On a spectrum of philosophical topics, one might be tempted to put mathematics and morality on opposite ends. Math is one of the most pristine and rigorously-developed areas of human thought, while morality is notoriously contentious and resistant to consensus. But the more you dig into the depths, the more alike these two fields appear to be. Justin Clarke-Doane argues that they are very much alike indeed, especially when it comes to questions of “reality” and “objectivity” — but that they aren’t quite exactly analogous. We get a little bit into the weeds, but this is a case where close attention will pay off.
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Justin Clarke-Doane received his Ph.D. in philosophy from New York University. He is currently Associate Professor of philosophy at Columbia University, as well as an Honorary Research Fellow at the University of Birmingham and Adjunct Research Associate at Monash University. His book Morality and Mathematics was published in 2020.
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0:00:00.0 Sean Carroll: Hello everyone, and welcome to The Mindscape Podcast. I’m your host, Sean Carroll, and on the podcast in various episodes in various guises, we’ve talked about morality, right? The ideas of right and wrong, and ethics and so forth, the philosophy behind it. It’s a notoriously tricky subject, people don’t agree either on ethics, what is right, what is wrong or on meta-ethics, how we even decide what is right and what is wrong. So, the whole subject of morality, even though it’s very important, we need to make decisions about what to do in our lives, it’s notoriously difficult to reach consensus, to find agreement on what things are true. It’s not like math, where you say two plus two and everyone says four, everyone is in agreement about that. So, turns out, guess what, spoiler alert, there are a lot more similarities, if you really dig in between the ways that we can and should and do talk about morality and the ways that we can and should and do talk about mathematics.
0:00:58.1 SC: We have a feeling that math is enormously more rigorous and agreed upon, only because we don’t think about it that hard. Once you really approach the philosophy of mathematics, the foundations of the subject, you realize people don’t agree, as Kurt Godel famously proved, there are apparently true statements that you can’t actually prove as theorems and so forth. And you can actually draw a quite an elaborated analogy between the philosophical status of mathematics, which is reportedly very clear and certain and rigorous, and the philosophical status of morality, which is notoriously fuzzy. And that’s exactly what is done by today’s podcast guest, Justin Clarke-Doane, who is a philosopher at Columbia University. He doesn’t say the morality in mathematics are exactly analogous, but he does make the case that they are more analogous than you might think, especially when it comes to the extremely tricky question of what is real, are mathematical objects real, or are moral beliefs real? Do they reflect something objectively out there in the world?
0:01:58.7 SC: And I’ve already made a mistake, because the Justin’s whole point is that being real is different from being objective, and we’ll get into that. And so, spoiler alert again, this is one of the hard ones, this is one of the podcasts, and I say this with great affection and it’s all completely intentional, I’m glad it turned out that way, and you should absolutely listen. But we get into the weeds because these are really difficult issues, it’s very easy to treat them superficially and not do them the justice that they deserve, because two plus two equals four, what else do you need to know, right? Killing babies is wrong, how hard can that be? Well, these are really hard. Neither math nor morality are things that we fully understood. And as we talk about it at the very end of the podcast, the fact that we can compare them at this level of really, really taking them seriously, foundationally is something that is difficult to do in our modern world where we’re hyper-specialized, right?
0:02:54.8 SC: The people who are experts in the philosophy of mathematics are not experts in moral philosophy and vice versa. So, whatever you might think about the substantive claims that Justin and/or I might make over this podcast, the project is, I think really crucially important, this effort to really dig in, be careful, be responsible, be intellectually honest in more than one field at a time, to bring things together and see what connections work, what connections don’t work, and so forth. Helps us understand the world in new ways. That’s what we’re all about here at The Mindscape Podcast, so let’s go.
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0:03:46.7 SC: Justin Clarke-Doane, welcome to The Mindscape Podcast.
0:03:49.0 Justin Clarke-Doane: Thanks so much. Thanks for having me.
0:03:50.8 SC: So, you’ve written a book called Morality and Mathematics, and trying to think of a way into the podcast, I think the first question to ask is, what the hell? [chuckle] What is the punch line of this? Why in the world are we motivated to write a book or think about the intersection of morality and mathematics?
0:04:08.6 JC: Yeah, the main reason is that superficially, these areas can seem totally different, on the other hand, under a microscope, there’s surprising similarities between the case for realism about one or the other, that is the case for the existence of independent facts of the matter. And it is a real… It turns out to be an important question for naturalists, people who think that there is only basically the world delivered by science, whether or not one can consistently be a mathematical realist, and a morally anti-realist. So that’s how I got started.
0:04:57.5 SC: So, I guess, there are both mathematical realists and anti-realists and moral realists and anti-realists, but…
0:05:05.9 JC: Right. Yeah.
0:05:07.3 SC: Maybe what you’re saying is that there might be the closest thing we have to a consensus, might be mathematical realism and moral anti-realism?
0:05:14.9 JC: Yeah. If there… I don’t know the poll numbers, but my guess is that certainly among philosophers and scientists who self-identify as naturalists, they would, if pressed say, well, look, I guess I gotta believe in independent mathematical facts, not just because. It seems hard to believe that if everyone were to die tomorrow, it would no longer be the case, that two plus two is four. But also because, science is up to its ears in math and it’s hard to even know what it could mean to say something like, I believe in our best physical theories, but I don’t believe in math as an independent reality. So, on the other hand, science seems to have no need for independent moral facts. At first glance, I suspect a naturalistically inclined philosopher or scientist like yourself…
0:06:12.1 SC: Yup.
[chuckle]
0:06:12.9 JC: Would be inclined to say that come on, ethics is something like a matter of convention, there’s obvious evolutionary reasons why it might have benefited our ancestors to establish norms of conduct that roughly conform to the ones we’ve arrived at. And that’s kinda all there is to it. But no need to postulate in addition to these causal stories about how we came to have the moral beliefs we came to have, independent facts that we either get right or don’t as in the mathematical case.
0:06:43.8 SC: Good, okay, that makes sense. So it’s not so much a statement about, like you say, the poll numbers overall, but there is a specific attitude that certain of us might have scientific naturalist, etcetera, that might go tend… Tend to go hand-in-hand with moral realism and… Sorry, moral anti-realism to mathematical realism. I think… So my prediction is, we’re gonna actually be talking more about mathematics than morality in this podcast, because everyone agrees that…
0:07:11.0 JC: Yeah, sure.
0:07:11.4 SC: Morality is tricky, but [chuckle] I guess the insight that you… One of the insights you wanna share is that the cases in each case for math and morality are more alike than you might think, even though they’re not exactly alike, is that fair?
0:07:27.6 JC: Right. Correct. Exactly, that’s right. Yeah.
0:07:30.1 SC: So let’s talk about mathematical…
0:07:30.9 JC: So… Yeah, yeah.
0:07:32.6 SC: Let’s talk about mathematical realism. Is that the same thing as Platonism? I don’t know what either one of those means, but what do those mean and how should you think about them?
0:07:41.0 JC: Yeah, right. Unfortunately, Platonism has come to take on a bunch of connotations that are sorta distracting in this context. So, I suspect that many people who aren’t familiar with the detailed literature will be inclined to say, look, I think there’s an independent fact about whether the twin primes conjecture is true, but don’t get me wrong, I’m not some kinda crazy Platonist, you know? I don’t think there is like Platoplasm out there in [chuckle] Plato’s heaven. Look, all I mean by mathematical realism is that mathematical statements of the sort that you encounter in an ordinary mathematics class, whether it be a number theory or geometry or set theory, these have truth values. They are the kinds of things that can be true or false. And whether they’re true is not up to us, you know?
0:08:41.8 JC: We don’t get to decide whether or not the twin primes conjecture is true. We don’t get to change our convention and make it false. We can of course, mean different things by the same symbols, but that won’t change the truth value of the thing we actually express presently with the twin primes conjecture sentence. So, that’s what I mean by mathematical realism. Now, I should say that it’s not just a coincidence that people often use the word Platonism to describe such a position. The reason people use that word is because it’s very hard to see how anything answering to the constraints I just gave could be anything other than what you might call Platonic objects.
0:09:26.3 SC: Okay.
0:09:27.0 JC: It doesn’t seem like numbers, for instance, are the kinds of things that we’re ever gonna observe through a telescope, that building a bigger particle accelerator is gonna discover… So if there are such things as infinitely many prime numbers and specifically Twin Prime numbers, then evidently they’d be something strange [chuckle] from the standpoint of an ordinary naturalist.
