# 2021-08-07 - ToKCast - David Deutsch answers a question about Gödel and undecidability. A question for David number 6 [Official](https://brettroberthall.podbean.com/e/ep-82-david-deutsch-answers-a-question-about-godel-and-undecidability-a-question-for-david-number-6/) [YouTube](https://www.youtube.com/watch?v=Km-mfQTy-4g) [Apple Podcasts](https://podcasts.apple.com/us/podcast/ep-82-david-deutsch-answers-a-question-about-g%C3%B6del/id1447087694?i=1000531313839) Duration: 00:12:15 ## Transcript ### Brett Hall 00:00:00 - 00:03:00 Welcome to ToKCast and my latest question for David. This is question number six in the series of questions that I asked David Deutsch during our conversation. This one is about inherently interesting versus uninteresting questions and the question of whether or not if certain things are undecidable, specifically in mathematics, that makes them inherently uninteresting. I'd kind of thought this was the case to some extent. I mean, of course mathematics in and of itself is intrinsically interesting. However, because of Gödel's results, we know that certain things cannot be proven to be true or proven to be false given mathematical systems. Now this idea of decidability or completeness and so forth, this is a part of mathematical logic. If you talk about basic logic, it's something called sentential logic or propositional logic. You can demonstrate, you can do a proof. In fact, this proof is one of the exercises that they make you go through if you do something like logic, mathematical logic, philosophical logic at university. They make you go through the process of demonstrating that propositional logic, for example, is both sound and complete. Sound means that everything that can be proved within the system is true. Complete means everything that is true within the system has a proof. This works for a bunch of different logics, but in particular the simplest of all logics, as I said, propositional logic or sentential logic. And Gödel himself, I think, showed the completeness of the next most complicated logic, which is called predicate logic, which also is both sound and complete. Now if you get to something slightly more complicated than that, which is a logic which includes something like simple arithmetic and you can use Peano's axioms of arithmetic, then you can show that although they're sound, anything that you prove within the system is true within the system, the opposite doesn't hold true. You can actually write down statements in the system for which you have no proof. In other words, you cannot decide, you cannot prove that the statement that you've written down is either true or false. It's a statement that is undecidable, that remains undecidable. This was Gödel's great insight and of course had significant consequences for the foundations of mathematics. And so part of my question to David today wasn't me acting like an interviewer, it was me genuinely asking a question about the interestingness, if that's a word, of these undecidable things. It's like, what's the point of these undecidable things anyway? Who really cares? Yes, it exists. It's interesting insofar as now we know that there can't be a single set of axioms from which you can prove all of mathematics for all time. And that means that mathematics is a creative exercise. Yes, that's great. ### Brett Hall 00:03:00 - 00:04:26 That's interesting. But if you went and found any particular one of these undecidable propositions, it doesn't have any effect in the physical world, does it? And I was laboring under a misconception there because David brings up Penrose using undecidable concepts within physics, within a certain kind of cosmology. And so that's really interesting. That's something I learned. Now, I didn't go and investigate it any further. I think it would be over my head. But it is interesting to think about the fact that there could be physics which has undecidable terms within it that relate directly to the physical world, in which case that would be very interesting because that could have consequences for technology and our understanding of physical reality. Anyway, without any further ado, let's get into the next question for David. And you have the great dichotomy of saying problems are inevitable, but problems are soluble. And here I have heard over the years people push back against this, especially the second part that problems are soluble. For example, they argue we can't know everything. And they will invoke that it's provably the case that given Gödel's result, there are true statements we cannot prove are true. So there is some and there are some mathematics that's not decidable. So some problems are not soluble. Isn't that correct? What's the argument against that line of reasoning? ### David Deutsch 00:04:26 - 00:07:08 I think that this line of reasoning is reaching for a standard of knowing that is impossible for anything. It's not possible to know even provable things by the standard that they're asking to know unprovable things. When we say that something is provable, we mean that it's provable from certain axioms. And Gödel's theorem is about provability, which means provability from axioms. Now, if you have an