 But can we talk about things that don't exist? That's the question I'm trying to answer on the 33rd episode of Patterson in Pursuit. My guest is a professor of philosophy at Tufts University in Massachusetts, and he takes a controversial position. He says not only can we talk about things that don't exist, we talk about them all the time, including mathematics. My guest is Dr. Jody Azuni, who's been arguing for quite some time that mathematical objects don't exist, and our conversation turned into one of my favorites so far. If you remember some of the earlier conversations I had on this topic, like with Dr. Williamson and Dr. Isaacson in Oxford, they took the position that mathematical objects have a real existence separate of our conception of them. They believe in the so-called Platonic universe. My position has been, well, I don't actually think those mathematical objects exist out there separate of our conception, I think they're constructed objects, they have a mind-dependent existence. Well, Dr. Azuni rounds out the spectrum and he says, no, they don't have a mind-dependent existence, they don't actually exist at all. So before we dive into it, I want to tell you about the sponsor for the show Praxis. Praxis is a company that is looking for young, enthusiastic talent, people that are in college and realize that, ooh, the college experience is not living up to the height, or people that want to avoid college altogether because they see academia is slowly crumbling from the inside. They take those individuals and they give them a nine-month program. It's three months of a professional boot camp that's followed by six months of a paid apprenticeship, and after you graduate from the program, they guarantee you a $40,000 a year job offer. As I've been saying for several weeks, Praxis is exploding in popularity and there's a good reason why. So if you want to be part of it, check out discoverpraxis.com and on their homepage, they have a button that says Schedule a Call. Click it, set up an appointment, and see if Praxis is right for you. So this interview is yet another that I honestly can't wait to break down. There's so much material here, there's so much here to talk about. For some reason, metaphysics gets a bad rap as a philosophic topic. I don't understand why I think these kind of conversations are a blast. So please enjoy the interview with Dr. Joni Izuni of Tufts University. Dr. Izuni is the author of eight books and growing, and I'm sure you're going to find what he has to say interesting and provocative. We like to use numbers all the time as humans. We use mathematical reasoning constantly. Mm-hmm. Let's call it indispensable. Indispensable. I think it's absolutely indispensable. Yes. We think about mathematics, so we try to formalize mathematics. We have all of these, but seem to be, self-evident truths, two plus two equals four. Everybody goes, oh yes, of course, that's a truth and they move on. But what I want to know is when we use numbers and we make mathematical claims, what are the actual objects that we're talking about? What is a number? When we say two plus two equals four, what is two in the first place? I'm going to describe my position. My position is that numbers don't exist. I'm a nominalist. My position is that lots of things don't exist, but my first focus and my earliest work was mathematical objects. And I became very interested in the main reasons that philosophers were pushing for why they did exist, which was a kind of indispensability argument. The indispensability argument turned on the idea that when you formulate a scientific theory to make it work and in fact just to express it, especially in physics, you end up crucially relying on numbers in the following sense. You have all of these, there are statements that come up. And so the idea was that if you were going to use these theories in a straightforward way, assert them, use them to describe, the whole package comes with it, including the mathematics. And that motivated other kinds of nominalists to say things like, well, we can sort of factor the mathematics from the physical stuff and exhibit a pure physical theory, which would then, then the mathematics would be shown to play a kind of instrumental role and we wouldn't be committed. Those programs all fail in my view. But there's a more direct way of going after this, which is in terms of the logic, there is an item called a quantifier that comes up in formal logic. And the thought is what's driving this indispensability argument is that that item plays a kind of committing role. You use it, it corresponds to the there is. And I suggest that as matter, as far as the formalisms are concerned and as far as natural language is concerned, this is simply not true. Okay? This is just false. In natural language, I tend to have lots and lots of examples of statements that we make that are true, that use there is statements, but we don't commit ourselves and we don't intend to. Can you give a few examples of that? Oh, absolutely. There are as many Greek goddesses as Greek gods. There are cartoon characters and animators who resemble one another. Okay? And that particular one that there is is, as it's sometimes put, quantifying over both something I take to exist, namely animators and something I don't take to exist, namely cartoon characters. You and I might be in the peculiar position of dreaming of the same imaginary woman night late. So we would say there is an imaginary woman. We dream of her. We might compare notes. Yeah, she was dressed and wearing the same thing the other night, blah, blah, blah. There are what are called hobnob puzzles kind of due to a geach, which say things like knob thinks that a certain witch has poisoned his pig and hob worries that the same witch has killed his cow, where the speaker doesn't think there is a witch. And so there's lots of examples like this where items that are playing these quantifier roles are nevertheless fun and the sentences are non-committing, but true. And that's key. The truth is important. And you have to, there's a lot of philosophical flanks to fill here. How do these things get