0:09:56.0 SC: Why don’t you remind us what the twin primes conjecture is?
0:10:00.6 JC: It’s just the claim that there are infinitely many prime numbers, P, such that P plus two is also prime.
0:10:07.1 SC: And have we proven this yet? Do we know if it’s true?
0:10:08.7 JC: No. Yeah, an interesting fact about it is that people seem to be getting closer and closer to proving it, in other words, they are establishing that there are infinitely many prime numbers, P, such that P plus blah, blah, blah is also prime, [chuckle] where blah blah, blah gets lower and lower. But they have not proved that there are infinitely many number… Prime numbers P such that P plus two is also prime. So, it’s a claim that my general senses number theorists think is very likely to be true, but it hasn’t been proved.
0:10:44.1 SC: Right. But we… For the examples that we’re using here, it’s nice to pick an example where we haven’t proven it so that we can sort of sensibly talk about, well, what if it weren’t true, but the standing of all of these claims is just the same as if we talked about one plus one equals two, right?
0:11:01.7 JC: You mean the standing of the unproven conjectures or the standing in general of these kinda fancy mathematical claims, whether proven or not?
0:11:09.4 SC: I just meant that when you were talking about the truth, there exists a truth value of a claim independent of human beings, etcetera, we might as well just use one plus one equals two as an example of this, right?
0:11:21.3 JC: Sure, sure. Yes, there are some distracting features of the super rudimentary cases, which is one reason I prefer not to use them. I’ll give you a simple example. So, right, ask a philosopher for an example of a kind of claim that you can tell just by thinking that’s also necessary, that is, it couldn’t have been otherwise, and they’re likely to cite something like one plus one is two.
0:11:55.1 SC: Yeah.
0:11:55.7 JC: It’s a little tricky though, because the sentence one plus one is two, arguably goes proxy a lot of the time in ordinary language for a claim of pure logic. So, it arguably goes proxy, say, for a claim like, if there’s a book on the table and there’s a book on the chair, and notebook on the table is a book on the chair, then there are “exactly” two books on the table or the chair, where exactly two there, even though I’m using the expression two can actually be defined without any reference to numbers. So, this was a famous accomplishment of Frege’s to show that sentences like that can actually just be understood as claims of pure logic. Where by pure logic for those who are familiar with this language, I mean, pure first order logic, which is kinda like the least controversial common core of logic that every philosophy student and analytic philosophy gets introduced to for instance.
0:13:11.7 SC: This is why talking to philosophers is great. You can really dig into one plus one equals two in a principled way, which is why. [chuckle] But still, I want to keep that simple example in mind, because let me put forward a naïve point of view, which I think is the point of view, but both that I have and that you think I have, which is that [chuckle] there’s no such thing… Well, it’s not true that one plus one equals two, that’s not the thing that is true, the thing that is true is…
0:13:43.7 JC: Right.
0:13:44.3 SC: The statement one plus one equals two with certain conventional understanding of what those symbols mean and assuming certain axioms about number theory is true. And if I had assumed different axioms, like if I were doing addition Modulo 2s with a one plus one equal zero, then it would not be true. And so, there’s a point of view that says that all mathematical truths are sort of conditional statements of the form, if you accept these axioms, then you get these theorems, or something like that.
0:14:14.3 JC: Great, yeah, yeah, yeah. So, a couple of things to say about that. So the first thing to say is the thing I said a minute ago, which is that you gotta distinguish between the truth of a sentence under some interpretation or another and the truth of what it expresses. So look, I could use the sentence, there are tall buildings to express the claim that there are numbers, and under that interpretation, if we think there are numbers, then we’ll agree that there are tall buildings before going out and checking whether there are any tall buildings. But obviously, in ordinary language to say that there are tall buildings has to do with buildings, not numbers. So we’re not interested in the truth value of that sentence under some interpretation or another, we’re interested in what it expresses. And one plus one is two in ordinary language, expresses that one plus one is two. And so, what we’re interested [chuckle] in is the status of that claim, that’s the first point.
0:15:14.2 SC: Yeah, okay.
0:15:15.4 JC: But the second point more importantly concerns the prospects for a kind of view which is extremely tempting and was quite popular at the turn of the 20th century, and I don’t wanna make it sound like it was all due to Godel, but cartoon narrative is, the simplistic narrative is basically after Godel’s theorems, this became a whole lot less persuasive of a position. Here’s the following position, which I take you to be sketching, which has a lot of appeal. Look, I distinguished a minute ago between Mathematics and Logic, so I distinguished the claim that one plus one is two, which on a face value reading says basically there are numbers one and two, and the one bears the plus relation to itself and to two, from the logical truth that blah, blah, blah, and I gave you that logical truth.
0:16:14.1 SC: Yeah.
0:16:15.5 JC: But why not now, when we move to fancier things like the Twin Prime Conjecture or something, also have take this to be surrogates for logical truths. Now, in this case, we can’t take the surrogate logical truths to be the same sorts of logical truths. There’s no logical truth corresponding to the twin primes conjecture of the sort I gave you for one plus one is two, but there is the following kinda thing that will work perfectly, generally. Whenever I say P for some mathematical proposition, what I really mean is, if the axioms are true, then P.
0:16:50.8 SC: Right.
0:16:52.5 JC: And that is… If that is… If you can in fact prove P from the axioms, that’s a logical truth. And now we’re just in the business of logic, there’s no special math, you don’t need to believe in whatever objects P refers to. Okay, there’s a number of issues to deal with here. Here’s the first issue. If we think the goal presumably is to be able to, so to speak, do everything we wanted to do previously and believe all the sentences we wanted to believe previously without really believing in numbers and worrying about objects outside of space time, how could we know them and so forth. So that means we wanna be… We don’t wanna take a stand on whether the axioms themselves are satisfied. That is, whether they in fact have a model, whether or not there are in fact numbers such that you got zero and you’ve got the successor of a number whenever you’ve got the number and you’ve got the induction scheme. But we don’t wanna worry about whether or not there really is such a thing out there, satisfying those. We just wanna say those axioms, whether satisfied or not, imply this other thing.
0:18:02.2 SC: Yeah.
0:18:03.9 JC: Okay, here’s the problem, if we take the claim if the axioms, then the twin primes conjecture to be what’s called the material conditional, basically just to claim if P then Q, which is equivalent to either it’s not the case that P or Q. Well then, if the axioms aren’t satisfied, that whole thing comes out true no matter what.
0:18:26.5 SC: Yeah.
0:18:26.9 JC: So it will be equally true that the twin primes conjecture is true on this reading and equally true that it’s false if there’s no numbers to satisfy the axioms. The deeper problem, if you replace the conditional with a logical relation is that it turns out, this is Godel’s famous second incompleteness theorem, to be… That it’s consistent to say false things about consistency. So it’s consistent to say false things about what follows from the axioms. So, if Peano arithmetic, standard arithmetic is consistent, then so is Peano arithmetic conjoined with a claim that it proves an inconsistency. Because if that weren’t consistent, then arithmetic could prove its own consistency if it were consistent, and it can’t by Godel’s second incompleteness theorem. So, this is why the… These are some reasons, there’s a bunch of reasons, but these are some of the reasons why the very attractive thought that lots and lots of people were drawn to, including Russell, Hilbert and others, before the ’30s, doesn’t seem to stand up to scrutiny.
0:19:39.0 SC: Good. So, just to… This is exactly where my head starts hurting, and I love this stuff, I read Gödel, Escher, Bach when I was a kid, and I want to understand it better, but there’s some aspect of it where I can get to the point where I am sure that I understand what’s going on, and then the next day, it’s completely gone, so I’m just gonna keep trying over and over again. So is the essence of what you just said, the idea, at least post-Gödel, even if it’s not completely his fault, but that you can’t just associate truth with theoremhood, that there’s some… That the notions of being a theorem in an axiomatic system and the notions of being true are different. Is that…
0:20:19.3 JC: That would be a kind of rough and ready slogan way of putting the up-shot of the second argument I gave. Yeah.
0:20:30.4 SC: Yeah. Okay, and there were some things sneaked in there about first order logic, which implies that maybe under second order logic, it might be a different situation.
0:20:41.8 JC: Right. Okay, so second order logic, there’s a famous debate in philosophy about whether second order logic deserves the title logic…
0:20:53.4 SC: You’ll have to do the best you can at explaining what first order logic is and what second order logic is, sorry about that.
