Episode 95: Gödel on Math

0:00 Mark: Before we get started, we'd like to invite you to a special celebration. We will be doing a live in front of an audience taping of episode 100 on Plato's Symposium near Madison, Wisconsin on Sunday, July 20, 1 to 4pm to get details and reserve your spot, look for the announcement at partiallyexaminedlife.com seating is very limited. You're listening to the Partially Examined Life, a philosophy podcast by some guys who at one point set on doing philosophy for a living, but then thought better of it. Our question for episode 95 is something like Are mathematical entities real?

And why should philosophers bother to study them? And we read two articles by Kurt Godel, some Basic Theorems on the Foundations of Mathematics and their implications from 1951 and the modern Development of the foundation of Mathematics in the Light of Philosophy from 1961. You can join the discussion, get the texts and lots More information@PartiallyExaminedLife.com this is Mark Linsenmayer speaking to you from Madison, Wisconsin.

0:58 Wes: This is Wes Alwan in Boston, Massachusetts.

1:01 Dylan: This is Dylan Casey in Middleton, Wisconsin.

1:03 Adi: This is Adi Habu in New York, New York.

1:06 Mark: Welcome, Adi, welcome.

1:08 Dylan: Welcome.

1:08 Adi: Great to be here. Thanks so much.

1:10 Mark: So you are someone who studied math and philosophy as an undergrad, but then decided not to go into that, right?

1:15 Adi: That is true. Loved it. But it was time to do something else after undergrad.

1:19 Mark: So you're a law Prof. I saw.

1:22 Adi: I am actually not technically a lawyer anymore. I've joined a bank on the trading desk now, but I do still teach a little bit on the side.

1:30 Mark: So was your passion as an undergrad, was this philosophy of math stuff or. Among other things?

1:35 Adi: Among other things. But philosophy of math was and will continue to always be close to my heart.

1:41 Mark: Adi had, I think, contributed some money to us and revealed his background, and so we started hitting him up for ideas. So for the many times where I hit people up, oh, you should be a guest eventually. Sometimes it actually happens. Ground rules for our discussion include. Do you remember one? Adi, do you remember one of the ground rules? Have you listened to enough episodes?

2:01 Adi: Oh yeah. My favorite one is no name dropping.

2:05 Mark: Well, that's an obsolete one

2:09 Wes: because Mark made it obsolete through practice.

2:11 Dylan: Especially today.

2:13 Mark: The modern version of that is don't make arguments that hinge on something other than what we've agreed to read. Don't say you know what I was talking about. If only you'd made love to the soundtrack of the proof to Fermat's Last Theorem. One of the other ones is try not to assume that our audience has read what we're talking about or has any other background in philosophy or really in higher mathematics, we should say we're not going to just drop Goldbach like

2:39 Adi: statements, oh, wait, we're going to. How are we going to do that then? I mean, I didn't know that was a Goldbach.

2:43 Wes: Easy.

2:44 Dylan: That's easy to explain.

2:45 Mark: We're just going to explain what we're talking about, that's all. You can't just drop it and assume that people know what we're talking about. I know this is math. And so the regular listeners that don't just skip this one altogether might be a little apprehensive. They'll be looking for, how can I apply Godel's theorem to Buddhism? How can I apply it to my life?

3:06 Adi: Yes, I'd argue that incompleteness impacts everybody's life. That's just.

3:11 Mark: Well, all right, yeah, we'll have a nice debate about that because I brought in an additional source that I'm going to not talk about very much, but I'll just mention that I read it just to refer to it. Godel's Theorem, An Incomplete Guide to Its Use and Abuse by Torkel Franzen. I read about the first half of that and his conclusion was almost wholly negative, that there's nothing of wider philosophical interest that comes out of the theorem at all. The incompleteness theorem.

3:41 Dylan: How about we just talk about what Godel says?

3:44 Mark: Well, we can try, but Godel is actually very sketchy. He's very fragmentary in these two articles. He didn't write a lot of actual philosophy. He wrote a lot of technical papers, I guess, many of which had philosophical things sprinkled around in them. And the couple that we're reading now were not even published.

4:01 Wes: I thought he had some pretty clear arguments, though.

4:04 Adi: No, that's right.

4:05 Dylan: Yes. I think that we can say what he said and we can talk about how he seemed to understand himself and we could try to make some progress into evaluating that as best we can. And that is good enough.

4:16 Mark: Can we be rigorous and exact in all that we say? Unless doing otherwise would be potentially more entertaining. Can we do that?

4:22 Dylan: Yes, we can.

4:23 Wes: Perhaps you can get entertaining towards the end of the show. How about that?

4:29 Adi: So I will throw out this. I think something that does come out that is important here is that whatever the gross or philosophical import of the incompleteness theorems may be, their impact on just the history and philosophy of math itself, which is important to all of our lives, is pretty clear.

4:45 Mark: All right, so let's throw out right off the bat what? The incompleteness theorem. Actually there are two of them, but the conjunction of the two is just called the incompleteness theorem. But it's actually two separate theorems that were proved separately. So let's just explain what they are.

4:59 Adi: I've got two simplistic statements, I think the simplified forms here for the first incompleteness theorem. Again, this is an informal statement. Any system powerful enough to express basic arithmetic contains true statements that are not provable by that system.

5:15 Mark: Okay. And the way he proves that is through one of those paradox statements. Right. Which is not to say that this is the only kind of statement that is true in the system, that is not provable in the system, but at least one, the one that he actually shows in the proof is the statement, this statement is not provable.

5:32 Adi: Right, that's right.

5:34 Dylan: Yep. Right there. Just him playing around with that should immediately make people think about Russell.

5:41 Adi: The Russell arguments are what made this a problem for math.

5:44 Wes: He alludes to that sort of turn of the century crisis. Right. In set theory.

5:48 Adi: Right.

5:48 Mark: I will refer listeners back to. We actually have a past episode on Russell on Math. If you want to hear more in

5:54 Wes: detail about this and actually to even give more basic background when we talk about these systems, listeners can go back to our Frege episode as well. We're dealing with the idea that arithmetic is reducible to set theory or to some sort of explication in terms of logic.

6:11 Mark: Well, just what a system is in the first place. It's just that it is a bunch of statements. So you have to define what counts as a symbol in the language and what doesn't, which in the case of the ones we're talking about is just the numbers, the integers, it's actually only integers. And what he was concerned with here and then you've defined some basic axioms. So in case of Peano arithmetic, it was basic things like kind of the definition of zero, the definition of plus. He actually lists Peano's axioms in that Russell episode. You can go back to that or just look it up and then you've got some rules of reasoning by which you can get from the basic axioms to other provable statements which become the theorems. So any statement in the language should be true or false. Right. Presumably that's what you want if it's a consistent system.

6:59 Adi: Yeah, well noted that. So something being true or false in the system doesn't necessarily have to do with its consistency. You just assume that a system, you'd be able to prove anything one way or the other.

7:09 Dylan: Yeah. So that's one of the things is in this sort of nether region. Right. It's not exactly an axiom, but it's a presumed condition of reasoning.

7:20 Mark: If the basic axioms are inconsistent, then you can prove anything from them. That's just a basic law of logic that if you start with a contradiction, then there's just a standard proof that says you can derive something even completely irrelevant from it.

7:35 Dylan: That property is not an explicit axiom, but it is a rule of reasoning that you bring to the table to do it.

7:43 Wes: Yeah, it's not an axiom. What axioms you're going to choose, Right.

7:48 Mark: For a given system, you can attempt to prove that is inconsistent just by deriving a contradiction. Right. Systems are not just these naturally occurring things. You can take any old set of axioms and say this is my formal system and then you can start testing it. It's just that the ones that are actually mathematically interesting are the ones like Peano arithmetic that have very simple starting points or natural or seemingly true starting points.

8:12 Dylan: Yeah.

8:12 Wes: Godel, ultimately he's going to want to say those starting points are arbitrary, Right? It's going to have something to do with a mathematical reality. But. Well, he's accusing some of his opponents of thinking that the starting points are arbitrary and system is hypothetical and you simply postulate some axioms and then see what follows from it. I think we can give a simple definition of consistency. Right. For a system to be consistent is just to say we can't derive both a proposition and its negation and then for it to be complete, because he's going to be talking about completeness here too, is to say for any proposition that's relevant to the domain of objects we're talking about, we can prove either the proposition or its negation.

8:53 Unknown speaker: Right.

8:54 Mark: So every sentence in the system, if it's consistent, will be either true or false. Right. If it's not consistent, then you could derive both this sentence and its negation. So it doesn't make sense to say any given statement is true or false, but you should be able to, if it's complete, not only say whether it's true or false, but actually derive it. You might have to really work hard to figure out what the proof is, but derive it from the basic axioms. So every sentence becomes a theorem in the system. Right. Either a theorem or a basic axiom, of course.

9:23 Adi: Yeah, I think that sounds fair.

9:25 Mark: So the whole starting point of this then, the immediate impetus besides Russell was, as you mentioned on the Precog Adi, that David Hilbert was one of the folks trying to put mathematics on a firm ground and thought that you should be able to derive any truth of mathematics from the basic laws. Right. Mathematics was of course complete and consistent and we should be able to therefore to come up with proofs for all these things like Goldbach's conjecture, which is any even number above two is the sum of two primes. That was one that it seems to be true. You know, when you test individual cases, okay, those are the two primes. But to prove that that's going to be true for every single number going up, there's no obviously mechanical way to prove that. Correct. So this was one of the examples of one that's at least we're not sure that it's unprovable, but at least nobody has proven it yet.

10:16 Adi: Right? Right. And just to back up for a moment, I am not sure if Hilbert was as concerned about completeness. What he was really worried about was consistency. Incomplete system may be. Okay. A system in which the possibility of incompleteness was there is very scary. And of course divorcing those two is very, very hard. But his real concern was consistency.

10:40 Wes: Okay.

10:41 Dylan: There had been, in the history of math and Adi, you surely know this better than I do, problems during the 19th century that were essentially consistency problems in the ways in which different parts of number theory and the calculus were founded.

10:58 Adi: That's exactly right. And that's a lot of where Hilbert and Frege and Russell's big concerns came from. All these var inconsistencies started popping up in mathematics. And so to kind of attack this, they wanted to start axiomatizing math, put math on a solid foundation, and ultimately have proved that the axioms were consistent so that they no longer had to be worried about the fear that math might be inconsistent.

11:24 Dylan: Do you have a relatively simple example off the top of your head, Adi, of one of the problems in 19th century mathematics that was a consistency problem that was addressed?

11:36 Adi: It's the barbershop example. It's the Russell example, which I think you guys have discussed before. This is the a male barber shaves all and only men who don't shave themselves problem. That statement, if you start parsing that the barber cannot shave himself or the barber cannot be a barber, which makes it inconsistent. And that is kind of a colloquialized version of a problem that you can actually formalize in set theory and create a similar problem.

12:04 Mark: And we should say why set theory matters here, because for basic Peano arithmetic, it doesn't seem like the notion of a set comes in at all. But at Least if you follow Russell and listen to our episode on that, when he was trying to explain where we get numbers from, what's the logical foundation for numbers, he ended up using sets. So you have to do some sort of analysis of set theory. And this was a problem that came up with set theory, right?