to be true if there's no objects that are being talked about and you have to tell a story? And how do these truths operate with sentences that are this way? How can we have the kind of attitudes we have towards these things if they don't exist? Sure. I tell stories about all these things. This is why I spend so much time writing philosophy, because the nominalist position, if you go this route, opens up a lot of other things you have to tell stories about. I focus, and sometimes I go on the offensive in the following way. Is it okay that I keep talking this way? Certainly, of course. Okay, I'm just making sure. Yeah, yeah. So there's something I call the aboutness illusion, which I think very powerfully drives not just the need to believe in mathematical objects, but the need to believe in fictional objects. And the aboutness illusion takes the following form. Let's say you and I agree there's no Hercules. And let's say you and I agree there's no Pegasus. There's no object in any sense at all. So there's nothing that has properties. Because if it doesn't exist at all, it doesn't have any properties. Nevertheless, we have a rather unavoidable cognitive impression that if I say Pegasus doesn't exist, I'm speaking of something specific that doesn't exist. And if I say Hercules doesn't exist, I'm speaking of something else specific that doesn't exist. Not the same as the Pegasus. But if you're going to strictly say these things don't exist, that's not correct. Okay? Now that cognitive illusion I think drives a lot of very weird philosophical positions. My Nongianism views that fictional objects do exist because how else could, or the hallucination, is that a dagger I see before me? He's thinking of a specific non-existent object that's floating right there. This is very powerful. Okay. So can I ask you a question? Yeah. Anytime. I have to ask you. I'm very tempted by this because if what you say is true, you can reduce the amount of existence. Ontological clutter. Exactly. Exactly. However, we have to ask you what do you mean by exist? Because if we're talking about something, isn't a necessary part of something being a something is that it is. It is existent. How can you have something that isn't? Well, that's right. But keep in mind what's going on here is when you say, how can I have something that isn't? I can't. How can I use the word something to talk about what doesn't exist? I can. Okay. But let's not put it that way. Let's put it a different way, a way that's going to push back on this temptation, which is let's just talk about the fact that some of our names don't refer and some of our terms don't refer. Okay? Okay. So I use the word Hercules and I use the word Pegasus, these don't refer to anything. And now what we're really focused on is the question of how can a sentence like Pegasus is depicted in ancient Greek mythology as a flying horse, how can that sentence be true if there's a word in it that doesn't refer? And that's a different point. And now I have to tell a story and I do tell a story which involves something I call truth inducing, which is we do fiction and we do mythology and then we talk about the mythology and in particular we talk about the contents. And the way we do it is we formulate terms that don't refer to anything and we give them a truth value based on the nature of the story so that we can say correctly Sherlock Holmes is depicted in the Arthur Colin Doyle stories as a detective living in Victorian England, okay, Victorian London let's say. So we can say something like that. And now I'm talking about a sentence, I am not talking about an object. Now psychologically we experience it a certain way. We're thinking about just as we do with novels, we recognize these things don't exist and yet we think about them. So would you say then you can't reference Sherlock Holmes outside of a sentence or outside of a story? Well no, you don't reference him period, but there are sentences in which the word occurs which have truth values, okay, and those sentences get their truth values in some derivative sense from the stories. You can compare Sherlock Holmes across stories, you can talk about Sherlock Holmes in movies, you can compare Sherlock Holmes, I can say Sherlock Holmes is depicted in all the fiction he appears in as far smarter than Donald Trump is depicted in anything that talks about him, perfectly good sentence. One that's not true because of, as it turns out. Well when you say stories that's depicted in stories, does a story have an existence? For the sake of our conversation let's take stories to exist. What they actually are might be words on paper, interpreted sentences, etc. I have very radical views about what ends up existing at the end of the day. Okay, well I have a book which unfortunately I don't have here so I can't give it to you, but I can always send it to you if you can give me the address called Semantic Perception, How the Illusion of a Public Language Arises and Persists. So that also focuses on certain other class of what I call cognitive illusions. What's going on in Semantic Perception is the idea is you and I have basically language organs, I'm going to use Chomsky style language here, I'm very sympathetic to a lot of what Chomsky has to say about this. We have, as it were, each of us has a faculty, a language faculty. What it produces is we end up competent in an idiolect, an individual idiolect. I have mine, you have yours, that's going on here. They overlap sufficiently that we enjoy successful communication, but they're not exact. But that's not our experience, our experience is very different. Our experience, you're a native English speaker, you have an involuntary experience of meaningful words on a page, there is no object like that in the world. There is design, there is just ink on a page here. Nevertheless we involuntarily project that into the world and we furthermore have the experience. I have the experience that this has a meaning as well as, you know, a grammar, but I'm really focused on its meaning and I am experiencing you in the successful communication as having a seeing the same thing that I am, just