0:21:01.4 JC: Sorry, right, of course, yeah. Of course, yes. So first order logic is the logic that allows you to talk about things and predicate properties of them, but not talk about the properties you’re predicting of them, roughly speaking, and I say roughly, because you could always let properties be in the realm of things you get to talk about with the original variables that you’re ascribing predicates to, but roughly speaking, that’s first order logic. Second order logic… And the key terms here, for those of you who want to pursue the matter more, the key terms are… The key term is quantifiers, so in first order logic, you’re allowed to quantify over things, but not quantify into predicate position. So you can say there’s an X such that X is tall, but you can’t say there’s an X, such that… There’s an X and there’s an F, such that X has F, where F is some property, like being tall. In second order logic you can do that, you can predicate, you can quantify into predicate position.
0:22:12.2 SC: Yeah, I missed the distinction you just drew there between saying that X is tall and saying that X is F and F means being tall.
0:22:21.4 JC: Okay. So in first order logic, you can say there’s an X and X is F, where you don’t get to quantify into the place of the predicate F, so you don’t get to let that be a variable, so F is some constant predicate, it corresponds to… In the technical set up it corresponds to a set of things. And that sentence is true, if there is something in that set. In second order logic, you get to quantify into the place of that constant predicate F, so you get to say, let’s think of a small x and big X, just to make it clear that we’re talking about variables. You get to say there’s a small x and there’s a big X, such that big X applies to small x, but you don’t get to do that in first order logic, that’s the idea. Does that make sense?
0:23:23.3 SC: It does actually make sense, roughly speaking, or amazingly, I should say. And is something, do I recollect correctly, that number one, second order logic is not universally accepted as a good way to go by logicians or philosophers, and number two, but if we help ourselves to it, then we can recover a notion of logical… Of a truth being close to theoremhood, or at least axioms picking out structures in the world?
0:23:58.4 JC: Yeah, the rough… Okay, right. So as I was saying, there’s this debate about… Famously, Quine said, a second order logic is just set theory in sheep’s clothing, and you can kinda get a sense of where he would have been coming from from what I just said about what second order logic is, ’cause you get to quantify over sets basically. Now, there is some debate about whether that’s really the only way to read the second order quantifiers, but on its face, that’s what you’re doing, and it really looks like this is just set theory. But here’s the key point about why people haven’t been impressed by appeal to second order logic in trying to recover the idea that math is just logic in disguise.
0:24:44.7 JC: The key idea is that second order logic is incomplete. And what I mean by that is you can come up with proof systems such that if the things that… If you can prove the thing, then it’s valid, second order valid, but you can’t come up with something such that if it’s second order valid, you can prove it, and so people feel like, surely logic has to correspond to provability. I mean, you can use the word logic for some system of truths, some collection of truths, but if it doesn’t correspond to things that could in principle be proved using some proof system, that’s a very misleading way of putting the point. So that’s why people have felt like second order logic is not going to vindicate the original hopes, both, it seems to me, ontologically, that is to do with what exists, it seems to be committed to basically just what math was committed to, sets and objects constructed out of them.
0:25:51.3 JC: But even if you don’t want to worry about ontology, at the epistemological level it doesn’t seem any easier to figure out how we could be reliable detectors of second order consequence than it would be to tell how we could just know the mathematical truths understood at face value.
0:26:08.4 SC: Okay, I think I do understand that. And for the people listening out there, I think that’s as weird as it’s gonna get. I really… I kind of wanted to sort of dig into this particular area, which is just as recondite and difficult for me to understand, you get it out there on the table because it is just fascinating to me and it’s a rabbit hole to go down. But as you said before, this was sort of… All of this was… All this Gödel, etcetera, stuff and truth being theoremhood or otherwise, a set of ideas was one argument, but there’s another argument about books on chairs, and I didn’t want to let that completely slip by because I think that this should be graspable to us all. So did I understand correctly that the example of the books being on chairs was somehow meant to illustrate the idea that there was a logical truth about the idea of one plus one equals two?
0:27:00.1 JC: Yeah, so remember at the beginning, I started out with this example, the twin primes conjecture and… And you pointed out, rightly, but the view is that the twin primes conjecture has the same kind of status as two plus two is four, so why don’t we just use two plus two is four? And I said, well, that’s a little… That can be a little confusing in this debate because when you’re talking about elementary arithmetic truths like two plus two is four, say one plus one is two, because already that becomes complicated when you try to translate the surrogate logical truth into ordinary language, but… Whereas one plus one is two, say, there’s a claim in the neighborhood that arguably in a given context is what we’re really trying to express when we say that one plus one is two, it has nothing to do with numbers, and it’s just a claim of pure logic that everybody has to agree is true.
0:28:01.1 JC: There’s no surrogate claim in the case of the twin primes conjecture or even the claim like you can’t tile a certain floor a certain way because the number of tiles would be prime or something. What I’m saying is that when you move beyond the most rudimentary arithmetic claims, this simple translation scheme gives way and you no longer can pull it off, but the original point was, if you take something as simple as one plus one is two, well, and you ask what the person on the street means by one plus one is two, my guess is there’s no clear answer to that question. They probably mean something like, if there’s a bike over there and there’s a bike over there and… No bike over there, there’s a bike over there, then there’s two bikes, where two bikes again, is actually shorthand for this complicated logical expression that doesn’t involve reference to numbers.
0:29:03.7 SC: I see, okay, so this is an argument for one plus one equals two not being the paradigmatic example we should have in mind, because maybe what you’re thinking of…
0:29:14.1 JC: Yeah, once you want to get into the question. Yeah, if you want to have a debate about whether to be a realist or a Platonist about math, I think it’s distracting to focus on the real rudimentary cases, because there’s, I think, ambiguous and natural language between claims about numbers and claims that everyone accepts, whether they’re Platonists or not.
0:29:35.8 SC: Well, maybe the other example that I personally will always have in mind and I want to see how it fits in here is geometry and the parallel postulate, which is an obvious example, because back in the day, the postulate that two initially parallel lines stay the same distance from each other was given by Euclid as a postulate, but people thought, oh, come on, we should be able to prove that, and eventually we realized we couldn’t prove it from the other axioms because, in fact, you could replace it with different axioms and get hyperbolic geometry or Riemannian geometry or whatever. So given that, given the fact that in geometry we have a clear case where there are alternate choices about what axioms to pick that give rise to different truths following from those axioms, well, how does that fact stand in relation to ideas about realism about these claims? Like what is supposed to be real, what happens in the actual world or just some logical concatenation of statements.
0:30:35.7 JC: Good. So there’s a distinction one has to accept between different areas of math, at least prima facie, and Stuart Shapiro, I think, calls the kind of areas like geometry, algebraic areas, just like it would be clearly misconceived to ask whether the axiom of commutativity of groups is true, period. It would be misconceived to ask whether the parallel postulate is true, period. On the other hand, claims of, say, arithmetic don’t seem to have that flavor, and you could think that we could just sit tight there and say, okay, they don’t seem to have that flavor, but on inspection, the best overall philosophy of math is gonna be one where it’s all like geometry everywhere.
0:31:39.1 JC: But as I said a minute ago, that doesn’t seem to be tenable, because arithmetic claims are really… Or I should say, claims about consistency are really arithmetic claims by Gödel’s second incompleteness theorem. So in other words, suppose somebody wants to say, here’s a very… Let’s go back to your original view that sounded pretty good, something like, look, there’s just different axioms you can have, and then you can ask what follows from the axioms. That’s just logic, and there’s no need to get into the business of speculative cosmology, [0:32:20.1] ____ numbers in order to accept that.
0:32:22.5 JC: Here’s the problem. Claims about consistency… If you want to take a stand on those not being like the parallel postulate, so you can’t just flip their truth value by flipping the system you’re working in, you’re taking a stand on arithmetic not being like the parallel postulate, ’cause those are arithmetic claims. So just to repeat the example I gave, if standard arithmetic is consistent, then so is standard arithmetic conjoined with a claim that says it’s not consistent. It codes that claim. So you’ve got two… You’ve got two theories, then, which are consistent if standard arithmetic is. One says, it says the axioms, and then it says it’s consistent.