12:29 Adi: Yeah. The idea of set theory was almost created so that it could be a more fundamental analysis of numbers. So the idea was that once you have set theory, you can kind of build up mathematics from set theory, because it's kind of a form of meta mathematics was the idea of set theory. And part of what the foundation of mathematics program was about was finding a place to build all of mathematics from. So presumably, if you had put down axioms from which you could derive set theory, everything can be derived from set theory. And so you can rebuild all of mathematics on solid foundations. The theory.

13:06 Wes: So the problem in set theory. Right. Is that the one you just described in terms of the. I guess the Barber's Paradox is the notion that. And this I call Russell's paradox, I think. That's right.

13:17 Adi: That's right.

13:18 Wes: If you have a. What is it? The set of all sets that don't contain themselves, then that set must contain itself.

13:28 Adi: That's right.

13:29 Wes: That's the paradox. And Russell's solution was his theory of types. And in a way, Godel's incompleteness theory that we'll be discussing in more detail will be the same sort of problem cropping up again at a different level. So Russell's theory of types was a way of pushing that problem aside so you could get a nice foundation to mathematics via set theory and logic. But it turns out that that foundation won't be. You won't be able to supply that, according to Godel.

14:00 Adi: Well, actually, no. So the idea is that the problem that pops up in Russell's paradox ends up being a problem of not having the right axioms in place. So essentially, after Hilbert set out the beacon and told everybody to start working on this, a couple of different axiomizations showed up. One of them was the Zermelo-Fraenkel set of axioms. And one of the axioms there, I believe it's the axiom of foundation, directly addresses the issue that Russell pointed out in Russell's paradox. So once you had an axiom to deal with issues like that, you were able to kind of get rid of a lot of these peripheral inconsistencies that seem to be popping up.

14:37 Dylan: I want to just slow down on this point, which is one solution to the problem of inconsistency is to just say you don't have the right axioms. You either don't have enough of them or you don't have the right ones, that they conflict with one another in some way. And so the way to get around that is to find the right ones and find enough of them. And the general goal is to find the fewest so that you don't have any redundancy, and then have them be enough so that they are inconsistent. And the ultimate goal would be that they can account for everything, that they are complete.

A good example of this kind of thinking or approach is the presence of the parallel postulate in Euclid. So in Euclid's geometry, there are a bunch of definitions, essentially axioms. And one of them that comes in is colloquially called the parallel postulate, that basically you have a straight line and you can only draw one line through a point off of that line that doesn't intersect it ever. And the reason that's interesting is that for a long time a lot of people thought that it can't possibly be the case, that that needs to be an axiom in the system, that it must be derivable from the other axioms.

And it turns out that that's not true, that you can actually show that to do Euclidean geometry, you have to

16:06 Wes: include that, and you can deny that axiom and build a different kind of geometry on top of that that's consistent.

16:13 Dylan: Yes, Both those things end up being really interesting. On the one hand, you have a system, Euclidean geometry, for which you have an axiom that seems like maybe you don't really need it, that it's more complicated, but you actually do need to have it in order to say all the things that you want to say or can say in Euclidean geometry. And another interesting thing about axioms is that what the axioms are define the kinds of truths and the kinds of things you can say. So just as Wes said, you can deny that axiom or put in different axioms, and you can end up with a different set of relations and a different set of truths.

And then that becomes interesting because whether that correspondence, those mathematical truths based on that particular set of axioms, how they correspond to the real world, then becomes a very interesting question. This becomes now, later on, what do you mean by a distinction between mathematical truths and physical truths? Just the whole notion of having a set of axioms. And the goal is I can talk about whether or not they are consistent or not, and whether or not I need more of them or not.

17:21 Wes: Well, Adi, I think you were about to explain the gist of the ZF axioms. I mean, my impression of the way Russell solved the problem with his theory of types was that it was kind of jerry rigged. And I think that's what Dylan was sort of getting at was that you define the problem out of existence with your axioms. And so I'm wondering if the ZF axioms, for instance, you know, prohibit self reference or something like that. That would prohibit you from falling into these little paradoxical traps in the first place.

17:49 Adi: That is right. One of the axioms does try to address these types of self referential problems. Again, I believe it's the Axiom Foundation. Just to clarify though, what Godel ends up using in the first incompleteness theorem isn't a violation of that axiom. It's just the type of problem that was created in Russell's Paradox is a different problem. It just so happens that they look kind of similar. So with respect to Dylan's previous point though, on the axioms, I think it is a very, very interesting question of where the axioms come from. There's, you know, these, when we talk about the Zermelo-Fraenkel axioms, did they come from essence of math in and of itself?

That's kind of what maybe Godel would want to say. He's very committed to this idea of mathematical objects being definitely realizable in existence. Or do they come out of maybe some form of social construction? Maybe it is part of what becomes popular in mathematics of the time period. I think it's a very interesting question.

18:49 Mark: Isn't this whole thing not concerned with figuring out which axioms are best? I mean, that's what Hilbert was concerned with in terms of making the foundations of mathematics solid. But the original proof by Godel was for not PA Peano arithmetic, but for pm some version of the Principia Mathematica. In other words, what Russell was working with, a sort of simplified version of that, but that very soon he himself and others were able to generalize these theorems so that they would apply to any axiomatic system whatsoever.

19:19 Adi: Wait, are you talking about Godel's theorem in particular?

19:21 Unknown speaker: Yes.

19:21 Adi: Yeah.

19:21 Dylan: No.

19:22 Adi: Well, the way that Godel's theorems work is that he proved them for very, very simple languages so that it naturally captured anything important essentially.

19:31 Mark: Right. Any system that could support basic arithmetic. This would affect any axiomatic system.

19:36 Adi: That's right. And anything more complicated, obviously.

19:38 Mark: Right. Well, okay, let's say more about what the proof actually is. You gave the first version already, that there's some sentence in the system that is true but is not provable. And it's done with this self reflexive maneuver of this sentence is not provable is the sentence in question. Right.

19:53 Adi: And then proving it. That's right.

19:55 Mark: Well, but you can't. No, it's just.

19:58 Dylan: Don't you end up saying that this sentence is not provable as a true statement.

20:01 Mark: Right. You say it's true. If it were false, that means it is provable, which would mean it's true. Right. At least it follows from the basic axioms of the system. So you say instead that it is true, which means it is false, that it is actually provable.

20:17 Adi: That's right. Right.

20:18 Mark: Because the statement again says this statement is not provable.

20:22 Adi: Correct. And the idea is that we know

20:24 Mark: it's true as long as the system is consistent.

20:26 Adi: Right.

20:27 Mark: So we don't know in the abstract actually that it is true. We just know that it follows from the system. That's really what the result says.

20:35 Adi: Now to be fair, you can't make that extra step until you get to the second incompleteness theorem.

20:39 Mark: Alright, go ahead. Let's give the second one then.

20:41 Adi: Any system with the power of basic arithmetic that also has certain basic logical properties cannot prove its own consistency. And with this proof, Godel kind of put an end to Hilbert's dream of having proofs that would show that mathematics was entirely consistent.

20:59 Wes: And he gives a kind of informal justification for this in this reading. Right? Yeah.

21:03 Mark: Let's give a little bit of his informal proof of the second theorem there.

21:07 Dylan: It's on page 308, the bottom of

21:09 Mark: that of the 1951 article. Yes, also called the Gibbs Lecture.

21:14 Dylan: He says the second theorem has to do with the concept of freedom from contradiction. For a well defined system of axioms and rules, the question of their consistency is of course itself a well defined mathematical question. Moreover, since the symbols and propositions of any one formalism are always at most denumerable, everything can be mapped onto the integers. And it is plausible and in fact demonstrable that the question of consistency can always be transformed into a number theoretical question, to be more exact into one of the type described below.

Now the theorem says that for any well defined system of axioms and rules, in particular the proposition stating their consistency, or rather the equivalent number theoretical proposition, is undemonstrable from these axioms and rules, provided these axioms and rules are consistent and suffice to derive a certain portion of a finitistic arithmetic of integers. It is this theorem which makes the incompletability of mathematics particularly evident, for it makes it impossible that someone should set up a certain well defined system of axioms and rules and consistently make the following assertion about it.

All these axioms and rules I perceive with mathematical certitude to be correct, and moreover, I believe that they contain all of mathematics. If someone makes such a statement, he contradicts himself.

22:37 Wes: Or if he perceives the axioms under consideration to be correct, he also perceives with the same certainty that they are consistent.

22:44 Dylan: Hence he has a mathematical insight not derivable from his axioms. Right.

22:48 Wes: This seems to me where he's relating consistency and completeness. Am I wrong about that? He's saying because if you know that you don't have consistency, then you can't say that they contain all of mathematics. Right. You can't say that I'm going to get completeness as well.

23:03 Mark: Well, if it's inconsistent, then you can derive anything from it.

23:06 Dylan: Yeah.

23:06 Wes: No, I'm just saying that he's drawing out the consistencies thing to say that the lack of consistency implies a lack of completeness.

23:14 Adi: Well, I think what he's doing here is he's using the result from the second incompleteness theorem to make his point about math being inexhaustible.

23:23 Wes: Right.

23:24 Mark: So this is not a statement of the proof of the second incursion.

23:26 Wes: Right. I was wrong about that. He's using it. He's not giving an informal justification of it. He's justifying this larger point about incompleteness or inexhaustibility, however you want to put it. Following from the inability to prove consistency,

23:38 Dylan: maybe we should do what Mark wanted to do in the first place and just say the first and second theorem one after the other so we get them straight.

23:46 Mark: Sure.

23:47 Adi: All right, let's do first incompleteness theorem. Any system powerful enough to express basic arithmetic contains true statements that are not provable by that system. So this is the theorem that involves that self referential statement that we talked about earlier. And what it shows that you've got systems in which there are true statements that cannot be proven by the system.

24:10 Dylan: It's incompleteness.

24:12 Adi: That's right. The second incompleteness theorem. Any system with the power of basic arithmetic and certain basic logical properties cannot prove its own consistency. All right.

24:23 Mark: And can we say a little more, just a few sentences of how he shows that to be the case, since it doesn't involve that self referential liars paradox, Russell paradox looking thing. Right.

24:34 Adi: Yeah. It's actually a rather intricate proof. I don't remember it off the top of my head. It's not as easily accessible as the first one, the first incompleteness theorem. You know, you can kind of show it very straightforwardly. The second complete theorem, if I remember correctly, is a little more intricate. I don't think there's an obvious way to summarize it, if I remember correctly.

24:50 Wes: One thing to note here though is that the two are actually intimately related. Right, the two theorems.

24:55 Adi: Exactly.

24:55 Wes: And that was kind of the point of the quote we just read at length. Because the idea is if you can't get consistency, then you could always perceive something to be true. The consistency of which you couldn't prove, which means that it's also incomplete. Right.

25:10 Mark: If you can't come up with all the true statements, you certainly can't show that they're consistent. You could take any subset, any finite subset of theorems in your system and look at them all individually and show that they are consistent. But you can't just say for the system as a whole, because there needs to be an infinite number of unprovable things in it.

25:29 Wes: The point of it is if your system is not consistent, you can find a statement that's true but not provable.

25:34 Mark: If your system's not consistent, you can find anything in it.

25:37 Wes: No, I mean if. I'm sorry, if you can't prove consistency.