as I see you as seeing this pen just as I do. But that is a, almost I want to call it a collective hallucination. That object is not out there. So there are several things I want to ask you, and then we'll get back to mathematics, but this is so directly related. Is your claim that what you could call concepts, maybe, that those don't actually exist? I was focusing specifically on public language, a concept, if we think of it as a mental entity. I mean, concept and philosophy, there is a use of it where it's a kind of public entity. But you know, concept, actually concept is this mongrel concept that's used in all sorts of ways. But if you're thinking of it as a mental entity, there is this, we have a psychological theory, a folk psychological theory, which talks about all sorts of entities, images, etc., etc. Not that, if that theory actually refers to anything, or not, I'm not prepared to say. Okay, okay, it's the reference part that I'm getting at. Right. What I'm focusing on specifically here is that when you talk about a public language, which we do, and we talk about English, and we talk about the grammar of English, and the meanings of words that are held in common, and we talk about, if we talk about the practices that we engage in, where we defer to others sometimes on the meaning of a word, I had this tendency to think of a tomato as a vegetable, but in point of fact I'm wrong and it's a fruit, right, we're deferring to certain botanical, I'm not going to say that right, experts. That object does not exist, is what I want to say. So the sentence though, that object does not exist, makes me think, well then it isn't an object. No, it isn't an object, but we experience it as an object, and we talk about it, and we rely on it, and we communicate with one another. And so we have certain experiences which we indispensable have to describe a certain way. But if I were to step back and do linguistic theory about this kind of public object or pragmatics or whatever, my theory would talk about public objects that would quantify over them just as if I were writing an essay on a novel of Dickens, I would talk about the characters and quantify over them, and in both cases those things don't exist, even though I say the sentences will be true. Okay, so moving from fictional characters, do you also have the same perspective on something like government? Does government exist? Governments, countries, yeah, I have not yet started to write about social ontology, I'm approaching it, I'm intending to write about it very shortly. But the answer is yes. The situation is a little more complicated because there's a sense in which we, it's only a sense though, it's not absolute, in which we take countries or think of them at times as composed of people or constituting of territory. Again, it turns out it's a very complicated notion. In fact, just a notion of a city like London is very complicated. This is the sort of thing that Chomsky has pointed out, and that it's playing a multiplicity of ontological roles and it's very shifty. But in point of fact, and so when you analyze what we would be dealing with here, there's going to be the only way to put it now, just informally, is that there are aspects of it that exist and aspects of it that do not. Okay, so. Okay, whereas in a certain sense the way I want to analyze fiction is it just doesn't, the terms don't refer, and I'm going to say the same thing about mathematical entities. Okay, so what I'd like to do is present to you something that I want you to tell me why I'm wrong, because I don't like the idea of Platonism, and I don't like the idea, well, I like the idea of nominalism, but I don't like sentences like we can say true things about non-existent, non-odd. That's a misleading way of putting it. The right way to put it is that we have sentences that are true with terms in them that don't refer. That's the way to, you're right, the other way of putting it, well it's not that the other way is imprecise, it's that the other way invites a position, which philosophers have adopted that, well there are two kinds of objects, there are ones that exist and ones that don't, or there are ones that have being and ones that don't, or maybe then none of them have being but then they are in some sense or have properties even though they don't exist and don't have being, and I don't want to say any of this, all of this to me is crazy metaphysics. Okay, so, but I think there's another option here. Okay. This is my own position. All right. That mental objects have a mind-dependent existence. So when we're talking about something like government or we're talking about something like Pegasus, Pegasus exists but what I mean by that word is a mental unit in my mind. Don't want to say that. Okay, so why is that, why don't you like that? Why you don't want to say that is that sounds like what's called a use mention error and so it goes something like this. Let's start with the keys on my desk over here, okay? I certainly have a concept of them, I have an idea of them and that's a mental entity and that's probably dependent on my mind in the sense that you're describing, although, you know, I'm going to have problems with that, but let's not worry about that because that's not important. The crucial thing is that we want to distinguish the keys, the physical keys from the mental entity. So there's at least two things that we're talking about here. And when I say the keys exist or the keys are heavy or any other number of things about the keys, I am not talking about the mental entity. The mental entity has a different set of properties. So here's how I might respond to that. Right, and so I want to say with Pegasus, you've got the mental entity, but you don't have the Pegasus. So that's the difference between the keys and the Pegasus. What do you think about something like this? That there are different types of existence. So you have a spatial existence. Yeah. The keys have the keys. Keys have spatial existence, sure. And fictional objects have fictional existence. Or what I would say is they have a conceptual existence. They have a mental existence. They're