0:33:06.9 JC: Here’s another theory, standard arithmetic, the axioms plus the claim, by the way, I’m not consistent. Those are both consistent theories, so if the consistency claim is like the parallel postulate, then we can’t even take an objective stand on what’s consistent with what, and that seems like a little bit too much to swallow. It seems like it’s turtles all the way down at this point, and we’ve lost control of what we were trying to say, which was there’s facts about consistency and what follows from what, but there’s no additional facts about what axioms are really true.
0:33:39.8 JC: So the big picture is, right, realism is a question about what’s true independent of us, but there’s this other dimension of the extent to which different areas of math are like algebra, where you can just choose different axioms, and there’s no real debate. And the mathematical realism debate, and it’s normally closely tied to a debate about how much our foundational theories are theories in which you can carry out metatheoretic reasoning and talk about things like consistency, the extent to which those are like geometry. And the key theory there is set theory, and there are different places to draw the line.
0:34:22.9 JC: And so for example, and then I’ll shut up about this, in the set theory case, for example, a very conservative view would be to say that any arithmetically sound set theory… Set theory is just the theory of sets, I know that’s an uninformative sounding claim, but it’s just the theory of collections of things, basically, and it turns out that you can take all claims of math basically to be short, to be… Really just another way of saying things about sets, this is a surprising fact, ’cause set theory has only one non-logical predicate it’s a member of. But anyway, so a conservative view would be to say that any arithmetically sound set theory is as good as any other.
0:35:05.7 JC: And by arithmetically sound, I mean a theory that doesn’t prove a false arithmetic sentence. But many people say, look, once we’re in the business of taking a categorical stand on the truth of the axioms, then why should we draw the line at arithmetic? Shouldn’t there be objective facts about the axiom of choice? Shouldn’t there be objective facts about what are called large cardinal axioms and so forth? So there’s a question of where you’re gonna draw the line, once you grant that, it can’t all be like geometry. So sorry, that was a lot, but the point is, there’s two issues, one is independent truth, other is how much are different branches of math like geometry.
0:35:46.6 SC: Yeah, no, actually, I’m gonna… I want to dwell on this a little bit because this is amazingly fantastic, this is clarifying things for me that I have been confused about for decades. So let me see if I can restate it in a way, and then you can tell me how close I am to having captured it. In my mind, I have this paradigmatic example of geometry where there are different choices about what axioms to start with that get you different theorems you can prove, different true statements you can make within those axiomatic systems. And you’re saying, well, you might be seduced into thinking that that is a model for all of mathematics, but here’s a reason why it’s not, because if you look at the example of arithmetic, the question of which axioms to use in arithmetic gets tied up with the question of the consistency of those axioms in a way that it’s not in the case of geometry. And so our willingness to just say, oh, pick whatever axioms you want about arithmetic is lower than it is for the corresponding geometrical axiom.
0:36:52.0 JC: Yeah, so let me try to be a little more explicit about that. That’s correct. So it’s not just that the truth of arithmetic claims gets tied up with the truth of claims about the consistency of arithmetic claims, it gets tied up with the truth of consistency claims, period. So for example, the claim that like set theory is consistent is really an arithmetic claim, it’s a claim that there is no natural number that codes a proof of zero equals one from the axioms of set theory. And it turns out that if set theory is consistent, it’s also consistent to say that there is a number that codes the claim of zero equals one from the axioms of set theory.
0:37:39.4 JC: And what’s going on there, is that a model of that overall theory is wrong, basically, about what counts as a natural number. From our perspective, if we want to be objectivists, you might say, about this and not think it’s like geometry, what we’re gonna say is what’s going on is that model thinks that a number that is not finitely many steps away from zero is actually finitely many steps away from zero, and that’s the number witnessing the proof of zero equals one. So that you said is basically right, I just wanna be clear that it’s a very general point, not something about just arithmetic, is that here’s a slogan form of it, if you want to take a stand on the… If you wanna take a stand on the consistency of theories, then you’re already taking a stand on arithmetic. There’s no way of saying I accept that there’s a fact of the matter about which theories are consistent in general while saying, but let a thousand flowers bloom, any set of axioms is as good as any other, ’cause some of those axioms are gonna disagree about consistency.
0:38:45.0 SC: Right, right. And the connection, it seems be coming right up to this connection to realism, so you’re saying that exactly because arithmetic has this set of implications for consistency, we want to think that a certain viewpoint… I don’t know if it’s right to say a certain set of axioms, but at least a certain version of arithmetic is the right one, the real one, the true one. Have I gone too far now?
0:39:16.0 JC: That is certainly… That is certainly the initially appealing view. So look, there’s a bunch of issues in the background that once again, kinda like with the two plus two is four versus twin primes case, that can be confusing, so maybe I should just distinguish. Look, there’s well-known debates about which logic is correct, so evidently those debates bear on what’s consistent, but I wanna be clear that this is an orthogonal issue, this is basically a debate about what whether there’s an objective fact about what’s finite. So a way to put it is, if you were to deny that there is an objective fact about the claim that codes the consistency of piano arithmetic, if you wanted to deny that, what you’d be denying is that there’s an objective fact of whether something’s a proof in a system of logic, where we’re allowed to look at different logics.
0:40:24.8 JC: So one debate is, do we know what we mean by finite? And what I’m saying at the moment is it sure seems like we’d better. Another debate is, once we think we know what we mean by finite, what’s the right logic? So that’s a complication. Back to the issue of realism, realism is once again sort of like an independent parameter, in the sense that one could in principle be a realist, I think there are independent facts to which our mathematical theories answer, but so to speak, it is all like geometry, and in fact, one could even be a realist and think that logic… The question of what logic is correct is like geometry. One could have this view.
0:41:08.3 JC: So for example, there’s not many people who are willing to go in print with the view that rides this train to the last station, as I’ve been discussing, but there’s a couple. And so, for example, Joel David Hamkins, in his work on the multiverse, has at least flirted with the idea that even questions of consistency in a logic are like the parallel postulate question. Now, most people feel like I’ve totally lost my bearings and now there’s not even gonna be a fact about what’s in the multiverse, ’cause it’s gonna depend on from what perspective in it you take. But I don’t wanna make it sound like that’s all the way off limits, there are respectable and interesting defenses of that position. Most people feel like you gotta draw the line before that, but that view, and this is the last thing I’ll say on this, is a realist view.
0:42:06.3 JC: So Joel describes his view as a Platonist multiverse view. He thinks that it’s almost like Platonism on steroids, ’cause where you might have thought there’s just the set theoretic universe and the natural numbers and so forth, he thinks that there’s, so to speak, a set theoretic universe for any set of axioms you might want to entertain, including ones that disagree with us about consistency.
0:42:32.7 SC: Right, good, I think I got that. I think I got that. And I’m always attracted to the sort of maximally pluralist crazy view, but maybe there’s good reasons against it, but… So let me sort of bring it back down to earth a little bit by kind of retreating… ‘Cause I think you’ve already answered this, but I’m just gonna ask you to answer it once again, now that we’ve laid out so much ground work, here is a view that I might have had, and you’re not under the obligation to give me everyone’s response to it, just tell me what you think is the best response to it. The view is, there’s a world. There’s a real world, there’s reality, and I’m gonna be Humean.
0:43:14.1 SC: For those of us who are listening to this, I also… I talked with Ned Hall a while ago about Humeanism about laws, so there’s not… We don’t think of the laws of physics as things that have separate existence, we just think of them as descriptions of the world. Okay, so I’m a reality realist, I think that the real physical world is real and that exists, but I don’t wanna be… And again, this is just the straw man position that we’re gonna squash once and for all, I don’t wanna be a mathematical realist, I wanna think… I just wanna say that all that exists is the world, and mathematical language can be a convenient way for we subsets of the world to talk about the world, but that doesn’t imply that it has any separate real existence. So once and for all squash that.
0:44:01.8 JC: Okay, well, look, I won’t claim to do that, but here’s a few thoughts to have. Okay, so the initial thought is a familiar thought from this literature. For people interested, it’s called the literature on the indispensability argument. So just a little background. WVO Quine famously wanted to be what’s called a nominalist about all abstract objects, all things like numbers and universals and things that wouldn’t be part of the ordinary physical world that Sean, you have an admirable belief in. He wanted to be a realist about that and kind of that’s it. And so he tried to figure out how to formulate all of basically our physical theories in a way that doesn’t make any reference to abstract objects.