25:39 Adi: Now just to be clear, the first incompleteness theorem doesn't imply the second.

25:44 Wes: Well, I'm saying the second implies the first. Am I wrong about that?

25:48 Adi: Yes, I believe that's fair.

25:49 Mark: I had read as well. Okay, so one of the unprovable statements would be this whole system is consistent, right? It has to be true. If the system is consistent, then that statement about it being consistent has to be true. But it is one of those. Just like the self referential statement, this statement is not provable. It is one of those that is not provable either.

26:10 Adi: But the difference is that with that statement you will never know if it's true.

26:14 Mark: Sure, right. You could prove conclusively that it's inconsistent. Right? You just find one contradiction. But you could just keep searching forever. And since you're never going to get through, this is something that's not obvious just from considering the self referential case. This statement is not provable. But once you allow one unprovable statement, you actually allow an infinite array of unprovable statements.

26:38 Adi: You certainly open the door to it.

26:40 Dylan: Unprovable truths.

26:41 Adi: Yes, unprovable truths. Now there's a deep question there about. And some people maybe argue that, listen, this is silly. You found this one silly statement and that's the only one and that's really the end of it. And I think what's kind of fun about that is that in the first reading, Godel actually talks about this idea of these supposed inconsistencies, maybe not actually being inconsistencies, but maybe just being illusions. The same way that we have illusions when we look at a stick in the water and see another stick. So maybe what we perceive as inconsistencies aren't inconsistencies. Maybe they're just kind of mental or image based illusions that we just don't fully understand what we're looking at.

27:22 Dylan: Wouldn't the implication of that be that it's really incomplete? When you brought up that image of the stick bending in the water, that to me meant that, well, I need to invent new physics in order to account for that. If that doesn't make sense to me, I have to come up with the paths of light vary based upon the density through which the light travels, yada, yada, yada. And so in that typical rough and ready way, when I have a paradox that means that I don't understand everything, that is, it's incomplete because of that inconsistency, just like Zeno's paradox.

Why is Zeno's paradox really interesting? Because you end up inventing calculus in order to solve it. And when you state the problem, you say, oh, you end up stating it as a problem because you've based your calculation on a certain understanding of numbers and the relationships between them and fractions. And by thinking that through, you end up coming with a conclusion that doesn't jive with the real world. And so in this, you're testing your numbering of the world and your understanding of numbers against the world itself, and you come up with a paradox.

And the end result of that is that because you're trying to line up numbers with the real world, that is number the world, you end up having to invent new numbers. Invent a new way of understanding numbers is a better way of saying it. It's not new numbers, but I end up adding numbers that I never thought were numbers before.

29:04 Adi: Well, I think that's a really interesting point. The one thing is that you're not really looking at the real world. You're looking almost internally at the concept of numbers, right? So there is no real world to go check an irrational number, to check what, you know, negative numbers look like. You're left with the concept of numbers to think about and reflect on, and then tinker with your axioms in such a way that Gets you to a point where you understand the new problems you run into.

29:33 Dylan: Are you specifically speaking in Godel's conclusion that this only points to the richness of mathematics and then the secondary conclusion that there is a mathematical world out there, his Platonism sort of world of concepts that exists to be probed apart from the physical world? Or are you speaking specifically about the generation of irrational numbers?

29:56 Adi: No, I think I'm speaking generally about kind of the way math kind of works. It's never constrained in a lot of ways by the real world. It's not constrained the real world in the way that physics or biology is. And I think that's where kind of Godel's concept of the inexhaustibility of math comes from. You can keep adding these axioms and making the world of math larger because there's nothing stopping you in some sense.

30:20 Mark: It's not just there's nothing stopping you, but you have to do it to, if you're searching after completeness, at least so that if you find, hey, I created this system. Oh, there are truths in it that aren't approvable from the basic axioms. So I'll create a new system that either. Well, I could just call those previously improvable things axioms as well, but that's a little artificial. Maybe I should just build a more advanced. I should throw in some set theory that you have to go sort of go out to the next higher level of set theory in order to problems

30:53 Wes: undecidable on one level can always be decided at a higher level with a stronger set of axioms.

30:59 Adi: Yes, that may be true, but then there'll be other problems that you can't

31:02 Wes: solve, and then there'll be a new set of problems and then you keep going forever.

31:05 Adi: Yeah, that's right.

31:06 Wes: That's one of the ways he gets at the inexhaustibility. But I think with his Platonism, he does sound like there are constraints. Towards the end of both these articles, it sounds as if they're real mathematical objects and we can have some sort of intuition or observe them in some way in ways that they constrain our concepts. So that as we'll see, he's going to argue that our mathematical concepts don't simply come out of the syntactical rules of the system. They're not tautological, but higher level concepts like induction, for instance, and then his talk of Husserlian phenomenology, as if you can reflect on concepts and see things about them. And it's not simply arbitrary, it's not I'm making this up myself, but that I'm discerning some reality about the world where we. Not the empirical world, but the world in some broader sense. So that's the Platonism at work there.

32:00 Mark: I'm glad you're making that connection right here because. So that is exactly the opposite conclusion that is usually used when Godel's incompleteness theorems are thrown out in philosophical discourse on the Internet by people that don't really know what they're talking about. As they say, it is some kind of under determination of reality that oh, any given set of axioms is you're not going to be able to prove all the truths of the system. So that means there's some. You could pick this truth or you could pick the opposite truth. And as far as the axioms go, you could go either way.

And so you'd have a multiplicity of different mathematical systems. Just like in geometry, we said you could take the parallel postulate or take the opposite and develop a different geometry. But actually it's not worked that way in arithmetic and you don't have multiple arithmetics taking advantage of this incompleteness in that way. That's just not the way that it's grown at all. That at least the way Godel's describing it here. Actually, as you meditate on these concepts, it sort of gets more and more firm, according to him, because you're contemplating this stable objective world that is outside of time and space.

It might just be determined by our concepts, but in a deep way. The concepts themselves, once you introduce them, are inexhaustibly rich in and there to explore. It's not something that we can just change details about them on a whim.

33:22 Dylan: The role of axioms that Godel's talking about, there's something analogous going on with something like the different sorts of geometry. But the way different geometries end up working have to do with the kinds of basic relations that you allow in them. You could think of those as axioms of the geometry, but they're maybe a little bit closer to the governing laws that you decide to put in place for that geometry. Like the difference between two different forces, say in physics, or two different fundamental interactions that you say, well, this is the kind of interaction that characterizes my geometry.

I will constrain it to admitting of the parallel postulate. Or alternatively I'll say, eh, we'll investigate things that don't admit of the law. The parallel postulate. The rules that are going on that Godel's talking about are in some ways more fundamental and deeper than the choice of my geometry.

34:24 Adi: Let me try and clarify your point. What you're saying is that what Godel wants to talk about is the system constrains itself. And maybe in so doing, he's also referencing one of his arguments for Platonism. There's not enough freedom in the system of mathematics such that we can just made it up. Is that kind of where you're going?

34:45 Dylan: Well, Mark was rightly bringing up a kind of conventional misunderstanding of Godel's theorem, which is the incompleteness and inconsistency theorems just throw the whole world open so that any kind of claims of truth are equally correlative. And a kind of offhand way, say, well, you know, it just happens to do with whatever truths I happen to pick out. But Godel's papers point out that there is a kind of infinite richness, but that isn't a richness that is absent of deep constraint. At some level. It's like there are two different kinds of infinite possibility going on.

One is the infinite possibility of axioms that Godel is most excited about. And then there's the other infinite possibility of the ways in which you can arrange a given set of axioms and come up with more and more new truths. And those truths would be consequences of them and perfectly part of the system. But those are a different class of truths than the truths that are outside the system that Godel thinks really point to the infinite possibility in mathematics.

35:53 Adi: When Godel is talking about this idea of the infinite number of axioms you can add, the axioms he's interested in adding are basically larger and larger types of infinity. There's a natural direction to the axioms you can add, namely larger and larger types of infinity.

36:09 Wes: Right.

36:09 Adi: So with a Mahlo cardinal, which is one really big type of axiom of infinity, there's even a larger axiom infinity you can add that's bigger than a Mahlo cardinal, for instance.

36:19 Mark: He thinks this is comparable to the inexhaustibility of the natural world.

36:25 Adi: Right.

36:25 Mark: You could always look farther and divide things up more and look deeper and look in more detail at particular sections that that's the kind of inexhaustibility there is in an accurate axiomization expanding out from number theory.

36:39 Dylan: It seems to me you would have on the one hand more and more consequences, say, of the physics that we have today, so more and more material science. And then there would be the question of what new forces are there, what new fundamental actions that I have to include in order to account for new phenomena, and those are like the new axioms, and those are two different ways of progressing to two different kinds of infinity. The one with new axioms is the larger and larger and bigger and bigger infinities. And the one developing material science and better transistors and stuff like that is articulating the given size of infinity more and more and more.

In either case, you have an infinite amount of work to do. Godel seems to be the most excited about the notion of the first one. As Adi said, the bigger and bigger and bigger infinities, and that's the great conclusion out of the incompleteness, is that you actually have bigger and bigger infinities lying ahead of you in mathematics. And that really undergirds his Platonism. To him, that's partly the evidence that the conceptual world exists outside of us.

37:55 Mark: I'm glad you distinguished those two types of infinities there, because then it means that if you try to use Godel's incompleteness theorem, which actually points toward this first type of infinitization into greater and greater levels of abstraction, greater higher and higher forms of set theory, that doesn't have any direct bearing on anything except the arithmetic component of natural science.

38:23 Adi: Right.

38:23 Mark: It doesn't follow from anything. Godel says that there's always more to discover in the natural world. That's just a separate topic from anything that he was considering. If you think that Godel has pointed out something fundamental about all systems, like the system of nature. Well, no, because the system of nature is not a system of axioms. Except insofar as you're talking about arithmetic as it is applied to measure nature.

38:48 Dylan: It depends on the question of the correspondence between mathematical thinking and the physical world. Even if you had a set of axioms that completely describe the physical world mathematically?

39:01 Mark: So basic scientific laws you're going to call axioms. Yes, which is not something Godel does. But this is a fine way of trying to think about what the connection actually is.

39:11 Dylan: Yeah. Okay. And so it seems to me that there are two questions that'll loom. One is, are there truths about the physical world that you end up having to bring in an act of judgment that is outside of those axioms? Are you going to have to bring in concepts that are outside of those axioms? That's something like the incompleteness argument. And the other is the question of have we excluded physical actions, physical properties of the universe that require new mathematical axioms that we hadn't used before that are basically point us to a mathematics that's outside of the mathematics.

Now, so say we have mathematics a that describes all of our physical world. The question of whether or not that completely describes the physical world is partly an empirical question and a question of, well, will I discover new interactions, new fundamental properties of the universe that require me to have more math?

40:17 Adi: Basically, any math you would need for even very, very complex physics is way, way, way behind anything that Godel and these axioms are ready to take care of. Any math that is needed to deal with standard descriptions of the world is real basic compared to what the Zermelo-Fraenkel axioms are taking care of. So when you start talking about adding infinities, you definitely don't need any of that to describe the real world. And I don't think we expect there to be a situation in which you would.