not spatially located. There are philosophers who at least sound like this. As I said, you'll find a philosopher for any position because logical space, surprisingly, logical space is as rare and expensive as apartments in Manhattan for whatever reason. So you'll find every space you'll find some philosopher. I'm a person who thinks, and now I'm going to try to try to out linguistic evidence for this, that that's not how the word word exists works. Exists does not have many meanings. Exists does not have many uses. It is not ambiguous. Now, one of the tools I use for this is I believe it's called the conjunction reduction test, which is if a word is open to multiple uses, you can't combine them in a way. In a single locution, you get oddities. So here's an example. Getting beer and getting up, different uses of get. And it's hard to say he got beer and up, right? Now, however, recall the example I gave to you a little earlier where I said there are cartoon characters and animators who resemble one another. One use of there are covering both two different kinds. And you can also say the same thing would exist. Yes, I totally agree. But what I would say is all that's reflective of is an imprecision of language that comes up in circumstances like this. So the way we clear that up is not by saying, oh, exist is intrinsically ambiguous. We just say, oh, well, we have to specify what we mean by the term. So if I say something like Harry Potter has square glasses, that's a false sentence. I could meaningfully say, no, that's false. Harry Potter has round glasses. What I mean by Harry Potter seems like a very clear concept. It's some kind of a mental unit that I can describe in various true or false ways. Other people might share that concept in their own minds. There's going to be this overlap between your concepts and my concepts. But we don't have to say Harry Potter doesn't exist. What I'd say is Harry Potter doesn't, that term has no spatial referent. That term is just a pure conceptual. Look, this is a view. This is a view. And you can even inoculate the view against the linguistic evidence. You don't have to call language imprecise, which seems mean. You can say, well, we're going to revise the language or regiment it. We're going to allow exist to work this way. Your metaphysics in a way is going to drive your revisions in language. That can happen. I'm going the other way. I'm going to say, you know what? I can get by without messing with existence. Now, parentheses, I have other arguments about this. I have trouble understanding different notions of existence. To me, what you're describing is something that exists in the same way but has different properties. It's mental or it's in space and time. And so I'm saying, you know, that's actually your position. Your position is actually just good old fashioned. You believe in different kinds of objects. Some of them are mental. Some of them are. And some of them are numbers. And you're labeling it. Now, I want to, I, in my back pocket, I haven't mentioned this. I have a criterion for what exists, which is not is in space time. But which would sound like it would just blatantly beg the question against you. I have a position, which is something like if something exists, it's mind and language independent. We have to, we don't get to dictate its properties. It gets to tell us what its properties are and we have to find out. So you reject the category of fiction. All mind dependent, you say mind dependent existent is those things. Yeah, that's not. There is no such thing as a mind dependent thing except in the following sense. And this is why I wasn't ruling out mental entities necessarily. Look, I make a chair. The chair is in a certain sense, carpenter dependent. That is to say the chair would not have existed unless the carpenter would have brought it into existence. I don't think that makes its existence dependent in any way except causally in the same way if there was no big bang, there would be no me. In that way, I can see mental events under a different description perhaps as brain episodes of a brain. Brain neurons fire, blah, blah, blah. That's an episode. And it may be, although I'm not positive this will ever work out, I think it might not that mental, the folk talk about ideas and in prayer we'll all turn into episodic descriptions of episodes and then that depends on what episodes are. So that would be the way I go. Now when it comes to therefore fictional entities, again I want to say what makes the Potter claims true or false is the movie and storytelling discourses that have been created and then do these sentences describe things correctly or not that are taking place in them? Not do they describe Harry Potter correctly because there is no Harry Potter but do they correspond in the right way to the sentences which occur in the Harry Potter stories? That's the story I'm going to tell. So what I'm saying is that at the end of the day I think you can tell your story, I can tell my story, where I would want to apply pressure against you in argument would be with my criterion. Okay, so when you think of an idea in your metaphysics, ideas don't exist. Let's say, I might think of them as, as I said, episodes and then- Okay, I guess that would mean- That they might exist depending on how I feel about episodes. Right, so what would something like that be? Is it something like a brain state? Yeah, or a perhaps a brain state or perhaps a particular event of the brain because one of the things that's going on in the brain is that things are happening and those are events and I think strictly speaking, for example, everything that I'm seeing, et cetera, et cetera, this can be characterized, not everything that I'm seeing because I'm seeing tables, chairs, et cetera, but the act of seeing as it were is a certain, in part, an event of my brain over a certain period of time, neurons firing, et cetera, et cetera, if that kind of language is still neurophysiologically accurate. So that's an event of the brain. Now, the brain does a bunch of things. There are events that occur in it. It goes over a process of development