0:45:04.2 JC: He figured, look, if this is just a language, just a shorthand way of saying something strictly about the physical world, then I should be able to say it in principle when the chips are down, even if for purposes of ordinary calculation, no one should waste their time pulling off the translation. And he and Goodman wrote this famous paper, Steps Towards a Constructive Nominalism, where they started that project. Well, long story short, Quine came to think, this is not gonna happen, this is not gonna be able to work. And he became a “reluctant” Platonist. So Quine’s view about math is sort of an awkward component to his overall empiricism.
0:45:53.8 JC: He was a kind of radical empiricist, he didn’t accept all sorts of stuff that ordinary philosophers who are quite empiricist-minded philosophers do. But he felt like I got to at the end of the day accept math and be a Platonist about it, because I want to accept science and what in the hell could it mean to say I accept the standard model of particle physics and I don’t believe in math? What does that mean? So that’s the first point. The second point is, even if you thought that either you could in principle pull this project off, and by the way, people continue to try. So Hartry Field famously inaugurated a kind of new approach to it.
0:46:39.0 JC: Even if you thought you could pull this project off, it’s hard to see what you’re really gaining, and here’s a short version of why. Everybody needs to be able to talk about consistency. I just said a minute ago that talk of consistency can be understood as just an arithmetic claim. Now, people like Hartry Field or people who want to make no reference to mathematical entities still need to talk about consistency, ’cause they wanna be able to say things like their physical theory that makes no reference to mathematical entities is consistent. So what they do is they introduce this primitive bit of what’s called modal ideology, modality is just the theory of possibility and necessity, and you can think of consistency as a kind of possibility claim, logical possibility.
0:47:28.1 JC: So something’s consistent if it’s logically possible. And what they do is they introduce, this is a primitive bit of terminology in the formal framework, and they take a stand of course on what’s consistent with what, and they claim that physics is consistent, but that doesn’t talk about numbers. My feeling is, what is gained by that? So okay, let’s suppose you could pull this off, let’s suppose that you could do away with any talk of numbers, and in fact, there’s cheap ways of doing this too, which I won’t go into right now, unless you want me to. But there’s cheap ways of avoiding any reference to mathematical entities too, but you just trade all that ontology, that is all that reference to mathematical entities, for beefed-up what Quine called ideology, that is basically operators in your basic language, and where the original question is, how in the world am I supposed to know these arithmetic facts about things that I can’t bump into or do experiments on, you’re trading that for a claim of the form diamond, that’s the symbol used.
0:48:37.1 JC: And then follow that with the claim P that you wanna claim is consistent, where exactly the same epistemological problems arise. How am I supposed to know that? There’s no experiment, there’s no observation. It has exactly the same kind of epistemological status as the status of the arithmetic claim that we wanted to do away with. So the big picture is, on the one hand, it’s very hard to see how to pull off the project because it’s hard to see what you could even mean to say that you believe and say… I mean, you’ve written a bunch on quantum field theory and things like this, and general relativity, it’s hard to see what you could even mean by saying you really believe all that, but don’t get me wrong, I’m not a mathematical realist.
0:49:22.9 JC: But the second thing is, even if you could make sense of what you mean, you’re gonna be committed to claims which seem just as good as arithmetic claims from the standpoint of epistemology, and so I don’t really see what’s gained from the standpoint of the problems that have traditionally motivated the desire to get rid of mathematical entities, namely how could we know about these things outside of space and time? Surely, we just believe in the physical world and what we can know about via observation and experiment. So I don’t know if that permanently dissolved with you, but there you go.
0:49:56.9 SC: Only temporarily, I’m afraid. Nothing permanent in this world. But at least temporarily, I get the thrust of it, but I also… So that’s sort of the straw man, anti-realist position, but let’s also squash the straw man realist position that says there is… I forget what you call it, Platoplasm, that there are triangles in the sky, and we’re all participating in triangleness or something like that, that’s not necessarily what is meant by mathematical realism, it’s not the kind you need to talk about the standard model of particle physics, it’s more something about the mind independent truth of mathematical statements, yeah?
0:50:39.5 JC: Yeah, that’s exactly right. Look, the triangle in the sky stuff, presumably, yeah, was never really a part of the realism that anyone advocated that I know of after Plato with his forms and the things in the world that mimic them. But as I said at the beginning, Platonism, as the view that there are abstract objects, that view does seem to be a little hard to deny if you’re a mathematical realist, because let’s just understand mathematical realism as the following view, mathematical claims of the sort you encounter in, say, number theory class, they’re true or false, independent of us, independent of our conventions. Again, of course, we could mean different things by the same sentences, that’s irrelevant, that’s like saying that you could meet… That banks might not be monetary institutions, ’cause you could mean instead that they’re land masses on a body of water, that doesn’t mean anything about banks, it means something about the word bank.
0:51:49.5 JC: So the idea then would be that we accept these claims as true independent of us, but what are they talking about? I mean, they’re talking about numbers and metric tensors and whatever, and so on an ordinary understanding of how truth works, and by truth, I don’t mean something heavy duty like capital T, I mean basically just what it takes for a sentence to get it right, like a very minimal idea, stemming from Tarski. If the sentence “there are prime numbers greater than a million” is true in just this very minimal sense of truth, then there’s gotta be something witnessing that and it’s gotta be a prime number.
0:52:37.2 JC: And what would that be? So I don’t wanna make it sound like the cartoon. If the cartoon view is supposed to be just the view that there really are abstract objects and that’s part of belief in realism. I don’t wanna make it sound like that view is silly, ’cause it actually is pretty hard to see how you can be a realist and not think that. But of course, that view is not the view that there’s triangles in the sky, right?
0:53:03.9 SC: Yeah, well, but I guess we’re… Well, part of the problem here is that we are using natural language words like exist or to be or are real to apply to tables and chairs and quantum wave functions and triangles, and maybe… I’m sure someone has, but is it worth exploring the project of putting subscripts on these different meanings of the word real when we’re talking about these different realms?
0:53:31.3 JC: Yeah, sure. So look, there is a lively debate about whether existence might in some sense be relative. The literature that is most developed on this and most taken most seriously today is the literature on what’s called quantifier variants, but here’s the standard kind of… Without getting into the technicalities of why those arguments might not be compelling, here’s the standard alternative point of view, for example, it’s Quine’s. Look, something either exists or it doesn’t, what you really mean to be saying when you say something like numbers exist in a different sense than electrons is just that they’re different.
0:54:27.8 JC: Numbers aren’t in space and time. They don’t have energy. Blah, blah, blah. But the notion of existence is just supposed to be the thinnest possible notion. It’s supposed to be, basically, there are truths about the thing. And you could introduce by stipulation subscripts to different uses of the word exist to mean. So for example, there would be no objection from this point of view to say, whatever I say exists up to, what I mean is it exists in its abstract. Okay, fine. But then all we’re just saying is, yeah, it’s just something that we can now say in the original language using the one and once and for all exists.
0:55:20.1 JC: If there’s anything like a consensus on this, the view is like, look, we can introduce by stipulation a thinnest possible notion of exists, it’s the sense in which if there’s a God, it exists, it’s the sense in which there are numbers they exist, if there are wave functions they exist. If there’s space time points, they exist. If there’s moral values, they exist. It’s the same sense. Of course, all these things are wildly different things, and so they all exist or not, but they’ll have different properties, but it’s just kind of like… It’s just kind of like confusing the debate to start using that one word, which is one of the few things we thought we had a grip on in philosophy in different ways, that’s the kind of standard alternative view.
0:56:06.1 SC: I got it. Yeah, I think actually that’s less immediately compelling to me than the other things you’ve said, but I do appreciate the force of the argument. So you’ve just made a reasonable case against slicing the concept of reality too thinly, but one of the moves you do make in the book is distinguishing the concepts of reality versus being objective, right? This is a major emphasis of what you’re trying to say, so naively, you might mix up those two things, what do you mean by being objective if other than being real?
0:56:42.1 JC: Yeah. Good sir, this is exactly where your example of the parallel postulate comes in real handy. So by being real or realism, I just mean that there’s an independent fact about the thing. By being objective, I mean that to a question from the area only one answer can be right, and the parallel postulate gives us an interesting case to think about in teasing these concepts apart, ’cause suppose you’re like a Platonist like the cartoon Platonist is Kurt Gödel. And you might ask what do Kurt Gödel think about geometry, not set theory or arithmetic, geometry. As far as I know at least, and it would be shocking to learn, he didn’t think that there was one true geometry as a matter of pure mathematics. Of course, there’s one true physical geometry, most people would assume, but it’s not like he thought that… Sadly, hyperbolic space doesn’t… It wasn’t allowed into Plato’s Heaven, but Euclidean space was. Okay, so there’s some sense in which there’s no objective fact of the matter as to whether the parallel postulate understood as a question of pure mathematics is right or wrong.