40:53 Dylan: It wouldn't surprise me if that's true. It's an interesting side question about what it means to talk about the flexibility or the coverage of a given mathematical theory and to show that that box is bigger than we could possibly ever need. That's what you're describing. I have a box of tricks that's just bigger than I could ever possibly even conceive of.

41:17 Mark: Possibly needing to describe things in the physical world. Yes.

41:20 Dylan: And the reason that's interesting is that historically in mathematics, it's not very long ago that that was simply not true.

41:28 Adi: I think that there's a real tendency to believe that math exists to basically power the sciences, power physics, and that's the value and that's the purpose of math. And I think what's interesting, and one that comes out pretty clearly from Godel, is that that's not what math is. For the people that do this, math is its endeavor in and of itself. And if I were to take it a step further, I would go as far as to saying it's like an aesthetic in and of itself. So I think that the desire to make math something that is going to be useful to the sciences is not the only way to look at it. How about that?

42:11 Dylan: Well, Godel certainly doesn't think about it that way. I agree. As a historical question, it was jiving the world with mathematics, essentially doing a kind of physics that moved math for a long time.

42:27 Adi: Right.

42:28 Mark: We need calculus because we need to calculate the area under a curve.

42:31 Adi: Right.

42:32 Wes: We need to account for motion.

42:33 Dylan: Yeah, we need to account for motion. Yes. And in fact, non trivial geometries predate by a lot general theory of relativity and other kinds of geometrical solutions in physics in the 20th century. Maybe that marks the transition where Math gets way out ahead of any consequence for the physical world. That's a separate question about trying to articulate what it means to talk about that kind of coverage, like what it means to say that a mathematical account covers empirical questions about the world and how you even formulate that kind of question. It's a different kind of completeness question.

43:10 Mark: Let me consider a more brute attempt to apply Godel's theorem to physics, which doesn't work. One of the things you said in that string, Dylan was talking about the basic axioms in arithmetic as comparable to the basic laws in a scientific explanation. So the basic laws of physics, for instance. So if you said, for instance, oh, hey, I came up with these basic laws, hey, but there's some truths that I see empirically, and they're not covered by those basic laws. Well, if you try to apply Godel's theorem literally, you might say, well, oh, that just shows that that axiomatic system that I came up with, that is incomplete.

There are some truths in here that are not provable by the system. But no, it actually just means that your model, your basic axioms, which you really shouldn't even call axioms, but the scientific laws you came up with are just inadequate. You need to modify those, you need to tweak those. It's not that every time you come up with a set of basic laws, that's going to entail some empirical results. That is true, yet conflicts with the basic axioms. That's confusing the model with the thing it's modeling.

44:17 Wes: Yeah, when you're coming up with scientific laws and models, you're trying to account for empirical data. And what the data is in mathematics, I think, is less clear. We don't, as Godel would want to say, have this complete freedom with our axioms. He sort of draws this parallel himself. Right? He draws that analogy. The two phases of childhood development, where in one direction, you're learning somatosensory skills, you're dealing with objects in the world, and then the systemization of that is science. But then there's language and logic, the ability to make inferences.

And our systemization of that still needs to go a lot further, but for him, it works on the same analogy of observing something, not in the same sense. And this is where we get into the Husserlian phenomenology. But observing, let's say, abstract objects, if we want to think on analogy, that's our data, those abstractions, our concepts, and our abstractions are our data. And we build models including these axioms in order to try and account for those. I don't know how well that holds up or works in the end, but that, I think, is the parallel.

45:22 Dylan: Yeah, I agree, Wes. And the disjunction of mathematics from the real world sort of looms behind this. And the process of that happening in the development of science and the development of mathematics, I think, is an interesting one in some ways. It's there from the beginning in things like Euclidean geometry, where you have the question of the ideality of a triangle and the fact that the triangle that you're talking about when you do a demonstration in Euclid is not a triangle that exists in the world. And so there's something like those mathematical objects that are apart from it.

That's why we have. We even talk about it being Platonism. But you also have this kind of consistent move to mathematize the world that especially is characteristic of the modern era. And that attempt to number the world leads to tremendous developments in mathematics to the point where it seems like it's not unreasonable to think, well, whatever I discover in mathematics is actually something that's true about the world. In fact, this happens in math over and over again. I mean, it's like the birth of theoretical physics is that you end up not just abstracting mathematical properties out of the physical world, but you do math and come to mathematical conclusions that you never thought of or never measured or never witnessed.

And then you go find them, and that correspondence is incredibly powerful. And then it gets one upped on you. You reminded at least that there are mathematical consequences that have nothing to do with the world, that they don't get observed in the world. Everything from something simple like a negative distance to geometries and mathematical constructions that it's just an open question about whether or not they correspond to the world at all.

47:19 Wes: Those are good points. I wanted to take us back to the point where he starts talking about the finite machine, because one of the interesting consequences of incompleteness, and this gets us into the Platonism as well. Basically, there's no finite machine or Turing machine, which I just take to be a computer or algorithm, which could be used to output only in all the propositions that are provable. So ultimately, we get this consequence that either the human mind surpasses the powers of a finite machine, either it's not computational, or there are problems about natural numbers that are not decidable by any proof the human mind can conceive.

47:55 Mark: Well, the machine would put out all and only the things that are provable. Right. Because that's the way the machine works is it applies.

48:04 Wes: Sorry, I mean true. All of your true.

48:07 Mark: Right. So it applies algorithmic operations to whatever set of axioms it starts with, and it comes with all the theorems of the system. So Turing thought that the mind was like that. But Godel is saying, maybe we should just read the quotes from him on 309, where I think we stopped quoting last time. Right, the end of the other quote. All these axioms and rules I perceive with mathematical certitude to be correct, and moreover, I believe that they contain all of mathematics. If someone makes such a statement, he contradicts himself. For if he perceives the axioms under consideration to be correct, he also perceives with the same certainty that they are consistent.

Hence, he has a mathematical insight not derivable from the axioms, because he can't prove consistency. However, one has to be careful in order to understand clearly the meaning of the state of affairs. Does it mean that no well defined system of correct axioms can contain all of mathematics proper? It does if by mathematics proper is understood the system of all true mathematical propositions. It does not. However, if one understands it by the system of all demonstrable mathematical propositions. He's just getting out this terminology he wants to use of objective mathematics, which is all the true statements, and subjective mathematics, which are the ones that can be proven.

And so he compares that to, if our mind is like a computer, it could only churn out subjective mathematics, it can only churn out the answers that are in its basic programming. But yet he thinks that his incompleteness theorems show that we have this extra bit of knowledge. Right, of the consistency of the system. And you can't prove the consistency of the system. And that's not something that comes out of the axiom. So therefore, what does he say? What's the. Either mathematics is incomplete in this sense that it's evident axioms can never be comprised in a finite rule.

That is to say, the human mind, even within the realm of pure mathematics, infinitely surpasses the powers of any finite machine. Or else there exist absolutely unsolvable Diophantine problems of the type specified, which we should say what that means.

50:03 Dylan: Yeah, and this is. He calls it his disjunctive conclusion.

50:06 Wes: The upshot of this is that either the workings of the mind cannot be reduced to the workings of the brain, which I take him to reject, or mathematics cannot merely be our own creation. So it's not that the mind has these extra powers. That's not the point here that gets us to Platonism. It's that there's mathematics beyond what is simply conceivable to us. And that in itself shows that mathematics isn't simply the creation of our own minds. Because he wants to argue against those people who are going to say that.

50:36 Mark: Well that's not the interpretation that George Boolos who writes.

50:39 Wes: No it is, that's his interpretation as well. But it's also pretty, it's clearly in what Godel writes. How does Boolos not interpret him that way?

50:48 Adi: Yeah, I think I interpret it a little differently as well. But Mark, can you share your interpretation?

50:51 Mark: I thought Boolos was saying that from other things that Godel wrote he actually agreed with Hilbert that there are not unsolvable Diophantine problems. And that just means like X's and Y's. You're trying to figure out which integers fit in here. And so you're saying 3×4 y0, which integers of all the integers make that work. That's a Diophantine problem. And to say that you've solved it is to say that you can come up with the set of the ones that work and the ones that don't work. So in other words it is decidable for any given set of integers and you can enumerate.

51:24 Wes: This guy gives evidence that mental procedures cannot go beyond mechanical procedures. He's quoting Godel as saying that's Turing's view. That's Turing's view?

51:34 Mark: Yes, that's Turing's view. And Godel was actually rejecting that. That Godel actually did think that there's no reason to think that there are absolutely unsolvable problems of that sort, that there are lots that we haven't figured out yet. But that's actually where the Platonism, where the open endedness that there's always room to search for a proof that we can't just say oh this is just absolutely insoluble.

51:58 Wes: Okay, yeah, so I'm wrong about that. Right.

52:00 Mark: He doesn't give us any more than the disjunction in this essay. He doesn't actually indicate which one he wants us to go with. He just says either the mind transcends any machine in that it can know these truths that the machine couldn't figure out or there are these unsolvable equations.

52:15 Wes: But he does say when he gets into section four on realism and Platonism, he does give us a new version of this disjunction and this is on page 311. So he says, yeah, if the first alternative holds. This seems to imply that the working of the human mind cannot be reduced to the working of the brain, which to all appearances is a finite machine with a finite number of parts. So apparently one is driven to take some vitalistic viewpoint. On the other hand, the second alternative, where there exist absolutely undecidable mathematical propositions, seems to disprove the view that mathematics is only our own creation.

This is why Mark I was taking him to accept the second horn of the dilemma. So on just what I read in Godel, I thought he wanted to reject vitalism. And he's going to accept the idea that there are these unsolvable problems because that shows that mathematics is not only our own creation. Doesn't it seem here that that's what he's doing? How am I misreading that?

53:08 Adi: No, no, I think you're right. I think I. When I was reading this, I thought he. His preference was for the second part of the disjunction. I think that's where he was headed. Mark, did you have a different interpretation there?

53:17 Mark: I think we need to sketch out a little more why that disjunction would follow from what he said earlier.

53:24 Dylan: Well, just before where you jump to the section. So the following disjunction of conclusion is inevitable.

53:29 Adi: Yep.

53:30 Dylan: And before that, where he's talking about the finite machine on page 309, it says, Evidently no well defined system of correct axioms can comprise all of objective mathematics, since this proposition, which states the consistency of the system, is true, true, but not demonstrable in the system. However, as to subjective mathematics, it is not precluded that there should exist a finite rule producing all its evident axioms. However, if such a rule exists, we with our human understanding could certainly never know it to be such. That is, we could never know with mathematical certainty that all propositions it produces are correct.

Or in other terms, we could perceive to be true only one proposition after the other for any finite number of them. The assertion, however, that they are all true could at most be known with empirical certainty on the basis of a sufficient number of instances or by other inductive inferences.

54:27 Mark: So so far, he's just giving that explanation of consistency that I was talking about before, that you can take any given, you know, subset of the system and look at them and say, oh, yeah, these true sentences, all consistent with each other. But you can't just do that for the whole, except by induction, right? Something that seems like you shouldn't apply it in math this way.

54:46 Wes: That.

54:46 Mark: That's the way we do it with the. The sun will rise tomorrow because it rose all the other days, well, that. That's how you'd have to say, oh, well, probably the whole system is consistent because all these subsets of the system, the finite subsets that I looked at, they're consistent. So probably the whole one is consistent, but you can never know for sure.