when new memories occur. There is neurophysiological changes. All of that would be the stuff I would end up talking about. Okay, so when you are thinking about creativity, human creation, because I have this wiggle room in my metaphysics for mind-dependent things, I would say something like, I can actually conceive of, let's say, a song that hasn't been written or a song that's in my head and it's never been fully there but I can almost, in a sense, hear it. I can say all those things with a language of like, yeah, well, in some sense, that song exists. That's what I'm referencing. That's what I'm super, like, so pseudo-hearing but its existence doesn't have any spatial reference yet. How do you think about creative objects? Yeah, creative objects, it's going to depend on the object. Let's assume that an object is a certain, the objects we're talking about here, there are such objects are time-bounded. They come into existence at a certain point. They go out of existence at a certain point. And so, and let's say these are the things we create. This is a very simple toy model because I think there are things we create that don't exist in a certain sense. Again, I'm speaking in this terrible way but I can anticipate and there are things we say, like, you know, the house I am going to build is one I can barely afford, okay? And that's a statement about a future house, okay? I am able to refer to, there are real questions about, am I actually referring to it? Because it exists in the future, let's say it really does exist and then I, maybe I'm able to refer to something in the future at the present moment and that would be an actual object that I was referring to. It may be the house never comes into being. In that case, I'm not referring to something, okay? But nevertheless, what I'm saying is true because one of the reasons the house never got built was because I couldn't afford it, right? So what I'm going to do in these cases and if you think about the creation of fictional characters which I treat as creation, those are the creation of things that don't exist and what we're going to describe instead if we say, well, what did the person create metaphysically? Well, it depends if it's a story, it's a bunch of sentences or it's ink on paper that's interpreted or whatever and then it's something that we allow to accrue over time the way we would treat it. In other cases, we're just going to talk about a practice that we can carry on of a certain sort, a way of talking and that's what we're actually creating when we're creating these objects. I have one more question for you and then we'll bring it back to mathematics. Okay. For me, because I can, I feel like I have a little easier of a time talking about mental objects in this realm and in this kind of non-spatially-existent realm I can also throw something like consciousness. Oh, okay, I just have a big basket of non-physically-existent things and I'm just going to put consciousness in there so that results in any kind of problem. When you think of consciousness or the experience of awareness or how does this fit into your metaphysics? Is this, am I referencing anything in the world? That's really complicated and it's a little like government. What's going to happen? I have not written about this. I have plans if I live. The book you will write in the future. The book I will write. Well, the way I've been going, I'm like at eight now so there's a chance I'll get there. Consciousness, I'm going to say, in some, we talk about it in a lot of ways and we have certain kinds of experiences and there are these issues about first person and about the way it seems to be, et cetera, et cetera. There are enormous puzzles here and not all of those puzzles are puzzles about metaphysics. But as far as the metaphysics is concerned there's going to be aspects which exist and aspects which don't. It's going to be the same thing as government. Having said that, I want to stress again, this goes very little towards saying anything very distinctive about what's going on with consciousness or addressing the issues that arise. So do you see any qualitative categorical difference between consciousness as a metaphysical something and the regular kind of stuff, the keys, the physical stuff? What I want to say as far as the metaphysics is concerned there is nothing else going on. That's right, as far as the metaphysics is concerned. And I have a book coming out in 2017 called Objects Without Borders which already says that a great deal of what we take to be in the world actually is not there. We project it, okay? This isn't even going near consciousness. This is worrying about things like what are called individuation conditions for objects and saying we project the individuation conditions. And that projection has no actual existence. There's nothing in the world that corresponds to it, structurally or otherwise. That's basically the thesis of that book. So what's happening here is a great deal, what I'm engaging in in metaphysics is a process that a lot, an earlier generation of philosophers thought was impossible and dating back to Kant everybody thought was impossible which is I'm actually trying to factor what we're projecting onto the world from what is actually there. I'm claiming we can do it, okay? The Kantian style view is something like, look, mind and world are all bundled together. You can't figure out which part is one and which part is the other. I don't think that's true, okay? I mention this simply because in saying that what's going on in consciousness there's nothing more than what's already in the world. I had to add, perhaps I was cheating a little bit in advertising a book but the stress I was trying to put on this was there's a lot less out there than you think. That was the idea. Okay, so now I'm coming full circle back to numbers. If it is true that numbers don't exist, then what is your explanation for the incredible explanatory power and use of these numbers? Well, that's a big, big, that's what I concede. If you're going to be a genuine nominalist of the sort that I am, you have got to tell a story about what is so