we certainly can NOT get around.
You might think we can get around Godel by adopting all true statements as axioms. That would certainly allow us to prove all true theorems. But we can’t do that, because the rules of the game say that if you’re going to adopt a bunch of new axioms, you have to be able to specify a procedure by which any idiot can tell what the axioms are.
If I say “Hercules always beats the hydra” is a new axiom, that qualifies, because any idiot can look at a sentence and see whether or not it is identical to the sentence “Hercules always beats the hydra”.
But if I say “all true statements are axioms”, then even a non-idiot will be unable to determine whether, for example, “Every even number greater than 2 is a sum of two primes” is an axiom.
So: Yes, you can always “get around” Godel in the sense of adopting one new axiom that lets you prove the theorem you want. The boring way to do that is to take the theorem itself as an axiom. The more interesting way to do it is to take some statement everyone (or almost everyone) believes as an axiom, and then do some hard work to get to your theorem.
But no, you can never “get around” Godel in the sense that no matter how many axioms you add — as long as they follow they “any idiot” rule — you’ll only be able to prove SOME additional true statements, never ALL of them.
Thank you, and apologies if “getting around” sounded flippant. Gödel’s theorem is one of my favorites in Mathematics. There is something mindbogglingly profound about it. I was a little relieved to hear that there was still a lengthy path from “consistency of Peano axioms” to proving the Hydra problem, as that indeed does not seem related.
Speaking of Goedel’s theorem (although way off topic), here’s my conjecture regarding Goedel’s reported discovery of a logical contradiction in the U. S. Constitution: https://www.ssrn.com/abstract=2010183
Steven, Oops. Thanks.
Steve,
You say that existence of an object means that properties of it have definite truth values, and you give “every unicorn has a blue stripe” as an example of one that doesn’t. I wonder about “every natural number has a blue stripe”. It also seems not to have a definite truth value. You could try to rule it out of order as being silly, but then we need to define “not silly” appropriately.
Dan
Daniel R. Grayson: It seems to me that “every natural number has a blue stripe” is objectively false, because having a blue stripe entails things like being visible, which in turn entails things like interacting with photons, which is something that natural numbers clearly do not do.
I will ponder whether there’s any more here for me to worry about, but my first-blush response is that this doesn’t seem like a problem.
I do very much agree that we should not be allowed to refuse to engage with meaningful questions just because they might “sound silly”.
Re: “Is every even number the sum of two primes?”
and: “David Henderson : 4 is equal to 2 plus 2. Notoriously so.”
Well, ahem, what does one mean by *two* primes? Do 2 and 2 count as two primes? Many would count 2 as a single prime.
Steve,
Re: “It seems to me that “every natural number has a blue stripe” is objectively false, because having a blue stripe entails things like being visible, which in turn entails things like interacting with photons, which is something that natural numbers clearly do not do.”
I would prefer to say “It seems to me that “every natural number has a blue stripe” is objectively *silly*, because having a blue stripe entails things like being visible, which in turn entails things like interacting with photons, which is something that natural numbers clearly do not do.”
Daniel R. Grayson: I’m pretty sure this comes down to the same issue as assigning a truth value to the statement “The present king of France is bald”. Bertrand Russell famously chose to interpret this sentence to mean “There exists an x such that x is the present king of France and x is bald”, making the statement false. But English is a sloppy enough language that one could write down alternative plausible interpretations that make the statement either true or meaningless. I’m not sure that anything important hinges on this choice (though of course Russell’s point — I’m pretty sure — was not that one choice is better than another, but that that there was a choice to be made).
Daniel R Grayson: Regarding the two 2’s, I’m pretty sure that many mothers can testify that “a set of two twins” is not the same thing as “a set of one twin”, even when those twins are identical.
Cosmologist Max Tegmark promotes this idea that mathematical structures are the same as physical ones.
https://en.wikipedia.org/wiki/Mathematical_universe_hypothesis
Can you please provide a citation, for the undecidability of the Hercules-Hydra problem? The original paper, or something equivalent?
Richard D: Kirby and Paris, “Accessible Independence Results for Peano Arithmetic”.