55:03 Dylan: So if it were so, this would mean that the human mind, in the realm of pure mathematics, is equivalent to a finite machine that, however, is unable to understand completely its own functioning. This inability of man to understand himself would then wrongfully appear to him as it's the mind's boundlessness or inexhaustibility. So that is what Mark just said, that it would seem like your mind was inexhaustible, but that's just because you had this kind of progressive one after the other process of proving things. True, but please note that if it were so, this would in no way derogate from the incompletability of objective mathematics.

On the contrary, it would make it particularly striking. For if the human mind were equivalent to a finite machine, then objective mathematics not only would be incompletable in the sense of not being contained in any well defined axiomatic system, but moreover, there would exist absolutely unsolvable diophantine problems of the type described above, where the epithet absolutely means that they would be undecidable not just with some particular axiomatic system, but within any mathematical proof the human mind can conceive. So the following disjunctive conclusion is inevitable.

Either mathematics is incompletable in this sense, that its evident axioms can never be comprised in a finite rule. That is to say, the human mind, even within the realm of pure mathematics, infinitely surpasses the powers of any finite machine, or else there exist absolutely unsolvable problems of the type specified. And isn't it the case that he in the end comes down on it, the mind surpassing a finite machine?

56:48 Wes: No, I think. Well, I'm arguing that he's going to say the second. Yeah, Mark, I thought was leaning towards the first.

56:55 Mark: Let's skim further. We don't have to keep reading.

56:57 Wes: Well, we could just go down to the point where he's now transformed the. This junction. You know, I just read it right, it's on 3.

57:04 Adi: 11.

57:05 Wes: If the first alternative holds, this seems to imply the working of the human mind cannot be reduced to the working of the brain. Get vitalism. On the other hand, the second alternative, where they exist absolutely undecidable mathematical propositions, seems to disprove the view that mathematics is only our creation, but the Creator necessarily knows all the property of his creature.

57:22 Adi: And I think he backs into that in part because to get to a point where you could accept the first part of the disjunction, you need to basically be able to cognize an infinite number of axioms that get you everything you want. And I think that's where you kind of are able to move into, or you're able to ultimately accept the second part of the statement. I think that's what powers the disjunction.

57:47 Mark: Okay, so for the rest of the papers, then, considering possible objections to this Platonism.

57:53 Adi: Right.

57:53 Mark: That he's saying there are things that we just can't know really about our own concepts, our own system that we stumbled across. I was going to say set up, but I'm not sure that there's a distinction between those two. As he's describing it, that once we define this mathematical system, there are parts of it that are inaccessible to us. That shows that it is objective in a strong sense. Right. And he says on 3 12, as far as the second alternative is concerned. Right. With the unsolvable equations, one might object the constructor need not necessarily know every property of what he constructs.

For example, we build machines and still cannot predict their behavior in every detail. But this objection is very poor, for we don't create the machines out of nothing, but build them out of some given material. If the situation were similar in mathematics, then this material or basis for our constructions would be something objective and would force some realistic viewpoint upon us, even if certain other ingredients of mathematics were our own creation.

58:48 Wes: So the unpredictability of machines comes from their materiality. So if someone's going to say, well, you can't predict all the properties.

58:54 Dylan: It only plays to what he wants to say.

58:57 Adi: Right, Right.

58:58 Mark: It's a variation of the point I was making before about if you think you've axiomatized a physical system and then you find something weird about it, well, that's just because your axiomization is wrong or something. It's not taking something into account. So let's say I thought I understood the laws of physics, and so I make a machine that it's supposed to do something, but it doesn't do what I expect. Well, there's something about the laws of physics I wasn't taking into account.

59:20 Dylan: But part of your point earlier was that that problem in physics might not be a question of bigger versus smaller infinities. It might be a question of the laws that you picked up. I mean, this points to the disjunction between the world of concepts in mathematics and the physical world that some portion of mathematics might be describing. Going to Adi's point earlier, that the world of mathematics and the size of the encompassing concepts in mathematics is so much bigger than what could possibly be used in the natural sciences, and that ends up being evidence in some respect for there being this world of concepts.

1:00:02 Mark: I'm not completely getting the connection. I mean, I was just trying to make the fairly simple point that like he's saying if I built my machine and it's out of steel and I didn't really know all the physical properties of steel, and then it might not perform according to my specifications because they're just things about the physical environment that exceed my planning. Whereas in math you've planned the whole thing, supposedly you've set up the axioms. And so if something unexpected happens, then it's either a matter of that the complexity of what I've created exceeds my grasp of it, which is one possibility, or it could be that I'm actually in a manner very comparable to building machine out of steel, when I build mathematical axiomatic systems, I'm building it out of some objective material that it's going to exhibit some properties that I didn't necessarily predict.

1:00:50 Dylan: That analogy has two parts to it. One is that it points to the existence of a material world, and two, that that material world is distinct from the material world of the natural sciences. That it's the conceptual world of mathematics that is distinct from that.

1:01:08 Mark: Yes. So I wouldn't call it. But an objective world.

1:01:13 Dylan: Sorry, sorry.

1:01:14 Mark: Objective realm of ontology.

1:01:16 Wes: Yes, the figuratively material world.

1:01:22 Adi: I found this idea actually really interesting because in some ways he wants to say that this idea is very ridiculous, but I don't know if we have any other examples. Right. The steel example is a really good one. Right. Because if I make something out of steel, if the machine operates in some way that I don't understand, then it must be that maybe I don't know enough about steel. What he wants to say is that in the world in which mathematics is entirely made up, it's nothing more than a mental concept that we've created and then we've constructed something on top of it using these just singular concepts.

Because we've created them, we should know exactly how they interact with each other. But I can't think of any other field of study in which the concept could be entirely humanly created so that I don't know if I'd have a counterexample to talk about. Like in the steel Example, does that make sense?

1:02:17 Dylan: Well, it does make sense to me, because at some level, there's always an input and something you're comparing your concepts to.

1:02:25 Adi: Right. And with this example where I can imagine a world in which, like we talked about earlier, it isn't necessarily ever really constrained by the physical world. It could be nothing but mental creations, concepts that are then strung together. And it seems at least fairly possible to me that mankind could make up mental concepts that, when strung together, do something that's different than what they understand,

1:02:49 Wes: or that we could be creating concepts, that we simply create them unconsciously, or that we're unaware of what we're doing, and then we have to go back and figure that out.

1:02:58 Mark: Cantor's second infinity could just lead to a giant phallus. We don't know we've unconsciously created these things.

1:03:04 Wes: No, that's not necessarily a reference to psychoanalysis. That's a reference to Kant. Right. When we construct the world, we don't consciously do it so that we know everything that we're going to find in the world. It's a surprise to us. The construction is unconscious.

1:03:16 Dylan: And at some level, the construction is richer than we even expected it to be or can be. Both are true. The world is richer than our concepts, and our concepts can be richer than we thought they were. They could have consequences that we didn't anticipate even within the concepts, not just with respect to how they jive with the world.

1:03:36 Adi: And, Dylan, this is, I think, what I find really interesting on this particular point. I can't think of another field I can look at where I can better understand how concepts that are entirely created inside the human mind would react once kind of unleashed on the world. Because I think math is this very interesting field that, again, could be entirely created without a physical world and might have some very specific properties. But I don't know because I can't compare it to any other field.

1:04:09 Dylan: That point you said right there, that math could have been created without any interaction with the physical world. I doubt that that's actually true.

1:04:19 Mark: Godel doesn't give any evidence for believing that. Right. I just was listening to our podcast on Russell and that he, too, thought that once you create this edifice, then it has an autonomy about it, but yet we get the basics still from experience. We get the numbers because we see objects in front of us. But then to actually get to number theory, you have to throw in infinity, which is something we don't experience. And so it quickly gets out of the empirical realm. And I'm not sure that Godel would disagree with that.

1:04:48 Wes: There's two questions here. There's one of if we weren't prompted by our interaction with the world historically, mathematics is unlikely to come about. But the second, and I think the one thing that Adi was pointing to, is in principle you don't need empirical observation. And this is absolutely true. Or at least maybe there's people who are still arguing the other side of this. So Mill tried to say you could get mathematics out of empirical propositions, but actually you don't need any empirical proposition, right? You just need logic proposals.

So some reductionists might just say that logic is all you need. And Godel's saying, actually you need a little more, I think, right, you need this intuition reduction, but it's still all you need is abstractions and concepts. So in principle you can get mathematics from just that. I think that would be very hard to dispute. But as a matter of contingent fact, yes, we have to be historical beings in the world, and probably if we didn't need to count sheep, it never would have gotten started.

1:05:47 Mark: He does mention Mill specifically in the second paper. He doesn't really give a criticism of the this empirical view of mathematics, but just says these attempts, like Mills, to ground it entirely in the empirical just haven't taken off.

1:05:59 Dylan: Without the world. I don't see how you have the notion of a concept that you even get logic out of it. That the very notion of identity, of two things being identical, being equal to one another, that this and not this are mutually exclusive, seem to me to come out of thinking about the world. And the only way that you get convinced that those things are true, that those deep down axioms that you don't even want to call axioms, you won't even enumerate them because you use them to think about the world, come out of the world, come out of your experience.

1:06:39 Wes: We're talking about genesis versus demonstration, right? Justifications versus justifications. I don't think we're in disagreement. You're talking about that they couldn't be generated outside of our historical context. But the other thing is also true, or possibly true, which is that you don't need any facts about those circumstances in order to ground them or prove them or whatever.

1:06:59 Mark: I would go further, that you can't actually ground them. That just like you don't see the items of geometry in nature, you know, lines with no width that you don't actually see A equals A in nature, you get things that approximate that kind of thing. So you have, well, either it's Raining or it's not raining. You know, you get ideas that would bring to mind the law of non contradiction, but just the fact that the law of non contradiction or any other logical law involves a universal quantifier, Right. For every statement, either it's true or false, you know, that is something that we don't experience.

It is something that at best we could get by induction from experience. That's the basis for Kant and Frege and many others for thinking that these are fundamental tools of the mind with which we approach the world. They could not be learned from the world, they could not be justified by the world.

1:07:50 Wes: Yeah, but certainly elicited by the world and shaped by evolution.

1:07:53 Adi: I think that's right. I think elicited by the world, but then not at all shaped by it. At the end of the day, the concepts can be developed so fully from very basic, I guess, empirical analysis.

1:08:04 Dylan: So maybe at some point we should read part of Aristotle's metaphysics or the prior analytics. I mean, the reason that Aristotle points to for why the principle of non contradiction that we have that is that without it we simply can't speak. He roots it in the fact that we communicate it all, that we have access to the principle of non contradiction and that that concept and the consequences of that out of logic comes out of that.

1:08:29 Wes: Yeah. The next objection that he tackles is the one right, that undecidable propositions about integers are meaningless, neither true nor not true. So how does he dispatch with that

1:08:40 Mark: one, in other words, to deny this distinction between objective mathematics and subjective mathematics, to just define it. And this sounds like if we go back to like William James on truth from our old episode there, and this was very explicitly said by the logical positivist, something along the lines of truth is verification. And so if something can't be proven either way, then it actually isn't meaningful, either true or false. It's not meaningful at all. It doesn't have a truth value. Where meaningful just means has a truth value. The individual words that make it up could sound like they mean something, but the phrase put together does not actually refer. And so it is meaningless in that sentence.