valuable about geometry, about numbers and about so much other higher mathematics and there's a story that has to be told. It's a long story, it's not quick and obvious. There's a way in which the platonic story made life easy. There are these guys, in fact Plato's version made it especially easy and the rest of the world kind of approximates it and that's why you've got application. Plato's story had an application story built right into it. I don't have an application story so I have to tell a much more complicated story and one of the things I wanna say is it's a piecemeal story. So when you're explaining it, it really turns on the application. So like if you're looking at your application of you cling geometry to a table, the story you're gonna tell about the success of the application is going to turn on the physical properties of the table and how they allow you to extract the information and extract descriptions of the table that are approximately right from the mathematical formalism. So you've got to tell a story that looks at the mathematics, looks at what the application of the mathematics looks like in that case and why what pops out is close to what is going on on the table. So that's generally the kind of story I tell and it's gotta be told in a way that doesn't presuppose that these things exist. So in particular, I can't tell a story of the following story. Draw a triangle on the blackboard and that approximates a Euclidean triangle. I can't tell that story because I've just invoked the Euclidean triangle. What I have to do is tell a different story which focuses on the theorems of Euclidean geometry and then are interpreting those in the physical application and then cranking out results which are close to the results that are empirically predicted. Am I at all communicating? Because as I said, this is a long story. Yeah, so let me ask you. Geometry is a great place to talk about this. Because in a certain sense, the success of the application is so visible in a certain sense because you've kind of got the physical item, big and large macro object to do the comparison with. That's what makes it an easier case for me than other cases. So in geometry, something like the Pythagorean theorem. Right. What I really don't like is the Platonist approach where we're talking about the ideal triangle that is separate of our conception of it that has these properties and sometimes correlates to this world, sometimes it doesn't. I like the middle ground of thinking of what I mean by the Pythagorean theorem has to do with my own conception. It's constructed objects in the mental world. Now in the mental world, then you can preserve kind of the explanatory power of Platonism and you can even say things like universal mathematical truths with this constructed object. But when we go to the nominalist perspective. You don't have that. You don't have. Do you have any? You still have concepts. Do you have any universality of mathematics in nominalism? You don't have it in principle. The universality of the applicability. I mean, look, mathematics as a whole is applicable. Okay, and that looks universal. But it's an illusion to think numbers are always applicable. Not if you're dealing with jelly-like things that are mutating into one another. Is that so, though? If there's nothing to count, then you're not going to apply numbers. Some mathematics will be applicable. I'm not denying the applicability of mathematics. I'm just saying, keep in mind you've got lots of different mathematical theories and they are often applied in different ways. And they're not necessarily the same. That's all I'm saying. So I'm trying to tame a kind of impression that you can have as like, oh my god, look at this miracle. Mathematics that applies to the world. And the right thing to say is, a chunk of it applies here. A chunk of it applies here. An enormous amount of mathematics applies nowhere. 20th century mathematics and 21st century mathematics is full of tons of mathematics that has no application. Yes, and I agree with you on that point, but for different reasons. Well, I understand. It's perfectly good mathematics. And often applications can emerge later. I can tell stories about that. There are cute stories. But let me not focus on that part. Let me point out something that's very unusual about mathematical concepts, which is the interesting thing is, if you think of the Earth, you can have an empirical concept of a triangle, which is, I draw this thing with chalk. The lines are irregular. They have a certain thickness. They're kind of thin, but not really very thin. And the angles, et cetera. And then what you realize is that if you use those concepts, you can prove very little. They are implicationally intractable. The reason being, the angles don't add up to a on something well, it's not going to be quite flat. And it's not going to add up to 180 degrees. It's a little more, a little less. What you can actually prove is very little. And most of our ordinary concepts, even if they have rich semantic structure like house, like London, it's very hard to prove much with them. The crazy thing about the original set of Euclidean concepts is that they're implicationally tractable. All sorts of things can be proved. And that was what made it valuable. You could prove that the interior angles of a triangle sum to 180 degrees. So one aspect of the advantage of mathematical concepts, there's nothing to do with whether they refer or not, is that they are implicationally tractable. And what we don't realize is that that is a rarity in ordinary language. Most of our concepts are implicationally intractable, which is why we find ourselves arguing with each other so much, even with perfectly sincere people. Again, I don't mean Donald Trump. I mean people with the best intention using concepts are going past each other, finding it very difficult to tease out implications. And in philosophy, the same thing, but not in mathematics. And what's your explanation for that, though? It's just a very specific to the concepts. In the case of one of the things that you do is you are narrowing, think about it this