0:57:54.6 JC: You can sort of have it either way. If you want it true, well, then I’ve got euclidean geometry for you. If you want it false, I hereby give you hyperbolic geometry. There’s no real objective matter of fact, but of course once we fix the meanings of the terms if we’re like Gödel and we’re realists, there’s a perfectly independent fact. It’s not like we get to make up the facts about the hyperbolic space or euclidean space. So what you might of is objectivity is kind of like a matter of uniqueness, it’s a matter of the richness of the reality. Are there lots of different realities kind of all answering to this discourse or is there a unique reality? Whereas, realism is just the question of, what is the nature of the reality? Is it mind dependent, is it convention dependent, or is it totally independent of us in every interesting sense? That’s the idea.
0:58:50.4 SC: Okay, now that actually does make sense. That sounds like a useful distinction and maybe one of the uses is when we finally get to compare all of our new found mathematical sophistication to morality, right? Now, given that we now understand math morality should be easy, right? So what are the options on the table when it comes to moral philosophy, are we even gonna worry about moral realist positions, or are we just gonna say, “We’re good naturalists, we’re not moral realists, we just wanna compare it to math.”
0:59:18.5 JC: Okay, so the point of the first chunk of the book until the last chapter is to say that when it comes to the question of realism, moral realism is on every bit as good footing as mathematical realism, as surprising as that would sound. So for example, all the obvious dis-analogies I argue, don’t hold up to scrutiny like that, well, we settle things in math once and for all with proof, whereas in ethics just people argue endlessly or in math, our theories get applied to the physical world where there’s no analog in the moral case or in ethics, we can explain why we have the moral beliefs we have without appealing to their truth via say, evolution or something, whereas there’s no analog in the mathematical case and so on and so forth, so we haven’t talked about that stuff, but just to let you know, I go through all that stuff and the argument is supposed to be on none of those… When it comes to none of those issues, is there actually a deep dis-analogy as surprising as that sounds.
1:00:30.4 JC: Okay, we then get to the issue of objectivity, and the position that I advocate, and this is controversial of course, about math is the one I mentioned earlier when I said a conservative view would be, that is, it’s the furthest view in the direction of it’s all like geometry, that you can go without losing a grip on what you mean, because of this stuff we talked about with Gödel and consistency and how that’s tied up with arithmetic. So the final view about math is realism, we don’t get to make up the mathematical facts I think that one simple reason to think that is we don’t get to make up the physical facts and the physical facts are up to their ears in mathematical facts, so there’s no clear way to be a physical realist, a naturalist and not a mathematical realist. On the other hand, what the debates that have occupied most people in the foundations of Math for most of the last 100 years, debates about axioms, like the axiom of choice, debates about axioms like extensions of ZFC, what are called large cardinal axioms, strong axioms of infinity.
1:01:56.5 JC: Those debates, I think, are all like a debate over the parallel postulate would be, if we were so misguided as to have one understood as a pure mathematical debate. So Math, I think in a slogan is as non-objective as you can be without getting into the problems I mentioned at the beginning, but it’s perfectly real and independent of us, and we don’t get to make up the truth. Okay, but here’s the trouble in the moral case, moral realism, I think, as I said, the argument of the book is that it’s on as good footing as mathematical realism, and actually there’s a case to be made that I won’t go into right now, that it’s on better footing. Once you really get clear about what moral realism is saying, it’s kind of hard to see how it could fail to be true.
1:02:43.1 JC: Here’s one simple idea for why that might be so, math has its own special subject matter, or in the lingo of philosophy, it’s own ontology, but morality actually doesn’t, it’s about things like you and me and societies and so forth, and all it does is predicate new properties of them, but it turns out to be an enormously difficult question to be clear about what it means to say there are new properties, that is to say, it turns out to be an enormously difficult question to be clear about what it would mean to be say a realist about some properties over and above some other properties, unless you want to be what’s called a Platonist about properties, which nobody does in these debates, ’cause that’s a different issue and one that certainly a naturalist shouldn’t be inclined towards.
1:03:32.4 JC: So that’s kind of like just a sketch of a reason to think that it’s not just that moral realism is on as good footing as mathematical realism, the case is stronger. Actually, it’s hard to see how not to be a moral realist, you might say. Okay, but here’s the issue, whereas in the mathematical case, with the exception of these arithmetic sentences that I think that we really can’t be pluralist about, we really can’t say it’s like geometry, ’cause they’re so tied up with meta logic in the ethical case, we don’t seem to have the option of that kind of pluralism because ethics, unlike math or physics or geography, is supposed to tell us what to do. If I’m wondering to use a tried case from the utilitarianism versus de-ontology debate whether to kill the one to save the five, I can’t say, well, it’s morally good someone to kill the one to save the five, it’s morally too bad to kill the one to save the five, and that’s all there is to it, ’cause now I gotta figure out whether to do what’s morally good sub one or morally good sub two. The question gets delayed.
1:04:41.0 JC: And what I think this shows is that though moral realism in the strict sense is on just as good footing as mathematical realism, and in effect, it comes cheaply, and so in fact does what you might call moral pluralism, the idea that moral truths aren’t even unique, it’s like geometry and ethics also, but what it shows is that the final question before action, the question that we are asking ourselves when we ask ourselves what to do, the deliberative question is not a question of fact, even if there were moral facts, it’s not that question. So just to be clear, then I’ll shut up about this. [laughter] Hume famously said, “You can’t derive an ought from an is,” that is pretty uncontroversial. You can’t learn that something is natural and then infer from it that therefore it’s good or something. I’m saying that you can’t even derive what to do from a question of ought. You can’t even derive a… Okay, so this is the right thing to do, or this is the good thing to do, that won’t even settle the question of what to do as a question of action. So questions of action are objective, but they’re totally not real, they’re the things that remain when the facts are settled, whereas mathematical facts are totally real but totally not objective or as non-objective as you can be without getting into the quick sand of the good old stuff.
1:06:13.5 SC: I guess I’m a little bit confused because some of the things you said seem to take implicitly to beg the question of whether there is objective moral statements, I mean, if we just believe to again, be super straw man about it, that what we call morality is just a gussied version of people’s preferences, I prefer that you not do that, I prefer that you do do that, etcetera, then I don’t see how the claim that morality… How can I then be forced into believing in more realism, even if I bought into your claims about mathematical realism?
1:06:56.7 JC: Okay, good, so there’s a verbal, in the pejorative sense issue going on here that we should clear up, and that is… One question is just, what is the right natural language semantics of moral claims?
1:07:14.0 SC: Sure.
1:07:15.0 JC: And my feeling is that there’s probably no determinate natural language of moral claims, people probably mean lots of different things, and once you start reading a lot of philosophy, your meaning probably changes and so forth. So look, at the turn of the 20th century, as you well know, there was a project of trying to show that all we mean in saying that something is good or bad, is something like boo that thing, or yay… Or sorry, yay that thing or boo that thing. We’re not saying something that can even be true or false, ’cause boo and yay can’t be true or false, we’re doing something more like venting emotion like you do at a football game.
1:07:53.8 SC: Yeah.
1:07:55.8 JC: So certainly in that case, there’s no moral facts, there’s not even relative facts, we’re not even in the business of stating any kind of fact, that is a question of natural language semantics, that’s a question of what people are actually doing when they utter some sounds. There’s another question of set aside natural language semantics, I hereby stipulatively introduce some language that will concern how we ought to live our lives independent of what anybody thinks, and what I’m saying is belief in those things, whether or not that’s what ordinary people are talking about, that’s on as good footing as belief in the… Belief in, say numbers, but that’s not what settles practical questions, what settles practical questions is more like the stuff you believe in, it’s more like stuff like questions that don’t have truth values.
1:08:49.0 JC: So when I ask myself the question of what to do, I’m not asking myself the question of what I ought to do, because questions of what I ought to do, if the argument is sound, don’t settle questions of what to do, ’cause I ought say utilitarian, kill the one to save the five. I ought, deontological not. And now the question is just whether to do what I ought utilitarian or ought deontological to do, and that can’t be a question of whether I ought in some other sense do it on pain of regress. So you might be right about natural language semantics, but I don’t see how that settles the metaphysics, which is what seems to me to be the important point.