1:09:21 Wes: So he's going to say something like, well, yeah, you can conjecture the truth of a universal proposition even if you don't have a proof for it. So there's plenty of times when we can think of some proposition before we've established a proof and think that it's true and then go figure out that it's true and then go demonstrate it.

1:09:37 Mark: It can be an open question Forever.

1:09:39 Wes: So it's not the case. It's simply just because we haven't demonstrated something that is not meaningful.

1:09:44 Mark: Yep. The logical positivist, of course, would respond, well, you can hypothesize that it's meaningful, but if you find that, you just try and try to find a proof you can't find. I mean, maybe if you actually prove that there is no proof, then you've proven that it's not meaningful. You thought it was meaningful, but you were mistaken. I don't think that Godel gets out of positivist trap so easily here.

1:10:05 Wes: Yeah, he doesn't give much more of an argument. Would I just, you know. And then after that, he's going to get into some positive things. Arguments for Platonism.

1:10:12 Mark: Right. The activity of the mathematician shows very little of the freedom a creator should enjoy.

1:10:16 Adi: Yes, right.

1:10:17 Mark: We should be able to. If it was a mental creation, it should be something that we could arbitrarily change. Just like if we. I invented the land of Oz. And so I. Damn it. I can have, you know, everything in the Emerald City actually be green. Or I could have it so that it only looks green because you have green glasses on. I can just change it between books without any freaking explanation of why that's the case. But that's not the way math is. I'm sure I'm the only one in history that was incensed by inconsistencies between different Oz books.

1:10:47 Wes: No, I doubt it. There's probably a convention you can go to, Mark, or a website.

1:10:52 Adi: Jesus.

1:10:53 Mark: I was very analytical as a kid about these things, and I would put. Why does Toto talk in one book and he can't talk in the other? Just these things would bother me.

1:11:01 Dylan: Do you love cinema sins, Mark?

1:11:04 Wes: That would bother any rational person, Mark.

1:11:07 Adi: So the third argument kind of goes back to another one of the concepts we were talking about a little earlier, which is that if you have two concepts that are created strictly by the mind, why should they be related to each other? He wants to say that integers and sets are constrained by one another. That makes sense in a world in which you believe in Platonism. Otherwise it doesn't make sense. And I think that kind of dovetails with the argument earlier about what the power of concepts that are entirely created with the mind would actually be, and how we don't have many other data points for it. Yep, that may be right.

1:11:41 Mark: Are we done with this first essay finally?

1:11:43 Wes: No. Then we have the whole thing against mathematical propositions as being reducible to syntactical relations or Linguistic expressions right at the bottom there. Tautology.

1:11:56 Mark: Can we sum that up in one minute? Because it's freaking carnap and people are going to be bored.

1:12:03 Dylan: He's against carnap.

1:12:05 Mark: Yes.

1:12:06 Wes: Well, this is important. I mean, Ryan.

1:12:08 Mark: So the extreme version of empiricism, this nominalism, is just to say mathematical equations are not about anything real, they're about language. And I think from what he's already said, it's pretty freaking obvious.

1:12:20 Wes: Or they're reducible to the syntax of your language and the rules for establishing the truth and falsity of sentences, which is to say, reducible to tautology is. Everything in the end is reducible to the law of non contradiction. I don't know if that would follow, but.

1:12:36 Mark: Right. And he says this would work with simple arithmetic. That you could say two plus two equals four is tautological in that you could just substitute one for the other and then it's four equals four.

1:12:46 Wes: Yeah.

1:12:46 Mark: That's a tautology. Right. They're all reducible to tautologies, that they're just linguistic conventions.

1:12:51 Wes: Yeah. For a certain subset of arithmetic, you can reduce everything to tautologies. But ultimately he's going to say, yeah, first of all, it's very clear you can derive the truth of the axioms from such rules. But in that derivation, you have to use the axioms themselves and the logical concepts. So it's not reducibility because you're using it. You're using it in your proof. You can't just simply eliminate it in syntactical relations. You can't simply build up the concepts that way. And proving that they lead to the concepts, you use those concepts, you buy this argument. I'm not sure I understand it enough to buy it or not buy it.

1:13:25 Mark: I just thought that the carnap suggestion here is so silly that it doesn't need an argument.

1:13:30 Wes: What do you think, Adi?

1:13:31 Adi: I wasn't entirely convinced by it. I mean, I think a lot of people pride themselves on mathematics being kind of this very rigorous analysis of tautology after tautology. And that's what makes it kind of this very wholesome, internally consistent project. I'm not sure if I entirely understood the force of the argument, and it may be that I just was not quite grasping what he wanted to get at.

1:13:53 Wes: I'd like to see an example of someone if you're building from the. Well, does he. Maybe he does give an example.

1:13:59 Dylan: Actually, he does give an example.

1:14:02 Mark: I think he's just referring elliptically to a longer argument that he made against Carnap in Another work. And to understand that, we have to get into carnap. But go ahead and give the example. I just don't think we're going to get too deeply into this.

1:14:14 Dylan: Well, on page 319 he goes on to. He refers to the syntactical bit and.

1:14:21 Wes: Yeah, it's down. Yeah, this is what I said.

1:14:23 Dylan: It says, this explains the well known but misleading fact that formulas like 5, 7, 12 can by certain definitions be reduced to explicit tautologies. This fact, incidentally, is misleading also for the reason that in these reductions, if they are to be interpreted as simple substitutions, the plus is not identical with the ordinary plus because it can be defined only for a finite number of arguments. If, on the other hand, plus is defined contextually, then one has to use the concept of finite manifold already in the proof of 2, 2, 4. Similar circularity also occurs in the proof of P. Or not P is a tautology because disjunction and negation and their intuitive meanings evidently occur in it. So he's basically saying you have to bring in other things than the syntactics in order to account for the claims in those proofs that they're not really pure tautologies.

1:15:18 Mark: Right. That's another point we made in the Russell podcast, that for plus to mean what it actually does, or for the number one to mean what it actually does, you have to have infinity, right? You have to have the universal quantifier that applies it over everything. That if you are dealing with a much more limited system that only has numbers 1 through 10 in it or something. Well, yeah, you could still do what looks like basic arithmetic, but those symbols, the numbers and the operators would actually not mean the same thing that they do in number theory, where the numbers are the integers that go all the way up to infinity.

1:15:48 Adi: Let me ask you this. I get that part of the argument. I just don't know what's the point. Finally, like, I get the point that there's something more than just tautologies happening in math that I'm fine with. But what is he trying to say with that?

1:15:59 Mark: Yeah, does it really mean that you can't do a syntactical analysis of this stuff, and so therefore you couldn't program it into a computer and artificial intelligence is impossible? That whole line.

1:16:09 Adi: I think that's what I'm just going to.

1:16:10 Wes: Well, that would follow. Well, if we look at the positive side of this, he's not saying it's not analytic, Right? So that's another interesting part of this. He's saying, yes, mathematical propositions are true by virtue of the meanings of terms and not by reference to physical reality, but the meaning. He's going to want to say the meanings are not man made. This is really the critical point. It's not just something we made up. It's not a matter of convention. The concepts have an objective reality of their own that can be, in some figurative sense, perceived.

So, yes, it's analytical, yes, it's true by virtue of the meanings of the terms, but it's not true by virtue of some arbitrary definition, bachelor meaning, unmarried man. It's true by virtue of what he calls objective relations of concepts that are not tautological, including, for instance, the meaning of the concept set will have a lot of work to do. And it's again, analytic, but not tautological and not devoid of content. He also thinks of tautologies as devoid of content.

1:17:05 Adi: Yep.

1:17:06 Wes: So I don't know how to evaluate that. It seems, you know, I'm willing to go with it intuitively, but I don't know how to. Adi, did you have any more.

1:17:15 Adi: No, I think you kind of nailed it, which is that I understand what he's trying to get at. I'm just not sure if it proves what I think he wants.

1:17:22 Mark: All right, do we want to throw in the couple of additional points that come from the second essay? The 1961 essay, which is more straightforwardly about the Platonism bit, you know, throughout it. And actually this is a lecture that not only was not published, the first one, the 1951, that at least was given as a lecture at Brown University, but then it seems like he intended to fix it up for publication and didn't quite get there. So this is one of the technically unpublished. But the Second one, the 1961 essay, which is called the Modern Development of the Foundations of Mathematics in the Light of Philosophy, was not presented anywhere.

There's some speculation by the commentator Dagfinn Føllesdal about what occasion it probably was written to be submitted to, but it looks like it didn't even get to a final form. This is an often referred to one because it's shorter, it uses everyday language, it doesn't dwell on the math so much. It looks like the most straightforwardly philosophical thing that you might run across by Godel. But it's a rough draft and it

1:18:26 Wes: dovetails well with what we were just talking about. Right, because it looks like ultimately what he's trying to say is that intuition, and we can say what that means, has to play some role. So the background of that is Kant famously tried to Say that mathematical propositions that you might think are analytic actually involve intuition. So when you're trying to derive some geometrical proposition, you would rely on intuition. And the implication of that is you couldn't axiomatize that you couldn't reduce geometry to a set of logical relationships between axioms and theorems.

1:19:00 Mark: Well, an analytic, as you were just objecting to it, means just in virtue of the arbitrary definitions of words, that bachelor is an unmarried man. So that's an analytic statement. If we pick different words for those things, the truth value would change. But if you say, are making a basic arithmetic statement. Well, yeah, of course, if you use different words, if the words have different meanings, you're reading off the meanings of the words in some way, but they actually stand for something. You're talking about the concepts, and those have an objective truth to them.

1:19:30 Wes: Yeah. So this axiomatization of geometry that led into the kinds of stuff we saw with Russell and Frege and Godel, that all sort of went against the idea that you needed intuition. Right. The whole project was to reduce mathematics to logic, and so you wouldn't rely on intuition in the way Kant thought. But Godel's now coming back and saying, actually, we do need some sort of intuition. There are these concepts which again are not simply reducible to tautologies and so on.

1:20:00 Mark: It seems a lot of the purpose of this essay is to show that you might have thought that advances in math, like my own incompleteness theorems, would make you skeptical, would make you say, oh, I can't prove the consistency of any of my systems. Even math, which we thought was on a solid foundation, is no longer a solid foundation. Surely nothing else is on a solid foundation. But he wants to argue that, no, because of this mathematical intuition and more generally the phenomenological method of Husserl, who, according to him, is just doing Kant the way Kant should have been done, that he just was a particularly skillful systemization of things that were already going on in Kant, that if you took Kant's words literally, they were false.

But if you got at the correct interpretation of Kant, which he thinks Husserl stumbled upon, then this would work not just for mathematics, but for other essences of things as well, that there's an intellectual intuition at work in thought.

1:20:54 Dylan: His last sentence of the whole paper is awesome. But now, if the misunderstood Kant has already led to so much that is interesting in philosophy and also indirectly in science, how much more can we expect it from Kant understood correctly?

1:21:10 Adi: Yeah, that may be the most pretentious thing I've ever as a third written. It's just so amazing.

1:21:18 Mark: Well, and maybe that's. He wrote that sentence and then said, okay, I'm putting this in the drawer.