way, the empirical notion of a triangle includes so much more than the Euclidean triangle does. And one of the things you do, if you narrow down the scope of your concepts, you increase the chances that they're going to be implicationally tractable, because they simply cover fewer cases. The advantage is that you don't want to get something that has not even a hope of an approximation empirically, because then you'll have stuff that you can pull out lots of implications, but there's no use. And as I again want to stress, there's lots of mathematics like that. You get something that's implicationally beautiful. You get lots of interesting results that you can show, but it has no application. And the people have to realize there's a lot of mathematics like that. Modern mathematics, I think, in particular, yes. Well, it really started in the 19th century. And there's nothing wrong with it, as I want to say. This is not a plea to cut back practices and mathematics programs by no means. It's to point out something special about an aspect of mathematics. And it's one of the reasons why mathematics became the backbone of physics, because physics, based on that mathematics, inherits the implicational tractability. Again, comparatively speaking, compared to ordinary concepts, which are just implicationally, they're utterly opaque. So something as simple as just a geometric triangle, which I have this, I have a, seems like there's some kind of a mental unit that I'm talking about that I can make true statements about. Think of it rather that you're using a mental unit, which is enabling you to infer, and you've got an image in your head, but that's not the object. There is no object. You can do that. If it is the image, though, can't we just say that the image is the object? No, because there's a bunch of ways in which we talk about mathematical objects that doesn't make the image the suitable rillata. In particular, we're both talking about triangles and Euclidean geometry. But my images are my images. Your images are your images. They're not the same images. That's true. But that's not what we're talking about. We're not talking about anything. But can't we at least talk about the image? Doesn't the image? And that's a subject for psychology. And one of the interesting things is that, broadly speaking, when psychologists study the concept of a number, for example, they do not get the numbers. They get something else. I mean, there are lots of literature on this very interesting stuff where we have a number of concepts of numbers that are operating in us. We have an analog notion. We have tiny 1, 2, 3, and 4 that we can recognize without subtilization. We've got, as I said, we can sort of estimate large groups of people and other objects. That's a concept of number. We have something that's coming out of the number quantifiers. But the actual mathematical numbers are not instantiated in anybody's head. Well, so let me ask you. So I'll start with an analogy and we'll bring it back to mathematics. I were a masterful artist, which I'm not, by any stretch of the imagination. And I were to share with you a technique for creating some amazing drawing on a wall. And I describe it as, hold the paintbrush this way and make this motion and blah, blah, blah. You might be able to construct something very similar to what I might have constructed. That's correct. Could this not be the same thing that's going on with mathematics? So what I'm talking about. It is the same thing that's going on in mathematics. But if that's true, though, if that's true, it seems like the image is the object. The triangle is like the painting that I'm. The reason it cannot be is because of this element that kicks in, which is if you're talking about the image, there can't be any right or wrong. The image is what it is. But with mathematics, we treat there being a right or wrong that's going on. We've got a language practice and a proof procedure, which in mathematics, that puts conditions on what the objects are supposed to be like. But I think I can explain that. So let's say we were talking about the image on the wall, the painting on the wall. I can make false claims about it. I could say, you know. No, no, no. But this is different. This is different. Of course, you can make false claims about it. But the point is, you're not making false claims about your image. If your image, for example, condenses the number line so that numbers that are larger are actually closer together than numbers that are smaller, which is how our number line works, that's correct about the number line in our heads. That is the image. That's not incorrect. That's correct. Yes. Well, but when you say the number line in our heads, what I'm saying is the way I think of you of mathematics, it is if somebody is a skilled mathematician, as they're a skilled linguist, they can tell rules and procedures and techniques to somebody to help them create a similar construction in their mind as the mathematician holds in his head. Actually, what they'll get them to do, probably they will not get them. If I'm a very skilled mathematician, and in part, there's a certain phenomenology accompanying that. So I have ways in which, when I'm thinking about certain higher dimensional objects, I slice and dice them a certain way so I get certain three-dimensional images, which are nicely connected to the properties of the thing. And I write down proofs. Now, my proofs don't echo that phenomenology. And then I teach somebody how to do these proofs, and they get the hang of it. They're likely to have a very different phenomenology, but they're going to be able to do the proofs the same way. Well, the core features would be the same, though. Just like if we're going through, and I like to do the artist example, the rules I tell you are about making an image of a horse. But I don't specify the color of the horse. We can still make true and accurate claims about that constructed image of, I could say, oh, you did this part wrong. You drew him, and he's wearing a top hat. No, you've done that wrong. You see, in that particular case, you've got a drawing on a wall, but