1:09:29.1 SC: Nope, sorry, you just lost me completely. [laughter]
1:09:34.6 JC: Oh, sorry.
1:09:35.1 SC: I go along… And I mean, I really, really wanna get this, so I completely buy that there is something called what I ought to do under utilitarianism, there’s something else called what I ought to do de-ontologically or probably many different sub-versions of all those things.
1:09:49.0 JC: Yes, yes.
1:09:51.0 SC: But then I don’t see any extra thing from that, then sometimes what you’re saying seems to presume that there is some just capital ought thing I ought to do. Are you saying that or are you denying that?
1:10:06.4 JC: What I’m saying is, even if there were such a thing, and the key question is what you could mean by that, but even if there were such a thing, it wouldn’t settle what to do, where the what to do is the question of action. So here’s the point, let’s go back to the mathematical case where we’re not dealing with arithmetic. The position of the book, and I think you’re sympathetic to this, but let me know if not, is that the question of whether every non-empty set has a choice function that’s just the action of choice is like the parallel postulate question, we don’t need to take a stand. We can say, every set sub-one has a choice function, but some set sub-two don’t, and that’s all there is to it. Everybody can have their own set theoretic universe and go home happy so long as we stick to arithmetic objectivity. Okay, question, could you hold that are Ethics are like that. Could you hold that the question of whether we ought to kill the one to save the five is like the parallel postulate question? I don’t think so.
1:11:07.5 JC: You could hold the question of whether we ought to realistically construe it is, but that would just show that questions of what we ought to do realistically construed don’t settle questions of what to do, that is questions of action, questions that we’re deliberating over because… Or as in the set theoretic case, you can sort of have all the trees laid out before you and you can help yourself to one or another depending on your interest and aims. In the ethical case, we’re supposed to appeal to ethics to figure out whether to kill the one to save the five. And we can only do one or the other. We can’t have it both ways. We have to take a stand in ethics or an action, I should say, in a way that we don’t have to take a stand in math. So what I’m saying back to your question is, even if the notion of Kapitalot [1:11:56.1] ____ made sense, and I don’t know what you mean by that, beyond just saying that there are some independent truths about what you ought to do, which I’m fine with, even if that notion made sense, it wouldn’t actually do what you want it to do, ’cause it wouldn’t tell you what to do.
1:12:14.0 SC: Well, is there a distinction between… So I think you’re drawing a distinction, but I’m not sure I’ve heard it drawn before, and if so, I wanna separate out the questions between the judgment aspect of morality, just attaching the label of rightness and wrong-ness to different perspective actions versus the action question. How to operationalize that and do something in response to it, are you saying that these are just two separate questions?
1:12:43.2 JC: Well, those are definitely separate questions. What’s a little less clear is whether there’s a question in between, so here’s three different questions, I think.
1:12:55.3 SC: Good.
1:12:55.6 JC: One question is, what ought I do? Understood as like a factual question about ethics. Another question is what I will do. Understood as a descriptive question about predictive psychology. I’m claiming there’s a question in between which is what to do, and sometimes, sadly, it might be that I deem that the thing to do, roughly speaking, is X but I’m weak-willed, so I don’t do it. So the answer to the question of what I will do is that I won’t do X. The question of what I ought to do, we could have it be either you ought to do X or you ought not, but the intermediate question of what to do is X, do X. You might think of the answer to the question of what to do as an imperative rather than a factual claim, so when I ask myself… I could ask myself the factual claim, whether I ought to kill the one to save the five, and I ought utilitarian and ought not de-ontological, and as you say, there’s virtue ethics and a million other sub-scripts we could add.
1:13:56.4 JC: And these are just all laid out before me like the different mathematical universes, the different set theoretic universes. Now there’s the question of which to use, that can’t be the question of which I ought to use, where ought is subscripted to one of these objects in the moral pluriverse on pain of triviality. So there’s the question of which to use. Now, the answer that’s gonna get is not a question you ought to use this or that, ’cause ought is gonna be subscripted to one of these oughts in the moral pluriverse, it’s gonna be something like, use that one where that can’t be true or false, and it doesn’t pretend to describe anything. But of course, it could be that, use that one, even though I don’t end up using that one ’cause I’m weak willed.
1:14:39.6 SC: [laughter] Well, but okay, good. I’m coming closer. I’m actually inching closer. It’s slow, it’s gradual, it’s painful, but I’m getting closer to the goal here, so good. The question of what ought to do is kind of like the geometry question, we could choose different systems and get different notions of ought in there.
1:14:55.8 JC: Yes, exactly.
1:14:57.7 SC: But in the case… But I don’t wanna do this, but I could imagine someone saying that there’s an analogy between the mathematics of geometry where you have many different axiom choices and many different possible consequences thereof, and the real single physical world where there is an actual one that is realized in nature, and that’s analogized to the many different ethical systems that have different notions of ought and the real one, the one that is actually the true one. And again, I don’t wanna be that person, but there are people like that, right?
1:15:34.0 JC: Yeah, so the million-dollar question is what though they could possibly mean. And so here’s what they… If you wanna get into the nitty-gritty of the debate, here’s what they actually say they mean. Okay, so famously, Nelson Goodman in the ’50s pointed out that it’s… So Nelson Goodman in the ’50s pointed out that not just you can’t induct on any predicate. Some predicates are what he called project-able and others aren’t. This was a kind of radicalization of Hume’s point that you can know that all Fs to date have been G, but not be certain that the next F will be G. There’s no deductive argument from that fact to the next step. Goodman’s point is there’s not even an inductive argument from the claim that all Fs to date have been G, so the next F will be G for lots of choices of G. The choice of G needs to be what’s called project-able. Now, what does that mean? Is there any informative gloss on the good predicates and the bad predicates? And one view that came to be adopted is that the good predicates are what you might call natural kinds. They carve nature at the joints. They’re not the gerrymandered things.
1:16:56.1 JC: And this view came to be adopted in lots of areas besides natural science. And one area where it can be adopted is ethics. You might say, “Yeah, there’s the moral pluriverse, but one of these moral-like properties is like the natural kind in the moral pluriverse. It’s the analog to the right way of carving up the physical universe or something.” And this is in fact exactly what people like David Enoch, people like Tristan McPherson, other realists say about this pluralist issue. It’s hard to deny that there is a property of maximizing utility, if there’s properties at all. The problem is that if utilitarianism is false, that’s not the natural kind, so to speak. That’s not the one that glows. Okay, but here’s the problem, and this problem in one form or another was pointed out by Shamik Dasgupta in the case of natural science. The problem is that either the claim that these properties are natural kinds is in itself a normative claim, a claim that carries with it information about that you ought to use it or not. If it’s not, then it’s just like the original Hume case where you learn that it’s natural, but who cares? That is, it’s like learning that it’s brown. It’s neither here nor there from the standpoint of practical deliberation. So maybe it’s the moral natural kind, I’m wondering which one to use, and it doesn’t pretend to speak to that.
1:18:28.9 JC: Okay, if it is a claim with normative import telling us which we ought to use, then the problem of pluralism re-arises at the level of natural kinds, so we ought sub-one use utilitarian ought. We ought sub-two use de-ontological ought or another way to put it is, it’s a natural kind sub-one, that utilitarian… Sorry, I should say utilitarian ought is a natural kind sub-one. De-ontological ought is a natural kind sub-two, and so on and so forth. So the problem just re-arises at the meta-level, it re-arises at the level of what counts as what you’re calling the one true moral universe. And so it seems to me that that doesn’t actually get off the ground if the problem is a problem to start with, it’s equally a problem with the alleged solution.
1:19:18.6 SC: Yeah. So this is… I actually am making progress, I think so. I’m almost hesitant to say what I think you’ve said because I’m sure I’m not gonna get it 100% right, but let me put it this way, I was initially confused by, on the one hand, you give a good sales pitch for mathematical realism on the other hand, you say that moral realism is on a similar ground, and I didn’t see that, but I think the missing part was it’s a thin kind of mathematical realism in some sense, or rather maybe it’s just mathematical realism, but the analogous thing and the moral domain is a thin kind of moral realism compared to what some people might want. You’re saying that you’re a moral… You’re defending moral realism in the same sense as mathematical realism, and of course, in the mathematics universe, we could have different axioms for geometry, but then there’s some reality to the relations or to the truths or whatever. So even though you say you’re a moral realist, or I don’t wanna say you’re a moral realist, but you’re defending more realism as analogous to mathematical realism, it does not go so far as to tell us what to do, it does not give us unique comparatives to act, is that… Right, okay.