1:21:22 Dylan: Well, apparently the last. The whole last half of the paper he crossed out.

1:21:28 Adi: Well, the funny thing about this paper is that it's also probably the most clear thing I've read by him ever.

1:21:34 Mark: Right.

1:21:35 Adi: Can you imagine having heard, had to listen, God forbid, the 1951 essay. I mean, I'm not sure if I would have followed anything.

1:21:43 Mark: Yeah, it's a little unclear how much it was changed after he actually gave it as a speech, what we're actually looking at for the 1951.

1:21:50 Adi: But yeah, so I thought this idea of intuition in math and importing this kind of phenomenological approach was kind of cool and also kind of true to practice in some way. There's kind of some controversy about how infinity kind of crept its way into math. And the story kind of goes that, you know, some mathematicians started using it a little bit, ended up in calculus, and then, you know, kind of crept into a few other fields of math. And pretty soon it became so ingrained in mathematics, throughout all fields of math, that everybody kind of got used to it and we started using it.

And then that was the end of it. It kind of crept into math. A bunch of mathematicians started using it accidentally. By the time somebody questioned it, everybody was kind of over being. Being skeptical of it. So that's kind of a clever move. Although if you're trying to avoid skepticism, I'm not sure how helpful it is, frankly.

1:22:43 Wes: Why is that?

1:22:45 Adi: Because then we're left to a point where we're relying on what we as a group get comfortable with. Sounds to me like the essence of skeptical concern.

1:22:55 Mark: Sounds like cultural relativism. This is a historically contingent body of knowledge, which seems to be exactly what the Platonist has to argue against is to say, I don't care what, what culture you come from, what time period you are, if you are engaging in this sort of phenomenology, you're going to come up with the same result as anybody else.

1:23:14 Adi: But interestingly, that's an odd conclusion.

1:23:16 Mark: Right?

1:23:16 Adi: Because if you're using a phenomenological approach, doesn't it almost necessitate needing some type of actual object to study? I mean, doesn't it almost imply Platonism?

1:23:25 Wes: Yeah. There's the two kinds of intuition, Right. For Husserl, there's a sensory perception of objects, Right. And then there's what he calls the eidetic perception, where you're treating the abstract nature of the object, let's say.

1:23:36 Mark: Right. You're reading the essence off of the perception.

1:23:39 Wes: Yeah, I think Adi's point is good there. Do you need a particular object? There aren't the same particulars motivating that phenomenological activity in mathematics, or are there?

1:23:49 Adi: Well.

1:23:50 Dylan: But isn't the conceit in phenomenology that the world exists outside of us and it is by careful examination of it that we can be ensured of our thinking about it will be appropriately rich? And that Godel's going to resonate with this because of his Platonism, that the mathematical world exists for him the way in which the physical world exists for the physicist, that it's there to be probed and examined and is just as rich. And that's why he would resonate with Husserl's phenomenology and read all of Husserl's books.

1:24:29 Mark: It's the world of essences for Husserl that you could say has an independent enough existence that we can then sort of examine it from all sides, that we actually are even putting aside the external existence of physical objects, we're just considering them as appearances and reading essences off of appearances, which would definitely include then if you abstract far enough talking about mathematics.

1:24:53 Wes: But I don't think we're doing the same thing with math, though.

1:24:55 Mark: I just don't know how Husserl treats with math. And he didn't read any of that. And it's just. It would be pointless speculation. That's an episode that we should not have.

1:25:05 Dylan: We could read his geometry book.

1:25:07 Mark: One could do that.

1:25:09 Wes: One could do that on one's own time. But it's an interesting idea, this idea of there's mathematical evidence akin to the kind of evidence that we would receive in sense perception.

1:25:23 Dylan: To me, that resonates with the notion that our sense perception comes along with our thinking. And so that at the very least there are sort of two sides of the same coin, that what we normally think of as a sense perception is utterly laden with our intellectual activity. And so therefore the flip side is also going to be true, that things that seem like utterly intellectual activities have deep aspects of sense perception in them, if only because they happen in our minds that are physical entities, or at least you might say, the mechanisms that those happen with. It reminds me of the argument that Socrates makes about love of philosophy being joined with erotic love. I mean, it's the same kind of thing that our physical being is joined up with our mental existence and it's not mere analogy, Dylan.

1:26:14 Adi: I think that's a really interesting point because I think what comes out of that idea of it being a form of sense perception is that when you don't have a physical object to touch so that you can experience a sense that your fingers have when they touch something cold, the sense perception of a mathematical object like infinity is something akin to an emotion. Right. You have to feel comfortable or something like that. Right. It's very interesting because it does a lot of work in that it intuitively makes a lot of sense that I want to have a sense perception with a mathematical object where I kind of come to understand it in a better way. But if I'm having an emotional reaction to a mathematical object that almost goes against the whole idea of a Platonist element to that math object, I'm not

1:27:01 Wes: sure is it an emotional reaction. I mean, we're dealing with concepts, right? And so they're not imagistic. And this is kind of a point made by both Plato and later Wittgenstein, that when you have a concept, it's not that you have any image of something. That's not the concept. When I talk about a cat, it's not that I imagine some particular cat, and that's my concept. The concept is without that kind of content. In a way, it's a potentiality. Some nominalist might just want to come and say, well, it's just a word, it's just a sign. And what makes it a concept is the function that it can play in various propositions.

Let's say, on that view, when one is doing an eidetic analysis of a mathematical concept like infinity, it might bring the thought to mind of an ever increasing quantity or something like that, or a limit. But I think at the point where you're dealing with it mathematically. Right. And the different Cantor infinities, it's about these inferential relationships to other truths. And that's really what the content of the concept is.

1:27:57 Adi: I don't know if that's exactly what's going on here, though, because I think that what Godel wants to do here is he wants to leave room for something more than just a relational.

1:28:09 Wes: I agree. Yeah.

1:28:09 Adi: And when he says something more than relational and he wants to use a term like intuition or phenomenology, and again, when I'm not talking about some physical object that I can touch and feel and experience, I don't know how to get away from a concept that my feelings or emotions or something is not being imported in there to make Myself comfortable with this new concept. Who wants to become comfortable with.

1:28:29 Wes: Yeah, I think you're right. I wasn't outlining Godel's position there.

1:28:33 Dylan: Well, but emotion is. I guess I would want to use the term intuition. But maybe in the end it ends up not being so different from emotion. I think of the phenomenon of just figuring something out. And that emotional or intuitional experience of deciding that I figured it out. And in fact, the pleasure that I get out of figuring something out. I mean, to me that's the kind of thing that Godel is pointing to and that even Husserl's pointing to in the activity of the mind of probing through concepts and coming to know them and figuring them out. And that maybe it's not so crazy to talk about that as an emotional response.

That I get to know these ideas and I get to know my way around them and they become familiar. And it's not as simple as I created them because in some ways they exist outside of me, but they become more my own. And there is some aspect of it that I know that I am closer to understanding it the more I have this emotional reaction of clarity. And what stimulates that is probably a little bit complicated.

1:29:43 Wes: My point is that being clear on a concept comes back more to potentiality and use and being able to. I'm getting at the weirdness of this. Even though we're talking on the analogy of perception. It's not ever like perception in the sense that we're talking about sensory content in our intuitions of. If you want to say we have intuitions of concepts. It's more about having a know how of how you would use it. So even if we talked about the concept of a set and what follows from that, it's a very ineffable sort of thing to have an intuition of it and to say, okay, these things flow from that concept.

The concept of a set is not something concrete that I can hold before my mind's eye. Certainly I think both Husserl and Plato would agree there's an analogy to sensation, but it's very, very different. That's all I'm trying to get at. There's an invisible aspect. It's the invisible visible. Say there's something on analogy of sight going on. But of course, then it's not. It's like seeing invisible things. I'm getting at Platonic analogies here.

1:30:38 Mark: Well, and one of the big differences between this intellection and experience of sensory objects is that it goes in the other direction on the spectrum that he's sketching out in this essay. So the whole thing is structured around leftward empiricist and hence skeptical philosophies, and then rightward rationalist, eidetic, certain philosophies, theological and. Yeah, theological. He compares the two. And just looking at the history of this, a lot of what he says in here is just to argue against the spirit of the time, which is in favor of the leftward empirical and hence skeptical philosophy, to say, no, that actually we do have something like eidetic intuition regarding mathematical matters, and so we can have certain knowledge, in other words, sure knowledge, unquestionable knowledge of certain mathematical, at least basic things.

And if you've done the proof correctly, of pretty advanced things. But then he wants to, even though he spends a lot of the paper asserting that over the spirit of the times and saying, you know, again, people that tried to do empiricist math like Mill were just full of it and obviously wrong and it didn't catch on. But then he wants to sketch a middle way, right? He's not a pure empiricist and he's not a pure rationalist. So what is this middle way?

1:31:52 Wes: The middle way is, you know, he says, he acknowledges that his sympathies are more with the right. But then, you know, he forges this middle way, which is to say we make a concession to the right that every clearly stipulated mathematical problem with a yes or no answer must have a clear answer. So you sort of ensure the objectiveness of mathematics that way. But on the other hand, the concession you make to the skeptic is that these truths are not always derivable from axioms. That's the middle way.

1:32:21 Mark: No incompleteness.

1:32:23 Adi: We've come full circle.

1:32:26 Mark: Yes, it should be obvious, this kind of stuff that if you look in the Stanford Encyclopedia article or in this, if you get this Franzen book, which is pretty darn entertaining, as well as giving a lot more detail on the math, yet in still a very readable manner. But it really is just merciless about the various uses that Godel's theorems have been used by postmodernists. So, for instance, here's a quote from a guy named the simplest observation of how Godel's theorems create a postmodern condition begins with the first incompleteness theorem. This theorem says, in effect, that a consistent axiomatic system strong enough to prove some weak theorems from elementary number theory, requiring only the operation of addition and multiplication, but not either operation separately, will be incomplete.

There will always be mathematical sentences formulated in the syntax of the system under consideration that are neither provable nor refutable in the system. And these systems are said to be undecidable with respect to the system. Since an undecidable proposition and its negation are each separately consistent with the base system, one can extend the old system to two mutually incompatible new ones by adding on the undecidable sentence or its negation as a new axiom. The classical example of this procedure is the generation of non euclidean geometries by adding the negation of the parallel postulate.

But this is just the example I already gave. The new system so constructive also have new undecidable sentences different from the originals in the process of constructing new undecidable sentences and then new systems incorporating them. Or the negation goes on ad infinitum, like a branching tree which never ends. So this is how a postmodernist is saying Godel has something to say about really our overall epistemic situation. Can you find the obvious flaw there?

1:34:00 Adi: Aside from its nonsense, or, well, just

1:34:03 Mark: the fact that it only applies to formal axiomatic system. So our overall epistemological situation, all the knowledge that we have, is not a system of axioms.

1:34:13 Adi: I mean, Mark, speak for yourself.

1:34:15 Wes: Also, there are plenty of axiomatic systems which we can prove the consistency and completeness for. They're not ones strong enough to give us arithmetic. But you and I, Mark, were required to, unless I'm misremembering this, prove the completeness and consistency of first order logic as part of our training at ut, right? We had to do it on a test. We had to remember how to do it.