you're actually going to say false and true things about. In this case, you've got a phenomenology that never goes public. And when psychologists study that thing, the only way it goes public is psychologists study it to some extent and get some idea. But when I'm teaching you mathematics, or you're teaching me mathematics, we are not sharing the phenomenology. Now, that's not, I'm not invoking another mind's problem. I never get access to your from. That's not the point. The point I'm making is that you have to get on to doing the mathematics. And there's no reason to think that you get on to it by getting the same phenomenology in your head that I get. In fact, if you end up being a really creative mathematician using the stuff that I started with and coming up with all these new proofs, you probably have a very different phenomenology. OK, OK. So let me, you're correct to point out the error of the horse example. However, I have another example. Just like a real philosopher, he's just, OK, you don't like that example? Here, look, I've got six more in my pocket. So let's say it's not drawing the public thing. Let's say it's storytelling. OK. And let's say I'm trying to communicate to an apprentice storyteller. And I say, look, here are the absolutely essential features of constructing the story. Here's what you do with the plot. Here's how you get the characters to be liked. Here are their names. And this is the storyline. If I want to construct the same phenomenological experience in you, here are the key details. But I say, there are things that you, if the color of the jacket that the person is wearing is different than it is in my phenomenal little experience, it doesn't really matter. Yes, so we would disagree. We'd have a different phenomenology, but the same core meat of the story is still there. If I've done a good job in communicating, the core construction of that story is going to be. I told you how to make the story. And now you go on to do the same thing I was doing with the story. Yes, there's some irrelevant details. Right, but none of this is really tracking the, again, it's the same thing. The phenomenology is not really necessarily tracking it that closely. It may turn out that, I mean, what's going on in this case is we are learning how to create a certain kind of external object, which is a story, which is a bunch of sentences following one another. And that's what, often at the end of the day, that's what counts. See, that's a good point because what I've done is smuggled in my metaphysics because I wouldn't have considered the story an external object. I'm considering it as a, that's why I gave the examples to say, oh look, this is not an external. Even if it's not written down, even if it's an oral, it's still an external object. And that's the kind of thing that's being passed to one another and that we're learning to make. And I agree that the phenomenology is crucial to pulling that off. It may be, for example, that someone who cannot psychologically entertain imaginary beings of a certain sort and start to attribute to things to them psychologically is not gonna be able to write a story. Or if they do, it's gonna be a really bizarre kind of thing. But nevertheless, that phenomenology enables them to do it but is not showing up. So in conclusion, because this is an excellent note to end on, I was having a conversation with a mathematician at Oxford. And we were talking about this subject. It came up fairly briefly. And he was saying, there is a school of thought. I don't know if you've heard of this because I don't know very much about it, but if you do, I'd like to follow this up just on my own research. There's a school of thinking in mathematics and in metaphysics called fictionalism, which essentially says that numbers are fictional objects. They're in the same, which is, that is my position. They're like, they're different versions of the story. Yes. Of this position. And what I would say is I really like that position because depending on the nature of the fictional object, the numbers or the concepts, you can create the story in such a way where you can make true claims about the spatially existent world. So when I talk about the idealized sphere, to the extent I make that a very clear concept, that applies to the real world. And that's how we get the amazing explanatory power of mathematics. But have you, is this a position that is not? Yeah, no, it's out there. There are different things called by fiction, called fictionalism. Usually the fictionalist position doesn't want these things to be true. At least among many philosophers, the fictionalist thinks that fictions are pretended true rather than true. So it's often, it's not invariably, but one way to think of it is as a pretence view. I'm not a pretence theorist. I think that these things just are true. And it's the fact that they're true that contributes to their value, okay? And what I'm saying is these sentences can be true. In fiction, just as in mathematics, without referring, but it's the truth of them and it's the nature of the truths that makes them successfully applicable. So I want to avoid the fictional world picture. In fiction, let alone in mathematics, okay? So I don't give the kind of explanation that you just mentioned. I'm gonna give a different one. As long as I'm successful in giving a different one, we can then get into a fight over the specific issue of should we, do we need to have a kind of fictional object for our story or not? And I'm going to say no we don't. Well that's an excellent note to end on. Thank you so much for this conversation. Yeah, thank you. All right, that was my interview with Dr. Jodi Azuni of Tufts University. I hope you guys liked it. I certainly did when I edited this interview up. I thought, damn, this was a good one. I got lots more coming down the pipeline. My wife and I are moving to New Zealand to continue our travel around the world. And I got some new articles and videos that are coming out soon. So the next few weeks are gonna be super exciting. Make sure to stay tuned. All right, that's it for me. Enjoy the rest of your day.