undecidable statement in some formal system, and if it was true, you could always add that as another axiom, and then it would be provable. So you might ask, yeah, but how do we know that that statement you're adding is true if we can't prove it? Well, unfortunately, that objection applies to any set of axioms, and it applies to the original set of axioms as well. And this is a mistake that has been made by many mathematicians, even after Gödel. I think it was made by virtually all mathematicians before Gödel. But even after, mathematicians tend to think that there is a class of axioms that don't need proof or something, that they are self-proving or self-evident, or they come from mathematical intuition, which is infallible or something like that. But in fact, the axioms, the rules of inference in logic, that, for example, if the proposition A and B is true, then it is necessarily the case that B is also true. Now, it seems self-evident, but it is not provable. So what are you going to do? Are you going to say that this is a problem that is insoluble and therefore it's not true that problems are soluble? No, people don't bother saying it with axioms because they have this misconception that axioms are somehow enshrined by God or something. We can know for sure that they are true. But that's not what knowledge is about. We are trying to solve problems. We're not trying to establish truths. And Gödel's theorem is not about establishing truths either. It's about deriving things from other things. Yes, yes. ### Brett Hall 00:07:08 - 00:07:57 And this is one line of discussion I often pursue here and correct me if I'm wrong, but this vast class of undecidable or unprovable statements, which must vastly outnumber the number that are provable statements, they're in a sense uninteresting because that's not what the claims of science, physics, for example, consist of. And so they don't have, and again, I would appeal to your better knowledge on this, they can't possibly have any effects on our world. They're not going to help us to solve problems, as you would say. We know they exist. But beyond that, are they interesting in any way? ### David Deutsch 00:07:57 - 00:10:33 Well, they're certainly interesting because mathematics is interesting. And we're interested in abstractions as well as in physical problems. And the reason that it seems implausible at the moment that undecidable propositions might be relevant to some practical problems like how to feed the world or how to eliminate viruses and so on is that the undecidable propositions are all about infinite sets. Infinite sets of things like the set of all numbers and that kind of thing. And physics, as we currently conceive of it, the infinite sets that appear in physics are very tame and don't have these problems in them. But that's like, you know, I can conceive that maybe one day somebody invents a different physical theory where the undecidability of certain propositions about the real numbers would tell you something about whether well about how to do and how not to do certain things that you want. I think Roger Penrose once pointed out that certain theories of quantum gravity do depend on Turing undecidable propositions. Basically, it's like you if you're working out a Feynman diagram to work out how what the probability is that a certain gravitational wave will interact with another gravitational wave, you know, this is very far from ordinary experience. Then if you want to work that out accurately, you have to do a sum, you have to sum the amplitudes for all possible processes over all geometries and for all possible topologies. And to enumerate all possible topologies in the relevant sense, I mean, I'm not a mathematician. I don't know about this stuff. But apparently that's an undecidable task. And therefore you couldn't work out this probability amplitude for the gravitational wave doing something or other, because you would have to have a mathematical expression for all topologies in a given sense. So that's an example of how in principle, laws of physics might have undecidable consequences. ### Brett Hall 00:10:33 - 00:10:39 Yes. And therefore that the undecidable consequences are actually interesting. ### David Deutsch 00:10:39 - 00:12:15 Well, it would be physically interesting. Yes. Yes, you know, in principle, something that you wanted to do might depend on working out this thing. But I don't see any particular reason for making a distinction as long as we have the necessities of life and comfort. I don't see any reason for making a distinction between problems in engineering and in physics and in pure mathematics. I mean, they're all problems. They're all conflicts between theories that interest people. People want to know what the resolution is. And one thing I say in the book about pure mathematical problems is that sometimes discovering that something is undecidable is the solution to the problem in the sense that it's the problem. And that explains why you could have these two conflicting intuitions about it. And then you can go and criticize your intuitions in other ways, because you know that you're not going to be able to resolve it by proving one of them true or false.