1:34:35 Mark: And then one of the things, if you were talking about, oh well, one that's strong enough to involve this basic arithmetic, wouldn't any physical system be stronger than that? Well, no, because if you're talking about stronger than that, you're actually talking about being able to found more of math. You're not saying that. You're also throwing things about politics into it. That doesn't make it a stronger system in the relevant sense. So you would only be proving the incompleteness theorem with regard to the mathematical component of whatever the larger system you know, whether it be the overall state of science, your overall epistemic situation as you walk around day to day, or anything else.

And so likewise, much more ridiculous things are pointed out in this book of people that try to prove that you just have to have faith, you know, that, oh, there are these undecidable things by the axiom. So you just have to have Faith. Well, way before Godel it was admitted that every argument must start somewhere. There have to be some postulates that are unexamined, that ultimately don't have any justification other than their self evidence. So that's not anything Godel invented. If you want to use that as your argument for faith or something, you can do that, but don't bring Godel into it.

He's not relevant at all.

1:35:37 Adi: I think also what kind of makes these arguments interesting is that he's importing Godel into philosophy and politics or language or whatever it may be. I don't think I needed Godel to understand that there's certain political questions which I'm not going to be able to derive from axioms. I just don't think that was ever on the table.

1:35:57 Dylan: The only way in which it might be on the table is the conceit that a kind of empiricist might have regarding moral laws. Right. And if you make the argument that, well, all of morality really is reducible to logic and I just have to come up with the right ones, it would be that, you know, that kind of maybe a little bit loose argument where you say that the constraints on my thinking are really constraints of logic and then something like Godel's theorem would seem to come to play there.

1:36:30 Adi: Yeah, that's interesting. I mean, we run into all the other criticisms that Marcus Freddie map.

1:36:34 Dylan: I guess it amounts to the extent to which you map other kinds of problems onto problems of logic. If you say, well, clearly politics is illogical and consistently inconsistent, then you would sort of reject out of hand any bearing that Godel's theorem might have on it. But if you said, well, you know, at the end there really ought to be laws of morality, consistent laws of morality, then it might have some bearing on it.

1:37:02 Mark: Yeah, you can use this as inspiration or analogy for things you could say, you know, I've set up all the basic rules for my governmental system, but there are always going to be some things that we would consider, like a just outcome in a particular civil case that are going to be under determined by the rules. Of course any system of laws is going to be incomplete. They're not going to determine every single thing. And so, you know, that's why we have judges to make the tough calls. So you could use that kind of stuff sort of more generally as an analogy. But just don't take it so literally because none of these things, your moral system or your governmental system are going to be formal systems in the sense that Godel was concerned with.

1:37:38 Wes: Yeah, I Mean, in the legal case, I think what we're saying when it's incomplete there is that we can never supply it with enough assumptions to do everything we want algorithmically, let's say, rules

1:37:49 Mark: of how to apply the rules.

1:37:50 Wes: Yes. New cases come up which require. I mean, well, Adi is a lawyer, right? The body of law just expands to accommodate all these different exceptions and it becomes Byzantine. But it's a different sense, it's a different meaning of incompleteness than what we were discussing here.

1:38:07 Mark: Addie, has Godel ever inspired you in your interpretations of the law?

1:38:11 Adi: It's funny you should say that. I actually was struggling to try and see if there was an application for Godel in a strictly logical legal system. That was something I was playing with at some point and ran into some of the criticisms or some of the concerns that Mark, you brought up earlier. The other thing that kind of comes up occasionally is this concept. As long as we're on this kind of little bit of a tangent, what's the problem with inconsistency? Why is it such a big deal? There have been various types of math that have actually tried to wrestle with whether you can actually cabin off inconsistencies and whether that's it doesn't actually lead to complete nonsense, which of course can lead to other. All sorts of fun, as you might imagine.

1:38:50 Mark: Just to read a couple more fun things from the Franzen book here, this is actually from an encyclopedia. By Godel's theorem, a system is either incomplete or inconsistent. Thus, logically speaking, it is impossible for us to fully prove any proposition. And he says the occurrence in such remarks of phrases like logically speaking is a noteworthy feature of many startling non sequiturs inspired by the Incompleteness theorem.

1:39:11 Wes: What was the word? Logically speaking, we what?

1:39:13 Mark: By Godel's theorem, a system is either incomplete or inconsistent. Thus, logically speaking, it is impossible for us to fully prove any proposition.

1:39:21 Adi: Well, that's a far reaching proposition.

1:39:25 Dylan: He said that he was French, I'm sure.

1:39:27 Mark: This is from the Encyclopedia Britannica. That is Encyclopedia Britannica quoting Carl Boyer's and Uta Merzbach's History of Mathematics. He actually sourced the quote from Encyclopedia.

1:39:38 Wes: It is mathematically impossible to prove any proposal proposition.

1:39:42 Mark: It is impossible for us to fully prove any proposition that's crazy in any system, not even mathematics. That's not even mentioned there. He also found 13 articles invoking Godel's theorem in the Bibliography of Christianity and Mathematics, first edition 1983. For instance, non standard models and Godel's Incompleteness Theorem point the way to God's freedom to change both the structure of knowing and the objects known.

1:40:05 Adi: Well, Mark, I think you've got your next podcast.

1:40:07 Wes: Man, exactly what Godel would have hated.

1:40:11 Mark: Theologians can be comforted in their failure to systematize revealed truth because mathematics cannot grasp all mathematical truths in their systems either. Awesome.

1:40:20 Wes: Well, roughly speaking, can't grasp all mathematical truths if that means can't prove.

1:40:24 Mark: It is argued by analogy from Godel's theorem that the methodologies, tactics and presuppositions of science cannot be based entirely upon science. In order to decide their validity, resources from outside science must be used. All right, you could go on endlessly with these things that are inspired. I think the one that is the most serious, which I actually didn't read the chapter on that in Franzen so I can't comment on, but are the ones corresponding to the first horn of the disjunction from the 1951 paper that the incompleteness theorem shows that the mind is not a computer, that it exceeds anything that could be known algorithmically.

Apparently Roger Penrose is a famous name associated with actually claiming that is very much the case and that is what Godel proved. I don't know that I have anything more to say on that other than what we've already said here. I will repeat the point that Godel's second incompleteness theorem only proves a conditional. If you know that a system is consistent, then you know that the consistency of that system can't be proven. In the system that whole conditional can be proven. What you don't know for sure is the antecedent that the system is consistent.

You may not have any reason to doubt that, but you can't point to knowing the consistency of the system as the thing that you know, even though it cannot be proven, as the proof that the mind can know something that cannot be proven because we simply don't know it for sure.

1:41:38 Adi: It sounds like Godel is very prolific according to these authors.

1:41:43 Mark: Yeah, it may well be. You know, we haven't talked even a bit about the use made of this image of recursion. Right. That comes out of the first proof. You know, that's if you read Godel Escher Bach by Douglas Hofstadter. That's probably the most famous book book with the word Godel in it. Not any good. You know, he's just fascinated by systems that deploy that kind of sort of turning themselves inside out, quoting themselves self referential systems and thinks that there's a lot to be made out of that. And you can explain consciousness and things with that. And I think you know all that stuff is just shows that it's a cool proof that can be inspirational in a variety of areas. But you don't want to say that Godel's proof actually proves anything, right?

1:42:25 Wes: There are some suggestive analogies. Yeah. That self referential nature suggestive analogy, human consciousness and how that what role it has to play in consciousness as we know it. Yeah, that's important. But how related it is to Godel. Yeah, probably only by way of suggestiveness or yes.

1:42:41 Mark: Don't say it's a Buddhist koan or something. The incompleteness theorem. I suppose any if you think contemplating the Liar's paradox would help you reach nirvana, then okay. But that's pre Godel.

1:42:54 Adi: Maybe it's just food for thought is maybe the takeaway. Godel allows people to think about things in different ways and then abuse it to their own ends.

1:43:01 Mark: Free your mind. Free your mind from the axiomatic system.

1:43:04 Wes: Man, I resisted adding the man.

1:43:06 Mark: I was freeing myself from my own cliche.

1:43:09 Wes: Sign, sign everywhere sign.

1:43:15 Dylan: The thing that I like most about Godel's theorem is the way in which he formulates a way in which to try to tackle the question of something talking about itself. Yeah, that's at the heart of philosophy and it's really cool that he was able to get purchase on that in some way within mathematics. And that's what I think is awesome about it.

1:43:39 Mark: Thank you, Adi, for joining us. You're an invaluable resource.

1:43:43 Adi: Thank you for having me, guys. This was a pleasure and I'm looking forward to even more discussions from you all on incompleteness in the future.

1:43:53 Mark: We can keep looking forward there.

1:43:56 Dylan: I think Mark blew his wad right now.

1:44:00 Mark: Next time we're going to have guest Lynda Walsh from University of Nevada Reno.

1:44:07 Dylan: That's it. Scientists as Prophets Hardcover. Lynda Walsh, Oxford University Press all right,

1:44:12 Mark: so we're going to talk about rhetoric and science and it's going to be awesome. Folks should go to partiallyexaminedlife.com and perhaps make a donation, perhaps support us. Big donors since last time have included Dee Bradley Baker, who's a famous voice actor and he might come and do silly voices on the show sometime. Douglas Matthews, Will Parker, Corey Moeller, Leland Gregory, Joseph McIntosh, Josie Nielsen, Mark Kleinman, Billy Pritchett, Richard McManus, Andrew Griggs, R. Mark Phillips, Charles Wysotski, Tanziel Ahmed, Bruce Jones, Daniel Sundet, Rex G.

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1:45:36 Dylan: That's right, be like Montel.

1:45:40 Mark: Do you know who Montel Williams is?

1:45:42 Adi: I do. And he follows you on Twitter.

1:45:44 Wes: Yeah, that just came up.

1:45:47 Adi: Really? That's. That's quite surprising.

1:45:50 Mark: Seems very random.

1:45:51 Adi: Next. I mean, what are you going through them all?

1:45:55 Mark: I hope so. I can only.

1:45:57 Wes: So, yeah, we've got Alec Baldwin, Lucy Lawless and Montel Williams.

1:46:02 Mark: All right, good night everybody.

1:46:03 Adi: Good night, guys.

1:46:05 Mark: Start now. Cough me up and brush me off and mean it if it's a new time then treat it as such it doesn't matter that much if you believe

1:46:28 Unknown speaker: it and the misery is axiomatic for the likes of you and me Beside it school and all if you'll ever Floyd smelling it choose now down or

1:46:49 Mark: up Just pick all the all right. And mean it if it's a nose dive then aim your luck to brush

1:47:05 Unknown speaker: the ground and clean it and a place I stand like a tree look solid as can be it's called all that good work that were to burn us sparingly and I know that some things happen to me But I'm missing what I know and I know that if you hold on to me that you will have someplace to fall from and to come back to life now sticking up grab it and massage it Time is up lets you and me

1:48:16 Mark: acknowledge it

1:48:19 Unknown speaker: and misery is axiomatic for the likes of you and me let's do it fine and still let's try to visit sparingly and a misery as automatic for the likes of you and me Sight it's cool and call it your and use it fruitfully and it reserves faith as non specific in a way it should not be Just grab me dear cause I'll be here to listen faithfully and get the it's all just axiomatic there's still much to do Axiomatic like my need for you Axiomatic that will stay long.