1:20:34.4 JC: Exactly. Exactly, and here’s… I just wanna be clear about the strength of the claim. The claim is not just that like, here’s a position one could have moral realism plus the claim that it doesn’t tell us what to do, the claim is supposed to be, for better or worse, that no moral realist view could have that effect. So the view is that moral realism is in a sense mixing up two issues, one issue is like reality and independence of some truths, that I think is pretty cheap. Another issue is deliberation and action, that issue cannot be kind of packed in to the objects out there ’cause we can always ask ourselves whether to consult those objects or other objects, and that can’t be another factual claim on pain of regress. So it’s not that I think there is some other view one could have that it’s just way, way out there, and I don’t believe it according to which moral facts tell us what to do to, it’s that I think they can’t tell us what to do and the arguments supposed to show as much. On the other hand, if all you’re concerned with is, are there truths out there that are independent of human minds and conventions about what we ought to do? Well, I think, yeah, that’s pretty cheap actually, that’s not hard to establish.
1:21:56.3 SC: This must be very frustrating for the traditional moral realist, who I think not only believe in moral realism, but they think that they know what the right truths are, and you’re sort of granting them more realism in saying, but it means nothing for when we actually get out there and need to flip the switch on the trolley.
1:22:11.9 JC: That’s exactly right. It’s like analogous to the view that, look, maybe there’s a God, maybe there’s not. In fact, maybe there’s a trivial argument for the existence of God, though I don’t think so in this case, but suppose there were, but still who cares? Who cares what God thinks, it would be like the kind of view. Yeah.
1:22:29.0 SC: Is there any practical implication for the question of what we are to do? At some point we still need to answer that question, right?
1:22:41.5 JC: Absolutely, so this is not the view that everybody should start lighting cats on fire, because there’s no answer to the question of whether do I light cats on fire or not, by the way, that example’s from Gilbert Harman’s famous discussion. I haven’t been thinking about lighting cats on fire. So the claim is that all the action is had at what you might call the non-cognitive level, there’s a whole another domain of deliberation, it’s the final deliberation we have to do before action, and it has nothing to do with facts, and this kind of deliberation has in fact been investigated at some length by, for example, Alan Gilbert in Thinking How To Live, and his framework deals with things called hyper-plans, which are these kind of idealized… He takes the question of what to do to be basically the question of what hyper-plan to adopt, which is kind of like an idealized plan, so that question all remains. The key point is none of those questions and the way to resolve them have anything to do with what facts there are in the world, whether moral or not moral, they have nothing… Of course, just to be clear, they have to do with what descriptive like whether something makes people happy or sad or whether it contributes to suffering or not, obviously that is relevant, but once you’ve settled all the descriptive facts, there’s no further facts that have anything to do with the question we need to resolve.
1:24:21.9 SC: I think that we’ve earned the right to kick back for the last question here and take a big picture view, [laughter] because one of the points you make, I forget whether it’s just in discussion or in the book, but there aren’t that many people writing books about morality and mathematics, which is in some sense a shame because the goal of philosophy is to be as broad in scope and fundamental as possible, and yet we’ve hyper-specialized, right? How do you think about… You’ve given us a worked example of an engagement between two usually distinct areas of philosophy, is there hope for more of that, or do we need to be more active in bringing it about, or is it always just gonna be something a few kooky individualists choose to look at?
1:25:16.6 JC: Yeah, no. I mean, I wish I had some super useful practical advice for how to make this happen, the problems nobody’s fault, it’s just like… I assume the same is true in physics, but you can let me know if I’m wrong. As the years have gone on, the discussions have become more involved and sophisticated, and it’s hard to stay on top of a lot of different literature, so you really have to make an effort and you really have to be looking for connections, but I do think that for example as someone who’s real interested in both ethics and philosophy of math and goes to conferences of both clients and stuff, I do think that it’s striking how often the same kinds of issues arise, and there’s a bit of re-inventing the wheel across different areas of philosophy in my experience. And so it does suggest to me that it would be useful every now and again to have kind of cross disciplinary conferences where it’s not expected that you know the other area, but the people in each part are trying to explain to the others where they’re coming from and you realize points of overlap and points where one conversation could inform the other.
1:26:29.8 JC: And if nothing else, I hope the book shows that that’s not just an academic exercise, it’s obviously an academic exercise, but it’s not just an academic exercise because in my view, I’ve been thinking about ethics and philosophy of math for a while and sort of what you to take of each, and the comparison is what allowed me to arrive at positive views of each. I don’t think I would have arrived at the view that I ended up arriving at about ethics or philosophy of math without thinking about the other, so it really can… It seemed to me enrich just the research one was already doing in their field, but I don’t have any magic pill for how to make it easier, ’cause just like any academic field, the literature has exploded and it’s got more and more technical and all that stuff, and yeah, I wish that… I wish I knew how to avoid the work required, I guess, to really dig in.
1:27:34.5 SC: I think one very simple thing that I can say is that it’s absolutely going to be necessary to have individuals who are willing to put in the work in multiple areas, but at the same time, we should not forget the value of not just doing the hyper-specialized technical work, which is intrinsically valuable, and I’m all in favor of it, it’s a good thing, but also packaging the best and most important parts of it in a more widely accessible form. In some form, that’s what we’re doing here in the the podcast, right? We’re trying to get very different areas in ways that are very broadly accessible, and then at least if you want to make original contributions to the field, you’re gonna have to dig in more deeply, but at least you might be able to have some informed judgement about whether or not it’s worth your time to dig in more deeply to one field or another.
1:28:28.1 JC: Totally, absolutely, no. I really consider you kind of a savant at this, and I think that… Yeah, that I should have said something about that, I’m not so great at that myself as maybe this podcast reveals, but I think that’s enormously important. And it’s even important… So for example, last example I’ll use in set theory, set theory is a relatively fringe area of math, of pure math. And there’s this very important kind of development that happened a long time ago in the ’60s, the development of forcing, which is a certain kind of independence proof technique. And at some point along the way, people started saying, “We have an open… ” what’s called an open… What they called an open exposition problem. We need to figure out how to package forcing in a way that other mathematicians can understand what forcing is, and that is a genuine problem as much as kind of the question of whether the twin primes conjecture is true. How do you put it? And I completely agree with you from the purpose of the general scientific enterprise, trying to figure out how the world is, that has to be part of it, and it’s as important as just the on the ground work that you get trained to do in grad school, so I’m 100% agreement, and I love that you are contributing.
1:29:57.2 SC: Well, as at the end of so many podcasts, the final thought is, “Oh, good. There’s plenty more work to do.” I’m never gonna run out of good work you do. So Justin Clarke Doane, thanks so much for being on the Mindscape podcast. This was great.
1:30:09.2 JC: Thanks so much for having me. Thanks.
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I feel like he’s actually trying to pull the wool over our eyes by talking about PA+!Con(PA). (Meaning Peano Arithmetic plus the axiom – written in Goedel numbers – that Peano Arithmetic itself is inconsistent.)
If you’ve never read a textbook on model theory, this seems super weird. How can you even add an axiom to something that says the thing is inconsistent, without introducing an inconsistency?
But what model theory makes you realize is that it’s totally okay to have a separation between provability and truth. There doesn’t have to be only one unique truth value assigned to each statement, even holding the axioms fixed! There are what are called different “models” of axioms, where the model is like a big lookup table telling whether any given statement is true or false. Any provable statement is true in all models, but unprovable statements will have some models where they’re true and some models where they’re not, and that’s totally okay.
It feels like Justin is trying to wring work out of the intuition that there is only one “real” model of the axioms of arithmetic. But that intuition is simply wrong, and it’s not even that hard to discard if you take the right math class.
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Great conversation..refreshing to hear Sean needed clarification. I thing he’s on the right track. Like the chicken vs egg ethical morality preceded mathematics. The Great American quote, “All Men Are Created Equal” or human beings were the most real things but some held higher value than others. In Mathematics we can argue whether numbers and triangles are real but what creates their reality is the relationships of order and points in space like the social relational properties of kings, fathers, mothers, children, in-laws etc. The Trolley Problem rears its head but individuals rarely make such decisions vs military, governments, corporations make these decisions every day; like the stretch of road Tiger Woods drove on was a compromised decision etc.